Facing the Future: Agents and Choices in Our Indeterminist World

Facing the Future

This page intentionally left blank

FACING THE FUTURE Agents and Choices in Our Indeterminist World

NUEL BELNAP MICHAEL PERLOFF M I N G XU

With Contributions by Paul Bartha Mitchell Green John Horty

OXPORD

UNIVERSITY PRESS

2001

OXFORD U N I V E R S I T Y PRESS Oxford New York Athens Auckland Bangkok Bogota Buenos Aires Cape Town Chennai Dar es Salaam Delhi Florence Hong Kong Istanbul Karachi Kolkata Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi Pans Sao Paulo Shanghai Singapore Taipei Tokyo Toronto Warsaw and associated companies in Berlin Ibadan

Copyright © 2001 by Oxford University Press, Inc. Published by Oxford University Press, Inc 198 Madison Avenue, New York, New York 10016 Oxford is a registered trademark of Oxford University Press All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Oxford University Press Library of Congress Cataloging-in-Publication Data Belnap, Nuel D , 1930Facing the future • agents and choices in our indeterminist world / Nuel Belnap, Michael Perloff, Ming Xu, with contributions by Paul Bartha, Mitchell Green, John Horty. P cm Includes bibliographical references and index. ISBN 0-19-513878-3 1 Agent (Philosophy) 2. Choice (Psychology) 3 Free will and determinism I Perloff, Michael II Xu, Ming III Title B105 A35 2001 128'4—dc21 00-064995

9 8 7 6 5 4 3 2 1 Printed in the United States of America on acid-free paper

Preface This is a book about the causal structure of agency and action. It frames a rigorous theory by using techniques and ideas from philosophical logic, philosophy of language, and metaphysics with a small "m." This theory, which we sometimes call "the theory of agents and choices in branching time," describes agents as facing a future replete with real possibilities, some of which various agents realize by making choices. It is central to our theory that choices and the actions that they ground are radically indeterministic: Before an event of choosing, there are multiple alternatives open to the agent. Furthermore, since the choice is real, so must be the alternatives, and each alternative must be as real as any other. All we can say before the moment of choice is that the agent will make one of the open choices, leaving behind the unchosen alternatives. After the choice, it is correct to say that they were once possible, but are no longer possible. None of the possible choices is a mental or linguistic figment, nor is any a mere ghost image of "the actual choice." Given that the possibilities relevant for action are always possibilities for our future, the theory also refrains from appealing to "possible worlds" other than the one and only world that we all inhabit. These ideas are in some part rooted in common sense. Without help, however, common sense cannot seem to pull them together into a coherent whole. One of our principal aims is to carry out that job by articulating them in a completely intelligible exact theory. The resultant theory of agents and choices in branching time pictures the causal structure of our world as made up of alternative courses of events branching tree-like toward the future. Each branch point represents a choice event or chance event. On the one hand, each continuation from a branch point is individually possible; on the other hand, it is impossible that more than one of these continuations should be realized. If that sounds obscure, we agree: It is, we think, almost impossible to speak clearly and accurately about indeterminism except in the framework of a rigorously fashioned theory such as the one we propose. The theory of agents and choices in branching time is of real but limited interest without its application to understanding the language of action. We propose a certain linguistic form, a "modal connective," as being unusually helpful to anyone who wishes to think deeply about agents and their actions.1 1

Our usage here is common but not universal. A connective is any grammatical construe-

vi

Preface

The form is "a sees to it that Q," where a names an agent and Q holds the place of a sentence; for example, we think of "Ahab sailed the seven seas" as "Ahab saw to it that he sailed the seven seas." The form is so important to our enterprise—all but three of our eighteen chapters are devoted to its study or use—that we give it an abbreviation, "[a stit: Q]" with "stit" as an acronym for "sees to it that." Stit theory explains the meaning of [a stit: Q] in the idiom of philosophical logic. In doing so, stit theory invokes a certain melding of the Prior-Thomason indeterministic semantics with Kaplan's indexical semantics. The combined semantics make [a stit: Q] roughly equivalent to "a prior choice by a guaranteed that Q." The stit idea is many-sided. We explore its grammar, semantics, and proof theory as logicians do. We delight in the fact that stit does not treat actions primarily as "things" to be counted or named. We explore some of the linguistics of stit (especially how and why its status as a modal connective lends itself to usefully complicated constructions), we consider some applications to difficult conceptual problems, and we argue the ability of stit to illuminate agency in a variety of ways. We look at some ways in which stit might be modified or generalized. In all of this, however, we try never to forget the central constraining thought: There is neither action nor agency nor doings without real choices, choices that find their place not merely in the agent's mind, but within the (indeterminist) causal order of our world. To see to it that Q, an agent must make a real choice among objectively incompatible future alternatives. When we say that an event may have many possible but incompatible outcomes, we thereby come down on the side of "hard" indeterminism as against determinism. There is no consensus on these ideas. Since the eighteenth century became understandably awed by the success of Newtonian science, the presumption of determinism has guided most of the philosophical and scientific explorations of both agency and nature. Taking determinism to be delivered by science as an unquestioned "fact," philosophers since Hume and Kant have worked at developing "compatibilist" theories that hold agency, in the guise of moral responsibility, to be compatible with what James called the block universe. Such theories have often taken possibilities as unreal: as arising from the mind, or from social practices, or from language—for example from consistency with the bits of language called "scientific laws." Our contrasting indeterministic presumption is eloquently expressed by the eminent paleontologist Stephen Jay Gould. I don't think that any deeper or more important principle pervades nature, and lies at the heart of all historical sequences, than this central but underappreciated notion of "contingency"—the great and tion that maps one sentence (or several sentences) into another sentence. For example, when you put sentences (i) "Ahab is captain" and (ii) "Ishmael is not captain" into the blanks of " and ," the result is another sentence, (iii) "Ahab is captain and Ishmael is not captain." That makes the "and" construction a connective. A truth functional connective is one like " and ": If for example you know whether each of (i) and (ii) is true, then you automatically know whether (iii) is true. A modal connective is defined negatively as being one that is not truth functional.

Preface

vii

liberating truth that tiny inputs, virtually invisible and risibly impotent in appearance at the outset, can cause history to cascade down any route in a vast array of entirely different pathways. (Gould 1999, p. 30) This book neither argues for indeterminism nor tries to pick holes in arguments for compatibilism. Our project assumes the indeterminism of the causal order in which agency is embedded, it assumes that actions are based on real choices, and it assumes that choices are therefore not predetermined. Our goal is not to persuade, but to make these ideas intelligible. Although numerous philosophers share our general point of view, not many exact theories share these assumptions and aims. Our strategy is to concentrate almost exclusively on the objectively causal side of indeterminism and agency, which already presents enough difficulties without bringing in noncausal concepts. We therefore lay aside many deeply important aspects of agency and choice that involve intentions, propositional attitudes, or other mental phenomena. We look for ways in which applications of stit theory can engender a better understanding of agency. Seven examples: (i) an analysis of refraining that clarifies how it can be both a doing and a not-doing; (ii) an analysis of imperatives that emphasizes their agentive content; (iii) an extended treatment of deontic logic that insists that obligations and permissions (a) are directed to agents capable of making choices, and (b) are embedded in the indeterministic causal order of our world; (iv) fresh analyses of promising and of assertion, analyses that argue the unwisdom of the doctrine that among all the objective possibilities, a unique course of events constitutes the one and only "actual future"; (v) an exploration of the causal side of the requirement on action that the agent "could have done otherwise"; (vi) the causal structure of joint agency; and (vii) a generalization of stit theory to strategies considered from a causal point of view. In sum: Holding the extra-mental and extra-linguistic status of incompatible possibilities as given, and supposing that the future sometimes depends upon an agent's choices among incompatible options, we offer a tense-modal theory intended to describe some causal aspects of agency in our indeterministic world. Guidance on reading this book. In the spirit of Carnap, each chapter begins with some introductory remarks in order to permit easy skipping of topics not of present interest to the reader. We append the following largescale structural notes as additional guidance. The book divides into six parts of varying degrees of technicality, followed by an appendix. Each of these may be characterized as follows. Part I introduces stit theory. We intend the chapters in this part to be accessible to all those with an interest in our topic. Portions of chapter 2 and chapter 5 do, however, involve willingness to put up with some elementary logical constructions, and chapter 2 offers a brief explanation of the theory of agents and choices in branching time that underlies stit theory. In this part we introduce key grammatical and semantic features of stit, including the distinction

viii

Preface

of the deliberative stit from the achievement stit. Largely concentrating on the achievement stit, the various chapters of part I suggest applications with the help of many pictures, make comparisons with some other work on agency, and, beginning with Anselm's work in 1100, give a little history of the modal logic of agency. This part also contains applications of stit theory to imperatives and to promising. Part II supplies precisely and in detail the nuts and bolts—or, more aptly, roots and branches—of the theoretical structure that supports our account of agents, actions, and our indeterminist world. The three chapters of this part are foundational in character, and involve substantially more rigor, though not much more mathematics. They stress conceptual analysis rather than theoremproving. This part more than any other focuses on the problems faced by any indeterministic theory. Here we argue against the beguiling but harmful doctrine of "the actual future," which says that among the many courses of events that might come to pass, there now exists a privileged such course that will actually do so. This foundational part examines, postulate by postulate, the theory of agents and choices in branching time, and explains in detail the semantic subtleties required of a language spoken in an indeterminist world. Part III offers two applications of the achievement stit: One chapter aims to illuminate the dark idea of "could have done otherwise," and another considers the causal aspects of joint agency. These chapters are a little more technical. Part IV is of the same level of technicality as part III: It offers applications of the deliberative stit, chiefly to help in elaborating such deontic concepts as obligation and permission, which, we believe, are in much need of a theory of agency. Part V uses the already-established theoretical structure of agents and choices in branching time in order to develop an austere (causal, not normative) account of strategies as a kind of generalization of stit. One chapter in this part connects our theory of strategies to Thomason's deontic kinematics. Part V proves a theorem or two, though much of it is, again, conceptual analysis. Part VI provides the technical backbone of stit theory, including proofs of decidability, soundness, and completeness. The chapters of this part are required reading for those who wish to investigate or develop the logical and mathematical properties of theories of agency similar to ours. The appendix gathers, for easy reference, most of the various theses, structures, postulates, definitions, semantic ideas, and systems that are introduced elsewhere and are employed throughout the book. We use boldface to refer to certain of these items: Look in the appendix for its sections §l-§9, for stit theses Thesis 1-Thesis 6, for postulates Post. 1-Post. 10, for definitions Def. 1-Def. 20, and for axiomatic concepts Ax. Conc. 1-Ax. Conc. 3. Although in each section of this book we feel free to refer to any other section, it may be useful to indicate the following dependency-structure among the various parts. Parts I (introduction to stit), II (foundations of indeterminism), and VI (proofs and models) are almost entirely independent. The reader's primary interests may therefore be allowed to determine with which of parts I, II, or VI it is best to begin. Part III (applications of the achievement stit) and part IV

Preface

ix

(applications of the deliberative stit) are mutually independent, whereas each presupposes familiarity with part I. Part V (strategies) requires chapter 2 of part I as minimum background.

This page intentionally left blank

Acknowledgments Our contributors deserve special thanks for giving us permission to use their materials: P. Bartha for contributing chapter 11 and for co-authoring chapter 12; M. Green for co-authoring chapter 6; and J. Horty for co-authoring §2A of chapter 2, which has its source in Horty and Belnap 1995. This essay is also used in a substantially different way in Horty 2001, a book elaborating the normative aspect of choice. Horty 2001 should be consulted by every reader of this book. We add that each contributor has also helped in other ways too numerous to count. Many other people have helped in the research reported in this book. We apologize for the simple alphabetical listings that we offer as sincere but inadequate thanks. The following assisted with the development of ideas and with the preparation of the manuscript during their respective terms as Alan Ross Anderson fellows: M. Allen, D. Bruckner, C. Campbell (who caught many dozens of errors and infelicities during a final proof-reading), K. Davey, U. Ergun (who created many of the figures), C. Hitchcock, C. Jones, J. MacFarlane (who produced the initial draft of the index), J. Roberts, L. Shapiro, A. Staub, V. Venkatachalam (who was responsible for the initial draft of the bibliography), and M. Weiner. In addition, a number of others have helped over the years, especially by commenting on or offering us fresh information in connection with the separate articles on which this book is based: G. Antonelli, L. Aqvist, C. Bicchieri, R. Brandom, M. Brown, D. Davidson, D. Elgesem, R. Gale, A. Gupta, D. Henry, I. Humberstone, D. Kaplan, M. Lange, W. Lycan, D. Makinson, J. Moore, R. Neta, I. Porn, K. Schlechta, K. Segerberg, S. Sterrett, J. Thomson, D. Turner, D. Vanderveken, F. von Kutschera, D. Walton, H. Wansing, and anonymous referees for several journals. Finally, we thank two anonymous OUP referees for encouraging us to turn a heap of articles into a book.

This page intentionally left blank

Contents I

Introduction to stit 1 Stit: A canonical form for agentives 1A Agentives 1B Stit: Simple cases 1C Grammar of the modal logic of agency 1D Mini-history of the modal logic of agency 1E Conclusion and summary

. . . . .

3 5 9 14 18 26

2 Stit: Introductory theory, semantics, and applications 2A Theory and semantics: The two stits 2B Applications of stit, with many pictures

. .

28 29 39

3 Small yet important differences from earlier proposals 3A Von Wright 3B Chisholm 3C Kenny 3D Castaneda 3E Davidson 3F Conclusion

. . . . . .

59 60 65 68 74 78 81

4 Stit and the imperative 4A The theory of fiats 4B Ross's paradox and stit 4C Chellas's theory 4D Agentive constructions 4E Negations of imperatives 4F The many varieties of imperatives 4G Embedding imperatives 4H Conclusion

. . . . . . . .

5 Promising: Stits, claims, and strategies 5A From stit to promising 5B From RR to promising 5C Strategic content of promises and word-givings

97 . 98 . 107 . 116

82

82 84 85 87 89 92 94 96

xiv

II

Contents

Foundations of indeterminism

6 Indeterminism and the Thin Red Line 6A Preliminary considerations 6B Parameters of truth 6C The assertion problem 6D The Thin Red Line 6E Time's winged chariot hurries near

. . . . . . , .

133 134 141 156 160 170

7 Agents and choices in branching time with instants 7A Theory of branching time 7B Theoretical reflections on indeterminism 7C Theory of agents and choices 7D Domain

. . . .

. . . .

177 177 203 210 219

8 Indexical semantics under indeterminism 8A Sources 8B Structure parameters: The "world" of the speakers 8C Interpretation and model: The "language" of the speakers . 8D Points of evaluation, and policies 8E Generic semantic ideas 8F Semantics for stit-free locutions 8G Clauses for stit functors

, . . . . . .

220 . 221 . 226 . 227 . 228 . 234 . 239 . 247

9 Could have done otherwise 9A Could have been and might have been 9B Could have done and might have done 9C Might have been otherwise 9D Might not have done it 9E Could not have avoided doing 9F Could have prevented 9G Could have refrained 9H Might have refrained 9I Had available a strategy for not doing 9J Summary

. . . . . . . . . .

255 . 257 . 257 . 259 . 260 . 261 . 262 . 263 . 265 . 268 . 269

10 Multiple and joint agency 10A Preliminary facts 10B Other-agent nested stits 10C Joint agency: Plain and strict 10D Other-agent nested joint stits

. . . .

271 . 272 . 273 . 281 . 290

III

Applications of the achievement stit

Contents

IV

xv

Applications of the deliberative stit

11 Conditional obligation, deontic paradoxes, and stit 11A Technical preliminaries 11B Semantics of obligation 11C Completeness 11D Conditional obligation 11E Oc-statements versus cO-statements 11F The Good Samaritan 11G Contrary-to-duty obligations 11H Problems with the proposed semantics of obligation . . . .

295

12 Marcus and the problem of nested deontic modalities 12A The parking problem 12B The form of obligations 12C The Anderson/dstit simplification 12D The form of prohibitions 12E Generalized prohibitions 12F Generalization on agents 12G Temporal generalization 12H The outer ought

318 318 319 321 322 325 333 335 335

V

296 298 303 304 306 309 312 314

Strategies

13 An austere theory of strategies 13A Nature of austere strategics 13B Review of choices in branching histories 13C Elementary theory of strategies 13D Favoring 13E Application to finding a strategy for inaction

. . . . .

14 Deontic kinematics and austere strategics 14A Basic concepts 14B From Thomason's deontic kinematics to austere strategics 14C From austere strategics to Thomason's deontic kinematics 14D Remarks

. . . .

VI

341 342 344 345 356 359 364

365 368 370 376

Proofs and models

15 Decidability of one-agent achievement-stit theory with refref 15A Preliminaries 15B Companions 15C Soundness: Validity of refref equivalence 15D Companion sets

381

. . . .

382 385 390 392

xvi

Contents 15E Alternatives and counters 15F Semi-ref-counters 15G Completeness and finite model property

. . 397

. . 401 . . 408

16 On the basic one-agent achievement-stit theory 16A Preliminaries 16B Soundness 16C Companion sets and their alternatives 16D Construction of preliminary structures 16E Completeness

. . . . .

415 . 415 . 417 . 419 . 424 . 428

17 Decidability of many-agent deliberative-stit theories 17A Preliminaries 17B Soundness 17C Completeness and compactness 17D Finite model property

. . . .

. . . .

18 Doing and refraining from refraining 18A Preliminaries 18B Main results

451 . . 452 . . 454

Appendix: Lists for reference 1 Stit theses: Thesis 1-Thesis 6 2 Structures 3 BT + I + AC postulates: Post. 1-Post. 10 4 Branching-time-with-instants definitions: Def. 1-Def. 9 . . 5 Agent-choice definitions: Def. 10-Def. 14 6 Basic semantic definitions: Def. 15-Def. 16 7 Derivative semantic definitions: Def. 17-Def. 20 8 Grammar 9 Axiomatics concepts: Ax. Conc. 1, Ax. Conc. 2, and Ax. Conc. 3

435 435 439 441 445

459

. . . . . . . .

. 459 . 460 . 461 . 463 . 466 . 468 . 469 . 471

. . 472

Bibliography

475

Index

483

Part I

Introduction to stit

This page intentionally left blank

1 Stit: A canonical form for agent ives Among the topics of discussion in this world, none are more common than those concerning the achievements and refrainings, obligations and prohibitions, successes and failures of the agents with whom we share a common space.* "What happened when so-and-so did that?" we ask, or "What should have been done?" Biographies and narrative histories, which form a sizable segment of our reading materials, have as their central concern the doings of agents, their obligations and prohibitions, the outcomes of their choices, and the range of things from which they refrain. Philosophers have long sought to find a distinction between those sentences which attribute agency and those that do not in the verbal configurations we commonly use to talk about such matters. If such a distinction existed it should be a relatively simple matter to uncover some general principle in everyday speech to differentiate Joshua fit the battle of Jericho from Joshua survived the battle of Jericho. We find, to our dismay, that we are no closer now to a linguistic litmus test for agency than was Aristotle. After all, it would have been of some importance for Aristotle's peers to decide between an agentive interpretation of Alexander succeeded to the throne of Macedonia and a non-agentive interpretation. For if Alexander was agentive in the matter of his becoming king, then there was a prior choice of his which led directly to * With the kind permission of Theoma, Belnap and Perloff 1988 is the basis of each section of this chapter except §1D. That section is drawn from Belnap 1991, for the use of which we thank the International Phenomenological Society.

3

4

Introduction to stit

that outcome, and he was likely guilty of regicide; while if he was not agentive in the matter of becoming king, then there was no choice of his that guaranteed his succession to the throne. J. L. Austin said that "The beginning of sense, not to say wisdom, is to realize that 'doing an action', as used in philosophy, is a highly abstract expression—it is a stand-in used in the place of any (or almost any) verb with a personal subject ..." (Austin 1961, p. 126). In his essay he tried to throw light on the question of "doing an action" by looking at the range of cases in which excuses are offered both in everyday usage and in the law, and to arrive at a proper vocabulary for action by "induction" on the proper uses of words. Many years have passed, the lesson has been learned, and it is time for philosophy to go beyond the mere beginnings of sense and progress toward a deeper understanding of an agent doing an action. How then should we proceed? The Austin legacy is of course a particularly rich one that is being carried on in a variety of ways. One way attempts to explain certain facts about language in terms of propositional content and illocutionary forces: "speech act theory" as represented for example by Searle and Vanderveken. That is not our program, as we make clear in many places hereafter, suggesting that a theory of "speech acts" should begin with a prior theory of acts. Accordingly we follow in the wake of Austin's suggestion that when faced with the question of the meaning of a word or a term such as "doing an action," we should "reply by explaining its syntactics and demonstrating its semantics" (Austin 1961, p. 28). A suggestion of this book is that the next step in the progression toward greater sense and wisdom is to have available the sort of clean and well-honed linguistic resource that Austin, and other philosophers, have realized to be necessary. We think that the most promising path to a deeper understanding of an agent making a choice among alternatives that lead to action is to augment our philosophical language with a class of sentences whose fundamental syntactic and semantic structures are so well designed and easily understood that they illuminate not only their own operations but the nature and structure of the linguistic settings in which they function. An example of a doing-an-action sentence that Austin might have had in mind is Ahab sailed in search of Moby Dick.

(1)

It has a personal subject, "Ahab," and an action-like verb, "sailed," and seems to be describing an action in Austin's deliberately wide sense. We take (1) not only to be true, but to be agentive for Ahab, for Ahab's sailing in search of Moby Dick was a direct result of a choice he made among alternatives available to him. On the other hand, although the perfectly ordinary sentence, The Pequod sailed in search of Moby Dick

(2)

is surely true, and though we may be hard pressed to say exactly why, we are not hard pressed to say that it is not agentive: It does not even have a personal subject. Consider now

1. Stit: A canonical form for agentives Ishmael sailed in search of Moby Dick.

5 (3)

Ishmael signed on as a member of the ship's company in total ignorance of Ahab's vengeful purpose; are we then to say that (3) is agentive for Ishmael, but false, on the consideration that both "Ahab" and "Ishmael" are appropriately "personal" subjects? Or is (3) true on analogy with the example about the Pequod? English is not to be trusted in these waters. Since English fails to serve us as an adequate pilot, what we want is a resource sensitive to the difference between those cases using an action-like verb in which agency is ascribed, and those cases in which there is merely the appearance of agency. With decent Austinian respect and regard for the structure of the language, we propose, insofar as is possible, to locate such a resource within English itself, a resource that will also allow us to become clearer about the relation between agentive sentences in their declarative uses and agentives in their imperative uses; for surely Mr. Starbuck, hand me yon top-maul, and

(4)

Mr. Starbuck handed Ahab the top-maul

(5)

are, in context, more than accidentally related. In fact, except for contextdetermined indexicals, they are the same agentive sentence in two different appearances: (4) is an imperative issued to Mr. Starbuck by Ahab, while (5) is an agentive declarative whose truth or falsity is intimately connected to the satisfaction or failure of satisfaction of the imperative.

1A

Agentives

Let us accordingly begin with the following convention: The agentive form that we are about to introduce shall be set off with square brackets [ ... ]. It shall have two open places as indicated, the first to take an agent term, the second to take a declarative sentence (the declarative complement of the new form). The point about the second open place is nontrivial: Having noted that declarative sentences can either ascribe agency or not, we specifically include as possible declarative complements for the second open place both those sentences that do ascribe agency and those that do not ascribe agency. The resulting squarebracket sentence is to say that the proposition expressed by the declarative complement is guaranteed true by a prior choice of the agent. So [the carpenter ... Ahab has a new snow white ivory leg] is to be agentive for the carpenter, and is to say that he is the agent in the matter of Ahab's having a new snow white ivory leg. With what verb or verb phrase shall we replace the ellipsis in that sentential form (it was after all only elliptical)?1 Among the candidate English verb phrases that history suggests are the following: 1

Numerous other philosophers have considered this or similar questions, including at least Anderson, Aqvist, Bennett, Chellas, Chisholm, Danto, Davidson, Fitch, Hamblin, Hilpinen,

6

Introduction to stit i. brings it about that ii. makes it the case that Hi. causes it to be the case that iv. is responsible for the fact that v. lets it be the case that vi. allows it to be the case that vii. takes steps in order that

viii. behaves so that in consequence ix. sees to it that As you can see, these are all grammatically acceptable; but items (i)-(iii) suggest that causal processes are either at work or hovering in the background, whereas we wish to de-emphasize this suggestion. Items (iv)-(vi) give the impression of moral judgment and ethical responsibility, and whereas those are important ideas (none more so), it is inappropriate to build them into the foundation of this enterprise. Items (vii) and (viii) are closer to the mark in their straightforward association of an agent and something made true by the agent, but even these candidates might suggest to some ears that we want you to concentrate on a second, prior action performed by an agent. The English verb form (ix), sees to it that, has, to our ears at least, fewer of the obvious defects of the others, and sees to it that has the definite advantage of suggesting alternatives and choices. So our preferred sentence form is [a sees to it that Q], which for logical emphasis we abbreviate as [a stit: Q], referring to such a sentence as a stit sentence. We intend that "Q" here take up a sentential position, as is proper for a complement of "sees to it that." (Note: In early chapters we use "Q," whereas in later chapters we use "A" or "5," and so on, with exactly the same intention. The slight difference in notation derives from the diversity of the origins of the various chapters, and in order to avoid fresh mistakes, we have thought it best not to enforce uniformity in this particular regard.) It is a consequence of what has just been said that a sentence [a stit: Q] with a as its subject is always agentive for a. Let us enter this important idea as the "agentiveness of stit thesis." Hohfeld, Humberstone, the Kangers, Kenny, Lindahl, Makinson, Melden, Needham, Porn, Talja, Thalberg, and von Wright. Bibliographic access to the literature with which we are familiar can be gained through Thalberg 1972, Aqvist 1984, Bennett 1988, Makinson 1986, and the bibliography of this book. Although we do not undertake to discuss all of these contributions, we do discuss some of them, especially in §1D and chapter 3. Above all, we should be remiss if we failed to note that NB and MP were each introduced to this family of ideas by our teacher, Alan Ross Anderson—see Anderson 1962 and Anderson 1970.

7

1. Stit: A canonical form for agentives AGENTIVENESS OF STIT THESIS. (Stit thesis. Reference: Thesis 1) Q] is always agentive for a.

[a stit:

This is the first of six "stit theses"; you can find a list in §1 of the appendix. By this thesis we intend the descriptive claim that the English sentence "a sees to it that Q," which we abbreviate as [a stit: Q], always describes a as an agent carrying out an action. The complement, Q, may be agentive for a or not. This almost equally important claim is our "stit complement thesis." STIT COMPLEMENT THESIS. (Stit thesis. Reference: Thesis 2) grammatical and meaningful for any arbitrary sentence Q.

[a stit: Q] is

The two theses together emphasize the double aspect of stit: Q in [a stit: Q] may be any sentence at all, but [a stit: Q] is always agentive for a—perhaps difficult to interpret, perhaps contradictory, perhaps merely false, but always agentive for a. So [Ishmael stit: Ishmael sailed on board the Pequod] is agentive for Ishmael and a true sentence, while [Ishmael stit: Ishmael refused to share a room at the Spouter Inn] is equally agentive for Ishmael, but false. The stit sentence [Queequeg stit: the Pequod is fitted out for its voyage], which has a non-agentive as its declarative complement, is agentive for Queequeg but false, for it is not due to Queequeg, but to Peleg, that the Pequod is fitted out for its voyage. At this point you may fairly ask how stit treats other agentive sentences, that is, sentences other than stit sentences that ascribe an action to an agent. Our answer is that if such a sentence is truly agentive, then it can appropriately be paraphrased as a stit sentence. Furthermore, if it is not truly agentive, then to attempt such a paraphrase will tend to reveal this fact. We are therefore led to advance the "stit paraphrase thesis." STIT PARAPHRASE THESIS. (Stit thesis. Reference: Thesis 3) Q is agentive for a just in case Q may usefully be paraphrased as [a stit: Q]. Therefore, up to an approximation, Q is agentive for a whenever Q [a stit: Q}. Given this thesis, we may use the equivalence Q [a stit: Q] as a helpful test. Contrariwise, the helpfulness of the test tends to support the thesis, rough as it is. In any case, we intend this strategy, clarification by paraphrase, as neither definitional nor reductive. It is rather an attempt to isolate, by way of a canonical form, a particular set of English sentences in order to study them more closely as they interact with each other and with other parts of language

8

Introduction to stit

in different linguistic environments, just as a biologist might tag a particular organism to follow its activities as it interacts with members of its own species and with other species in various physical environments. Analogically, most of us are comfortable in saying that an English sentence is conditional if it can be paraphrased into the canonical form "if Q1 then Q2," and then revealing that the sentence The Prophet will tell Queequeg and Ishmael about Ahab provided they stop a minute is in fact conditional by paraphrasing it as If Queequeg and Ishmael stop a minute then the Prophet will tell them about Ahab. EXAMPLE. The sentence (1) is agentive since it can be paraphrased by prefixing an "Ahab saw to it that": [Ahab stit: (1)].

(6)

By Thesis 3, then, since (1) "Ahab sailed in search of Moby Dick" is correctly paraphrased as (6) [Ahab stit: Ahab sailed in search of Moby Dick], both the original and its longer transform are not only true but agentive for Ahab. EXAMPLE. It seems a travesty to paraphrase the sentence about the Pequod, (2), by [Pequod stit: (2)]. Indeed, on their ordinary readings, (2) seems true, while its attempted stit paraphrase seems false. Furthermore, it appears to us that this difference helps us see the non-agentiveness of (2)—helps us to see, that is, that (2) does not ascribe an action to an agent and is therefore non-agentive. EXAMPLE. It strikes us that (3) has two quite ordinary readings, an agentive reading in analogy with the Ahab example, (1), and a non-agentive reading in analogy with the Pequod example, (2). For many purposes the difference is irrelevant or at least of negligible importance; but if one is discussing serious matters of, for example, ethical responsibility, then the difference counts. It matters in such cases whether or not (3), as employed in a particular discussion, can be paraphrased by [Ishmael stit: (3)]. If the non-agentive reading of (3) seems forced, take others. Ishmael sailed over the seas

(7)

is presumably agentive, while Ishmael sailed over the side of the Pequod,

(8)

in its most natural use, would not be agentive. Our agreement on these verdicts tracks our agreement on the effort to paraphrase with stit: It seems right to claim that (7) [Ishmael stit: (7)], and it seems wrong to claim that (8) [Ishmael stit: (8)]. Observe also that there are certainly circumstances under which one would take

1. Stit: A canonical form for agentives [Ishmael stit: (8)]

9 (9)

as true (perhaps he deliberately saw to it that he went flying over the side). Nevertheless, this possibility should not be confused with passing the stit paraphrase test for agentiveness. The truth of the stit sentence (9) does not imply that its complement, (8), is agentive, nor that the stit sentence is equivalent to its complement. Only when (8) is taken as meaning the same as (9) is it plausible to take (8) as ascribing agency to Ishmael.

1B

Stit: Simple cases

We now use the strategy of clarification by paraphrase to demonstrate how stit sentences can help us better to know our way around in some areas in which agency counts. Consider for example Ahab found the White Whale.

(10)

Certainly it is true, but is it agentive? In order to answer we have to address ourselves specifically to the question: Did Ahab see to it that Ahab found the White Whale? As we have seen already, Ahab was agentive in his search for Moby Dick, but was he agentive in finding it? We think not. Although Ahab was a participant in guiding the Pequod to the ultimate outcome, and the major participant, his actually finding the White Whale was due in large part to chance, to natural forces beyond his control. "Time and tide flow wide," remarks Mr. Starbuck. "The hated fish has the round watery world to swim in." Thus, though (10) is true, it is false when paraphrased as [Ahab stit: Ahab found the White Whale]. Therefore, insofar as (10) is true, it is non-agentive; and insofar as it is agentive, it is not true. Consider the following pair: Queequeg struck home with his harpoon.

(11)

Queequeg's harpoon struck home.

(12)

Some careful speakers of English might always use (11) as an agentive and (12) as a non-agentive, whereas most of us are liable to use them interchangeably, sometimes in one way and sometimes the other. But [Queequeg stit: Queequeg struck home with his harpoon]

(13)

[Queequeg stit: Queequeg's harpoon struck home]

(14)

and

come to much the same thing. Such differences between (11) and (12) as there are disappear when they are embedded into stit contexts; the stit sentences (13) and (14), regardless of the uncertainties of their complements, are transparent with respect to agency.

10

Introduction to stit

1B.1

Imperatives

"Clear away the boats! Luff!" cried Ahab. Treatment of such imperatives is by no means tangential to our concerns. We endorse the view that C. H. Hamblin expresses in his masterful study: Imperatives are not only among the most frequent of utterances; they are also, surely, among the most important. If the human race had to choose between being barred from uttering imperatives and being barred from uttering anything else, there is no doubt which it should prefer. (Hamblin 1987, p. 2) Hamblin additionally reports that a full twenty percent of Shakespeare is in the imperative mood. (We happily acknowledge that a study of Hamblin's book set us under way and helped us avoid some threatening reefs.) Imperatives, in our usage, constitute a grammatical category. Following an established tradition, however, we think of each use of an imperative as having a force and a content. With regard to force, Ahab's imperatives may have been orders or commands, which many think the only possibilities; but Ahab might instead have been inviting, requesting, suggesting, advising, ... the helmsman to luff. Putting force to one side, however, we are after the content, about which stit theory has a definite and (as it happens) helpful opinion that we sum up in the "imperative content thesis." IMPERATIVE CONTENT THESIS. (Stit thesis. Reference: Thesis 4) Regardless of its force on an occasion of use, the content of every imperative is agentive. For example, Luff! can have its content represented as The helmsman luffs, which in turn, since it is agentive, can be paraphrased as The helmsman sees to it that he luffs. Thus, Luff! can be paraphrased as Helmsman, see to it that you luff! In this case the application of our thesis is easy because The helmsman luffs, which looks to be the most plausible content for the imperative, is already agentive. Still, there is more to be learned. To luff is to see to it that the bow of the boat is heading directly into the wind. Accordingly, the content of the imperative Luff! can be put into the canonical form [Helmsman stit: the helmsman luffs]

(15)

or equivalently, [Helmsman stit: the boat is headed into the wind].

(16)

The two stit sentences are equivalent, but while the complement of the former is agentive, the complement of the latter doesn't mention the agent at all.2 Unlike Luff! the imperative 2

"Complement" versus "content" sometimes sounds confusing. The (grammatical) complement of [a stit: Q] is Q. Its (semantic) content is an agentive proposition.

1. Stit: A canonical form for agentives Be on deck at dawn!

11 (17)

does not show its content so obviously; the stit apparatus, however, increases in value as problems become more complex. Consider (17) as addressed to the helmsman. The obviously non-agentive The helmsman is on deck by dawn

(18)

cannot, by our thesis, represent its content; we need an agentive form, and fortunately there is already something at hand: the stit sentence, [The helmsman stit: the helmsman is on deck at dawn]. This, in turn, can be transformed back into the imperative Helmsman, see to it that you are on deck at dawn, which we take to be an accurate paraphrase of the original imperative (17). This is important because you might be tempted to think that (18), inasmuch as it displays an agent as its subject, is the content of the imperative (17); but there are many ways in which it might be true that the helmsman is on deck at dawn, while false that he sees to it that he is. Because the content of every imperative must be agentive, a helpful paraphrase will not merely have the name of the addressee as its subject: That name will appear as subject of a sentence of the form [a stit: Q}. We therefore have the following: Since every imperative has agentive content, and since every agentive can be paraphrased as a stit sentence, it must be that every imperative will have a stit paraphrase of the form [a stit: Q]—sometimes with Q an agentive (hence a possible content of an imperative) and sometimes not. These brief remarks do not suffice. We return twice to the topic of imperatives, once in §1C, and again in chapter 4.

1B.2

Deontic contexts

The traditional account (see, e.g., Aqvist 1984) has it that deontic statements have one of the forms

Oblg:Q,i.e. it is obligatory that Q, Frbn:Q, i.e. it is forbidden that Q, or

Perm:Q, i.e. it is Permitted that Q where Q is a declarative. 1-1 REMARK. (On notation) The standard deontic notation is Op, Fp, and Pp. We depart from the standard for good reason: We shall in the course of this book deal with many other modalities, several of which would have an equal right to single letters such as 0, F, and P. We therefore use abbreviations for deontic and other modalities that we hope are short enough for the eye to take

12

Introduction to stit

in at a glance and long enough for the mind to remember. (It helps that we do not often use these longer expressions in the course of complicated calculations in which extreme brevity is a decided advantage.) Among its many virtues, this approach allows such great latitude that it can be connected to every possible action by any agent. On the other hand, agents have in this grammar no distinguished place; if invoked at all, the agent's name is only an accidental feature of the declarative complement. It is an easy mistake to think that Oblg:Q says more than it actually does say. It does not say, for example, who, if anyone, is obliged to see to it that Q. Consider It is forbidden that cooks be on the bridge. It is tempting to read this as though it imposes a prohibition on the cook, but it doesn't, as we can see from the analogous It is forbidden that dogs be on the bridge; in either case the sentence form is not fit to tell us who is to see to it that no cooks (or dogs) are on the bridge. While a deontic language of declarativesentence complements may be satisfactory for impersonal oughts, many within the tradition have seen that agents need to be treated with more care. Perhaps foremost among those who have argued that these standard alternatives are inadequate is Castaneda, for example, in Castaneda 1974 (see additional discussion in §3D). Some deontic logicians have suggested the step of changing the grammar so that Oblg:, Perm:, and Frbn: are taken to be adjectives modifying actionnominals. The fundamental forms are then taken as Oblg:a, i.e., a is obligatory, Frbn:a, i.e., a is forbidden, or Perm:a, i.e., a is permitted, where a stands in place of a term designating an action. So Oblg:a might be instanced by Sailing is obligatory, or Frbn:a by Bringing the boat into the wind is forbidden. Since actions are always the actions of agents, this step is in the right direction, but it still fails appropriately to recognize the agent. Furthermore, there is a considerable loss in expressive power, since clearly understood declarative sentences, Q, are so much easier to come by than are clearly understood action-nominals, a. In order to regain the headway lost in the move from sentences to action-nominals, some have tried adding negative doings such as not-sailings, or disjunctive doings such as luffing-or-flensing; but even with these additions, there remain two features missing from the overhauled model: first, the flexibility that follows from permitting an arbitrary declarative within the scope of a deontic statement, and second, the grammatical means to identify and keep track of the agent. We propose a combination having the strengths of both the declarativesentence complement plan, Oblg:Q, and the action-nominal complement plan,

1. Stit: A canonical form for agentives

13

Oblg:a. We propose to focus on deontic statements that have only agentives for their complements and thus can always be paraphrased into one of the following forms. Oblg:[a stit: Q], i.e., a is obligated to see to it that Q Frbn:[a stit: Q}, i.e., a is forbidden to see to it that Q Perm:[a stit: Q}, i.e., a is permitted to see to it that Q Though Perm:[a stit: Q], for example, is not agentive (in our technical sense), we intend it as quasi-agentive in the loose sense that it involves both an agent and an agentive, and in the stricter sense that like an agentive, it has the agent itself as a recoverable part of its intension. Our proposal calls, then, for a deontic language enriched by the restriction that the complements of obligation, permission, and prohibition be limited to stit sentences, a proposal that forms part of the "restricted complement thesis." RESTRICTED COMPLEMENT THESIS. (Stit thesis. Reference: Thesis 5) A variety of constructions concerned with agents and agency—including deontic statements, imperatives, and statements of intention, among others—must take agentives as their complements. Recall that (i) stit sentences always express an action, (ii) stit sentences never lose or misplace the agent, and (TO) there are no grammatical or metaphysical or semantic restrictions on the declaratives that may be put in place of Q. With these features of stit in mind, applying Thesis 5 to deontic constructions retains, we think, the most valuable features of both the declarative-sentence complement account, Oblg:Q, and the action-nominal complement account, Oblg:a. For example, in context the burden of the prohibition of cooks (or dogs) from the bridge might be Oblg:[the Third Mate stit: no cooks (or dogs) are on the bridge]. Or we might have Frbn:[the cook stit: the boats are lowered]. Question: Does this imply Oblg:[the cook stit: ~(the boats are lowered)]? The application of stit theory to deontic concerns is a topic deserving treatment at length. We return to it explicitly in §2B.9, chapter 11, chapter 12, and chapter 14, as well as more indirectly in chapter 4 (via imperatives) and in chapter 5 (via the idea of promising).

14

1C

Introduction to stit

Grammar of the modal logic of agency

The formal grammar of stit is easy: A singular term goes in one blank, and a sentence in the other. The grammar of English agentives, especially that of imperatives, is more delicate. The first topic is embedding. We all know that declarative sentences can either occur stand-alone or be embedded in a larger context. One may encounter the following standing alone: The Prophet will tell Queequeg and Ishmael about Ahab. One may also find the selfsame declarative as the consequent of a conditional: If they will stop a minute, then the Prophet will tell Queequeg and Ishmael about Ahab. One should bear in mind that exactly the same possibilities exist for the two other "moods" of English, the interrogative and the imperative. With respect to interrogatives, one has for example the two stand-alone interrogatives Did Queequeg and Ishmael give their informed consent to sailing after the White Whale? and

Whom did the Prophet tell about Ahab? One also has these selfsame interrogatives embedded in a (declarative) "dependence" construction: Whether they gave their informed consent to sailing after the White Whale depends on whom the Prophet told about Ahab. When interrogatives are embedded, we tend to call them "indirect questions," a terminology that distracts from the chief point: The purely grammatical phenomenon of the very same interrogative sentence occurring now as stand-alone and now embedded is precisely analogous to declaratives occurring in these ways. Finally, and crucially for our purposes, the English grammar of imperative sentences exhibits precisely the same behavior. For example, start with the stand-alone imperative Mr. Starbuck, hand me yon top-maul. (Of course the "Mr. Starbuck" may be dropped when context makes clear the addressee, and "me" must be supplanted by "Ahab" when the imperative is used by some third party.) Then each of the following is a way to embed this selfsame imperative in a larger context. Mr. Flask, request Mr. Starbuck to hand me yon top-maul.

1. Stit: A canonical form for agentives

15

Ahab ordered Mr. Starbuck to hand him yon top-maul. Mr. Starbuck carried out Ahab's order to hand him yon top-maul; or at least Mr. Starbuck handed Ahab yon top-maul. Did Ahab advise Mr. Starbuck to hand him yon top-maul? Ahab demanded that Mr. Starbuck hand him yon top-maul. Ahab demanded that yon top-maul be handed to him by Mr. Starbuck. Mr. Starbuck refused to hand Ahab yon top-maul. Mr. Starbuck refused Ahab's request (order, advice) to hand him yon top-maul. Mr. Starbuck is obligated (permitted, forbidden) to hand Ahab yon topmaul. There is clearly no doubt, then, that imperatives can be either stand-alone or embedded. In English, embedding any of declaratives, interrogatives, or imperatives is a complicated matter requiring at least inversion and context-driven switching of pronouns. Our deepest comment is this: Embedded imperatives are in truth embedded imperatives. That is, they are constituent or embeddable grammatical transforms of the very same imperative sentences. In a logically perspicuous language, they would be the very same sign designs. Furthermore, agentive declarative sentences are, we think, also syntactic variants of imperatives, whether stand-alone or embedded. Both Mr. Starbuck handed Ahab yon top-maul and the clause embedded in That the top-maul was handed to Ahab by Mr. Starbuck was of no concern to Mr. Flask represent variants on exactly the same theme. The intimidating variety of syntactic structures is a good reason for thinking about stit. In all these cases the paraphrase into canonical form allows us to see that whatever the complications of natural language, we have one and the same agentive sentence, its surface grammatical form varying from context to context. We may put this as a thesis, our last: the "stit normal form thesis." STIT NORMAL FORM THESIS. (Stit thesis. Reference: Thesis 6) In investigations of those constructions that take agentives as complements, nothing but confusion is lost if the complements are taken to be all and only stit sentences. Our recommendation is that both imperatives and agentive declaratives be normal-formed as stits. What, then, of the grammar of stit itself? We think that the best thing to say is that a stit sentence itself is both a declarative and

16

Introduction to stit

an imperative.3 A stit sentence can be embedded wherever a declarative or an imperative can be embedded. We have seen the variety of English sentences that are helped by paraphrase as stits. Let us consider the other direction. Given a stit sentence, [a stit: Q], we should expect to read this piece of notation differently in English, depending on how it is used. Here are some paradigmatic examples of readings of [a stit:

Q}.

• As a stand-alone imperative: a, see to it that Q! • As a stand-alone declarative: a sees to it that Q, a is seeing to it that Q, a saw to it that Q. • As an embedded imperative: a to see to it that Q, for a to see to it that Q, that a see to it that Q, a's seeing to it that Q. • As an embedded declarative: a sees to it that Q, that a see to it that Q.4 These subtle alterations required by English grammar are really required; but they tend to obscure rather than reveal the fact that we can locate the selfsame agentive in a variety of contexts: sometimes as a stand-alone declarative, sometimes as an embedded declarative, sometimes as a stand-alone imperative, and sometimes as an embedded imperative. Our paraphrase of agentives into normal form clarifies that situation. A stit sentence, [a stit: Q], since it displays its agent and appropriate declarative complement publicly and obviously, is the appropriate picture of the underlying agentive partly because it remains recognizably the same in any and every context. A further important feature about the grammar of imperatives, aside from their embeddability, is that they display an agent. As Castaneda (1975) and others have urged, they have the deep grammatical form a to verb. In this respect an imperative is unlike a declarative in general, which may or may not express an agentive proposition and even when it does may not wear its agent on its surface, as the linguists say. But imperatives must show forth an agent, at least in the sense that to be understood, and (the point is crucial) to be used in larger contexts, the agent must be uniquely recoverable from the surface (for example, as the addressee of a stand-alone imperative). Those engaged in the descriptive grammar of English have and are entitled to different views on this matter. Perhaps the work most pertinent to our concerns is Badecker 1987. Badecker surveys some Chomsky theories, which by deriving 3 We are not sure that this use of overlapping grammatical categories is for the best. Let us emphasize again that the aim of applying stit notation is to clarify, not to analyze, and, in particular, not to provide a syntactical criterion for when a certain surface form must be considered an imperative rather than a declarative. Let us also emphasize that "force" versus "content" has no role here, since the topic is purely grammatical. 4 Note that this "see" is not indicative, but rather subjunctive, like "hand" in "Ahab demanded that Mr. Starbuck hand him yon top-maul."

1. Stit: A canonical form for agentives

17

all infinitive constructions by transformation of declaratives are deeply at variance with the spirit of the present book, though our aims are so different from his that it is hard to call the variance a conflict. In healthy contrast, the lexicalist theory that Badecker offers in his chapter 3 awards infinitive constructions independence from declaratives, and thereby more nearly shares our direction; however, there remains the question of whether in agentive infinitive constructions, such as Mr. Starbuck refused to hand Ahab yon top-maul, we should or should not take it that there is a "trace" of Mr. Starbuck heading the infinitive phrase. We certainly need Mr. Starbuck to get the semantics right, but that far from settles the syntactical question for English. In any event, Badecker supplies a helpful framework for addressing this and related questions. A consequence of the fact that a stit sentence is a declarative as well as an imperative is this: It can be embedded wherever a declarative or an imperative can be embedded. For example, with regard to the former, a stit sentence can be embedded under a negation. The result of such an embedding is on the face of it not itself a stit sentence; in the special (and, to a logician, prominent) case of negation, the result of embedding looks at least on the surface like

not like some instance of

But more deeply, for the special case of negation, the result of embedding is not always any kind of agentive; that is, by the paraphrase test for agentives suggested by Thesis 3, to which we hope you have agreed, the declarative it is false that a sees to it that Q is not invariably paraphrasable (or indeed equivalent in truth value with) a sees to it that it is false that a sees to it that Q. Thus, by Thesis 3, (19) is not itself an agentive. In more colorful language that speaks against taking a naive approach to inventing a "logic of imperatives," we may say that the negation of an imperative is not always an imperative. We later discuss the interaction of negation with stit in a number of places; see for example §2B.6 and §2B.8 on "refraining." Finally, let us recall that when concerned with the modalities of agency, it is similarly helpful to use stit sentences to keep track of the agentives. The English Starbuck could have handed the top-maul to Ahab is helpfully paraphrased as [Starbuck could-have-stit: Starbuck hands the top-maul to Ahab], while

18

Introduction to stit Starbuck is obligated to hand the top-maul to Ahab

is paraphrased as Oblg:[Starbuck stit: Starbuck hands the top-maul to Ahab]. In all these cases it is crucial that the content of the order and the obligation, the ability and the action, be all the very same, all captured by a single stit sentence.

1D

Mini-history of the modal logic of agency

Throughout this book we constantly refer to "the action," "the obligation," "the ability," "the order," and so on. This is because English, through its love of the subject-predicate construction, drives us to such usages. We do not believe, however, that English also drives us to wax ontological about actions, obligations, abilities, orders, and so on. We may wish to fashion theories about entities taken to fall under these common nouns. Stit theory is not to be taken as arguing against such a wish. On the other hand, we may not. Stit theory has the advantage that it permits us to postpone attempting to fashion an ontological theory, while still advancing our grasp of some important features of action, obligation, and so on. It does so by invoking a "modal" construction in place of a subject-predicate construction that requires an ontology. By a modal construction we mean an intensional grammatical construct having sentences as both inputs and outputs. A "modal logic of agency" intends that some such construct express agency (or action), as for example our own favored English construct "a sees to it that Q." The modal logic of agency is not popular. Perhaps largely due to the influence of Davidson (see the essays in Davidson 1980), but based also on the very different work of such as Goldman 1970 and Thomson 1977, the dominant logical template takes an agent as a wart on the skin of an action, and takes an action as a kind of event. This "actions as events" picture is all ontology, not modality, and indeed, in the case of Davidson, is driven by the sort of commitment to firstorder logic that counts modalities as Bad. The project has had some successes, all of which we shall ignore, and some failures, most of which we shall ignore. (See Bennett 1988 for an indispensable perspective.) Certain of its failures, however, are to be attributed to the neglect of the modal features of agency. We compare the ontological approach to action with our modal approach in §3E. Here we pursue the modal history. The earliest modal logic of agency of which we have learned was formulated around the year 1100 by a Dominican trained at a famous Norman institution, the abbey of Bec. (The date of c. 1100 is implied by the sources that were conveniently available to us without scholarly digging, namely, the discussion on p. 120 of Henry 1967 together with the "Anselm" entry in the Encyclopedia Britannica of 1968.) We refer to St. Anselm, who succeeded Lanfranc as prior and then abbot of Bec, and then later as archbishop of Canterbury. No mere pale

1. Stit: A canonical form for agentives

19

theoretician nor private saint, the archbishop was deeply involved in controversy with the tyrant William Rufus and later his brother Henry in regard to the matters of lay investiture and clerical homage; he vigorously opposed the former. These controversies were heavily freighted with the concepts of promising and commitment and agentive powers. In order to make clear that his authority in matters spiritual was not at the pleasure of the king, Anselm refused to accept the papal pallium from the hands of William Rufus. Partly in consequence, the archbishop was in effect exiled by the king. Anselm's brief notes on the modal logic of agency were, we think, composed during this bitter exile. In the document that Henry 1967 calls N, Anselm writes: Quidquid autem 'facere' dicitur, aut facit ut sit aliquid, aut facit ut non sit aliquid. Omne igitur 'facere' dici potest aut 'facere esse' aut 'facere non esse.' (p. 124; from N 29.8.10) Paraphrase by Henry: For all x, if 'x does' is true, then x does so that something either is so or is not so. Hence the analysis of 'doing' will in fact be an analysis of x's doing so that p, and of x's doing so that not-p [where 'p' is a clause describing a state of affairs, and 'not-p' is short for 'it is not the case that p']. (p. 124) Anselm goes on to describe a kind of square of opposition that clearly indicates he had in mind a modal logic of agency (to the extent to which that can be said without anachronism), but his work seems to have remained unnoticed until after the stirring of modal logic in this century. If you promise to accept our remark as merely helpful rather than authoritative, we will hazard that Henry 1953 is the first reference to Anselm that appreciates his work as modal, and that Chisholm 1964a (who cites Henry 1960) is the earliest reference to Anselm by an active researcher in this field (see §3B for further discussion of Chisholm). Other references to Anselm on this topic: Danto 1973, Humberstone 1976 (the reference does not appear in the published abstract, Humberstone 1977), the perceptive Walton 1976b, 1976a, and 1980 (which cites Dazeley and Gombocz 1979), and a sterling account in Segerberg 1989. It is amazing (and perhaps a little sad) that over eight hundred years elapsed between Anselm's invention or discovery and the next contributions to the modal logic of agency. The first modern desire for a modal agentive construction seems to have been felt by philosophers working their various ways through the embedding requirements of legal and deontic concepts. One can certainly see the need expressed in the pioneering work of Hohfeld 1919, who introduces locutions such as the following (p. 38). X has a right against Y that he shall stay off the former's land. This Hohfeldian construct plainly embeds the modal agency construction in a "has a right" context. This is true of all of Hohfeld's work: The modal agency construction is always embedded in additional constructions imputing legal rights, duties, powers, and so on. Agency has not yet received a separate modal treatment. (We discuss Hohfeld a bit in §5B.l.)

20

Introduction to stit

The next place we know a modal agentive construct to crop up, much more explicitly but still embedded in the context of a normative expression, is in Kanger 1957:

Ought(Y sees to it that F(X, Y)). (p. 42) Although the locution "sees to it that" is displayed only in a normative context and wholly without comment, it is clear from the general tenor of Kanger's methodology that he intended to be isolating a norm-free concept of agency. The explicit grammatical breakthrough for the logic of agentive modality comes in Anderson 1962, who, reflecting on Hohfeld, introduces for the first time a separate form of expression intended to disengage the concept of agency from normative considerations.5 When on p. 40 Anderson takes

M(x, p, y) to represent the case "when x executes what is regarded as an 'action' ... and y is the recipient or patient of the action executed by x," he suddenly gives us a clean target for some analytic questions that otherwise seem confusing. Anderson sometimes reads M(x, p, y), with perhaps too little attention to the connections between formal and English grammar, as "x does p to (for) y." Evidently here agency is, for better or worse, not separated from patiency. And certainly there is in Anderson no semantic theory of agency or patiency, and only a trace of a deductive calculus (e.g., Anderson points out that the implication between ~M(x, p, y) and M(x, ~p, y) goes only from right to left). That is, Anderson pioneers in isolating agency and patiency, but he does so only immediately to recombine them with deontic concepts. In 1963 each of two logicians, Fitch and von Wright, advanced modal theories of agency, each of them stressing syntactic developments. Fitch 1963 defines "does A" in terms of two other modalities, "striving for" and "causes," and offers a deductive calculus. The work has not been taken up by later logicians and is seldom cited in the published literature. Indeed, although NB was Fitch's admiring and fond student and colleague, he regrets to say that he had to be reminded of this essay by Segerberg 1989, which contains a maximally useful account. Von Wright, beginning in 1963 and continuing at least through 1981, was, we think, the other logician to be a first to treat agency (or action) as a specific modal or quasi-modal topic, always with that specially honest von Wrightian insistence on the lack of finality of the formulation in question, including attending to nonmodal formulations in which complements are taken as terms signifying specific or generic actions, rather than sentences. As in other cases, the work keeps a close eye on deontic logic, to which he contributed so much. We think von Wright did not succeed in disentangling agency from change, 5 Very likely the breakthrough for Anderson came about after correspondence or conversations with his friend Kanger. Somewhat later Anderson visited Manchester, where Henry was. Henry remarked in personal correspondence that during this year of 1965 there was a colloquium involving a number of persons interested in agency, including, e.g., Hare and Kenny.

1. Stit: A canonical form for agentives

21

evince interest in the general problem of embedding of agentives. For instance (but only "for instance"), von Wright 1963 took as a convenient primitive the notation

d(p/p), which is to be read as expressing some such idea as "the agent preserves the state described by p" (pp. 43, 57). Like Anderson, von Wright tends to leave to the reader the task of putting bits of logical grammar together with bits of English grammar. In contrast with Anderson, however, agency here has been separated out from patiency. (We further discuss von Wright in §3A.) Kanger and Kanger 1966 introduce as a separate locution X causes F, where F is supposed to be a sentence, but in a fashion like Anderson's, they logicize about it only by setting down that F may be replaced by its logical equivalents, and that the proposition that X causes F implies that F. Three influential lines of research began about the same time as that of the Kangers, each of which highlighted the separate existence of agentive modalities, namely, those initiated by Castaneda, by Kenny, and by Chisholm.6 Castaneda, whose views concerning deontic logic have informed both philosophers and logicians for many years (since at least Castaneda 1954), has much to say that is relevant to agency as a modality. Though his philosophical concerns led him to pursue goals other than the formulation of a modal logic of agency, he repeatedly urged the fundamental importance of the grammatical and logical distinction between "propositions" and "practitions" (a distinction put as clearly as anywhere in Castaneda 1981); but because there is no possibility of constructing a Castaneda "practition" from an arbitrary sentence, in the way for instance that Anderson's M(x, p, y] or von Wright's d(p/p) each permits an arbitrary sentence in place of p, Castaneda practitions cannot themselves serve as the foundation for such a modal logic of agency. (§3D expands our consideration of Castaneda.) Kenny 1963, in the course of initiating a rich literature on the verbal structure of our causal and agentive discourse, says that any "performance" in his technical sense is describable in the form bringing it about that p. And Chisholm 1964b takes the following as a basic locution on which to found an extensive series of definitions and explanations in the vicinity of agency: There is a state of affairs A and a state of affairs B, such that he makes B happen with an end to making A happen, 6 Of course other work on the theory of action has also influenced the modal logic of agency, but that literature is unsurveyably vast. We note as a passing example that there is hardly a one of our past or present departmental colleagues who has not contributed.

22

Introduction to stit

where the letters stand in for "propositional clauses," and where the subject of "makes happen" can be either a person or a state of affairs. The discussions of Kenny and Chisholm, though relevant to logical questions, are themselves not directed toward the formulation of either proof-theoretical or semantic principles governing their respective basic locutions. They are sufficiently closely connected to our project, however, that we return to Kenny and Chisholm respectively in §3C and §3B. This is as accurate a record as we can manage of the early history of the modal logic of agency. If this story is right, then the following gives its gist. History of the modal logic of agency prior to 1969 Anselm c. 1100

facere esse (x does so that p)

Hohfeld 1919

X has a right against Y that he shall stay off the former's land

Kanger 1957

Ought(Y sees to it that F ( X , Y ) )

Anderson 1962

M(x,p, y) (x does p to [or for] y)

Fitch 1963

Does A

von Wright 1963

d(p/p) (the agent preserves the state described by p)

Kanger & Kanger 1966

X causes F

Castaneda 1954ff, Kenny 1963, Chisholm 1964ff

relevant discussions

As we note in several places previously, in chapter 3 we extend our discussion of certain among these figures by more closely relating their thoughts to stit theory: von Wright, Chisholm, Kenny, Castaneda, and Davidson. Also in §5B there is a little more consideration of some Hohfeldian themes. But for now we leave this early part of the history. The first modal logic of agency with an explicit semantics is, we think, that of Chellas 1969. The primitive locution is to be read as "T sees to it that O," where T is an agent and O takes the place of a sentence (pp. 62-63). Chellas only deploys this locution in one context, namely, as the argument of an imperative operator.7 7

But Chellas does not restrict the complement of an imperative operator to sentences having the form A-TO as is required by our Thesis 4.

1. Stit: A canonical form for agentives

23

As for semantics, Chellas takes as a paradigm the technique made famous by Kripke not long before Chellas was writing; we mean deployment of a binary relation between "worlds" in order to clarify modal concepts. Chellas in particular gives a semantic clause for ATO: is true at the present world just in case O is true at all those worlds under the control of—or responsive to the action of—the individual which is the value of T at the present world. (p. 63) The language that Chellas uses in this pioneering explanation, like the "relative possibility" language of Kripke a few years earlier, is neither familiar in itself nor further clarified by Chellas. Perhaps this is the reason that, like his predecessors, Chellas in practice confines his agentive locution to the imperative context from which his need for it sprang, and does not pause to investigate its separate properties. 8 After Chellas there is a substantial group of logicians all of whom have deployed a binary relation or a pair of binary relations in an effort to generate a semantic understanding of an agentive modality that might be used as the complement of an imperative or of a deontic operator; we know of Porn 1970, 1971, 1974, 1977; Needham 1971; Aqvist 1972; Kanger 1972; Hilpinen 1973; Humberstone 1977; Lindahl 1977; and Talja 1980. For a critique of the line of research being described, with special reference to Porn 1970, see Walton 1975; also of note are Walton 1976b, 1976a, 1980, which develop some insights in an independent and more nonsemantic fashion. The earlier Porn articles and that of Aqvist use only a single binary relation; the idea of using two binary relations seems to be independently due to Needham 1971, Kanger 1972, and Hilpinen 1973. (Unless we have overlooked it, there is no cross-mention; we have not seen Needham's M.A. thesis, but make the inference from Porn 1977.) The reason for the second binary relation is given as this: Agency has not only a sufficient condition aspect but a necessary condition aspect (Kanger 1972, p. 109; Hilpinen 1973, p. 119), and one needs a separate relation for each. The later workers in this mini-tradition play variations on this theme. In our judgment this line of investigation, although initially promising, and although producing some useful insights, has not been much followed up for the following reason: It has remained obscure what one is to make of the binary relations that serve as the founding elements of the entire enterprise. Kanger 1972 says, for example, that one of the relations holds between a person and a couple of worlds or indices when everything the person does in the second world is the case in the first; and the other relation holds when the opposite of everything the person does in the second is the case in the first (p. 109). That is far from clear, and no one in the tradition is, in our judgment, any clearer than that. For a final example, we describe and quote at length from Porn 1977, which among those we mention is the most developed grammatical and semantic 8

We discuss Chellas a bit more in §4A, and we rely on his agentive operator, ATO, in several places—writing it, however, as [a cstit: A] in order to conform to our standard symbolism for the stit sentence.

24

Introduction to stit

treatment of agentive modality. (All page references are to Porn 1977 and all words not inside quotation marks are ours.) Dap is read "it is necessary for something which a does that p" (p. 4). It is said (pp. 7-8) that an equivalent concept is found as the definition of "a sees to it that p" in Chellas 1969, chapter 3, section 4, and in Porn 1971. "Consider all those hypothetical situations u' in which the agent does at least as much as he does in u. If v is such a situation, it may be said to be possible relative to what the agent does in u. ... if p is necessary for something that a does in u, then there cannot be a situation which is possible relative to what a does in u and which lacks the state of affairs that p. ... A natural minimal assumption is that the relation [of relative possibility] is reflexive and transitive." (PP. 4-5) D'ap is read "but for a's action it would not be the case that p" (p. 5), and also "p is dependent on a's action." (p. 7) "For the articulation of the truth of D' a p at u we require all hypothetical situations u' such that the opposite of everything that a does in u is the case in u' ... [the relation must be] irreflexive and serial." (pp. 5-6) Further, to connect the two modalities D and D', a condition is imposed that "requires that worlds which are alternatives to a given world under the relation [for Da] be treated as equals in contexts of counteraction conditionality." (p. 6) C' ap is read "p is not independent of a's action." (p. 7) Eap is defined as the conjunction of Dap and C'ap, and read "a brings it about (causes it to be the case that, effects that) p" (p. 7). Porn says (p. 8) that an equivalent concept is found in Needham 1971, p. 154, an essentially equivalent concept in Hilpinen 1973, section 6, and explicitly in Porn 1974, p. 96. Porn finds unacceptable (p. 7) an alternative E* a p, defined as the conjunction of Dap and D'ap, which Porn says is equivalent to a definition of Kanger 1972. There are three points to be made about this extract. The first is that given the available apparatus, Porn 1977 seems to us to offer the best explanations of and the most detailed working out of the modal logic of agency as based on abstract binary relational semantics. Second, even these best-possible explanations seem difficult. The conclusion one might draw is that one should doubt the likelihood that the abstract relational-semantic point of view itself can continue to serve in the way that was hoped. But third, however, and counting against this conclusion, is that Porn 1977 is evidently formulating, in the context of the relational semantics, the very combination of "negative" and "positive" conditions that much later were built into stit theory (see §2A.2 and §2A.3). Aqvist 1974, 1978 provide a much less abstract and more intuitive semantic setting; these articles are the first of which we know that make the fundamental

1. Stit: A canonical form for agentives

25

suggestion that agency is illuminated by seeing it in terms of a tree structure such as is familiar from the extensive form of a game as described in von Neumann and Morgenstern 1944. Aqvist's account of agency is in some respects akin to that described in this book, in some respects less flexible, and in some respects richer. His aim is not strictly to provide a modal logic of agency; for example, the primitive of Aqvist 1978 is "DO(a, Pa)" to be read "a does, or acts, in such a way that he Ps," and where "Pa" must be an atomic sentence (rather than an arbitrary sentence), and like von Wright, Aqvist wraps agency together with change. But his goal is close enough to warrant (i) a comparison (which is not attempted here) and (ii) a suggestion that the reader consult these sources. A notable relevant article is Mullock 1988. Of decisive importance is the uncommonly rich joint work Aqvist and Mullock 1989, which applies insights derived from the tree structure to serious questions in the law. Aqvist and Mullock 1989 is, as we later say also of Hamblin 1987, required reading. There is one later commentator on the tradition just described who is of special excellence and interest: Makinson 1986. In a series of more than a dozen articles beginning with Segerberg 1980, and including among others Segerberg 1981, 1982, 1984, 1985a, 1985b, 1987, 1988a, 1988b, and 1989, a distinguished modal logician develops a richly motivated and intuitively based formal approach to action by taking a routine as the guiding concept. Segerberg explicitly bases some of the intuitive and formal aspects of his work on studies that in computer science have come to be called "dynamic logic," the influence on Segerberg being primarily through Pratt. Consult Elgesem 1989 for a sympathetic yet critical and penetrating account of Segerberg's line of research. The work is not fully in the modal logic of agency, since it stresses a grammar of (i) terms (including complex terms) for naming "actions" and (ii) predicates for expressing properties of "actions," and thus self-consciously avoids a grammar of nesting connectives. But instead of a complaint this is intended only as a reason for limiting ourselves to a mere mention of what may indeed turn out to be not only valuable in itself but a useful link between the ontological and modal points of view on agency. Mention of Pratt calls attention to the existence of a large and interesting formal literature that we fail to cite as part of this mini-history except insofar as it has influenced Segerberg, namely the work on "dynamic logic" and its cousins that has been done by Floyd. Hoare, Pratt, and other computer scientists (see Segerberg 1989 for a brief entree via Pratt that is written especially with the logic of action in view, and see Pratt 1980 for an excellent fuller account). There are three reasons for excluding this line of investigation from the present survey: (i) we are very far from familiar with the literature, so that making it accessible is best left to someone else. Further, what we know of it (ii) stresses the ontological rather than the modal approach, whereas the latter is the topic of this mini-history, and (in) what we know of it is relevant to action only in the wide sense of "action" that encompasses mechanical action, that is, the sense of "action" that encompasses the action of programs and starter motors. In fact the present modal point of view makes it arguable that this literature is no more relevant to agency than is the literature of any other discipline that gives us

26

Introduction to stit

ways to fill out the sentential complement of "sees to it that": An agent can see to it that the starter engages and passes through various stages, or that a certain recursive program runs, or ... . But it seems best to make explicit our failure to more than barely mention such a large literature just because so many persons think that although it may be arguable, it certainly isn't plausible that it has no special relevance to agency. On the other hand, NB once asked a well-known computer scientist/mathematician after a lecture on parallel processing if he had meant his use of "actor" and "agent" to be anything but an idle metaphor; he was aghast that one should need to inquire. The articles Brown 1988, 1990, and 1992 represent a sustained and important investigation of the modal approach to ability and its connection with action. Brown initially proposed a modal operator that has something of the force of "can do." Horty 2001, which uses [a bstit: Q] for this Brown connective, explains Brown's ideas and relates them to stit theory. Penultimately there is von Kutschera 1986, which articulates in one form or another nearly all of the essential underlying ideas concerning agency on which we base the semantics offered in subseqent chapters. We can describe the extent of von Kutschera's priority only by using some phrases not defined until later. At the very least, one must credit von Kutschera 1986 with the no choice between undivided histories condition, with generalization beyond the discrete, with generalization to multiple independent agents (including the independence of agents condition), with attention to strategies, and with semantics for the "deliberative stit" that we study in Definition 2-5 and §8G.l. It also needs to be remarked that von Kutschera 1986 cites the earlier von Kutschera 1980. Finally there is Hamblin 1987, which in the context of a study of imperatives provides a rich source of formal, informal, and semi-formal ideas on the topic at hand, many of which have influenced the present work; in particular, collegial reflection on Hamblin's "action-state semantics" was the immediate context of the beginning of the research reported in this book. Our own recommendation is that no one ought to try to move deeply into any part of the theory of agency without reading Hamblin.

1E

Conclusion and summary

Although not so grand as Ishmael's, we think our story well worth the telling, and though we have not sailed far, our mainsail has been unfurled and we have caught the first breeze. What we propose is an augmentation of our current linguistic resources with a linguistic form, the stit sentence [a stit: Q], that (i) leads us carefully to attend to the agent of an action, (ii) is capable of taking any English declarative as its complement, (iii) is recoverable as the same stit sentence either as a declarative or an imperative, and (iv) is grammatically suitable for embedding within wider contexts. Among its other virtues, the stit sentence sheds light on refraining and helps to clarify some of the agentive modalities. This linguistic addition, attentive to grammatical form and semantic

1. Stit: A canonical form for agentives

27

structure, promotes greater clarity in the way we talk and think about the phenomena of our world, and thus justifies its added complexity. In succeeding chapters we strive to deepen our understanding of agency by providing stit sentences with careful and well-motivated semantic analyses, and we apply and generalize on it in a variety of ways.

2

Stit: Introductory theory, semantics, and applications In chapter 1 we followed and extended the idea, going back at least to Anselm, of treating agency as a modality—a modality that represents through an intensional operator the agency, or action, of some individual in bringing about a particular state of affairs.* We proposed that using the stit construction as a normal form, when we are confused, is a happy way to clarify some aspects of action and agency. In this chapter we sharpen our understanding of stit—and thereby, if we are right, of agency. The central idea is that the concept of action must be understood in relation to an open future, and we formulate a rigorous theory that tries to understand how action is compatible with and indeed requires irideterminism. In this way we essay a contribution to what Kane 1998 calls ''the intelligibility question" (p. 105). We often label the chosen approach stit theory, because it concentrates on the linguistic form "a (an agent) sees to it that A," which we abbreviate simply as [a stit: A]. Part of the theory, however, has nothing to do with language. This nonlinguistic part instead purports to articulate in a general way how agency fits into the overall causal structure of our world. For reasons that will emerge, we often ponderously refer to this theory as "the theory of agents and choices in branching time,'' or, more briefly but less memorably, as BT + AC theory— or, when endowed with instants or times, as BT + I + AC theory. Stit theory, including its nonlinguistic part, provides a precise and intuitively compelling semantic account of the stit operator within an overall logical framework of indeterminism; the account is then used as a springboard for investigating a number of topics from the general logic of agency, such as the proper treatment of certain concepts naturally thought of as involving iterations of the agency "This chapter draws on several sources. We thank John Horty for co-authoring §2A, which, with the permission of Kluwer Academic Publishers, is drawn from Horty and Belnap 1995. §2B, except for §2B.10, finds its source in a portion of Belnap 1991, for the use of which we thank the International Phenomenological Society, while §2B.10 is based on part of Belnap 1996a, with the permission of Kluwer Academic Publishers.

28

2. Stit: Introductory theory, semantics, and applications

29

operator, as well as interactions of this operator with other truth-functional and modal connectives. The theory of agents and choices in branching time supports more than one semantic candidate for stit. We mention several of these briefly in various later chapters, but only two are primary in this book. One of the two derives from the early work of two of the authors, NB and MP. The other, which is simpler and for certain purposes easier to use, first appeared in von Kutschera 1986, prior to the work of the authors of this book; later one of the authors of §2A of this chapter independently suggested it, in Horty 1989, explicitly as an alternative to the account of stit put forth by NB and MP. The first major purpose of the present chapter, which we carry out in §2A, is to describe in §2A.l the underlying theory of branching time and the tense logic that is appropriate to that theory, and then in §2A.2 and §2A.3 to describe the individual semantics of each of the two stits, together with the theory of agents and choices on which they rest. These discussions are all explicitly intended to be preliminary; later chapters treat B T + I + A C theory and each of the two stit operators with more rigor and in more detail. A second purpose of this chapter, accomplished in §2B, is to offer an equally preliminary exploration, with many pictures, of some applications of the first, more complicated of the two stit operators. In order to distinguish between the two agency operators under discussion, and for other reasons that will soon become apparent, we describe the operator featured in the early work of NB and MP as the achievement stit, represented in this section as "astit"; and we describe the alternative suggested by von Kutschera and Horty as the deliberative stit, represented here as "dstit." When in this book we speak simply of a stit operator—or use "stit" alone as an operator in some sentence—we often mean to generalize over both the deliberative and achievement stit operators, and perhaps others of the same family. Other times, however, we use plain "stit" as meaning one of astit or dstit, context making it clear which stit is at issue.

2A 2A.1

Theory and semantics: The two stits Background: Branching time

Stit theory is cast against the background of an indeterministic temporal framework, in particular, the theory of branching time due originally to Prior 1967, pp. 126-127, and developed in more detail in Thomason 1970 and Thomason 1984.l 1 From time to time in this book we insert the reminder that although we use the phrase "branching time" because of its fixed place in the literature, we never, ever mean to suggest that time itself—which is presumably best thought of as linear—ever, ever "branches." The less misleading phrase, which we occasionally use, is "branching histories," with an essential plural to convey that it is the entire assemblage of histories that has a branching structure. A single history cannot branch, but two histories can branch from each other. As a further occasional reminder, we observe that although in this book we idealize each history as linear, there is a more adequate treatment of branching histories in Belnap 1992. There each history in the branching assemblage is idealized not as linear, but instead as a four-dimensional spacetime. For similar ideas see also McCall 1994, Rakic 1997, and Placek 2000.

Introduction to stit

30

Figure 2.1: Branching time: Moments and histories

The theory.is based on a picture of moments as ordered into a treelike structure, with forward branching representing the openness or indeterminacy of the future and the absence of backward branching representing the determinacy of the past. Such a picture leads, formally, to a notion of branching temporal structures as structures (we sometimes say BT structures) of the form (Tree, <), in which Tree is a nonempty set of moments and < is a treelike partial ordering of these moments—an ordering such that, for any m1, m2, and m3 in Tree, if m1 < m3 and m2 < m3 then either m1 < m2 or m2 < m1. We routinely use m1 < m2 when m1 < m2 but m2 < m1. A maximal set of linearly ordered moments from Tree is a history, representing some complete temporal evolution of the world. If m is a moment and h is a history, then the statement that m 6 h can be taken to mean that m occurs at some point in the course of the history h. Because of indeterminism, a single moment might be contained in several distinct histories: We let H(m) = {h: m & h} represent the set of histories passing through m, those histories in which m occurs. These ideas can be illustrated as in Figure 2.1, where the upward direction represents the forward direction of time.2 This diagram depicts a branching temporal structure containing five histories, h1 through h5. The moments m1 through m4 are highlighted; and we have, for example, m2 E h3 and H(m4) = {h4, h5}.

In evaluating sentences against the background of these branching temporal structures, it is a straightforward matter to define a notion of truth-at-a-moment that is adequate for the truth-functional connectives, and even for the opera2

We represent the directionality of time in pictures by means of up-down, which led us in earlier publications to the language of "downward" and "upward." Many presentations of tense logic, however, use a left-right representation, which is evidently equally helpful. In order to use language that favors neither picture over the other, in this book we always use the neutral words "backward" and "forward." Or almost always: We retain "upper bound" and "lower bound" as too entrenched to abandon.

2. Stit: Introductory theory, semantics, and applications

31

tor Was: that represents the simple past tense: The definitions from standard (linear) tense logic suffice. 3 Since these structures allow alternative possible futures, however, it is not so easy to understand the operator Will:, representing future tense. Returning again to Figure 2.1, suppose that, as depicted, the sentence A is true at m3 and at m4, but nowhere else. In that case, what truth value should be assigned to Will:A at the moment m1? On the approach developed by Prior and Thomason, there is just no way to answer this question. Evidently, Will:A is true at m1—A really does lie in the future—if one of the histories h2, h4, or h5 is realized; but it is false on the histories h1 and h3. And since, at m1, each of these histories is still open as a possibility, that is simply all we can say about the situation. In general, in the context of branching time, a moment alone does not seem to provide enough information for evaluating a statement about the future; and what Prior and Thomason suggest instead is that a future-tensed statement must be evaluated with respect to a more complicated point of evaluation consisting of a moment together with a history through that moment. We let m/h represent such a point: a pair consisting of a moment m and a history h from H( m ) . Since future-tensed statements are to be evaluated at moments and histories together, semantic uniformity suggests that other sentences must be evaluated at these more complicated indices as well. We therefore define a BT model (branching time model) as a pair m = , in which G is a BT structure , and 3 is an interpretation function mapping each propositional constant from the background language into the set of m/h pairs at which, intuitively, it is thought of as true. Where = represents, as usual, the relation between a point of evaluation belonging to some model and the sentences true at that point, the base case of the truth definition for branching temporal models tells us simply that atomic sentences are true where 3 says they are (see §8F.l). 2-1 DEFINITION. (Truth for atomic sentences) for A an atomic sentence.4

m,

m/h

= A iff m/h E J(A)

And the definition extends to truth functions, past, and future as follows (see §8F.2 and §8F.5). 2-2 DEFINITION. (Truth for truth functions and tenses) and 3

We remind the reader that, as explained in Remark 1-1, we use nonstandard notation for various connectives. Taking into account the large number of connectives used in this book, our purpose is to be as easy as possible on both the eye and the memory. 4 It is not usual for languages of this kind to admit the possibility that even atomic sentences might be true at one point of evaluation m/h but false at another point m/h', for different histories h and h' belonging to H(m). What we have in mind are situations such as the following. If, in a restaurant, Karl is offered cake or pie for dessert, it seems that "Karl chooses pie," which is at least not obviously non-atomic, might be true relative to one history through m, but false relative to another. In any case, whether or not relativizing truth for atomic sentences to both moments and histories is actually necessary for evaluating statements of this kind, allowing for the possibility at least does no harm.

32

Introduction to stit

Was: A iff there is an

such that

Will:A iff there is an

such that

and

As usual, we say that a sentence is valid if it is true at every point of evaluation— in this case, every m/h pair—in every BT model. It is easy to see that, as long as we confine ourselves to Was:, Will:, and truth-functional connectives, the validities generated by this definition in branching temporal models coincide with those of ordinary linear tense logic, for the evaluation rules associated with these operators never look outside the (linear) history of evaluation. However, the framework of branching time allows us to supplement the usual temporal operators with the additional concept of settledness, or historical necessity, along with the dual concept of historical possibility. Here, Sett:A is taken to mean that A is settled, or historically necessary; Poss:A, that A is still open as a possibility. The intuitive idea is that Sett:A should be true at some moment if A is true at that moment no matter how the future turns out, and that Poss:A should be true if there is still some way the future might evolve that would lead to the truth of A. The evaluation rule for historical necessity is straightforward (see §8F.4). 2-3 DEFINITION. (Truth for historical modalities) Sett

for

• Poss:A is defined in the usual way, as ~Sett:~A. It is convenient to incorporate this concept of settledness also into the metalanguage: We will say that A is settled true at a moment m in a model m iff m, m/h = A for each h in H(m), and that A is settled false at m iff m, m/h = A for each h in H(m) (see Def. 17). Once the standard temporal operators are augmented with these concepts of historical necessity and possibility, the framework of branching time poses some technical challenges not associated with standard tense logics. On the other hand, it is directly applicable to a number of philosophical issues, such as the representation of indeterminism, for which standard tense logic is no help. Details and references can be found in Thomason 1984 and Zanardo 1996. In this book, chapter 6 and chapter 7 contain extended discussions of branching time as a representation of indeterminism (summarized in §3), while chapter 8 explains the required semantics in exact detail.

2A.2

The achievement stit

The stit operator introduced in chapter 1 is designed to approximate the idea of seeing to it that. More exactly, a statement of the form [a astit: A] should be taken to mean something like:

2. Stit: Introductory theory, semantics, and applications

33

the present momentary fact that A is guaranteed by a prior choice of the agent a. And it is for this reason, because it is used to describe the present momentary outcome of an agent's prior activity, that we characterize this operator as the achievement stit. In order to capture the meaning of the achievement stit, we must be able to speak of an individual agent's choices, and also, evidently, of the present. As a means of representing these concepts, the basic framework of branching time is supplemented with three additional primitives. (See chapter 7 for details, and §3 for summaries.) The first is simply a set Agent of agents, individuals thought of as making choices, or acting, in time.5 Now what is it for one of these agents to act, or choose, in this way? We idealize by ignoring any intentional components involved in the concept of action, by ignoring vagueness and probability, and also by treating acts as instantaneous. In this rarefied environment, the idea of acting or choosing can be thought of simply as constraining the course of events to lie within some definite subset of the possible histories still available. When Jones butters the toast, for example, the nature of his act, on this view, is to constrain the history to be realized so that it must lie among those in which he butters the toast. Of course, such an act still leaves room for a good deal of variation in the future course of events, and so cannot determine a unique history; but it does rule out all those histories in which he does not butter the toast. The second primitive introduced in earlier chapters, then, is a device for representing the constraints that an individual is able to exercise upon the course of history at a given moment, that is, the acts or choices open to him at that moment. Formally, these constraints can be encoded through a choice function, mapping each agent a and moment m into a partition Choiceamof the histories .H(m) through m. The equivalence classes belonging to Choiceamcan be thought of as the possible choices or actions available to a at m; and the idea 5

This book, while expressly designed to contribute to our understand of agency, and therefore of action, has tried to avoid using language that might suggest that the authors understood the ontology of actions. (The relation of stit semantics to some of the previous philosophical work on agency and action is discussed in §1D and chapter 3.) In the present section, we have been somewhat more relaxed in informal passages about using devices such as singular terms that purport to refer to actions as things in the world, but we should nevertheless be understood in exactly the same spirit. For example, when we say "moment of action," we certainly mean to be calling the reader's attention to a particular moment, but we do not intend to suggest that we understand what, if anything, could be meant by saying that there is an x such that x is an action and x is located at that particular moment. Roughly the same remarks hold for "moment of choice," which we are using as interchangeable with "moment of action" in spite of the following: Literary convention easily permits using "moment of choice" for an earlier moment of indecision, while tending to reserve "moment of action" for a later moment shortly after "the action" has commenced. This literary distinction—reminiscent also of Zeno—suggests to us the importance of highlighting the transition from "not-having-acted (or chosen)" to "having-acted (or chosen)" as essentially involving two regions of branching time, not just one, a thought spelled out in §7A.4, and briefly taken up from time to time elsewhere in this book.

Introduction to stit

34

Figure 2.2: An agent's choices

is that, by acting at m, the agent a is able to determine a particular one of the equivalence classes from Choiceamwithin which the future course of history must then lie, but that this is the extent of his influence. As additional notation, we let Choice am (h) (defined only when h E(m) represent the particular possible choice from Choiceamcontaining the history h. And of course, in order for this choice information to make any sense, we must require that any two histories in H( m ) that have not yet divided at m must lie within the same possible choice; the choices available to an agent at m should not allow a distinction between two histories that do not divide until some later moment. The information represented through these choice functions can be illustrated as in Figure 2.2, which depicts a structure containing six histories, and in which the actions available to the agent a at three moments are highlighted. The cells at the highlighted moments represent the possible choices or actions available to a at those moments. For example, a has three possible choices at m1— Choice am = {{h1, h2},{h3},{h4,h 5 ,h 6 } } — a n dtwo at m2. Because h1 and h2 are still undivided at m1, they must fall within the same partition there, and likewise for h4 and h5. At m3 the agent a effectively has no choice: Histories divide, but there is nothing a can do to constrain the outcome. (It may be that the outcome can be influenced by some other agent whose choices are not depicted here; or perhaps it is something that just happens, one of nature's choices.) At such a moment, it would be possible to treat the choice function as undefined for a; but it is easier to treat it as defined but vacuous, placing the entire set of histories through the moment in a single equivalence class. Little more is needed to provide for multiple agents: Given a moment, m, there is Choicea1m, representing the choices available to a1 at m, and there is Choicea2mwhich represents the choices available to a2 at exactly that same moment. The overall box in Figure 2.3, which looks like and is supposed to look like a von Neumann normal form of a game, is a blowup of the single moment m. The vertical split represents two possible choices for a1 at m, while the horizontal split represents two possible choices for a2. Since the choices of a1

2. Stit: Introductory theory, semantics, and applications

35

Figure 2.3: Multiple agency

and a2 are simultaneous, however, one must impose the additional constraint that Choicea1mand Choicea2mare "independent": Every combination of a choice by a1 at m and a simultaneous choice by a2 is an open possibility. In other words, no choice by a1 can make it impossible for a2 to make a simultaneous choice. This is illustrated in Figure 2.3 by the fact that from each (inner) box there proceeds at least one history. The independence-of-choices constraint is, of course, to be taken to apply to the simultaneous choices of any number of agents. Gathering the individual primitives into a tuple gives us {Tree, <, Agent, Choice>, which we call an "agents and choices in branching time structure," or, more briefly, a BT + AC structure. (The exact definition is given in §2.) The final primitive is a set Instant of instants partitioning the moments of Tree horizontally into equivalence classes. Intuitively, an instant represents a set of contemporaneous moments from each of the various histories, with the different moments belonging to a single instant thought of as occurring at the "same time" in the different histories. The instant containing the moment m is represented as i(m). It is supposed that each instant meets (intersects with) each history at exactly one moment, and that instants respect the temporal order of histories in the following sense: If the moment at which an instant i1 meets a history h is later than the moment at which i2 meets h, then the same relation holds between the moments at which the instants i1 and i2 meet any other histories. These suppositions about instants amount to strong restrictions on the structure of Tree, satisfiable only if all histories share an isomorphic temporal ordering, which is then inherited by the instants themselves. The point of the restrictions, of course, is to allow for temporal comparisons between moments from different histories. When the basic framework of agents and choices in branching time is supplemented with the Instant primitive, the result is an "agents and choices in branching time with instants" structure, or BT +1 + AC structure, of the form (Tree, ^, Instant, Agent, Choice>, with BT + AC as before; and we can define BT + I + AC models as structures of the form m = , in which 6 is a BT + 1 + AC structure and 3 an interpretation mapping each propositional constant, as before, into a set of m/h pairs. Also, when we combine branching time with instants, but without agents and choices, into a tuple (Tree, <, Instant),

Introduction to stit

36

Figure 2.4: [a astit: A] true at m/h

we have a BT + I structure. It is these structures that provide the backdrop for the semantics of the achievement stit; the claim is that the structures are not just mathematical curiosities, but describe—up to a legitimate idealization—the world in which agents act. Before stating the evaluation rule for the achievement stit, we first require an auxiliary definition. Suppose that the moments m1 and m2 occur at the same instant (i( m 1 ) = i(m 2 )), and consider some moment w prior to both (w < m1 and w < m2). If m1 and m2 lie on histories belonging to the same Choiceawpartition, these two moments are then said to be Choiceaw-equivalent (the underlining to suggest that the equivalent moments occur at the same instant). The idea behind this definition is that, through his choice at w, the agent a can guarantee that whatever moment occurs at the instant i( m 1 ) (= i(m 2 )) will lie within some particular choice aw-equivalence class, but there is nothing he can do to determine which of the moments within that class it will be. Using this auxiliary concept, the rule for evaluating an achievement stit, [a astit: A], at a point m/h of a BT + I + AC model m can now be set out as follows (§8G.3): 2-4 DEFINITION. (Truth for the achievement stit)

9m, m/h = [a astit: A] iff

there is a moment w < m such that (i) for all moments m1 Choice aw-equivalent

to m, we have m, m1/h1 = A for all h1 E H( m 1 ) ; and (ii) there is some moment m2 E i(m) such that w < m2 and m, m2/h2 = A for some h2 E H( m2 ). This formidable definition (which is equivalent to that of §8G.3) can be grasped more easily by reference to Figure 2.4, depicting a situation in which [a astit: A] is true at m/h, as a result of an action by a at the prior moment w, known as a witness, which action is determined as effective by A being possibly false at the moment m2, known as the counter.6 The evaluation rule embodies two requirements, positive and negative, captured by clauses (i) and (ii}. The positive requirement is that, as a result of a prior choice by a at the witnessing moment w, things have evolved in such a way that A is guaranteed now, at the instant 6

A convention for interpreting these figures: When a sentence is written next to a moment, it should be taken as settled true at that moment. Thus, for example, the sentence A is taken as settled true at the moment m in Figure 2.4.

2. Skit: Introductory theory, semantics, and applications

37

of TO, to be true. (In a generalization described most thoroughly in §8G.4, the witness for [a astit: A] may also be a chain of moments.) Of course, since a is unable to determine a single history through his choices at w, he could not have guaranteed that we should now have arrived at TO; but he was able to guarantee that we should now have arrived at some moment Choiceaw-equivalentto TO, and A is settled true at all of these. The negative requirement is that it was not yet settled at w that A should now (at i( m )) be true, so that a's action at w did have some real effect in bringing about the present truth of A.

2A.3

The deliberative stit

The primary conceptual difference between the achievement stit and the deliberative stit is this. The truth of an achievement stit, [a astit: A], depends on two separated moments, the first being the moment at which both the stit sentence and the outcome A are evaluated, and the second being the required prior moment of choice or action, at which a guarantees the outcome. By contrast, the deliberative stit is referred only to a single moment: A sentence of the form [a dstit: A] is evaluated at the moment of choice or action, the very moment at which the agent a sees to it that A. We might say that the deliberative stit should be interpreted by something like that A is guaranteed by a present choice of Because it is only the future that can be affected by our actions or choices, it is usually natural to take the complement of a deliberative stit as future tensed; and it is for this reason also that this stit concept is characterized as deliberative. The terminology echoes most immediately the notion of deliberative obligation from Thomason 1981a, but it goes back to Aristotle's observation in the Nichomachean Ethics that we can properly be said to deliberate only about "what is future and capable of being otherwise" (1139b7; see also 1112al9-blO). Since it involves reference only to a single moment-history pair, the truth conditions for the deliberative stit can be stated easily (see §8G.l for a little more detail). 2-5 DEFINITION. (Truth for the deliberative stit) m, m/h = [a dstit: A] iff (i) 971, m/h' = A for each h' E Choice am(h), and (ii) there is some h" E H(m) for which m, m/h" = A. Evidently, clauses (i) and (ii) here are analogous to the positive and negative requirements from the achievement stit. In the present case, the positive requirement is simply that a should act at m in such a way that the truth of A is guaranteed; a should constrain the histories through TO to lie among those on which A is true. The negative requirement, again, is that A should not be settled true, so that a's actions can be seen as having some real effect. In addition to the primary, one-moment/two-moment contrast between the achievement and deliberative stits, there are two other differences that should be mentioned at once.

38

Introduction to stit

The first concerns the role of histories. Although the indices at which an achievement stit is evaluated contain both moments and histories, the histories are nearly idle in the evaluation rule, needed only as objects of quantification in deciding whether A is settled true or not settled true at some moment. An achievement stit, if true at some moment-history pair, must be true at every history through that moment; this fact is reflected in the validity of

which tells us that any true achievement stit is settled true. For similar reasons, the theory yields also the validity of

the complement of any true achievement stit must itself be settled true. By contrast, since deliberative stit statements are evaluated at the very moment of an agent's choice or action, histories must play a more central role in their evaluation; they provide our only access to the outcome of the agent's action. The theory therefore yields validities such as the following:

The first of these tells us that a deliberative stit is never settled true; the second that it can be true only if its complement is contingent, again, by contraposition, reflecting Aristotle's idea that there is no deliberating about matters settled by necessity. The final point of contrast between the achievement and deliberative stits concerns the role of instants. These play an essential role in the semantics for the achievement stit, but no role at all in the deliberative stit. Because of this, models for evaluating deliberative stits alone can be simpler than the stit models described earlier: They need not contain Instant as a primitive, and so do not require us to assume a notion of "same time" across different histories in order to make sense of agency. They can be based on BT + AC structures instead of on BT + I+AC structures. One way of understanding the semantic differences between the achievement and deliberative stits that result from the reliance of the former on instants is by considering the following two sentences:

These sentences may seem to express plausible principles of interaction between the two stits, but in fact, both are invalid, as we can see from the model depicted in Figure 2.5. Here [a astit: A] is settled true at the moment m2, with m1 as witness: The positive requirement is satisfied because A is true at every moment Choiceam-equivalent to m2, and the negative requirement is satisfied because there is a moment m4 in i(m2) at which A is not settled true. However, [a dstit: Will:A] is not true at m1h1: Although the positive requirement that Will:A

2. Stit: Introductory theory, semantics, and applications

39

Figure 2.5: A countermodel to some stit relations

should be true at m1/h' for each h' in Choiceam1 ( h 1 ) is satisfied, the negative requirement, that there should be some h" in H( m1 ) such that Will: A fail at m1/h", is not. It is easy to see also that [a dstit: Will: B] holds at mi/hi, but that there is no point in the future of m1 along h1 at which [a astit: B] holds. In spite of these contrasts, it is well to keep in mind that each of the achievement stit and the deliberative stit represents, in its own way, a transition from indeterminacy to a determinate outcome due to the choice or action of some agent. In the case of the achievement stit, the settled outcome is at some temporal remove, whereas in the case of the deliberative stit, the outcome is immediate. The sign of this for dstit is the validity of the following:

Even though neither [a dstit: A] nor Will:[a dstit: A] is ever settled true, Was: dstit: A] becomes a determinate, permanently settled fact immediately after the choice is made. (We use the idea of (*) later, on p. 186.)

2B

Applications of stit, with many pictures

We devote the rest of this chapter to using the achievement stit to answer some hovering questions, using pictures whenever we can. It is in fact the possibility of meaningful pictures, we think, that constitutes an advantage of these semantics over those mentioned in §1D. The pictures are a way to give force to the Braithwaite slogan, "no calculus without calculation." So that the pictures can be efficient conceptual supports of calculation, and at the same time convey intuitive significance, please give them the benefit of the doubt by treating as not relevant what is not explicitly drawn. In the following we shall be dealing exclusively with the achievement stit. Since therefore there is no danger of confusion with dstit, we shall write just "stit" instead of "astit."

40

Introduction to stit

Figure 2.6: Stit not closed under logical consequence

2B.1

Stit not closed under logical consequence

2-6 QUESTION. (Closure under consequence?) Is what a sees to closed under logical consequence? As a paradigm, is it possible that a sees to the conjunction of A and B but does not see to it that B? Of course it's possible, as Figure 2.6 makes obvious. In this and later figures, we make clear which values are given, and which are calculated. The point of Figure 2.6 is that [a stit: A & B ] holds at the moment m1, but even though B is a logical consequent of A&B, nevertheless [a stit: B] fails there. The crucial fact is that the negative condition for [a stit: A&.B] is satisfied in virtue of the counter, m2, since A&B fails there because of the falsity of A. In contrast, at the only potential witness mo, B is already settled true where each history through mo intersects the relevant instant, i( m1 ). The negative condition for [a stit: B] is therefore violated. B is true, all right; it is just that B is not something to which a has seen. So what a sees to is not closed under logical consequence, and obviously so. There is not the slightest paradox in saying, there is neither "funny logic" nor grammatical subtlety required in calculating, that from the fact that you see to it that there is at least one injured man who is bandaged, it does not follow that you see to it that there is at least one injured man, even though that there is at least one injured man who is bandaged logically implies that there is at least one injured man. To the contrary, it is deeply built into the real-choice-based idea of agency that such cases should be typical.

2B.2

Refraining

2-7 QUESTION. (Refraining) Is refraining both doing and not doing (in the same respect, at the same time, etc.)? The trouble with refraining, as most people appreciate, is that it is often hard to do. The trouble with refraining, as most philosophers appreciate, is that it is hard to pin down because it is both acting and not acting; discussions of this topic in the recent literature show how difficult it is to avoid being confused. Consider the two imperatives:

2. Stit: Introductory theory, semantics, and applications

41

Turn not thy back to the compass. Refrain from turning thy back to the compass. These stand-alone imperatives seem to have the same content. That is, they seem to offer the same advice, or create the same obligation, or make the same demand, ..., depending on the force with which they are uttered. Maybe that will help us see what it is to refrain. To refrain from turning thy back certainly entails not turning thy back, that is, not acting in a certain way. But we cannot accept that refraining from turning thy back is the same as not turning thy back, for not turning thy back does not entail refraining from turning thy back: There are plenty of things in the world out there that are neither turning their backs nor refraining from turning their backs. Further, since many things that aren't turning their backs are not even agents, it cannot be that not turning thy back is necessarily acting in a certain way. A standard but often useless analytic technique when faced with this sort of situation is to suggest that therefore refraining from turning thy back is: not turning thy back plus "something else." But what else? One might think the something else is mental, and certainly the mind may come into it, but let us leave it out as long as we can, for surely thou canst refrain from turning thy back without paying attention, without having a plan not to turn thy back, even without intending not to turn thy back. Also one might think that the something else has something to do with obeying or refusing an order, piece of advice, demand, request, and so on, but to think the thought is to reject it: Thou canst refrain from turning thy back to the compass without there being any such injunction anywhere about. If, however, we look for the solution not in something external to the canonical form, but as something already contained in it, we take the first step toward the appropriate solution. As von Wright 1963 says (p. 45), refraining is a doing, it's a kind of action. But of course the action in question is not the action of turning thy back! So maybe, someone might think, refraining from turning thy back is the action of not turning thy back; that sounds kind of right. But it cannot be exactly or clearly right, since we just said that not turning thy back does not entail refraining from turning thy back, and is not even an action, and that sounds right, too. So the problem is still with us: Refraining seems to be not acting (not turning thy back), but also acting (not turning thy back). No wonder von Wright posited refraining (or forbearing) as indefinable in terms of action alone.7 Our perplexity makes us see why it is so tempting to start trying to distinguish Not acting: not turning thy back to the compass 7 Von Wright 1963 tried a definition of the "something else" in terms of "ability," an attempt rejected by Brand 1970, pp. 234-235. The intuitive facts are difficult to disentangle. Even the technical facts are a little complicated. For the achievement version of stit and of refraining explained in this chapter, refraining cannot be defined by combining stit and ability in the way suggested by von Wright. In contrast, if stit is taken as the deliberative stit, then as shown in Horty 2001, von Wright's suggestion is exactly on target.

42

Introduction to stit

from Acting: not turning thy back to the compass by beginning to make noises about external versus internal negation, even when we know perfectly well that we do not know what we mean by this distinction; and why it is tempting to invent the-act-of-not-turning-thy-back-to-thecompass, thus enriching our ontology, also without having any sense that we know what we are talking about. In the framework of stit, with its clear recognition that embedding is not only possible but encouraged, there is a natural solution to this puzzle about refraining. It involves a simple series of steps: • "a is turning his back to the compass" is equivalent to "[a stit: a is turning a's back to the compass]." Indeed, this equivalence is, by Thesis 3, our test for a sentence agentive for a, a test that will clearly distinguish "Stubb is turning his back to the compass" from "Queequeg is a native of Kokovoko." • "a is not turning his back to the compass" is equivalent to "~(a is turning s back to the compass)." • Conclude that "a is not turning his back to the compass" is equivalent to "~[a stit: a is turning a's back to the compass]." is turning a's back to the compass]" is not agentive for that is (by a negative use of the stit paraphrase thesis, Thesis 3), it is not in general equivalent to "[a stit: ~[a stit: a is turning a's back to the compass]]." • Recall that refraining is agentive, so that the content of a refraining sentence must, by Thesis 4, have a canonical form [a stit: Q], and equally keep in mind that the negative imperative "Turn not thy back to the compass" must also have an agentive canonical form. • The following are then seen to be equivalent: a refrains from turning his back to the compass and

is turning a's back to the compass]]. • Now you can (literally) see that refraining from turning thy back to the compass is agentive and also (literally) involves an inaction: Refraining from turning thy back to the compass is (actively) seeing to it that thou dost not (actively) turn thy back to the compass.

2. Stit: Introductory theory, semantics, and applications

43

But let us continue more generally. We know that any agentive proposition can be expressed in the form [a stit: Q], where Q might or might not itself be agentive. We now add that any refraining proposition can be expressed in the form

whether Q is agentive or not. (English will always demand that the subject of Q is a, but it does not care whether or not Q is agentive for a.) The "Refrain from turning thy back to the compass" example chose Q as agentive, but there is also "Refrain from being (or don't be) on deck at dawn", or "Don't get caught by the cook." Because in these cases Q is not agentive, they have simpler, nonnested analyses, just "[a stit: ~(a is on deck at dawn)]," or "[a stit: ~(a is caught by the cook)]." Without postulating new "negative acts" as ontological items or strange undefined internal versus external negations, all is clear: The agency of "a refrains from seeing to it that Q" flows from the agency of [a stit: ~Q], and the negation comes from what it is that a sees to; namely, that not Q. When Q is itself agentive, this can of course be further filled out, as in the "turning thy back to the compass" example. Now we see the drive for the double use of "a isn't turning a's back to the compass". If taken as non-agentive, it is canonically just is turning a's back to the compass]; but if it is to be taken in an agentive sense, then you will have to read it as: stit: a is turning a's back to the compass]]. English does not have a short and precise way to make the distinction; but the stit locution does the job. Who would have thought that refraining from acting involved an embedding of a non-acting within an acting, a non-agentive within an agentive? Only through attending to the grammar of the canonical form [a stit: Q], which promotes such embedding by its very design, does the truth of the matter become accessible. The concept of refraining is confusing and difficult to think about without the aid of theory and pictures. Figure 2.7 vividly illustrates the differences required, and permits the essential calculations. (This and succeeding pictures represent choices only for a, since only those happen to be relevant to the particular points to be made here.) i. Easiest to see is that [a stit: ~Q] holds at mo, with witness Wo and counter at m1. ii. It will also be useful to note of Figure 2.7 that [a stit: Q] holds at m1, with witness w1 and counter at my,. (Pause to observe that Wo cannot serve as witness for [a stit: Q] at m1. Reason: Q fails at m3, which is choice equivalent for a to m1 at Wo, so that there is a violation of the positive condition for Wo to witness [a stit: Q] at m1.)

44

Introduction to stit

Figure 2.7: [a stit: ~Q] versus ~[a stit: Q] versus [a stit: ~[a stit: Q]]

iii. It is clear that at m2, one cannot attribute a guarantee of the fact that Q holds there to any prior choice of a, for that fact was up to nature. The same is true of m3 and m4: At all of m2, m3, and m4, holds, which is the mere absence of seeing to it that. It is worth noticing that we can make this statement about m4 without even knowing whether or not Q itself holds there; all we need to observe is the failure of Q at moment m3, which is choice equivalent to m4 at Wo for a. It follows that the positive condition for Wo to witness [a stit: Q] at m4 fails in virtue of the failure of Q at m3, and since Wo is the only potential witness for [a stit: Q] at m4, it must be that [a stit: Q] fails at m4, which is to say, it must be that ~[a stit: Q] holds there. iv. Moments m2 and m3 on the one hand, and m4 on the other, are quite different with respect to refraining. Figure 2.7 shows that the moment w1 does stand witness to a's responsibility for his or her own inaction with respect to Q at m2 or m3: Not only does the right-hand choice for a at w1 guarantee that a does not see to it that Q, but the left-hand choice from w1, at which a does see to it that Q, testifies that at w1 a had a real choice concerning his or her seeing to it that Q. The moment m1 stands, that is, as the "counter" required for the truth of the claim that at m2, or m3, a saw to it that he or she did not see to it that Q. v. In contrast to moments m2 and m3, you can tell that in fact at moment m4, a did not actively refrain from not seeing to it that Q. The only potential witness is Wo; but since a did see to it that Q at m1, and since m1 is choice equivalent to m4 at Wo for a, the positive condition fails, and thereby the claim to agency. At 7774 not only does a fail to see to it that Q, but he or she also fails to see to it that he or she fails to see to it that Q. At 7774 you can therefore observe the difference between mere

2. Stit: Introductory theory, semantics, and applications

45

not seeing to something on the one hand, and positively refraining on the other, for, as we have calculated, at m4 there is "not seeing to it" without "refraining."

2B.3

Could have A short rushing sound leaped out of the boat; it was the darted iron of Queequeg. Then all in one welded commotion came an invisible push from astern, while forward the boat seemed striking on a ledge; the sail collapsed and exploded; a gush of scalding vapor shot up near by; something rolled and tumbled like an earthquake beneath us. The whole crew were half suffocated as they were tossed helterskelter into the white curdling cream of the squall. Squall, whale, and harpoon had all blended together; and the whale, merely grazed by the iron, escaped.

Queequeg missed. His harpoon did not strike home. But could Queequeg have struck home with his harpoon? Questions of this kind have been notoriously difficult to answer, partly because of an inability to tie down the relevant sense of "could have." 2-8 QUESTION. (Ability) What are we to make of "could have done it"? Consider the vexed example (11) on p. 9, and the consequent result of applying "could have" thereunto: • Queequeg struck home with his harpoon Queequeg could have struck home with his harpoon • Queequeg's harpoon struck home Queequeg's harpoon could have struck home. If you are not clear whether Queequeg's striking home with his harpoon comes to the same thing as Queequeg's harpoon striking home, then you will not be clear whether Queequeg's ability to strike home with his harpoon is about Queequeg's agency or the powers of nature. The one appears more a question of an agent's ability and the other appears more a question of the squall; but paraphrase into canonical form settles the issue: Queequeg could have stit: Queequeg struck home with his harpoon and

Queequeg could have stit: Queequeg's harpoon struck home are both unambiguously concerned with agentive powers. A recipe emerges: Take any sentence, whether agentive, non-agentive, or unclear as to agency, feed it as complement to the stit construction, and the result is bound to be an agentive; feed that intermediate result as the complement of a "could have" and you are bound to have a claim about agency. (Exercise: Carry out this plan for "Queequeg missed.") As notation we suggest

Introduction to stit

46

Figure 2.8: Could have done otherwise

[a could-have-stit: Q] which makes "could have stit" sentences look like the quasi-agentives that they are. The point is that when approaching questions about "could have," first become clear about the complement of "could have," because that is essential to being clear about "could have" itself; further, to focus on the "could have" of agency, paraphrase into canonical form, that is, into a stit sentence. You may think that such canonical forming is too much trouble, and sometimes it is; if you are concerned, however, to give a general account of agentive "could have," then the availability of a canonical form is exactly what is required: (i) Since every stit sentence is an agentive, you will not have explained "could have" unless you explain "could have seen to it that Q" for arbitrary Q. (ii) Since every agentive can be paraphrased as a stit sentence, if you do explain "could have seen to it that Q" for arbitrary Q, you will have done the whole job. Next, refine the question of "could have done it" by letting the "done it" be specialized to "done otherwise." 2-9 QUESTION. (Doing otherwise) How should we understand "could have done otherwise" ? Our chief thought, expanded in chapter 9, is that there are hidden complexities in the use of the phrase "do otherwise." Figure 2.8 helps with the following discussion. Evidently a stit: Q] holds at TOO, with Wo as witness and m3 as counter. The rest of the picture is just like Figure 2.7. Observe in the first place that the other choice available to a at Wo obviously does not guarantee that ~Q, so on that ground alone it was impossible for a

2. Stit: Introductory theory, semantics, and applications

47

to see to it that ~Q. That other possible choice only risks ~Q, but does not guarantee it. So in this diagram the agent is in no position to "do otherwise" in the sense of seeing to ~Q. In the second place, the other choice available to a at Wo does not even guarantee that [a stit: ~[a stit: Q]], that is, that a refrains from seeing to it that Q. It is possible, but it is not guaranteed, since nature can take us straight to m4, where we calculated that a does not refrain. If this is what "could have done otherwise" means, then ''could have done otherwise" is by no means a consequence of taking the future as open. One can also see, however, that at Wo there exists a strategy for a such that if (i) a knows about that strategy, and if (ii) a wishes to follow it, and if (iii) a does not run into problems of weakness of the will, then a is in a position at Wo, in this somewhat Pickwickian or conditional sense of (or absence of sense of) "guarantee," to guarantee that he or she does not see to it that Q, that is, that ~[Q stit: Q]. The strategy is simply to make the right-hand choice at each of Wo and w1: Then, no matter what nature has in store, the issue is bound to be ~[a stit: Q]. The exact statement and proof of this principle, which is a much more complicated matter than one might have expected, is the main theorem of chapter 13. Consult also the applications of the idea of a strategy to promising in chapter 5. But proof aside, one should allow that this weakened "strategic" sense is a long way from what your average expert on free will might have meant by "could have done otherwise," though perhaps it is what the most subtle dialecticians of the topic were implying. In any event, the pictures seem to make the discussion easier to follow.

2B.4

Closure of stit under D

2-10 QUESTION. (Closure under c) Is what one sees to closed under material "implication"? No.8 Oddly enough, as Gupta pointed out to us, it depends on the relative order of the witnesses provided for [a stit: P] and for [a stit: P c Q } . If the witness for P is earlier (or the same as) the witness for P c Q , the "detachment" is guaranteed. The detachment might, however, fail when the witness for Pc Q is earlier. This is clear from Figure 2.9 and Figure 2.10: In Figure 2.9 the witness w1 for the seeing to it that P at m0 is earlier than the witness Wo for the seeing to it of the conditional atTOO;in that circumstance one is bound to have [a stit: Q} at TOO witnessed by W0, and with the same counter at m1 serving for and both In Figure 2.10, however, the witness w0 for the seeing to it that P atTOOis properly later than the witness w1 for the seeing to it of the conditional at m0In this case it can be atTOOthat one sees to it that P and sees to it that PcQ without seeing to it that Q. In particular, w1 cannot witness [a stit: Q] at TOO because of the failure of Q at m2, which is choice equivalent toTOOat w1 for a. 8

But see (12) in chapter 11 for the opposite result for dstit.

48

Introduction to stit

Figure 2.9: Do [a stit: P] and [a stit: P c Q ] imply [a stit: Q}7

Figure 2.10: Do [a stit: P] and [a stit: P c Q ] imply [a stit: Q]?

2. Stit: Introductory theory, semantics, and applications

49

Figure 2.11: The ten-minute mile

And W0 cannot witness [a stit: Q] at mo, because by then the fact that Q at i(m 0 ) is settled—there is no "counter."

2B.5

The ten-minute mile

2-11 QUESTION. (Ten-minute mile) How is it possible to be the agent of one's own run of a ten-minute mile? The answer is in Figure 2.11. Suppose that a has been steadily running at a ten-minute pace, and at frequent moments (of which there is no last—this is the critical condition) a has the option to drop out of the run. Consider [a stit: Q} at mo as "a sees to it that a finishes the mile in just ten minutes." Evidently [a stit: Q] should be true at mo, but it is equally evident that no single prior moment such as W0 is adequate as a witness. The reason that W0 cannot serve as a witness is not just intuitive. As the picture shows, the positive condition is violated, for Q fails at a moment that comes out of a right-hand side of a box that is later than W0, and hence Q fails at a moment that is choice equivalent to m0 at W0 for a. We therefore need to complicate our semantics (the underlying extra-linguistic theories remaining unchanged) by permitting chains as well as single moments to count as witnesses. The details are a little delicate, but you can catch the idea. It is the whole chain of choices coming right up to the finish line that stands as witness to the truth at mo of "a sees to it that a finishes the mile in just ten minutes." One has only to generalize the positive and negative conditions appropriately; see the discussion of "witness by chains" in §8G.4. Figure 2.11 shows, incidentally, that the successful ten-minute miler is, in a sense just about to be defined, a "busy chooser"!

2B.6

Refraining from refraining I

2-12 QUESTION. (Refraining from refraining) Suppose you want to refrain from

50

Introduction to stit

refraining from seeing to it that you juggle sixteen balls in the air. Is there a way in which you can do that without seeing to it that you juggle sixteen balls in the air? We are reminded of the imperative with which Davidson sometimes began a delivery of his intriguing lecture on whether animals can think: He used to say, "Don't bother me with your stories!" Now that there is a picture and a calculation, that is what we say: Don't bother us with your stories about locked rooms or making yourself drunk or having your friend or enemy bind you hand and foot, for except in what are impressively complicated circumstances, it is just not possible for you to refrain from refraining from seeing to it that you juggle sixteen balls without your actually seeing to it that you juggle sixteen balls. Here is the hard fact underlying our Davidsonian impatience with storytellers: If you can tell a story in which [a stit: ~[a stit: ~[a stit: Q]]] holds but [a stit: Q] does not, then it is going to have to be part of your story that it contains a busy chooser, that is, an agent a for whom there is an infinite chain of moments occurring in some definite interval, that is, a chain that is bounded both above and below by moments, and at each of which a has a nonvacuous choice, Def. 14. (That this concept is the relevant one was pointed out by M. McCullagh.) But if you allow us to impose on your stories the condition that there are no busy choosers, then you cannot tell a story in which a refrains from refraining from seeing to it that a juggles sixteen balls but nevertheless fails to see to it that a juggles sixteen balls. Without busy choosers, refraining from refraining from seeing to indeed implies seeing to. In fact the implication goes both ways. We call the statement of equivalence between [a stit: ~[a stit: ~[a stit: Q]]] and [a stit: Q] "the refref equivalence," Ax. Conc. 1. (refref)

We discuss this striking equivalence a number of times in this book, with the mathematical facts being elaborated in chapters 15 and 18. Although a full proof of the refref equivalence is not appropriate to this sketch, Figure 2.12, which serves as a kind of partial "semantic tableau," may help you to appreciate the flavor of verifying the implication from left to right. Suppose that [a stit: ~[a stit: ~[a stit: Q]]] holds at m0. It needs a witness w0 and a counter [a stit: ~[a stit: Q]], which we write at m0'. This in turn needs a witness W0' and a counter [a stit: Q], which we write at m0"; a definite argument is needed, however, that W0' is correctly drawn as later than (or identical to, a possibility expressed by the double lines) W0. A similar argument as to the need for (and placement of) a witness and a counter justifies the remainder of the right side of the diagram. Then a reductio argument permits us to argue that [a stit: Q] can be "moved over" to a moment m1 on the left that is choice equivalent toTOOat W0 for a. The left side of the diagram is part of a subsidiary reductio: The picture as drawn places the witness w1 for [a stit: Q] at m1 as properly above W0, which can be shown to be impossible, provided a is not a

2. Stit: Introductory theory, semantics, and applications

51

Figure 2.12: With no busy choosers, [a stit: ~[a sht: ~[a stit: Q]]] implies [a sht: Q]

busy chooser. In fact, if there are no busy choosers, the witness for [a stit: Q] at m1 must be W0 itself. This easily implies that [a stit: Q] must be true at mo, as desired. (See §15C for a full proof.)

2B.7

Modes of action and inaction

Since stit represents agency via a modality, we may ask stit theory to tell us the following. 2-13 QUESTION. (Modes of action) How many (non-equivalent) stit modalities are there? By a "modality" in this context we mean a sentence that can be made from a fixed sentence, Q, by means of stit and negation. With this understanding, the answer is "ten"—provided that there are no busy choosers. The five "positive" modalities are these:

(i) implies (ii) implies (iii), (i) also implies ( v ) , and (iii) is also implied by (iv). Since (with no busy choosers) refraining from refraining from doing collapses

52

Introduction to stit

to doing, there are no more positive modalities. (The deliberative stit, §8G.l, has exactly the same structure; see the diagram in Horty 2001.) The other five modalities—the "negative" modalities—are just the negations of these. One can use the idea of modalities to obtain some further insight into "why" the refref equivalence holds. Let Q <-> The Pequod was scrubbed, and let a = Queequeg. Now think of yourself as building a "semantic tableau." Start at the bottom with Q, and consider what you can consistently add. First, did Queequeg see to it that the Pequod was scrubbed, or not? Evidently either is consistent. Let us conjoin to Q the statement that Queequeg was not responsible for that fact,

Now consider whether Queequeg was causally responsible for the fact that he did not see to it that the Pequod was scrubbed. It could go either way. It might be that Queequeg actually and agentively refrained from seeing to the scrubbing: [a stit: ~{a stit: Q]]. Let us, however, examine the other possibility. Perhaps Ahab confined Queequeg so severely that he had no more causal responsibility for not seeing to the scrubbing than a table has for not seeing to it that the moon rises: ~[a stit: ~[a stit: Q]]. Let us add this conjunct, yielding, so far, the consistent story,

We are now ready for the last step. Queequeg either saw to the last conjunct or not. In our particular story he did not, since his confinement by Ahab stripped him of all agency in the matter of the Pequod being scrubbed. The question, however, is whether on any consistent story you can add that Queequeg saw to the third conjunct. Can you consistently add [a stit: ~[a stit: ~[a stit: Q]]], giving the total story as

The refref equivalence, Ax. Conc. 1, says "no," since it makes that last conjunct equivalent to [a stit: Q], which openly contradicts the second conjunct, ~[a stit: Q}. And contrariwise, if you cannot think of a consistent story for (1), then you are marshaling evidence for yourself in favor of the plausibility of the refref equivalence.9

2B.8

Refraining from refraining II

2-14 QUESTION. (Refref with busy choosers) What happens to refraining from refraining when one does allow a story to make reference to a busy chooser? 9 See §2B.8 for a hint and Xu 1995a for a detailed consideration of what happens to the "modes of action and inaction" when the refref equivalence fails due to the presence of busy choice sequences.

2. Stit: Introductory theory, semantics, and applications

53

Figure 2.13: With busy choosers, [a stit: ~ a stit: ~[a stit: Q]]] does not imply [a stit: Q]

Figure 2.13 reveals the failure of the implication from [a stit: ~[a stit: ~[a stit: Q]]] to [a sizt- Q]. There is not enough room to show all relevant assignments; the idea is that aside from the top right-most moment, those jumping out of the right-hand side of a box have Q, and those that are limit points of an infinite chain of busy choosing have ~Q. You can tell that the example of Figure 2.13 deeply involves a busy chooser. We are not going to discuss this awesome garden of forking paths other than to refer the reader to the mirror game of the Example with busy choosers 9-1 on p. 268 and to the elaborations of chapter 18. Also see Xu 1995a for a consideration of deeper levels of complexity of busy choice sequences and how they relate to the modal structure of the logic of agency.

2B.9

Deontic contexts again

As a further illustration of the beneficial consequences of using the canonical form of stit for agentives, we produce a surprise in the course of exploring an aspect of our suggestion in §1B.2 that deontic statements be put in their own canonical form as follows: Oblg:[ stit: Q], i.e., is obligated to see to it that Q Frbn:[ stit: Q], i.e., is forbidden to see to it that Q Perm:[ stit: Q], i.e., is permitted to see to it that Q The standard equivalences for forbidden/permitted continue to make sense, for example,

Frbn:[ stit: Q]

Perm:[ stit: Q],

54

Introduction to stit

but it is not equally easy to find either a sensible forbidden/obligation equivalence, or a sensible permission/obligation equivalence, that both satisfies the restricted complement thesis Thesis 5 and is also in accord with the tradition. The first two lines of the following list are clearly false and the second two involve bad grammar: is not agentive for

That (2) fails is illustrated by the following example: Though the cook is forbidden to see to it that the boats are lowered, he is not thereby obligated to see to it that they are not lowered. The point about (Hi) is that since to be obligated is always to be obligated to do something, and since not doing something is not doing something, by our restriction that the complements of deontic modalities be limited to stit sentences, it follows that Oblg:~[a stit: Q] makes no sense. But the solution is obvious: To be forbidden to see to it that Q is to be obligated to refrain from seeing to it that Q, and we know what refraining means. Accordingly, the proper equivalences, forced on us by the restricted complement thesis, are just STIT DEONTIC EQUIVALENCES. (Axiomatics concept. Reference: Ax. Conc. 2) and

The first is illustrated as follows: If the cook is forbidden to see to it that the Pequod is headed up into the wind, then the cook is obligated to see to it that he does not see to it that the Pequod is headed up into the wind, and conversely. Symmetry suggests as well that to be obligated to see to it that Q is to be forbidden to refrain from seeing to it that Q, so that we add the following to the stit deontic equivalences Ax. Conc. 2: Oblg:[a stit: Q] <-> Frbn:[a stit: ~[a stit: Q]]. If the helmsman is obligated to see to it that the Pequod is headed up into the wind, then the helmsman is forbidden to see to it that he does not see it that the Pequod is headed up into the wind, and conversely. And now the surprise: If both the Oblg: and the Frbn: equivalences hold, then by substitution (of ~[a stit: Q] for Q) and transitivity, then must also be added to the stit deontic equivalences Ax. Conc. 2. The equivalence (2) says that the only way to be obligated to refrain from refraining from luffing is to be obligated to luff. Right or wrong? It would seem that the only way the equivalence could possibly hold or fail to hold were if as a matter of (nonnorrmative) stit fact itself, the following—namely, the refref equivalence, Def. 14—held or did not hold:

2. Stit: Introductory theory, semantics, and applications

55 (refref)

We are therefore led to consider yet once again whether we can think of a case in which the right side holds without the left. Does refraining from refraining imply doing? The mathematics for the achievement stit is disposed of in §15C and chapter 18, and Horty 2001 shows that the refref equivalence holds for the deliberative stit; but we take it that our general point of view commits us only to trying to get clear on the matter, not to already having accomplished that desirable end. In any event, given the refref equivalence, (2) already follows from the "normality" of the obligation modality.

2B.10

Contemplation and action

We briefly consider a thesis of Prior. The thesis of chapter 4 of Prior 1968 is that contemplation and action are incompatible in the sense that no one can find out what anyone is going to decide—under the assumption that there really is action and not just its appearance. We think Prior's argument is absolutely good; all we do is lay out what we take to be its essential elements in present terminology. The contribution of this section is to reveal how little philosophy of mind or epistemology is needed. One only needs to situate the contemplation and the action within the causal structure of the events of our world. As we have seen, if our world permits an open future, then an adequate semantic theory needs to require that truth (as opposed to settled truth) be relative both to a moment and to a history. Suppose that we have a connective $ with at least the following two properties:

The first condition tells us that $ is history-independent (in the natural sense, which we define explicitly in Definition 6-4), so that since truth of c (A) at m/h implies settled truth of c (A] at m, it is enough to say of o (A) that it is settled true (or false) at a moment m. Because of their history-independence, we do not need to mention a history for these sentences. The second condition says that o is a "success" locution. It follows easily from these two provisions that If o (A) is true at m/h, then A is settled true at m/h. Or with equal propriety: If o (A) is settled true at m then Sett:A is settled true at m. o (A) cannot, that is, be true, hence settled true, without the complement, A, being settled true. We are now ready to turn to Prior's argument. Let us substitute for his "finding out" the philosophically more familiar verb "to know," and let us furthermore fix the knower as Autumn Jane. Having made these substitutions, consider the claim that

56

Introduction to stit At 3:00 Autumn Jane knows that at 5:00 her father will have seen to it that her dress is clean.10

We introduce just two postulates governing the semantics of knowledge, both of which we think are implicit in the Prior argument. The first postulate about knowledge is one whose point can only be appreciated by keeping in mind the branching of the future: When we have knowledge, it is settled true that we have it. If at a given moment-history, m/h, Autumn Jane knows that P, then this does not depend on the history, h, but must be equally true for every history passing through m. This postulate says that knowledge is history-independent: regardless of P, for each moment, m, "Autumn Jane knows that P" is true on any pair m/h just in case "Autumn Jane knows that P" is true on every pair m/h1 (with h, h1 E H( m ) ). In this way knowledge is quite an ordinary property, like "red" or "six feet tall," and it would appear that the history-independence of knowledge hardly needs a special argument. Many of the standard epistemic, doxastic, and psychological attitudes have exactly the same character of history-independence. If Autumn Jane believes (wonders, asserts, predicts, or guesses) something, then that she has that belief (is in that state of wonder, is making that assertion, prediction, or guess) does not depend on what the future may bring; it is a settled truth about her.11 The second postulate is that knowledge implies truth, by which we mean this: For every moment-history pair, if "Autumn Jane knows that P" is true at that pair, so is P. Given that at 3:00 she knows that at 5:00 her father will have seen to it that her dress is clean, then in fact at 3:00 it is true that at 5:00 her father will have seen to it that her dress is clean. The problem with which Prior is wrestling does not appear to arise unless we have this one among the philosophers' senses of "knowledge" in mind. If knowledge only conveys justified—but perhaps false—belief, or only some justified level of certainty or only some justified certainty as to high probability, or if it merely tends to signal truth but does not betoken an exception-free universality, then there is 10 Please read this English sentence as having the form "Autumn Jane knows that: At-inst5.00:[a stit: Q]" in the sense of §8F.5. The "3.00" in the example only indicates the time at which the knowledge ascription is to be evaluated, and the "will have" is intended as a logically redundant concession to idiomatic English, inserted in this example because 3:00, which is the time of evaluation, is prior to 5:00. It is perhaps worth noting in this regard that At-instl:[a stit: Q] is by no means equivalent to [a stit: At-mstt:Q], even when Q is history-independent. The former requires the witness to be prior to i (regardless of the moment of evaluation), whereas the latter requires the witness to be prior to the moment of evaluation (regardless of i). In the formulation used, the witness—if any—must be prior to 5:00, but in the absence of further considerations—such as those adduced below—need have no special relation to the moment of evaluation. 11 There is some plausibility in taking some psychological attitudes as history-dependent. It may for example be that whether a person intends to do something depends on what the future brings. It takes some doing to see why the same should be true for knowledge, and although we do feel entitled to claim that the plausible thing is to treat knowledge as historyindependent, the alternative is not without interest. As we imply at the very end, one is indeed welcome to see the argument of this section as providing independent support for the idea that knowledge is history-dependent.

2. Stit: Introductory theory, semantics, and applications

57

no puzzle in supposing that Autumn Jane "knows" what someone is going to do. The puzzle only sets in when it is supposed that knowledge implies truth. We have just given knowledge the properties (i) and (ii) ascribed to o in (3). This licenses us in the first place to speak of the truth of "Autumn Jane knows that P" at moment m1, dropping reference to a history h. And in the second place, it follows that if "Autumn Jane knows that P" is true at m1, then P is settled true at m1. It is therefore impossible that Autumn Jane know something that is not settled true. She can consistently be said to believe it, wonder if it is so, assert it, guess it, and so on, for although each of these shares historyindependence with knowledge, none of these has the "success" property (ii), that of implying truth. But since knowledge does have this second property, Autumn Jane cannot consistently be said to know what is not settled true. Thus, if on Monday Autumn Jane knows that on Tuesday the path will be muddy, then on Monday it is settled true that on Tuesday the path will be muddy. Given this intermediate conclusion, it is plain what we need about the concept of decision: nothing about the torments of the mind in process, but only the colorless fact that at or before the moment of decision it is not settled what decision is made. In making this point, we will use the achievement stit, as we have been doing right along. (Permit us to observe, however, that a deliberativestit version of the following argument is easier to follow, since dstit identifies the moment of choice with the moment of outcome.) On the present theory the required true premiss could be stated as follows: Let m, m1, and w be moments. Let w be the witness for [a stit: Q] at m. Suppose m1 < w. Then at m1 it is not settled that [a stit: Q] will be true at i( m ) "Yes" where some histories through m1 cross i ( m ) , but "no" on others. This is the most elementary of consequences of the negative condition. By contraposition, if it is settled at m1 that [a stit: Q] will be true at i(m), then m1 < w. Further, since "Autumn Jane knows that" has the properties (i) and (ii) of o, and so implies the settled truth of its complement, for any history h through m1, if Autumn Jane knows that it is true at m1/h that [a stit: Q] will be true at i ( m ) , then it is settled true at m1 that [a stit: Q] will be true at i( m ). Therefore, Autumn Jane cannot know that a stit: Q] will be true at i( m ) before the witness for [a stit: Q] at w. "At 5:00 Autumn Jane's father will have seen to it that her dress is clean" is not knowable by Autumn Jane—or anyone else— until after the witness. Note, incidentally, that stit theory certainly allows that at 3:00 Autumn Jane can know that at 5:00 her father will have seen to it that her dress is clean. For example, her father may have taken the dress to an ideally reliable launderer at some point earlier in the day. In this case the witness precedes the moment at which we are evaluating Autumn Jane's state of knowledge. Just to double underline the point we are urging, which is that the argument does not depend on interesting commitments about the mind, we repeat the

58

Introduction to stit

premisses and the conclusion, (i) Like many standard propositional attitudes, knowledge is history-independent. (ii) Knowledge (in the sense that gives rise to Prior's puzzle) implies truth. (iii) No stit is settled to be true at an instant until after its witness. Therefore, (iv) it is impossible to have knowledge of a stit until after its witness. Of course the essential content is that given (iii), the "epistemic" propositions (i), ( i i ) , and not-(iv) constitute an inconsistent triad. For certain other senses or uses of "knowledge" it may be preferable to resolve the inconsistency by keeping not-(iv) while denying (i) or ( i i ) . The important thing is to follow Prior in knowing about the inconsistency before settling matters.

3

Small yet important differences from earlier proposals In chapter 1 we introduced [a stit: Q], the stit sentence, solely to represent agentive sentences.* We stated some theses about stit and agency, and we presented a "mini-history," going back to St. Anselm, of earlier proposals to treat agency through a modal construction. Then in chapter 2 we provided a theory of branching alternative histories as a foundation for two semantic accounts of [a stit: Q]—the achievement and deliberative stits—and we explored some applications of the achievement stit. Here we propose to deal in more detail with some earlier discussions to which we alluded in our "mini-history." Among several proposals in the neighborhood of the stit sentence that we inventoried in §1D, we will concentrate on those of von Wright, Chisholm, Kenny, and Castaneda. As we shall see, their influential proposals differ in small yet important ways from the stit sentence. These differences offer an opportunity for additional discussion of some of the stit theses proposed in chapter 1 and listed in §1: the agentiveness of stit thesis, stit complement thesis, stit paraphrase thesis, imperative content thesis, restricted complement thesis, and stit normal form thesis. We will not here argue for the stit paraphrase thesis, Thesis 3. Rather we will show its value throughout as we use it to distinguish between sentences that are agentive and those that are not. Two of the theses, the agentiveness of stit thesis, Thesis 1, and the stit complement thesis, Thesis 2, are considered, in one form or another, in every section. Von Wright, Chisholm, Kenny, and Castaneda have all pressed for the exclusion of some sentences as complements in agentive sentences. Again and again we shall see the ways in which our understanding of agency and action is increased when we insure that [a stit: Q] *With the permission of Kluwer Academic Publishers, this chapter draws on Perloff 1995.

59

60

Introduction to stit

is grammatical and meaningful for any arbitrary sentence Q, and concomitantly that [a stit: Q] is always agentive for the agent We will consider the restricted complement thesis, Thesis 5, specifically for deontic contexts in connection with the proposals of both von Wright and Castaneda. As we shall see, deontic logic is enriched and clarified by the restriction that deontic statements such as obligations, prohibitions, and permissions must take agentive complements. About the stit normal form thesis, Thesis 6, we will say nothing more, except to express the hope that by the end of the chapter you find that its truth and value go without saying. As for the imperative content thesis, Thesis 4, although imperatives come up here from time to time, we defer their direct discussion until chapter 4. Before concluding this chapter, we will have a few words to say about Davidson, whose approach is very different from that of stit theory. Davidson rejects modal treatments, crafting instead a linguistic structure that is obviously extensional, a structure that invites us to focus on an ontology of actions as events.

3A

Von Wright

In §1B.2 we argued the inadequacy of the two standard accounts of deontic statements. The one allows any declarative sentence, no matter how divorced from agency, to occur as a complement in a deontic account. The other lets the complements in deontic statements be action nominals. Take for example a permission statement symbolized by Pa. On the sentence-complement account, one would replace "a" by a sentence such as "Persia is conquered," so that Pa could be instanced by "It is permitted that Persia is conquered." According to the action-nominal account, however, permission would be treated as an adjective that modifies action nominals. So with "a" replaced by the actionnominal, "conquering Persia," Pa might have as an instance "Conquering Persia is permitted." Von Wright's program, along with both Castaneda's practitions and the stit sentence, has, among its goals, providing an acceptable alternative to these two unsatisfactory approaches. In this section we explain just enough of an early proposal of von Wright's to compare it to the stit sentence. Page references are all to von Wright 1966. Both von Wright's sentence form and the stit sentence restrict the complements of deontic statements to a well-defined class of agentives. Both reflect the fact that time and change must be reflected in the structure of agentive sentences. Von Wright, however, incorporates time and change into his language by restricting the complements of agentive forms to T-sentences. As we shall see, that decision, in contravention of the stit complement thesis, significantly weakens his account, particularly in its failure to illuminate the connections between acting and refraining. The T-sentence, which stands at the center of von Wright's language of actions and agents, takes, as complements, two propositions—each of which describes a state of affairs—and yields a sentence that describes an event. An event sentence such as p Tq "describes the transformation of or transition from

3. Small yet important differences from earlier proposals

61

a p-world to a g-world" (pp. 25-26), where the transition is either the alteration of a state into its negation, or the maintenance of a state just as it is. Von Wright tells us that transitions are to be thought of as ordered pairs of states of affairs, whose ordering "is a relation between two occasions which are successive in time" (p. 27). Since between two successive moments a state of affairs may either remain unchanged or be transformed into its negation, there are four elementary state transformations: PTP PT~P ~PTP ~PT~P.

In von Wright's language, just as sentences describing states of affairs appear as complements in T-sentences, so T-sentences appear as complements in sentences of achievement or forbearance. Sentences describing agentive achievements are constructed by prefixing the d operator to a T-sentence, while forbearings or refrainings are described by prefixing the / operator to a T-sentence. Since we began with four elementary T-sentences, there will be eight elementary forms, an agentive doing for each elementary state transformation, and an agentive refraining for each elementary state transformation. For example, we have d(~PTP)

as von Wright's notation for sentences of the form the agent sees to it that a world in which not-p obtains is transformed into a world in which p obtains, while f(~PTP)

is the official notation for sentences of the form the agent refrains from seeing to it that a world in which not-p obtains is transformed into a world in which p obtains. (We ignore truth-functionally compounded T-sentences; that development makes no difference to our story. See Belnap 1999 for additional discussion of T-sentences in connection with the theory of concrete transitions and with special reference to branching space-time.) In order to begin the comparison, let us recall that according to the stit paraphrase thesis the English sentence Alexander tamed Bucephalus, when it is agentive for Alexander, can be represented as a stit sentence of the form [a stit: Q], specifically [Alexander stit: Bucephalus was tamed].

62

Introduction to stit

The sentence is settled true just in case there was a prior choice of the agent Alexander that guaranteed the truth of "Bucephalus was tamed." On von Wright's account that same sentence, represented by d(~pTp), is true just in case (i) "... the state described by ~p prevails on the first of the two successive occasions" and (ii) "... the change described by (~pTp) does not happen, as we say, 'of itself, i.e., independently of the action of the agent" (p. 43). That is, on the first of two occasions "Bucephalus was not tamed" is true, on the second occasion "Bucephalus was tamed" is true, and the change did not happen independently of Alexander. Von Wright's agentive sentence forms, with T-sentences as their complements, are an important step forward in our understanding, insofar as they explicitly introduce at the heart of the discussion of agency considerations of time and change. Such considerations focus our attention on the fact that a language adequate to the description of agents and their achievements must deal with the past, present, and—above all—the future. Let us notice that the agent, whose importance is acknowledged in the original English sentence—as well as in von Wright's statement of the truth conditions— is lost in the official idiom; when the agentive sentence occurs as the complement of another sentence the agent is not easily retrievable. The stit sentence, on the other hand, is specific in locating the agent as someone who makes choices among alternatives and by those choices sees to it that a proposition is settled true. The agent is essential in the things which he brings about; if "Alexander tamed Bucephalus" is settled true, it is so because Alexander made a choice, a choice that guaranteed the truth of "Bucephalus is tamed." For these reasons, the stit sentence gives a distinguished place to the agent, by insuring that the agent term always remains easily and obviously recoverable. Further, the agent is essential in the things that he or she forbears from bringing about. Forbearing, or refraining, presents an interesting challenge because forbearing is not doing. As von Wright puts it, "forbearing is not the same as not-doing simpliciter" (p. 45). The problem then is to talk about something which seems to be both a doing and a not-doing and also seems to be neither a doing nor a not-doing. As we saw earlier, von Wright's device for introducing agentive refrainings or forbearings into the language is structurally analogous to his introduction of agentive accomplishments; he simply prefixes each of the four elementary change statements with a refraining operator. Recall that f(p T~p) is the official version of a refraining sentence where the agent refrained from seeing to it that a world in which p obtains was transformed into a world in which p didn't obtain, so when Alexander refrained from setting free Darius's wife is true, von Wright would have us represent it as Alexander forbore to destroy the world in which Darius's wife was held captive (which world does not, on the occasion in question, change of itself into a world where she was not held captive).

3. Small yet important differences from earlier proposals

63

It is surely to von Wright's credit that he faces the problem of refraining directly. For the most part, it is a problem that philosophers interested in the topic have, at best, briefly mentioned before going on to talk about something else. Refraining, as von Wright has clearly seen, is equal in importance to other agentive achievements and equally worthy of philosophical attention. Because the two operators d and f are introduced independently, we have lost the deep and important connection that holds between seeing to it that and refraining. Because that connection cannot be clarified on von Wright's account, he is forced to treat refrainings and agentive accomplishments as entirely different species in the same genus. There is nothing in either the syntax or the semantics of von Wright's sentence forms that helps in any way to clarify or explain their relationship. Exploiting the ability of the stit sentence to clarify the linguistic environments in which agentives are found, we can use the sentence Alexander refrains from freeing Darius's wife vividly to exhibit just how refraining (§2B.2) is connected with doing, how an agent sees to it that something is not done. First, notice that Alexander frees Darius's wife is agentive for Alexander since equivalent to [Alexander stit: Alexander frees Darius's wife]. In contrast, Alexander does not free Darius's wife is not agentive. It is not agentive because it is a mere denial of agency. A denial of agency such as "a does not free Darius's wife" fails quite generally to be equivalent to [a stit: a does not free Darius's wife], since many things, including some that aren't even agents, are not seeing to it that they do not free Darius's wife. Because refraining is agentive, "Alexander refrains from freeing Darius's wife" must have the form [a stit: Q]. It can thus be represented as [Alexander stit: ~[Alexander stit: Alexander frees Darius's wife]]. Using the stit sentence, we can see exactly how it is that refraining is agentive and how it differs from simply seeing to it that some state of affairs obtains. To refrain, an agent a must see to it that a, that very same agent, is not agentive in seeing-to-it-that some proposition holds. Thus, to refrain from freeing the captive, Alexander must see to it that he, Alexander, is not agentive in the matter of freeing Darius's wife. That, of course, is consistent with someone else seeing to it that the unfortunate woman is freed. It is also consistent with her seeing to it that she is freed, that is, with her escaping. The general account of refraining from doing is thus

64

Introduction to stit

which exactly accords with §2B.2. This picture of refraining, in which an agent sees to it that he or she doesn't see to it that some proposition holds, shows in its structure the connection between seeing to it that and refraining. It is equally at home with more modern constructions such as "Don't smoke in bed" and "Don't sit under the apple tree with anyone else but me." When the declarative complement of the refraining statement is not itself agentive, yet has an agent as its grammatical subject, a simpler account is available. For instance, the imperative "Refrain from getting captured by Alexander," as addressed by Darius to his wife, has a simpler non-nested analysis:

in particular [Darius's wife stit: ~Darius's wife gets captured by Alexander]. As it happened, she failed to refrain from getting captured, but Alexander didn't fail to refrain from freeing her. Within deontic contexts, the conceptual gain provided by the stit sentence in accord with §1B.2 and §2B.9 is just as pronounced. In harmony with the restricted complement thesis, Thesis 5, deontic statements must have one of the following forms: Oblg:[ stit: Q]: is obligated to see to it that Q Frbn:[ stit: Q]: is forbidden to see to it that Q Perm:[ stit: Q]: is permitted to see to it that Q.

The equivalences Frbn:[ stit: Q] Perm:[ stit: Q]

Perm:[ stit: Q and Frbn:[ stit: Q]

continue to hold, but as indicated also in §2B.9, what to take as the forbidden/obliged equivalence is not so obvious. Frbn:[ stit: Q]

Oblg[ stit: ~Q]

can be false when ~Q is not agentive for a. For example, a common soldier may be forbidden to see to it that the prisoners are fed without being obligated to see to it that the prisoners fail to be fed. Furthermore, Frbn:[ stit: Q]

Oblg[ stit: Q]

is grammatically unacceptable because, as we just saw, ~[a stit: Q], though the negation of an agentive, need not itself be agentive. The following equivalence, as suggested in §2B.9, seems likely to be the appropriate one: Frbn:[ stit: Q]

Oblg:[ stit: ~[ stit: Q]].

3. Small yet important differences from earlier proposals

65

"The guard is forbidden to see to it that the prisoners are set free" is true just in case "The guard is obligated to see to it that he does not see to it that the prisoners are set free" is true. That is, the guard is forbidden to free the prisoners just in case he is obligated to refrain from freeing the prisoners. Symmetry suggests that whenever the guard is obligated to set the prisoners free, he is forbidden to see to it that he does not set them free. Thus the equivalence Oblg:[ stit: Q]

Frbn:[ stit: ~[ stit: Q

also seems correct. (In §3C we say a bit more about refraining and negation.) In summary, von Wright's form marks a step forward in our understanding as it properly restricts the complements of deontic structures to agentive sentences. But because his agentive forms are constructed by affixing the d operator or the / operator to T-sentences, there is no possibility of the nesting of agentives, something the stit sentence promotes. With the nesting of agentives, we gain the ability to express the connections between such agentives as refraining and doing. With the ability to express these connections, we can demonstrate the value of restricting the complements of deontic constructions to agentives when the complements of agentives are all declarative sentences.

3B

Chisholm

Von Wright, as we have seen, constructs his language of agency and agentive achievements upon a foundation of changes and transitions. One difficulty von Wright faced was providing an adequate account of the connection between doing and refraining. A natural response is to introduce agentive intentions into the syntactic structure of the language. Chisholm, in a series of essays, has elaborated upon just such a sentence form, one whose foundation includes the explicit introduction of agentive endeavors. Such an undertaking, involving as it does the introduction of intentions within the syntactic structure, invites comparison to the stit sentence because stit incorporates all mental considerations into the prior choice of the agent. Chisholm's work on this matter, beginning in 1964, covers more than a decade and is to be found in more than a dozen different places. Though there are terminological changes and subtle conceptual alterations, the basic ideas and their modes of presentation remain sufficiently unchanged to be presented in a unified fashion. For simplicity of exposition we follow most closely the account in Chisholm 1969. Chisholm's enterprise is built on the sentence form He made it happen that

in the endeavor to make it happen that ...

that transforms the ordinary English Joan of Arc joined the Dauphin's forces to defeat the English into

66

Introduction to stit Joan of Arc made it happen that she joined the Dauphin's forces in the endeavor to make it happen that the English are defeated.

This sentence form—Chisholm calls it an undefined locution—has as its official abbreviation (M _,...). There are three open places in the undefined locution: The first is the subject term of the sentence; the second, the complement of "makes happen," is nonintentional; the third, the complement of "in the endeavor to," is explicitly intentional and teleological. We will discuss the grammar and metaphysics of the first open place, then go on to speak about the grammar and metaphysics of the other two open places in the undefined locution. We conclude this section with a brief discussion of when an agent "could have done otherwise." Because the agent term in the undefined locution has no distinguished place either within the structure of that form or in the official abbreviation, it may not be obvious that there are three open places in the undefined locution. There appear to be only two, one represented by the dots and another by the dashes. Chisholm's consistent use of the word "he" as the subject of the undefined locution reinforces this misperception, as does the fact that the agent term is dropped altogether in the official abbreviation (M , ...). The stit sentence, in contrast, insures by its structure that the agent term is always recoverable no matter where it occurs. Though it is not clearly marked, there is no doubt that there is a place in the undefined locution for an individual term. While Chisholm doesn't tell us much about it, he does say that "[t]he subject term of 'makes happen' may designate either a state of affairs or a person" (Chisholm 1964b, p. 615). By allowing terms that refer to states of affairs to appear in the subject place, Chisholm is trying to exploit the similarity of structure between saying that one state of affairs makes another happen and saying that an agent makes-it-happen or sees to it that a certain outcome eventuates. He says in identifying the two that "... any instance of our locution will refer to the agent as a cause, and it will imply that he makes something happen" (Chisholm 1969, p. 206). It may not be a good policy to allow states of affairs to appear in the first open place. For, while it is plausible to say that one state of affairs makes another happen, it will never be plausible to say that one state of affairs makes another happen in the endeavor to make something happen. So though it may be true to say that The French attacking the English at Patay made it happen that the English were routed at Patay, it will never be true to say that The French attacking the English made it happen that the English were routed at Patay in the endeavor to drive the English out of France,

3. Small yet important differences from earlier proposals

67

because no states of affairs endeavor or intend or aim to do anything. Therefore, since there are no true instances of the undefined locution with a subject term that refers to a state of affairs rather than an agent, and since allowing both terms designating states of affairs and agent terms to appear in the same open place offers no conceptual advantage, it is a mistake to allow terms referring to states of affairs to appear at all. Let us turn now to the grammar and metaphysics of the second and third places of the undefined locution. In these two places Chisholm allows the "blanks to be filled by well-formed sentences or, in the case of the second blank, by the subjunctive version of such a sentence" (Chisholm 1971, p. 36). Because he wants to speak of an agent making happen, in addition to an agent makes it happen that , he also allows gerundive expressions to appear in those two places. Chisholm justifies this grammatical decision as follows: For every well-formed sentence, there is a corresponding gerundive expression. Thus for "Socrates is mortal" there is "Socrates being mortal" as well as "Its being the case that Socrates is mortal"; and for "Socrates is mortal and all swans are white," there is "Socrates being mortal and all swans being white" as well as "It being the case that Socrates is mortal and all swans are white." The gerundives of well-formed sentences may be said to designate states of affairs. (Chisholm 1971, p. 39) The stit approach, by contrast, recommends grammatically restricting the complements of agentives to all and only declarative sentences because that approach avoids unnecessary difficulties. Nested gerundives are more confusing and more difficult to understand than are their sentential counterparts. Additionally, gerundives, whether nested or not, pose serious difficulties in deciding their reference. The stit sentence avoids such problems by allowing all and only declarative sentences to stand as complements. Concerning the metaphysics of those two places, Chisholm suggests a restriction on the sentences (or gerundive expressions) that are allowed to occur there; those that refer to actions, he tells us, may not always be acceptable substituends. He argues that because "we are attempting to describe action ... in terms of making things happen in the endeavor to make things happen," we should not normally allow "expressions which themselves refer to actions" (Chisholm 1966, p. 32). In contrast, the stit sentence doesn't rule out agentive sentences as complements; it encourages finding uses for agentive complements. As we saw in the discussion of refraining, such a practice can be actively enlightening. To sum up the last two points: Insofar as Chisholm is willing to have all and only indicative sentences as substituends in those places, it is a point of similarity with stit; insofar as his position is either less restrictive (in that it allows gerunds) or more restrictive (in that it disallows agentives) it differs from stit. These differences aside, both Chisholm's undefined locution and the stit sentence are designed to help clarify such questions as when it is true to say that

68

Introduction to stit

an agent "could have done otherwise." On Chisholm's account, when we say that an agent could have done otherwise, we mean that, although at eight o'clock this morning I did not make it happen that I arranged things in such a way that I would be in San Francisco now, nevertheless it was then within my power to make it happen that I so arranged things. The 'could,' therefore, is 'constitutionally iffy.' And the proposed explication is consistent with saying that for some time there has been a sufficient causal condition for my not being in San Francisco now. But the freedom of the so-called 'actus voluntatis elicitus' is preserved. For we may say that there was no sufficient causal condition this morning for my not undertaking to arrange that I would be in San Francisco now; and from this it would follow that such an undertaking was then within my power. (Chisholm 1969, p. 217) There is in chapter 9 an extensive discussion of various concepts in the vicinity of could-have, with proofs and pictures. Here, we only note that Chisholm has clearly described one clear case of when an agent could have done otherwise. Where Q = "Chisholm boards the plane for San Francisco," then the example has both Q and ~Q as agentive, the point of evaluation at the moment "now" where [Chisholm stit: ~Q] is settled true, and there is a choice for Chisholm at 8:00 A.M. alternative to the one he made that guarantees Q to be settled true at that same instant. That is, at 8:00 A.M. Chisholm had a choice either to see to it that he was in San Francisco that afternoon or to see to it that he was not. But the situation is more complicated than that. This is not all we might mean by "could have done otherwise." Because, by the negative condition, Chisholm need only risk the falsity of an outcome, only one of Q or ~Q needs be agentive for us to correctly picture when an agent could-have-done otherwise. Notice that in each of the cases the freedom of the 'actus voluntatis elicitus' is preserved. We can make these calculations because the stit sentence (i) distinguishes between agentive and non-agentive sentences, ( i i ) focuses attention on the agent, (lii) insists that all and only declarative sentences appear as complements in agentive sentences, and (iv) has a precisely drawn semantics that pictures agents making choices against a background of branching time.

3C

Kenny

Kenny presents his important and influential exploration of agents and their achievements through his discussions of emotion and the will in Kenny 1963. Unlike von Wright, whose language is founded upon a metaphysics of elementary state transformations, and unlike Chisholm, whose starting point involves endeavors and intentions, Kenny's account of agents and actions begins with

3. Small yet important differences from earlier proposals

69

language itself. In this section we shall see how the stit sentence, which is very like Kenny's canonical form, "bringing it about that P" can be used to clarify some important problems in the vicinity of agency, problems that require the full power of the stit sentence for their solution. Kenny proceeds by "considering the special logical properties of the finite verbs we employ to report actions" in the attempt "to isolate a simple and fundamental pattern of description of human activity, which reports of emotional states and reports of voluntary action alike exemplify" (Kenny 1963, p. 151). The finite verbs used to report actions—verbs of action—are those verbs "which may occur in the answer to a question of the form "What did A do?" (p. 154). Because the answers to that question may vary widely, Kenny focuses more specifically on "transitive verbs of action, such as occur in the sentences 'Brutus killed Caesar', 'Wren built St Paul's', 'Mary roasted the beef, and 'Shaw admired Caesar'" (p. 154). After locating a class of sentences in which each member has an agentive subject, a transitive verb of action as its main verb, and an object of the verb, Kenny argues that such sentences, that is, agentive sentences, ought not to be accounted simply as relations of the form Rab. There are at least five important differences, Kenny argues, between action sentences and ordinary two-place relations. We will consider Kenny's five ways later in this section. Having established, to his satisfaction, the distinction between action sentences and other relations, Kenny proceeds to a discussion of the verbs that occur in action sentences. Throughout the discussion Kenny remains constantly aware of the grammatics of his project, that is, of the important connections between the grammar of English sentences to be studied and the grammar of the logic to be used to represent them. Kenny locates three species of verb in the genus verb of action: static verb, performance verb, and activity verb. Static verbs, Kenny tells us, such as "love" "understand," and "fear," do not have continuous tenses, and "where '(o ' is a static verb 'A has od' implies 'A os'" (p. 173). Performance verbs such as "discover," "learn," and "find" do have continuous tenses, and "where 'o' is a performance-verb, 'A has od' implies 'A is not oing'" (p. 173). Activity verbs such as "listen to" and "keep a secret" do have a continuous tense, but " 'A is oing' implies rather 'A has o d ' " (p. 172). Kenny's canonical form, "bringing it about that p," represents any sentence whose main verb is a performance verb. He reasons that "... the primary form of description of a piece of voluntary behavior takes the form of a performance verb ... [and] any performance verb can be replaced by an expression of the form 'bringing it about that p.' where '(p)' describes the state of affairs which is the result of the process" (p. 236). We can see from the quotation that Kenny has already decided to restrict the complement in "bringing it about that p" to sentences that describe states of affairs, more particularly states of affairs that are the results of processes. Once again, the stit complement thesis specifically rejects that restriction. Kenny's own example,

70

Introduction to stit ... when I learn French, I bring it about that I know French, i.e., that I have the capacity to speak French (p. 183)

can be used to reinforce the importance of the thesis. All of the following stit sentences are, in this context, equivalent: [Kenny stit: he learns French] [Kenny stit: he knows French] [Kenny stit: he has the capacity to speak French], though the complement of the first is agentive, the complement of the second describes a state of affairs, and the complement of the third is explicitly dispositional. By allowing any declarative sentence to appear as complement, the stit sentence changes the focus from actions to agents. It is the agent, in this case Kenny, who learns French, is responsible for paying the tuition, or is required to take the examination. The stit sentence reflects this fact by insuring in its semantics that the agent term remains recoverable no matter how deeply the stit sentence is embedded within wider contexts. Exploiting the stit complement thesis yet again we can disentangle Kenny's claim that "'John is taller than James' cannot be rewritten in the form 'John is bringing it about that ...'" (p. 182). Doubtless he means that "John is taller than James" is not agentive for John, because it is not equivalent to "John sees to it that John is taller than James." But he might be interpreted to mean that the sentence "John is bringing it about that taller than James" is incoherent because ungrammatical. Or he might be interpreted to mean that [John stit: John is taller than James], though grammatical and meaningful, will usually be false. "Usually" because if John has tied his legs to a rack and is stretching himself to gain the needed height, the sentence is bizarre but true. The stit sentence provides a way to distinguish the cases. Exploration of the modalities of agency requires breaking away from a picture of actions as events. Kenny, in his perceptive discussion, is considering just that alternative. He tells us that there are two ways of reporting the same event ("A od B" and "B was od by A"), and it is indeed possible to regard the notion of an event as an abstraction from these two forms of expression, designed to enable us to consider an occurrence without commitment to a special interest in either A or B.1 (p. 180) Restating the point with stit, we may say that there are two ways of reporting the same action, both of which can be paraphrased as [a stit: Q] : a sees to it that Q or 1

The symbols "A" and "B," as Kenny uses them, take individual terms for either the subject or the object of a "transitive verb of action."

3. Small yet important differences from earlier proposals

71

that .Q was seen to by a, and it is indeed possible to regard the notion of an action as an abstraction from these two forms of expression. Consider now the five ways Kenny claims that sentences of agency and action differ from ordinary relations of the form Rab: i. "Actions exhibit a variable polyadicity which is foreign to relations. [In sentences of action] we can pass from a dyadic relational proposition to a proposition made up of a single-place predicate and a name" (p. 156). ii. "The reverse of the first one. A sentence reporting an action not only can be shorn of one of its terms without making nonsense; it can also have further terms added to it in various ways" (p. 159). iii. "There is always more than one way in which a sentence containing a transitive verb of action may be false" (p. 162). iv. "Whatever is capable of occurring as the first term of a given n-adic relation is capable of occurring as the second or the nth term of the same relation. No distinction can be made between the various terms of a relation corresponding to that which can be made between an agent and the object of an action" (p. 167). v. "If Leonardo had not painted the Mona, Lisa, then one of the terms of the 'relation' would not exist. A relational sentence is equally 'about' either of its terms" (p. 168). Concerning the first and second ways: The stit sentence does not, nor is it meant to, shed light on Kenny's problems about variable polyadicity. One part of the problem is tractable. We can use the stit sentence to "pass from a dyadic relational proposition to a proposition made up of a single-place predicate and a name." We can make the inference, for example, from the canonically formed agentive [Leonardo stit: the Mono Lisa is painted] to the monadic "the Mona. Lisa is painted." This is an instance of the more general truth that [a stit: P] implies P. Further, adding sentential terms in the complement place of a stit sentence is grammatically unproblematic because there is no restriction on the declarative complement that can appear there, but it is logically tricky. Since all declaratives may be used as sentential complements in stit sentences, sentences of any logical complexity can equally stand in that place. But not all such additions yield valid arguments, [a stit: P or Q] does not follow, for example, from [a stit: P] or [a stit: Q]. Where Q is ~P, the case is clear since the tautology (P V~P) is never false. Because its falsity is never at risk, it is not guaranteed to be settled true by a choice of the agent. For example, it is true that [Leonardo stit: he paints the Mona Lisa] implies (by propositional logic)

72

Introduction to stit [Leonardo stit: he paints the Mona, Lisa] or [Leonardo stit: he doesn't paint the Mona Lisa],

but, as Porn 1974 lucidly argues, it is never true that [Leonardo stit: he paints the Mona Lisa or he doesn't paint the Mona Lisa]. Concerning the third way, Kenny says A sentence may be false in more than one way, in this sense, if and only if more than one state of affairs which would make it false may be described merely by the use of terms occurring in the sentence itself along with quantifiers, variables, and the negative operator (p. 163). Kenny's point here has less to do with the differences between agentives and other relations than with the adequate representation of negations in agentive sentences. As we noted in §2B.2, English does not clearly distinguish among various negations of a sentence such as The Mona Lisa is completed. The stit sentence, because it has sufficient structure to accommodate negation, is up to the task: From the comparatively simple ~[a stit: Q], It is false that Leonardo saw to it that the Mona Lisa was completed to the more complicated [a stit: ~ Q ] , Leonardo saw to it that the Mona Lisa was not completed or still more complicated [a stit: ~[a stit: Q]], Leonardo refrained from seeing to it that the Mona Lisa was completed. Because stit encourages nesting, and because stit has the appropriate structure to manage any number of negations, it can even deal with such sentences as [a stit: ~[a stit: ~[a stit: ~Q]]], Leonardo refrained from refraining from seeing to it that the Mona Lisa was not completed. Nor are we left up in the air about the interpretation of this sentence. In §2B.6 and §2B.8 we discussed refraining from refraining in a preliminary way, and in §15C we prove that in the special finite case that we sometimes call "no busy choosers," (see Def. 14), the sentence [ stit: ~[ stit: ~[ stit: ~Q]]] is equivalent to

3. Small yet important differences from earlier proposals

73

[a stit: ~Q]. We prove, that is, that the absence of busy choosers implies the validity of the "refref equivalence," Ax. Conc. 1, expressing the idea that to refrain from refraining from seeing to it that ~Q is the same as seeing to it that ~Q. Furthermore, chapter 18 proves the converse, that the presence of busy choosers implies the invalidity of the refref equivalence. Kenny's claim in the fourth way is that any term in a relation can stand equally in any place of the relation, but that the same is not true of agentive or action sentences. But there is no real difference, in this regard, between agentives and other relations. In "The painter of Mona Lisa is of Italian nationality," or "The element hydrogen has an atomic weight of 1," either of which might be symbolized as Rab, confusion may result when the terms are interchanged. The same kind of confusion might result if someone were to say "That the Mona Lisa is painted sees to it that Leonardo." Though inequality of terms is not distinctive of agentive sentences, Kenny has perceptively highlighted the contrast in such sentences between the agentive place and the sentential complement place. With regard to the fifth way, concerning existence and aboutness, the point seems to be that very often individuals are identified and described by reference to an antecedent. Though the antecedent is often agentive, it need not be. From the relational sentence "The annual flow of mud down the Nile doesn't deposit mud in the Nile Delta," we may infer that one of the terms of the relation, the Nile Delta, does not exist. Kenny's canonical form "bringing it about that P" is put to work in his theory of volition. The account has three parts: (i) the agent brings it about that P, (ii) the agent wills that P, and (iii) it is in the agent's power not to bring it about that P. With respect to ( i i ) , stit theory avoids explicit discussion of volition or will. Instead stit deals with an approximation of sees to it that by having [a stit: P] be settled true just in case a choice of the agent, a, guarantees the truth of P. Questions of volition or will are thereby postponed by incorporating all mental considerations into the agent's prior choice. With respect to ( i i i ) , if it means that it was possible at the time the choice is made for P to be false, the claim is correct. For if [a stit: P] is settled true, then it was possible for P to be false at an alternative moment. On the other hand, if (iii) means that it must be in the agent's power to see to it that he or she doesn't see to it that P, then (iii) is wrong. The considerations are similar those at the end of the preceding section. When Leonardo sees to it that the Battle of Anghiari is completed is settled true, it is not a necessary condition that there was a prior choice point when it was possible that Leonardo sees to it that the Battle of Anghiari is not completed is also true. It might have been that Leonardo's students were going to complete the painting at any point at which Leonardo stopped working. Further, if the

74

Introduction to stit

force of (in) is that it must be in the agent's power to see to it that ~P, where P is specifically agentive, it is also wrong. When Leonardo sees to it that he paints the Mona, Lisa is settled true, then there was a prior choice point at which it was possible for Leonardo paints the Mona Lisa to be false, but it is not a necessary condition that there was a prior choice point when it was possible that Leonardo sees to it that he doesn't paint the Mona Lisa is true. Further discussion can be found in §2B.2, including a pictorial demonstration of these results in Figure 2.6. The point is that when [a stit: P] is settled true, then at the prior choice point it was obviously possible for P to be false, therefore possible that [a stit: P] would turn out to be false, but neither of these involve a'S power. Therefore, neither the possible falsity of P nor the possible falsity of [a stit: P] need be due to the agency of a. As we have seen, because the complement of the canonical form "bringing it about that P" is restricted to sentences describing states of affairs, that form cannot shed sufficient light on the differences between agentives and other relational sentence forms. Nor can it shed sufficient light on the various negations of agentive sentences. Further, because it does not have a precise semantics, "bringing it about that P" cannot make the fine distinctions between agentive powers and temporal possibilities. The stit sentence, although it can help clarify all those issues, remains only subtly different from Kenny's canonical form.

3D

Castaneda

Von Wright begins his study with considerations of change and alteration, Chisholm with intentions and endeavors, and Kenny with the language of agency and action itself. Castaneda, to whom we now turn, sets out to clarify matters by considering practical reasoning. On Castaneda's account, the language of agents and their achievements takes actions to be the finished products of causal sequences that originate in practical reasoning. Practitions, the contents of practical reasoning, are markedly similar to stit sentences. This similarity is not an accident. It arises from the fact that both practitions and stit sentences are intended, among other things, to function as the descriptive core of deontic constructions. The schematic account that follows merely skims the surface of Castaneda's extensive theories. It is meant to provide just enough background to allow us to make some comparisons with stit. For further discussion of Castaneda's language of action see Castaneda 1974 and Castaneda 1975, as well as the commentaries of Bratman 1983 and Tomberlin 1983a. Castaneda separates practical thinking from theoretical thinking, giving each a different characteristic activity and a different characteristic content. The

3. Small yet important differences from earlier proposals

75

characteristic activity of theoretical reasoning is contemplation and its characteristic content is the proposition. The characteristic activity of practical reasoning is deliberation and its characteristic content is the practition. Practitions are "the nonpropositional purely practical contents of practical thinking"; they appear either as prescriptions or intentions (Castaneda 1976, p. 106). Prescriptions are second or third person contents, while intentions are first person nonpropositional contents of practical thinking. Prescriptions "are, on the one hand, the cores of commands, requests and acts of telling someone to do something; on the other hand, a practition is what one intends, plans, or decides to do" (Castaneda 1976, p. 106). For the purposes of this discussion, we think nothing is lost by talking only about practitions. Castaneda's canonical form for practitions, X to A, with X open for "a name or description of an agent or agents," and A open for "a description of the act or acts to be performed" (Castaneda 1974, pp. 39-40), is very similar to the stit sentence, except that it restricts the second open place to verbs that describe acts. Thus, even though all practitions are agentive, not all agentives are practitions. Castaneda offers an argument for adopting the narrower approach. (The asterisks are a quotational device that produce the name of a content of practical reason, that is, they produce the name of a practition; see Castaneda 1975, pp. 19-20. An obvious typographical error has been corrected in this passage.) Some philosophers, including myself during some periods of reflection, have proposed to analyze all actions as bringings about of some state of affairs. On one view of such a type, the attribution to John of the action of opening the door made by the action proposition *John opened the door* is really the proposition *John brought it about that the door became open*. This view is, however, glaringly incorrect. Clearly, John can bring it about that the door becomes open, without himself opening it, for example, by having Marilyn open it. (Castaneda 1975, p. 98) By now we know that according to the stit complement thesis (Thesis 2), [a stit: Q] is agentive for any arbitrary sentence Q; according to Castaneda's objection some sentences, those that produce obviously false agentives, should be excluded from the complement place. The clear truth of Castaneda's premiss, that there are some complements in agentive constructions that produce absurdities, ought not to lead us to accept the conclusion that such constructions should be excluded from our language. Consider that conjunctions in propositional logic are true when all their conjuncts are. We do not, on that account, exclude those conjuncts that are the negations of one another. We have trivialities and absurdities in the language of our theories because it is good logical practice to include as well formed all those constructions that promise to be illuminating, even if they are absurd, even if they are trivial. Thus, in a stit sentence there is neither grammatical nor metaphysical nor semantic restriction on the declaratives that may be put in place of Q. While

76

Introduction to stit

some resulting stit sentences may be distinctly odd and some obviously false, such cases turn out to be revealing. They further our understanding of agency, of agentive sentences, and of the wider environments in which agentives are found. We have already seen that when Q is a tautology, [a stit: Q] will never be true. The negative condition will never be satisfied, since the tautology is settled true on every history, so cannot be true due to the prior choice of a. Similarly, when Q is a contradiction, [a stit: Q] will never be true. In this latter case, the positive condition will always be violated since there is no history on which a contradiction is ever settled true. We also noticed in previous sections that having these forms available is a significant aid to the understanding of inferences involving agentives. Castaneda's example in his objection to the stit complement thesis does not happen to concern complements that are tautologies or contradictions, but rather a class of English sentences involving two agentive sentences with different agents as subjects. Castaneda's conclusion is that neither one should be allowed to appear as the complement of the other. Again, because there is much to be said about such cases that cannot be said if such sentences are deemed to be ungrammatical, it is not good policy to exclude them. Consider the two agents John and Marilyn and the sentence "The door is open." Let us assume that the English sentence John opens the door, when it is agentive for John, is strongly equivalent to [John stit: the door is open], and is settled true just in case "the door is open" is true due to a prior choice of the agent John. Let us also assume that [Marilyn stit: the door is open] is equivalent to [Marilyn stit: Marilyn opens the door]. When John and Marilyn act jointly, perhaps by jointly heaving a heavy door that neither could open alone, it is settled true that [John and Marilyn stit: the door is open]. What drives Castaneda's intuition is that when John opens the door and

Marilyn opens the door are both agentive, then neither

3. Small yet important differences from earlier proposals

77

[John stit: Marilyn opens the door] nor

[Marilyn stit: John opens the door] will ever be true. We prove as Fact 10-2 that stit theory implies this result; here our aim is merely to make it plausible. Consider just the last example: If John's choice guarantees that The door is open is true, then there is no room for Marilyn's choice. She cannot be agentive in the matter of the open door. On the other hand, when Marilyn's choice guarantees that "the door is open" is true, then [John stit: the door is open] must be false. Marilyn might convince or cajole or seduce John into opening the door. In those cases, she does not see to it that the door is open; instead she sees to it that John is more or less likely to open the door. Notice how much different it is to deny another agent a choice. John can see to it that Marilyn doesn't open the door by barring her way or opening it before she does. She in turn can see to it that he doesn't open the door by locking it. If, however, Marilyn rigs up a Rube Goldberg device on which she pulls a lever that throws a switch that drops a ball that shoves an unconscious John against the door, whereupon his body pushes the door open, then "John opens the door," taken as summarizing this event, is not agentive. On that reading of "John opens the door," [Marilyn stit: John opens the door] is settled true but unproblematic. For when the complement of a stit sentence is not agentive, and the stit sentence is true, there is no difficulty. On the other hand, when the complement Q of [a stit: Q] has an agent-term different from a for its grammatical subject, if the complement is agentive then the stit sentence is always false (see §10B for details). That is an interesting and informative fact, made clear by allowing all declarative sentences to appear as complements of stit sentences. There is a place for the exclusion of complements, not in stit sentences but in deontic constructions. On that point we are in full agreement with Castaneda. Practitions are what agents intend, plan, or decide to do; practitions are the contents of requests and tellings. The stit sentence is also meant to represent all such contents. It is an agent who is obligated to see to it that something is the case. An agent a fulfills an obligation to see to it that Q just when [a stit: Q] is settled true. Castaneda tells us that one of his oldest philosophical theses is that "deontic concepts are modal operators that transform the purely practical, nonpropositional contents of practical thinking into propositions" (Castaneda 1976, p. 106). The restricted complement thesis makes the same point in its insistence that deontic constructions must take agentive complements. One exciting application of this thesis is found in Castaneda 1981: the attempt to solve all the

78

Introduction to stit

deontic paradoxes by enforcing restrictions on deontic complements. A similar project, using the stit sentence as the foundation, is reported in chapter 11. Notice that the restricted complement thesis taken together with the stit complement thesis insures completeness because any declarative whatsoever can occur within a deontic construction. Non-agentives occur only as the complements of agentives. As we have just seen, the stit sentence differs from a practition in one important detail: Whereas a stit sentence can be constructed with any arbitrary declarative as its complement, practitions are designed to be less catholic. As we have also seen, where the two forms agree they are equally useful as complements in deontic constructions, clarifying these and other contexts in which agency is important. Where they differ, however, the restrictions imposed by the practition tend to limit, rather than advance, our understanding of agency and the constructions in which it is important.

3E

Davidson

Because Davidson 1966 has given rise to a substantial literature, our commentary will not be extensive. Several remarks are in order concerning the difference between the stit sentence and Davidson's approach. Though our goals are similar, Davidson intends to clarify the inferential structures in which action sentences appear by looking at the parts from which they are constructed. We have in mind the intention made explicit in the initial description of Davidson 1966 of its own task: I would like to give an account of the logical or grammatical role of the parts or words of [simple sentences about actions] that is consistent with the entailment relations between such sentences — From the modal point of view, it is striking that the aim set out in this passage includes only half of what is needed for a compositional account of meaning. Davidson 1966 sets out to show how "the meanings of action sentences depend on their structure," but does not begin with the aim of showing how the meanings of sentences that contain action sentences depend on their structure. The stated aim does not include, for instance, telling how the meaning of "Jones refrained from buttering the toast" or "Mary demanded that Jones butter the toast" or "Jones, butter the toast!" or "How speedily did Jones butter the toast?" or "Jones brought it about (or saw to it) that Jones buttered the toast" depends on the meaning of "Jones buttered the toast," or perhaps telling how it doesn't if it doesn't. Half the compositional problem has been left out of the initial statement of purpose. The modal logic of agency tends to strike the other way. The modal logic of agency should be interested in larger contexts containing agentive sentences. The fact is that with regard to embedding agentive sentences in larger contexts, it makes a difference that they are agentive, and it makes a difference who the agent is. Embedding contexts care about these things. The reason

3. Small yet important differences from earlier proposals

79

that the action-as-event paradigm has not contributed to our understanding of the embedding of agentives is, perhaps, that its resources do not permit it to do so. Here is a slogan that gives a smallish part of the explanation why: Propositions and sentences have negations, but—in spite of formal tricks to the contrary—actions and events do not have negations. That is why the stit analysis of "refraining," §2B.2, seems so clear in contrast to analyses that invoke, for example, "negative acts." See also the beginning of chapter 5 for an example of how "actions as events" seems to interfere with efforts to analyze the idea of promising. The following observation may also be part of the explanation. At the end of Davidson 1966, the question is raised of the intentionality of action. With clear recognition that what is being addressed is a part of the other half of the compositional problem, Davidson proposes that we introduce intention by means of an explicitly embedding expression such as it was intentional of x that p, where V names the agent and 'p' is a sentence that says the agent did something. Although the proposal presumably requires for its coherence that p display a term for "the agent" in some recoverable fashion, it seems to borrow no other feature from the earlier thesis that the logical form of the embedded sentence involves an ontology of events or actions. The indicated lack of influence of the earlier thesis on the later proposal is contrary to expectations, since generally in compositional semantics a view about the logical form of a "part" of a certain kind both constrains and is constrained by a view about the logical form of an expression that embeds just that kind of part. We think that this is another indication of the difficulty of using the picture of actions-as-events as a guide in understanding the role of agentive sentences when they are embedded in larger contexts. Davidson 1966 canvassed proposals similar to ours and found them wanting for a variety of interesting reasons. We take up four. First, his essay objects that modal constructions do not solve his problems, for example variable polyadicity. We concur, noting only that we are concerned with different problems, for example, as noted earlier, the problem of providing a form for agentive sentences that will show how they contribute to the larger contexts—imperative, deontic, and so on—to which they are peculiar. Second, it is surely correct that "no grammatical test ... in terms of the things we may be said to do, of active or passive mood, or of any other sort, will separate the cases here where we want to speak of agency" (p. 94). But rather than being an objection, that is exactly a reason to promote the stit paraphrase thesis, Thesis 3. Since there is no grammatical test for agency, we gain the most advantage by determining that a sentence Q is agentive whenever it can be paraphrased as [a stit: Q]. Third, Davidson objects that "the doctor may bring it about that the patient has no appendix by turning the patient over to another doctor who performs the operation; or by running the patient down with his Lincoln Continental. In

80

Introduction to stit

neither case would we say that the doctor removed the patient's appendix" (p. 86). But as we noted in §1A, these are not reasons to reject the stit sentence. Rather they are reasons to embrace it, for we want to distinguish between agentive and non-agentive interpretations of the same English sentence. By now it is clear that The doctor removes the patient's appendix is agentive for the doctor if, and only if, it is equivalent to [The doctor stit: The doctor removes the patient's appendix]. Objecting to Chisholm's undefined locution, Davidson suggests that we ... take as our example "Jones batted an eyelash." In this case I think nothing will do but "Jones made it happen that Jones batted an eyelash" (or some trivial variant), and this cannot be called progress in uncovering the logical form of "Jones batted an eyelash." (p. 87) On the contrary, we contend that progress has already been made, and that the stit sentence will likely continue the progress. In English when "Jones batted an eyelash" is not agentive, the appropriate verb is "blink." When "Jones batted an eyelash" is agentive the appropriate verb is "wink." It may be that Davidson is right that "when we understand the verb we recognize whether or not it includes the idea of an agent" (p. 94); but since natural languages are fluid and changing, someday that nice distinction might be lost. Nice distinctions, comfortable enough this year, may be covered by linguistic barnacles, gobbled up by more aggressive phrases, or simply lost in a sea of idle chatter. (Consider that in our time English is losing a word to describe young men of a particularly lighthearted demeanor.) Idealized constructions, on the other hand, keep firm control of their environment as well the individuals and structures that reside there. This permits us to maintain the distinction between [Jones stit: Jones batted an eyelash], which is agentive, and Jones batted an eyelash, but he didn't see to it that he did, which isn't. Apart from philosophical clarification, the difference is important when we are trying to decide whether to respond with a wink of our own or with an eyedropper. Fourth and finally, with respect to Davidson's objection to a "bringing it about that" proposal of Reichenbach 1947, and by extension to any stit-like analysis: If [The astronaut stit: The astronaut flew to the Morning Star] is settled true, then assuming that the Morning Star is necessarily identical to the Evening Star,

3. Small yet important differences from earlier proposals

81

[The astronaut stit: The astronaut flew to the Evening Star] is also settled true, though the astronaut knew that he flew to the former and didn't know that he flew to the latter. On the other hand, if the identity is not necessary, the inference is suspect. (This objection is suggested in Lemmon 1966.)

3F

Conclusion

None of the differences between stit and its predecessors is extensive. Rather they are small and subtle. But, as Reichenbach reminds us, little deviations often lead to deep insights. "It is," he says, "as if nature discloses its fundamental relationships in the minute errors of current theories" (Reichenbach 1980, p. 32). Von Wright's sentence form, Chisholm's undefined locution, Kenny's canonical form, Castaneda's practition, and the stit sentence all attempt to provide the foundations for a modal language of action. The stit sentence advances that project. The stit sentence has no restrictions on declarative sentences to occur as complements. The stit sentence is sufficiently rich to serve as a canonical form for all agentive sentences. The stit sentence equips itself with the resources to maintain focus on the agent. The stit sentence is rich enough to display the value of restricting the complements of deontic statements to agentives. The stit sentence has a semantics that pictures agents as making choices against a background of temporal logic and branching time. As we have repeatedly seen, these minute differences, taken together, yield significant theoretical advantages.

4

Stit and the imperative Suppose that we humans were to be deprived of all forms of speech, except just one that we might choose to keep.* Which would it be? On p. 10 of chapter 1 we cited and endorsed, as a much-neglected truth, Hamblin's confident assertion that we should choose imperatives over all other forms. The central concern of the following is to explore ways that stit can help in the understanding of this important category. Among the factors underlying the relationship between stit and imperatives is the fact that on the one side stit is a theory of agents seeingto-it-that, while on the other issuing and receiving imperatives are distinctive forms of agentive behavior: Agents use imperatives to tell other agents what to do, or what not to do, to advise them about how to act and what to wear, to invite other agents to dinner and to the theater, to request favors and to demand attention. The concerns of this chapter are narrowly defined, focusing primarily on three previous studies of imperatives: Hofstadter and McKinsey's theory of fiats, Chellas's exploration of the logical form of imperatives, and Hamblin's in-depth study of the logic and grammar of imperatives. Our aim is to show both how stit has been informed by these works, and how stit, in turn, attempts to carry forward the projects there begun.

4A

The theory of fiats

We begin our expedition with the pioneering work Hofstadter and McKinsey 1939. They present the fiat as the appropriate form for the representation of imperatives. Although they consider a stit-like form called a directive, Hofstadter and McKinsey choose instead to focus their attention on the fiat. They have located an important difference between a directive, which provides a place for the agent who is to carry out the imperative, and a fiat, which "contains no reference to the agent who is to carry it out" (p. 446). Their decision to *With the kind permission of the American Philosophical Quarterly, this chapter is drawn from Perloff 1995. 82

4. Stit and the imperative

83

focus on the fiat, which makes "the satisfaction of an imperative ... analogous to the truth of [an indicative] sentence" (p. 447), yields a simple and powerful theory of imperatives. The syntax is uncomplicated, using the exclamation point as the only special symbol. The theory is expressively complete in the sense that every indicative can be associated with a corresponding imperative by having an exclamation point appended to it. The semantics incorporates an already developed truth-tabular account developed for indicatives. Imperatives are assimilated to indicatives via the notion of satisfaction. "We understand an imperative to be satisfied if what is commanded is the case. Thus the fiat 'Let the door be closed!' is satisfied if the door is closed" (p. 447). So, the ordinary language imperative chosen by Hofstadter and McKinsey as their paradigm, Close the door!

(1)

is represented by the fiat Let the door be closed!

(2)

Since the semantic configuration of the fiat is designed to be structurally isomorphic to the semantics for classical logic, the fiat (2) is satisfied iff the indicative sentence The door is closed

(3)

is true, and fails to be satisfied iff (3) is false. Extending the semantics to accommodate conjunction, the fiat Let the door be closed and bolted!

(4)

is satisfied iff both (3) and The door is bolted

(5)

are both true; whereas (4) fails to be satisfied if either (3) or (5) is false. Moving to disjunction, the disjunctive fiat Let the door be closed or bolted!

(6)

is satisfied if either (3) or (5) is true, not satisfied iff both are false. One of the items in Ross's searching examination of the theory of fiats in Ross 1941 is his famous counterexample. The counterexample points out that the truth conditions for (6) bring unwanted consequences. For example, the fiat Post this letter or burn it! follows from Post this letter!

(7)

84

Introduction to stit

Ross comments that this result shows us that the theory of fiats "is surely not a logic of such content which we have in mind in the case of practical inferences" (p. 61). The linguistic setting in question, the setting Ross describes as a case of "practical inference," is just the sort of setting where agency is essential. Since the stit sentence is designed to provide a normal form for agency, let us reconsider Ross's paradox with that machinery at our disposal. (For an alternative approach to the constellation of issues surrounding imperatives, see Segerberg 1988b and Segerberg 1990.) Our discussion will depend both on the stit theses listed in §1 and the semantics for stit offered for the deliberative stit, §8G.l, and for the achievement stit, §8G.3 (here it will make little difference which stit is at issue).

4B

Ross's paradox and stit

We may, if we choose, decide that certain constructions are not agentive. In such a case, we may rest content with the classical implication, since, if the nonagentive The letter is posted is true, then The letter is either burned or posted is also true. The imperative content thesis, Thesis 4, however, recommends that we treat the content of any imperative sentence as agentive. (We return to the topic of agentive constructions in §4D.) By the stit paraphrase thesis, Thesis 3, we know that an agentive construction is always appropriately paraphrased as a stit sentence. Further, since stit extends us the opportunity to clarify by normal forming, we are in position to verify that [a stit: The letter is posted] does not imply [a stit: The letter is either burned or posted]. The details are subtle enough to make a demonstration worthwhile. Let us begin by noticing that although

follows from

4. Stit and the imperative

85

the disjunction of agentives portrayed by (8) is not itself agentive. It is not agentive because, by the stit paraphrase thesis, it is not in general equivalent to a sentence of the form [a stit: P\. Since (8) is not agentive, it cannot, by a "negative" application of the imperative content thesis, be the appropriate canonical form for a disjunctive imperative. Further, by a "positive" application of the thesis, there is an alternative close at hand. Stit theory permits us to take

as the appropriate representation of the complement of the imperative (7). But now the situation is entirely changed. The offending inference is no longer valid. Here is an indication that (10) does not follow from (9): Suppose that [a stit: P\ is settled true at a moment mo, and (for the purposes of this proof) suppose that there is only one relevant choice point for a, say w. We call the set of moments later than w that lie on the same instant as mo "the horizon from w at i(m 0 )" (see Def. 9). Suppose Q is settled true throughout the horizon from w at i(m0)- In such a case (PV Q) is also settled true throughout that horizon. But then [a stit: (PV Q)} could not be true at any of those moments, since the negative condition Definition 2-4(ii) would be violated. This account of "Ross's counterexample" is not ad hoc. It is part of the very foundations of stit theory. But that is not the only account accommodated by stit. Another alternative takes [ stit: [ stit: P] V [ stit: Q as the appropriate representation for (7). In such a case, the proof that [a stit: [a stit: P] V [a stit: Q]] does not follow from [a stit: P] relies on an example in which, among all the moments in the horizon from w at i(mo), Q is false somewhere and Q is true wherever [a stit: P] is not true. With Ross's paradox now seeming less paradoxical, and with the need for a less restrictive sentence form in mind, we turn to Chellas's study of the logical structure of imperatives.

4C

Chellas's theory

The theory of fiats sets the study of imperatives on the right path and with a good footing. Fiat theory shares a helpful point of view with stit, and also with Chellas 1969, who puts the matter as follows: ... imperatives are a species of sentence the logical form of which can be investigated by examining the logical properties of their counterparts in a suitably articulated, well-defined language (p. 4). It is to Chellas's forceful and influential study that we now turn. (Horty 2001 provides additional discussion of Chellas's contributions.) We saw earlier that while Hofstadter and McKinsey consider both fiats and directives, they provide

86

Introduction to stit

a formal account only for fiats, and we saw that their semantics is classically based. Chellas supplies imperatives with a modal semantics and is emphatic in the need for adding a temporal structure. The "recognition of temporal elements," he says, "compels us to acknowledge the dependence of imperative obligation upon time; an (imperative) obligation to the effect that such-andsuch be the case may hold at some times and not at others" (p. 77). In stit theory, the temporal structure is portrayed by branching histories. Chellas's logical form for the directive,

is easily seen to be a notational variant of a stit sentence, with the interpretation that T sees to it that O where T is an agent and O takes the place of a sentence. On consideration, however, there are important differences between Chellas's directive and the stit sentence. First, Chellas tells us that ATO is true in this world iff O is true at all those worlds responsive to the action of the agent T. The comparable idea in stit is stated by the positive condition: that [a stit: Q] is settled true when Q is settled true at all those moments choice equivalent to mo (the responsive worlds) for a from the perspective of m (this world). Stit's negative condition for the truth of [a stit: Q} would, in Chellas's terms, say that ATO is true in this world iff O is true at only those worlds responsive to the action of the agent T. This condition was not part of Chellas's original program, and Chellas 1992 reiterates his explicit rejection of this condition as part of a critical evaluation of the stit program.1 Further, in contravention of the restricted complement thesis, Chellas does not restrict the complement of his imperative operator to directives, allowing either fiats or directives to occur as complements for imperatives. Let us now consider one of Chellas's central ideas: "The primary thesis maintained in this essay is that sentences in the imperative mode express obligations" (p. 1). There is, without doubt, a deep and important connection between imperatives and obligations. But, as we shall see in §4F, there are many kinds of imperatives, including advice, suggestions, invitations, and requests, most of which do not establish obligations at all. What, then, shall we say of the relation between imperatives and obligations? By the restricted complement thesis, the complements of both imperatives and deontic constructions must be agentive. Further, by the agentiveness of stit thesis, Thesis 1, we are guaranteed that [a stit: Q] is always agentive for a. Chellas's primary thesis leads naturally to the following: If an imperative obligates a to see-to-it-that Q, then a has been ordered to see-to-it-that Q. When Kathy says to her son Ben Clean your room! 1 Although we retain our conviction that the negative condition is an important part of the concept of agency, in later chapters we find a number of contexts in which it seems good to use the form, [a cstit: Q}, with the no-negative-condition semantics of Chellas 1992. We call it the Chellas stit.

4. Stit and the imperative

87

and by that utterance creates an obligation, in this case the obligation that Ben clean his room, she has succeeded in ordering him to clean his room. When Ben utters the same sentence, he does not succeed in ordering his mother to clean her room. Even if he intends it as an order, he succeeds, at best, in offering a piece of advice, or perhaps making a joke. Without the ability to impose obligations of one sort or another, imperatives must fail to be orders. This approach evidently differs essentially from that of Searle 1965 and Searle and Vanderveken 1985 in its concern with what is accomplished with an utterance rather than with what an utterance is intended to accomplish. We learn some important lessons from Chellas: (i) the need for an agentive form as the complement for imperatives; (ii) the need for a modal semantics for imperatives; (iii) the need for temporal elements in the semantics and (iv) the close relation between obligations and imperatives. All of these lessons are taken account of in the syntax and semantics of stit. Still further progress, however, is possible if we widen our understanding of imperatives. We proceed toward that destination with Hamblin as our guide.

4D

Agentive constructions

In the last several sections, the notions of an agentive and an agentive construction have been invoked repeatedly. The agentiveness of stit thesis guarantees that a stit: Q] is always agentive for a. The stit paraphrase thesis tells us that Q is agentive for a iff Q is appropriately paraphrased as [a stit: Q}. The restricted complement thesis requires that imperative and deontic constructions, among others, take agentives as their complements. We now propose to show how the stit understanding of this important idea continues along the trail blazed by Hamblin 1987. The most striking difference between stit and Hamblin is that Hamblin's tests for agentives look to features of ordinary usage and ordinary language, while stit utilizes the benefits of its theoretical apparatus to separate agentive from non-agentive constructions. (Hamblin credits the distinction between agentive and stative to Vendler 1957 and the application to imperatives to Kenny 1963.) That is, Hamblin is trying to uncover the relevant differences between agentives and non-agentives in the ways we ordinarily speak about such matters. Stit, on the other hand, specifies the distinction with the aid of a theoretical structure. The point in both is the same: to discriminate between those constructions that are, and those that are not, agentive. Let us first look at Hamblin's five tests (pp. 54-56) for distinguishing agentive from stative predicates and then proceed to the stit versions. i. "Agentive, but not stative, predicates can take a 'continuous' tense"; so Ben is running is acceptable, but not ordinarily Ben is knowing the answer.

88

Introduction to stit ii. "Agentive, but not stative, predicates fall under the genus do something." In answer to a question about what Jessica did, you may answer Jessica cleaned the house but not

Jessica was a student. Hi. "Agentive, but not stative, predicates can be augmented with the adverb deliberately, or with adverbs of manner such as carefully, enthusiastically"; We may properly say Mark worked deliberately, but not

Mark rested deliberately, or

Mark slept carefully. iv. "Someone may be persuaded or reminded in respect of an agentive predicate such as to go to bed, but not in respect of a stative one such as to understand the instructions or to be asleep." v. "The present tense of an agentive, but not stative, verb can be used with future reference"; so we might have Kathy leaves for school soon but not

Kathy sleeps soon. Because his approach relies on the vagaries of ordinary speech, Hamblin recognizes that these tests are, at best, indications of the distinction between agentive and non-agentive constructions rather than precise markers. Notice, for example, that "Kathy sleeps soon" passes the first test, fails the third, fourth, and fifth, and remains in doubt with respect to the second. It is perhaps inevitable that attempts to find absolute distinctions in the ways we ordinarily speak are doomed to imprecision. Stit, by contrast, specifies the difference between agentive and non-agentive constructions in terms of the theory itself and validates that account in application. Consider the sentence Jessica trips.

(11)

Is it agentive or not? The stit view is that (11) is agentive whenever Jessica's tripping is guaranteed true because of a prior choice of the agent, in this case Jessica. If (11) is true because of Jessica's prior choice, then it will be appropriately paraphrasable as

4. Stit and the imperative [Jessica stit: Jessica trips].

89 (12)

The same holds true for any sentence Q (by the stit complement thesis, Thesis 2) and for any agent a (by the agentiveness of stit thesis, Thesis 1). Further, by the positive condition, whenever a choice point, w, witnesses that (12) is true at a mo, (11) is settled true at all the moments choice equivalent to TOO (with respect to Jessica and w); and, by the negative condition, there will be a moment on the horizon from w at i(mo) where (11) is not settled true. The relevance of these theses for imperatives is displayed by the imperative content thesis, Thesis 4, that an imperative must have an agentive as its content. Earlier, in §4C, we saw how stit accommodated Chellas's primary thesis. Let us now look at what Hamblin calls the "most fundamental principle behind [his] treatment of imperatives." Hamblin's addressee-action-reduction principle is to the effect "that the meaning of any imperative can be spelt out in plain-predicate agentives" (p. 58). By the stit paraphrase thesis, any sentence Q is agentive iff it is strictly equivalent to [a stit: Q]. So the meaning of any imperative, by this combination of ideas, can be spelled out with stit sentences; and that, as we have repeatedly seen, is the intended application of the imperative content thesis.

4E

Negations of imperatives

Having in hand the basics of stit theory, and a fuller account of the concept of an agentive, we are now prepared to continue our exploration of imperatives, concerning ourselves with negation. Before presenting the stit versions, let us survey Hamblin's classification of imperative negations (pp. 64-71). Hamblin's type 1 negation is the first of five distinct kinds of negations for imperatives. This negation, he tells us, applies to statives as opposed to agentives. Type 1 negation is "the one closest in spirit to propositional negation." In this sense the negation of "Let the door be closed" is "Let the door not be closed." Type 2 negation applies to agentives. Negation of this sort changes an order "from an instruction to carry out that action to an instruction to refrain from doing so." In English, one has the passage from Close the door

(13)

Don't close the door.

(14)

to

For negations of type 3, Hamblin supposes that there "is a natural order of events that will prevail provided nobody, or nobody relevant, interferes." This sort of negation, Hamblin tells us, can be expressed with "the use of emphasizer words" as in Don't actually close the door.

90

Introduction to stit

The idea is that this delicate use of emphasis is supposed to convey an order to let the event occur. Negations of type 4 comprise those "locutions whose role it is to negate ... pressures, enticements or encouragements to undertake a course of action." Hamblin characterizes such negation as what some deontic logicians have called 'external' negation. Type 4 negations transform the order to close the door into the permissive You may refrain from closing the door. Finally, negations of type 5 "may serve to repudiate or withdraw corresponding imperatives," in case the issuer of an imperative wishes not to be committed to the imperative. Thus, I do not order you to close the door is a negation of the fifth type and "different from explicitly permitting noncompliance." Type 2 negations present stit theory with a nice problem. In stit theory there is only one primitive form of negation, sentential negation. It can be used to negate any sentence, whether agentive or not, whether imperative or not. Let us notice that because a stit sentence takes any arbitrary declarative as complement, each stit sentence naturally admits a sentential negation in either of two places. For example, [ sitt: the door is closed]

can be negated either by

or

In (15) the scope of the negation is the declarative complement and the resulting construction remains agentive. That's good, but nevertheless (15) cannot represent the content of the type 2 negation, (14), since (15) evidently answers instead to "See to it that the door is not closed," or perhaps even, in context, "Open the door." In (16) the negation is external, but (16) is no longer agentive. Although (16) expresses not doing, because it is no longer agentive, it cannot transform an order "from an instruction to carry out that action to an instruction to refrain from doing so." (16) cannot be the type 2 negation of (13). Something more is needed, and it is a decided advantage of stit that it gives us that something more. The stit sentence

4. Stit and the imperative

91

is both agentive and expresses not-doing. (See also the discussion of refraining in §2B.) Now we can see why the appropriate normal form for (14), "Don't close the door," is (17) rather than either (15) or (16). Type 3 negations invoke a distinction between letting something happen and making it happen. Negations of type 3 assume "a natural order of events." Because in stit the future is open, replete with real choices and real possibilities, it is a matter of policy and not oversight that this type of negation is not accommodated in the foundations of stit. Note that it is not inconsistent with stit theory to characterize some outcomes as "seen to" and others as "allowed to happen" by adding additional concepts to the basic structure. Type 4 negation, characterized by Hamblin as "external negation," transforms an order into a permissive, or may transform an imperative such as (14) into "You may close the door." But these reflect the standard deontic equivalences, equivalences likely to be represented in stit theory, with the obvious abbreviations, as STIT DEONTIC EQUIVALENCES. (Axiomatics concept. Reference: Ax. Conc. 2) Frbn:[ stit: Q] Perm:[ stit: Q] Oblg:[ stit: Q]

Oblg:[ stit: ~[ stit: Q ~Oblg:[ stit: ~[ stit: Q]] ~Perm:[ stit: ~[ stit: Q]].

(There is extensive discussion of agentives in deontic contexts in chapters 5, 11, 12, and 14.) To understand negations of type 5, where the issuer retracts a commitment, we focus on the fact that issuing an imperative is an act, a speech act, something the agent sees to. Therefore, even without a theory of ordering, we may describe Kathy's order that Ben close the door either by Kathy orders that Ben sees to it that the door is closed,

(18)

or, using the stit paraphrase thesis, by the canonical form, [Kathy stit: Kathy orders that [Ben stit: the door is closed]].

(19)

Evidently (19) is a sentence having the form The negation of (20), using Hamblin's negation type 5, is that is, Kathy sees to it that she does not order Ben to close the door. In short, Kathy refrains from ordering Ben to close the door. In stit theory there is only one negation, sentential negation. As we have just seen, however, that negation together with stit can help to represent all the negative forms described by Hamblin that don't depend on a distinction between making and letting. We may go a little further if we permit "ordering" to be paraphrased, roughly, as "seeing to it that the recipient is obligated," so that (20) comes to

92

Introduction to stit

a sees to it that (3 has an obligation to see to it that Q. The next step is to (redundantly) double the stits in (22), just as an analytic device:

The point of this redundant doubling is to permit us to call attention to some possible blanks, one in front of each ingredient sentence:

This normal form of an order makes it evident that in each of the numbered blanks there is room for a negation. A worthwhile stit exercise is to see how instructive it is to fill these blanks with negations in various combinations, thus pulling together into a single schema some otherwise confusing observations. For example, filling 3 and 5 with negations describes the positive granting of permission or authorization, while filling 2 and 5 with negations describes the positive act of refraining from laying on an obligation, which is sometimes thought of as a kind of permission, Finally, filling 1 and 5 with negations describes the non-act of not laying on an obligation, which some also might think of as falling within the precincts of permission. This suggestion is to be compared with the more refined idea of Wansing 1998, according to which agency is made intrinsic to the concept of obligation; see §11H.

4F

The many varieties of imperatives

Having followed Hamblin through the difficulties of agentive constructions and the nature of imperative negation, let us continue our progress by attending to the wide variety of imperatives that he locates. No work that we know has been so sensitive to, or elaborated so thoroughly upon, the varieties of the imperative as has Hamblin 1987. It is, as we saw earlier, easy to think that Close the door!

(25)

has its natural home as an order given by one agent to another. But there are other constructions as common as commands and equally representative of imperatives. Hamblin's list includes demands and requests, as well as advice, instructions, suggestions and recipes. Of course, if an agent has the requisite authority, then, as we said earlier, that agent may use (25) as a command. Absent the appropriate authority, the agent might utter (25) as a request or even a demand. Demands, Hamblin tells us, "lack the authoritative backing of commands, but are altogether too peremptory for requests" (p. 9). The category of imperatives consisting of commands, demands, and requests is, according to Hamblin, willful imperatives. Willful imperatives need no rational justification apart from the social context in which they are issued. Willful imperatives require only that the issuer have the proper social standing or be in the proper

93

4. Stit and the imperative

social relation to the addressee. Nor do willful imperatives need to be acted upon to be imperatives. Hamblin also locates a contrasting class of imperatives, nonwillful imperatives: Advice and suggestions are two examples. When Kathy advises, not orders, Ben to Close the door! her advice is naturally expressed in the imperative. Advice is supposed to serve, at least putatively, the interests of the addressee. It may be rationally justified, argued for, and considered as either serving those interests or not doing so. Suggestions, instructions, and recipes too, are advice-like and nonwillful; that is, they are supposed to be in the interests of the addressee, and can be rationally justified or criticized as serving or not serving the addressee's interests. "Interests" here may be more widely or narrowly interpreted. We might advise you to see to something, not simply because it is prudent, but because it is the wiser or more moral course for you to choose. In either case, the justification for the advice concerns the addressee rather than the issuer of the imperative. Notice further that Kathy's utterance remains a piece of advice whether or not it is ever acted upon, whether or not Ben even seriously considers acting on it. The examples so far given do not exhaust the range of imperatives. There are invitations as well as wishes, maledictions, and exhortations. An invitation, like a command or an order, is willful in that it requires that the issuer have the appropriate social relation to the addressee. For example, Kathy can invite Ben to her party only if she has the ability to give Ben the requisite permission to attend the party. Further, invitations may be made without either the hope or expectation that the addressee will accede to the invitation. So, we can imagine that while Kathy hopes that Ben will attend the party, she invites Jessica to the same party without in any way wanting Jessica to attend. Nor need Jessica attend the party, nor even consider going, for the invitation to be in force. These considerations point to the need for a semantic account of imperatives distinct from either the satisfaction of fiats or descriptions of the psychological states of issuer or addressee. Consider the illumination possible using the stit sentence where the semantic structure is designed to accommodate (i) agentive choice, (ii) the openness of the future, and (iii) the fact that imperatives embed one within another. For example, if, when Kathy says to Ben, You should come to my party!

(26)

and we interpret Kathy's speech act in (26) as "[Kathy stit: Oblg:[Ben stit: Ben comes to her party]]," then, as we learned in §4C, the utterance (26) is an order. If, in contrast, the appropriate interpretation of what Kathy did with (26) is the agentive [Kathy stit: Perm:[Ben stit: Ben comes to her party]], then she has allowed Ben to come to the party. When speech acts such as (26) are properly characterized as advice, as in

94

Introduction to stit [Kathy stit: Kathy advises that [Ben stit: Ben comes to her party]]

then as Hamblin tells us, they "are accountable" (p. 13). nonwillful imperatives, like advice, are accountable as they involve the issuer and addressee in the asking for and giving of reasons. Kathy may, upon giving such advice, be asked to show how going to the party serves Ben's interests. Her advice may then be judged as good or bad, appropriate or inappropriate, sound or unsound. Of course, she might be asked to give reasons for an order, but then, as Hamblin points out, because an order is a willful imperative, she may properly reply that she has given an order or issued an invitation and that's all there is to it. (If Kathy's speech act constitutes an invitation, we need a more complex analysis. We consider that account at the end of §4G.) Such analyses are not the only ones available using the machinery of the stit sentence; they are only some examples of the possibilities the stit sentence opens up to theorizing.

4G

Embedding imperatives

We have seen that the stit sentence not only accommodates Ross's counterexample, but illuminates the conceptual territory surrounding that inference. We have just seen, in §4F, how the richness and diversity of imperatives is better accommodated in stit theory than by a logical grammar of fiats. There is yet another feature of stit that recommends it: Imperatives embed, and the stit sentence not only allows embedding, it encourages it. 2 Consider the following from Kathy to Ben I advise you to order Jessica to close the door.

(27)

Though we can extract from (27) the fiats Ben is advised to order Jessica to close the door! Jessica is ordered to close the door! The door is closed! there is, in the structure of fiats, neither grammatical nor semantic clarification of the relations among the embedded imperatives. Let us see how stit's positive encouragement of embedding contributes to our understanding of (27). As we already know, the stit sentence facilitates regimenting both the act of issuing an imperative and the content of the imperative. So (27) might be represented as [Kathy stit: Kathy advises Ben to order Jessica to close the door].

(28)

Next, by the restricted complement thesis, the complement of Kathy's advice in (28) must be agentive, represented as 2

See §1C for additional consideration of this fact. That section emphasizes that imperatives embed in exactly the same sense as declaratives and interrogatives.

4. Stit and the imperative [Ben stit: Ben orders Jessica to close the door].

95 (29)

Again invoking the imperative content thesis, we regiment the complement of Ben's order in (29) as [Jessica stit: Jessica closes the door]. Let us now have a look at the full stit canonical form for (27): [Kathy stit: Kathy advises [Ben stit: Ben orders [Jessica stit: Jessica closes the door]]]. You may think that such normal forming is too much trouble, and no doubt it often is. If, however, you want to provide a logical grammar for imperatives, then the availability of such a normal form is just what is required. The stit sentence incorporates an inferential structure appropriate to agents and choice, accommodates the wide variety of imperatives, and is fully adequate to the fact that imperatives may be embedded one within another. Notice too that normal forming can be of significant help in clarifying such commonplace phenomena as issuing invitations. As we promised, we now reconsider Kathy's utterance (26), interpreting her speech act as an invitation. Issuing an invitation is more complicated than simply giving permission. It is a notion sufficiently complex to require attention to both the varieties of imperatives and to the ways in which they embed. If an agent issues an invitation, the addressee not only receives a permission, but also is allowed to obligate the issuer of the invitation in some way. When an agent accepts an invitation, a right has been conveyed from the issuer to the addressee. Contrast that with the case where one agent has the appropriate social standing to give a permission to another, without also being able to transmit the right to obligate. For instance, Kathy might give Ben permission to attend Jessica's party, but because Kathy cannot (usually) convey the appropriate obligations, she might not be able to invite Ben to Jessica's party. When Kathy succeeds in inviting Ben to her party, she bestows a right on Ben. He has the right, upon accepting her invitation, (at least) to obligate Kathy not to deny him access to the party. Notice that this right is important because while Kathy is able, in general, to deny others access to her own party, she is not usually able to deny other agents access to Jessica's party. Some vagueness is surely unavoidable in the issuing and accepting of invitations. For example, when you invite us to dinner it may not be clear, without further discussion, whether or not you are offering us the right to obligate you to pay for the meal. Though invitations may in the rights conferred upon other agents be sometimes more vague, sometimes less so, sometimes stronger, sometimes weaker, we nevertheless make progress when we picture (26) as an invitation having the canonical form [Kathy stit: Perm: [Ben stit: Oblg: [Kathy stit: she does not deny Ben access to the party]]].

96

Introduction to stit

As we saw in §4C, when the issuer of an imperative succeeds in creating an obligation, the imperative is an order. Now we have seen that when the issuer of an imperative, in this case Kathy, simultaneously succeeds in granting a permission and obligating herself, the imperative is a permission.

4H

Conclusion

In the preceding, we have carried forward the exploration into the nature and structure of imperatives by looking at some of the ways in which stit, especially via the stit theses listed in §1, incorporates the insights of previous philosophers concerned with the same issues and builds upon them. With the help of the imperative content thesis, we have seen that the stit sentence is a notational variant of the directive, a construction that (by the agentiveness of stit thesis and the stit paraphrase thesis) is suitable to represent agentive constructions, a task the fiat is unsuited for. Because stit provides a modal logic of agency, it is able to make headway that was not possible so long as the study of imperatives remained more narrowly confined. Imperatives are thus treated as a special case, albeit an important special case, of the language and logic of agency. We have tried to show the appropriateness of the stit normal form thesis, Thesis 6, with respect to imperatives; that is, since imperatives are among the constructions whose contents must be agentive, nothing but confusion is lost if we take those contents to be expressed by all and only stit sentences.

5

Promising: Stits, claims, and strategies To promise is generally to speak; certainly all promisors are speakers, and one supposes that all fully competent speakers are promisors. The work on "speech acts" dating back to Austin's work in the 1940s and 1950s (Austin 1961, 1975) and deepened especially by Searle and Vanderveken (see, e.g., Searle and Vanderveken 1985, Vanderveken 1990, and Vanderveken 1991) has exploited the fact that promising can be located among the many things one can do with words. But promises are not always made by means of speech (they are not always "speech acts"), nor is speaking words such as "I promise" always to promise. What are we to conclude from this combination of "generally but not always" ? We suggest that while the truth of the "generally" warrants the extended consideration of promising as a "speech act" such as one finds in the work of Searle and Vanderveken, the double failure of the "always" also warrants looking for a complementary "speech-independent" account. To promise is to act. This suggests taking "action theory" as a candidate for a speech-independent basis for an account of promising. The suggestion is momentarily confirmed when we observe that the content of a promising, for example, the content of "promising to smile," also seems to be an act, evidently a different act from the promising. The matter is complex, and we may reasonably look to the "acts are events" approach to help us in sorting out the complexities in a "speech-independent" way. Such approaches generally postulate that an action is a kind of event, as in for example the quite different studies Davidson 1966 and Thomson 1977. Versions of the acts-are-events approach often depend on giving names to actions or quantifying over actions. Those reifying approaches, however, (need not but) may easily lead to avoidable questions such as the following. Is the act of promising to smile a "basic" act? If not, is the act of smiling a part of the act of promising to smile? Or is the act of smiling caused by the act of promising to smile? If the act of smiling is the content of the promising, then what was promised when there was no smile 97

98

Introduction to stit

and hence no act of smiling? And so on. Conclusion: It might not be a good idea to assume that the only way to articulate the complexity of a promise is in terms of structures appropriate to events, for example, part-whole or causal relations. All that reification of acts as events may begin by sounding scientific and hardheaded, but after one or two steps it may lead directly to perplex. An alternative idea represents some aspects of the complexity of a promise by structures provided by stit. This gives us a speech-independent approach to the idea of promising that is even more hardheaded, but that does not have to stop to worry about the perils of reification. Without the intent to minimize the potential value of an "acts as events" analysis of promising, we urge that the light of stit shine in corners not—or not yet—illuminated by reifying actions. The purpose of this chapter is to explore that idea. We begin with a kind of triple pincer attack on promising. In the first part of the chapter, §5A, we use stit theory to suggest approaches to the structure of promising. In the second part, §5B, we begin with Thomson's approach in The Realm of Rights, asking how the modal logic of agency might contribute to what is there said about promising, and, more critically, how the point of view of that book can enrich a stit-based account of promising. Then finally, in §5C, we call on the theory of strategies of chapter 13 to make good on what seem to be the common deficiencies of both of the first two sections.

5A

Prom stit to promising

The modal approach of this book features "a sees to it that Q," abbreviated "[a stit: Q]," where a is an agent and "Q" holds the place of a declarative sentence. We repeatedly argue that concentrating on this construction is a good idea. The argument is pragmatic: To do so is helpful. How? And how much? Part of the purpose of this chapter is to help answer these questions.

5A.1

Stit slogans

We have loosely defended a series of grammatical or interpretive "stit theses," as listed in §1. In this chapter we rely especially on the stit complement thesis (any declarative sentence is fair game for stitting), the stit paraphrase thesis (paraphrasability as a stit is a good criterion of agentiveness), and the restricted complement thesis (many constructions need agentives as their complements). Imperative slogans, however, are perhaps more fitting than declarative theses for carrying the stit approach to agency. We consider some on our way to promising. These are intended for application in the course of philosophical rumination—not for use when ordering croissants. Here is the first, and most fundamental. 5-1 SLOGAN. (Feature the agent) Always give pride of place to agents.

5. Promising: Stits, claims, and strategies

99

This is a grammatical slogan intended to be philosophically helpful. Natural languages make it easy to lose track of agents by means of their passives and their impersonal rewordings. Example: Shoplifting is forbidden.

(1)

On the surface this looks as if we are classifying the abstract activity of shoplifting as among those that are forbidden rather than permitted. Stit will not let you get away with such superficialities. In order even to express the agentive idea of shoplifting, you will have to put the agent term in its proper place, writing a form such as stit

shoplifts].

The grammar of stit, with its requirement that the agent term be ever present, will force you to keep in mind that which is certainly central; namely, that it is some agent that is being forbidden to do the shoplifting. Let us peel one more layer from example (1). We noted that in (1) the nefarious agency of shoplifting is hidden in the impersonal gerund. It is equally true that the casual adjective "forbidden" itself conceals agency. If shoplifting is forbidden, then there is agency in the forbidding. Stit gives an approach to this tangled topic by suggesting that we start by making the agent of the forbidding explicit: stit: Forbidden([ stit: shoplifts])]. You will naturally and rightly object that a single agent "B" is not plausible here. Two comments are appropriate. First, stit theory from the beginning has made room for joint agency, as is more than likely present in this case. See chapter 10. Second, the point of a philosophically useful grammatical form isn't to reveal all or even the most interesting features of a situation. It only needs to be helpful. And you can see that even without encoding the ins and outs of political science, the stit form makes manifest at least this, that there is more than one agent involved when taxpaying is forbidden. Stit helps.1 The second portion of advice might be put as follows. 5-2 SLOGAN. (Agentives versus non-agentives) Worry about the difference between agentives and non-agentives, and use the stit paraphrase to keep track. Consider the thesis that if you are morally responsible for something, then you could have done otherwise.2 On the stit approach, this falls into two linked parts: If you are morally responsible for something, then you did it; and

(2)

1 Wansing 1998 explores an improvement on this paraphrase, noting that it treats "forbid" in terms of a non-agentive state of affairs. He suggests instead that we build agency into the making of the deontic state, so that the underlying form always features the source as well as the target: "a1 forbids a2 to shoplift"; then we shall never have an impersonal "forbidden" as a primitive notion. See §11H for a little discussion of the idea. 2 We discuss this question in a somewhat different context at the beginning of chapter 9.

100

Introduction to stit If you did something, then you could have done otherwise.

(3)

We discuss the complexities of (3) in chapter 9. Here let us consider (2). Stit theory complains about the following ambiguity: Is the "something" in (2) agentive or not? Why is this important? Well, it is nearly a logical truth that if you are morally responsible for the fact that you saw to it that Q, then in fact you saw to it that Q. There needs to be no discussion (or not much) about (2) when "something" is taken agentively, as made explicit in the stit paraphrase. It borders on incoherence to hold someone morally responsible for seeing to something that he didn't see to. But consider the case when "something" in (2) is not agentive at all, but a non-agentive state of affairs. The form is this: If you are morally responsible for the fact that Q, then in fact you saw to it that Q. This, as we also argue in chapter 9, is far from a nearly-logical truth. Our use of italics should make the matter obvious. Consider an example in the vicinity of promising. Your friend entrusted you with some money. You chose to go to the race track. You are morally responsible for the non-agentive fact that you were at the race track, and indeed you did, in this story, see to it that you were at the race track. Further, you are morally responsible for the non-agentive fact that you are in the way of temptation to gamble with your friend's money, and in fact you did, in this story, see to it that you were in the way of temptation to gamble with your friend's money. But do not generalize from this example. Continue the story. You bet your friend's money on a very fine horse with every chance to win, and at wonderful odds. You made a good bet with your friend's money. However, against all odds, the horse lost, and so was your friend's money. Here, because you deliberately put your friend's money at risk, there is reason to say that you are morally responsible for the fact that your friend's money is gone. But that remains true even though you did not see to it that your friend's money is gone. You put the money at risk, with full agency, which is a ground of your moral fault; but its loss was due in part to chance, not entirely to your agency. Some moralists prefer to confuse this issue. They prefer to say "Well, in a way you did see to it that your friend's money is gone." They should resist this temptation. The reason is this. To speak sloppily in this way is to interfere with the enterprise of clearly explaining the connection between the facts that (i) your friend entrusted you with some money, that (ii) the money is gone, that (in) you were fully agentive in putting it at-risk-from-the-chances-of-nature while not fully being the agent of its loss, and that (iv) nevertheless you are morally responsible for the fact that the money is gone. Moralists should stay aware of what an enormous difference there is between moral responsibility for what we do and moral responsibility for complex non-agentive or partly agentive states of affairs. The stit approach helps keep us honest.

5A.2

Preliminary stit account of promising

The third slogan is a good one.

5. Promising: Stits, claims, and strategies

101

5-3 SLOGAN. (Try stit) If a form of speech takes a complement that concerns action, try thinking about its complement in terms of the stit paraphrase thesis. That may help in understanding how the complement contributes to the larger sentence in which it is embedded. We take up promising as an extended illustration of Slogan 5-3. What gets us started is that to promise is to do something; "a promised ..." is agentive. Promising is a paradigm "speech act," therefore an act. That it is an act implies that it should or might be enlightening to consider promising as a stit:

[a sitt:_] What goes in the blanks? It seems always best to begin in the safest way, by putting in the selfsame thing we started with:

[aa siit:a promised ...] That may seem like "no advance," but in fact the stit normal form does encourage us to zero in on promising as an accomplishment. What is it that was actually done when the promise was made? How did the world change due to the agency of the promisor? It is no mean beginning to notice that promisings occur at concrete moments with a "before" and an "after," and with alternative possibilities for the future. As a stit, a promise involves a choice that rules out alternative possibilities. These remarks give us a way of asking what occurs when a promise is made: How is the world different, after an occasion on which a promise is made, from what it would have been after some other choice available on that occasion? We come back to this. In shedding light on this question, stit can provide further assistance. When one promises, as we earlier recorded, one does something. In addition, what one promises itself pertains to agency. A helpful but too-strong statement of this is that promising is promising to do, which is to say that the complement of the promising is itself an agentive. This is the essence of the restricted complement thesis, Thesis 5, as it applies to promising. Here is an easy example: Paul promised to pick up Marie at the station at 9:00.

(5)

To see how stit helps (if it does) even in this easy case, let us repeat the applicable version of the restricted complement thesis, which we stated earlier in ordinary philosophical prose: Promising is promising to do.

(6)

Now let us refine that version by applying the stit paraphrase thesis, Thesis 3, which suggests that we word the "promising to do" thesis (6) as follows: 5-4 THESIS. (Stit-complement thesis for promising) We can always paraphrase the complement of "promise" as a same-agent stit.

102

Introduction to stit

Our verdict on this thesis is that it is helpful for the present, but falls short in the end. Here we are exploiting the extent to which it can help. In (5), the paraphrase suggested by Thesis 5-4 is so easy as to not yield much information: Paul promised that [Paul stit: Paul picks up Marie at 9:00].

(7)

The rephrasing forces on us that we need Paul as a double agent, once of the promising and once of the up-picking. With stit he never gets lost in some philosopher's impersonal reductionist neverland. Here is an example where stit helps a little more. Let us start with Paul promised to be at the station at 9:00.

(8)

Here it looks as if the promising is not a promising to do at all, but a promising to be. Stit allows us to put the distinction with maximum clarity. The difference between (5) and (8) is that in the latter, the complement sentence Paul is at the station at 9:00 describes an agentless "state of affairs." That makes it look as if one can promise such a state of affairs in an agent-free sense, and perhaps so. Thesis 5-4, however, bids that we not start at that place. Until we are forced by the data, as we will be, we refrain from wondering what it is to promise a state of affairs. It is more natural, for this example (we'll later come to more difficult examples), to begin with the thought that to promise to be at the station at 9:00 is as much a promise to do something as is the more agentive-sounding promise (5) to pick up Marie at 9:00. Stit makes it clear: Paul promised to see to it that he would be at the station at 9:00,

(9)

or, with the stit made even more explicit, Paul promised [Paul stit: Paul is at the station at 9:00].

(10)

That's clearly a useful paraphrase of (8), is it not? Even though being at the station at 9:00 is not at all the same as seeing to it that one is there at 9:00, nevertheless promising "to be at the station at 9:00" does pretty much come to the same thing as promising "to see to it that one is at the station at 9:00." The stit complement thesis for promising, Thesis 5-4, nicely explains why we should have one paraphrase and not the other. Furthermore, paraphrasing reduces (but does not remove) the temptation to make up a philosophy that tries to construe promising as a relation between a person and an agentless state of affairs. By concentrating on the stit paraphrase, we at least postpone the necessity of such a construal. Let us put the conclusion a little more sharply. Thesis 5-4, which we introduced as a good thing to mean by the thesis (6) that promising is always promising to do, was threatened by promises having the form "promise to be ...," as in (8). We met that threat with the claim that a "promise to be" is at least

5. Promising: Stits, claims, and strategies

103

sometimes properly paraphrased by a "promise to see to it that one is ...," as in (9), or, more explicitly, as in (10). To promise to be at the station at 9:00 just is to promise to see to it that you are at the station at 9:00. There is no absence of agency, except on the not-to-be-trusted surface. 5-5 REMARK. (Will-stits are not really agentive) This account is helpful at this dialectic point, but also deeply misleading, as we later see. The content of the promise of (9) is not really a same-time-as-promise stit, as suggested by (10); instead it must be future-tensed in order to make sense. So there is "will-stit" rather than simply "stit"; and "will-stit" is not quite agentive. It is something that you can promise, but it is not something you can do. The stit paraphrase thesis provides the relevant test. Given "Paul will see to it that Q" there is no paraphrase having the form "Paul sees to it that ...." In particular, the best candidate, "Paul sees to it that Paul will see to it that Q," must fail, since according to stit theory, Paul can never make a choice today that guarantees what choice he will make tomorrow (no choice between undivided histories, Post. 8). This way of clarifying thesis, objection, and reply, however, makes it obvious that there is a new and deeper threat to the original thesis that promising is always promising to do. Consider the following. Paul promised Marie that the train would be at the station at (11) 9:00. We intend this as an example in which the complement of the promising is not only non-agentive but entirely outside the control of the agent, Paul.3 It is inevitable, however, that our imaginations stray to ways in which the agent might after all manipulate the time of arrival of the train, which on our account would tend to lead to consideration of cases of a different kind. For this reason we postpone important but complicated cases, and substitute the following example. Paul promised Marie that the sun would shine at 9:00.

(12)

The point of the example (12) is that "what is promised" cannot plausibly be imagined to be a matter in which Paul is agentive. Nevertheless, (12) is perfectly acceptable English. It is not only grammatical, but properly conversational. It is exactly the kind of thing that one person might say to another; and yet it is not conceivable that Paul is promising to make his agency relevant to the shining of the sun. This, then, is a threat to the thesis (6) that promising is always promising to do. It is a threat that cannot be met by an easy insertion of a "see to it that" in analogy with the passage from (8) to (10), for it would be silly to promise to see to it that the sun shines at 9:00, whereas (12) is far from silly. (The conceptual problem evidently concerns the form (12) in its serious 3 It is easy to confuse the question of whether or not a proposition is or is not in the control of an agent with the question of whether or not it is agentive with respect to that agent. The ease with which such distinctions confuse us is exactly a good reason to appeal to logic for some help with agency.

104

Introduction to stit

uses; of course one might also use that form in speaking to a child as a kind of pretense.) The reader who continues on to §5B will recognize that we have in (12) a good case of word-giving in the sense of Thomson 1990. Our aim here, however, is to do as much as we can with the stit apparatus, even though it may temporarily lead us astray in ways that will need to be corrected. And certainly we are not yet at the end of our resources for defending the helpfulness of the thesis that a promising is always a promising to do, that is, a promising to see to it that something. Let us first try insisting that yes, promising that the sun will shine is promising to see to it that something. What could that something be, given that it obviously isn't that the sun will shine? We do not think that stit gives an automatic answer to this question. Nevertheless, consider a concrete situation featuring the promise reported in (12). Imagine some circumstances. What was the point of the promise? Even without supposing that there is anything like a uniform answer to this question, the generalized shape of the story is not so difficult to make out. Say that Marie had been wondering whether or not to take her umbrella. Paul in promising that the sun will shine in effect says "don't bother." That, however, is not precise. There are many ways to say "don't bother" that are not promises. What is special about the promise? Well, as we might say, and as Thomson explicitly makes clear, the promising entitles Marie to rely on Paul's declaration that the sun will shine. We are nearly in a position to break out of the circle. The stit approach suggests looking for something that Paul promises to do when Paul entitles Marie to rely on his word that the sun will shine. Put this way, it is natural to look at the situation when the sun fails to shine, as Paul promised. So suppose that 9:00 comes, and there is Marie huddled in the pouring rain without her umbrella. What then? The first thing to observe is that there is no hint of Paul's "breaking" a promise in this situation as so far described. Presumably "breaking" involves agency, whereas in our story the sun failed to shine without any help from Paul. And yet we cannot let Paul entirely off the hook. To make the example easy to follow, suppose Paul himself had a umbrella, but refused to share it with Marie in the downpour. Suppose he said that all along that was his plan; in promising Marie that the sun would shine he had no idea of helping her out if his declaration turned out false. Then, even though promise-breaking is not to be attributed to Paul, we ought to say that he made a lying promise. We ought to say that the act carried out in uttering a sentence (12) is constituted in part by a conditional promise to do, namely, a promise that Paul see to it that Marie is helped or that her troubles are alleviated or that she receives restitution or recompense for damages (or something) in the case in which the sun does not shine. At the very least, Paul is promising to see to it that he is graceful in taking blame if things go wrong. It is not only that he, as patient, receives demerits if the sun doesn't shine, but that he, as agent, sees to it that he accepts those demerits, even though, as seems important, he might have done otherwise. Clarification: In labeling Paul's act a "conditional promise to do," it is obvious that we intend the condition to apply to the "doing," not to the "promising."

5. Promising: Stits, claims, and strategies

105

Warning: A conditional stit is not always a stit (see chapter 11). Further warning: The conditional stit is just about bound to be in the scope of a future tense, and so subject to Remark 5-5 on p. 103. For both of these reasons, and inspired by Thomson 1990, this account of (12), while imbued with some of the spirit of Thesis 5-4, tries not to confuse the spirit with the letter. More of this later. One last example, with a complement of a sort that lies between the agentive complement, "that Paul picks up Marie at 9:00," and the agentless complement, "Paul is at the station at 9:00." Suppose Paul promised to see to it that Marie buys a train ticket.

(13)

This of course makes eminently good sense. According to stit theory, however, the complement should not be rendered by "[Paul stit: [Marie stit: Marie buys a train ticket]]," since one person cannot guarantee the choice of another. In chapter 10 we discuss various ways of understanding such examples, pointing out the many differences among them. According to one of these ways, (13) is usefully paraphrased using some idea of "probability": Paul promised to see to it that it is highly probable that Marie buys a train ticket,

(14)

for the probability is something that Paul can, according to stit theory, reasonably see to. If, however, we think about many choices by Paul, as we did in considering (12), it is natural to point out that there are and will be available to Paul many choice points at which he can render it more or less likely that Marie buys a train ticket. A minimal such notion is this: Paul never makes a choice that excludes the possibility of Marie buying a ticket, if there is available to him another choice keeping that possibility open. If we think about the content in this way, we make at least some objective sense out of saying that Paul's agency is relevant to Marie's buying a ticket. We can do this even though it is (conceptually) impossible for Paul to guarantee that result. So this case lies properly between the cases (5), in which Paul can, at least in approximation, see to what he has promised, and (12), in which Paul's agency is irrelevant. Let us summarize what is easy to see about promising when viewed from the point of view of stit. • The paradigm case of promising, we say, is explicit promising to do, such as when Paul promised to pick up Marie at the station. In this case the complement of the verb "promise" appears to be already agentive, and can therefore apparently be well paraphrased as a stit. Later we deal with the fact that the stit will be embedded in a future tense. • There are also important cases of promising expressed with superficially non-agentive complements, for example promising to be somewhere. In these cases, too, the promise is a promise to do, the sense of which is aptly given by inserting a stit. That is, given "promise that Q," one finds a useful paraphrase in "promise to see to it that Q." Observe that it is not

106

Introduction to stit Q itself that is being paraphrased; the non-agentive "to be at the station" is miles away from the agentive "to see to it that one is at the station." Rather, we paraphrase the entire promising construction, changing "Paul promised to be at the station" to "Paul promised to see to it that he would be at the station." In semi-symbols, we paraphrase "Paul promised that (Paul is at the station)" as "Paul promised that [Paul stit: Paul is at the station]." It is a benefit of the imperfect Thesis 5-4 that it leads us to see things in this way.

• There are cases in which the complement of the verb "to promise" articulates something to which the agency of the promisor is relevant but not in the simplest stit-like way. In a word, these cases involve consideration of more than a single choice by the agent. Such cases are perhaps the most frequent in occurrence, and certainly they are far and away the most challenging to a theory of choices in our indeterministic world. One minimal subcase: The content of the promise is to keep open the possibility of the complement. And this is something that the promisor can do. • There are cases in which the complement of the verb describes an entirely non-agentive situation that is obviously beyond the control of the promisor. Even here, it seems that it is of the essence of the promise that the promisor promises to do something. Certainly the promise is not that the promisor see to it that the complement be true; in this case such a thing is impossible. We suggested instead that the act of promising standardly initiates a conditional promise to act appropriately (protect against loss, make restitution, etc.) in case the promised complement turns out to be false. Therefore agency is crucially involved in the content of such a promise. We cannot, however, claim accordance with Thesis 5-4, since, as we have said, we cannot identify future-oriented conditional stits with plain stits. Let us return to the original paradigm case of promising, namely, promising to do. Can the stit approach help any further? Have we gone as far as we can go when we paraphrase (5) as (7)? In fact we can do more, not yet so much with the complement, but with the speech act of promising itself. The clue is that a speech act is or should be an act, and so should be paraphrasable as a stit. To promise is to see to something. Of course there is the easy paraphrase: To promise is to see to it that you promise. But once one starts thinking in terms of stit, it is natural to try to find other more-or-less equivalent complements of the stit that may be more informative. The history of philosophical analysis tells us that none will be absolutely equivalent, but that historical lesson should not prevent us from trying to find helpful alternatives. The way stit helps is by encouraging us to ask: Just what is it that Paul can do when he promises to pick up Marie at 9:00? On the stit account each exercise of agency is a transition, a transition that changes the world. Stit thus encourages asking, How is the world different after Paul has promised to pick up Marie at 9:00? Deeply embedded in the stit theory of agency is the truth that

5. Promising: Stits, claims, and strategies

107

one cannot choose today what can only be chosen tomorrow (no choice between undivided histories). For this reason, it is certain that the promise does not guarantee that on the morrow Paul chooses to pick up Marie. So what can Paul guarantee that wasn't guaranteed before his promise? One answer is this: He guarantees that he is obligated to pick up Marie. He was not so obligated before the promise; he is obligated after the promise; and it is the promising that is responsible for this change in the world. It therefore seems helpful to articulate (7) as follows: [Paul stit: 06/0: [Paul stit: Paul picks up Marie at 9:00]].

(15)

In short, Paul's promise is, in approximation, the act of laying an obligation on himself. On this account, that it is done with words is irrelevant. Perhaps promises are typically "speech" acts, but this account points us away from the speech and toward the act, namely, the act of creating an obligation where none existed before. (For much more on mixing deontic logic with stit, see chapters 11, 12, and 14.) The formulation of (15) is natural given only the stit theory as so far elaborated. It is, however, deficient. One of the several things that emerge from the second part of this chapter, to which we soon turn, is that not only Paul but also Marie is agentive in the creation of the obligation on Paul ("uptake"). Also it becomes clear that "obligation," while giving the right (deontic) logic, is not the right word.

5A.3

A final stit slogan

We have gone a long way without mentioning a key theme of the modal logic of agency: 5-6 SLOGAN. (Exploit nesting) Exploit the recursive power of modalities; that is, look for ways in which to clarify by embedding one stit in the scope of another. We recommend this slogan here because it is as important as any in the stit program. We defer its illustration, however, until §5B.3, where we offer an impressive triple stit.

5B

From RR to promising

Thomson 1990 (The Realm of Rights, henceforth "RR") shares with stit theory a reliance on a modal agentive construct that finds its source in Hohfeld 1919. This makes for enough commonalities to have allowed us already, in §5A, to have relied on some of the key ideas of RR, and we shall suggest that conversely the use of some central ideas of stit theory can help to refine the RR account of promising. (When we quote from RR, we make a few small changes in symbolism.)

108

5B.1

Introduction to stit

Claims and duties in RR

RR adopts or adapts Hohfeld with the help of the following modal locutions (our italics): Cx,yP, read: X has a claim against Y that P (p. 41).

(16)

DY,xP, read: Y is under a duty toward X, namely the duty that Y discharges if and only if P (p. 41).

^

Px,vP, read: X has as regards Y a privilege of letting it be the case that P (p. 44).

t18)

>

In doing so RR endorses the following Hohfeld equivalences: Cx,YPittDY,xP. Px,yP iff Not-(Dx,x,YNot-P).

In this representation of claims there are two places, marked by X and Y, for singular terms and one, marked by P, for a sentence. Thus, RR carries the ideas of claims, duties, and privileges with the help of modal operators. The example RR gives for the usefulness of the modal form is that this form makes it easy to express the generalization that claims are closed under entailment. The stit approach helps by suggesting that this generalization may not be straightforward when one mixes agentive and non-agentive complements. The stit approach, even without the introduction of stit notation, makes it natural to question entailment-closure. In analogy with a favorite stit example (see §2B.l), one wonders whether that X has a claim against Y that Y bandage at least one injured person does in fact imply that X has a claim against Y that there is at least one injured person. RR explicitly states the following principle. 5-7 THESIS. (RR's unrestricted complement thesis for claims, duties, and privileges) Any sentence whatsoever may be put in for "P" (see p. 41). This thesis is to be compared with the unrestricted stit complement thesis, Thesis 2, and with Thesis 5-4 for promising. We especially note that even though the two announce different targets, Thesis 5-4 and RR's Thesis 5-7 appear to conflict in spirit, and we will say something about that shortly. We shall see that for promising RR seems to adopt, or nearly to adopt, not indeed our Thesis 5-4, but some restricted complement thesis. We shall wind up endorsing this line, and thereby suggesting, for promising, an account somewhere between our Thesis 5-4 and RR's Thesis 5-7 (see §5C). Our immediate thrust, however, will be to take the foundations of RR as they are, its unrestricted complement thesis, Thesis 5-7, and all, and to follow up what RR makes of promising on that basis. It nevertheless seems best to indicate at this point how the stit approach can help make us wonder about Thesis 5-7, and especially about the quite different ways that agency figures in the three English readings (16), (17), and (18).

5. Promising: Stits, claims, and strategies

109

• Claims. The English reading in (16) adds no agentive words or ideas beyond any that analysis may find in the idea of claiming itself. Keep this in mind for contrast. • Duties. The English in (17) uses a definite description, "the duty that Y discharges if and only if P." This use seems to require that for each Y and P there is exactly one duty that Y discharges if and only if P. If this is right, then after all a kind of restriction of complement is being imposed, though certainly not necessarily a restriction to stit complements. One nevertheless observes the presence in (17) of an apparently agentive or quasi-agentive expression, "discharges." • Privileges. In contrast to (17), the English reading provided in (18) presupposes nothing whatsoever about P. Instead, it provides an explicit restriction on the complements of "has a privilege." Not just any complement will do; we are given only a complement of the form "X lets it be the case that P," where P itself is unrestricted. Now "letting" reads as a kind of agentive or quasi-agentive idea, so this is a kind of restricted complement, not indeed a restriction on PX,Y , but only on complements for the English word "privilege." For instance, the form does not allow in any immediate way that X has as regards Y a privilege of eating X's salad, since eating salad is—at least on the surface—not the same as a letting it be that — Sure, that's confusing. That's the point: There is work to be done, and it might not be easy. The considered choice by RR to use the locutions "discharge" and "let" in connection with duties and privileges imply that a simpleminded insertion of stit is not going to suffice. One may nevertheless hope to be guided by the suggestion that arises in applying stit theory to deontic constructions, namely, that being permitted to see to it that P is equivalent to not being obligated to refrain from seeing to it that P (see §2B.9):

Note the delicate placement of the internal negation: neither on the outer stit nor on P. One can imagine that something like the equivalence (19), which unquestionably relies on the restricted complement thesis for deontic constructions, might connect privileges and duties. Certainly this seems to work in the simplest case, the case where the complement of both claim and privilege is restricted to stit. That is, the following seems illuminating. X has as regards Y a privilege of seeing to it that P <-> Not-( Y has a claim against X that X see to it that Not-(A' see to it that (20) P). Since stit theory says that "refraining from doing" is "seeing to not seeing to," stit theory allows the equivalence (20) to be dramatically shortened for an agentive P, as in the following Hohfeldesque example.

110

Introduction to stit X has as regards Y a privilege of crossing Y's land <-> Not-(Y has a claim that X refrain from crossing Y's land).

We want to make it plain, however, that although one may reasonably hope that the simple case will provide guidance, and that stit theory will clearly explain to you how you are cheating yourself if you interchange "refraining from crossing" with "not crossing," it would be optimistic to hope for a quick solution to more difficult cases without extensive work.

5B.2

Word-giving in RR

Leaving aside foundational questions as to the relation between agentive or quasi-agentive concepts and those of Hohfeld, we go on to promising. That RR explicitly prefers unrestricted complements helps to explain the conceptual order in RR according to which promising is taken as a species of word-giving, which itself is taken to be a special case of assertion. Here is a sequence of examples that accords with RR. Paul asserts to Marie that he will be at the station at 9:00. Paul gives his word to Marie that he will be at the station at 9:00. Paul promises Marie that he will be at the station at 9:00. This sequence minimizes—and we think is intended to minimize—the need for a restricted complement thesis for promising. In this respect the entire movement is opposite to the spirit of stit theory, in the context of which one would be happy to construe word-giving as a kind of promising (as in our treatment of promising that the sun will shine at 9:00), and asserting as a kind of word-giving.4 But that is not here to the point. Among these three, RR takes assertion—and especially assertion to someone—as a primitive among the array of concepts it develops. It is consonant with the thrust of RR that assertion should be taken as seeing to the existence of some kind of normative state, as we also explicitly suggest in §6E. It is, however, a feature of the RR analysis that definiteness about this seems not required. In any event, the other two members of the triple are given sharp characterizations. Postponing for a while any consideration of promising per se, we begin with two theses of RR, the most important of those that govern word-giving (our italics). The Assertion Thesis: Y gives X his or her word that a proposition is true if and only if Y asserts that proposition to X, and (i) in so doing, Y is inviting X to rely on its truth, and (ii) X receives and accepts the invitation (there is uptake). (p. 298) The Word-Giving Thesis: If Y gives X his or her word that a certain proposition is true, then X thereby acquires a claim against Y to its being true. (p. 302) 4

There is an elementary treatment of assertion in §6E, with emphasis on the special problem of an assertion whose complement is in the future tense.

5. Promising: Stits, claims, and strategies

111

This account of word-giving relies not only on the concept of assertion-tosomeone, but also on invitation and acceptance, and on acquiring a claim. The RR account of claims is rich, and we leave it alone. In other passages RR summarizes the account of word-giving as follows: For Y to give X his or her word that a certain proposition is true is for Y to invite X to rely on its truth, which invitation X accepts; and if you issue an invitation to a person to rely on the truth of a proposition, which invitation the person accepts, then surely the person thereby acquires a claim against you to its truth, (p. 302) The arguments for these two theses are sharp, interesting, and persuasive, but here our focus will be more on the form and content of the theses themselves. First, what is it for X to rely on the truth of a proposition? RR does not give us a separable account of this, nor is it obvious that such would be appropriate. The problem about inviting might be raised in this way. If Paul invites Mary to come to a party, then he is inviting her to do something, namely, come to the party, that she could do without an invitation. If in contrast Paul invites Mary to accept his offer of a contract, then he is inviting her to do something that she could not normally do without the invitation. In the former case, we can easily have an account of coming to the party that is separable from the invitation; in the latter case, it is not obvious that we can easily have an account of accepting an offered contract that is conceptually separate from our account of the offer. It looks as if inviting reliance is more easily understood in the second way, not the first. And RR gives the following account of what I do when I invite-yourreliance, not indeed quite generally on the truth of a proposition, but anyhow a reliance on "an expectation about my future behavior": I thereby take on myself a complex responsibility, namely to make that expectation true, or, if I will for some reason be unable to, to take reasonable steps to see that you do not lose by virtue of having accepted my invitation to rely. (p. 95)

(21)

From the stit point of view this passage would give insight into the "doings" that are involved in an invitation to rely. RR, however, emphasizes a different aspect of the situation. When a person accepts an invitation to rely on the truth of a proposition, "then surely the person thereby acquires a claim against you to its truth" (p. 302). This is the connection between the Assertion Thesis and the Word-Giving Thesis. RR puts the matter as follows: In sum, a word-giver B alters the world in such a way that an act of the word-receiver A (uptake) makes A have a claim against B to the truth of the word-given proposition. In other words, the wordgiver B gives the word-receiver A a power by the exercise of which A makes himself have a claim against B. (p. 320) This summary abstracts entirely from the communicative situation (invitations given and received), and puts the matter in a language that is apt for stit.

112

Introduction to stit

We are led to see that on the RR account, at the end of the process what X does when X accepts the invitation to rely is simply this: X sees to it that he or she has a claim against Y that P. Let's write this down: [X stit: X has a claim against Y that P]. That expresses what X "does" in accepting the invitation: X binds Y. So what does Y do? On the RR account, Y's invitation amounts to giving X the power to create the claim. But we do not need to invoke the metaphors "give" and "create." Both are at bottom stits. Working from the inside out, the power is nothing but a can-do: Can:[X stit: X has a claim against Y that P}. That expresses exactly the power that X has when X is in a position to accept the invitation.5 Furthermore, "giving" X the power is also nothing but a stit: [Y stit: Can:(X stit: X has a claim against Y that P]].

(22)

That is what Y does when Y invites X to rely on his or her word: Y sees to it that X can see to it that X has a claim against Y that P. Having gone so far with RR, one can see the likelihood that the time-sequence, represented in (22) by the nesting, has been overspecified. It seems not to matter that Y goes first and X second, that is to say, that Y empowers X. It seems equally all right if X empowers Y: [X stit: Can:[Y stit: X has a claim against Y that P]}.

(23)

Why not? If for instance X asks a solemn question (p. 295 of RR gives an example involving keys on a bureau), there seems no call to ask for an additional act on X's part, after Y's answer, in order to construe Y as having been a wordgiver in the full sense, including uptake. Nor does it seem wrong to think of a situation in which neither act precedes the other, so that we should have a stit that is founded in the simultaneous choices of two agents, a "strict stit" in the sense of chapter 10: [{X, Y} sstit: X has a claim against Y that P}.

(24)

What these three, (22), (23), and (24), have in common is that the complement "X has a claim against Y that P" has come true due to the prior choices of X and Y, both being essential. This important agentive idea, although not attached to any of the stit locutions so far treated, seems to be a common generalization of [a stit: Q] to [F stit: Q], where F is a set of agents, and of [a stit: Q] as witnessed by a moment to [a stit: Q] as witnessed by a chain. Let us describe it by means of an alternative semantic clause for strict stit that builds in both witness by chains and the essentiality of each of a group of agents. (By 5

"A power is an ability to cause, by an act of one's own, an alteration in a person's rights" (RR, p. 57). We have separated the "power" into its two natural components: A can-stit, which represents "an ability to cause," and an expression of a claim, which is one sort of right. The ability to do so is an advantage of stit.

5. Promising: Stits, claims, and strategics

113

"alternative" we mean that what follows is an alternative to the "principal" concept of strict joint agency introduced in Definition 10-4 as a definition of [F sstit: Q].) 5-8 DEFINITION. (Alternative account of strict joint stit) [F sstit: Q] is true at a moment-history pair m/h (on this new account) iff there is a chain c of moments that is (i) nonempty and entirely prior to m, and such that (ii) through each moment m1 in c there is a history h1 such that Q is false at the moment coinstantial to m lying on h1; (m) unless a history h2 intersecting c be separated from h by a definite joint choice of the agents in F at some moment in c, then Q is settled true at the moment co-instantial to m lying on h2; and (iv) clause (iii) fails for every proper subset of F: For every agent a in F, there is a history h2 intersecting c that is not separated from h by any joint choice of the agents in F— {a} such that Q is not settled true at the moment co-instantial to m lying on h2. A paradigm instance is the creation of a contract between Paul and Marie by their joint agency, but without any specification of who offers and who accepts. What is required is an antecedent chain of choices separately or jointly by Paul and Marie that witness the creation of the contract—the negative condition (ii) ensuring that the completion of the chain is needed to ensure the contract, and the condition (iv) ensuring the essentiality of each of Paul and Marie. Observe the following delicacy: Clause (iv) refers to the very same chain as is introduced at the beginning of the definition. Thus we leave open the possibility that some other chain may witness the agency of a proper subset of F. It seems to us that although it is perfectly true that word-givings do not require the time sequence imposed by RR, still, nothing of relevance to the themes of RR hangs on this, and indeed the analyses of RR are easier to follow if we simply assume that the time order is as RR presumes.

5B.3

Promising in RR

We move from word-giving to promising. RR almost seems to make a point of not taking the complement of "promise" as an agentive. We think, however, that the details of RR can be improved by taking seriously Thesis 5-4 for promising, even while rejecting it. What, on the RR account, is the difference between a mere word-giving and a promise? We quote four additional requirements from scattered locations on pp. 299-301. 5-9 THESIS. (When a word-giving is a promising) • First, a word-giving is a promising only if the proposition asserted is in the future tense. • Second, a word-giving is a promising only if the proposition asserted has the asserter as its subject.

114

Introduction to stit

• Third, a word-giving is a promising only if the proposition ascribes an act ("B will go to A's office at 4:00"), or a refraining from acting ("B will not go to A's office at 4:00"), or a limited range of states ("5 will be in A's office at 4:00"). • [F]ourth ..., unless the word-receiver cares whether the proposition asserted is true, and indeed wants it to be true, then even if there is uptake, so that a word-giving does take place, what takes place is not an instance of promising. We have two suggestions for modifying the details of the RR account of promising. First, we suggest dropping the fourth condition about caring as beside the point. There is meticulous argument in RR that caring should not be placed as a requirement on word-givings, but there seems to be no parallel argument in RR explaining why this condition should be laid on promises. As far as we can see, though the matter is far from obvious, the same reasons that would lead one to withhold the condition in the one case are equally operative in the other. Certainly the caring condition seems tacked on, and our analytic life seems simpler if we drop it. As RR says in another context, we do not need to build the caring condition into the concept of promising in order to keep it in mind. Our discussion gives it no further consideration. Our second suggestion is to fold the first two conditions into the third, something like Third'. A word-giving is a promising only if the proposition ascribes to B a future act ("B will go to A's office at 4:00")

(25)

We intend this as but the slightest of variants, adding only that the proposition in question ascribes a future act to B—regardless of the syntactic shape of the sentence (if any) that expresses the proposition. There are two connected reasons favoring the suggested changes. (i) The "first" and "second" conditions as formulated in RR pertain to sentences (words), not propositions. It is sentences, not propositions, that are or are not in the future tense (first condition), and it is sentences, not propositions, that have subjects (second condition). The policy of RR, however, is that word-givings concern propositions, not sentences (p. 295). This policy should not be given up when it comes to promises. As everyone knows, the same proposition can often be expressed either in the future tense or otherwise, and can be expressed with one subject or with another. Such differences of modes of expression should not make a difference between word-giving and promising. For example, for many sentences P one finds that the logician's future-tense sentence "It will be the case after an interval of a day that it was the case prior to an interval of a day that P" is altogether equivalent to P. And for many sentences P, one finds that nearly any contained singular term is the subject of an equivalent sentence. But if "Paul promised Marie that she would become taller than Paul" is more like a word-giving than a promise, then so is the equivalent "Paul promised Marie that he would become shorter than Marie." And if "Paul

5. Promising: Stits, claims, and strategies

115

promised Marie that he is a bachelor unless he is married" is more word-giving than promising, then so is "Paul promised Marie that he will be a bachelor unless he marries." (ii) For a proposition to ascribe to B a future act, it does not suffice that the proposition be expressed by a sentence that has B as its subject and is in the future tense and ascribes an act to B. We therefore do not think that the first and second conditions of RR accurately express its intent. We think that a good way of expressing that intent in linguistic terms is instead as follows. A word-giving is a promising only if, where B is the promisor, the proposition can be expressed by a sentence

There you have your future tense, your promisor as subject, and your ascription of agency (the stit). The only difference is this: In (26) the three pieces are taken together, not separately. Our replacement is obvious, and worth trying. This replacement is a significant modification of the spirit of RR's unrestricted complement thesis, Thesis 5-7, for it insists that the complement of "promise" must be a future-tensed agentive with the same agent as the promisor. That may sound like the "same subject" condition of RR, but it is not. From the beginning stit theorists have supposed that the agent is a distinguishable or recoverable "part" of the "content" of an agentive. This is no mere verbal matter. See especially §1C and §3A. The replacement, however, is also a significant modification of a restricted complement thesis, Thesis 5, in the sense of Thesis 5-4. In order to appreciate the exact form of (26), one probably needs to read the stit as a "deliberative stit" rather than an "achievement stit," though neither of these constructions is entirely apt except for simple cases. The present point is this: The choices that are available to the agent, and not just the outcome of those choices, are to be envisaged as in the future of the promising. It is easier to get straight on this with the deliberative stit. It is a separate point that Q itself is likely to be well represented by a sentence in the future tense, so that the outcome represented by Q follows after the choice that witnesses the stit. The stit part of the sentence (26) expresses agency. But Thesis 5-9 (its third part) gives three disjuncts. There is in the reach of that third condition not only a promise (i) to do, but also a promise (ii) to refrain; and there is a promise (in) to be. Of these only (i) seems to be covered by (26). What shall we do about the other two disjuncts of the third condition given by Thesis 5-9 as to when a word-giving is a promise? Luckily, however, stit already covers both of these extra cases—and in a unified way. With regard to ( i i ) , a promise to refrain, it is an explicit thesis of stit theory that refraining is expressible by a stit, and is therefore very different from mere not-doing. For example, that X not trespass can be true of nearly everything, whereas only sometimes is X in a position to refrain from trespassing, that is, in a position to see to it that X does not trespass. For the latter, but not for the former, X must certainly be an

116

Introduction to stit

agent in the matter. With regard to ( i i i ) a promise to be, for example a promise by B to be at A's office at 4:00, we think that here above all the distinction that RR intends between (mere) word-giving and promising comes to the fore. If it is merely a matter of where B is. then we have word-giving. But it becomes a promise if it is a matter of B exercising some agency in getting to the office, that is, if it is a matter of B's seeing to it that B is at A's office at 4:00. The central point is this: To the extent that there is a difference between (mere) word-giving and promising, the complement needs to be or at least involve an agentive. A mere agentless state-description will not do. In short, we suggest that a simple promise, when represented in the same style as before, would involve a claim of X against Y that Y will see to it that P. In full dress: Y promises X to see to it that P against Y that will:[Y stit: p]]]

[Y stit: Can:[X stit: X has a claim

Hey, this is a three-stit extravaganza! Its very complexity indicates, in accord with Slogan 5-6, the prospective utility of careful theory in assisting us to find our way. On the other hand, we see by following RR that the simple version of the stit complement thesis, Thesis 5-4, which looks to be exactly right for the English version, must give way at least to the future tense when all is spelled out; and a future-tensed agentive is not, definitely not, itself an agentive. To suggest that it was would be to fly in the face of the principle of no choice between undivided histories, Post. 8.

5C

Strategic content of promises and word-givings

We, following RR, have advanced the future-tensed same-agent stit form as appropriate to the content of a promise in simple paradigmatic cases. This amounts to a slight variation on the restricted complement thesis, Thesis 5, as applied to promises. Because even these cases are complex, we treated no others, even though we noted that simple cases, although revelatory, are perhaps not all that frequent. The dividing line is crisp: In the most complex cases one must attend to many choices, whereas in the simplest cases a single stit-warranting choice suffices. To build a theory of multiple choices and their outcomes is inevitably more difficult than constructing a theory of single choices. How to go on? Our conjecture for a better approximation—one that does not forfeit gains already made—would be that the content of more complex promises can be usefully expressed in terms of the "austere theory of strategies" developed in chapter 13. Briefly, a strategy for a considers a "field" of moments, and is defined for some or all of the moments in its field. The moments at which it is defined are said to be in its "domain," and for every moment in its domain, a strategy for a picks out from among the choices available to a at that moment one or more choices that count as in accord with the strategy. As

5. Promising: Stits, claims, and strategies

117

a verbal convenience, we say that the strategy "requires" that the agent choose in accord with the strategy, even though the austere idea of a strategy does not depend on deontic ideas. That, austerely, is it. It is basic to the theory of stit and of strategies that you cannot stit the content of a strategy. This is an elementary consequence of the principle of no choice between undivided histories. It therefore seems that we have a sound theoretical basis for modifying the stit complement thesis, Thesis 2, according to which the complement of stit must be a declarative sentence. We have not, however, laid sufficient groundwork for such a modification. The stit complement thesis is at least in part linguistic, and although in chapter 13 we suggest an abstract theory of strategies, we have not suggested any linguistic devices with which to carry the content of a strategy. Perhaps none are necessary; perhaps it suffices just to consider, at least for a first approximation, sentences, Q, for which there is a strategy such that Q is true at the moment of promising paired with each member of the smallest set of histories that the strategy really guarantees (Definition 13-5). Some such suggestion would appear to cover, for example, future-tensed stit sentences, and also combinations of conditional stits such as "if the sun doesn't shine, I will apologize, and furthermore if it rains, I will lend you my umbrella." It is not, however, sufficient to consider just this first approximation. The difficulty is that the approximation will only work for the simplest cases such as Paul's promise to pick up Marie, (7). It will not work to cover our understanding of Paul's promising something that Paul can influence but cannot guarantee, for example, (13), nor of something over which he has no control, for example, (12). We offer a better approximation, in §5C.2, that is intended to cover these cases uniformly.

5C.1

Promising versus word-giving I: Example

As a first run at the problem, we work through an example that suggests that strategic ideas can help us to judge if there is a difference between word-giving and promising that you do something. Consider the following. (Giving your word that you will do something) Paul gave Marie his word that he will see to it that he is at the station at 9:00.

(27)

(Promising that you will do something) Paul promised Marie that he will see to it that he is at the station at 9:00.

(28)

On the RR analysis, as we have represented it, there is no difference whatsoever, since a promising is analyzed as a word-giving with a future-tensed agentive complement. So long as the complement is a future-tensed same-agent stit, there is, on our version of the RR account, no difference between word-giving and promising. And maybe that is the most useful account. It is just common sense to observe that in many cases we use "I promise you that" and "I give you my word that" interchangeably, with no felt difference of meaning; and it is certainly a theme of stit theory that English grammar is no sure guide. One

118

Introduction to stit

wonders, however, if one can use the theory of strategies to draw a distinction in an interesting and helpful way. Such a theory might permit one to refine the "complex responsibility" of (21) in two different ways, ways that distinguish (27) from (28). Speech act theorists sometimes distinguish the two by saying that assertion or word-giving embodies a fit of words to the world, whereas a promise embodies a fit of the world to the words. (The idea comes perhaps from Anscombe 1957, p. 4.) We shall show that the theory of strategies can help give definite meaning to the distinction drawn by these metaphors. The theory of strategies would allow the following, which we present in rough form. For (27), the word-giving, the content might be represented by a strategy that requires no choices whatsoever that are relevant to Paul's seeing to it that he is at the station at 9:00, but that (29) prescribes that Paul compensate Marie if he does not see to it that he is at the station at 9:00 (see Definition 5-15). This is a "words-to-world-fitting" strategy. Although it does call for some choices involving compensation when the words do not fit the world, the central point is that the strategy does not call on Paul to make choices favoring (objective) possibilities fitting his words. In other words, Paul can follow this strategy without doing or choosing anything that favors his seeing to it that he is at the station at 9:00. We propose this as an explication of the metaphorical phrase, "fit the words to the world." Note also that the word-giving strategy is "primary" rather than "secondary" in the sense of Definition 13-7. That is, it says nothing whatever if Paul makes an excluded choice, for example, if Paul chooses not to compensate Marie. For (28), the promising, the content might be represented by a strategy that (i) prescribes that, if possible, Paul choose in a way that guarantees that he see to it that he is at the station at 9:00, (ii) prescribes that, if possible, Paul keep open the possibility of Paul's later guaranteeing that he sees to it that he is at the station at 9:00, and (iii) prescribes that Paul compensate Marie if in fact he "infringes his promise" by arriving at a moment at which it is no longer possible for him to see to it that he is at the station at 9:00 (see Definition 5-13).

(30)

This is a "world-to-words-fitting" strategy. It calls for choices relevant to getting Paul to the station at 9:00. We propose this as an explication of the metaphorical phrase, "fit the world to the words." Observe that this strategy is, crucially, "secondary" rather than "primary," since it imposes requirements on Paul (that he compensate Marie) when he violates the strategy by infringing his promise. As secondary, the strategy contains a "contrary to duty" (so to speak) component, namely, the compensatory sub-strategy. There is evidently a large difference between the strategies described in (29) and (30), which (inconclusively) suggests that the theory of strategies can help

5. Promising: Stits, claims, and strategies

119

clarify the view of those of us who feel that there is an important difference between (27) and (28). It offers hope that there is a useful distinction to be made between word-giving that an action will be performed and promising to carry out that action. It provides a way to try to make rigorous and objective sense out of the metaphorical difference between "fit of words to world" and "fit of world to words."

5C.2

Promising versus word-giving II: Strategic accounts

We put the matter more rigorously, invoking in a preliminary way some concepts not fully developed until chapter 7 and chapter 13. Needless to say, increase in rigor will cost increase in complexity of detail. Consider a promising that P. We already know that not only P, but also the promisor and promisee, play a conceptually essential role. The easiest way to keep this firmly in mind is to consider not just "a promising that P" but something like "X promises Y that P." Even this, however, although enough for English, is not enough for analytic clarity. We must also keep in mind that a promising is a concrete act that occurs at a specific moment in our world. We must keep in mind that there is always a "moment of promising," for we shall find that it plays an important role in the analysis of the strategic content of a promise. For this reason, "X promises Y that P" does not suffice; we make the moment of promising explicit in our analytic target by considering the following instead. (Making the moment of promising an explicit argument is strictly an expository convenience. One could as well take "X promises Y that P" by itself, keeping track of the moment of promising as a (mobile) semantic parameter in the sense of §6B.3.) 5-10 TARGET. (Analytical target) At mo, X promises Y that P. The following is designed to serve as a helpful if greatly oversimplified uniform account of the content of Target 5-10 in terms of a strategy—with the full understanding that the particularities of the account involve a number of decisions on our part. Our aim is to give one more or less helpful account, not to survey a variety of possibilities. The account will be given entirely in the austere language of choices amid branching histories; in particular, there will be no reliance on "beliefs" or "wants" or any other mind-dependent concepts. As elsewhere, the point is not at all to suggest that our objective account suffices. Helpful, yes. Enough, no. This is also the place to observe that we do not in this section aspire to an understanding of the act of promising represented by Target 5-10. Our more modest target is a structural description of the strategic content of that promising. It is as if we undertook to analyze the propositional content of an assertion without aiming to explain the act of assertion. It will be plain that our account has its roots in the "agent-action reduction" program of Hamblin 1987.

120

Introduction to stit

5C.2.1

Double time references

Our account acknowledges that a satisfactory account of the strategic content of Target 5-10 will need to involve "double time references." Let us observe that although we speak here only of promising, the double-time-reference ideas should play a role in understanding the content of a wide variety of "speech acts." Perhaps they even play a role in all, to the extent that (objective) indeterminism enters the picture via either choice or chance.6 So what about "double time references" with regard to Target 5-10? A natural guess is that one of the two references will be to the moment of promising, mo; and that is right. It is perhaps equally natural to guess that the other reference will be to a time at which P is evaluated. It turns out, however, that things are not so simple. The other reference is indeed to a later moment, say m1, at which we are considering the status of the carrying-out of the promise. Easy examples show, however, that there is no simple evaluation of P itself at such later moments. For example, let P be "X will pay $5 to Y" or, with the future tense rendered explicitly as a Prior connective, as follows.7 It will be true that X has paid $5 to Y.

(31)

Feeding (31) into our generalized Target 5-10 gives a more special target: 5-11 TARGET. (Promising to pay $5 in the future) At mo, X promises Y that it will be true that X has paid $5 to Y. In considering whether or not the promise has a "carried out" status at some future moment m1, it seems clear that we do not wish to evaluate the actual complement, (31), of the promising. Instead we want to consider only its ingredient, X has paid $5 to Y.

(32)

When it is settled that (32) is true, the promise has been "carried out." At such a moment, when X has already paid $5 to Y, the future-tensed complement, (31), is doubtless settled false. The former matters; the latter does not. This example, and others that crowd the mind, might suggest that a uniform and nonsyntactical account of Target 5-10 is beyond hope. A close-to-home example such as "X will see to it that X pays $5 to Y" seems especially confusing, no matter how the "sees to it" is handled. We suggest, however, that a natural account can be found in the doctrine of branching histories and its accompanying semantics. If we pay careful attention to these semantics, we 6 Double time references are important, for instance, in understanding bets on objectively indeterminate future events, e.g., "I bet that this coin will land heads." But such ideas may contribute little or nothing to understanding bets on the settled past, e.g., "I bet that the Normans invaded England in 1064." 7 In (31) we use "has paid" instead of "pays" for two reasons. First, the achievement is easier to think about than the process, and just as useful in illustrating a promise. Second, the important feature of the example is a transition from "not-yet-paid" to "now paid," rather than some idealized instant of payment.

5. Promising: Stits, claims, and strategies

121

can see exactly how "double time references" work without presuming anything whatever about the various syntactic ingredients of P. The fundament of those semantics is this: Because of indeterminism, the truth value of a sentence may depend both on the moment of evaluation and also on the history of evaluation.8 That is, the metalinguistic predicate "true on" must be relativized both to a moment and to a history, so that one says "A is true on m/h," where h is one of the histories to which the moment, m, belongs. It follows that the sentential complement, P, of Target 5-10 has a rich semantic content in terms of its truth value at each moment-history pair, m/h. In principle, the entire semantic content of P is available to us in explaining the contribution of P to the content of Target 5-10. An essential feature of our account, however, is that promising uses only a small part of this richness; namely, promising uses only the pattern of truth values of P at pairs that feature a single moment,TOO,the moment of promising. The sentential complement, P, of Target 5-10 contributes nothing but its pattern of truth values at the various pairs m o / h , where mo is fixed once and for all as the moment of promising, and where h ranges over all the histories to which this moment belongs. It is striking that this be so. One would have thought that one would have to consider the truth value of P at moments after the moment of promising in order to get the right "world to words" fit. As we saw in considering Target 5-11, however, in looking at these later moments, we are not considering the truth value of P itself. In this special example, we look instead at the truth value, at later moments, of an ingredient of P, namely, (32). We made a point out of the fact that we do not consider the truth value of P = (31) itself. By attending to the pattern of values of P only at the moment of promising, we obtain a uniform account that works for arbitrary P, regardless of syntactic structure. (Naturally, in the context of this book, we presuppose that P exhibits a structure for which it is illuminating to speak of a rigorous branching-histories semantics. We offer no contribution to other aspects of the theory of promises.) We claim that it is possible to forget the grammatical complexities of P, and consider only its pattern of truth values at m0/h, whereTOOis the moment of promising and h is any history through that moment. Our claim is that this pattern supplies exactly the right semantic content for the promising to be a promising that P. We claim that in analyzing the content of Target 5-10, we need use only "P is true on m0/h," where mo is the moment of promising, and h is one of the histories to which that moment belongs. 8 This idea was introduced in §2A, and will be spelled out at length in chapters 6 and 8. If context-dependencies are present, and they must be considered for an accurate understanding of the language of indeterminism, then truth also depends on the moment of utterance, and perhaps also on the place, speaker, etc., in a manner indicated in those chapters. (There is, given indeterminism, no such thing as "the" history of utterance; see §6D and Question 8-4.) The moment and history "of evaluation" must be seen to be separate from these. In these terms, observe that the moment of promising represents a moment of evaluation, not the moment of utterance of a promise-report such as (9) or Target 5-10. With regard to Target 5-10, mo need not be "now," X need not be "I," and Y need not be "you." Compare the treatment of direct-discourse assertion in §6E.

122

Introduction to stit

It may seem as if we are suddenly abandoning the "double time references" theme. If P is evaluated only at the moment of promising, mo, and if we are not allowed to look at specific syntactic constituents of P, what could possibly be the role of later moments? Our (necessarily abstract) answer is this. It is the job of a later moment, say m1, to pick out some among all the histories through mo to be the range of h.9 Typically, we shall wish to know whether or not it is the case that, for all, none, or some of the histories to which m1 belongs, P is true at m0/h. This version of the "double time reference" idea is so central (and so confusing) that we shall postpone illustration in order first to introduce some jargon. 5-12 DEFINITION. (Double time reference ideas) • Sett-true(m 0, m1, A) iff (i) m0 is earlier than (or identical to) m1, and ( i i ) for every history, h, to which m1 belongs, A is true at m0/h. • Sett-false(m0, m1, A) iff (i) mo is earlier than (or identical to) m1, and (ii) for every history, h, to which m1 belongs, A is false at m0/h. • Poss-true(m0, m1, A) iff (i) mo is earlier than (or identical to) m1, and ( i i ) for some history, h, to which m1 belongs, A is true at m0/h. The structure of English tenses makes it difficult to restate these ideas informally. Let us position ourselves at m1, so that we may refer to the earlier moment, mo, with a past tense. Then, approximately, Sett-true(mo, m1, A) iff it is settled true at m1 that A was true at m0. For example, let only one payment of $5 be at issue, let mo be the moment of promising, and let A be "X will have paid $5 to y." Take a later moment, m1, at which X has paid $5 to Y—so that certainly at m1 it is settled that X has paid $5 to Y. Then on the one hand, A is oddly but certainly false at m1, since only one payment is at issue. On the other hand, at m1 it is settled that "X will have paid $5 to y" was true at m0.10 And that is exactly the right idea: Sett-true(mo, m1, X will have paid $5 to Y). In a Prior-like language that permits nesting of tense connectives and modalities, we may put it this way: At m1 it is settled true that at mo X will have paid $5 to Y. In terms of formal Prior-like connectives (see §8F), the sentence is this: At-mom m1 :Sett:At-mom m0 :Will:(X has paid $5 to Y). It is a pity that English, lacking this easy expressiveness, makes the key idea seem more difficult than it is. A picture may help. With reference to Figure 9 By the principle of no backward branching, Post. 3, the family of histories to which a later moment m1 belongs must be a subset of the histories to which mo belongs. In other words, any history that represents a possible continuation from a moment in the future of possibilities of mo already also represents a possible continuation from m0. 10 In any decent tense logic, including that applicable to branching histories (indeterminism), the following is a logical truth: If A, then it has always been true (with mo as a special case) that it will be true that A. If X has paid $5 to Y, then it has always been true that it will be true that X has paid $5 to Y.

5. Promising: Stits, claims, and strategies

123

Figure 5.1: Double time references

5.1, suppose that m1 represents a (possible) moment at which X has paid $5 to Y, and that m2 represents a (possible) moment at which it is a settled fact that Y will never be paid. The histories h1 and h2 represent possible continuations after X has paid $5 to Y; observe that these continuations cannot change the settled fact that X has paid $5 to Y. Similarly, h3 and h4 represent possible continuations after nonpayment becomes certain, continuations that cannot change that settled fact. Now let A be "X will have paid $5 to Y." A A A A

is true at m0/h1 is true at m0/h2 is not true at m0/h3 is not true at m 0 /h4

In these circumstances, Sett-true(mo, m1, A), since A is true at m0/h for every history through m1; and Sett-false (mo, m2, A), since A is false at m0/h for every history through m2. The critical point is that the role of a later moment, for example m1 or m2, is not as a place to evaluate A. Instead, the later moment supplies a group of histories over which "h" is to range while evaluating A at m0/h. In double time reference, the earlier moment supplies the moment of evaluation, while the later moment supplies a group of histories of evaluation.

5C.2.2

Strategic content of a promise

Given the ideas of Definition 5-12, we are in a position to explain the content of Target 5-10 in terms of a strategy. We shall also rely on a number of strategic concepts that are rigorously defined in chapter 13. Using these two sources of concepts, we give the explanation in terms of a definition of "strategic content of X's promising Y that P at mo," making plain the four required arguments, X, Y, P, and m 0 —the last being the moment of promising. 5-13 DEFINITION. (Strategic content of X's promising Y that P at m0) We are thinking of mo as the moment of promising; and we are letting S0 be the

124

Introduction to stit

promising-that-P strategy (more fully: the X-promising- Y-that-P-at-m0 strategy) that we are defining. In these terms, we define that S0 is the strategic content of X 's promising Y that P at mo iff the following. First, the "field" for the strategy, S0, must be mo together with all moments, m1, in its future of possibilities. (A promising-that-P strategy is, however, far from being totally defined on that field, or even complete in the sense of Definition 13-14, as will be clear from the clauses that follow.) It must be that s0 is a strategy for X whose domain of definition is some portion of that field. Second, in defining so, we divide up the various moments, m1, in its field as follows. i. Suppose X arrives at a moment m1 such that Sett-true(m0, m1, P), so that no matter the historical continuation of m1, P was true at the moment of promising with respect to that historical continuation. (Reminder: If P is a future-tensed sentence, e.g., "Will: A," and if in this example A is a settled fact at m1, then Sett-true(m0, m1, P).) We say at that moment, m1, that the promising-that-P strategy has been "satisfied." (To say that the promise has been "kept" is to say more than is warranted. Even "carried out" perhaps suggests something extra beyond the causal.) At any such moment m1, the strategy, S0, says nothing: m1 does not lie within its domain of definition. Observe that it already follows that once a promising-that-P strategy is satisfied, it remains so forever. ii. Suppose X arrives at a moment m1 such that Sett-false (mo, m1, P), so that no matter the historical continuation of m1, P was false at the moment of promising with respect to that historical continuation. Then at that moment m1, we say (following Hamblin) that the promising-that-P strategy has been "infringed." (To say that the promise has been "broken" is to say more than is warranted.) At any such moment, the strategy; s0, prescribes that X "compensate" Y in a way that we do not spell out, but that would certainly involve a sub-strategy that would work toward compensation until completed, or perhaps until some "deadline" was reached.11 Hi. Suppose X arrives at moment m1 at which neither Sett-true(mo, m1, P) nor Sett-false(m0, m1, P); that is, whether or not P was true at the moment of promising is historically open at m1. Here is where there might be room for X's agency directed toward P. What the strategy, S0, directs depends on circumstances at m1, which may be one of three kinds. a. There is available at m1 a strategy, s, for X that guarantees the set of histories with respect to which P is true at mo. Then the strategy, S0, must dictate that X make some (any) choice that is in accord with some strategy, s, for X that is available at m1 and that guarantees the set of histories with respect to which P is true at mo. 11

It is not plausible that an account of compensation can be built on nothing more than the semantics of an unarticulated complement, P. It is also not plausible that there should be any rigorous and general theory of compensation.

5. Promising: Stits, claims, and strategies

125

b. There is no P-at-m0-guaranteeing strategy available to X at m1, but there is, in the future of possibilities of m1, some moment, m2, at which not only do we have Poss-true(m0, m2, P), but furthermore, at m2 there is available to X a P-at-m0-guaranteeing strategy. Then the promising-that-P strategy, S0, must dictate that X make some (any) choice that keeps open the possibility of arriving at a moment m2 at which there is available to X a P-at-m0-guaranteeing strategy. c. Neither at m\ nor at any moment m2 in the future of possibilities of m1, is there available a P-at-mo-guaranteeing strategy for X. The promising-that-P strategy, S0, must dictate that X make some (any) choice that keeps open the possibility of P-at-m0; that is, some (any) choice that contains a history with respect to which P is true at m0- (These words are meant to include the possibility of a vacuous choice, Def. 14, as in a situation in which there is nothing that X can do.) The most significant feature of this definition is on the surface: In case ( i i i } , when the truth of P at mo is still open, when things could go either way with respect to the truth of P at TOO, then there is a call for agency. X must do something in the direction of making P true at TOO, if possible. X must not choose to guarantee that P is false at TOO. This is the heart of the strategic unpacking of the speech-act idea that a promise involves "fitting the world to the words." This promising-that-P strategy, S0., is "tough" in that it requires compensation for P turning up settled false even if that happens in spite of X's best efforts. Example: "I promise you that Billy will brush his teeth," and Billy fails to cooperate. On the other hand, the promising-that-P strategy we have just defined has a certain looseness in that even when X has a chance to make a single choice guaranteeing that P will be settled true in the immediate future, it permits X to make another choice on which, depending on later choices by X, P might turn out settled false. A more "wholehearted" strategy (Hamblin) would require even more agency on the part of X. The theory of strategies permits us to make many such distinctions. More sophisticated strategies might require, for example, that X pay attention to probability distributions on the various choices available. It is easy to see that there are a variety of different ways for X to "fall off" the promising-that-P strategy defined. The primary way occurs as described in (i), when X makes a choice that fails to be in accord with any strategy that guarantees that P, at a moment at which such a choice is available. There is no special word for this kind of failure; we could invent one, but it seems that which word is appropriate might depend too much on mentalistic considerations. For example, suppose X has the money promised to Y, but gambles it, and wins, and then pays Y after all. X was in a position to pay the promised money, but, by choosing to gamble, X fell off the prescribed promising-to-pay strategy. We

126

Introduction to stit

should not be likely to say in this case that X "broke" the promise, since Y was paid after all, and indeed it seems likely that we might say that X "kept" his promise. Nevertheless, putting at risk the formerly secure ability to keep the promise seems to deserve some sort of evaluative expression. 5C.2.3

Strategic content of a word-giving

In any case, the present account of promising seems already enough to force a division between (i) a word-giving-that-P of the word-to-world-fitting kind, and (ii) a promising-that-P of the world-to-words-fitting kind, even when P has the form of a future-tensed agentive. Since this is a vexing topic, it will be worthwhile to spell out the details. Evidently we should begin with an account of the strategic content of a word-giving. 5-14 TARGET. (Analytic target) At m0, X gives Y his or her word that P. (Later we consider the particular case in which P is a future-tensed agentive.) Here is our account of the content of Target 5-14 in terms of a strategy. Much will be familiar, since we have deliberately copied as much as possible from Definition 5-13. 5-15 DEFINITION. (Strategic content of X's giving his word to Y that P) We are thinking of mo as the moment of word-giving; and we are letting S0 be the word-giving-that-P strategy (more fully: the X-word-giving- Y-that-P-at-m0 strategy) that we are defining. In these terms, we define that S0 is the strategic content of X 's word-giving Y that P at mo iff the following. First, the "field" for the strategy, S0, must be mo together with all moments, m1, in its future of possibilities. (A word-giving-that-P strategy is, however, far from being totally defined on that field, or even complete, as will be clear from the clauses that follow.) It must be that S0 is a strategy for X whose domain of definition is some portion of that field. Second, in defining S0, we divide up the various moments, m1, in its field as follows. i. Suppose X arrives at a moment m1, such that Sett-true(m0, m1, P), so that no matter the historical continuation of m1, P was true at the moment of word-giving with respect to that historical continuation. We say at that moment, m1, that the word-giving-that-P is "vindicated."12 At any such moment m1, the strategy, S0, says nothing: m1 does not lie within its domain of definition. Observe that it already follows as a matter of logic that once a word-giving-that-P strategy is vindicated, it remains so forever. 12 In (ii) we say that a word-giving may be "impugned." We choose "vindicated" and "impugned" for word-givings as parallel to "satisfied" and "infringed" for promises. As we suggest in §6E, the vindicated/impugned pair seems also felicitous for assertions that are less serious than word-givings. In any case, the structurally important points, regardless of English words, are (1) that a matter that is unsettled at the moment mo of word-giving becomes and forever after remains settled at m1, and (2) that some sort of norms kick in at m1.

5. Promising: Stits, claims, and strategies

127

ii. Suppose X arrives at a moment m1 such that Sett-false (m0, m1, P), so that no matter the historical continuation of m1, P was false at the moment of word-giving with respect to that historical continuation. Then at that moment m1, we say that the word-giving-that-P has been "impugned." At any such moment, the strategy, S0, prescribes that X "compensate" Y in a way that we do not spell out, but that would certainly involve a sub-strategy that would work toward compensation until completed, or perhaps until some "deadline" was reached. iii. Suppose X arrives at moment, m1, at which neither Sett-true(mo, m1, P) nor Sett-false (m0, m1, P); that is, whether or not P was true at the moment of word-giving is historically open at m1. Here is where there is room for X 's agency directed toward P. Since, however, it is only a matter of word-giving-that-P, and not of promising-that-P, nothing is prescribed for X to choose. No effort of fitting world to words is called for. It just doesn't matter. The last clause, the one dealing with moments at which it is open whether P is true at mo, is where the two types of strategies differ. Here is where the present analysis puts the essential difference between promising that P and word-giving that P. Promising typically requires that the promisor do or choose something at some such moments, moments after the moment of promising and before it is settled whether the promise is satisfied or infringed. Word-giving never imposes any such requirement. The analysis of these two concepts via their strategic content emphasizes this. 5C.2.4

Promising versus word-giving a future-tensed agentive

Let us now consider the special case in which P is a future-tensed agentive, X will see to it that Q.

(33)

Does there remain a difference between the following? At mo, X promises Y that (X will see to it that Q).

(34)

At m0, X gives Y his word that (X will see to it that Q).

(35)

English is not of as much help as we should like. Indeed, a frequently encountered "analysis" has it that promising to do something just is giving your word that you will do it. We may see, however, that even on the oversimplified and partial accounts of promising-that-P and word-giving-that-P, the sharp difference remains. Let the strategic content of (34) be given by Definition 5-13, and let the strategic content of (35) be given by Definition 5-15. Suppose, for a very simple case, that Figure 5.1 on p. 123 pictures the world, and all its possibilities, choices, and chances. Suppose that mo is the moment of promising, or word-giving. Let the splitting just after mo represent some choice or chance beyond the control of X. Let m1 and m2 be (nonvacuous) choice points for X. Suppose that truth values for Q are distributed as follows:

128

Introduction to stit Q Q Q Q

is true at m1/h1. is not true at m1/h2. is true at m2/h3,. is not true at m2/h4.

In this simple case, one can use stit theory (the dstit variety) to calculate (or just see) that at each of these four pairs, "X sees to it that Q" has exactly the same truth value as Q: "X "X "X "X

sees sees sees sees

to to to to

it it it it

that that that that

Q" Q" Q" Q"

is true at m1/h1. is not true at m1/h 2. is true at m2/h3. is not true at m2/h4.

Also, since m1 and m2 are the only moments at which X has a nonvacuous choice, "X sees to it that Q" is false everywhere else. Therefore, by ordinary tense logic, "X will see to it that Q" must follow along, history for history, when evaluated at the moment of promising or word-giving: "X "X "X "X

will see will see will see will see

to to to to

it it it it

that that that that

Q" Q" Q" Q"

is is is is

true at m0/h1. not true at m0/h2. true at m0/h3. not true at m0/h4.

The strategy for the promise represented by (34) therefore dictates that X must not "turn right" upon reaching either m1 or m2- If he does so, if he chooses h2 upon reaching m1, or if he chooses h4 upon reaching 7712, the strategy for (34) will be infringed, and will call for compensation of Y. We might well say that X has broken his promise to Y since by "turning left," X could have guaranteed satisfaction of the promise. The point is that if X throws away a "last chance," X has violated the strategy that represents the content of a promise, even if later X compensates Y. The strategy for the word-giving represented by (35) is different precisely with respect to what it dictates at m1 and m2. At these moments, it says nothing at all. X is free to choose as he likes, so far as the word-giving strategy goes. If X chooses at m1 or m2 to "turn right," X has not violated the strategy that represents the content of the word-giving that he will see to it that Q. If at mo X has merely given his word to Y that he will see to it that Q, then on the account we have given, X undertakes no responsibility for guaranteeing the truth of (33) at m0. For avoiding violation of the word-giving strategy, it suffices if X compensates Y even after choosing to guarantee that (33) is false at mo!13 13 We have taken the word-giving target from RR. It is therefore essential to observe that the analysis of word-giving to be derived from RR via quote (21) does not give the present result for word-giving. The quote from RR fits our theoretical account of "promising" better than our account of "word-giving." Our aim in choosing to speak of "word-giving" is not, however, to saddle either RR or indeed ourselves with a particular analysis. Its sole purpose is to exhibit how the theory of strategies provides a chance to help sharpen any such analysis—without invoking mentalistic concepts.

5. Promising: Stits, claims, and strategies

129

By giving us insight into the difference between two ideas naturally (but not inevitably) associated respectively with the English expressions "word-giving that one does" and "promising that one does," the proposed application of the theory of strategies passes a critical test.

This page intentionally left blank

Part II

Foundations of indeterminism

This page intentionally left blank

6

Indeterminism and the Thin Red Line Indeterminism is essential to the idea of agency as we conceive it.* For this reason we drop our concentration on agency for the space of three chapters in order to give primary attention to some foundational questions concerning indeterminism itself. There are many such; the key problem of this chapter is how to make sense of assertions about future events made at a time when the outcomes of those events are not yet determined. One tempting solution to this "assertion problem" is the "Thin Red Line" of the chapter title, a version of indeterminism that combines two doctrines: (i) the doctrine of an open future filled with real (incompatible) alternatives, and (ii) the doctrine that one of these futures is the real or actual future, the distinguished one among the alternatives that will "in fact" happen (que sera, sera). We argue that this solution will not do: Although "actuality" makes sense for the past and the present, there is no such thing as "the actual future." Our argument is based on a close examination of the basic ideas of a Kaplaninspired indexical adaptation of the Prior/Thomason semantics, which we take to be required of a language to be used in an indeterminist world—such as we believe ours to be. Abandonment of the Thin Red Line leaves unsolved the problem of making sense of assertions about the indeterministic future. We propose a natural solution to this problem based on the point that assertion has normative consequences for the future, no matter what happens, a solution that implies that the Thin Red Line is useless. In the chapter following this one, we go over our grounding theory of the open future postulate by postulate; and then in chapter 8 we explain our semantic point of view, also in detail. There is some repetition between this chapter and those; this one should be taken as introductory. Furthermore, readers not so *Mitchell Green is co-author of this chapter. With the kind permission of Ridgeview Publishing Company, it draws on Belnap and Green 1994.

133

134

Foundations of indeterminism

interested in rigorous foundations may wish to skip chapter 7 and chapter 8 altogether, or perhaps return to them later.

6A

Preliminary considerations

Before turning to the exact analysis that we think is required in addressing the "assertion problem," we briefly lay out some preliminary ideas: a generalization of McTaggart's A- and B-series, a sketchy account of the Thin Red Line, a clarification of the idea of indeterminism with which we work, and a rehearsal of the basic postulates of branching time theory. Throughout it is essential to keep in mind that we are not considering "second order" indeterminism, that is, indeterminism as a feature of our mind or of our language or of our theories. We are always taking our questions to concern "first order" indeterminism, that is, indeterminism as a feature (if it is one) of the causal order of our world.

6A.1

The A series and the B series

Central to the idea of indeterminism is this: At a given moment in the history of the world there are a variety of ways in which affairs might carry on. Before the toss of the coin there are two things that could happen, either heads up or tails up. This possibility is not merely epistemic, but in re.1 In spite of this it seems clear that the outcomes heads and tails could not be temporally related. Although heads, if it happened, would be subsequent to the coin toss, and tails, if it happened, would be subsequent to the coin toss, surely neither of heads or tails precedes the other, nor are they simultaneous. One may hold that being present, past, or future is an intrinsic property of moments, and thus be a champion of the past-present-future "A-series" of McTaggart 1908. We aim to avoid this question altogether. Instead in considering indeterminism we concentrate on a generalization of McTaggart's "B-series," which describes the before/after relations that moments bear to one another, with no reference to whether any of those moments is past, present, or future. Both sides of the debate about the A-series admit that the B-series captures some of the truth, and surely much of our thinking about our world derives from the presumption that it has the B-series shape. That is to say, thinking of "order" as a generalization of "series," it is natural to think of the B-order of our world as a series, defined by the condition that every distinct pair of moments m1 and m2 are such that either m1 is prior to m2 or m2 is prior to m1. Yet if the B-order is a series, it follows that the events heads and tails cannot both be in our world, even if the coin toss can. (For an introductory account of the Aand B-orders, see Gale 1968, pp. 65-86. For important work on indeterminism 1

Throughout this chapter we shall use hackneyed examples such as coin tosses and the occasional sea battle. In each case the example will serve its purpose only if understood as indeterministic in the sense delineated in §6A 3. Not wishing to prejudge any questions about which concrete situations are really indeterministic, we invite the reader to supply her own examples if the ones that we employ tread on her sensibilities.

6. Indeterminism and the Thin Red Line

135

from the point of view of an A-theorist, consult McCall 1976, McCall 1984, and McCall 1994.)

6A.2

Introduction to the Thin Red Line

One common way of conjoining the aspect of indeterminism just formulated with an understanding of our world deriving from a construal of the B-order as a series is to hold that at a given moment in the history of our world from which there are a variety of ways in which affairs might carry on, one of those ways is privileged over against all others as being what is actually going to happen. Not only is it true that the coin could either land heads or tails, and therefore true that it will be the case that either the coin lands heads or the coin lands tails. Not only is it the case that either the coin will land heads, or the coin will land tails. What is furthermore true is that there is, at the time of the toss, a directly referential, rigid, absolute specification of "what, at the time of the toss, is actually going to happen." This specification breaks the symmetry, picking out either heads or tails. Only our limited minds keep us from knowing which. One might express this by saying that our world does not itself branch, leaving open the ontological status of the alternative possibilities required by indeterminism. The thought is bolstered by the fact that when we see heads we are inclined to say, "Aha! So this is what was going to happen. It was therefore true before the toss that the coin was going to land heads." (See Ryle 1954 for motivation of the view that whatever happens was to be.) It used to be said of the British Empire that it was maintained by a thin red line of soldiers in service to the queen. The Russians dashed on towards that thin red-line streak tipped with a line of steel. (W. H. Russell to the London Times from the Crimea, 1854) We shall express the view just sketched by saying that from among the lines along which history might go, subsequent to an indeterministic moment, one of those lines is the course along which history will go, and it is both thin and red, much as it might be inscribed on a blackboard. You may think of the hue as infrared, to capture the idea that the Thin Red Line does not imply that mortals are capable of seeing the future. Our tendency to believe that there is a Thin Red Line is powerful. When we presume that there will be a sea battle, in spite of knowing that there might not be, what we seem to presume is the following: There will in fact be a sea battle even though there need not be. When we wonder, "What does the future hold? Will there be a sea battle or not?" even when we know that there are two possibilities, we seem to be wondering whether there will in fact be a sea battle. Further, belief in an actual future, a Thin Red Line, appears to be consistent with believing that there is indeterminism in the world. There seems nothing untoward in supposing that while there are two things that might become of the coin about to be tossed, it is in fact going to land heads. It would seem to be consistent to say, "The coin will not land tails, but it could. What is

136

Foundations of indeterminism

more, the coin will land heads, but it needn't do so." (See OhrstrOm 1981 for an explicit call for a Thin Red Line; Brauner, 0hrstr0m, and Hasle 1998 for discussion; and Barcellan and Zanardo 1999 for formal development, discussion, and a suggestion of applicability in computer science.) We shall argue that in spite of its stalwartness the Thin Red Line should be forsworn. We shall argue, against an established opposition, that one can make much better sense of an indeterministic, branching structure for our world by abandoning the idea of an actual future as distinguished among the possibilities. We shall furthermore argue that the Thin Red Line doctrine turns out, on closer scrutiny, to have unpalatable consequences. We shall call the view that in spite of indeterminism one neither needs nor can use a Thin Red Line the doctrine of the open future. We intend this terminology to make contact with the intuitions (but not the style) of many decades of thinking about indeterminism, and to make contact with an analogy that we develop in detail between the "openness" of expressions like "the coin will come up heads" and the well-known "openness" of "x1 is brindle." We first lay down constraints upon the notion of indeterminism relevant to the present discussion (§6A.3). Explaining the open future under those constraints involves two related tasks. The first and conceptually prior task is extra-linguistic, requiring the elucidation of a notion of indeterminism that applies to our world (§6A.4). This account will embody an understanding of a B-order that is not a series, but instead is free from any assumption of linearity, an understanding that will explain how this very world, if indeterminism is ever true, is replete with possibility. The second task is a linguistic one, involving the careful development of a semantic theory for temporal discourse in an indeterministic world (§6B). At this point we will have finished our positive development of the open future doctrine. A natural basis for doubt about that doctrine is that it appears unable to make sense of one who asserts that, bets that, or wonders whether there will be a sea battle even when it is clear that there might not be. A special case of this problem is formulated in §6C as the "assertion problem." One superficially reasonable response is to postulate a Thin Red Line. In §6D, however, we argue that the Thin Red Line doctrine, in any of the versions of that doctrine that we consider, has unacceptable consequences, ranging from a mistreatment of actuality to an inability to talk helpfully about what would have happened had what is going to happen not taken place. In §6E we "solve" the assertion problem by arguing that our framework does make sense of such acts and attitudes once we come to see that to assert that A is to do something that has a normative significance no matter how history carries on.

6A.3

Concepts of indeterminism

Among all the numerous concepts of indeterminism, we are concerned with one that is local, pre-probabilistic, objective, feature-independent, de re, existential, and hard.

6. Indeterminism and the Thin Red Line

137

We need a local concept of indeterminism. The contrast is with global. Perhaps our entire world is indeterministic or not, or perhaps a law or even an entire scientific theory is indeterministic or not, but we take it to be essential to be able to describe a certain specified transition as indeterministic or not. Here and from now on by a transition we mean an ordered pair of events, the first entirely preceding the second. The earlier is called the initial of the transition, while the later is called the outcome. EXAMPLE. (Throw of a die) This morning we threw a die. It showed six; but there were five other possible outcomes. Then we loaded the die, and threw it again. Again it showed six; but this time, that was the only possible outcome. The first throw-to-six transition was indeterministic. the second was not. We need a concept that can be used in this local and particular way to describe individual transitions from one event (the initial of the transition) to another (its outcome). In this study we simplify by worrying only about the temporal dimension of locality. We will suppress the relativistic considerations arising from the spatial aspects of indeterministic transitions. (See Belnap 1992 for an exploration of indeterminism in a relativistic setting.) The idea of locality just described should rightly make you think of "single case probabilities." But we need a pre-probabilistic concept of indeterminism. Perhaps when the first throw was made there was a 1/6 chance of six. Before the numbers come into it, however, there is the idea that given the throw that in fact came up six, there were five other possibilities. To say that does not require any probability measure; only possibilities are needed. These possibilities are themselves "local." The actual transition was throw-to-six, the other possibilities were the transitions throw-to-one, throw-to-two, and so on. Nothing as global as a "world" or "theory" or even "law" comes in this early, and no numbers representing probabilities are part of this concept. Quite to the contrary, any concept of probability must rest on a concept of possibility (sometimes called "the probability space"). We need an objective concept of indeterminism. We mean that the question of how many possible outcomes there were for a certain throw shall be classed with the question of how many ears there are on a certain Scottie, and contrasted with questions that are explicitly about who thinks what about what, and whether it is reasonable to do so. Our aim is to theorize about a concept of indeterminism that does not require simultaneous explicit theorizing about people and their thoughts or norms or culture. Thus, we are after a concept of indeterminism that does not put the number of possible outcomes of a certain throw in anyone's head, or make it relative to laws or theories, or have it depend on the status of a conversation, or depend on what people care about. All to exactly the same extent as the number of ears on a certain Scottie. The most explicit contrast is with "epistemic indeterminism," also called epistemic possibility, as codifying a form of ignorance. There is a difference between calling a transition indeterministic relative to some feature, and calling it feature-independently indeterministic. By definition a transition is feature-independently indeterministic if there is more than

138

Foundations of indeterminism

one possible outcome for its initial, and otherwise it is feature-independently deterministic. In this study we shall be content with these "feature-independent" ideas because of their foundational role. The feature-relative ideas are, however, of great importance. When every possible outcome of an initial has a certain feature, we may well say that the transition was deterministic with respect to that feature. Given the throw, for example, not only did the die come up six, but it landed on the floor. Since it landed on the floor on every one of the other five outcomes, the transition was deterministic with respect to landing on the floor. This sort of feature-relative determinism (for a particular feature), which is doubtless of enormous importance both for science and common sense, is evidently consistent with feature-independent indeterminism. We want a concept that can apply to a transition de re, without requiring some description under which the initial falls. No matter how you describe the throwing of the die, even if you are confused enough to refer to it as the holding of a martini by that man in the corner, still, it has just six possible outcomes. Take the ancient example: EXAMPLE. (Rest or motion) She remained at rest. But she wasn't tied up. So she could have moved or remained at rest. On the other hand, given the exact history of her beliefs and desires, there was evidently but one possible outcome, not two. Here the initial has been described in two different ways. The first patch of rhetoric suggests free will, the second suggests instead iron-clad determinism. To insist on a concept of indeterminism that is fundamentally de re is to disbar pretending to plausibility of contradictory phenomena via colorful redescriptions. You can't have it both ways. Either her concrete situation, no matter how described and no matter what was "similar" to it, admitted two possible outcomes or it admitted only one. Thinking de re prevents you from evading the problems of indeterminism (or determinism) by switching descriptions—as if you could change the number of ears on a particular Scottie by describing it as very like a whale. For our continuation to have interest we need to consider only the weakest possible indeterminist claim, namely, the existential claim that some transitions are indeterministic. The existential claim is obviously consistent with the commonsense view that numerous transitions are more or less feature-relative deterministic in interesting ways: The stars in their courses bravely run, fire burns here and in Persia, and although the ways of men are various, the same motives are followed by the same actions. A concept is hard if you have a rigorous theory for it, or at least if you wish you had such a theory and are miserable to the extent that you don't. If in contrast you are happy with some interesting stories, some "paradigm cases," or with a sketch of an outline of a skeleton of a never-to-be-supplied theory, the concept is soft. Wittgenstein's concept of game is soft; von Neumann's is hard. We are interested in indeterminism as a hard concept. We want a rigorous theory.

6. Indetcrminism and the Thin Red Line

139

We will not further discuss these seven characteristics. In the remainder of this chapter, however, and indeed throughout this book, we wish always to be understood with this characterization in mind.

6A.4

Our World and its causal order

Branching time is a rigorous theory about indeterminism. It is also an account of how to talk sanely in an indeterministic world. We introduced the theory in §2A. Here we describe the theory briefly and informally, reserving our detailed treatment for chapter 7. Branching time is not itself an indeterministic theory; instead, it says what indeterminism is, and it says what determinism is, but branching time does not choose between them. Because there is not much point in all the fuss if universal determinism be true, this is almost a quibble, but not quite. It implies a recommendation that one become clear on the difference between indeterminism and determinism before taking a stand. The theory was first adumbrated in print by Prior 1967 (pp. 126-127) in the course of his work on the logic of tenses; 0hrstr0m and Hasle 1995 (p. 189) report that Kripke suggested an early version in a 1958 letter to Prior. It was first carefully formulated in Thomason 1970; see Thomason 1984. We need sharply to distinguish the portion of the theory—the "ontology" or perhaps "metaphysics" or (less pretentiously) "extra-linguistic" portion—that precedes and underlies the semantic development. The theory of this section is therefore quite independent of linguistic concerns. There are three fundamental ideas, already employed in earlier chapters: moments, the causal ordering relation, and Our World.2 First there is the idea of a moment (we use "m"); a moment is an instantaneous concrete event with unlimited (presumably infinite) spatial extent. Among moments, this very moment is what Whitehead called "all Nature now," idealized to a zero temporal thickness. The concept of a moment (we borrow jargon from Thomason) is of course a Newtonian idea. It is distant from our everyday conceptions, and it is nonrelativistic. It inherits from Laplace's demon the implausible presupposition that the fundamental terms of the causal order shall be entire instantaneous world-slices, instead of smallish local events or point events. It is, however, a worthwhile approximation to the truth. Accordingly we shall pretend that expressions such as "this actual moment" have definite worldwide meanings. The second idea is the causal ordering relation, also called the earlier-laterthan relation, m1 \<m2. This is a B-order relation, which we postulate to be branching rather than linear because of indeterminism. We use "<" for the companion proper relation, so that m1 < m2 iff (m1\< m2 & m1 /= m2). The whole idea of branching time as a theory of indeterminism is that there can be incompatible moments each of which might follow upon a given moment, though there are never incompatible moments in the past. Thus, given m1 < m2, it is 2 The term Our World is new to this book, but not the idea, elsewhere we say Tree. We use Our World when emphasizing the thought expressed in the name, whereas we use Tree when we are thinking more abstractly.

140

Foundations of indeterminism

right to think perspectivally of m1 as in the past of m2. On the other hand, one should say that mo is in the "future of possibilities" of m1—not simply in its "future." The reason is that it is intuitive that a future of possibilities, unlike a future history, can contain incompatible moments. The final idea is Our World. Start with this very moment (yours or ours; at this level of idealization it doesn't matter). Now form the set of all moments that are connected to this very moment by means of any zigzag combination of the causal ordering or its converse. That is, include all moments that you can reach by means of a "causal path," no matter how complicated. That is what we mean by "Our World" construed as a set of moments.3 The theory of branching time is, as we give it, a theory only about Our World and not about any other. Thus by "moment" in what follows we always mean "moment of Our World." Moments are thereby not to be confused with creatures of the mind. Moments are mundane. The theory technically has only two primitives, Our World and the causal order. The theory uses these primitives in four postulates: nontriviality, partial order, historical connection, and no backward branching. These, which we introduced in §2A, we summarize in proper form in §3 and study in chapter 7. Here, for easy reference, we briefly repeat their meaning. Nontriviality is a technical postulate that says that Our World is nonempty. Partial order describes the causal relation, /<, as reflexive, transitive, and antisymmetric. This last signals that moments of Our World must be understood as concrete, unrepeatable events rather than abstract "states" or "times." Historical connection says that every two moments in Our World have a common historical ancestor. It is the postulate that describes Our World as one world. Finally, no backward branching says that all branching is forward, never backward. No backward branching reflects the uniqueness of the past. Starting from any moment there is exactly one chain of moments in the backward direction, its past history. The conviction is that although Our World contains alternative incompatible possibilities in the future of possibilities of a given moment, there are no incompatible moments in the past of any moment. A moment may have more than one possible outcome, but not more than one possible "income." A history (we use "h") is defined as a maximal chain of moments, a complete possible course of events stretching all the way back and all the way forward. We may consider a new "same-time" primitive, Instant, that renders all histories isomorphic, so that it makes sense to have a doctrine of linearly ordered instants of time to complement the theory of branching moments. An "instant" of linear time can be defined as a set of same-time-mated moments; see Post. 5. Given Instant, it makes sense to ask what might have happened "at this very instant." The primitive seems convenient both for scientific approximations and for fashioning persuasive illustrations, and earlier chapters have relied upon it in conceptualizing the achievement stit. One may question whether it is finally respectable either scientifically or from the point of view of common 3 In the theory of branching space-time, Belnap 1992 uses "Our World" for a similarly motivated set of point events, instead of moments. In both cases the defined term names a set that exhausts—to the extent admissible for an idealization of the given kind—our world.

6. Indeterminism and the Thin Red Line

141

sense. Furthermore, it is not needed for an elementary account of indeterminism, and it may confuse us into thinking that we understand things that we do not understand, such as what it means to identify stretches of time across histories. Nevertheless, for its convenience in making up examples, we include Instant as one of our primitives. Finally, we add a domain, Domain, taken as a set that is big enough to represent (without contradiction) all that there is. There is no reason relevant to our work on indeterminism not to assume that Our World and Instant are subsets of Domain. In sum, we are looking at a "branching time with instants and domain" structure (a BT + I structure) (Our World, /<, Instant, Domain) that is identified with our very own world.

6B

Parameters of truth

As we indicated in §6A, there is a problem about speech acts using futuretensed sentences in the language of branching time. We take up assertion as a special—but surely central—case. Crudely put, "the assertion problem," as we call it, arises because given indeterminism, it would seem as if future-tensed statements "have no truth value." (This dangerous phrase receives a technical definition and an accompanying warning in §6B.7.) The reason that this is a problem is that it seems to make sense to assert a predictive statement even in the face of indeterminism. Since, however, such a statement "has no truth value," how can it make sense to do so? In order clearly to see the nature of the assertion problem, we adumbrate a perspective on a variety of linguistic devices that have been considered by philosophical logicians. All of these devices involve relativization of truth to one or more parameters such as domain, interpretation, and assignment of values to variables as in first-order logic, and to further parameters such as moment, history, world, time, place, speaker, addressee, demonstrated object, presupposition-set, and so on. By speaking in this general way about the relativization of truth to parameters, we hope (i) to elucidate the assertion problem, (ii) to sort out proposals to solve that problem by introducing a Thin Red Line representing the "actual future," (iii) to explain our own discontent with those proposals, and finally (iv) to suggest our own solution. We do not, alas, know how to do all this without introducing some auxiliary semantic ideas, chiefly ideas that are derived from Prior 1967, Thomason 1970, and Kaplan 1989—and of course Tarski. In chapter 8 we go over many of these ideas in detail; our treatment here, which is casual, is intended as a modest introduction.

6B.1

Grammatical and semantic presuppositions

The following grammatical and semantic presuppositions are helpful to our account, where to avoid circumlocution we let L be the language in question.4 4

Because we are not discussing the mathematics of L, it suits our purposes to describe L with some looseness.

142

Foundations of indeterminism • We presuppose that the grammar of L is compositional.5 There are some individual constants and some individual variables and some operator and predicate constants, and there are some predications Ft\...tn serving as "atomic" sentences. Although L has both operators and predicates, for simplicity of exposition, we suppose in this chapter that sentential connectives (counting Vxi as such) constitute the only interesting means of composition. Indeed, most of what we say involves only one-place connectives. We let O be a one-place connective, so that when A is a sentence, O ( A ) is also a sentence. • Truth needs to be relativized somehow or other. In first-order logic, for instance, one needs to relativize truth to a domain, Domain, to an interpretation of constants, 3, and (for open sentences) to an assignment of values to the variables, a. We call these relata parameters of truth. When we assemble parameters into a "legal" sequence, for example (Domain, 3, a), we call the result a point (of evaluation). (For a candidate sequence to be "legal" is for the parameters to satisfy conditions such as "the value that a gives to each variable must belong to Domain") • Speaking generally, then, a point of evaluation contains information coded as a legal sequence of parameters, (z1, ..., zn), that is enough to fix a truth value for each sentence of L. To express that A is true relative to (z1, ..., Zn). we write

• Tedious convention: We use "parameter" in two ways, both for a member, z1, of the sequence and also for a position in the sequence. For instance, we might say that some domain, Domain1, is a parameter of truth, but we also think of the first position in (Domain1, 3, a) as "the domain parameter," a position that can also be filled by, say, Domain2. In abstract discussions we let Zi be an arbitrary parameter qua ith position, and we let zi be one of the ways that Zi can be filled. • Semantically, we presuppose that "z1, ..., zn\= A" is explained or defined recursively in the manner due to Tarski. — We suppose that "atomic" sentences Ft1...t m have or are somehow awarded a truth value at each point, (z1, ..., zn), so that "z1, ..., zn \= Ft1...t m " is well defined. — We suppose that there is associated with each sentential connective, say O, a recursive explanation or definition of the truth value of O(A) at each point in terms of the truth value of A at each point. That 5

We are neither arguing nor assuming that English, for example, is compositional. The presupposition instead embodies our view that noncompositional features of language do not bear on the assertion problem in the following limited and revisable sense: If we cannot make headway in the simpler context of a compositional language, then adding noncompositional devices will not help in addressing that particular problem.

6. Indeterminism and the Thin Red Line

143

is, for each point, ( z 1 , ..., zn), whether or not z1, ..., zn depends on the following: for each point ( z 1 , ..., z'n), whether or not z1, ..., z'n \= A. In other words, we suppose that our semantic understanding of, for example, O(A) in terms of certain parameters is a well-determined function of our semantic understanding of A in terms of those parameters. In short, the semantics of L is given by a recursive explanation or definition of "z1, ..., zn \= A" Easily lost amid the abstractness is the central idea, derivative from Tarski, that if you wish to give your semantic explanations or definitions by means of a simple compositional recursion that follows grammar, then denotation and truth must be relativized to parameters. It will be important to our treatment of the assertion problem that we keep track of the individual nature and purpose of these various parameters. We do this as an approach to a key question: Given that some philosophers have introduced the "Thin Red Line" (the "actual future"; see §6A.2) to help with understanding the future tense, exactly how does the Thin Red Line work as a parameter of truth? Treating parameters in general will provide a firm foundation when we turn to answering this question. From the point of view of the assertion problem, the most profound division of parameters is into the "mobile" versus the "immobile," a rigorously definable division that we now informally and roughly explain. 6-1 DEFINITION. (Mobile versus immobile parameters) • If L has any connective whose recursion clause requires shifting some parameter, Zi, then Zi is mobile with respect to L. More ponderously, if for some connective, O, there is a point ( z 1 , ..., zi, n ) such that the recursion clause determining whether z1, ..., zi, ..., (A) requires considering whether or not z1, ..., z'i, ..., z'n \= A for some point such that zt is not identical to z't, then the parameter, Zi, is mobile with respect to L.6 • If, however, the value zi of Zi can be held fixed throughout the entire recursion, then Zi is immobile with respect to L. More ponderously, suppose that for every connective, O, of L, "z1, ..., can be defined without considering whethe except perhaps when zt = zi. Then Zi is immobile with respect to L. This division into mobile versus immobile parameters, even though difficult to keep in mind, is essential to understanding the assertion problem as we conceive it. We return to this point. Now we lay nearly all of our cards on the table, all at once, by listing the various parameters of truth whose understanding we think is critical to understanding indeterminism. We divide parameters for L primarily into mobile versus immobile, and then we subdivide them into types that depend on the underlying purpose of each parameter. 6 This (informal) definition takes into consideration that some connective might need to vary some Zi along with Zi.

144

6B.2

Foundations of indeterminism

Immobile parameters of truth

Semantic values need to be relativized to the following immobile parameters of truth. These parameters are essential to the recursion, but we emphasize their immobility: They can be held fixed throughout the entire recursion. We enumerate them according to their purpose. Structure parameters. The purpose or function of the structure parameters is to represent as much as is needed of the world (or worlds?) of the speakers of L. This representation must of course be described using language, but what is represented should be altogether independent of language. (All of the other parameters are more closely tied to language.) In this chapter, in contrast to the more abstract chapter 8, we are thinking of ourselves as the speakers of L. and so it makes sense for us to fix the structure parameters as 6 = (Our World, \<, Instant, Domain), We are thus here thinking of 6 as a single intended "standard" structure. Each and every structure parameter is immobile, since no structure parameter is ever shifted in the recursive account of u z1, ..., zn \= A." For example, in first-order logic, the domain never needs shifting. What happens to A in other domains or on other interpretations is irrelevant to the recursion. Of course we need to shift structure parameters in defining useful forms of "implication" or "logical truth"; it is just that this shifting plays no role in the recursion clauses defining (parameterized) truth. Interpretation parameters. The purpose of the interpretation parameters is to interpret the various "nonlogical" constants of L, and in this way to represent some of its important semantic features. We collect the interpretation parameters together into "the interpretation parameter," writing 3 as its value. The interpretation parameter is immobile, never being shifted in the course of the truth recursion. In this chapter we think of 3 as fixed, giving a single intended or "standard" value to the constants of L. (We may of course vary 3 if we wish to define something like "A is valid in ©.") Model parameter. The difference between & and 3 is that 6 describes language-independent features, while 3 gives some of the semantics of L by interpreting its constants. We need both, and so use "model" as jargon for an acceptable combination of a language-independent structure and an interpretation: A model m is defined as an appropriate structure-interpretation pair (6, 3). As noted, one may think of m as varying in order to define "implication" or such. We do not, however, need to vary m in our efforts, in this chapter, to become clearer on the nature of the assertion problem: m remains fixed throughout the chapter as representing the intended or "standard" combination of structure and interpretation, the "standard" model. The model, m, thus represents both our world and the constant features of our language.

6. Indeterminism and the Thin Red Line

145

Context parameters. The purpose of the context parameters is to represent relevant features of the context of a certain speech act. the context in which a certain expression of L is used. Elaboration of this idea is due to Kaplan, on whom our entire discussion relies. Whenever we refer to Kaplan's ideas by page number, the reference is to Kaplan 1989. We will often say "context of use" in referring to the Kaplan concept, which we represent with "c." Even holding the model, m, fixed as "standard," we should naturally think of many contexts of use; that is the reason that we do not count the context, c. as part of the model. How much information c supplies depends on how rich we take L to be. The individual items determined by c are called "context parameters." For example, if the indexical "I" is present, we need a context parameter for the speaker. The context of use provides what is in fact determined by an idealized speech act using the given sentence as vehicle. The context provides only what Kaplan calls "fact-of-the-matter parameters" (p. 593). They are what they are. and the logician is not entitled to make them up. Context parameters have two dramatically different functions. The first function, which Kaplan entirely makes clear, is to provide a semantic value for context-dependent expressions, such as the indexical, "I," as already mentioned, for other indexicals such as "here" and "now," for demonstratives such as "this," and for connectives anchored in the context such as "actually." Exactly what we may realistically count as a context parameter is one of the principal questions to be addressed when we consider the assertion problem. In order to answer this question, we shall need to ask what counts as the context of an act of assertion. But that must wait. In the meantime, we record somewhat gnomically that for understanding the assertion problem, the context of use will in addition have an "initialization" role, which we explain in due course. In spite of the fact that there are many intended or "standard" contexts of use, context parameters are just like structure and interpretation parameters in one key respect: They are one and all immobile. The context of use is never shifted in the course of the truth recursion. We summarize: No recursion clause needs to shift either model or context in the course of obtaining a parameterized truth value for O(A) recursively based on a semantic account of A. Of course one could invent a connective that requires shifting some structure parameter, interpretation parameter, or context parameter; but we think Kaplan's remark about the last case is applicable to all three cases: Such a connective would be a monster (p. 511). (As indicated in §6E, however, treating "direct discourse" as part of L leads to shifts of context, and therefore renders the context parameter mobile.)

6B.3

Mobile parameters of truth

If L were confined to non-context-dependent atomic sentences and truth functions thereof, then 3)1 would provide enough parameterization for truth for any A without free variables. We could give an adequate account of truth with the help of a locution such as

146

Foundations of indeterminism

read "A is true on m." When context-dependent expressions are part of L, however, (2) is not enough. We need to add the context-of-use parameter, c:

As we learned from Tarski, however, if L has quantifiers, neither m nor (m, c) is enough. And this point must be generalized: (Almost) any non-truthfunctional connective requires additional parameters of truth—each of which will be mobile. This observation leads us to the mobile parameters, made so by the presence in L of some non-truth-functional connective whose recursion clause requires shifting that parameter. As relevant to our discussion of the assertion problem, we include the following types of non-truth-functional connectives: (i) quantifiers such as Vx1, (ii) tenses such as Will:, and (m) historical modalities such as Sett:. These three types of non-truth-functional connectives force us to include the following mobile parameters, each to be shifted by the recursion clauses for one of these connectives: (i) quantifiers shift the assignment parameter, (M) tenses shift the moment-of-evaluation parameter, and (m) historical modalities shift the history-of-evaluation parameter.7 Assignment parameter. The assignment parameter provides an assignment of values to the variables of L in connection with universal quantification or another variable-binding device. Letting a be an assignment, one must define m, c, a \= A. The newly added assignment parameter must be mobile. For instance, as Tarski demonstrated, the value of x1 must be shifted in the recursion clause that explains the manner in which whether or not m, c, a1\= \/x1A depends on whether or not m, c, a2 \= A. In order to fix the truth value of Vx1A (e.g., Vx1-Fx1) at a point (m, c, a1), it is by no means enough to consider the truth value of A (e.g., Fx1) only at the point (m, c, a1) itself. One must consider also points (m, c, a2) at which x1 is assigned a different value. So, since one cannot hold the assignment parameter fixed throughout the Tarski recursion, there must be an assignment-parameter of truth, and it must be mobile—paradigmatically so. See §8A.l for how this observation relates to Tarski's own language of "satisfaction." Moment-of-evaluation parameter. The moment-of-evaluation parameter is needed for tenses. It is not enough to consider only points (m, c, a) in the truth recursion. One must instead define m, c, a, m \= A, where m is the 7

To say that "Vxi " is a non-truth-functional connective may sound strange. We think that avoidance of this plain fact is driven by the desire to support with grammatical rhetoric certain views on ontology or language, perhaps "nominalism" or "extensionalism." If you know what you mean when you say that quantifiers are extensional, and if you find it good to say so, fine. But that is no reason to ignore the fact that quantificational connectives are not truth functional—nor to conceal this otherwise obvious fact by gerrymandering grammar so that one avoids making it explicit that, e.g., Vx1 is a connective.

6. Indeterminism and the Thin Red Line

147

moment of evaluation. The moment-of-evaluation parameter must be shifted, and you cannot shift a parameter that doesn't exist. Even when structure and interpretation and context are uniquely fixed, if you have tenses, you must have a parameter to shift in their recursive processing. For instance, the moment-ofevaluation parameter must be shifted in the recursion clause that explains m, c, a, m1 \= Was:A, since that evidently depends on whether or not m, c, a, m2 \= A for some m2, where the moment of evaluation, m2, is earlier than the moment of evaluation, m1. Therefore, there must be a moment-of-evaluation parameter, and it must be mobile. History-of-evaluation parameter. The history of evaluation parameter is needed for the historical modalities. It is not enough to consider only points (m, c, a, m) in the truth recursion. One must instead define m, c, a, m, h \= A. For instance, using Sett: for historical necessity as in Definition 2-3, the history-of-evaluation parameter is shifted in the recursion clause that explains how the truth value of Sett:A at a point (m, c, a, m, h1) is a function of the truth value of A at points (m, c, a, m, h2) encoding a history of evaluation, /i2, that is another history passing through m. Therefore, there must be a history-of-evaluation parameter, and it must be mobile. In line with earlier usages, we employ a just slightly different notation: We shall write

Since we have finished adding parameters, this is our "final" notation. We bind m and h with a slash to remind us that m must be a member of h, and also to emphasize how closely moments and histories are intertwined in our understanding of indeterminism. In terms of pure logic, the three parameters a, m, and h appear to have quite the same status: We are driven to add them because of their role in understanding non-truth-functional connectives, and they are mobile. Because of the first point, we may well say that, unlike the model parameter (which combines structure and interpretation) and the context parameter, they are "auxiliary." We shall see, however, that they are related to assertion in quite different ways. Sorting that out will help us with the assertion problem.

6B.4

The context parameter and the context parameters

As we have said, the purpose of the context parameter is to represent relevant features of the context of a certain speech act, the context in which a certain expression of L is used. As Kaplan makes plain (p. 591), for each speech act there is only one context, c, but this one context has many features. Sometimes we refer to the context (or context-of-use) parameter, referring to the one context. Sometimes, however, we shall think of many context parameters (plural), each representing some one of the many features of the one context. Context parameters have two entirely different functions.

148

Foundations of indetenninism

6B.4.1

First function: Context-dependent expressions

The first function, thoroughly explored by Kaplan, is to provide a semantics for context-dependent expressions, such as the indexical "I," as already mentioned, for other indexicals such as "here" and "now," for demonstratives such as "this," and for connectives anchored in the context such as "actually." How much information the one context-of-use parameter, c, supplies—or, equivalently, how many context parameters there are—depends partly on how rich we take L to be. For example, if the indexical "here" is present in L, the context parameter must supply a place of the context of use, and if L contains "now," it must determine a time at which the speech act occurred. You cannot, however, make up features of the context as you go along. You can reasonably decide to treat "now" or "here" as context-dependent, but that is only because there is in fact a time of use and a place of use. The same cannot be said for "the integer of the context of use," since there isn't one. Nor does "the finger of the speaker of the context of use" make sense, since there are in all likelihood many such fingers. We are insisting that the context parameter provides only what Kaplan calls "fact-of-the-matter parameters" (p. 593), doubtless idealized. We are going to omit all demonstratives and most context-dependent expressions from the mini-language, L, presented here as a help in understanding indeterminism. For this reason, we shall rely very little on the first of the two functions of context parameters. Let us proceed to the second function in order clearly to see what context parameters may sensibly be deployed. 6B.4.2

Second function: Initialization

The assertion problem worries over how we can see the practice of making future-tensed assertions as consistent with indeterminism. We envisage such an assertion as made by means of a sentence considered as stand-alone. There are in contrast sentences considered as embedded (or embeddable) in some construction. The distinction is important because the two call for different semantic treatments. It is, however, difficult to keep in mind since L, in common with the typical languages for which we have a semantics, and in contrast to English, has no syntactic distinction that marks stand-alone sentences from those that are embedded. That is why we have to use semantics in order to make the contrast between sentences considered as stand-alone and sentences considered as embedded (but see Green 2000). The usage is, however, clumsy, and we use "stand-alone sentence" as short for "sentence considered as stand-alone." The semantic difference is as follows. Stand-alone sentences. We always start the evaluation of a stand-alone sentence at the context of use. If a mobile parameter can be fixed by the context, then we automatically do let the context fix it for stand-alone sentences. For example, let us parse "Meg was hungry" as Was:(Meg is hungry),

(5)

6. Indeterminism and the Thin Red Line

149

and let us consider (5) as stand-alone. We know that the past tense moves evaluation of "Meg is hungry" backward to an earlier moment; but we need an initial moment at which to start this backward motion. This initial moment is naturally identified as the moment determined by the context of use. We will say that the context initializes the moment of evaluation for sentences considered as stand-alone. This is the second role of the context parameter, adumbrated by Kaplan, p. 595, in (of course) his own vocabulary and in a slightly different connection. The first role of the context parameter is to help with context-dependent expressions, and its second role is to start the process of semantic evaluation of sentences considered as stand-alone at the very context of the speech act that uses that sentence as a vehicle. Embedded sentences. We often evaluate embedded sentences, even the very same sentences that can be used stand-alone, when the mobile parameters take us away from the context of use. For example, suppose that we parse "Meg will have been hungry" as "Will: Was: (Meg is hungry)." Then even though we start by letting the moment of evaluation be fixed by the context of use, by the time that we reach "Was:(Meg is hungry)" (which is exactly the same sentence as (5), but now considered as embedded rather than stand-alone) we are working with moments of evaluation that are in the future of the (unmoved) context of use. 6-2 POLICY. (How to treat stand-alone sentences) Technically, we express initialization-by-context by refusing to evaluate sentences considered as standalone at arbitrary points. Instead, we insist that stand-alone sentences be evaluated only at context-initialized points, where every mobile parameter that can be initialized by context in fact is initialized by context. (That is intended as a definition of "context-initialized point.") The other side of the coin is that we must permit evaluation of sentences considered as embedded at arbitrary (non-initialized) points. Otherwise the mobile parameters would be useless.

6B.5

Classification of mobile parameters

In order to approach the assertion problem more closely, we use the idea of initialization by context as a way of refining our classification of the parameters of truth. We already have the division into immobile versus mobile. Now we ask for each mobile parameter whether or not it can be initialized by context. Here is our answer: • Moment of evaluation, m: mobile, and initialized by context. • Assignment of values, a: mobile, and not initialized by context. • History of evaluation, h: mobile, and not initialized by context.

150

Foundations of indeterminism

At this point we have merely stated how the three mobile parameters are to be classified. Since this classification is crucial to the assertion problem as we eventually describe it, mere statement is not enough. Here is our reasoning. Moment of evaluation mobile, but is it initialized? Yes. The moment of evaluation is initialized by the context of use. We do not want to begin evaluation of a stand-alone sentence, A, at an arbitrary point. Because we are considering a stand-alone sentence, A, as used in a particular context, we must take context of use into account. If A contains any occurrences of, for example, "I" or "here" or "now" or the like, to make sense of A obviously requires a context of use. It is somewhat less obvious that if A contains something like the tenses Will: or Was:, we must begin the evaluation of A at the moment of use. The reason is this. To think of A as stand-alone is to think of it as a vehicle for a possible speech act. A speech act is an act, and an act is a happening. A possible happening has a unique causal position in Our World, in terms of the causal ordering, \<, so that the same must be true for a speech act: Each possible speech act, like any other happening, has its definite causal past and its definite future of possibilities. In our idealization, we fix the place of an event in the causal order of our world by means of a moment. There is nothing conventional about this. The moment of use is a "fact of the matter." To encode this technically we take two steps. First, we introduce a "moment of use" parameter, the very moment determined by the context, c, of use. We let mc be the moment of use. Since the moment of use will be the only context parameter (feature of the context) that we employ in the mini-language, L, it is convenient to replace "the context parameter" by "the moment-of-use parameter." That is, we shall write

instead of "O, c, a, m/h \= A" as in (4) on p. 147. The notation (6) will serve for every sentence, including those considered as embedded. Second, we cater to the special needs of sentences considered as stand-alone. We do this by evaluating them only at points such that the moment of evaluation is identified with the moment of use:

Observe that the moment of evaluation is identical to the moment of use; that is what we mean by a context-initialized point, the kind of point at which we must evaluate A when it is considered stand-alone. Assignment of values is mobile, but is it initialized? No. Typically we don't assert sentences with free variables. Why? Because the context of a speech act does not provide an initial assignment of values to the variables. We are not making a technical remark; of course you can define "assignment of use" as a technical concept in any way you like. In this logicians' sense, one could use context to initialize the assignment-of-values parameter. Real contexts of use,

6. Indeterminism and the Thin Red Line

151

however, do not in fact provide an initial value for variables such as x1. Not only is there no fact of the matter, but it is probable that no language on Earth subscribes to a conventional context-of-use-determined assignment of values to its bindable variables.8 There just is no such convention; and if there were, it would have no discernible purpose. We mean that there would be no purpose for the users. It is not germane that pretending that such a convention exists might make things easier for some descriptive logician. We intend to be agreeing with Kaplan 1989, pp. 592-593, which contains the only pertinent discussion known to us. Our chief point, however, is that the context is supposed to be fact-ofthe-matter, and there is no fact of the matter regarding an initial assignment of values to the variables. History of evaluation is mobile, but is it initialized? No. The historyof-evaluation parameter is not initialized by the context of use: Given indeterminism, there is no such thing as "the history of use." The point is that there is a mismatch between the idea of a speech act as a happening, and the idea of a history. On the one hand, a speech act is an ordinary momentary contingent event with an indeterminate future. On the other, a history encodes a possible course of events in absolute concrete detail that stretches far past the break-up of our little solar system. It is therefore only common sense that a speech act has many possible futures, or, in our jargon, belongs to many histories, given indeterminism. The happening of an assertion that perchance uses "The coin will land heads" as its vehicle no more determines one of these histories than any other. A speech act determines "the moment of its occurrence," but not "the history of its occurrence." So since there is no "history of the context," the history-of-evaluation parameter cannot be initialized by context. More accurately, the history of evaluation cannot be initialized by a fact of the matter. Perhaps there could be a language community that adopted a convention according to which the history parameter was initialized as some named history, say Fred, or as some contextually salient history, perhaps (if it makes sense) the history most likely to occur, given what most people in Boston now believe. But we ought to conclude that for serious philosophical studies of indeterminism, evaluation of future-tensed statements should not be entrusted to such a conventionally specified history of evaluation. In appraising our claim that there is no history of the context, one must be aware of a source of false intuitions. When one is postulating possible worlds in the style of Kripke or Lewis, one normally also postulates that "there is no overlap" between worlds. That makes it sensible to speak of "the world of the speech act." Since histories resemble worlds, it is easy to forget that in this 8

We think we are not cheating when we disallow as a counterexample the use of, e.g., the pronoun "he" in English both as bindable by quantificational constructions such as in "If a mathematician is worth his salt, he can solve cubic equations," and also deictically as in "He can solve cubic equations" (said while pointing). By an "assignment" we mean an assignment to (all of) the variables, and by a "variable" we mean an expression that has only the bindable use. Of course a language can employ the same sign design in various capacities, as English does. But "xi", in contrast to English "he," has only the one function.

152

Foundations of indeterminism

crucial respect histories are not like possible "worlds": Unlike worlds, histories overlap, so that a single speech act will typically belong to many possible histories; and that is why the phrase "the history of the speech act" is impermissible. These points will become sufficiently central to our treatment of the assertion problem as to warrant highlighting. 6-3 SEMANTIC THESIS. (Initialization) (i) The moment-of-evaluation parameter is initialized by context, (ii) The assignment-of-values parameter is not initialized by context. (iii) The history-of-evaluation parameter is not initialized by context. In one sense, these statements are true by convention, since we built their truth into the design of our mini-language, L. We nevertheless label them "theses" because at a deeper level they are debatable, and we have given arguments in their favor.

6B.6

Semantic analyses

We still have not said all we need to say about the meaning and use of these parameters of truth, but the rest will make more sense if we first provide some illustrative semantic analyses using the scheme (6). These analyses are parts of a standard recursive definition of truth as given already in §2A.l, all of which are explained in more detail in §8F. Quantifiers. m, mc, a, m/h \= VxiA iff m, mc, a1, m/h \= A for every assignment a1 that does not differ from a except perhaps at the variable xi. Tense connectives. m, mc, a, m/h \= Was:A iff there is a moment m1 in the past of m such that SOT, mc, a, m1/h \= A. Also m, mc, a, m/h \= Will:A iff there is a moment m1 in the future of m along history h such that m, mc, a, m1/h \= A. (Tenses never move off h.) Temporal connectives. m, mc, a, m/h \= At-instt,:A iff there is an instant, i, in Instant such that the denotation of term t at the point m, mc, a, m/h is i, and such that, where m1 is the (unique) moment in i that lies on history h, m, mc, a, m1/h \= A. ("Chase up or down the history h until you hit the exact moment where h crosses the instant named by t; that moment is where you must evaluate A (with respect to h).") For this to make sense, Instant must be a subset of Domain—as indeed it should be. Settled truth and historical possibility. m, mc, a, m/h \= Sett:A iff for every history h1 to which m belongs (that is, for every possible future for m), m, mc, a, m/hi \= A. For Poss:A, change "every" to "some."

6. Indeterminism and the Thin Red Line

153

Context-anchored connectives. m, mc, a, m/h \= Now:A iff m, mC, a, m1/h \= A, where m1 is the unique moment at which the history of evaluation, h, meets the unique instant to which the moment of use, mc, belongs. ("Now" is the time of the moment of use.) Also m, mc, a, m/h \= Actuallyl:A iff m, mc, a, m c /h1 \= A for every h1 to which mc belongs. The actuality connective has a subscript to conform to §8F.7, where two distinct actuality connectives are introduced. Actuallyl:A could be read as "It is settled true at this actual moment that A." Examination of these analyses makes it easy to see which clauses force which parameters to be mobile.

6B.7

Open and closed for stand-alone sentences

One can better appreciate the nature of our three mobile parameters and the problems that they cause for assertion if we introduce a little more jargon. (We regret that some of the jargon is not as memorable as one should like.) 6-4 DEFINITION. (Independent, dependent, closed, and open in parameters) • A point, ( z 1 , ..., zn), bears witness to the Zi-dependence of A iff ( z 1 , ..., zn) is such that the truth value of A varies as Zi is (legally) shifted away from Zi while all other parameters are held fixed. A sentence, A, depends on a parameter, Z, iff some point bears witness to the Z-dependence of A. We also say that A is Z-dependent. A sentence, A, is independent of a parameter, Z, iff A does not depend on Z. We also say that A is Z-independent. • A sentence, A, considered as stand-alone is closed in a parameter, Z, in two cases: Either A is independent of Z or Z is one of the parameters that can be initialized by context. We also say that A is Z-closed. We frequently use the following terminology for the two different ways in which A, considered as stand-alone, can be closed at Z: Either (i) A is closed by independence or (ii) it is closed by initialization. A sentence, A, considered as stand-alone, is open in a parameter, Z, iff two conditions are satisfied: Both A depends on Z and Z is not initialized by context. We also say that A is Z-open. In still other words, A is neither closed by independence nor closed by initialization. Note a slippery but critical detail of our usage: "Independence" and "dependence" can apply to sentences however considered, but we reserve "closed" and "open" for sentences considered as stand-alone. Keep in mind that the definition of "closed" is disjunctive: by independence or by initialization. It is evident that virtually all interesting sentences depend on the immobile model parameter, m, and usually also on the context parameter, mc. For the assertion problem, however, we need be concerned only with the relation of stand-alone sentences to mobile parameters. To help the argument, we think of the following three easy sentences as part of L.

154

Foundations of indeterminism Meg is hungry.

(8)

x1 is brindle.

(9)

Will:the coin lands heads.

(10)

Referring to these sentences as paradigmatic examples, we claim the following. 6-5 SEMANTIC THESIS. (Open stand-alone sentences) i. A sentence such as (8), considered as stand-alone, is closed in all three of the mobile parameters of L. ii. A sentence such as (9), considered as stand-alone, is open in the assignment parameter. iii. A sentence such as (10), considered as stand-alone, is open in the historyof-evaluation parameter. These three semantic theses need arguments, but they can be short. Ad ( i ) . (8) is closed by independence in the assignment parameter (no free variables to interfere with independence) and the history parameter (whether Meg is hungry does not depend on what happens in the future), and (8) is closed by initialization in the moment-of-evaluation parameter. So (8), considered as stand-alone, is closed in all three mobile parameters of L. Ad ( i i ) . That (9) depends on the assignment parameter is evident from standard first-order semantics: Given an understanding of "is brindle" from m, there are context-initialized points such that the truth value of (9) at those points depends on the value of x1, and therefore depends on the assignment parameter. Furthermore, from Semantic thesis 6-3 we know that the assignment parameter is not initialized by context. In other words, (9) is not closed by independence and is not closed by initialization in the assignment parameter. Therefore, by the account of "open" given in Definition 6-4, (9), considered as stand-alone, is open in the assignment parameter. Ad (in). That (10) depends on the history parameter is what sets our "assertion" problem. Pick a context-initialized point such that the moment of use in question is earlier than the relevant toss of the coin, and such that the toss is genuinely indeterministic in the sense of §6A.3. We are also supposing that this indeterminism is well represented (up to idealization) by our branching account of Our World and its causal order as in §6A.4. Finally, we are assuming the account of "Will:" that we gave most recently in §6B.6. It takes all of this to justify the thesis that the truth of (10) depends on which of the many possible historical continuations, starting from the context of use, is given as the value of the history parameter. That is to say: (10) depends on the history parameter. Furthermore, Semantic thesis 6-3, for which we gave arguments, says that the history parameter cannot be initialized by context. Therefore, since (10), considered as stand-alone, is neither closed by independence nor closed by initialization in the history parameter (Semantic thesis 6-3), (10) is by definition open in the history parameter.

6. Indeterminism and the Thin Red Line

155

So the assignment and the history parameter are deeply alike: Each of these two parameters is mobile but uninitialized by context, and each offers examples of sentences that, when considered as stand-alone, are open. In the service of setting the assertion problem, we take our semantic claims just one step further. We claim: In the absence of a specific convention, if a stand-alone sentence, A, is open in a certain (mobile and non-initialized) parameter, Z, then there is at least one context-initialized point such that it makes no sense to say that A has a truth value at that point independently of Z. In carefully controlled but more colorful language, we express this claim by saying that if a stand-alone sentence is closed neither by independence nor by initialization with respect to some parameter, Z, then "it has no truth value" at some context-initialized point save relative to Z. More generally, if there is any parameter relative to which a stand-alone sentence is open, we say that by definition the sentence "has no truth value" at any context-initialized point that bears witness to the openness. Use of the expression "has no truth value" is dangerous; we try to minimize the danger of misleading by the upcoming "Observation," and by emphasizing here that the phrase is defined. 9 Because of the definition, we have the following. 6-6 SEMANTIC THESIS. (Stand-alones open in either the assignment or the history parameter sometimes have no truth value) Fix the model, m, and the moment of use, mc. This suffices to give (8), considered as standing alone, a truth value: "m, mc \= (8)" makes sense. So (8) "has a truth value." But m and mc do not suffice to give a truth value to sentences, considered as stand-alone, that are either assignment-open like (9) or history-open like (10), even if their evaluation is restricted, as is appropriate for stand-alones, to context-initialized points. Sentences like (9) or (10) may well "have no truth value." In symbols, i. "m, mc \= Meg is hungry" makes sense since "Meg is hungry" is closed in each of the mobile parameters. We could indeed define "truth" given model and context as follows: For every (or some, or most, or your favorite) a and h, m, mc, a, mc/h \= Meg is hungry. ii. "m, mc \= x1 is brindle" does not make sense since ux1 is brindle" is open in the assignment parameter. The definition proposed in (i) would of course be formally correct; our complaint is that it would not be "materially adequate." Note in particular that in contrast to (i), since "x1 is brindle" is open in the assignment parameter, here it would make a difference whether one framed the definition with "every" or "some" or "most" or "your favorite." In short, it can happen that (9) "has no truth value." iii. "m, mc \= WilL:the coin lands heads" does not make sense. (Ditto.) 9 We do not prejudge whether or not there is a useful connection between our defined idea of "no truth value" and other notions such as those used in cases of presupposition failure, vagueness, ungroundedness, etc.

156

Foundations of indeterminism

We intend Semantic thesis 6-6 to be an obvious consequence or rewording of Semantic thesis 6-5. We have put it this way in order to make an observation that helps us with our approach to the assertion problem. OBSERVATION. (No gaps) To say that open sentences such as (9) or (10) "have no truth value" is not to say that open sentences have some third truth value, or a third special status.10 It is even a mistake to think of open sentences as introducing "truth value gaps" that might sometimes be filled by introducing "supervaluations."11 No one makes these mistakes about assignment-open sentences. As is evident from our strictly parallel treatment of the semantics of assignment-open and history-open sentences, we think that they should never be made. As a preventative against the temptation to start speaking of "truth value gaps," it is healthy to intone that "given a model and a context, an open sentence always has a truth value—once a suitable value is supplied for each parameter in which it is open."12

6C

The assertion problem

In Semantic thesis 6-3, Semantic thesis 6-5, and Semantic thesis 6-6, we have stated some claims, for which we have argued, about the semantics of a language like L designed to illuminate indeterminism. We have stressed the semantics of open sentences considered as stand-alone. These claims are all relevant to assertion as we understand it, but observe that none of our semantic theses 10

It seems to us that Lukasiewicz 1920 and others since have made this mistake about future-tensed sentences, as 0hrstr0m and Hasle 1995 have argued. Consider also the view of Dummett 1991 that no theory is a form of "realism" if it claims that sentences about the future fail to be "determinately either true or false" (p. 7). If "determinately" is read as "settled," the present version of branching time—which was not available for Dummett's consideration—certainly involves that very claim; but it is hard to find a theory that is more explicitly realist than ours. 11 It seems to us that Thomason 1970 makes this mistake. We do not, however, concur with the diagnosis of 0hrstr0m and Hasle 1995, who find a problem in explaining how the disjunction of two "indeterminates" such as "Will:(the coin lands heads)" and "~Will:(the coin lands heads)" could fail itself to be "indeterminate." That article shares the common presupposition that, without a Thin Red Line, branching itself commits one to "the rejection of bivalence" and the inability to employ truth functions (p. 193). The analogy with quantification helps here to lessen the felt need for the presupposition: Evidently there is no problem in labeling "x1 is brindle" and "~(x1 is brindle)" each "indeterminate," whereas their disjunction, being an excluded middle, is infrequently said to be "indeterminate." And certainly there is seldom a call to suggest that the principle of bivalence, or the usefulness of truth functions, fails for the likes of "x1 is brindle." 12 It is all right and even beneficial to say, in ordinary quantification theory, that even given a model, an open sentence such as "x1 + x2 = x3" has no truth value. The mistake that (to our knowledge) is never made is to jump from this observation to think that quantification theory forces philosophy to formulate a theory of "truth value gaps," or of "supervaluations," or of a third truth value that could play a role in truth table calculations. A related mistake that is never made in the case of quantification theory is to think that because its open sentences "have no truth value," quantification theory is somehow wrong (or antirealist, or whatever). Our claim is that the semantics of history-open sentences should be equally free of these mistakes.

6. Indeterminism and the Thin Red Line

157

mentions assertion. They are purely semantic. Now we state some claims about how various kinds of open sentences relate to assertion. In the following, when we speak of "asserting A" we intend it as short for "using A as a vehicle of assertion, so that what is asserted is expressed by A." 6-7 ASSERTABILITY THESIS. (Which stand-alone sentences are assertable?) When we ask in the following whether or not a sentence is "assertable," we do not mean anything like "warranted" or "correctly assertable." These meanings are not what is wanted for our assertability thesis. i. Triply closed sentences such as (8) "Meg is hungry" are (paradigmatically) assertable. It makes perfect sense to use one as the vehicle of an assertion. ii. Assignment-open sentences such as (9) "x1 is brindle" are not assertable. It makes no sense to use one as the vehicle of an assertion. hi. History-open sentences such as (10) "Will:the coin lands heads" are genuinely assertable. It makes perfect sense to use them as vehicles of assertions. These claims do not go without saying; we argue for them. Ad ( i ) . The assertability of triply closed sentences is paradigmatic. Semantically, m, mc \= A makes sense for these sentences, so that they have a clear model- and context-dependent truth value, a fact that seems to warrant their eminent assertability. Ad ( i i ) . It makes no sense to assert an assignment-open sentence such as "x1 is brindle." It is literally senseless to use that sentence as a vehicle for assertion. To put the matter a little more circumspectly, such anassignme open sentence is not a proper vehicle for assertion. A purported assertion using such a sentence would be defective. The defect is radical: It is not just that no one could know what was asserted; more than that, nothing is asserted by making such an utterance. Observe that since (ii) is about assertion rather than just semantics, it does not follow from Semantic thesis 6-3, Semantic thesis 6-5, and Semantic thesis 66. We lay (ii) down as a new fact, a fact that describes our practice of assertion (as opposed to mere utterance). No one would understand you if you said, using direct quotation, "Jack confidently asserted the sentence, 'x1 is brindle'."13 13 Red herrings abound. There could be a special convention permitting, e.g., dropping universal quantifiers on stand-alone sentences used as vehicles of assertion, and indeed there is such a convention among some groups of mathematicians and other technical workers. The convention is, however, "merely conventional," requiring addition to our underlying agreements about semantics and the practice of assertion. Furthermore, it remains the case that the convention covers only a few cases. Indirect discourse would offer a separate topic. "Jack asserted that x\ is brindle" makes no sense as a stand-alone sentence (since that sentence is itself assignment-open), but it might make some sense if it is embedded in a quantificational connective: "For some x1, x1 is owned by Mary, and Jack asserted that x1 is brindle." Switching from asserting to uttering is yet another way to change the topic. Of course, using direct discourse, we can truthfully say that Jack uttered the sentence, "x1 is brindle,"

158

Foundations of indeterminism

Ad ( i i i ) . In contrast to the senselessness of claiming that an utterance of an assignment-open sentence could be an assertion, there is, we claim, no radical defect in asserting a typical history-open future-tensed statement such as "The coin will land heads," even under conditions, even known to the speaker, of radical indeterminism. In appropriate circumstances such an assertion is called a prediction. Although we might raise questions about justification and the like, what is predicted by the use of "The coin will land heads" is perfectly well determined by model and context alone, even though on our account the truth of the prediction depends on the history parameter. To put the point in indirect discourse, we are saying that "Jack asserted that the coin will land heads" makes good sense, even if we add that he was not justified in doing so. An analogy with betting may help. It makes no sense to bet on "x1 is brindle," while to bet on "the coin will land heads" seems a paradigm of intelligibility. We are saying that assertion is like betting. It makes sense to assert, or to bet on, the truth of a sentence that is open in the history parameter, but it does not make sense to assert, or to bet on, the truth of a sentence that is open in the assignment parameter. We are finally ready to formulate the assertion problem as we see it, as Assertion problem 6-8(iii). 6-8 ASSERTION PROBLEM. (Tension between semantic and assertability claims) Items (i) and (ii) are background; the problem is stated in ( i i i ) . i. Our semantic account of triply closed sentences jibes with our assertability claims: They do have a definite truth value (given model and context), and they are paradigmatically assertable. ii. Our semantic account of assignment-open sentences fits with our assertability claims: They "have no truth value" (given only model and context) and they are not assertable. iii. Our semantic account of history-open sentences seems severely in tension with our assertability claims almost to the point of apparent contradiction: Like assignment-open sentences, history-open sentences "have no truth value" (given only model and context). But if they have no truth value, it would seem that they would be no more assertable than assignmentopen sentences. After all—if we may be permitted language known to be untrustworthy—it is certain that to assert A is to assert that A is true. We nevertheless claim that, in spite of "having no truth value," it is indeed proper to assert a history-open sentence, and even to assert that it is true. The assertion problem is to eliminate the apparent tension reported in (in). but even in direct discourse we cannot truthfully say that Jack asserted that sentence. (It seems to us that the philosophers' use of "Jack said, 'x1 is brindle'" waffles between mere uttering and asserting, and should therefore not be permitted in discussions such as this.) Another topic would be what happens if we replace "x1" by an English "it," for in that case demonstrative or other conventions are likely to kick in.

6. Indeterminism and the Thin Red Line

159

In the end we confront the assertion problem head on, but first we discuss three tempting ways to avoid the problem. These all have in common the view that, contrary to Assertability thesis 6-7(iii), it is not in fact possible to assert a history-open sentence, precisely on the grounds that even given model and context, a history-open sentence has no truth value. Way number one admits that our language contains (something like) historyopen sentences, but argues that they are not assertable. The reasoning might be that the practice of assertion makes no sense unless assertors are reliable, so that it is not part of the practice to assert in the face of indeterminism. Or it might be that assertors should warrant what they assert, and they cannot do that when making a risky prediction. Or it might be that it sounds funny to the ear to say "The coin might or might not land heads, and I assert that it will." All of these suggestions, and their cousins, seem at first glance reasonable, and one could doubtless speak at length on one side and the other, citing details of our practice of assertion and (for the funny-to-the-ear argument) conversation. We do not, however, think the discussion would advance our investigation, since the concepts involved are too soft to help us with understanding a hard concept of indeterminism. The second and third ways of avoiding the assertion problem have something in common: Both claim that our language (or any sensible language?) is free of history-open sentences. Because of the conjunctive nature of our account of "open" in Definition 6-4, two such ways are possible. Way of avoidance number two works as follows: It claims that our language is entirely free of history-dependent sentences, whether stand-alone or not, even given indeterminism. The most common form that this claim takes is the Peircean view that future-tensed statements are intrinsically closed by independence. Put in our terms, it is the claim that Will:A really means Sett: Will:A. If that were true, one would have the advantage of being able to drop the history parameter altogether, since the Peircean future tense can be understood without it: 6-9 DEFINITION. (Peircean future tense) m, mc, a, m \= Will:A iff for every history, h, through m, there is a moment, m1 that lies on h and is later than TO such that m, mc, a, m1 \= Will:A. (There is quantification over histories, but there is no parameter for histories.) The trouble with the Peircean account of " Will:" is that it makes no sense of someone who purports to assert that the coin will land heads even though it might not, that is, who sincerely asserts both that Will:A and that Poss:~Will:A. See the discussion of "Antactualism" in Burgess 1978 for an elaboration of this view. A variation of the Peircean position is offered in McArthur 1974, according to which there are in general two things that could be meant by a use of Will:A: either the Peircean Sett:Will:A or Poss: Will:A. According to this view, which of these two things is meant depends upon the speaker's intentions. We believe this does not help.

160

Foundations of indeterminism

One of the most compelling arguments against the claim that no sentence in our language "really" is history-dependent comes from the analogy with betting. A strict Peircean would have to say that I bet that the coin will land heads

(11)

really means I bet that present circumstances determine that the coin will land heads.

(12)

Surely (11) and (12) are different bets. They are different because one who makes the bet expressed in (11) wins if, and only if, the coin in question lands heads. However, this is not true of the bet expressed in (12). One who makes that bet can lose even if the coin lands heads. If it can be shown that it was historically possible that, as of the time of his speech act, that coin not land heads, then the bettor who speaks in (12) loses. The third way of avoiding the assertion problem is also a way of denying that our language contains any history-open sentences. The third way also drops the history parameter. On this view, the particular history required for evaluating a future-tensed statement is supplied by the context of use, or even by our world itself, independent of context: the Thin Red Line (§6A.2). In the impending §6D, we argue at length against this tempting way to avoid the assertion problem. Then, in our final section, §6E, we directly confront the assertion problem, offering our positive suggestion for its solution. We show how a natural understanding of the speech act of assertion makes comfortable sense out of the practice of asserting historically open sentences, and does so in a fashion that hangs on both to Semantic thesis 6-5(m) that some stand-alone sentences are open in the history parameter and therefore have no truth value (Semantic thesis 6-6(iii)), and also to our Assertability thesis 6-7(iii) that history-open sentences are properly assertable. Having half an eye on the parts of those theses that claim that assignment-open sentences are for that very reason not assertable, we suggest exactly how, in relation to our practice of assertion, being history-open is fundamentally different from being assignment-open. To the extent that we are successful in these enterprises, attempting to avoid the assertion problem seems less inviting.

6D

The Thin Red Line

We turn now to consider the third way of avoiding the assertion problem, namely, the Thin Red Line. In §6A.2 we described and motivated the Thin Red Line as best we could with examples; it might be worthwhile at this point to consult those paragraphs. Here we need to be more abstract. Consider the following. Sure, there are many things that might happen, but only one of them is what really will happen. (Que sera, sera.)

13)

6. Indeterminism and the Thin Red Line

161

If one has objective leanings at all, it is easy to feel a need to treat both conjuncts with equal objectivity. It seems good to try to combine objective indeterminism with an objective actual future. One is thereby tempted to continue to represent objective indeterminism by postulating that our world (up to an idealization) is treelike, but to hold in addition that there is a distinguished history, the Thin Red Line, which we abbreviate as TRL. One may posit a TRL without shifting from an objective to a subjective construal of indeterminism (a contrast discussed in §6A.3), and we must understand the TRL proposal in this objective way. The proposal succeeds in avoiding the assertion problem by postulating an "actual history" in addition to the "moment provided by the context of use." Parametrically speaking, one has a point something like (m, TRL, mc, a, m), where we specifically note (i) that the TRL parameter is immobile, and (M) that the mobile history parameter is omitted. In the semantic theory of branching with a TRL, the future tense, Will:A, moves you forward along TRL, the past tense, Was:A, moves you backward along it, and the temporal connective, At-instt:A, moves you to the moment lying at the intersection of the TRL and the instant denoted by t. For instance, for the future tense we have 6-10 DEFINITION. (TRL account of Will.A) m, TRL, mc, a, m \= Will:A iff there is a moment, m1, that lies on TRL and is later than m and is such that m, TRL, mc, a, m1 \= A. The semantics of "actually" is likewise bound to the TRL itself. Talk of possibility or necessity or inevitability is explained without reference to a history parameter by means of quantifications that involve moments that lie off the TRL. For this purpose we introduce two new connectives, Settled-will:A and Possibly-wilLA that take over the jobs formerly performed by Sett: Will:A and Poss: Will:A. 6-11 DEFINITION. (Possibly-wilLA and Settled-will:A) •m, TRL, mc, a, m \= Possibly-wilL: A iff for some m1 such that m < m1 (whether or not m1 lies on TRL), m, TRL, mc, a, m1 \= A. • m, TRL, mc, a, m \= Settled-will:A iff in every history, h, through m, there is a moment, m1, later than m such that m, TRL, mc, a, m1 \= A. The definiens of Settled-will: mimics the definiens of the Peircean account of Will: given in Definition 6-9. (We might also have given that same definition to the connective, Inevitably:, of §8F.6.) Unvarnished Will: sticks with the TRL. The assertion problem is thereby avoided. Very likely there are endless possible versions of TRL theory; we shall consider only a few simple ones. In what follows we shall reserve plain "TRL" for generic use, which will permit us to distinguish the particular versions by subscripting the TRL acronym. In each case we try to make clear both our misgivings about the "logical" or linguistic explanations offered by the version, and our doubts about the extra-linguistic commitments of the version.

162

Foundations of indeterminism

Figure 6.1: Our World with TRLabs

6D.1

Absolute TRL

One version we call "branching with absolute TRL" or more briefly, "branching + TRLabs" TRL abs represents the actual history, the one and only actual history in all of Our World. If you metaphorically stand outside Our World, you will see it clearly marked in "red" as in Figure 6.1.14 The branching + TRLabs theory sounds all right, but it is not. It has the "logical" defect that it gives no account whatsoever of predictive speech acts occurring at moments of use that lie off the TRL abs and is by so much useless. Suppose Will: A is considered as stand-alone, and so evaluated at a contextinitialized point (m, TRL abs mc, a, mc). We are in trouble if mc does not lie on TRLabs'- By Definition 6-10, Will: A is always false at such points. (Evidently it does not help to make it a presupposition of Definition 6-10 that mc lies on TRL abs) We have no trouble with predictions that will be or have been made, but we have no way of understanding predictions that might have been made. We have no way of getting a grip on "Had things gone otherwise, Jack would have asserted the following: "It will (eventually) rain." Given the context of Jack's assertion, the TRL is no longer able to guide us in understanding his reference to his future. We put a more objective complaint in the form of a question: What in the structure of our world could determine a single possibility from among all the others to be "actual"? As far as we know, there is nothing in any science that would help. To the extent that scientific theories require objective possibilities for the future, there is no hint that those theories pick out a Thin Red Line. (Of course you can get a physicist to say (13); but that is not a piece of sci14

"To stand outside Our World" has a clear nonmetaphorical meaning, namely, to use only sentences whose truth value depends only on m = (G, 3), being totally independent of context as well as independent of the mobile parameters. Quantificationally closed sentences based on quantifying over moments and histories of Our World, using \< as the only nonmathematical predicate, and perhaps containing one or more proper names of moments or histories, would qualify for this class.

6. Indeterminism and the Thin Red Line

163

ence.) If we cannot find anything in our world to ground the idea of a uniquely "actual" history, perhaps we should go outside our world, say to a Leibnizian God who might have some basis for such a choice; and indeed it might seem as if theological concerns might well direct one's feelings toward the TRL. That would, however, be a mistake: TRL theory presupposes mdeterminism, and should never be confused with determinism. According to Prior, for example, it is determinism itself that Augustine, Luther, Calvin, Pascal, Barth, and Brunner all defended "in the name of religion." See Brauner, Hasle, and 0hrstr0m 2000, who also point out that a theologically inspired (and later abandoned) determinism is what first led to Prior's interest in tense logic. In any event, our own view is that if a theologian wrestles with what is required by indeterminism, he or she need not and should not invoke a TRL. The TRL abs theory also has troubles with actuality. (Thomason 1984, p. 145 and p. 160, makes remarks that the argument of this paragraph may be seen as elucidating.) As Lewis has argued (Lewis 1970), this world's being the actual world does not favor it over any others, but is just a reflection of the fact that this is the world at which we are conversing. To suppose that there is one from among the histories in Our World that is the absolutely actual history is rather like purporting to stand outside Lewis's realm of concrete possibilia and pointing to the one that is actual. But this is wrong in both cases. Additional dismay accrues from asking how we could know whether we are on TRL abs. How could we find out? Perhaps by seeing whether or not we eventually find out "what will happen next Tuesday"? That seems to make a certain sense, since it seems that, according to the theory, we might not find out what will happen next Tuesday. What will happen will of course happen, but does that recipe include a guarantee that TRL abs is our future?

6D.2

Context-dependent TRL

Having found wanting the TRL theory that adds to branching time a postulate that one history is absolutely actual, independent of context, it is natural to try next doing without the fancy that there is a Thin Red Line given once and for all. Replace this with a context-dependent theory asserting that each possible context of use has its own Thin Red Line, of course assuming that the moment of use belongs to the "TRL of use." We call this "branching with context-dependent TRL," or more briefly, "branching + TRLC." Technically, we can still parameterize truth in about the same way, (m, TRLc, mc, a, m); but now we move TRLC with the context, requiring that mc € TRLC. We may say that TRLC is "the actual history of the context of use." (Observe that the TRLC parameter continues to be immobile, as are all context parameters.) Making TRLC context-dependent will indeed solve the problem of predictions. When Will: A is considered as stand-alone, we are evaluating it at (m, TRLc, mc, a, mc). Hence the moment of evaluation, having been initialized as the moment of use, must belong to the TRL of use, TRLC. That sounds all right, but it is not. Our "logical" objection is this. If Will: A is a sentence, then it can occur not only stand-alone, but embedded. When it is

164

Foundations of indeterminism

embedded, we can no longer assume that the moment of evaluation is initialized as the moment of use, and therefore we cannot assume that the moment of evaluation lies on TRLC. The trouble occurs immediately if we embed Will: A inside, for example, Possibly-will:, a connective that according to Definition 611 is bound to carry us to a moment of evaluation that does not lie on TRLC. We shall then be unable to understand Possibly-will: Will: A. One might think of responding to this difficulty by redefining Will: so that on a moment off the TRL, Will: takes on the meaning of Possibly-will:. Adding that epicycle, however, will solve only the one embedding problem; it will help not at all with an endless variety of other embeddings that one ought to consider, so that also in this respect the TRL is useless. There are also sources of objective doubts. In §6B.5, which should be consulted, we gave our reasons against the idea of a "history of the context." Because the matter is important, we restate and add a bit here (see also Question 8-4). TRLC is supposed to be the TRL of use, a feature of the context of speech. It seems a mystery, however, just how the context of our modest speech act could determine the exact course of world history from now on, long past the dissolution of our galaxy. The phrase "our history" does not make sense, unless determinism be permanently true, for if it is not, our moment—the event that is the (idealized) context of this communication—is part of many possible histories, all of which are equally marked as "ours." A context of utterance can determine much, including a speaker, an auditor, a moment, a past, a focus of attention, and so on, but it cannot determine a unique history of which it is a part, for (unless determinism be permanently true) there are too many histories of which it is a part. No conversational device, whether artificial or rooted in ordinary language, can overcome this difficulty, any more than any charitable story about the use of "our child" can make it true that each couple has exactly one child. The point is not that we cannot take the trouble to uniquely specify some history or other; maybe we can, and if so we can certainly name it anything we like, say "Kronos" (taking a cue from Plantinga, God Freedom, and Evil, p. 43). Nor is the point that the conversational situation cannot render some one history "salient" for the participants. The point instead is that nothing can specify the (unique) history to which we and this context of utterance belong because (unless determinism be permanently true) no uniqueness is to be had. Furthermore, the TRLC theory also has difficulty with actuality. Do not be misled by the fact that the TRLC theory may be considered indexical. For suppose you agreed, in accordance with our remarks in §6D.l, that "the actual world" does not, in Lewis' framework for modality, privilege any one world over any other. Then we hope you will agree that from among the Lewisian worlds that are exactly alike up to a given time, no one of them has a firmer claim to actuality than any of the others. It would then be most natural as well to say that at a given indeterministic moment of use, mc, there is no privileged actual history or future from among those on which mc lies. Lastly, even though you can be sure from indexicality that you now live on TRLC, it seems a question whether the theory allows that you might not be on it tomorrow.

6. Indeterminism and the Thin Red Line

6D.3

165

Moment-of-evaluation-dependent TRL

There seems only one reasonable next try at a solution to the problem of embedded occurrences of Will:A; namely, to move to a theory that makes the Thin Red Line a function of the moment of evaluation. Technically, we change TRL from a simple name of a history to a function, TRLfcn, which picks out a unique Thin Red Line, TRLfcn (m), for each moment, TO. We call this "branching with a TRL function," or more briefly, "branching + TRLfcn." Parametrically speaking, we evaluate at points (m, TRLfcn, mc, a, m), thinking of TRLfcn as a function from moments to histories that is objectively determined by the very structure of our world, and thus as an immobile part of the "standard" model. (We do not need a contextual version of this theory, since TRLfcn(mc) already gives us a "history of the context of use.") We seem to have a theory that both avoids the assertion problem and also gives a good account of embedded occurrences of Will: A by tracing forward from the moment of evaluation, TO, on TRLfcn(m). Perhaps, however, that piece of problem-solving goes a little too fast. Note that as so far described, branching + TRLfcn theory has put no conditions whatsoever on TRLfcn. To solve the problem of how to evaluate Will: A at an arbitrary moment, it is evident that we must at least postulate that

If we read TRLfcn(m) as "the actual history from the perspective of TO," it seems both technically essential and intuitive to make m part of that history by means of (14). Then when we are given a future tense, Will: A, to evaluate at m, we trace forward on TRLfcn(m). Once we start thinking of conditions imposed (by the very nature of our world?) on the TRLfcn function, a dilemma seems to threaten. One either postulates that

or not. Suppose, first, that we postulate (15). Then let m1 admit two incomparable future possibilities, m2 and m3. First use (15) to calculate that

Also argue by (14) that m2 6 T R L f c n ( m 2 ) and m3 E TRL (m3). But then by (16), m2 € TRLf c n (m3). Therefore, both rn 2 and m 3 belong to TRLf c n (m3). But this contradicts that TRLfcn(m3) is a chain (since m2 and m3 were supposed incomparable), and therefore contradicts that TRLfcn(m3) is a history. Suppose, second, that we do not postulate that comparable moments determine the same actual history. Then when we attempt to nest tenses, we obtain several unreasonable results. For example, one loses the following natural and intuitive implications present when standard tense logic is applied to linear time, structured by standard properties normally attributed to time: transitivity, no first moment, and no last moment.

166

Foundations of indeterminism

Figure 6.2: Branching with TRLfcn

i. Will: Will: A implies Will: A, ii. A implies Was: Will: A. These both fail without (15). There is, however, an entirely natural way to regain (i) without conceptual cost. P. 0hrstrOm pointed out to us, in correspondence, that one can replace the offensive (15) by a weaker cousin:

This postulate is weak enough to avoid the trouble caused by (15), but strong enough to preserve (i). (17) is furthermore conceptually natural for a branching + TRLfcn theory, so natural that it seems best to think of it as an essential part of that theory.15 Figure 6.2 gives a sense of what the theory calls for. In this picture, TRLfcn(m) is indicated by a "red" line beginning at the various moments, m. Note that each moment belongs to exactly one "red" line, and that it is only after a "break" that a new TRLfcn(m) can be started. This means that Will: Will: is bound to behave properly (given "standard" properties of each history). Our "logical" complaint about branching with TRLfcn therefore reduces to the loss of ( i i ) , which still fails even in the presence of (17). To see why one should regret the loss of this implication, consider Figure 6.3.The picture indicates that the coin is flipped at mo, where heads will be the outcome (at m1) but tails might be the outcome (at m2) Now picture Jack at the moment of use, m2, where the coin landed tails at 2:00 P.M. It would seem that in order to speak truly at m2, Jack would be obliged to say The coin has landed tails, but this is not what was going to happen at 1:00 P.M. At 1:00 P.M. the coin was going to land heads. It's just that it didn't. 15

The point is clearly made and defended in Brauner, 0hrstr0m, and Hasle 1998. We are also reminded by these workers that (17) was explicitly introduced by Thomason and Gupta 1980, and used in another form in McKim and Davis 1976.

6. Indeterminism and the Thin Red Line

167

Figure 6.3: Tails does not imply Was: Will:Tails

In symbols:

This is an odd thing to be forced to say. As Diodorus would have it, if the coin lands heads, then the coin's landing heads is what was to be.16 McKim and Davis 1976 sketches another variation on the TRLfcn idea according to which each moment provides a future, rather than a history, where (putting the idea in our own terms) / is a "future history" in the sense of Def. 6 rather than a entire history, including a past. An analog of the problem that we posed for branching with TRLfcn arises for the McKim-Davis view. To make clear where we are so far, we have seen that shifting from branching with absolute TRL to branching with context-dependent TRL solves the problem of evaluating (stand-alone) predictions off the TRL; and we have seen that the additional shift to branching with TRLfcn solves the problem of evaluating embedded future-tensed sentences. We have also seen that the shift does so in a fashion that, under reasonable assumptions, saves much of linear tense logic, while failing to save all of it. That A implies Was: Will: A is left in the dust, as is that At-insti.oop M -'At-mst2 oo p.M.:A is equivalent to At-inst2 oopM :A. One might also point out that although on this theory one does not have that Will:A implies Settled-will:A (that something will happen does not imply that it is inevitable that it will happen), nevertheless one has that Will:A implies Settledwill: Was: Will: A. That something will happen does indeed imply that it is inevitable that it will be true that it was going to happen. (Please forgive our contorted English.) We do not wish to rest our case, however, entirely or even chiefly on logical or linguistic oddity. For one thing, the history of contemporary semantics testifies that someone can always cook up a definition of "point of evaluation" and some semantic clauses that will legitimate any desired technical result. We want our semantic investigation to be not like that. We are trying to stick to structure 16

It is simple tense logic, not "fatalism," that A implies Was: Will: A, given ordinary linear properties of histories. Equally a matter of simple tense logic is the fact that At-inst2.oop.M :A implies At-inst1.ooP M : At-inst 2-ooP M: A (added time references are vacuous; see §8F.5).

168

Foundations of indeterminism

parameters and context parameters that, in our opinion, are firmly rooted in the (idealized) nature of our world. We are trying to follow our own policy of not making things up as we go along. The objective reason that we do not agree that there is a "real" history determined by a moment is this: Our moments represent what is settled, what is a definite matter of fact, what is determinate; and the future course of history is still unsettled, is not yet a matter of fact, is indeterminate—if indeterminism be true. One ought not be taken in by a definite description such as "the future" or "what will happen" (as in (13) on 160). Given indeterminism, even when all parameters are initialized to context, the referent of these definite descriptions depends on what is going to happen. Que sera, sera is like "Whoever is x1 is x1—that's certain!" Our objective reservation to TRLfcn theory is that TRLfcn (and indeed each form of TRL theory) involves commitments to physical facts (for example, the coin will land heads) that do not supervene upon any physical, chemical, biological, or psychological states of affairs. The fact, if it is one, that at a given indeterministic moment m there is some history such that it is the one that will occur is not a state of affairs that supervenes upon what is true of particles, tissues, or organisms that exist at m. Those of us who do not postulate a Thin Red Line have no need of such a mysterious realm of physical fact. (We hope you join us in regarding as spurious a reassurance having the form, "but it's only a logical fact." That's bad logic. See also the argument of §8D.l.) The TRLfcn approach also has troubles with actuality. Thinking now about Our World, suppose we imaginatively locate ourselves at a certain moment. For a history to be actual would be for it to be the history to which the moment we inhabit belongs. It is not, however, in general the case that the expression "the history to which the moment we inhabit belongs" secures a referent, since uniqueness fails in the face of indeterminism. One does on the other hand always succeed in referring with the expression "the set of histories to which the moment we inhabit belongs," for which an alternative description might be, "the actual situation." Put another way, suppose you agree, as twice before, that "the actual world" does not, in Lewis's framework for modality, privilege any one world over any other. Then we hope you will agree that from among the Lewisian worlds that are exactly alike up to a given time, no one of them has a firmer claim to actuality than any of the others. It would then be most natural as well to say that at a given indeterministic moment m of Our World, there is no privileged actual history or future from among those on which m lies. The worry about knowing whether or not one is on the TRL might take the following form: A person might not know whether what is happening is what was going to happen (while still being sure that what is happening will have happened). That's a bit like Jane thinking "I don't know whether or not I am east of some point west of here (whereas I am sure that I'm west of some point east of here)."

6. Indeterminism and the Thin Red Line

6D.4

169

Epicycles

Permit us another epicycle or two before we engage in our positive argument. Suppose someone adds the absolute TRLabs not as an immobile part of structure, but as a mobile parameter. Either this parameter is used in exactly the same way that we use the history-of-evaluation parameter, or it is not. If so, we think the addition represents a rhetorically misleading change of philosophical terminology since, on the open future view, there is nothing "real" or "true" about the current value of the mobile history-of-evaluation parameter. And if there is a difference in use, in spite of our pessimism, we must necessarily reserve comment until the difference is made clear. Suppose instead one adds the TRLfcn function not as an immobile part of the structure authorized by the nature of our world, but as a mobile parameter shifted by some non-truth-functional connective. Will it be initialized by context or not? If not, there would be no TRLfcn determined by the context of use. In this case, it would seem that since this new parameter would be mobile and uninitialized, the assertion problem would return in exactly the same form as before. Contrariwise, if the mobile TRLfcn is initialized by context, then the TRLf c n must, we suppose, also exist as a context parameter. This causes no really new problems; it just deepens old ones. Recall that with TRLC we deemed it a mystery just how a little speech act could determine the exact course of world history from now on, forever. With TRLfcn the situation is even heavier: TRLc-fcn, "the TRL function of the context of use," even determines many counterfactual such courses, each course beginning with a moment that is perhaps far off in some causally distant region of our world, hidden among those possible events that might have happened had things gone otherwise in the past. The basis in reality of these counterfactually actual historical paths seems an additional mystery worth pondering. A special case is the proposal of Brauner, Hasle, and 0hrstr0m 1998 to add a TRLf c n parameter, not in order to help with the plain future tense, which they treat in the Prior-Thomason way, but instead as a basis for defining for each moment a set of TRLfcn functions that determines what is "immediately possible" at that moment (compare Def. 4 for our concept of immediate possibility). The proposal is too new for us to jump in with an opinion, but naturally questions concerning the objective meaning of the TRLfcn arise. We mean our questions, however, to be real questions, not rhetorical. We are entitled to ask with regard to all the TRL theories what their added parameters would mean, and how they would help us to understand (or perhaps hinder us from understanding) the objective features of, for example, agency, strategies, promising, assertion, betting, causality, and so on. Perhaps these additions could help on all fronts; if so, such helpfulness might support a claim for the usefulness of this new version. That is certainly our claim for the history-of-evaluation parameter. In pointing out these features of the TRL, we do not take ourselves to have proven that belief in the Thin Red Line is inconsistent or incoherent or even false. The chief thrust of our thinking is not so much that adding the TRL causes trouble, but that, as we argue in the next section, its addition does

170

Foundations of indeterminism

no serious work (beyond causing confusion). To help make our position clear, consider again a version of a TRL theory on which a point (m, TRL, mc, a, m/h) contains both a history-of-evaluation parameter and a TRL abs or TRLC or T R L f c n . Surely such a combination is mathematically consistent. Consistency, however, is not enough. You may take the arguments to follow in support of the prediction that when history-of-evaluation and TRL are combined, you will find that the TRL is mere idle filigree. Our conclusion in the following section is that the mobile history-of-evaluation parameter is useful for making sense of assertion (and much else), but that the TRL is not. Insisting on adding a TRL to branching time is exactly like insisting on adding a privileged assignment of values to the semantics of quantifiers. To do so would be technically consistent, but pointless.

6E

Time's winged chariot hurries near

We have argued that responding to Assertion problem 6-8 by denying Semantic thesis 6-6, that history-open sentences "have no truth value," either because they are always closed by independence (via something like the Peirce future tense) or are always closed by initialization (via the TRL) faces grave difficulties. In this section we shall argue that we can keep both Semantic thesis 6-6 (historyopen sentences have no truth value) and Assertability thesis 6-7 (history-open sentences are meaningfully assertable). More than that, we will argue that the property of being history-open positively helps our understanding of assertion in the context of indeterminism. This will augment our case for the doctrine of the open future partly by removing an alleged difficulty and partly by suggesting increased explanatory power. Using the term "branching" to refer to the open future doctrine, Lewis has objected to that doctrine in the following way: The trouble with branching exactly is that it conflicts with our ordinary presupposition that we have a single future. If two futures are equally mine, one with a sea fight tomorrow and one without, it is nonsense to wonder which way it will be—it will be both ways— and yet I do wonder.17 The theory of branching suits those who think this wondering is nonsense. Or those who think the wondering makes sense only if reconstrued: You have leave to wonder about the sea fight, provided that really you wonder not about what tomorrow will bring but about what today predetermines. (Lewis 1986, pp. 207-208; we discuss this passage again in §7B.2, where we also quote its continuation.) 17 "It will be both ways" does not apply to branching time. That doctrine always insists that there are alternative incompatible ways the future might be, which fit together not by "both ways happening," but precisely by having a branch point at which they both were possibilities. At the branch point, the future can be either way, but not both ways.

6. Indeterminism and the Thin Red Line

171

Consider Lewis's suggestion that on the open future view the only way to make sense of a person purporting to wonder what the future will bring is to construe her as wondering what the present predetermines. If this suggestion is right, then it is a strike against the open future view, since it would appear to be a mistake to identify wondering whether there will be a sea battle tomorrow with wondering whether it is settled true that there is to be a sea battle tomorrow. (We are also indebted to C. Hitchcock for formulating an objection along these lines.) We shall argue that it makes sense to wonder about what history has not yet decided so long as history will decide the matter. We shall also argue that it makes sense to assert A when A 's truth value is not settled at the moment of use; the idea is that assertion is an act that has consequences for the speaker no matter how things turn out. Assertion can be treated in either indirect discourse, "a asserts that the coin will land heads," or in direct discourse, "a asserts 'the coin will land heads'." Both are valuable approaches, and each has its own awkwardnesses and limitations. We won't try to stick to one or the other when conveying rough ideas, but when we are fine-tuning, we shall take L to have the direct discourse version of assertion. To this end, we endow the grammar of L with a quote-function such that when A is a sentence of L, 'A' is a term of L that denotes A.18 In order to avoid having to worry about the consequent threat of circularity or paradox, however, we severely limit the resources that L can bring to bear in this regard: A must must involve neither assertion concepts, nor any remotely semantic concepts. Furthermore, with the same end in view, when we introduce "a asserts " into L, we insist that the blank be filled only with a quote-name: a asserts 'A,'

where A is one of the semantics-free and assertion-free sentences of L.19 Let us note first that wondering, asserting, hoping, and betting are each of them history-constant affairs: Whether a person asserts (wonders, hopes, bets) A does not depend upon what history has not yet settled. For this reason (assuming that A is assignment-independent as well as history-independent),

depends only on model, context, and moment of evaluation. Observe that in (18), the moment of evaluation, m, has not been initialized as the moment of use, mc; in accord with §6B.4.2, we are being careful to evaluate "a asserts 18

Tedious remark: The single quotes belong to L; we mention them but do not use them. As for double quotes, our use of them is always casual, striking for easy reading rather than rigor. 19 The threat could presumably be met—instead of avoided—by employing revision-theoretic ideas such as those in Gupta and Belnap 1993 We readily admit, incidentally, that, e.g., "a asserts (bets on, wonders about, etc.) 'Meg is hungry'" is at best awkward English, a fact that we henceforth ignore. Certainly we do not mean to endorse the view that assertion is "really" of sentences.

172

Foundations of indeterminism

'A'" as embeddable, not merely stand-alone, so that we can make sense out of for example the past-tensed "Was:(a asserts '.A')." It is therefore necessary to observe that in (18), the "moment of assertion" is to be identified as the moment of evaluation, m, rather than the moment of use, mc. That is, as in any case of embedding, when the assertion-statement is considered as embedded, one must distinguish the moment of use of the (presumably) larger sentence in which the assertion-sentence is embedded from the moment at which the embedded assertion-statement is evaluated. There are additional complications when we turn recursively to evaluating A itself, (i) Direct discourse requires that we consider A as stand-alone, letting its moment-of-evaluation parameter be initialized by context to be the very moment of its use as a vehicle of assertion. (ii) Direct discourse requires that we throw away the moment of use, mc, of the larger sentence in which "a asserts 'A'" is embedded. Instead, direct discourse requires that we identify the moment of use (and also, by ( i ) , the moment of evaluation) of the asserted sentence, A, with the moment, TO, of evaluation of the (considered-as-embedded) assertion sentence, "a asserts 'A'." (m) Direct discourse requires that we do not consider assertion of assignment-dependent sentences; accordingly, except in certain later dialectic passages, we require that A in (18) contain no free variables. We shall therefore be interested in

where we mean to indicate, by means of position in the sequence of parameters, that the moment of evaluation, TO, of "a asserts 'A'," as in (18), is the proper moment to take as the moment of use of A—and, since now we are definitely considering A as stand-alone, also as the moment of evaluation. In addition, we are sure that A in (19) is assignment-closed, since A contains no free variables, so that a plays no role. (This is more intricate than we like, but not more so than we think necessary. When we wish to consider "a asserts 'A'" as stand-alone, we can—as usual—reduce complexity by initializing TO to mc.) Second, it is not the case that if "a asserts 'A'" is history-independent, then so is A. The paradigm example is precisely a asserts 'Will: (the coin lands heads)'.

(20)

The sentence reporting the assertion is independent of the history parameter, but the asserted sentence is not. One might nevertheless reason that since stand-alone "Will: (the coin lands heads)" requires a history if it is not to be without truth value, we will try in vain to evaluate (20) at a moment, TO, without requiring a specific history. Attempting to do so may seem like attempting to evaluate a asserts 'x1 is brindle'

(21)

without knowing what value has been assigned to x1. An analogous line of thought would apply to attempts to evaluate other attitude and performative verbs, such as "believe," "wonder" and "predict." The difficulty is that assertion (etc.), unlike mere utterance, relies on the meaning or semantic content of the

6. Indeterminism and the Thin Red Line

173

sentence asserted, whereas neither what is claimed to be asserted in (20) nor what is claimed to be asserted in (21) "has a truth value." It would be inadequate to attempt to be quick with the assertion problem by merely pointing out that "Will: (the coin lands heads)" (considered as assignment-closed, and asserted at m), although without truth value, has a "content," represented by the set of histories, {h: 3K, m, a, m/h \= Will:(the coin lands heads)}, a set of histories that can be made available for helping to define an assertion relation via (20). That is, it does not suffice to declare that the "content" of the assertion at m of " Will:(the coin lands heads)" is the set of histories on which, fixing other parameters and given m as moment of assertion, it is true. One should not think one has explained assertion of a sentence without a truth value by saying that assertion is a relation between a person and a "content." The problem is that even "x1 is brindle" (considered as history-closed, and asserted at TO), which is equally without truth value, has a "content" in some technical sense, represented, for example, by the set of assignments {a: yR, m, a, m/h N x\ is brindle}. It would be arbitrary for us to insist without discussion that a set of assignments cannot serve as the content of an assertion, whereas a set of histories can. After all, it is part of the assertion problem that assignments and histories are just alike in respect of being mobile parameters that are not initialized by context (Semantic thesis 6-3). We therefore are obliged to give some reason why the set-of-histories content of " Will: (the coin lands heads)" is the sort of thing that can be the content of an assertion, while the set-of-assignments content of "x1 is brindle" cannot. To this end we shall offer an account of assertion that undergirds a distinction between the "content" of "x1 is brindle" and the "content" of"Whill:(the coin lands heads)." This account is doubly skeletal because (i) it makes assertion out to be only intensional, rather than also intentional, which it surely is, (ii) partly for this reason it omits nearly all of the explanatory aspects of this concept to be found for example in Brandom 1994, and (in) even the Spartan intension that we here attach to assertion is approximate, ignoring much of the interesting subtlety. Since, however, our only aim is to show how the speech act of assertion, intentional though it may be, fits into and in part relies on the indeterminist causal structure of our world, sans TRL, this is enough. §5C.2.3 in effect already defuses what we may call "the word-giving problem," in which it may seem puzzling how one can give one's word concerning a proposition that is, owing to indeterminism, neither settled true nor settled false. Definition 5-15 explains the Thomson idea of word giving in terms of a strategic content representing that the word-giver owes compensation on each history containing a possible future moment at which his word-giving is impugned, and is otherwise free of requirements. It is evident that this recipe treats in one and the same way each history passing through the moment of word-giving; no room is left for a Thin Red Line.

174

Foundations of indeterminism

Assertion seems to be a broader category than word-giving: Not every assertion is a word-giving in the Thomson sense, requiring compensation upon impugnment. Asserting A merely opens the speaker to being either vindicated or impugned depending on whether her claim is borne out, with the understanding that vindication and impugnment can take different forms from case to case. The idea is that the act of assertion creates future-oriented liability to vindication or impugnment, since, provisionally, "a asserts 'A'" is true just in case, if A is true then a is vindicated, and if A is false then a is impugned. This is schematic, since vindication and impugnment come in many forms depending in part upon the subject matter and the conversation or situation in which the assertion is proffered. For example, sometimes vindication or impugnment involves owing some form of credit or discredit to the assertor, and sometimes not. Rather than addressing these issues, however, it is more to our purpose to ask when a person's assertion is vindicated or impugned. One idea is this. a's assertion of A at a moment, m, is vindicated or impugned on a history, h, as of the moment of assertion (provided A is assignment-closed), according as m, m, a, m/h \= A or m, m,

(22)

a, m/h ¥ A.

We are now in a position to see that on the present account of assertion, it makes sense to talk of asserting u Will:(the coin lands heads)" exactly because assertion consitutes a way of closing the history parameter—not indeed semantically (the semantics of the asserted sentence is unchanged), but pragmatically, by the very act of assertion. Let us turn (22) into a kind of skeletal semantic explanation of assertion as follows. 6-12 SEMANTIC ACCOUNT. (Vindication/impugnment account of "asserts") m, mc, a, m/h \= a asserts 'A: iff (i) A is assignment-closed, (ii) a utters A at m, and (ii) for every history h1 passing through m, if m, m, a, m/h1 \= A, then a's utterance is vindicated on h1 as of m, and if m, m, a, m/h1 ¥ A, then a's utterance is impugned on h1 as of m. This account, though in need of refinement, is adequate to our analytic purpose of making at least minimal sense of assertion of sentences that "have no truth value."20 Like word-giving, assertion involves closing the history parameter not in the sense that an assertion of A is an assertion that A is historically possible or settled true. Rather, assertion involves a closing of the history parameter in 20 We might be thought to have made a "monster" (§6B.2) out of assertion, because the moment-of-use context parameter is no longer immobile; in giving a recursive account of (20) in one context, we must shift context when we evaluate the embedded "Will:(the coin lands heads)." The monsterhood arises out of any treatment of direct discourse, and would disappear if we kept to indirect discourse, but we would not advise that course. As long as we are being explicit about direct discourse rather than "sneaking in a quotation device" (Kaplan 1989, p. 511), all seems well; and in any case, better a monster than a fog.

6. Indeterminism and the Thin Red Line

175

the sense that assertion is an act that has implications for the speaker and others no matter how things eventuate. That is why, in coming to an understanding of assertion, every history through the moment of assertion is treated equally, and why it is a step backward to bring in a Thin Red Line. Semantic account 6-12 is amenable to revision in two different directions. The first concerns the causal structure of assertion. A far better version would avoid claiming that vindication/impugnment occurs at the very moment of assertion. The revision would spell out that vindication/impugnment occurs only later, at moments m1 when it becomes settled that A was true (false) when evaluated (not at m1 but) at the moment, m, of assertion. (And if on h1 it never becomes settled one way or the other, then a is neither vindicated nor impugned anywhere on h1.) On the revised version, it would be clear that if one is making an assertion concerning a matter not yet determined, then one has to wait for vindication or impugnment. This better formulation would involve double time references, as in §5C.2.1 and §8F.5. Either formulation puts us in a position to explain the difference indicated in Assertability thesis 6-7 between assignment-open sentences, which have no truth value and are not assertable, and history-open sentences, which also have no truth value, but are assertable. In relation to assertion, the content of " Will:(the coin lands heads)" differs in principle, and not just technically, from that of "x1 is brindle." For the content of "Will: (the coin lands heads)," unlike that of ll x1 is brindle," is the sort of thing that can be borne out or not, depending upon what comes to pass. This is particularly evident for the double-time-reference refinement: Time will tell whether we arrive at a moment at which the truth value (at the moment of assertion) of "Will:(i\\e coin lands heads)" becomes settled—and whether we do or not determines whether the person who asserted "Will:(the coin lands heads)" is vindicated or impugned. On the other hand, finding an object that is brindle gives us no guidance whether or when one who purports to assert "x1 is brindle" is vindicated or impugned. In another direction, Semantic account 6-12 can be fleshed out by adding further concepts and conditions while still eschewing a Thin Red Line. In this way the vindication/impugnment skeletons help not only with assertion, but also with other speech acts that face the open future. One notices, for instance, that the right side of Semantic account 6-12 applies equally to assertions and to conjectures, even though these are evidently different: An assertion is liable to challenge by means of such words as "Why do you think so?" A question such as this is, however, inappropriate for a mere conjecture. We may envision an improvement that marks this difference by building in further norms for assertions, conjectures, and so on, that go beyond liability to impugnment and vindication, and that attribute normative requirements and entitlements at future moments other than those at which vindication and impugnment come into play. For instance, one could note that one who asserts A, in addition to being vindicated or impugned at certain moments, is also obliged to respond at any future moment at which there is an appropriate challenge having the form "How do you justify that claim?" There is, in contrast, no such requirement on one who conjectures A. (See Green 1999 for further discussion.)

176

Foundations of indeterminism

We turn to the case of wondering. (§7B.2 discusses some related aspects of this case.) It may seem that if at moment TO it is sensible to wonder whether A, then it must be that either A is settled true at TO, or that A is settled false at TO. More generally, it may seem that if one is to be able, at TO, properly to raise the question whether A, then A must be either settled true or settled false. If the truth of A depends on as-yet-undetermined future happenings, then it may appear that the question whether A is badly posed. Such a question seems to make no more sense than the question whether the round square is pink. It is indeed bad practice to ask questions that have no answer. We must, however, distinguish questions that have no answer now from questions that will never admit of an answer. A question that will never admit of an answer, such as "Is the round square pink?" is badly posed.21 We should not, however, tar with the same brush those questions that do not admit of a settled answer at the very moment at which they are asked, but which will be answered no matter how the future goes. No matter how things eventuate, the question posed on Monday, "Will there be a sea battle tomorrow?" will be answered. If there is a sea battle on Tuesday, then we may say, "The answer to the question is definitely 'yes; while if on Tuesday there is no sea battle, then we may say, "The answer to the question is definitely 'no'." We should therefore not reject the Monday question as badly posed. It is perfectly correct on Monday to say something like "We cannot yet provide a settled answer to that question, but must wait and see." Yet another option is to respond by taking a stand, by, for example, asserting that there will be a sea battle. The person who has posed the question, if she does not demur, will now have more to which she is entitled in her deliberations concerning how to act, and will know whom to vindicate or impugn depending on whether a maritime conflict ensues. At this stage one might attempt to mount a kind of ad hominem attack against us. One might first suggest that wondering whether A is, approximately, wanting to know whether A, and then one might remind us of our own argument, in §2B.10, that one cannot know, at m, whether there will be a sea battle. It would seem fair to conclude that one cannot wonder, at TO, whether there will be a sea battle. This attack, however, if it is to rely on §2B.10, would have to show that in order at moment m to want to know whether A, it must be possible, at that very moment, to know whether A. This, however, is not a plausible property of wanting. We all sometimes want things we cannot have. What is more to the point, it is typically sensible that one wants to have something not at the very instant of wanting, but after a time. In the case at hand, if a person wants, at m, to know whether A, then she wants something that she eventually will have. It would be impetuous to give up wanting relief from slooplessness upon being told that one will have to wait a bit. 21

We neglect the possibility that a denial of the presupposition of the question counts as an answer. For simplicity we also neglect consideration of questions whose answerability itself depends on future happenings; for example, "Will the sea battle be followed by a full-scale war?" may turn out to be badly posed (if there turns out to be no sea battle), and is therefore risky.

7

Agents and choices in branching time with instants We continue to discuss a variety of foundational matters concerning the theory of agents making choices in our indeterminist world.* Other chapters, especially chapter 2, have set out and briefly discussed a number of postulates—the BT +1 + AC postulates—on which stit theory and its semantics is based. In this chapter we consider these postulates one by one. We begin in §7A by treating at some length the postulates concerning branching time with instants—the BT + I postulates. The section includes a rudimentary discussion of the interplay between "propositions" and "events" in BT + I theory, and considers alternative theories, especially Tx W theory, that are relevant to indeterminism. We proceed in §7B to offer some reflections on indeterminism, to take up determinism and its denial, to consider arguments against branching, and to point out how "causality" arises naturally in branching time. §7C discusses those postulates of BT + I + AC theory that relate to agents and their choices. In §7D we discuss the postulate introducing the domain of quantification.

7A

Theory of branching time

C. P. Snow rightly says that our academic society falls apart into two cultures, that of the scientist and that of the humanist. There is nevertheless but one world, the common home of physical process and of agency. To say as much is not to become embroiled in reductionist or antireductionist disputes; it is simply to note quasi-geometrical facts such as that Jack's very human deliberation about whether to go to the beach preceded the very physical destruction * With the permission of Kluwer Academic Publishers, parts of this chapter draw on Belnap 1996a.

177

178

Foundations of indeterminism

of the beach by the hurricane, which happened far away from (and in causal independence of) Alfredo's finishing his pasta primavera, which was followed by a coin flip that by chance came up heads. It is in this spirit that we propose branching time with agents and choices as a high-level, broadly empirical quasi-geometrical theory of our world that counts equally as proto-physical and proto-humanist. Like geometry, it does not pretend to be that famous "theory of everything." It concerns above all the structural aspects of how the doings of agents fit into the indeterministic causal structure of our world. We postulate an underlying temporal-modal-agent-choice structure (Tree, \<, Instant, Agent, Choice, Domain) subject to certain constraints. You will find a list of all u BT + I + AC postulates," with BT + I + AC standing for "agents and choices in branching time with instants," in §3. We go over these postulates one at a time, concentrating on foundational questions.

7A.1

Nontriviality and partial order

We take up the two most basic postulates before going on to those that are distinctive of branching time as a theory of our world. The first postulate is especially trivial. NONTRIVIALITY. (BT + I + AC postulate. nonempty set: Tree /=O.

Reference: Post. 1)

Tree is a

Tree is intended to represent our world as a set of events, so that nontriviality says that something happens. Its role is technical; we do not care to argue for the thesis that there is something rather than nothing. We are explicit that this representation, as such, gives no information about other sorts of things that our world doubtless contains; in particular, it is irrelevant to the purposes of this book whether events, or entities of some other category, are "fundamental" or "derived," nor does this book tackle the problem of how events relate to, for example, enduring objects such as brains and persons. MOMENTS. (Definition. Reference: Def. 1) A moment is defined as a member of Tree. We let m and w range over moments; and we let M range over sets of moments. Each moment should be pictured as an instantaneous, spatially unlimited, really possible concrete event ("super event" in the language of Thomson 1977), taken pre-relativistically. We intend that each moment stretches across all of space-time. This makes no relativistic sense, and at least because it is not relativistic, branching time is not a true theory. It is, however, an approximation to the truth, and will give us some modest insight into how our world hangs together. A more adequate proto-physical theory called "branching space-time" is presented in Belnap 1992. That theory takes seriously that physical and human happenings are local events rather than universe-wide. We nevertheless remain here with branching time, since no study known to us has attacked the presumably more difficult problem of understanding agency in anything like branching space-time.

7. Agents and choices in branching time with instants

179

Tree, although a set of moments, may perhaps be taken to represent or supervene upon a concrete entirety or world in something like the sense of Lewis 1986. It may matter whether the fundamental notion represented by Tree is concrete or abstract or mixed, or indeed whether the distinction itself should be taken as intramundane, but not for present purposes. What does matter is the key difference between this concept of "world" and that of Kripke 1959 (etc.), of Lewis 1986 (etc.), or of the standard four-dimensional concept derived (we suppose) from Newton by way of Einstein and Minkowski: The world, our (only) world, contains real possibilities both for what might be and for what might have been. The fundamental idea is that possibility—real possibility, objective possibility—is in the world, not otherworldly. If there are alternative worlds, then they, too, come with their real possibilities. Other philosophers take possibilities to be what is consistent either with the laws of logic, or with the laws of physics, thus giving possibilities a fundamentally linguistic status. Still others think of possibilities as abstract or as creatures of the mind. We do not mean to lodge an objection against any of the many other concepts of "possibility" suggested by philosophers in the course of other studies. We only urge that fashioning a rigorous theory of agency and indeterminism is worthwhile, and that in doing so it is greatly useful to construe possible events as both concrete and objective. This study presupposes, but does not argue for, this point of view. Some thinkers doubt that it makes sense to think of a concrete event as necessarily (i) super-large and (ii) instantaneous. Of course none of our postulates guarantees this interpretation of "moment," but it is certainly what we intend. As for (i), although with Thomson 1977 we find super events entirely respectable, there are surely smallish events as well, to be studied in another theory such as the "branching space-time" theory mentioned earlier. We are careful not to claim insight into the special problems raised by considering local events. We take (ii) as either literally true or as a helpful idealization, following Euclid's use of "point." Either way, we have not the slightest objection to theories that work out a rigorous ontology of our world based on "intervals" or the like, provided they are rigorous and provided they are helpful. Evenhandedly, we do not count such theories as objections to ours. In a similar vein, it is critical to our enterprise, but not to other eminently worthwhile enterprises, that the theory advanced aspires to live up to the standards of rigor embodied in the work of Frege, with every concept sharply defined in terms of a family of primitives clearly stated to be such. Sometimes we say "Tree" and sometimes we say "Our World." We use "Our World" when emphasizing that our theory is about our world, and we use "Tree" when we wish to call attention in a more abstract way to the particular quasigeometrical properties of our world, its "shape," so to speak, that we soon begin postulating. Tree is not just a set; it comes with an order on it that we call the "causal order."

180

Foundations of indeterminism

CAUSAL ORDER. (BT + 1 + AC postulate. Reference: Post. 2) ordered by \<:

Tree is partially

Reflexivity. Transitivity Antisymmetryu. We immediately introduce < as the irreflexive and asymmetric cousin of \<. STRICT CAUSAL ORDER. (Definition. Reference: Def. 2) m1 < m2 is the strict partial order of Tree associated with the partial order \<; that is, m1 < m2 iff (m1 \< m2 & m1 /= m 2 ). We refer to either of these as the causal ordering of Tree. Picture the direction of m1 \< m2, which flows from past to future, as generally forward from m1 to m2 (see Figure 2.1). CAUSAL ORDER READINGS. (Definition. Reference: Def. 2) We read \< and its converse with the plain words earlier/later, below/above, lower/upper, backward/forward, and so on, and insert proper when we intend < or its converse. When m1 is properly earlier than m2, we also say, in a fashion much more revealing of our intentions, that m1 is in the (causal) past of m2; and m2 is in the (causal) future of possibilities of m1. Observe that this intuitive use of "causal" does not imply that the earlier moment causes the later one; the usage comes from those dealing with special relativity, a group that also does not pretend that propter hoc follows from a mere post hoc in the causal order. We need to use the language of causal order in part to enforce that we are not speaking of abstract "times," but rather of possible concrete events. Reflexivity of \< is, as always, a postulate of convenience; self-causation is not implied, and indeed is denied by the irreflexivity of <. Transitivity in Post. 2 is perhaps arrogant since it extends the causal order in a simple way to the furthest reaches of our world. That the postulate is something like the truth for the middle-sized portions of our world that lie close to hand, however, seems beyond doubt, so that unless one is engaged in studying either the very small or the very large, it seems a safe enough postulate. We certainly do not ourselves know how to fashion a simple theory that does without it. Antisymmetry already signals that the domain of application of branching time must consist in nonrepeatables (fully concrete events), not abstract situations or "states" of either "systems" or "times" such as nearly everyone with some training in physics is likely to think of. To ask which abstract "states" can follow which is not at all the same as asking which concrete events can follow which, and does not give rise to the same theoretical constraints. States can come again: s1 < s2 and s2 < s1, and hence si < s1, is admissible for states. Concrete events such as moments, however, are not "repeatables" and cannot properly precede themselves. The following easily defined concept underlies much of our work.

7. Agents and choices in branching time with instants

181

COMPARABILITY. (Definition. Reference: Def. 3) Moments m1 and m2 are comparable if \< goes one way or the other: (m1 \< m2) or (m2 \< m1). We do not postulate outright that there are incomparable moments, but that certainly must be the case to give point to this study, whose entire thrust is to make sense out of a world in which we face incomparable possible events in our future of possibilities—that is to say, distinct worldwide events neither of which lies in the past of the other. Next we introduce the notions of chain and history. CHAINS AND HISTORIES. (Definition. Reference: Def. 3) • A chain, c, in Tree is a subset of Tree such that every pair of its members is comparable: c C Tree & Vm1Vm2[mi,ma G c —> m1 and m2 are comparable]. We let c range over chains in Tree. • A history, h, of Tree is a maximal chain in Tree: h is a history of Tree iff h is a chain in Tree, and no proper superset of h is itself a chain in Tree. We let h range over all histories. • History is the set of all histories of Tree. We let H range over subsets of History. In branching time, chains represent certain complex concrete events, some of which we will categorize in §7A.4. History is, however, the essential concept. A history represents a single possible course of events. A history has run on from time immemorial, and will presumably run on forever. Nothing can consistently be added to a history, neither ahead, nor behind, nor in the middle. Histories take to the ends of time "a way events can go," and are in this sense exceedingly "long." Histories also take definiteness to the limit (they decide all disjunctions), and are, in contrast to a thickish family of histories, maximally "skinny." Note, however, that we do not take histories as a primitive idea; given Tree as a set of moments ordered by \<, the concept of history is already there—unless one suspects set theory, or has philosophical qualms with taking a commonsense idea to the limit. These suspicions or qualms may or may not in the end be warranted, but to take such (perfectly legitimate) philosophical concerns as blocking the road to even beginning an inquiry such as this one seems likely to interfere with one's understanding of agents and their doings. Perhaps the idea of a history as a set of moments should be taken to supervene on an underlying notion of history as a concrete whole that has parts instead of members, but it doesn't matter for present purposes. What does matter is that a history is not an entirety or world. Our world is chock-a-block with real possibilities, perhaps with chances, and certainly with actions, and therefore with choices among sets of incompatibilities, none of which find a home in a single history. Chains and other sets of moments can be bounded either above or below; for perhaps pedantic completeness, we assemble the standard definitions.

182

Foundations of indeterminism

BOUNDS. (Definition. Reference: Def. 5) • A moment, m, is a (proper] lower bound of a set of moments, M, iff TO is [properly] earlier than every member of M; and similarly for [proper] upper bounds of chains. • We let m < M iff m is a proper lower bound of M, and similarly in other cases. • We let M1 < M2 iff every member of M1 is earlier than every member of M2, and similarly in other cases. • Greatest lower bounds and least upper bounds of sets of moments are as usual in the theory of partial orders.

7A.2

Forward branching only

Now that we have introduced histories, let us consider the difference between their forward and backward branching. We do not, as we have said, postulate forward branching, but we certainly expect it. Looking backward, however, we postulate that there is no branching toward the past. After explaining the postulate, and giving some consideration to forward branching, we argue in favor of no backward branching. No BACKWARD BRANCHING. (BT +1 + AC postulate. Reference: Post. 3) Incomparable moments in Tree never have a common upper bound; or contrapositively, if two moments have a common upper bound, then they are comparable: (m1 \< m3 & m2 \< m3) —» (m1 \< m2 or m2 \< m1). This is the postulate that makes Tree look like a tree. In order to help see its significance, it is convenient to define ideas of "past" and "future" with a Dedekind-like abstractness that does not presuppose any special order-type on the histories (other than the fact that they are chains). CUTS, PASTS, AND FUTURES. (Definition. Reference: Def. 6) • A historical cut for a history, h, is a pair (p, f) of sets of moments such that (i) neither p nor / is empty, (ii) p < f, and (Hi) (pUf) = h. • p is the past history of / iff / is a future history of p iff (p ,f) is a historical cut.1 • M is the causal past of a future history, /, iff M is the set of all proper lower bounds of /. • We often say just past because given no backward branching, a causal past is the same as a historical past. 1

We use "p" for past histories and "f" for future histories, but also for other purposes; we believe, however, that no ambiguities arise in this book.

7. Agents and choices in branching time with instants

183

• M is a future of possibilities, or a causal future of a past, p, iff M is the set of all proper upper bounds of p. In branching time, a future history must be distinguished from a future of possibilities, so that in this book we never say merely "future." 2 • We say that (p, M) is a causal cut iff p is a past, and M is the future of possibilities of p. • All of these usages may straightforwardly be adapted to speak of pasts and futures of single moments. The postulate of no backward branching tells us that each future history has exactly one past history: If (p1, f) and (p2, f) are historical cuts, then p1 = P2. Where / is a future history, this uniqueness entitles us to introduce the phrase "the past history of f." Since no backward branching implies that the past history of / is the set of all moments preceding /, by strict analogy with special relativity, we may also call the past history of / the causal past of /, or just its (plain) past. As a special case of no backward branching, our causal past is a history in which everything has a place in terms of earlier/later; our past is not an assemblage of incompatible possibilities. If two possible events (i.e., members of Tree) each lie in our past, then one of them was causally earlier, and lies in the past of the other, giving rise to an actual linear causal sequence, earlierlater-now. This is not merely a matter of temporal dating; it concerns causal linkage. This discussion does, however, falsely but conveniently presuppose that causal linkage is between entire spatial "slices" instead of between small local events. As previously noted, this defect is remedied in Belnap 1992, but not in this book. Although a future history has but one past history, the converse is by no means true. If determinism is not permanently true, there will be past histories p to which more than one future history can be appended: One can have historical cuts (p , f i ) and (p, f2) with f1 /= fa. There is therefore no "rigid" sense to the expression "the future history of p." We may, however, reasonably speak of a unique "future of possibilities." Each future of possibilities will look something like a tree, and will be a subtree of Tree. When at a given moment Lee-Hamilton says "the past is stone, and stands forever fast," his use of the phrase "the past" safely refers to a past that is uniquely determined by the (idealized) context of utterance. 3 Furthermore, this past is a portion of each history of which his utterance is a part: No matter 2

In earlier publications we sometimes used "future" as a short form of "future of possibilities." That was an expository error; in delicate discussions of indeterminism, it is better to avoid this contraction because of its confusing conflict with ordinary English usage. One should be resolute in speaking of either "future histories," or "future of possibilities," never of just "futures." 3 It is a confusion to infer from this that "it was true that Q" implies "it is settled that it was true that Q"; instead, as we note at the very end of this section, one has only that "it was settled true that Q" implies "it is settled that it was settled true that Q."

184

Foundations of indeterminism

the future history, that past stands forever fast. Given no backward branching, talk of so acting as to "influence" the past is therefore just talk. When, however, at a certain moment Tennyson says "I dipt into the future, far as human eye could see, / Saw the Vision of the world, and all the wonder that would be," we must be careful, especially when we learn that the Vision involves a sea battle. There is no philosophical problem if we interpret the poet as denoting his future of possibilities, the tree that fans out from the moment of his utterance in intricate arborescent profusion—a proper source of wonder indeed. Nor is there a problem if the poet is using "the future" as "the future history," but with its "nonrigid" denotation relative to a poetically visionary history—as long as we do not forget this relativization and take proper care to disambiguate Tennyson's expression accordingly, just as we would in the case of an occurrence of "the river" in The Lady of Shalott. (This simply adapts to singular terms the sentential semantic insight of Prior-Thomason that we adopted and first explained in §2A.l, and that we make more explicit in §8E.) If, however, by the phrase "the future" the poet intends to denote a unique future history of which his moment is a part, then his intention cannot be carried out, for (unless determinism be true from that moment as far as human eye can see) there is no such unique future history, as from time to time we repeat. We believe (if that is the right word) in forward branching, and in the impossibility of backward branching. Sometimes one hears a philosopher or a physicist maunder on about distinct pasts that coalesce in a present moment, and doubtless it is good that our conceptual limits be tested. We confess, however, that we ourselves cannot follow these fancies. That we face alternative future histories seems to us right; that we are faced away from alternative pasts seems to us wrong. That starting with the concrete event that occurred yesterday morning there were incompatible possible events each of which might have transpired seems to us right; that more than one of these incompatible possible streams of events might have finished up in this very concrete situation seems to us wrong. In common with antisymmetry, no backward branching makes sense only for objective, concrete events. First, we advance no theory at all about what is possible (not objectively but) "for all one knows." A given concrete situation could obviously have been preceded by any of various inconsistent predecessors, "for all one knows." It is precisely to preclude this epistemic or doxastic use of "possible" that we so tiresomely repeat that our present concern is with "objective" possibilities. Second, no backward branching fails to apply to "states" or other repeatable carriers of partial information. There is no doubt whatsoever that a present "state" may be accessible from either of two earlier incompatible states. There is no doubt about this because there are so very many senses to the word "state." Surely there are physical "systems" with a favored family of "states" that branch only forward and not backward, others that branch only backward and not forward, others that are doubly deterministic in terms of their favored "states," and still others with more exotic structural properties or with no interesting properties at all. After all, everything happens. None of this, however, is relevant to our postulate of no backward branching. To discuss any of it is to change the topic.

7. Agents and choices in branching time with instants

185

Figure 7.1: Backward branching?

EXAMPLE. (Backward branching with states) Count position and acceleration as adding up to a "state," and, invoking an example suggested by Bressan, slide a disc along the floor. Friction will bring it to a stop. Its future is then determined as continuing in the stopped state, at least for a while, whereas from its stopped state there is no inference to when in the past it was thrown. True, but irrelevant. No backward branching does not imply that this particular definition of "state" gives us information as to when the disc was thrown; it only implies that regardless of the poverty or richness of any concept of "state" that is brought into play, there is a fact of the matter admitting no real alternatives. The concrete event of the disc coming to a halt has in its past a unique concrete event of its being thrown—a fact that is no less true for being absent from physical theories cast in terms of systems and states. One of the reasons we think no backward branching a necessity is that we do not know how to make sense of agency without it. We expose our puzzlement with two side-by-side diagrams. In Figure 7.1, the left-hand diagram represents our own theory of a moment of choice. Picture a castaway on a deserted island. Let us position ourselves at moment m1, and let us suppose that at 1:00 P.M. the castaway chose between lighting his signal fire or not. We may now say that ever after 1:00 P.M. it has been a settled matter which choice the castaway made: He chose to light the fire. Because of his choice, at 3:00 P.M. he was rescued. We can say that although before 1:00 P.M. no-fire was a real possibility for the castaway, that possibility was not realized. Of course we at m1 (perhaps in England) might not know which choice the castaway made, and we might not know whether or not he was rescued; but given our position at m1, there is a settled fact of the matter.

186

Foundations of indeterminism

In contrast, the right-hand diagram in Figure 7.1 represents backward branching. Let us position ourselves at moment m3 in this right-hand diagram. We can still say that the following is now settled at 2:00 P.M., and also at 3:00 P.M.: Either it was settled that the castaway chose fire or it was settled that the castaway chose no-fire. Accordingly, no matter which history through m3 you pick, at those times it was definitely settled whether or not the castaway was rescued at 3:00 P.M. This is in accord with our usual views about the consequences of choice: Choosing the signal fire settles the matter in favor of rescue, choosing no-fire settles the matter in favor of its opposite, no-rescue. That sounds fine. The trouble is that now at 5:00 P.M. it is no longer settled which choice the castaway made, and it is no longer settled whether or not he was rescued. There used to be a settled fact of these matters, but it has dissipated. There is no longer a settled fact as to whether he was rescued or not at 3:00 P.M. The diagram seems to show that the castaway had a real choice at 1:00 P.M., but the same diagram seems to say that the choice at 1:00 P.M., real as it is (or was), has no differential consequences for us at 5:00 P.M. The consequence of choosing fire at 1:00 P.M. was rescue at 3:00 P.M., and the consequence of choosing no-fire at 1:00 P.M. was no-rescue at 3:00 P.M. But whatever status "rescued at 3:00 P.M." has for us at 5:00 P.M., on this diagram "not rescued at 3:00 P.M." has quite the same status. Since the differential consequences of the choice have disappeared by 5:00 P.M., it may be that we should say that the choice has also disappeared: That the castaway had a choice at 1:00 P.M. used to be true (at 2:00 P.M. and 3:00 P.M.), but it isn't true any longer (at 5:00 P.M.). We hope that such talk makes you as nervous as it does us. We do not mean to have exhibited a contradiction, or even a manifest absurdity. We intend only to expose our puzzle about making sense of agency if there is backward branching. Perhaps backward branching is all right; we just don't see how. We do not know how to develop a sensible theory of how choices fit into the world except with the understanding that once a choice between options is made, for ever after it is a definite and settled matter which choice was made (see (*) on p. 39). Here is another example. According to the present theory, if you choose on Saturday to promise your neighbor to mow his lawn, then ever afterward it is a settled matter that you made the promise. You may forget, your neighbor may forget, everyone may forget, but the fact remains a fact. Even if you are released from your promise, it will be in virtue of later happenings. You cannot hope for release by its becoming true that you didn't make that choice on Saturday, or even its becoming "possible" that you didn't make it. You may reasonably hope that the promise will be voided in any one of a number of ways; but no matter how long you wait in hope, the settled fact that on Saturday you chose to promise to mow the lawn is not going to go away. If, however, we assume that histories can branch backward, then such a hope is (perhaps) reasonable. And if so, it can happen that on Monday it is a settled fact that on Saturday you made a promise, and then, on Tuesday, the "fact" that you promised can become "possibly false" (or "not definitely true"—we hardly know what words to use to describe this imaginary situation that in fact we are unable to imagine). Maybe

7. Agents and choices in branching time with instants

187

that makes sense; but we ourselves have absolutely no idea (in the absence of doublethink) of how to develop a theory of promises in such a setting. One pressure for backward branching comes from certain interpretations of certain physical theories. According to these interpretations of these theories, the world looks just the same upside down. We of course have no special expertise in physics, and so turn to the quantum-field theorist Haag 1990. I want to suggest here that—once one accepts indeterminism—there is no reason against including irreversibility as part of the fundamental laws of nature. ... It should be stressed that this picture does not touch CTP-invariance or detailed balancing. ... the term "time reversal" should be replaced by "motion reversal." (p. 247) ... the use of probabilities in statistical mechanics and quantum theory is necessarily always forward directed since the past is factual and the future open. If irreversibility is introduced on a fundamental level as proposed then the coincidence of the different "arrows of time" (psychological, thermodynamic, cosmological...) is immediate and in particular dissociated from any cosmological model, (p. 250; citations omitted) Evidently even physicists disagree about these matters; it is, however, surely respectable to take Haag as our governing authority. If we do so, backward branching receives no aid and comfort from any physical theory. The tense-logical principle that seems to us most closely associated with no backward branching is this: "If something was settled true, then it is (now) settled that it was true." This principle—which is not to be confused with its invalid sound-alike, "If something was true then it is (now) settled that it was true"—is the one whose loss would leave us with a feeling of having lost our grip. Settled features of past moments stay settled. In particular, if we look back at earlier choice points, it is and will always remain a settled matter which choice was made at that earlier point.

7A.3

Historical connection

The TxW theories discussed in §7A.6 demand from the outset that no event belong to more than one history, that no event have more than one possible future in the past of which it lies. The next postulate not only permits meaningful overlap of histories, but insists on it. HISTORICAL CONNECTION. (BT +1 + AC postulate. Reference: Post. 4) Every two moments have a lower bound: V m 1 E V m 2 [ m o < m1 & m0 < m2]. In other words, every two histories intersect. So every two histories share a common past: their nonempty set-theoretical intersection. Our world, as represented by Tree, has no loose or floating pieces; it constitutes a whole, bound together by the causal order. It is perhaps this

188

Foundations of indeterminism

postulate, above all, by means of which the theory insists that we not admit histories that are idle creatures of the imagination, or histories that could be "defined" by piecing together some arbitrary array of logical or conceptual or scientific possibilities. Instead, every possible history, h, has a definite causal relation to the very moment in which we converse, since h must share with it a common past. Historical connection is the postulate that endows the theory with a sense of robust reality. In other words, historical connection makes Tree a single world, Our World, instead of a mere collection of "worlds." In this sense it is historical connection that ensures that agents are actual and that our choices are real. Historical connection puts all moments of Tree "in suitable external relations," as Lewis 1986 says (p. 208). It gives content to exactly the sort of real possibility that is pertinent to an understanding of stit. Just to make things clear by an example, we are disallowing that it is or was really possible that there should be blue swans unless there is some definite moment in our past that has a moment in its future of possibilities at which there are blue swans. Naturally, as armchair philosophers of indeterminism, we do not claim special insight into what is really possible; that is a matter for common sense or science or metaphysics. The point of the example is only to express our doubt that it is easy to be sure that it is or was really possible that there should be blue swans. Of course something terminological is going on here: We are using "really possible" as what is or was determined as possible in the world of our context of utterance, and thus in a sense much narrower than that sought by, for example, Lewis 1986 through the idea of recombination. But there is also something nonterminological: We think that the Humean picture of enormous recombinational possibilities for the immediate future (e.g., blue swans on our desk one nanosecond from now— Lewis 1986, p. 91, says that "anything can follow anything") is not relevant to what can be seen to, and that instead what counts is only the current—much narrower—set of real possibilities. When the intersection h1 n h2 of two (distinct) histories h1 and h2 is not only nonempty, as promised by historical connection, but has a least upper bound, m0, we say that h1 and h2 split at mo (Def. 7). We write h1 Lmo h2 (Def. 4). And if every two (distinct) histories split at some moment, we say that the semi-lattice condition is satisfied (Def. 7), since in context this holds iff every two moments have a greatest lower bound. Is it true? Do each two two histories split at a moment? In the branching space-time theory of Belnap 1992 we postulate the principle of "prior choice," which says that if a point-event, e\, belongs to one history, h1, and not to another history, h2, then there is a particular point-event, e0, in the past of e1 such that h1 and h2 split at that point. In the context of branching space-time prior choice is, as far as we can see, a deep causal principle, and in fact in that context, we do not know how to carry on without it. The same words in the context of branching time express the semi-lattice condition. We are therefore certainly tempted to enter that postulate as a strengthening of historical connection, especially because there is no reason of which we know to suppose that the semi-lattice condition is false. On the other hand, by careful formulations that do not rely on that

7. Agents and choices in branching time with instants

189

condition, one can come to see that it does not make any difference to the theory of agents making choices in branching time. The fundamental reason for this seems to be that in our theory of choice, we postulate that regardless of the semi-lattice condition, when an agent makes a choice, there is always a last moment of indetermination, which is therefore the greatest lower bound of two moments belonging to different choices. In other words, if the fact that we are in one history rather than another is to be explained by choice, then those two histories, anyhow, split at a moment. All this abstruse talk is to explain why we waver when it comes to postulating the semi-lattice condition. In the end we decide somewhat arbitrarily to do without it. Wavering to one side, however, even in the rudimentary context of branching time, one ought to see historical connection, especially when strengthened to the semi-lattice condition, as a powerful causal principle that, like prior choice, says that a real cause for something being one way rather than another always lies in the past. If histories were disconnected, one could not say that without bringing in soft ideas such as "similarity." We also refrain from postulating that all the moments in Tree have a common lower bound, which would imply a kind of nonrelativistic Big Bang.

7A.4

Propositions, events, and their interplay

Branching time makes possible several important functional connections between concrete events on the one hand and "propositions" on the other, connections that seem difficult to make clear in other frameworks. This material is seldom used explicitly in this book, and may be skipped. Nevertheless, the ideas are implicit in many of our formulations, and explicit in a few, so that we take a bit of space to lay them out. Propositions. Let us begin by articulating the idea of "proposition." On our view, "proposition" is serviceable but inconstant philosophical jargon. That is, in spite of its usefulness, there should never be an assumption that we all use it in the same way, or that anyone knows what anyone else means. "What is 'grasped' by a mind" doesn't help to pin down what a philosopher might mean, since "grasp" is an unpacked metaphor. "Meaning of a sentence" helps only to the extent that a particular understanding of "meaning" is in play. We stick with some "intensional" notion of propositions closely tied to the idea of (parameterized) "truth-conditions," thereby giving up any pretense of using an "intentional" concept that is essentially suitable for mental "grasping." Having limited our target to propositions in a sense that resonates with parameterized truth conditions, we next observe that among those who think about time, there is sometimes a debate concerning whether a proposition can "change its truth value." We are sure that among the various concepts of proposition, some make such a change sensible, and some do not. • If the proposition is taken to be a "timeless" or "unchanging" proposition, one may represent it in branching time by means of a set of histories,

190

Foundations of indeterminism H, thinking of H as true in just its members. This already makes the conceptual role of histories in the theory of branching time in some deep respects analogous to the role of "worlds" in Kripke-inspired modal logics. This notion of "proposition" is best tied to sentences considered as standalone and as assertable, (§6B.4.2), since such sentences are open only in the history of evaluation (combine Semantic thesis 6-5 and Assertability thesis 6-7).

• If one wishes to think of a proposition, however, as changing its truth value over time, one represents the proposition well in branching time by a set of moment-history pairs, that is, a subset of Moment-History. This "time-dependent" or "moment-dependent" notion of "proposition" is suitable for thinking about sentences considered as embedded in, for example, tense connectives. • Indeed, when thinking of the contribution of sentences embedded in quantifier connectives, it can be helpful to advert to "assignment-dependent" propositions. All these representations and concepts of propositions are useful. As a practical matter, and with no intent to legislate, we will reserve the unmodified notion of "proposition" for the first case: a set of histories, H. A proposition, H, is said to be true in each of its members, and false in each of its nonmembers. By itself this is trivial; we shall see, however, that the usage coheres with a number of other ideas. First we say a few words about events. Events. Branching time permits only a rudimentary theory of events. That is because its primitive notion of a "moment" is already intended as something that is, although instantaneous, unbounded in spatial extent. Therefore no basis is provided for local events; for such events one would need to turn to branching space-time (Belnap 1992). It is good to think of a moment as a possible event, a possible momentary event. Momentary events automatically have their locus in the causal structure of our world, so that it makes sense to think of them as concrete. A more general notion of a (concrete possible) event is as follows: A concrete possible event is a nonempty set, M, of moments. This Quine-like construction yields a kind of de re concept, since it gives only the causal locus of the event in our world. On the other hand, by antisymmetry, moments resemble Leibnizian monads insofar as each moment determines its entire past and its entire future of possibilities, a property that in some sense is passed on to sets of moments. In any case, we shall see that the theory of events qua sets of moments is more fecund and explanatory in branching time than it would be in an objective determinist theory. What we say about a set of moments qua event depends on its causal "shape." Two cases are of special interest: "initial" events and "outcome" events. (Even when idealization is maximized, we do not propose that every set of moments

7. Agents and choices in branching time with instants

191

corresponds to an event in some intuitive sense. It is always the special cases that are of interest.) Consider the setting up or initializing of a situation that is indeterministic either by choice or by chance. One might have in mind, for instance, the process leading up to the flip of a coin. (A flip is of course a spatially local event, a feature that we must ignore.) There must exist some moments before it is decided whether heads or tails will eventuate, and also some moments after. Extend a series of pre-decision moments forward until further extension would have to involve a moment at which either heads is decided, or tails is decided. Let / be a set of moments resulting from this procedure; a set of moments such as / is what we call an "initial" event. (We use "I" as a mnemonic variable ranging over initial events.)4 Speaking abstractly, because, for example, heads-or-tails is by assumption eventually decided, the most essential condition on the causal "shape" of an initial is that it must be nonempty and properly upper bounded (by some moment at which the matter is decided), and therefore / must be a nonempty chain. For many purposes neither the "beginning" of such a setting-up nor its internal structure (e.g., denseness) makes a difference and may be ignored. For these purposes, we may therefore enter the following. INITIAL EVENTS. (Definition. Reference: Def. 8) I is an initial event iff I is a nonempty and upper-bounded chain. Fixing on an initial, /, as just defined, consider now a particular outcome of some indeterministic set-up, say "the coin's having come up Heads." Being an outcome, such an event, O, must be nonempty and causally lower bounded; so much is essential, for otherwise O could not come to be. OUTCOME EVENTS. (Definition. Reference: Def. 8) iff O is a nonempty and lower-bounded chain.

O is an outcome event

There are likely a number of enlightening notions of "outcome" obtainable from this beginning. Because the whole matter is, however, somewhat tangential to agency, we do not elaborate; we define only the following relational idea of one event being an outcome of another. 4 Suppose the semi-lattice condition. Then an initial event could be defined more simply as a single moment, the moment at which some tails history splits from some heads history. Such a moment would be the last moment of indetermination. If it seems farfetched that there should be such a moment, observe that when, e.g., a ball begins to move, elementary physics infers from continuity principles that there is a last moment of rest, but no first moment of motion. To give up this bit of physics is to give up idealizing motion and therefore to give up on rigorous physical theory; to give up the existence of a last moment of indeterminateness (given the semi-lattice condition) is analogous. A further analogy is also helpful. If we take seriously that the ball has parts that are space-like related in the sense of special relativity, then we cannot find a frame-invariant last moment of rest. A typical response to this problem is to idealize the ball as point-like, that is, as without space-like related parts. One does not need to believe that the ball "really is" point-like in order to profit from the clarity that such an idealization permits. We intend our idealized idea of a point-like upper end of an initial event to be just like that' You do not have to believe that initial events "really are" like that in order to profit from our proposed idealization.

192

Foundations of indeterminism

OUTCOMES OF EVENTS. (Definition. Reference: Def. 8) • O is a (possible) outcome of /, iff / is an initial event and O is an outcome event, and (every member of) / properly precedes (every member of) O. • O is an immediate outcome of I iff O is a (possible) outcome of I, and if furthermore no moment lies properly between / and O. The idea of a "transition" as an ordered pair of events now falls into place. TRANSITIONS. (Definition. Reference: Def. 8) is an [immediate] transition iff O is an [immediate] outcome of /. The suggestion that it is sometimes better to construe events qua "happenings" as transitions rather than as events in our technical set-of-moments sense is made by von Wright 1981. Here we see that we are forced in this direction by the example of immediate transitions, where there is a "happening" but no room for an "event" (in the set-of-moments sense) between initial and outcome. In kinematics, the transition from rest to motion is a "happening" of this kind: transition, not event. The same is true for our idealized concept of choice considered as a happening. (Belnap 1999 expands on this thesis.) Propositions from events. Having defined propositions as sets of histories and events as sets of moments, we briefly indicate some interrelations by means of definitions intended to be revelatory. The proposition that such-and-such an event "exists" or "occurs" is much thrown around, usually in a contingent sense. Either in determinism or in Lewis-like constructions, however, there appears to be no objective, rigorous theory on the basis of which to make sense of these words. With unavoidable artificiality, we will say that an event "exists" in a contingent but timeless sense that relates the existence of an event to histories. (By implication, we reserve "occurs" for a moment-dependent idea; but we do not cash in this reservation.) It turns out that the sense to be given to "exists" rightly depends on the "causal shape" of the event that is presupposed. We define three notations that encode three different ways of mapping events into timeless "existence" propositions. FROM EVENTS TO EXISTENCE-PROPOSITIONS. (Definition. Reference: Def. 4) • H(m) is the set of histories in which m lies (or the set of histories "passing through" m): h E H(m) iff m E h. • H [ M] = {h: M C h}, so that H[M] is the set of histories entirely containing M. For suitable M, this is the set of histories in which M passes away. • H<M> = {h: (Mnh) = O}, so that H<M> is the set of histories that pass through at least one member of the set of moments, M. For suitable M, this is the set of histories in which M comes to be.

7. Agents and choices in branching time with instants

193

H(m) represents the contingent proposition that m timelessly exists, a proposition that is true in all and only those histories that m inhabits. H[M] does not always make useful sense as a proposition. When, however, / is an initial event, H[M] is well interpreted as the (timeless) proposition that / is "completed," "finishes," or in Aristotle's phrase, "passes away." If you are thinking about experimental preparations or agonizing deliberations, this is a sensible thing to mean by "exists": A preparation-event or deliberation-event does not "exist" in a history unless it does so entirely. If some history, h, splits off in the middle of a deliberation, we decline to say that the deliberative event "exists" in h. (One would need explicitly to tackle the present progressive in order to have a rigorous account of a deliberation-event "existing" in histories in which it is not completed.) Note a comfortable interaction: The proposition that / is completed depends not at all on the causal "shape" of / in its nether region. Any initial cofinal with / toward the future will determine the same set of containing histories as does /. For example, if / is taken as a representation of the setting-up of an indeterministic experimental situation, whether or not the set-up is completed is insensitive to everything except the causal locus of its being completed. When O is an outcome event, the meaning of "exists" needs to be quite different. In that case, the proposition H is the right sense of "exists," since outcomes exist by beginning (not ending), and H is true in all and only those histories in which O begins, commences, or in Aristotle's language, comes to be. Suppose a coin is flipped. If we look only at the bare causal structure, it is natural (but not necessary) to think of the flipping as lasting as long as there is indetermination as to the outcomes heads or tails, no matter how that flipping may be related to the way that the coin dances in the air. In the same way, it is natural (but not necessary) to think of the heads outcome beginning whenever the possibility of tails is excluded. In other words, the heads outcome (not must but) may be taken to begin whenever that outcome is determinately settled, and the proposition that the heads outcome exists should be true in just those histories in which settled-heads begins to be. For purposes of causal analysis, it doesn't matter when the heads outcome ends. Consistency. Without belaboring the point, we note that the interplay between propositions and events in branching time generates a firmly based family of consistency/inconsistency concepts. We may take from possible-worlds theory the idea that when propositions are represented as sets of histories, consistency between them is definable as having some history in common, a history in which both propositions are true. We add that it is helpful to define the consistency of two initial events I1 and I2 by the consistency/inconsistency of the propositions H[I1] and -H[I 2 ], so that the question is whether or not there is a history in which both preparations are (not just started but) completed. Dually, two outcome events O1 and O2 are consistent/inconsistent iff the propositions H and H are consistent/inconsistent, so that the question is whether or not there is a history in which both outcomes begin to be. Even more delicately,

194

Foundations of indeterminism

one may ask whether a certain initial is consistent with a certain outcome by way of well-chosen propositions. Finally, one has the right definition of what makes a transition "contingent." CONTINGENT TRANSITION. (Definition. Reference: Def. 8) is a contingent transition iff is a transition, and if some history is dropped in passing from the completion of / to the beginning of O: H[I] -H = O. These remarks are intended to indicate with almost excessive brevity that branching time permits and indeed suggests a rigorous and modestly enlightening theory of the interrelations of (possible) concrete events and propositions. We mention that some additional ideas on initials and outcomes are offered in Szabo and Belnap 1996 and in Belnap 1996c, and in the unpublished set of notes Belnap 1995.

7A.5

Theory of instants

If "histories" are a way of making a sort of vertical division of Tree, then Instant, whose members are instants, is a kind of horizontal counterpart. In branching time, the doctrine of instants harkens back to the Newtonian doctrine of absolute time—and therefore is suspect. We use it, but we don't trust it. For that reason if for no other we try to be as clear about it as we can. Instants are perhaps not fully "times" because we are not in this study relying on measures or distances, but it is intuitively correct to think of i( m ) as the set of alternative possibilities for "filling" the time of TO. We need instants because we think that for the achievement sense of stit, in considering whether Autumn Jane stit she was muddy at a certain moment, it is relevant to consider what else might have been at the instant inhabited by that moment. Evidently our uses of "moment" (in which we follow Thomason) and "instant" are jargon not sanctioned in ordinary speech, although the distinction is certainly there to be drawn. Not all parts of stit theory rely on the theory of instants. Only the semantics for the achievement stit has need of these horizontal comparisons. For example, the theory of dstit, §8G.l, and the theory of strategies of chapter 13 are developed quite apart from the idea carried by Instant. In this important sense, the theory of instants is not a deep presupposition of stit theory. There is a contrast at this point with TxW theories as described in §7A.6. We nevertheless develop the theory of instants to the extent required by the semantics of the achievement stit. There are three postulates. Instant AND INSTANTS. (BT +I + AC postulate. Reference: Post. 5) i. Partition. Instant is a partition of Tree into equivalence classes; that is, Instant is a set of nonempty sets of moments such that each moment in Tree belongs to exactly one member of Instant. ii. Unique intersection. Each instant intersects each history in a unique moment; that is, for each instant i and history h, inh has exactly one member.

7. Agents and choices in branching time with instants

195

iii. Order preservation. Instants never distort historical order: Given two instants i1 and i2 and two histories h and h', if the moment at which i1 intersects h precedes, or is the same as, or comes after the moment at which i2 intersects h, then the same relation holds between the moment at which i1 intersects h' and the moment at which i2 intersects h'. We next offer some convenient definitions and simple facts, after which we comment on the postulates.

INSTANTS. (Definition. Reference: Def. 9) • The members of Instant are called instants, i ranges over instants. • i(m) is the uniquely determined instant to which the moment m belongs, the instant at which m "occurs." •

m

(i,h) is the moment in which instant i cuts across (intersects with) history h; that is

• Order preservation can conveniently be stated in the symbols just introduced: implies • Fact: m(i ( m 0 ) , h 0 ) , a function of mo and ho, is the moment on history h0 that occurs at the same instant as does mo: • i|>m = {m0: m < mo & mo E i}. We say that i|> m is the horizon from moment m at instant i. • Where i1 and i2 are instants, we may induce a linear time order (not a causal order!) by defining i1 < i2 iff m1 < m2 for some moment m1 in i1 and some moment m2 in i2. Instants can also be temporally (not causally) compared with moments, m: i1 < m iff m1 < m for some moment m1 in i1 and m < i2 iff m < m2 for some moment m2 in i2 . Post. 5(i) encodes that making same-time comparisons between histories is objectively sound. We do not pretend to understand the conceptual problems involved in making such comparisons. The problem becomes ironically clearer when it is made more difficult by transference to branching space-time, where it is same-place-time comparisons between inconsistent point events that is at issue. All we can add is a conviction that it will not be possible to make suitable advances without consideration of the work of Bressan 1972, Bressan 1974, Zampieri 1982, and Zampieri 1982-1983, for they are the only persons we know who have worked within what seems to us the only reasonable position, that identifying place-times across possible situations is neither trivially easy (perhaps Kripke thinks this) nor a matter of partial constraint and partial stipulation (perhaps Lewis thinks this) nor empirically insignificant (perhaps this is van Fraassen's view), but a matter of serious physics.

196

Foundations of indeterminism

Postulates Post. 5(ii) and Post. 5(iii) on Instant are very likely too strong (too oversimplifying); our justification is that agency is already hard to understand, so that it won't hurt to try to see what it comes to in circumstances that are not altogether realistic—as long as we keep track of what we are doing so that later we can try to move closer to reality. Thus, which most reality-oriented persons think not so plausible, but which greatly simplifies our picture of time, all histories are said to have exactly the same temporal structure. It follows that all histories are isomorphic with each other, and with Instant, which justifies the ordering on instants defined in Def. 9. On the other hand, no assumption whatsoever is made about the order type that all histories share with each other and with Instant. For this reason the present theory of agency is immediately applicable regardless of whether we picture succession as discrete, dense, continuous, well-ordered, some mixture of these, or whatever; and regardless of whether histories are finite or infinite in one direction or the other. The theory of Instant is not, as we have said, as fundamental as that of <, and perhaps it is too strong, even pre-relativistically. Certainly the present assumption that all histories have isomorphic temporal orderings is stronger than comparable assumptions of Thomason 1970 or Thomason and Gupta 1980, and probably it should ultimately be weakened. In the meantime, while it is good to be concerned about oversimplification, the justification of our procedure is that the assumption can be clarifying when it comes to thinking about certain aspects of agency. Instant gives us a theory of linear "time" (based on < between instants) to play off against the theory of branching "time" (based on < between moments).

7A.6

Times x Worlds and other alternative theories

Some of branching time can be modeled equally well on what might be called a "Tx W" theory (with T for times and W for worlds) after Thomason 1984, or might be called a "divergence" theory after Lewis 1986. (See Zanardo 1996 and Di Maio and Zanardo 1998 for authoritative explanations and developments; this entire section has benefited from Zanardo's advice.) The key idea is that histories (to use the present terminology) are taken as ontologically distinct each from each (no common parts), and that enough in the way of additional concepts and argument is added to render it credible that two histories can "perfectly match up through" (Lewis) or "differ only in what is future to" (Thomason) a particular time. (Thomason in the passage cited is describing, not endorsing.) The modeling, however, is specious. The same bundle of (Humean) concepts and arguments that makes it credible that two histories might match before a certain time makes it equally credible that they match after a certain time while failing to match before that time; there is no asymmetry in the nature of things when it comes to mere matching. For this reason, such an approach to understanding our world, unlike branching time, can well encourage backward branching babbling. Branching time, incidentally, does not deny forward matching of distinct histories; instead what it denies is their forward overlap. For example, it may be that one of the histories on which we turn left becomes

7. Agents and choices in branching time with instants

197

very like one of the histories on which we turn right. Perhaps this happens at some point after the heat death of the sun; why not? But "very like" is not the same as sharing particular moments (and is not even very like). The gap between identity/distinctness on the one hand, and similarity/difference on the other, is unbridgeable. A helpful analogy is to points of space, some of which are well known as both very like and very distinct. The analogy is particularly apt since the theory of branching time is, in its general spirit, geometrical. Every theory has merits and demerits. All Tx W theories share the following two demerits: They need for their foundation a prior story about (i) times and (ii) worlds. Both of these stories are likely to be tall. (i) The notion of a time as an entity independent of events gives pause. It would seem that in order to make sense out of indicating a particular time, one would have to explain quite a lot about clocks or cesium atoms or seasonal social practices, or something. This is something that Tx W theorists need to do. The problem is perhaps best expressed in worrying over the requirement that one can make sense of "same time" across different worlds, which is normally entered as an unexamined and unquestionable presupposition of Tx W theories. Of course in the theory of instants we rely on a postulate that is similar to this presupposition. That we introduce this postulate explicitly and separately after giving the fundamental ideas of branching time is conceptually significant, since we can develop the main ideas of tense and modality without it. The Tx W approach hides this independence of tense and modality from the "same time" problem. Further, our "same time" postulation is explicit rather than presuppositional, so that it is easy to give it up when it does not help, or when it does not ring true. In short, a Tx W theory obviously provides no way to avoid presupposing instants or times shared across histories. Therefore the extent to which the concept of agency does not presuppose same-time comparisons across histories is the extent to which we have a good reason for not using Tx W as a foundation for the theory of agency. (ii) The idea of a possible world is the idea of something Very Big and (we suppose) hard to understand. As possibilities go, there are (we suppose) none bigger than worlds. It seems somehow a pity to start with something so big when what we want to understand is Caesar's situation when deciding whether or not to commence crossing the Rubicon. It would seem better to begin with a theory about more local incompatible possibilities, such as those available within ten or fifteen minutes, or available within ten or fifteen seconds, or (best) available immediately. The argument is that it is easier to credit a foundation built upon small possibilities, such as "the alternative moves in an actual chess game that I did not finally make" (Marcus 1985/86), rather than a foundation built upon something as large as worlds.5 Different from Tx W is the framework employed in Chellas 1969, which one could call an "h:T—> S" theory: Each history, h, is taken as a mapping from 5

Marcus herself allows the point as persuasive, but argues against taking it seriously. It would be out of place to respond to her argument here.

198

Foundations of indeterminism

the set T of times to a set 5 of states of affairs. An h:T—> S theory allows that histories can "overlap" in the sense of sharing exactly the same states of affairs at exactly the same times, and perhaps, though we are unsure, this is genuine overlap. Still, on h:T—> S theories, states of affairs are repeatables that can in principle occur at different times; they are therefore quite different from moments, each of which belongs to but a single instant, as is appropriate to a concrete event. Perhaps one could obtain a useful h:T—> S theory from branching time by locating an interesting sense of "state of affairs" as partially characterizing moments. Otherwise the h:T—>S theory seems faced with the double demerit of needing to provide both an account of times, and an account of states of affairs, on pain of being without application. Perhaps, for instance, belief in "states of affairs" is wild-eyed metaphysics (we hope not). Thomason 1984 discusses the notion of a "Kamp structure" which is a variation on the TxW idea. (The literature uses the word "frame" more often than "structure"; there is no difference in concept.) In a Kamp structure, each world is provided with its own temporal ordering. No real ground is gained, however, since using these structures to simulate branching still requires the concept expressed by saying that two worlds "perfectly match up through" a given time. Perhaps ground is even lost, since the ontology of Kamp structures requires making sense of the possibility that the times of two worlds could be dramatically different in their ordering, while nevertheless sharing some particular entity ontologically classified as a "time." For example, the respective times of two worlds might each contain both 4:00 P.M. and 5:00 P.M. The first world, however, could put 4:00 P.M. and 5:00 P.M. in their natural order, while the second world reverses their order. In considering what this could mean, we seem to lose our grip. In any case, all of our philosophical objections to Tx W as a theory of the structure of our world apply equally to Kamp structures. (This discussion is indebted to correspondence with A. Zanardo.) Both TxW structures and Kamp structures can be defined with first-order conditions. Yet another such representation of branching times is the notion of an "Ockhamist structure" (X, <, ~ ) of Zanardo 1996. The idea is that one can represent the individual histories of Tree by letting < be the union of disjoint linear orders, and one can represent the moments of Tree by making ~ be an equivalence relation standing for "same moment as." We learn from Zanardo 1996 that Kamp structures, and Ockhamist structures, while exhibiting different ontologies, are mathematically equivalent representations. We shall complain about them equally, and about Tx W as well. The complaint is, however, easiest to understand in terms of "bundled trees" and "moment-history structures," each mathematically equivalent to Kamp and Ockhamist structures. "Bundled trees" are closer to branching time in their ontology. (The idea is from Burgess 1978, 1979, 1980, the phrase from Zanardo 1996.) Bundled trees take "branches" as fundamental, where a branch is defined, using our terminology, as a forward-maximal chain (a future history) having a moment as its initial. This change in ontological perspective, while useful, is not, we think, of great importance. The reason is that branches are evidently in one-toone correspondence with our moment-history pairs, and do approximately the

7. Agents and choices in branching time with instants

199

same work. There is, however, something about bundled trees that is indeed of considerable significance: Apart from perspective, bundled trees result from our trees in exactly the same way that "Henkin general models" of second order logic result from standard models of second order logic (Zanardo 1996). Like second order logic, general models take the domain of individuals as primitive. Quite unlike second order logic, however, general models also take the range of predicate variables as primitive, which permits them to license the omission of some of the subsets of the domain of individuals, provided certain constraints are satisfied. Analogously, bundled trees take the set of branches to be primitive. This permits bundled trees to license the omission of branches, provided constraints are satisfied. When one adjusts for ontological perspective, this is tantamount to yet a fifth representation: One takes both the set Tree of all moments and the set History of all histories as primitive, in what might be called a moment-history structure , subject to the requirement that each moment belongs to at least one history. In this way one is able to replace second-order conditions with firstorder conditions. Since first-order theories are always more technically tractable than second order theories, professional logicians tend to prefer to study them. This is by no means a misplaced "preference," since the study of these more general moment-history structures provides a great deal of illumination in the way that, for example, much illumination is to be had by studying nonstandard first-order models of arithmetic, or of second order logic. As a reading of Thomason 1984, pp. 151-152, suggests, however, there remains a question about the descriptive adequacy of the more general moment-history structures when taken as theories intended to be true to the facts of our world (or language?). In order to evoke the negative thrust of our opinion, we will label these more general structures as "missing-history structures." An example will provide the reason that we think missing-history structures are descriptively inadequate. (After articulating the example we will make some remarks that examine how our negative opinion looks in the context of Tx W, leaving it to the reader to draw conclusions about Kamp structures, Ockhamist structures, and bundled trees.) Let there be a specially interesting radium atom, a, such that as the seconds tick by after moment mo, the situation is as follows. As long as a has not yet decayed, (i) a might decay before the next tick, and (u) a might not decay before the next tick.

,..,

We don't need metrics in order to describe the situation, but it is essential to the story that the sequence of ticks has no upper bound. The situation is pictured in Figure 7.2, a diagram that we borrow from Thomason 1984. With reference to Figure 7.2, we let p <-> atom a has not (yet) decayed, so that

200

Foundations of indeterminism

Figure 7.2: Suppose hw is not a history

~p <-> atom a has (already) decayed. (Note that p is true or false independent of history, so that it makes sense to label the moments themselves with p or ~p.) At issue is whether this situation can be properly described if the rightmost sequence, hw, is not considered to be a history. The missing-history approach says "yes," whereas our no-missinghistory approach says "no." The following is clear from Figure 7.2, as well as from (1): Without exception, every no-decay chain (from mo) of length n can be extended to a no-decay chain of length n +1.

(2)

The "can" of (2) is not just mathematical. This "can" is to be taken in its usual historical-modal sense involving quantification over histories. In other words: At mo it is a settled fact that if a no-decay chain of length n will come to pass, then it is possible (but not guaranteed) that a no-decay chain of length n +1 will come to pass. The truth value of (2) does not depend on whether or not our world is missing the history, hw; (2) holds whether or not we count hw as a history. The truth value of the following, however, depends on precisely that. At mo, it is inevitable that a will decay after a finite number of ticks.

(3)

This may be restated in various ways. (i) At mo, it is inevitable that, sooner or later, the atom, a, decays. ( i i ) At mo, it is inevitable that the chain of no-decays terminates. (in) It cannot be that a never decays. (iv) That a never decays is impossible. (v) At m0 it is false that a might never decay.

7. Agents and choices in branching time with instants

201

In (3) and in all of its restatements, the modal words are to be taken as historical modalities, equivalent to quantification over histories. So it matters whether the maximal chain of moments, hw, which is determined by the rightmost sequence of moments, counts as a history. On our no-missing-history account, hw has to be a history merely in virtue of being a maximal chain in the tree: (Tree, <> has the histories that it has, and if (Tree, <> represents our world with its causal order, there is nothing more to say. Therefore the no-missing-histories account says that (3) is, in all its versions, false. Suppose, however, that hw does not count as a history, as is certainly allowed if a missing-histories structure, , can describe our world. Then all the versions of (3) are true. If you leave out hw as a history, then no matter what history, the sequence of no-decay terminates after a finite number of ticks. Does this quarrel finish in a draw between the missing-histories and nomissing-histories representations of our world? We don't think so. It seems to us plain, following an analogous verdict by Thomason 1984, pp. 151-152 (but contrary to the verdict of 0hrstr0m and Hasle 1995, pp. 268-269) that anyone who asserts both (2) and (3) has contradicted himself. Surely, we say, it is a real possibility, not to be ruled out by switching "logic," that the atom may never decay. There is also an argument against the legitimacy of the missing-histories representation that does not depend on intuitions concerning the validity of hardto-understand arguments. We introduced moments as representing concrete possible events, and < as representing the (indeterministic) causal ordering among them. This gives us as rooted in objective reality—idealized, of course. The set of all histories is uniquely determined in terms of Tree and <, so that we take that set as itself objective rather than made up to suit conversational context, or language, or the like. Let us now consider a missing-histories structure, (Tree, <, History>, where one or more histories is missing from History. How can we see as objective the separation of histories (maximal chains) into those that belong to History and those that do not? Using "chronicle" where we say "history," 0hrstr0m and Hasle 1995 argue that we must "assume that not all linear subsets [of Tree] are possible chronicles." But what objective property of our world could justify treating some maximal chains as real possibilities and others as not? These questions seem to us to have only implausible answers; consult the following observations. Observations. (i) Probabilities don't come into it, just possibilities. One may wonder, however, if a case could be made for missing histories by forging a conceptual identification of "impossible" with "zero probability." No, for standard reasons: It is all too likely that in our world, every endless branch has zero probability, but there is no sanity in letting all of them go missing. (ii) It is no good subtracting from a tree all those branches containing only points that belong to other branches. It is indeed true that there is only one of these in Figure 7.2; it is, however, all too likely that in our world, every branch contains only points that belong to other branches, ( i n ) Some persons might think that Omnipotent God can rig things so that both (2) and (3) are

202

Foundations of indeterminism

Figure 7.3: Missing history in Tx W perspective

true. God could arrange things, for instance, so that on the one hand, as long as the atom has not decayed, it is guaranteed that it has another chance not to decay, and nevertheless, God forbids that it should never decay. Or maybe lawful nature can rig things in this way. We don't understand it. How could even God, or nature, prohibit that the atom will never decay, given that each stage of nondecay can be prolonged? You can say it of course, but does it make sense? We doubt it. Coming back to an earlier point, each of the bundled tree and the Ockhamist structure and the Kamp-structure representations, being equivalent to the moment-history representation, deserves the same missing-histories complaint, as does Tx W. In the case of Tx W and Kamp structures, however, it is considerably less obvious that there is an unreasonable "omission" rather than a reasonable "resistance to addition" (A. Zanardo, correspondence). Let us spell this out. In, for example, Tx W, one might have the infinite collection of disjoint worlds {w1, ... w5, ...}, as in Figure 7.3. In this figure we represent a single moment by means of a collection of points, one for each world, that sit on the same level. A single point then corresponds not to a moment, but to a moment-history pair. The heavy dots represent branch points, and the topmost dots, marked "*", represent the last moment of no-decay in that world. The Tx W diagram of Figure 7.3, with or without ww, is representationally equivalent to the branching-time diagram of Figure 7.2, respectively with or without hw The point is that if you stare at just the infinite collection of worlds {W0, ...}, without ww,, you may well not feel that there is a "missing world." You may instead feel that in passing from {w0, ...} to {w0, ...; ww}, an unexpected world has been added. We certainly think that such a feeling is justified by the diagram. From that perspective, it seems an open question as to whether "limit worlds" such as ww should or should not be added in all cases. You may share in the feeling that

7. Agents and choices in branching time with instants

203

when History is the set of all histories, (Tree, <, History> is somehow unusual or special. You may therefore prefer to abandon the language of "missing histories" versus "no missing histories" that we have employed because it suggests that having all the histories is philosophically normal. You may instead prefer to speak of "moment-history structure" versus "complete moment-history structure," with its suggestion that having all the histories is special. Are we then back to seeing the quarrel as finishing in a draw? We don't think so. We think that the Tx W picture is a mere diagrammatic representation: There is more in the diagram than there is in our world. In particular, there seems to us no objective truth to all of those disjoint "worlds" in Figure 7.3 except to the extent that they jointly represent our one objectively real world with its concrete events in their indeterministic causal order giving rise to a system of overlapping, branching histories. We recommend not trusting diagrams like Figure 7.3 when they are not rooted in objective features of our only world. To put the matter another way, if what binds all the points on a certain level into the representation of a single concrete event is not sheer identity, then there is nothing else objective for it to be. "Matching" is a myth. One may take the fact that Tx W or Kamp diagrams can mislead as an additional complaint against them: They conceal the truth about the structure of our world by means of too many henscratches and too much loose play in their free-floating History parameter. On the other hand, we certainly recognize that to the extent that we are arguing from premisses, our argument is circular. Someone who believes that the TxW diagram of Figure 7.3 gives the right picture of the "facts" of the decay example of Figure 7.2 will draw quite the opposite conclusion about both.

7B

Theoretical reflections on indeterminism

Before passing on to consider the postulates governing agents and their choices, we sharpen our understanding of determinism/indeterminism, and clarify the idea of branching histories by considering some objections.

7B.1

Determinism and its denial

This theory about Tree does not contain an explicit denial of determinism, but at least we can give a slightly freshened account of what determinism means. You will note that the account involves neither laws nor theories nor any other human creations. It is in this sense "objective" or "natural." First we need a concept. UNDIVIDED. (Definition. Reference: Def. 4) h1 = m0 h2 iff m0 E h1nhh2 and there is an m1 such that mo < m1, and m1 E1 h1nh2 (unless there is no m1 such that m0 < m1). We say that h1 and h2 are undivided at mo. We adapt undividedness-at to pasts as well: Two histories extending a past, p, are undivided at p iff they share a moment properly later than p, so that they appear as a single line as p comes to a close.

204

Foundations of indeterminism

That is, not only must the histories share m or p, but they must also share some properly later moment. Now we can say with absolute clarity what it is for a concrete situation to be deterministic. DETERMINISTIC. (Definition. Reference: Def. 4) Tree is deterministic at a moment, m [or at a past, p] iff every pair of histories through m [or p] is undivided at m [or p]. There may be many histories through a deterministic past, p, but if so, they must become "many" after p is past; there is no branching as p itself comes to a close if p is deterministic. There is another way of saying this that involves the notion of immediate possibility. IMMEDIATE POSSIBILITY. (Definition. Reference: Def'. 4) A set, H, of histories is an immediate possibility at a moment, mo iff H is a subset of H( mo ) that is closed under undividedness at mo: (h1 E H and h1 = mo h2) —> h2 E H. So for Tree to be deterministic at mo is for there to be but a single immediate possibility at mo. Sounds right. Obviously Tree may be deterministic at m but not so at either earlier or later moments. On the present account of determinism, one can coherently believe that our world used to be deterministic but is not so now, although it may become so once again. We go on to say that Tree itself is deterministic if it is deterministic at every past. In that case there is obviously but a single history. A determinist is someone who believes that the tree (world) of which our moment is a part is deterministic. It would appear that many philosophers believe that anyone who is not a determinist is softheaded and probably needs therapy. Others believe that anyhow all respectable philosophical theories, including theories of agency, should at least be consistent with determinism. Determinism, however, is an extremely strong theory, going far beyond determinism of the present moment. In any event, we are not determinists, even though the denial of determinism is not a postulate of this book. But more than that, on the theory here offered, if anyone could ever see to anything, then determinism is false. So even though we do not lay down indeterminism as a postulate, since we believe that sometimes people have choices, we are indeterminists. Accordingly we think that any theory (of anything) should be compatible with at least a little indeterminism. We are "compatibilists" in the best sense. We agree with Kane 1998 that in particular the question "whether a kind of freedom that requires indeterminism can be made intelligible" (p. 105) deserves, instead of a superficial negative, our most serious attention, and indeed we intend that this book contribute to what Kane calls "the intelligibility question." Note, incidentally, that some situations in our world could, for all we know, be governed by indeterminist laws that are nonprobabilistic. Such a law might describe the sorts of possible outcomes for some type of initial event, but without carrying information concerning the relative probability of those outcomes. Agency may or may not be like that.

7. Agents and choices in branching time with instants

205

It is abstractly interesting to observe that our world could be strongly antidetermmistic in the sense that there is splitting of histories at every moment, so that H( m1 ) = H( m2 ) implies m1 = m2 for every m1 and m2 in Tree. It seems to us entirely possible that strong antideterminism is true—but maybe not. Certainly it would be mathematically attractive to be able sometimes to think of a moment as a set of histories (those that pass through it), just as we can think of a history as a set of moments (those that it passes through). With equal certainty, however, this is not a good place at which to argue from beauty to truth. As a final word on this topic, we note that even "strong" antideterminism is not all that strong. Even "strong" antideterminism is consistent with the truth of numerous and important determinist theories about "systems" and "states" and such. Indeterminism is not disorder.

7B.2

Arguments against branching

In a passage that we also quote on p. 170, Lewis 1986 advances persuasive arguments against branching time. The trouble with branching exactly is that it conflicts with our ordinary presupposition that we have a single future. If two futures are equally mine, one with a sea fight tomorrow and one without, it is nonsense to wonder which way it will be—it will be both ways—and yet I do wonder. The theory of branching suits those who think this wondering is nonsense. Or those who think the wondering makes sense only if reconstrued: You have leave to wonder about the sea fight, provided that really you wonder not about what tomorrow will bring but about what today predetermines. (pp. 207—208 of Lewis 1986; we quote the remainder of this passage later, on p. 208) In addition to our earlier comments, we respond to this argument in two quite different ways. The first response is that it takes our ordinary ways of thinking too seriously. We draw an analogy, due to Burgess 1978 (p. 165), between (i) objections to "some of our ordinary ways of speaking" based on standard relativity theory and (ii) objections to some of those ways based on branching time. In fact we sharpen the Burgess analogy by emphasizing the similarity of the roles of "many histories" in indeterminism and of "many frames of reference" in the theory of relativity. When taken as a piece of argumentation, the analogy could go like this. If the foundation of the Lewis argument in our ordinary ways were solid, then the following would be an easy reduction to absurdity of the theory of relativity. The trouble with the theory of relativity exactly is that it conflicts with our ordinary presupposition that we have a single present or "now." If two presents or "nows" are equally mine, one with a sea fight at Neptune's north pole and one without, it is nonsense to wonder which way it now is—it is now both ways—and yet we do wonder.

206

Foundations of indeterminism

That is a bad argument. Certainly many of us have a tendency to wonder what is going on at Neptune's north pole (or on the far side of Sagittarius) right "now." The presupposition underlying this wondering is "ordinary" and perhaps even "natural" for clock-aware and scientifically sophisticated persons born in the last few hundred years. But nevertheless it is false, as we are taught by relativity. We urge as apt the analogy suggested by this rewording of the quoted argument against branching time: Relativity insists that our world provides us with no uniquely natural spatially extended "now," although it may permit us to consider, if we wish, the limited family of all nows ("hyperplanes") to which a given utterance-event belongs. And indeterminism insists that our world provides us with no uniquely natural spatiotemporally extended actual history, although it permits us to consider, if we wish, the limited family of all histories to which a given utterance-event belongs. To the extent that common sense asks for a unique naturally given "now" to which a given utterance-event belongs, or for a unique naturally given "actual history" to which a given utterance-event belongs, to that extent, common sense is asking for something it cannot have. Nevertheless, the Prior-Thomason semantics, which explicitly recognizes the relativity of many statements to histories as well as moments, can give common sense a large amount of what it wants and can correct some parts of the Lewis formulation that are too hasty. "We have a single future." If this means that it is settled what will happen, for example, that either it is settled that there will be a sea battle or settled that there will not, it is false. If it means that it is settled that incompatible events will never happen, it is true. If it means that there is a single future history following upon this utterance, it is false. If it means there is a single future of possibilities, it is true. Branching time indeed claims that it can happen that "two futures are equally mine, one with a sea fight tomorrow and one without." But what does it mean to say that a future is "mine"? Branching time says that it means that it is among the futures now possible, where the "now" is indexically mine. To avoid making branching time look silly in a way that it surely isn't silly, the quoted description should be amended by insertion as follows: "two possible futures are equally mine, one with a sea fight tomorrow and one without." Lewis misdescribes the theory of branching time in saying of such a situation that "it will be both ways." Branching time is entirely clear that "Tomorrow there will be a sea fight and tomorrow there will not be a sea fight" is a contradiction. What is true and not surprising is that "It is possible that tomorrow there will be a sea fight and it is possible that tomorrow there will not be a sea fight" is eminently consistent. Nor is this merely a matter of formal tense logic. It seems to us deeply realistic to take it that if the captain is faced with two possibilities, sea battle tomorrow or no sea battle tomorrow, then those are possibilities for him, on that occasion. They are equally his, not one more than the other, exactly in accord with Lewis's account of (not his own theory but) branching time. Suppose the sea battle comes to pass. Then (after the sea battle) the two possibilities were his, and were equally his. In particular, branching time rejects the Lewis

7. Agents and choices in branching time with instants

207

shadow-theory according to which the captain himself is to be found in the world of the sea battle, whereas merely one of his "counterparts" can be located in worlds without sea battles. The point goes back at least to Burgess 1978, who reflects as follows on his earlier decision to go to the office instead of the seashore: "Tense logic insists, pace Lewis, that I am the very same person who could have gone to the shore; it's not just someone like me who could have gone" (p. 173). Suppose we "step outside of branching time." To do this is to confine ourselves to language that has no trace of indexicality, a perfectly proper thing to do (see note 14 on p. 162). Histories and moments and persons are then linguistically accessible only via (rigid) naming or quantification. Branching time then says that for suitable moment mo and histories h1 and h2, the captain lives through a sea battle the day after m0 on h1, and lives through no-sea-battle the day after mo on h2. Here seems a premiss for a reductio of branching time, for branching time then seems to say that the captain has it both ways, both living through a sea battle and living through no-sea-battle. The reductio is, however, an illusion. Omitting the relativization to histories is intolerable. What branching time says is that the captain "has it both ways" in the entirely innocuous sense that he lives through a sea battle on history h1 and lives through no-sea-battle on history h2. That just says that there are at mo two possibilities for him, a fact about our world that we must keep. It does not say that the two possibilities will each be realized, an absurdity that branching time denies. It does not say that these possibilities remain possibilities at moments after the sea battle has commenced. It only says that in the past of such moments the two possibilities were available. Current possibilities drop off (McCall 1994) with passage into the future, but not the fact that they once were possibilities. Once was-possible, always was-possible. What about "wondering" whether or not there will be a sea battle? Evidently our wondering is history-independent: The fact that we wonder is dependent on the moment but independent of the history parameter. So what sense can we make of wondering about a history-dependent complement such as "there will be a sea battle tomorrow"? Lewis points out one alternative, which he rightly presents as not very ordinary, namely, that the complement of the wondering is the history-independent question, whether or not it is now settled that there will be a sea battle tomorrow. We proposed in §6E a two-point understanding of wondering that lets its complement remain open in the history parameter. First, it seems natural to construe "wondering q," where q is an indirect question (e.g., "whether there will be a sea battle tomorrow"), as "wanting to know" (or perhaps "wanting to have") a true answer to the question of q. Second, it seems obvious that our wants are not normally satisfied immediately; we must in general wait. Just so, in order to obtain satisfaction of the want expressed in our wondering, we shall need to wait until tomorrow; for only tomorrow is it possible for us to come to know whether or not a sea battle comes to pass. We next comment on some ideas in the remainder of the passage the beginning of which we quoted once on p. 170 and again on p. 205. Lewis continues in the following way.

208

Foundations of indeterminism But a modal realist who thinks in the ordinary way that it makes sense to wonder what the future will bring, and who distinguishes this from wondering what is already predetermined, will reject branching in favour of divergence. In divergence also there are many futures; that is, there are many later segments of worlds that begin by duplicating initial segments of our world. But in divergence, only one of these futures is truly ours. The rest belong not to us but to our other-worldly counterparts. Our future is the one that is part of the same world as ourselves. It alone is connected to us by the relations—the (strictly or analogically) spatiotemporal relations, or perhaps natural external relations generally—that unify a world. It alone is influenced causally by what we do and how we are in the present. We wonder which one is the future that has a special relation to ourselves. We care about it in a way that we do not care about all the other-worldly futures. Branching, and the limited overlap it requires, are to be rejected as making nonsense of the way we take ourselves to be related to our futures; and divergence without overlap is to be preferred. (p. 208)

The first sentence refers to "the future," the one that "is truly ours." Branching time says that only the future of possibilities is uniquely determined by the moment of utterance, so that "the future" either refers to this, or else is not history-independent (is open in the history parameter). Branching time says that if indeterminism be true, then there is no more sense to "the actual future" than there is to "the actual distant instantaneous present" or to "the odd prime number." But what about the future that "is part of the same world as ourselves"? Assuming indeterminism, there is the following dilemma. • If we read "world" as "history," then it makes no sense to speak of "the world of which we are part." There are many such possible histories to which this utterance-event equally belongs. All of them are "connected to us by the ... spatiotemporal relations ... that unify a world," for there is, in our opinion, no more fundamental "natural external relation" than the causal ordering itself. It is to be borne in mind that even wholly incompatible moments are mediately connected by <; that is exactly the import of historical connection. It is why (or how) Tree constitutes a single world, our world. • If we read "world" as Tree in its entirety, then although it would make sense to speak of "the world of which we are part," it would not make sense to speak of "the future" history that is part of that world. On the other hand, invoking a familiar contrast, it would indeed make sense to ponder "the future of possibilities." Lewis then gives what are in effect three arguments that, contrary to branching time, "this very event" picks out a unique future history.

7. Agents and choices in branching time with instants

209

• The first argument is that we can define "the future history" as the one that "alone is influenced causally by what we do and how we are in the present." This sounds all right, but it is not. It is not merely that we cannot evaluate this proposal in the absence of an objective theory of "causal influence" and of "what we do." Nor is it just a matter that the argument won't tell against a theory such as ours that explicitly holds that if there is only one possible future history issuing from this present event, then, the future being settled, there is no "influencing" the future by what we do in the present. The difficulty is above all that the proposed definition of "the future history" is akin to defining "the odd prime number" as the one that "alone is not divisible by two." • The second argument seems to be that the future is the one we wonder about. As we indicated, however, at the very end of chapter 6, and again a couple of pages ago, to the extent that wondering is wanting to know, wondering is similar to other wants. Whether one wants to know what will happen as the open future becomes determinate, or one wants to have a dapple gray pony, one must bide one's time. • The third argument is about caring. Certainly branching time shares with this passage the premiss that we do not care about other-worldly futures. Indeed the passage suggests a certain mild ad hominem: What we care about is what alternatives there are for us to choose among. We do not (much) care what alternatives there are for other-worldly counterparts, should they exist as Lewis's theory requires. We don't see how counterpart theory can either make sense of or reject the demand that future (incompatible) possibilities be for us rather than for our counterparts. Suppose for example that we are choosing whether or not to start a sea battle. Surely we care about what will happen if there is a sea battle, and we also care about what will happen if there is not a sea battle, since these are possibilities for us. And given that we really do have a choice, and know that we do, such caring about incompatible alternatives makes sense. If these histories are (right now) all really possible, then we do now rightly care what is true on each. We really do care about what happens on more than one history—as long as the histories are ours.

7B.3

Cusp of causality

There is more than one theory of causality in branching time; see von Kutschera 1993 and Xu 1997. Here we ask: Suppose that A, which in general is dependent on the history of evaluation, is nevertheless settled true at m. A has become a settled "fact." When did this become so? If we can answer this, we shall have found a causal locus for the "effect" that A is settled true at m. If A is the sort of sentence that changes its truth value over time, then the question may be difficult (Belnap 1996a gives an unsatisfying answer). If, however, the truth of A is moment-independent, then the question becomes manageable. In this case,

210

Foundations of indeterminism

A is variable only in the history parameter. "Dated" sentences such as "At 4:00 P.M., A," or renditions of "The coin will land heads sometime tomorrow" have this feature: Their truth depends on the history of evaluation (and in the second example also on the moment of use), but not on the moment of evaluation. So suppose A is moment-independent. And suppose that A is in fact settled true at m. To look for a causal locus, we rely on the following. 7-1 FACT. (Settledness of moment-independent sentences) If the truth of A is independent of the moment parameter, then settledness propagates forward and lack of settledness (openness) propagates backward: If m1 < m2, then if A is settled true [false] at m1, it is also settled true [false] at m2; and if A is not settled true [false] at m2, it is also not settled true [false] at m1. Consider now the improper past of m, call it p. It could be that A has been settled true throughout p, in which case, since settledness propogates forward, A is "universally true," that is, A is settled true throughout all of branching time. Suppose, then, that m has not been settled true from all eternity. Then because of Fact 7-1, we may make a Dedekind cut of p into two nonempty chains: At every moment in the lower portion, call it copen, the truth of A will depend on the history parameter; while at every moment in the upper portion, call it csett, A will be settled true. Thus, where copen draws to a close is the very "point" at which the status of A changes from "not yet settled to be true" to "now settled to be true." We put "point" in shudder-quotes because copen may not end in a moment, so for definiteness we call all of copen the cusp of causality. The cusp of causality is where to look for the causal locus of the "effect" that A is settled true at m. It is well to keep in mind that m, where we have supposed A to be settled true, may itself minimally upper bound the cusp of causality. For example, if the A in question is that we are at the restaurant at 6:00 P.M., Murphy's Law suggests that it may well take right smack up to 6:00 P.M. in order to settle that fact in our favor; in which case the cusp of causality is the entire set of proper predecessors of the moment of our being at the restaurant at 6:00 P.M.. Some people think this is a defect in the theory; we think it is a defect in the world. Though certainly imperfect in that it is nonrelativistic, the concept of the cusp of causality is nevertheless an empirical, objective causal concept. The idea is that nothing can be an effect unless it changes from not-being-settledto-be-true to being-settled-to-be-true, and that the locus of causality must be at the "point" at which this change takes place: The cusp of causality is just where the effect comes to be settled true.

7C

Theory of agents and choices

We have finished discussing the postulates governing the BT + I part
7. Agents and choices in branching time with instants

211

possibility of making cross-history time comparisons. In this section we go on to consider the remaining parameters Agent and Choice, with the aim of characterizing how agents and their choices fit into the causal structure of our world as represented by , and in the next section we add a coda about Domain.

7C.1

Agents

Nowhere in this book do we offer thoughts that help much when considering the mental equipment or other aspects of the "real internal constitution" of agents. The topic is important, and important to us; it is just that our attempt at progress in this book takes us in a different direction. Alan Ross Anderson used to say that all progress in philosophy comes by simply assuming certain problems have been solved even though they haven't, and getting on with the investigation. We pursue this policy by entering a significantly uninformative postulate concerning agents. AGENTS. (BT +I + AC postulate. Reference: Post. 6) set.

Agent is a nonempty

We call the members of Agent agents, and we let lowercase Greek letters a and (3 range over agents. We intend that the concept of Agent is absolute in the sense of Bressan 1972 (or a substance sort in the sense of Gupta 1980), which means that we may "identify" agents across times and histories. In particular, there is no fission and no fusion of agents as we move from moment to moment. In this book, however, we do not happen to discuss questions such as de re versus de dicto ascriptions of agency, nor do we worry about when agents come to be or pass away. That is why we can get by with the simpler set-theoretical representation of Agent given by Post. 6. There is a brief discussion of Agent as an absolute concept in §10C.l, where we are worrying about joint agency, and a little more in §12F, where we are thinking about generalizing on the agent position of stit statements. None of this, however, is offered as a serious contribution to solving the problem of personal identity. On the other hand, we do think that consideration of agents in branching time may deepen that problem. BT + AC theory, for example when considering strategies, characterizes the same agent as making some (possible) choices sequentially, and also some (possible) choices under incompatible circumstances. That the idea of "same agent" seems essential to the idea of, for example, a strategy is, we think, a good reason to suppose that a concept of personal identity that does not essentially involve agency is much too partial. A concept of personal identity that depends on exclusively "passive" notions such as experiential content, or on only backward-looking ideas such as memory, is by so much inadequate.

7C.2

Choices

The penultimate parameter of a BT + I + AC structure is Choice, which tells what choices are open for each agent at each moment in Tree.

212

Foundations of indeterminism

CHOICE PARTITION. (BT +1 + AC postulate. Reference: Post. 7) Choice is a function denned on agents and moments. Its value for agent a and moment m is written as Choiceam. For each agent a and moment m, Choiceam is a partition into equivalence classes of the set H( m ) of all histories to which m belongs. The postulate may strike one as unhappy to the extent that it seems to insist that choosing is localized in a moment, and is therefore instantaneous. The opposite seems so natural: Choosing takes time. We therefore feel obligated to explain, in part, our reasons for giving a key role in BT + 1 + AC theory to such a prima facie counter-intuitive postulate. Jack has made a deliberate choice to go to the beach. There is, we think, no harm in postulating that there is a momentary event that entirely precedes his beginning to deliberate; let it be m1. There is also no harm in postulating a momentary event that lies thoroughly after his arrival at the beach; let it be m2. To find a plausible m1 and m2 we don't need to know exactly how Jack's deliberation relates to its outcome; we only need to know that his deliberation has a beginning that precedes his arrival at the beach. Having fixed m1 and m2 in this way, if deliberation is to have point, it must not be decided at m1, before the deliberation begins, whether or not Jack later arrives at the beach (Aristotle). And at m2, after Jack arrives, it must certainly be decided that he arrives (how else?). So now draw a chain of moments from m1 to m2. Since that chain has members, which we call moments, then by simple Dedekind analysis, there must either be a last moment of undecidedness or a first moment of decidedness, or both (a Dedekind jump) or neither (a Dedekind gap). Our theory of branching time does not say which; it only says that there is a transition from undecidedness to decidedness. Nevertheless, our thoughts go—we think harmlessly—with those who think of the flow of our world as continuous, and hence without jumps or gaps. Furthermore, it is certain that if our world is continuous, and can be accurately represented by a continuum of instantaneous moments, then there must be either a last instantaneous moment of undecidedness or a first such of decidedness. Since epistemology is not likely to help us choose between these two, we see no objection to our always thinking of a last instantaneous moment of undecidedness, a last moment at which it is still not decided whether Jack will arrive at the beach. (Evidently there will also be such a last moment of undecidedness if our world proceeds by discrete jumps.) Talk of "deliberation flowing into action" is all right unless it blinds us to these observations, and especially to the primacy of an objectively more-orless localizable transition from undecidedness into decidedness. It is this idea that is idealized in Post. 7. We theoretically identify "the moment of choice" as the last moment before the matter is decided, while still thinking of choice itself as fundamentally a transition from undecidedness to decidedness. (See §2A.2 for discussion.) Observe that nothing in the postulate denies that deliberating takes time. If deliberation is mere wheel-spinning, we say nothing about it. If, however, it involves, for example, a continuous (or discrete) ruling-out of alternatives, we should then represent it theoretically as a continuum (or succession) of choices.

7. Agents and choices in branching time with instants

213

Finally, we observe that the postulate says nothing about whether choice is localized in some homunculus in the brain, or whether it is always made by a neuronic "community," or indeed whether choice is in any way localized. Nor does any part of our theory say anything about what choice feels like, or whether every choice is a conscious choice. These are difficult and important questions. We remind the reader here, however, as we do elsewhere, that our explicitly and advisedly limited theory concerns only the causal structure of choice, to the exclusion of its "content." Nor will it hurt to insert our view that those who deal with the "content" of choice to the exclusion of its (indeterminist) causal structure can easily be led astray. The fundamental postulate Post. 7 on choice says that histories are divided evenly into equivalence classes; we discuss this aspect after we introduce the equivalence-relation notation for choices, together with some closely related notation that we use with considerable frequency. CHOICE NOTATION. (Definition. Reference: Def. 11) i. Choice represents all the choice information for the entire Tree. ii. Choiceam gives all the choice information for the agent a and the moment m; Choiceam should be thought of as a set of possible choices, and we call it "the set of choices possible for a at TO." By Post. 7 we know that Choiceam is a partition of H( m ) : Each history on which TO lies belongs to exactly one member of Choiceam. It is essential to keep in mind that a possible choice is a set of histories, not a single history. No one—of course—can choose their future down to the smallest detail. iii. Choiceam (h) is defined only when h passes through TO, and is then the unique possible choice (a set of histories) for a at TO to which h belongs. The notation is justified by the fact that according to Post. 7, each member of H( m ) picks out a unique member of Choiceam to which it belongs. iv. choiceam (m1) is defined only when m1 is in the proper future of the moment TO of choice. Pick any history, h, containing m1, a history that will a fortiori contain TO. Then Choiceam(m1)is defined as Choiceam (h). This definition is justified by no choice between undivided histories, Post. 8, which we have not yet discussed in this chapter. We include the notation in this list for reference. Its theoretical import is this: If we are at a moment m1, such that a moment of choice, TO, lies in its past, the "the choice that a made at TO" is uniquely determined; it is a settled fact. v. Choiceam (m1) is defined only when instants are present, and when m1 is properly future to m. Recall that i( m 1 ) is the instant on which m1 lies. Then Choiceam (m1) is defined as the set of all moments on the instant i( m 1 ) that also lie on some history in Choiceam (m1). In symbols, Choiceam(m1 ) = i(m1) n uChoiceam (m1). If we are at a moment m1, such that a moment

T of choice, m, lies in the past, then Choiceam (m1) picks out the "projection" of the uniquely determined choice made by a at TO onto the present instant or time.6 It is the intent of the postulate Post. 7 on Choice that when agent and moment are fixed, choice equivalence is an equivalence relation on the histories passing through the moment: reflexive, symmetric, and transitive. For example, to say that Choiceam(h1) = Choiceam (h 2 ) is another way of saying that h1 belongs to Choiceam (h2). The next definition describes the equivalence-relation notation for choices. CHOICE EQUIVALENCE. (Definition. Reference: Def. 12) and we say that h1 and h2 are choice equivalent for a at m. We say that h1 and h2 are choice separated for a at TO. is defined only when instants are present, and when TO < m1, We say that m1 is choice equivalent to m2 for a at m. hi |am h2 is contrary to h1 =am h2. It makes sense to think of a given possible choice as "separating" each history in the choice from each history not in the choice. So if we wish to describe separation in a sentence that makes "a" the grammatical subject, it is all right to say something like "a has a possible choice at TOO that separates h1 and h2." We should not give in to the temptation to say that "a can choose between h1 and h2." To put the matter as clearly and therefore trivially as possible, a can never choose between histories, but can only choose between choices. We now consider whether Choice is correctly described as a partition, or, what is technically equivalent, whether choice equivalence for a at TO is a reflexive, symmetric, and transitive relation on H( m ). Reflexivity is perhaps a "throwaway" postulate, except to the extent that it implies that every history belongs to some possible choice. Perhaps it makes sense to say that at some moment, if certain things happen a has chosen, but if other things happen, a has not chosen; J. MacFarlane has suggested some potential cases, and has urged that we keep our minds open to this possibility. Nevertheless, it seems to us that speaking causally, and regardless of how people evaluate a given agent or situation, we tend to side with those who find conceptual difficulty in an alleged situation involving the simultaneous possibility of choice and no-choice. We do, however, agree with MacFarlane that we know much too little to be warranted in discouraging the development of alternative theories along these lines. 6

Choiceam (m1) gives a set of histories and Choiceam (m1) gives a set of co-instantial moments, which is confusing. The underlining on Choice is a mnemonic intended to help keep the two concepts apart by calling to mind the horizontal picture of an instant.

7. Agents and choices in branching time with instants

215

Symmetry seems to us inescapable when it comes to choices. Transitivity, as usual, is the easiest feature of a partition-postulate to doubt. Given that we have no choices that distinguish h1 from h2, and no choices that distinguish h2 from h3, are we forced to conclude that there are no choices for the agent that distinguish h1 from h3? This seems to us at least a quasi-empirical question, not decidable by easy philosophical speculation. Is the nontransitivity of "not noticeably different" (e.g., for colors) a reason for doubting the transitivity of choice equivalence? Whether yes or no, it would be good to have a theory that did not presuppose transitivity of choice equivalence. We have none to suggest, and so are content to "admit" that in the end our ideas, which make heavy use of transitivity, only apply when transitivity either seems plausible, or at least seems to be an idealization that does not interfere with our modest progress in understanding agency. Especially for the theory of strategies, it is good to generalize choice equivalence and choice separability from concepts based on a single moment to concepts based on a set of moments. Think of M in these definitions as a collection of choice points lying in your future. Which histories can you tell apart by choices at these points? (See Def. 12 for the choice-equivalence notation used in defining these ideas.)

INSEPARABILITY/SEPARABILITY. (Definition. Reference: Def. 13)

The fundamental idea is carried by h1 =am h2, which says that no choice for a in M separates h1 and h2. "Inseparability" in this context is a geometrically suggestive rhetorical variation on "choice equivalence" that correctly intimates the ncmtransitivity of the relation. Its contrary is also conceptually important; two histories are "separable for a in M," for instance, if somewhere in M there is a choice for a that keeps one of the histories as a possibility while ruling out the other as thereafter impossible. Figure 7.4 provides an example of nontransitivity of inseparability, using a simple case of a chain, c, consisting of the two moments {W0,w1} represented by divided rectangles. The division of each rectangle represents that at each moment there are two possible choices for a. The history, h, that splits from the chain at the lower moment is choice inseparable for a by c from each of the histories h1 and h2 that split at the top moment; but obviously h1 and h2 are not choice inseparable for a by c from each other. That is, no choice that

216

Foundations of indeterminism

Figure 7.4: Nontransitivity of choice inseparability at a chain

a makes in the course of c separates h from h1, or h from h2, but there is evidently a choice (at the top moment) that separates h1 from h2. So choice inseparability for a at a chain is not and should not be transitive. There are two further postulates relating to choices by agents. The first amounts to a "new" principle relating agency to the causal order, while the second concerns multiple or joint agency.

7C.3

No choice between undivided histories

This postulate requires the previously given temporal-modal definition of "undivided," Def. 4(iv). No CHOICE BETWEEN UNDIVIDED HISTORIES. (BT +1 + AC postulate. Reference: Post. 8) If two histories are undivided at m, then no possible choice for any agent at m distinguishes between the two histories. That is, one of two histories undivided at m belongs to a certain choice possible for a at TO if and only if the other belongs to exactly the same possible choice. In symbols from Def. 4 and Def. 12: As reported in, for example, chapter 1, we learned the no choice between undivided histories condition from P. Kremer in 1987. All the postulates having to do with choices are to be found in one form or another even earlier in von Kutschera 1986. This postulate is perhaps the most "interesting" of the BT + 1 + AC postulates. As far as we know, the idea of relating choice to brute causal undividedness has no earlier rigorous expression, even though the relation is and must be of importance to the theory of action and to moral theory. An easy consequence is that from the point of view of a properly later moment m1, what choice an agent made at each properly earlier moment m, is uniquely determined. Or to say the same thing from the point of view of m, for each moment m1, properly later than TO, there is a unique possible choice for a at m that contains all histories passing through m1. For argument, assume m1 is properly later than TO. Then the two moments constitute a (two-member) chain that by Zorn's lemma can be extended to a maximal chain, that is, a

7. Agents and choices in branching time with instants

217

history, h, that contains them both. Existence is partly given by Choiceam ( h ) , which evidently contains a history, namely, h itself, that passes through both m and m1. To continue with existence, let h' be any history through m1, and hence, by no backward branching, through m. By no choice between undivided histories, h' must belong to Choiceam ( h ) , which then contains every history through m1. As for uniqueness, if any possible choice for a at m contained all histories through m1, it would also contain h, which would imply its identity with Choiceam ( h ) , as required. Provided m1 is properly later than m, this fact justifies introducing the concept, "the choice that was made by a at m," where the past tense is issued from the point of view of m1. In Def. 11 (iv) we introduced the notation Choiceam (m1) for this concept. It follows from no choice between undivided histories that, to the extent a choice objectively limits the future, you cannot today "choose to choose" to have potatoes tomorrow. You can say "Tomorrow I will choose to have potatoes" and you can intend today that tomorrow you choose to have potatoes, and you can stay well within the bounds of commonsense language by saying "I hereby choose to have potatoes tomorrow," even when you know full well that today's choice does not determine the matter, since tomorrow you can always change your mind. You may even be in a position to choose today that you, without further choice, have potatoes tomorrow. (Perhaps you put yourself on automatic pilot.) What you cannot do is both choose today and also choose tomorrow. You can have it both ways in conversation or in your mind or in your philosophical gloss, but you cannot have it both ways in reality. One way of putting the matter is this: Adopting a plan for the future is not executing that plan; and this fact derives from the fundamental principle that there can be no choice between undivided histories. A final thought: This fundamental principle surely has consequences for the concept of "weakness of the will." It does not speak to the mental side of that concept, but it does help to place weakness of the will in the proper causal context. There would be no weakness of the will if one could choose between undivided histories.

7C.4

Independence of agents

INDEPENDENCE OF AGENTS. (BT +1 + AC postulate. Reference: Post. 9) If there are multiple agents: For each moment and for each way of selecting one possible choice for each agent, a, from among a's set of choices at that moment, the intersection of all the possible choices selected must contain at least one history. In symbols: for each m £ Tree, and for each function fm on Agent such that Sometimes this is thought of as "independence of choices," which is a good thought. At any one moment m, the choices possible to each agent are indeed independent in the sense, for example, that any possible choice, Choiceam ( h i ) , that a1 makes is consistent with any possible choice, Choiceam2 ( h 2 ) , that a2 makes: The intersection of these two sets must be nonempty. Since the choices

218

Foundations of indeterminism

are simultaneous, it is certainly reasonable to think of them as independent. What might be confusing about the language of independence is this: The entire set of choices, Choiceam1, open to a1 at m is by no means "independent" of the entire set of choices, Choiceam2,, open to a2 at m. The very fact that each member of the first set must have nonempty overlap with each member of the second is itself a fierce constraint. It is easy to see, for instance, that no two agents can possibly have exactly the same possible choices at exactly the same moment (vacuous choice aside). Fierce or not, however, we think that this postulate is banal. If there are agents whose simultaneous choices are not independent, so that the choice of one can "influence" what it is possible for the other to choose even without priority in the causal order, then we shall need to treat in the theory of agency a phenomenon just as exotic as those discovered in the land of quantum mechanics by Einstein, Podolsky, and Rosen. (This point of view can be found in the consideration of the Prisoners' Dilemma by Green and Bicchieri 1997, where it plays an important role.) We are in effect postulating that the only way that the choices open to one agent can depend on the choices open to another agent is if the one agent's choices lie in the causal past of those of another agent. That almost concludes our one-by-one discussion of the BT +I + AC postulates. Before briefly considering one more postulate, we describe several concepts that prove useful in various contexts.

7C.5

Vacuous choices and busy choosers

Agents choose at "choice points," but happily not all moments are such. Although we propose no theory of the coming to be and passing away of agents, at least we can use the idea of "vacuous choice" to avoid suggesting that each agent is constantly and forever choosing: A vacuous choice (which is, incidentally, the only sort offered under determinism) is not a choice. VACUOUS CHOICES. (Definition. Reference: Def. 14) • A moment TO is a choice point for a iff there is more than one possible choice for a at TO. • A possible choice for a at TO is vacuous or trivial iff it is the only possible choice for a at m; and is otherwise nonvacuous. There can only be a vacuous choice for a at m when Choiceam = {H(m)}, in which case m itself is said to be a trivial choice point for a. The idea of a busy chooser is that of an agent that makes infinitely many choices in a finite period. It is odd that although the idea is altogether peripheral to our central ideas, in technical discussions it crops up with unexpected frequency. BUSY CHOOSERS. (Definition. Reference: Def. 14) • A chain c is a busy choice sequence for a iff (i) c is both lower and upper bounded in Tree, and (ii) c is an infinite chain of (nontrivial) choice points for a.

7. Agents and choices in branching time with instants

219

• a is a busy chooser iff some c in Tree is a busy choice sequence for a. We don't pretend to know if there are any busy choosers, or even if the idea makes sense. (We don't think anyone else knows either.) All we do is keep track, as well as we can, of places in the theory of agents and choices in branching time at which it makes a difference. In this regard, the following is worth noting: Sometimes what makes a difference is the existence of an unending forward series of nonvacuous choices, while sometimes it is the existence of an unending backward series.

7D

Domain

The last entry in the structure (Tree, <, Instant, Agent, Choice, Domain) is Domain, which we employ as a field of possible denotata, and as the range of individual variables. RICHNESS OF Domain. (BT +1 + AC postulate. Reference: Post. 10) The domain of quantification, Domain, must include Tree, History, Instant, and Agent as subsets. That is, we postulate that moments, histories, instants, and agents shall all be among "what there is." We enter this postulate partly for some technical reasons that emerge in chapter 8, and partly to emphasize how harmless and nonparadoxical it is to include these entities in the domain of quantification. These entities are not in the least "meta-linguistic," nor does their addition make the domain "too big" to be a set. Of course Domain must be subject to some artificial limitation or other in order to avoid paradox, but we think that this set-theoretical (or even ontological) problem has nothing whatever to do with agency or the causal structure of our world.

8

Indexical semantics under indeterminism Prior chapters of this book have offered bits and pieces of a description of agency in our indeterministic world. This chapter brings us to the topic of the semantics of a language taken to be used by those (ourselves) who live in such a world. Just to have a label, we sometimes call this topic either "the semantics of indeterminism" or "indeterministic semantics." Agency constructions we treat in §8G; until then we emphasize branching time itself. We include (i) quantificational devices, (ii) temporal constructions, (Hi) historical modalities, (iv) some mixed modalities, and (v) indexicals tied to the context of use. We include all of these items because they crop up in discussions of indeterminism, sometimes in a confusing way. We say what we have to say in the idiom of formal semantics. This amounts to an organized account of the semantics needed for a language spoken in an indeterminist world. First we go over and extend some of the foundational and generic semantic ideas broached in chapter 6. Then we go one by one through a large number of constructions useful for understanding indeterminism, in each case giving an exact semantic account. We emphasize points that we take to be important, and we draw out the analogies among and differences between the semantics of quantifiers, historical-modal and tense connectives, and indexical connectives, and indicate how they are to be combined. We briefly review the already-presented semantics for the achievement stit and the deliberative stit, and we explain how stits might be witnessed by a chain of choices (instead of a single choice). The chapter ends with a mention of an alternative agency construction, the transition stit. Let us note that in compensation for the inevitable tedium of processing henscratches, the early sections expand on the fundamental ideas that go into the formal semantics.

220

8. Indexical semantics under indeterminism

8A

221

Sources

As we briefly noted at the beginning of §6B, we draw on four sources for the key ideas needed for the semantics of indeterminism. • Quantification. Tarski's studies in the 1930s provide the foundation for all compositional semantics, and in particular are the source for our compositional understanding of quantifiers. • Tenses. Prior 1957 initiated the compositional understanding of linear tenses. • Historical modalities. Prior 1967 as made rigorous by Thomason 1970 (see also Thomason 1984) is the source of our understanding of the historical modalities, and also of how tenses should be understood given a representation of indeterminism by means of branching time. • Context of use. Kaplan 1989, much of which had circulated in typescript since Kaplan's 1971 lectures, provided the full-scale development of "indexical semantics" based on the idea that compositional semantics must pay delicate attention to the context of the use of an expression. The semantics of indeterminism relies heavily on all four of these sources, which we discuss in turn as ways of following out the fundamental Tarski idea of relativizing truth to parameters.

8A.I

Tarski's quantifier semantics

Tarski saw (among much else) that, if the account of truth for quantificational sentences is to proceed in a straightforward compositional fashion, one must relativize truth to new parameters that go beyond an explanation or translation of the various constant features of the language (including a specification of domain) . Those new parameters are assignments of values to the variables. Why did it take someone as exceptional as Tarski to attain this insight? Perhaps because most of us think primarily in terms of stand-alone sentences (vehicles of assertions), which should contain no free variables. In this case one does not need the assignment parameters. One needs them only when thinking of a sentence as arbitrarily embeddable, specifically including the possibility of embedding within the scope of a quantifier. And one must think of sentences in this way if one is to pass recursively from a semantic account of Fx1X2 to a semantic account of Ex2Fx1x2, and thence to an account of a potential stand-alone sentence, Ax1Ex2Fx1x2. It does not suffice to have an absolute "truth value" for the embedded open sentences Fx1x2 and Ex2Fx1x2. One needs to know the truth value of these embedded sentences relative to each appropriate family of assignments of values to the variables. In the deepest sense, a quantifier such as Ax1 is not "truth functional." Tarski himself perhaps makes this a little difficult to see by his linguistic detour through his "satisfaction" relation, but the underlying point shines through:

222

Foundations of indeterminism

Open sentences such as F X 1 X 2 do not have an absolute truth value. They rather have a truth value only relative to assignments of values to the variables, so that such assignments are parameters of truth. Let us be explicit about how we extract the assignment parameters from Tarski's account so that those familiar with Tarski may understand our stylistic departure. His fundamental locution is illustrated by

where the members of the sequence such as a and b are individuals drawn from the domain. To make this work, one must know which member of the sequence, (a, b, ..., ), goes with which variable. In this example, we have used subscripts whose sole purpose is implicitly to supply an ordering of the variables. This order is to be used in conjunction with the order of the sequence, so that we know that a goes with x\ and b with x2. The ordering of the variables is essential to the meaningfulness of the Tarski relation of satisfaction by sequence: One must understand which member of the sequence goes with which variable. As Tarski observes, however, one does not need for this purpose to suppose a primordial ordering of the variables such as supplied by subscripts. A more local ordering will do as well: One could, for example, take the order from "first occurrence in the sentence under consideration." In this case, a would go with £2 and b with x\. No matter: In any case, since what "satisfies" a sentence is a sequence of entities from the domain, one must have some ordering of the variables in order to be able to say which entity goes with which variable. The first step in our extraction of the assignment parameters is to observe that, if we take what satisfies a sentence to be a sequence, there is always a detour through some sort of imposition of an order on the variables. This detour has as its only purpose defining which entity in the sequence goes with which variable. We avoid the detour by letting the second argument of the satisfaction relation be not a sequence but a function, a, defined on the variables, such that a directly assigns an individual in the domain to each variable, without presupposing a sequencing of any kind. So after this first step we have, for example,

The second step is both linguistic and conceptual. Linguistically, we merely reword the very same relation. Instead of "a satisfies FX2X1," we write

The conceptual aspect of the second step is that in choosing (3) we are emphasizing that an expression such as "Fx-^Xi" is grammatically sentential. We know it is sentential because it is subject to exactly the same embedding operations as any other sentence: conjunction, negation, and the like. Therefore, for example, "Fx^Xi" deserves a sentential semantics, which is to say, an account of the conditions under which it is true, in this case not absolutely, but relative to

8. Indexical semantics under indeterminism

223

an assignment of values to the variables. The language of satisfaction obscures this. We call anything to which truth is relativized a parameter of truth (§6B.l). Once we see the need to relativize truth to assignment parameters, it is natural to consider other parameters. What in addition do we need to fix a truth value for, say, Ex2Fx1X2? Evidently we need both an interpretation of F specifying the ordered pairs to which it truly applies, and also a domain specifying the range of the quantifier, Ex2. These are additional parameters of truth. The assignment parameters are, however, quite different from the interpretation and domain parameters. What makes them different is that quantifiers such as 3x2 are translocal in the assignment-to-X2 parameter in the sense that in order to fix the truth value of Ex2A relative to a particular assignment to X2, one must sometimes look at the truth value of the embedded sentence, A, relative to assignments other than the assignment with which one starts. The assignment-to-x2 parameter is therefore mobile in the sense that the language contains operations, namely, quantifiers such as 3x2, that are translocal in the assignment-to-X2 parameter. In contrast to the quantifiers, the truth functions are one and all local in each assignment parameter: To fix the value of, for example, ~A at a family, a, of assignments to variables, one needs only the value of the embedded sentence, A, at a itself. And when one thinks of the domain and the interpretation of constants as parameters of truth, then these parameters are immobile if the language, as is typical or even universal, contains no operation that is translocal in them. The key concepts of the preceding paragraph will be useful in speaking clearly about the language of indeterminism. "Mobile parameter" and "immobile parameter" were previously defined; see Definition 6-1 on p. 143. Here we insert definitions of the relations "local in" and "translocal in" between connectives and parameters. In order to avoid excess detail, however, we offer only approximations to the exact concepts. 8-1 DEFINITION. (Local/'translocal and mobile/immobile) • A connective (or any grammatical operation), o, is local in a parameter if in fixing whether o (A) is true or false at a certain value of that parameter, one needs to consider the truth or falsity of A at only that value. Examples. Ax1 is local in the assignment-to-X2 parameter. Truth functions are local in every parameter. Every operation is local in the Domain parameter. • A connective, o, is translocal in a parameter if in fixing whether o (A) is true or false at a certain value of that parameter, one must consider the truth or falsity of A at other values of the parameter. Example. Ax1 is paradigmatically translocal in the assignment-to-X1 parameter. Before moving on to linear time, branching time, and indexicals, let us mention a parameterization in our neighborhood that we do not discuss. Modal logic

224

Foundations of indeterminism

characteristically adds one or two new immobile parameters, the set of "worlds," and perhaps a "relative possibility" relation on that set, and also a mobile parameter, namely, the "world" parameter. The familiar modal connectives are translocal in the world parameter. This relativization of truth to worlds is not needed for understanding indeterminism as we conceive it, and so we add no such parameters. On the other hand, the technical devices and conceptual ideas of modal logic due especially to Kripke are essential to both plain or linear tense logic, and to the historical modalities of branching time.

8A.2

Prior's linear tense semantics

Just as Tarski taught us how to interpret quantifiers by adding special parameters of truth (§8A.l), Prior has in the same way shown us how to understand much about tenses. Linear tense logic (generally called simply "tense logic") is due to Prior 1957. On the semantic side, linear tense logic characteristically adds three new parameters to which to relativize truth: an immobile set of "times," an immobile linear "temporal" ordering of this set, and a mobile "time of evaluation" parameter. All operations are local in the set of times and the temporal ordering, whereas the familiar tense connectives such as "it was true that," "it will be true that," and "it has always been true that" are translocal in the time parameter. We do not much discuss linear tense logic as a separate topic, since its ideas appear willy-nilly in the semantics for branching time. Reason: Each history is linearly ordered, so that we will see its workings in that context. For more information about linear tense logic, see Burgess 1984.

8A.3

Prior-Thomason's branching-time semantics

We have seen that we learn from Tarski how to interpret quantified sentences (§8A.l), and from Prior how to understand linear tenses (§8A.2), in each case by thinking of truth as relativized to certain parameters. In exactly the same way, we learn from Prior 1967 and Thomason 1970 that if one wishes to be coherent, one has to interpret the subtleties of English tenses with the help of yet additional parameters of truth based on branching time. No philosopher who wishes to argue either for or against branching time should remain ignorant of this work; the penalty is contamination of an otherwise responsible appraisal by an almost certain—but unnecessary—tendency to slip into tensed modal muddles. We spell out some of the details of the Prior-Thomason semantics. The indispensable idea from Prior-Thomason is that truth shall be relativized to moment-history pairs, where the moment belongs to the history. So, in addition to the immobile tree structure itself, there are two new mobile parameters to which truth (as well as denotation, etc.) is relativized: the moment of evaluation, and the history of evaluation. Linear tense operations are translocal in the former, shifting from moment to moment; and the historical modalities are translocal in the latter. The most difficult part of this Prior-Thomason idea, the part that goes beyond linear tense logic, is negative in character:

8. Indexical semantics under indeterminism

225

Given indeterminism, it does not suffice to think of truth (or denotation, etc.) as relative only to moments. One may wish that this were not so, but (in our opinion) it is. One must relativize truth to the history parameter as well. The reason is that only thus can we make sense, in branching time, out of plain (linear) future-tense sentences such as There will be a sea battle tomorrow.

(4)

Think of (4) as uttered before the admirals have made their decisions. Then the truth of that sentence (given indeterminism) depends not only on the moment at which the sentence is uttered. It depends in addition on which future course of events—which history—is being considered. (In §6D we investigated a variety of futile shortcuts.) To put the matter in easily understood words: Given indeterminism, what will happen (for example, whether or not there will be a sea battle tomorrow) depends on what will happen (i.e., on which history is being considered). In yet other words, a sentence such as (4) depends on the history-of-evaluation parameter in exactly the same sense that a sentence like x1 is an admiral

(5)

depends on the assignment-to-X1 parameter. In each case, one has an example of "dependence" on a parameter in the sense of Definition 6-4, p. 153.

8A.4

Kaplan's indexical semantics

Kaplan 1989 explains with maximum lucidity how indexical expressions should be semantically understood. (All page references are to this work by Kaplan.) The indispensable idea extracted from Tarski is that truth shall be relativized to assignments of values to the variables (§8A.l). The indispensable idea from Prior's linear tense logic is that truth shall be relativized to time (§8A.2). And as we have just seen, the indispensable idea from Prior-Thomason is that truth shall be relativized to moments and histories (§8A.3). In the same way, the indispensable idea from Kaplan is that truth shall be relativized to a context of use.1 Kaplan uses the context of use to determine much: the speaker of the context (used to obtain a denotation for "I"), the place of the context (used for "here"), the "world" of the context (used for "actually"), and the time of the context (used for "now"). These we can conveniently label as "context parameters." They are brand new, and not to be identified with the mobile history or moment parameters. The latter were introduced to help with some translocal connectives, but the context parameters are used for indexicals. To avoid complications irrelevant to indeterminism, we omit consideration of speaker and place. Of more significance is that we replace world and time with 1 Warning: Actual utterance is not what is at stake. For example, we wish to evaluate all the sentences of an argument in the same context of use. Kaplan explains this; see p. 546.

226

Foundations of indeterminism

a single context parameter, the moment of use. If someone utters something, then (ideally) there is a unique moment to which that utterance is tied. Each such utterance has a unique causal past, and a unique future of possibilities, the whole of which is summed up by the moment of use. Among indexicals to which the moment of use contributes are Now: and Actually:. So the assignment parameters come from Tarski, the moment-of-evaluation and history-of-evaluation parameters from Prior-Thomason, and the moment-of-use parameter from Kaplan. We need two more families of parameters: the "structure parameters" and the "interpretation parameters." Were our semantic purpose to stop with "truth," although we would in that case certainly require these parameters, we could leave them implicit, as does Tarski, and indeed as we do in many places in this book. Leaving them implicit in the recursive account of "truth" would be warranted by the observation that the structure and interpretation parameters are immobile, so that they never need to be varied in the course of the recursion explaining truth. "Truth," however, is not our only goal. In order to introduce ideas of "semantic equivalence" and "semantic implication" and "semantic validity," we shall need to quantify over values of the structure and interpretation parameters. For this further purpose, we use the next two brief sections to be explicit about them.

8B

Structure parameters: The "world" of the speakers

In order to fix truth, one needs some "structure parameters" to represent various features of the "world" of the speakers. We shall need parameters for the set of moments, for the ordering of these moments, for the set of instants, for the set of agents, for the choice function, and for the domain of quantification. When we gather these individual parameters, we call the result ''the structure parameter." A structure is a value of the structure parameter. A typical structure will look like (Tree, <, Instant, Agent, Choice, Domain), and we let 6 range over such structures. The key definition is this: & — (Tree, <, Instant, Agent, Choice, Domain) is a BT + I + AC structure iff 6 satisfies the postulates on branching time with agents and choices summarily listed in §3, postulates that it was the business of chapter 7 to discuss in detail. "BT + I+AC" stands for "branching time + instants + agents and choices," or, as we chiefly say, "agents and choices in branching time with instants" (leaving it to context whether or not Domain, needed only for quantification, is included). In other places in this book we also discuss other structures of various kinds. The chief cases are these: (i) a BT structure ("BT" for "branching time") has the form (Tree, <, Domain), (ii) a BT + I structure has the form (Tree, <, Instant, Domain), and (iii) a BT + AC structure has the form (Tree, <, Agent, Choice, Domain). See §2 for a list of all of the kinds of structure that we treat.

8. Indexical semantics under indeterminism

227

In each case we implicitly allow that Domain might or might not be present, depending on whether or not quantification is at issue.

8C

Interpretation and model: The "language" of the speakers

Parallel observations hold for the "language" of the speakers: What has elsewhere been implicit must here be made explicit. For this reason, we introduce "the interpretation parameters," the value of each of which fixes an appropriate "meaning" for a single "nonlogical" constant. We assemble values of all the interpretation parameters into a single value, called an interpretation, of "the interpretation parameter." We shall let 3 be an interpretation. J must represent the nonlogical atomic features of the "language" of the speakers by fixing a meaning of an appropriate type for each atomic nonlogical constant.2 We say what we mean by "appropriate type" only for BT + I+AC structures, leaving other cases to trivial adaptation. In order to carry out this task, we first offer a preliminary account of the grammar of the language(s) of indeterminism that we propose to discuss. This is intended to be the same mini-language that we introduced as L in §6B.l (with the caveat of note 4 on p. 141 that our account will be loose). In this chapter, however, we hardly ever use the denomination L. We call on some atomic features, and some modes of composition. Here we indicate only atomic features, composition by operators, and predication, as given in slightly more detail in §8. We leave the reader to infer the sentential modes of composition (the connectives) from the semantic clauses that we give in §8F. Typically any one of our discussions draws on only part of the following equipment: prepositional variables p; individual constants u (some of which are special terms for agents a, and one of which, t, denotes "the non-existing object" to be available as a throw-away value of definite descriptions); individual variables xj; operator letters f; and predicate letters F. INTERPRETATION. (Definition. Reference: Def. 15) 3 is an &-interpretation for a BT + I + AC structure & = (Tree, <, Instant, Agent, Choice, Domain) iff 3 is a function defined on propositional variables p, individual constants u (some of which are agent terms a and one of which, t, is to denote "the non-existing object"), operator letters /, and predicate letters F, such that 3 assigns to each propositional variable a function from Moment-History into {T, F}; assigns to each individual constant (including t) a member of Domain; assigns to each agent term a member of Agent; assigns to each n-ary operator letter a function 2 We tend to think of the "atomic" features of the language, except for individual variables, as "nonlogical," and the compositional features as "logical." In this study, however, we use "logical" and "nonlogical" without analysis and without serious commitment. We do refer the reader to MacFarlane 2000 for an argument that important features of the language of something like branching time should perhaps be classified as "logical" due to their proposed connection with the concept of assertion.

228

Foundations of indeterminism

from Moment-History into functions from Domainn into Domain; and assigns to each n-ary predicate letter a function from Moment-History into functions from Domainn into {T, F}. Representation of "world" and "language" must fit, and when they do, we call the combined representation a "model." MODEL. (Definition. Reference: Def. 15) m is a BT + I + AC model (based) on 6 iff m is a pair , where 6 is a BT + I + AC structure, and where 3 is an G-interpretation. We repeat that every one of the structure and interpretation parameters is immobile (the language has no operations translocal in any of the structure or interpretation parameters). We also repeat that their immobility is a principal reason that elsewhere we have been able to assume that the representations G of features of "world" and 3 of features of "language," encoded in the idea m of a model, are fixed implicitly in one particular way that makes it plausible that they are adequate idealized representations of certain features of our world and our language. Grouping parameters of truth. We now have all the parameters we need. We rely on Definition 6-1 in order to characterize them as immobile or mobile. As indicated there, the structure and interpretation and moment-of-context parameters are immobile, whereas the assignment, moment-of-evaluation, and history-of-evaluation parameters are mobile.

8D

Points of evaluation, and policies

Truth is relative to a specification of each parameter. Once we have fixed the values of the structure and interpretation parameters by specifying a model, m = , we still need to fix a number of other parameters: all the assignment-tovariable parameters, the moment-of-use parameter, the moment-of-evaluation parameter, and the history-of-evaluation parameter. Not just any fixing will do, as we indicate in the following definition of a "point." POINT. (Definition. Reference: Def. 16) A BT + I + AC point is a tuple <m, mc, a, m/h>, such that m is a BT + I + AC model, mc E Tree, a is a function from the individual variables into Domain, m E h, and h E History. Henceforth we assume that <m, mc, a, m/h> is a BT + I + AC point. We speak of the various parameters in ,<m, mc, a, m/h> in the following way: m is the BT + I + AC model, mc is the moment of use, a is the assignment (of values to the variables), m is the moment of evaluation, and h is the history of evaluation.

8. Indexical semantics under indeterminism

229

In this usage, "point" is short for "point of evaluation." A point encodes a specification of the structure and interpretation parameters , of the context, and of each of the three mobile parameters of truth. 3 Sometimes, in order to avoid too many symbols, we use the following convention that lets us use TT for <m, mc, a, m/h>. TT FOR <m, mc, a, m/h>. (Definition. Reference: Def. 16) We let TT be the point <m, mc, a, m/h>, and adopt the following convention: In any context in which we write "TT," we will understand the expressions Tree, <, Instant, Agent, Choice, Domain, G, J, m, mc, a, m, and h just as if we had written "<m, mc, a, m/h>." It is striking that each point contains two independent references to moments: the moment of use, mc, and the moment of evaluation, m. They play of course entirely different roles: The moment of use is immobile (unshifted by any operation), and may be used to fix indexical expressions, whereas the moment of evaluation is paradigmatically mobile, being shifted by a variety of tense constructions. In addition to their uses in connection with various special indexical expressions, some context parameters have another role to play (Kaplan, p. 595; see our overlapping discussion in §6B.4.2). In the present case the following, which restates Policy 6-2, is critical. 8-2 OBSERVATION. (Starting evaluation of stand-alone sentences) The moment-of-use parameter is used to start the evaluation of any stand-alone sentence considered as being uttered to some purpose involving the semantics of the sentence, for example, uttered as an assertion. We expand the discussion of §6B.4.2. If you want to evaluate Themistocles was surprised,

(6)

and you understand that "was" moves evaluation into the past, you need a place to start that motion. When (6) is itself being considered as stand-alone, the moment of use gives us that starting point. It works in a special way: The (paradigmatically mobile) moment of evaluation is fixed, to begin with, as the very moment of use. It is only when we come to the sentence Themistocles is surprised,

(7)

which is implicitly embedded in (6) by past-tensing, that there is a divergence between moment of use (which remains the same as for (6) taken as stand-alone) and moment of evaluation (which needs shifting, existentially, toward the past). 3 Elsewhere in this book, in order to minimize complexity of exposition, we not only keep the model, m, implicit; we also pass over recognition of either the assignment-to-variable parameters or the context parameters. In short, in spite of thinking of just one target language, we relativize truth in different ways depending on which problems we are attacking. For example, when we are not concentrating on quantifiers, we omit relativization to the assignment parameters.

230

Foundations of indeterminism

You will note that in expressing Observation 8-2, which is derived from Kaplan, we use loose language. It is difficult to do otherwise for the following reason: Although we have in mind that a sentence can be considered either as stand-alone or as embeddable, the symbolic language (unlike, e.g., English) makes no such distinction. In the symbolic language, there is no syntactic mark (such as initial capitalization and final period in written English, or intonation in the spoken language) that distinguishes sentences taken as stand-alone from those taken as embeddable. This lack of match between English and the symbolic language makes analysis more difficult. Here is the best we know how to do (without describing a new kind of symbolic language) by means of a definition and a policy. (See Green 1998 for a study of "illocutionary-force-indicating devices," including Frege's sign of assertion.) First the definition, which is a BT +1 + AC-specific version of the more general concept indicated in Policy 6-2. CONTEXT-INITIALIZED POINT. (Definition. Reference: Def. 16) We say that a BT + 1 + AC point, <m, mc, a, m/h>, is context initialized iff the moment of use, mc, is identical to the moment of evaluation, m. That is, context-initialized BT + I + AC points have always the form <m, mc, a, mc/h>. The idea is that in a context-initialized BT + I + AC point, the mobile moment of evaluation is "initialized by" the moment of the context of use. Next the policy. 8-3 POLICY. (Differential

treatment of stand-alone and embeddable sentences)

• We recommend and urge, on pain of confusion, that sentences considered as stand-alone may usefully have their evaluation restricted to contextinitialized points. • We recommend and urge, on pain of confusion, that sentences considered as embeddable shall not have their evaluation so restricted, but that they shall be evaluated also at points in which the moment of use, mc, and the moment of evaluation, m, diverge. The rationale for the first part of this policy is that each utterance should be conceived as tied to a concrete context, and that such a context determines a unique causal position, with a definite past and a definite future of possibilities. We idealize such a position with the moment of use, mc. This moment of use is the very moment at which we wish to evaluate a stand-alone sentence. That is why, for stand-alone sentences, we initialize the moment of evaluation with the moment of use. Keep in mind, however, that the moment of evaluation is mobile, and can be shifted by tense constructions ingredient in the stand-alone sentence. And that is the very reason for the second part of Policy 8-3. Since there is in the symbolic language (and indeed often in the English examples of philosophers) no syntactic difference between stand-alone and embeddable sentences, the definition of "context-initialized point" (Def. 16) and

8. Indexical semantics under indeterminism

231

Policy 8-3 represent the best that we can do. The definition and policy are, we think, exceptionally useful in discussing tenses and indeterminism, for in those ventures the failure to observe the distinction between stand-alone and embeddable sentences is especially harmful. Here is one way the definition and policy offer immediate progress. Kaplan (pp. 505-506) asks that in thinking about the "character" or meaning of a sentence, we first fix context, and then ask for the content of the sentence in that context. There should, however, be' two notions of content-in-a-context, depending on whether we are thinking of the sentence as stand-alone or embeddable. If we are thinking of it as stand-alone, then the moment of evaluation is initialized by the moment of the context. Since one cannot reasonably treat a sentence with free variables as stand-alone (Assertability thesis 6-7), it is obvious that for stand-alone sentences the history is the only mobile parameter that is left to vary when considering a stand-alone sentence. If one correlates time to moment and world to history, this explains the otherwise puzzling phenomenon noted by Kaplan on p. 546: ... the truth of a proposition is not usually thought of as dependent on time as well as a possible world. The time is thought of as fixed by the context. That is right for stand-alone sentences: Time (or moment) is fixed, while world (or history) is not. If, however, we are thinking of the sentence as embeddable by means of translocal connectives such as tense operators, then for "content in context" we must let (i) the moment of evaluation diverge from and vary independently of ( i i ) the moment of the context. This explains why Kaplan permits content to vary over times as well as "worlds." We shall remain unclear as to the point of our semantic constructions unless we bear this in mind. We note that Kaplan is working in a Tx W framework (see §7A.6). It is, we think, an indication of the relative helpfulness of the moment-history framework that it explains a phenomenon that from the point of view of Tx W seems just puzzling. 8D.1

History of the context?

Policy 8-3 arises partly in virtue of the fact that there is a pairing of two parameters that have the same range of variation, namely, the moment of context and the moment of evaluation. This is sometimes called "double indexing." The phenomenon in general is of no special interest; after all, each xj parameter in quantification theory has exactly the same range as any other, so that, for example, x1 and x2 exhibit "double indexing" of the domain. When, however, as in the case of moments, one of the paired parameters is a context parameter and the other a mobile parameter, we may speak more particularly of "context-mobile pairing." We deepen our appreciation for Policy 8-3 if we ask the following two "context-mobile pairing" questions, one about histories, and one about assignments to variables. (The following discussion expands on that of S6B.5.)

232

Foundations of indeterminism

8-4 QUESTION. (History of the context?) Because we shall have "modal" connectives that are translocal with respect to histories, it follows that there must be a mobile "history of evaluation" parameter. Furthermore, the mobility of the history of evaluation plays an essential role in our account of assertion (Semantic account 6-12); but why is there not also a "history of the context," to be a paired with the mobile history of evaluation, as context parameter? If there were, for stand-alone sentences we could initialize the history of evaluation with "the history of the context." It would clearly make technical sense to provide a context parameter ranging over histories, to be interpreted as "the history of the context of use." And if we had one, we could enlarge our definition of "context-initialized point" to recognize this pairing. Further, there is plausibility in the Kaplan intuition that with "a little ingenuity" one can always make sense out of pairing a context parameter with a content parameter (p. 511). Indeterminism, however, compels a view absolutely contrary to this: There is no "history of the context." When you utter something, you do not thereby uniquely determine the entire future course of history. Your utterance has many choices and chances ahead of it, and so belongs to many histories. The context of use determines a unique moment, but not a unique history. Just to be explicit, we mean to be challenging principles such as that suggested by Kaplan on p. 597: "Any difference in world history, no matter how remote, requires a difference in context." Turning this around says that the identity of the context of use is enough to fix the course of world history, both past and future. That, if indeterminism be true, holds well enough for past history: The past, though largely unknown, is fixed. But it fails for the future: A single, well-identified context of use is typically part of a large variety of possible future courses of history. There is no unique "future of the context." The note on pp. 334-335 of Salmon 1989 is similar to Kaplan, though more convoluted. Salmon specifies the "quasi-technical notion of the context of an utterance," to be distinguished from the "utterance" itself, by saying that if any facts had been different in any way, even if they are only facts entirely independent of and isolated from the utterance itself, then the context of the utterance would, ipso facto, be a different context, even if the utterance itself remains exactly the same. Salmon concludes that ... although a single utterance occurs in indefinitely many different possible worlds, any particular possible context of an utterance occurs in one and only one possible world. If, however, "facts" are what is fixed at the moment of utterance, whether they are "isolated" or not, they cannot fix what the future brings—if indeterminism be true. Distinguishing "utterance" from a quasi-technical notion of "context of utterance" cannot make it otherwise. From this perspective, it looks as if Salmon is wrong—if indeterminism be true. It is, however, a delicate matter to label the

8. Indexical semantics under indeterminism

233

difference between Kaplan-Salmon and ourselves as a difference of opinion. Here is a place in which there is a dramatic difference between the ideas of "world" and of "history." Kaplan and Salmon and, for example, Lewis seem to picture a "world" as something like a single space-time. We, in contrast, picture a world as something like an indeterministic tree. What is common to these two pictures is the use of "world" for a system of events connected externally by some sort of causal-type relation. It is for this reason that on either picture, it is plausible that a given utterance, or context of utterance, will determine its one and only "world." This seems close to common ground. If, however, histories can branch indeterministically in the way that we suggest, then a single utterance, together with all the most distant "facts," belongs to many histories, no one of which is specially determined as "actual" by the moment of utterance. Of course one could define "context of utterance" as a pair consisting of the moment of utterance together with a particular future history, the "actual" future history, and Salmon's note seems to suggest that he does in fact rely on the notion of an "actual" future history. He seems to rely, as do others, on the view that among all possible futures, one is marked out as a "Thin Red Line" in exactly the sense that we decried in chapter 6.

8D.2

Assignment-to-variables of the context?

As we discussed in §6B.5, there is still another family of mobile parameters, the assignment parameters, for which we do not provide a matching context parameter. We expand on that discussion. 8-5 QUESTION. (Assignment-to-x\ of the context?) Because we shall have "quantifier" connectives that are translocal with respect to the assignment-to-X1 (for example) parameter, there must be an "assignment-to-:ri of evaluation" to be something like a "content" parameter, in which the quantifier is translocal. But why is there not also an "assignment-to-rri of the context" to be a paired context parameter? If there were, we could initialize the mobile assignmentto-xi parameter with the context assignment-to-x1 parameter for stand-alone sentences. It would clearly make technical sense to provide a matching assignment-to-X1 as a context parameter. In some passages, indeed, Kaplan suggests that image (see especially p. 592), although he does not go so far as to provide both an assignment-to-x1 context parameter and an assignment-to-x1 mobile parameter in which the quantifiers can be translocal: There is only one assignment-to-X1 parameter, not two. Here the explanation lies not in the nature of things, but rather in (presumably universal) linguistic practice. Assignments to x1 are not anchored in the context in any serious way. As Kaplan clearly says (p. 593), there is no "factof-the-matter" that contextually determines a unique assignment to x1. So the symbolic language we are describing fails to provide an assignment-to-x1 as a context parameter because there is nothing in our language (or in any language we know) to which such a technical device would correspond.

234

Foundations of indeterminism

It is good to recognize, however, that we speakers of English (supplemented with variables) could adopt a convention that supplied each variable with a value in each context of use. We could then go on to insist that, for stand-alone sentences, the mobile assignment used for evaluation is initialized by the context assignment. Suppose, for example, we require that the context determine that x1 and x2 are both assigned zero. That seems as good a convention as any, since there is (as Kaplan says) no "fact of the matter." Then if we took something like

as a stand-alone sentence, and accordingly used Policy 8-3 to restrict ourselves to context-initialized points of evaluation, we should find out that (8) is automatically true. Ugh. Having realized that we could adopt such a convention, we are glad that we speakers of English haven't done so. And the symbolic language we describe is, in this respect, just like ours. All of its context parameters are "fact of the matter" parameters really and objectively determined by the context of use; none are subject to some doubtful convention manufactured by a logician.

8D.3

Points and policies summary

(i) The special nature of context-initialized points is recognized. (ii) Policy 8-3 is firmly in place for differential consideration of stand-alone and embeddable sentences, (iii) That a point of evaluation, <m, mc, a, m/h>, contains a context parameter for the moment of use, but none for "history of the context of use" and none for the "assignment of the context of use," is no accident or idle logician-imposed convention, (iv) Nor is the fact that there is no analogy to Policy 8-3 for histories or assignments.

8E

Generic semantic ideas

In the following section we detail recursive semantic clauses for the various operators that we treat. Here we outline generic features of the semantic concepts that are defined by that recursion. Except for the "in-context" concepts, these are all standard. We are supposing two kinds of categorematic expressions, terms and sentences. The semantic value of a term is always an entity in Domain, while the semantic value of a sentence is always one of the two truth values, T or F. Since we are going to include operations that are translocal in assignments, moments, and histories, we know that we shall have to relativize semantic value to these parameters. We further explicitly relativize semantic value to the moment of context, even though it is immobile, partly for explicit indexicality, partly to exploit the idea of context-initialized points and the attendant Policy 8-3, but fundamentally because we are thinking of evaluating terms and sentences in many different contexts. And we explicitly relativize semantic value to

235

8. Indexical semantics under indeterminism

the structure and interpretation parameters because we wish to have not only "truth," but also "equivalence," "implication," and "validity." We want a way of referring to the semantic value of any categorematic expression, be it term or sentence. Once we have "semantic value" for sentences, we automatically have truth, falsity, and the idea so important for indeterminism, settled truth. SEMANTIC VALUE, DENOTATION, AND TRUTH. (Definition.

16)

Reference: Def.

• Semantic value. For any categorematic expression, E, be it term or sentence, Val m , m c , a , m / h ( E ) , is "the semantic value of E at the point (937, mc, a, m/h>." Val m,mc a m / h ( E ) is defined recursively by clauses given in §8F and §8G. Note that by the earlier clause of Def. 16 that appeared on p. 229, we may write Valtt (E) in place of • Denotation. Where t is any term, Val m , m c , a , m / h ( t ) E Domain. Valtt(t), or V a l m , m c , a , m / h (t) is "the denotation of t at the point <m, mc, a, m/h>." Also, as before, Valtt(t) stands in for Val m, m c , o , m / h ( t ) . • Truth. Val m,mc,a,m//i(A) is "the truth value of A at <m, mc. a, m/h>." Where A is any sentence, Alternate much-used notation for truth and falsity: - m, mc, a, m/h tt= A iff Va/m, m c , 0 ,m/h(A) = is true at point <m, mc, a, m/h>."

T.

Either is read "A

- m, mc, a, m/h ¥ A iff Va/gjt,m c ,a,m/h(-<4) = F. Either is read "A is false at point <m, mc, a, m/h)." We sharply distinguish settled truth, which is not history dependent, from plain truth, which is. SETTLED TRUTH. (Definition. Reference: Def. 17) • A is settled true at a moment m with respect to m, mc, and a iff 971, mc, a. m/h 1= A for all h € H( m )- We may drop h, writing m, mc, a, m t= A. • A is settled true throughout a set of moments, M, with respect to m, mc, and a iff m, mc, a, m \= A for all m 6 M. In addition to dropping h, we may replace m by M, writing m, mc, a, M M A. • We most often write just m, M -= A and m,1, m t= A since we use these concepts most often when assignment and context are not relevant. Note that in order to avoid confusion, we do not introduce special notions of "truth" or "settled truth" for stand-alone sentences. We have two kinds of points, but not two kinds of truth or settled truth. We do, however, as indicated next, define two notions each of equivalence, implication, and validity, to be

236

Foundations of indeterminism

used depending on whether a sentence is considered as embedded or as standalone. That is, we want strong notions of semantic equivalence, implication and validity, notions that are suitable for application to embeddable sentences and terms, and we also want versions suitable for stand-alone sentences. We pick out the latter concepts with the help of the adjective "in-context," which always signals restriction to context-initialized points. We define the ideas only for BT + I+AC structures and models, leaving other cases for adaptation. EQUIVALENCE. (Definition. Reference: Def. 18) • Expressions EI and E2, either both terms or both sentences, are (semantically) equivalent iff for all BT + I + AC points (971, mc, a, m/h), Val<m,mc,a,m/h(Ei) = Valyx,m=}a,m/h(E2) (v^ry same semantic value at all BT + I + AC points). We write • Expressions E1 and £2, both terms or both sentences, are in-context equivalent iff for all context-initialized BT + I + AC points, (971, mc, a, mc/h), (the very same value at all context-initialized points). We write Semantic equivalence warrants replacement in any grammatical position, no matter how deep the embedding, and is thus of great importance. In-context equivalence, on the other hand, is suitable only for sentences considered as standalone, or occasionally for terms when considering replacement of one term by another in such a sentence. IMPLICATION. (Definition. Reference: Def. 19) • A set F of sentences implies a sentence A iff for all BT + I + AC points (971, mc, a, m/h), if 97T, mc, a, m/h 1= AI for every member A1 of F, then m, mc, a, m/h t= A (truth preservation at all points). We write T \= A. • A set F of sentences in-context implies a sentence A iff for all contextinitialized BT + I + AC points (971, mc, a, mc/h}, if m, mc, a, mc/h N AI for every member AI of F, then 971, mc, a, mc/h t= A (truth preservation at all context-initialized points). We write F |=OT"cfcr A. The strong notion of implication is important partly because it is monotone (or antitone) for each primitive connective in the language. That makes "logic," for instance natural deduction, easier. In-context implication is suitable only for stand-alone sentences. VALIDITY CONCEPTS. (Definition. Reference: Def. 20) • For 97t a model, A is valid in 971 (or VJl-valid) iff 9Jt, mc, a, m 1= A for every mc & Tree and a over Domain and m 6 Tree. We write 971 1= A. • For 6 a structure, A is valid in & (or 6-valid) iff 97t 1= A for every 6model 971. We write © 1 = ^ 4 . When & is understood, especially when & is taken as a representation of Our World, we say that A is valid.

8. Indexical semantics under indeterminism

237

• When K is a class of structures such as those listed in §2, A is valid m K (or K-valid) iff & t= A for every structure & in K. • Each of these validity concepts has also an in-context version defined by restricting points to context-initialized points. When symbols are wanted, we write ^n-ctx^ insteaci of « ( = » . Thus, A is (i] in-context OT-valid iff m \=m'ctx A, (ii) in-context 6-valid iff & \=m~ctx A, and (iii) in-context K-validiff K\=in-ctx A. Strong validity is interesting and useful, but perhaps not as intuitive as its incontext cousin. In-context validity is suitable only for sentences considered as stand-alone. Observations on in-context semantic concepts. It is tempting to think that an in-context version of, for example, truth for stand-alone sentences is "real" truth. The temptation is to be firmly resisted: Truth is essentially parameterized, and in the end, one needs all the parameters. The reason is this: There is much insight to be gained by tracing the semantic value of expressions "from the inside to the outside," watching for the way that the semantic value of complex expressions depends on the value of their parts. For this enterprise, one should treat expressions as embeddable, and if so, "real" semantic value— including "real" truth—has to be relativized to the assemblage of all parameters. None can be left out for "real" truth. 4 There is a difference between the strong semantic ideas and their in-context cousins for sentences that contain indexical expressions, such as Now:, which pick up their meaning from the moment-of-use parameter. Here is an example, after Goodman, that relies on the difference between "strong absurdity" and its in-context cousin, "in-context absurdity." (We are thinking of absurdity in an intuitive way; our discussion does not require anything more.) The boat Jack owns is larger than the boat Jack owns now.

(9)

Although (9) is not strictly contradictory, it sounds absurd, and, as a standalone sentence, it is absurd. The (tedious) semantic explanation is this. There are two definite descriptions involved. For brevity, let B be the phrase, "the boat Jack owns," and let Now-B be the phrase, "the boat Jack owns now." Semantically, the referent of B depends on the moment of evaluation, and is independent of the moment of utterance. The semantic properties of Now-B are just the reverse: The referent of Now-B depends on the moment of utterance (because of the "now"), but is independent of the moment of evaluation. In some sense, then, B and Now-B might refer to different boats. (B is not semantically equivalent to Now-B.) But if we are thinking of (9) as stand-alone, then we are to invoke in-context concepts: In checking for absurdity, we are to evaluate (9) at only context-initialized points. At context-initialized points, the moment 4

It is part of our meaning that one should not think of "supervaluation" as giving "real" truth. Here we depart from the recommendation of Thomason 1970.

238

Foundations of indeterminism

of evaluation is identical to the moment of utterance, so that at all contextinitialized points, B and Now-B must refer to the same boat. (B is in-context equivalent to Now-B.) No boat can be larger than itself; whence the in-context absurdity. This result is good, because when you are thinking of (9) as stand-alone, the refined notion of in-context absurdity seems in compelling correspondence with one's intuitive judgment of absurdity. If, however, you are thinking of (9) as embeddable, then you do not care that it is in-context absurd. You care only that it is not absurd in the strong sense that refers to all points, not just to the context-initialized points. If (9) were absurd in the strong sense, then the following, which embeds (9) within a future tense, would also be absurd. It will be true that the boat Jack owns is larger than the boat he owns now.

(10)

Since (10) makes perfectly good sense, it must be that its contained part, (9), is not absurd. The (tedious) semantic explanation is this. The denotation of B = "the boat Jack owns" depends on the moment of evaluation, and is independent of the moment of utterance. The semantic properties of Now-B = "the boat Jack owns now" are just the reverse: The denotation of Now-B depends on the moment of utterance (because of the "now"), but is independent of the moment of evaluation. The function of the future tense in (10) is precisely to move the moment of evaluation, while leaving the moment of utterance unchanged. ("Will" is translocal in the moment of evaluation and local in the moment of utterance.) You must evaluate (9) at points that are not context initialized: You must evaluate the embedded sentence, (9), with the moment of evaluation being future to the moment of utterance. And when you do that, B refers to the boat Jack owns at that future moment, while Now-B still refers to the boat Jack owns at the moment of utterance. Since it makes perfect sense that these boats be distinct, it is not at all absurd to suppose that one is larger than the other. Hence, although (9) is absurd when taken as standing alone, when (9) is considered as embeddable, for example, in (10), it is by no means absurd, a fact that we explain by considering points that are not context initialized. For indexical-free sentences, however, there is no difference whatsoever between the strong concepts and the in-context concepts. Suppose that A is being considered as stand-alone. Suppose that A is also without free variables, and is therefore assignment-independent. Then one could meaningfully omit both the moment of evaluation parameter and the assignment parameter, leaving only model, context, and history. One cannot, however, go on to omit the history parameter, even for stand-alone sentences. We need the relativization to histories if we are to understand a stand-alone sentence such as "There will be a sea battle tomorrow," which may well be asserted before the matter is settled true or settled false. See especially Assertability thesis 6-7.

8. Indexical semantics under indeterminism

8F

239

Semantics for stit-free locutions

To complete the semantics for the language of indeterminism, we need to give explanations of the semantic values of atomic terms and sentences, and of how the semantic values of more complex expressions arise out of the semantic values of their "parts." We follow Curry in calling a way of building new expressions out of old a "functor." We treat functors that build sentences from sentences (connectives), that build sentences from terms (predicates), that build terms from terms (operators), that build terms from sentences (subnectors), and that build sentences from a combination of terms and sentences (mixed nectors). Context will always make it clear which sort of functor is at stake. Mathematically we are defining "Valm,mc,a,m/h(E)n (the semantic value of E) by simultaneous recursion on terms and sentences. Conceptually it seems better (or anyhow at least as good) to take the following as explaining the meaning of the various features of the language of indeterminism in terms of some prior understanding of "Va/ O T i m c j Q , m / / l (E)." For brevity in stating upcoming semantic clauses involving variable-binding functors, we adopt the following. 8-6 DEFINITION. (Semantic abbreviations) • "Xj" can name theassignment-tore-xjparameter. (We also continue to use "ay' to name a piece of notation.) • We want a short way of "shifting" the values of the assignment parameter. Let z be any appropriate value of the x0 parameter, which is to say, let z G Domain; and let (OT, mc, a, m/h} be any point. Then is the point that is just like (371, mc, a, m/h) except that the value of the parameter, Xj, is shifted to be z. (If there is no such point, the notation is undefined.) In the rest of this long section we present a plethora of semantically explanatory clauses for the "stit-free" part of the language. Then in §8G we go over the stit ideas. We organize the work of the present section on the "stit-free" part as follows. After the clauses covering the atomic expressions of the language, we organize our explanations in terms of abstract properties of the functors, primarily considering whether they are "local" or "translocal" in various mobile parameters in the sense of Definition 8-1. Also we occasionally need the concept of "anchoring." Before proceeding, we give a definition of "anchoring" in rough terms, which suffices for present rough purposes. (The rigorous definition is too tedious to be helpful here.) 8-7 DEFINITION. (Anchoring) Let $ be a functor. • $ is anchored in a parameter if in passing from E to o (E), you require the very identity of the current value of that parameter, over and beyond the pattern of semantic values of E as the parameter varies.

240

Foundations of indeterminism •

o

unanchored in a parameter iff it is not anchored in it.

EXAMPLES. (Anchoring) Negation is not anchored in any parameter, since it cares only for the truth value of A, no matter the values of any parameter. The universal quantifier, Axj, is unanchored in the xj parameter (it needs only the pattern of truth values, not the actual value of xj); but Axj., is anchored in the Domain parameter (you need to know the actual domain). In modal logic, necessity is always anchored in the set-of-worlds parameter; in S4 it is also anchored in the world parameter (you have to know where you are in order to quantify over the relatively possible worlds), whereas in S5, necessity need not be anchored in the world parameter (since in the case in which you are quantifying over all worlds, you do not need to know where you are). And, paradigmatically, indexicals are anchored in the context of use. Having put the definitions of assignment-shifting and anchoring in play, we proceed to detailed semantic clauses, which will occupy us until the end of this chapter.

8F.1

Atomic terms and sentences; operators and predicates

We are considering a mini-language with the following equipment. • p (and sometimes q) ranges over propositional variables. • u ranges over individual constants, including two sorts of special terms: (i) a ranges over agent terms (and frequently over the agents themselves), and (ii) f is a term that artificially denotes "the non-existing object," to be available as a throwaway value of definite descriptions when existence or uniqueness fails. • Xj ranges over individual variables. • / ranges over operator letters. • F ranges over predicate letters. Terms and sentences. • t ranges over terms of any kind. f ( t 1 , ..., tn) is a term. • A ranges over sentences. We also sometimes use B, C, D, P, and especially Q as ranging over sentences. F(t1, ..., tn) is a sentence. The semantics of these features is as follows. • For A an atomic sentence, • For t an individual constant u (including agent-terms a and the term, f, for "the non-existing object"), Va/ O T i m c ! a , m / / l (i) = 3(i).

8. Indexical semantics under indeterminism

8F.2

241

Absolute functors

A functor is defined as "absolute" iff it is both local in and unanchored in every parameter. Identity, alone among predicates, has this property. Among connectives, only the truth functions are absolute.5 Here are the semantic explanations of a few absolute functors.

• Other truth-functional connectives are analogous, or may be introduced with their usual abbreviations. We occasionally use the following: V, D, =, T and

8F.3

Variable-binding functors: Translocal in an assignment parameter

• AxjA. READING: For all x3, A. SEMANTICS: • LXj(A). READING: A definite description: the sole x.j such that A. SEMANTICS: Val<w,mc,a,m/h(i<X3(A}) = the sole z such that [z / Xj](9Jl, mc, a, m/h) \= A if there is exactly one such. If not, ("the non-existing object"). • Other variable-binding operators are analogous. 5 That makes it plausible to guess that "absolute" is a good thing to mean by "extensional." That won't work, however, because, e.g., predicate letters (of the sort considered here) are thought to carry only extensional meaning, even though they are anchored in the model, m, and therefore not absolute. One might next guess that "local in all parameters" is a good meaning for "extensional." Observe, however, that if we were to take this as an explication of "extensional," then quantifiers would turn out just as "non-extensional" as, e.g., the modalities. Given all this confusion, and since it is difficult to determine what "extensional" means (van Bentham 1988, p. 109, suggests that "no general satisfactory definition seems to exist"), we feel that it is better to stick to the ideas of "local" and "absolute," which have clear and definite meanings that are logical rather than ideological.

242

Foundations of indeterminism Grammar LX-j(A) is a (categorematic) term, the definite description. Its value when existence-and-uniqueness fails is a mere throwaway. It corresponds to Kaplan's definite description. For his "dthat" one may use ix} (Actually1 :A), which is context-dependent, and independent of the moment and history of evaluation. (We use t. instead of its inversion for convenience.)

8F.4

Historical-modality connectives: Translocal in the history parameter

These are the "historical-modality" connectives, closely tied to some of our intuitive ideas of necessity and possibility. In processing these connectives, you do not have to consider any other moment than the current moment of evaluation: They are local in the moment-of-evaluation parameter. • Sett:A. READING: It is settled true, or historically necessary, that A. SEMANTICS: m, mc, a, m/h \= Sett:A iff V/ii[(m e hi and h1 E History) —> OT, mc, a, m/hi \= A}. • Poss:A. READING: It is historically possible that A. SEMANTICS: 971, mc, a, m/h t= Poss:A iff 3hi[m € h1 and h1 6 History and 971, mc, a, m/hi ^ A}. • Can:A. READING: It can be that A. DEFINITION: Can:A iff Poss:A. We use Can: as a mere stylistic variation of Poss:. This gives a version of the "all in" ability idea of Austin 1961 (p. 177) when combined with the deliberative stit as in §9G. For versions suitable for use with the achievement stit, see §8G.

8F.5

Tense and temporal connectives: Translocal in the moment-of-evaluation parameter

Next are "standard" linear temporal connectives of the sort introduced by Prior 1957. All of these connectives should be thought of as "temporal." They are—this is fundamental—local in the history parameter. That is, temporal connectives, including the tenses, always start on a given history, and the connective moves you, but only "vertically." These connectives never move you off the history on which you started—that is what "local in the history parameter" means. For this reason, they are entirely "linear," and have the same "logic" as standard tense logic. The use of these connectives is absolutely natural and familiar; they require no strange explanations—once one accepts that, in analogy with variables, there is always a current history of evaluation even though there is no history of the context. They interact, however, in enlightening ways with the historical modalities. • Was:A. READING: It was true that A. Or put the main verb of the reading of A—if it has one—into the past tense. SEMANTICS: 977, mc, a, m/h == Was:A iff 3mi[mi E h and m1 < m and 071, mc, a, m\/h 1= A].

8. Indexical semantics under indeterminism

243

Prior's "P." Shift along the present history, existentially, to earlier moments, and check them for A (with respect to the current history). Because histories are closed downward and thus form a chain, the clause "m1 £ h" is redundant. That the route of backward travel is uniquely determined, however, should not blind you to the importance of the nontriviality of keeping to the same (the current) history when you evaluate A. As Prior explained, Was:A does not imply Sett: Was-A—although Was-Sett.A does indeed imply Sett:Was:A. The point is that although the route traveled is unique, it is part of many different histories

Will:A. READING: It will be true that A; or put the main verb of the reading of A, if it has one, into the future tense. SEMANTICS: m, mc, a, m/h t= Will:A iff 3mi[mi € h and m < m\ and 971, mc, a, mi/h N A}. Prior's "F." Here you shoot forward, existentially, along the current history, checking each moment along the way. And look: In contrast to Was:, you cannot, in understanding Will:, get rid of a reference to the history parameter.

Was-always:A (Prior's H) and Will-always:A (Prior's G) have analogous clauses that shift you, quantificationally, along the current history. Sometime/A and Always:A are similarly local in the history of evaluation. At-instt:A. READING: That A was, is or will be true at (the instant or time) t. When English A is not too complicated, "A at t" or "At t, A," will often do. SEMANTICS: Instant and Grammar' t is any term, and A is any sentence At-mstt :A reflects English constructions such as "At 4.00 P M. the coin will come up heads." The clause tells you to shift W, mc, a, m/h by replacing the current value of the moment-of-evaluation parameter by a new moment, namely, the one (and the only one) in which the current history intersects the instant specified by the value of t. "Travel up or down the current history until you hit the instant t; that moment is where you must evaluate A (with respect to the current history)." This understanding of At-mstt '-A makes it false when t does not refer to an instant. We manage to live with this awkwardness. You can see that At-instt '• is translocal in and only in the moment-of-evaluation parameter, for that is the only parameter that is shifted. Also, as long as t itself is independent of the moment-of-evaluation parameter, At-mstt :A is bound to be independent of the moment-of-evaluation parameter. For instance, let t be "4:00 P M.," with some definite date understood. Adding additional temporal connectives to At-mst^-oopM :A has no more effect than (is just as vacuous as) nesting one xj-binding operator within another. For example, "It has always been true that at 4:00 P.M the coin comes up heads" just reduces to "At 4:00 P.M. the coin comes up heads." So does "At 2:00 P.M. at 4:00 P.M. the coin comes up heads" (see the discussion of Figure 6.3). Some philosophical logicians feel that including terms naming times (or instants) in a formal tensed language is wrong-headed, or at best inelegant. Our excuse for doing so is that such terms as used in "at" constructions play a role in many philosophical discussions of determinism versus indeterminism. Not everyone is clear on how they should work under indeterminism, which makes it worthwhile to clarify their logic. Of particular importance in understanding indeterminism is the fact that At-instfaogp M.'-A is not in general independent of the history-of-evaluation parameter. "At 4.00 P M. the coin comes up heads," or even "It has always been true that at 4.00 P.M. the coin comes up heads," can be just as dependent on the history-of-evaluation parameter as "It will be true that the coin comes up heads." "At" constructions pose quite the same problems as does the future tense. These matters are difficult to hear or see in natural language,

244

Foundations of indeterminism where we can (apparently) say, "At 4:00 P.M. it will be inevitable that Jack at I'OO P.M might be running at 3:00 P.M. at 5:00 P.M." We don't know if this makes sense or not. Our point is that those who make philosophical points explicit in a Prior-Thomason connective form have much less difficulty—especially with regard to scope, which is so ambiguous in English and so clear when all tense talk is carried by connectives.

• At-momt:A. READING: That A was, is, or will be true at the moment t. SEMANTICS: 9Jt, mc, a, m/h t= At-momt:A iff Valv(t) 6 E and (9Jt, mc, a, Valv(t)/h) t A. Grammar: t is any term and A is any sentence. This construction travels up or down the current history to the moment denoted by t, and evaluates A just there, still on the current history. It does not move off the current history (it is local in the history parameter), and is therefore a true temporal construction. Accordingly, it cannot be successfully used in connection with moments off the current history. (Contrast: At-mstt :A is always successful when t denotes an instant, since the current history intersects every instant.) The At-momt •' connective seems much stranger than the At-mstt '• connective, chiefly, we suppose, because we don't have names for moments, but we do have names for instants (or times). The At-momt: construction is nevertheless of great usefulness in untangling "double time references" such as are needed in order accurately to understand the following scenario. (Our account may sound a little awkward, because we omit several indexicals that would naturally be used.) It's 4:00 P.M. At 2:00, Themistocles said, "I promise that Themistocles will, (11) choose to fight a sea battle." So in direct speech, what Themistocles promised was this: Themistocles will choose to fight a sea battle.

(12)

We are interested in how one can use the semantic content of (12) in order to illuminate promise keeping. So let it be true at 4:00 that Themistocles has kept his promise; such is, at 4:00, a settled fact. What does this mean? Here are two candidates that don't work. - At 4:00 Themistocles has kept his promise iff (12) is settled true now, at 4:00. This is wrong. After all, now, at 4:00, what Themistocles promised, namely, (12), is settled false. The point is that when evaluated at 4:00, the future tense of (12) would reach forward into times later than 4:00, long after the sea battle. (We neglect as a distraction the possibility that there be another sea battle.) — At 4:00 Themistocles has kept his promise iff (12) is settled true at the moment of promising. This is wrong. At the moment of promising, the sentence, (12), was not settled true. That is, (12) was, at the moment of promising, true on some histories and not on others—for the choosing still lay in the future. What signifies that the promise has been kept is more complicated. - At 4:00 Themistocles has kept his promise iff at 4:00 it is settled true that (12) was true at the moment of promising. The key is the "double time reference": The "settled" is evaluated later, at 4:00, while what is settled is the truth of (12) at the moment of promising. See §5c.2.1 for a more extended discussion of double time references.

8. Indexical semantics under indeterminism

245

The upshot is that you cannot do without "the moment of promising," even though you certainly do not have a name for it. Here is how the analysis could go in indirect speech, where, as advertised, we can use the At-momt. connective. (Also we shall be using (12) instead of mentioning it.) We let the moment of promising be mp. At 4.00 Themistocles kept his promise iff — Wrong: At-mstn oo-'(12). - Wrong. At-mommp:Sett:(12). - Right: At-mst4 00.Sett:At-mommp:(12) None of these connectives can be shifted around; and it is essential that there be a "double time reference."

8F.6

Some mixed temporal-modal connectives

We add just a few "mixed" connectives. All these are definable, and in some cases we give a definition instead of a semantics. • Inevitably:A. READING: It is inevitable (sooner or later) that A. DEFINITION: Sett: Will:A. • Was-always-mevitable:A. READING: It has always been inevitable that A would be true at the present time. SEMANTICS: 97T, mc, a, m/h 1= Was-always-inevitable:A iff m, mc, a, mi/hi N A for every member m1 of i( m ) and every h1 to which m1 belongs. Note that Was-always-inevitable'A is not equivalent to Was-always'Inevitably:A, since the former makes reference to the present instant. It is useful in connection with the achievement stit. • Might-have-been:A. READING: That A might have been true at the present time. DEFINITION: ^Was-always-inevitable.'^A. • Universally:A. READING: It is universally true that A. SEMANTICS: m, mc, a, m/h t= Universally:A iff 3Jt, mc, a, m\/hi \= A for every member m\ of Tree and every h\ to which m1 belongs. This connective crops up only infrequently in this book. Accordingly, so as to be maximally mnemonic, we refrain from abbreviating "universally." Note that because of historical connection, Universally:A is equivalent to the following mouthful: Wasalways:Sett: Will-always:A. When A has a syntax appropriate for a "lawlike statement," and perhaps has also an appropriate social standing in the scientific community, the truth of Universally:A might be a sensible thing to mean by saying that A is a "law." If there are other worlds, perhaps one could term such a law "a law of our world," and imagine other worlds with other laws. In this sense, Umversally:A could well be an "imaginative contingency," an intentional idea for which we propose no theory.

246

Foundations of indeterminism

8F.7

Indexical connectives: Anchored in the context of use

Both the "now" connective and the "actuality" connectives are indexical in the sense that they are anchored in a context-of-use parameter. (These connectives are also translocal in the moment-of-evaluation parameter, but that feature is not shared by all indexical constructions.) • Now:A. READING: It is now true that A. SEMANTICS: Now:A iff Our indeterminist adaptation of Kamp's well-known explanation of Now:A comes to this: Stay on the current history, but travel up or down to where the current history intersects the very instant (time) of the context of use. Using that context-determined moment with the current history, check the truth value of A. The new moment will be "at the same time as" the moment of use, but since it lies on the current history rather than on any history containing the moment of use, it may well be inconsistent with the moment of use, as in the following: "I am not now rich, but if the coin had come up heads, now I would be rich." In symbols, with A1 for "rich" and A2 for "heads": Now.^Ai & Was:(Poss:A2 & Sett:(-A2 D Now:Ai)).

• Actuality connectives. Actuality is hard, partly because different thinkers have such different views. We are getting at "actuality" in an indexical sense, shared by, for example, Kaplan and Lewis, that anchors actuality in the context of use. Of this species, the easiest-to-understand actuality connective, Actually1:, ties actuality to what is settled true at the moment of use. There is, however, an interesting variant, Actually2:, that has a disjunctive nature. On this variant, one considers whether or not the moment of use sits on the current history. If so, the variant ties actuality to what is plain true at the moment of use on the current history. If, however, the moment of use does not sit on the current history, the variant ties actuality to what is settled true at the moment of use.6 We mark the easier-to-understand and the disjunctive variant with a simple "1" and "2" respectively because we have little to say about their different properties and applications. - Actually1:A. READING: It is actually true that A. SEMANTICS: 9

Actually2:A. READING: It is actually true that A. SEMANTICS: 971,

6 It would be in the spirit of Thomason and Gupta 1980 to add a parameter that makes sense out of picking a particular history through the moment of use, perhaps a history that is "very like" the current history, which passes through the current moment. Since, however, "very like" is not an idea that fits well with the more austere causal notions that we employ, we forgo adding such a parameter.

8. Indexical semantics under indetenninism

247

As always, Now: and the actuality connectives do very little work at the head of sentences considered as stand-alone, since anyhow, by Policy 8-3, the moment of evaluation is already initialized by the moment of use, so that the shifting called for by Now: or an actuality connective is vacuous. When embedded in translocal connectives, however, Now: and the actuality connectives really do shift the moment of evaluation. This completes our discussion of how various variable-binding, temporal, history-modal, and indexical functors work in branching time. Next we take up the stit functors.

8G

Clauses for stit functors

We give properly recursive forms of semantic explanations of the choice-based concepts—the stit concepts—that have been previously defined and used in various places in this book.

8G.1

Semantics for the deliberative stit

The simplest of our two major stit concepts is the "deliberative" stit, as defined first by von Kutschera 1986 and later by Horty 1989, and which we apply at length in chapter 11 and chapter 12. Recall Def. 16, according to which TT = (OT, mc, o, m/h). • [t dstit: A}. READING: t sees to it that A. SEMANTICS: 9ft, mc, a, m/h N [t dstit: A] iff the following conditions are satisfied. - Agency. Val^(t) 6 Agent. — Positive condition. For every — Negative condition. For some hi € -H\ m ), 9ft, mc, a, m/hi ¥• A. The history h2 on which A is not true is sometimes called a "counter." The agency condition is needed when the agent position of a stit sentence is available to every term. In this book we do not often enter the agency condition explicitly, since usually, such as when we reserve "a" for an agent-term, it is a presupposition that the term in the agent position denotes an agent. We remark that in normal applications, one would expect that the term, t, is independent of both history and moment, as is always true for our use of a. One will usually obtain what one wants by trading in [t silt: A] for 3xi[xi = t & \xi stit: A}}. In the semantics of dstit, there is no double temporal reference: The moment of evaluation of A and the moment of choice are identified, so that while dstit is an entirely viable candidate for helping to analyze agentive locutions, its expected properties are considerably different from those of the achievement stit. If one wishes to think of dstit as reporting an "action" or even a "choice," it is difficult to say in comfortable English just when the choice is made. The difficulty is that before or at m it is not yet settled which choice Val,r(t) makes, while at any later moment the choice has already been made. The source of the difficulty is that dstit reports an immediate transition, Def. 8, so that there is no room for an "action" qua "event" between initial and outcome.

248

Foundations of indeterminism The solution is to understand "the action" as being the transition itself. Since a transition consists in a pair of "events," initial event and outcome event, it is obvious that a transition cannot have a "simple location" (Whitehead).

8G.2

The Chellas stit

An even simpler stit concept, from Chellas 1992, omits the negative condition:

8G.3

Achievement-stit semantics based on witness by moments

In Definition 2-4, we presented a semantic explanation of the achievement stit. In this section we present those same semantics in just slightly different words. Then in §8G.4 we generalize in order to minister to a specific shortcoming, and in §8G.5 we suggest a potentially helpful alternative. • [t stit: A}. READING: t sees to it that A (the present fact that A is entirely due to a prior choice o f t ) . SEMANTICS: 971, mc, a, m/h N [t stit: A] iff there is a moment, w, that is a momentary witness to [t stit: A] at m relative to (971, mc, a, m/h), where that phrase is defined by the following conditions. — Agency. VaZ 7 r (t) € Agent. — Priority, w is properly earlier than TO: w < m. — Positive condition. 971, mc, a, m\jh\ t= A for every moment m\ in Choice^ "^'(TO), and for every history h\ in H(mi). — Negative condition, w must lie under some moment m\ in i(m) such that there is a h\ € #(mi) such that 971, mc, a, m\/h\ ¥ A. That is, A is not settled true at m\. Sometimes m\ is called a counter. is said to be choice eqvivalent to m for relative to w (Def. 12). Thus the positive condition is that A should be settled true at every moment choice equivalent to m for Val^(t) relative to w. In other words, the choice that Val^(t) made at w (where the past tense is from the perspective of m) "guaranteed" that A would be settled true at every moment of J( m ) accessible from w via that choice (where the subjunctive is also from the point of view of m, since we are considering alternative ways of filling the same instant). The negative condition ensures that the choice that Valv(t) made at w was not irrelevant to the truth of A at J( m ) in the sense that at w the falsity of A at i(m) is risked—its settled truth there is not already guaranteed at the moment of choice.

A moment in

Since none of the conditions mention h except in evaluating t as Valn(t), it is clear that [t stit: A] is history-independent if t is. Also worth emphasizing is that [t stit: A] has a kind of double temporal reference: There is a reference to where A is evaluated and there is a reference to where the witnessing choice occurs. By our semantics, the compound [t stit: A] is true (if in fact it is so) where A is evaluated, not at the moment of choice.

8. Indexical semantics under indeterminism

249

We next offer a representation, suitable for use with the achievement stit as in §9B, of the "all in" ability of Austin 1961 (p. 177), present or absent on a particular occasion for a particular agent and with respect to a particular complement. Such ability statements can be tensed either as of the moment of witness or as of the moment of evaluation of the complement. We use "can" for the former and "could have" for the latter: • [ii cant2-stit: A}. READING: Agent t1 can see to it that at instant £2, A. SEMANTICS: 9JI, mc, a, m/h 1= [ti cant2-stit: A] iff Valnfa) € Instant, and there is a moment mi in i >m (i.e., TOI lies on the horizon from m at z) such that m is a momentary witness to [ti stit: A] at TOI relative to (9JI, mc, a, m/h). • [t could-have-stit: A}. READING: t could have seen to it that, at the present time, A. SEMANTICS: SOT, mc, a, m/h t= [t could-have-stit: A] iff, where a; is a variable not free in [t stit: A], [i(m) I ^K^j fricj ai m/h) 1= Was:[t canx-stit: A]. These definitions illustrate the restricted complement thesis, Thesis 5. Because [t stit: A] occurs as a unit on the right sides of these definitions, you can see that the stit sentence is indeed a complement. You can also see that the right sides would make no sense if [t stit- A] were replaced by an arbitrary sentence—which is exactly to say that the complement position is restricted. See also Figure 9.1.

8G.4

Achievement-stit semantics: Witness by chains

A deficiency in the "momentary-witness" concept of stit just defined is that it makes it a matter of "logic" that if Autumn Jane saw to it that she was clean at 4:00, then there was some momentary choice that witnessed that fact. And maybe there was; but suppose instead that the witness to the outcome was a chain of choices by Autumn Jane with no last member. Picture her just prior to 4:00 as balancing on a board that crosses over a puddle, and award her the ability to choose to fall off at any of an unending series of moments approaching 4:00. Also permit our illustration to be simple by adding what is probably false: that the lapse from cleanliness occurs as soon as you like after the choice to fall off. Under this supposition the witness to her seeing to her own cleanliness at 4:00 was no single choice, but the whole unending chain of choices properly approaching 4:00. (See the discussion of the ten-minute mile, Question 2-11.) To keep us all on the right track, we explicitly note that the deficiency to which we point is not at all carried by stories in which in ordinary speech we would say that a long list of preliminary activities went into, say, Autumn Jane's setting of the table, or to change the illustration in order to make the point even more visible, a story in which in order for Autumn Jane to win at chess at 5:00, a complicated, temporally discrete series of moves was required. The chess illustration makes it clear that catering to these stories has nothing to do with an unending sequence of choices. Reflection on the conceptual problems raised by such stories is important, but these problems are so different from those that we are now discussing that they will require a different treatment. See chapter

250

Foundations of indeterminism

13 for some relevant developments via the theory of strategies, and §8G.5 for an additional suggestion. We turn to the quest for an understanding of the witness of a stit by a possibly unending sequence of choices. We use the notion of choice inseparability explained in Def. 13. • Five conditions must be satisfied by c in order to be a witness to [t stit: A] at m relative to (9Jt, mc, a, m/h). — Agency. Val^(t) G Agent. — Priority. All moments in c must be properly earlier than m. — Nonemptiness. The chain, c, must of course be nonempty. — Positive condition. OT, mc, a, m\/h\ 1= A for every moment mi and history h\ such that m-i 6 i(m)<^hi and mi is inseparable from m for Val^t) in c (i.e., mi =^al^^ m- see Def. 13). — Negative condition. Every moment, w, in c "risks" the falsity of A in the sense that above w there is some moment mi in i( m ) and some history hi e #( mi ) such that £01, mc, a, mi/hi ¥ A. In other words, for every moment, w, in c, it is not settled at w that A be true at i(m)-

The truth conditions for stit, as witnessed by a chain, are as follows. • [t stit: A}. READING: t sees to it that A. SEMANTICS: S01, mc, a, m/h N A iff there is a chain, c, satisfying the agency, priority, nonemptiness, positive, and negative conditions for c to witness [t stit: A] at m, relative to (yjl, mc, a, m/h). See also Definition 13-24 for an equivalent version of the chain-witness semantics for the achievement stit. The idea of chain witness is considered in more detail in Belnap 1996a. Chain witnesses are involved in this book whenever "busy choice sequences" are at issue.

8G.5

The transition stit

Every concept of agency must refer to alternative possibilities, to what might have been without the particular exercise of agency at issue. Which alternatives? Our approach (for better or worse) always seeks an objective answer to this question. In the case of dstit, the alternatives are entirely and uniquely determined by the causal structure of the world as encoded by Tree and ^, together with the choices possible for agents as represented by Choice. The critical point is that in the case of dstit, the "moment of outcome" (that is, the last moment at which the outcome has not yet begun) is identified with the "moment of choice" (that is, the last moment at which the choice is still hanging in the balance). (See the discussion in §7A.4 of immediate transitions, Def. 8.) The alternatives can then be objectively identified as the histories representing

8. Indexical semantics under indeterminism

251

the alternative futures of the moment of choice. When, however, we need to consider outcomes at some temporal remove from the moment or moments of choice, we can no longer be so simple. The achievement-stit solution was to bring in instants, via Instant, with which to make "horizontal" comparisons. The alternative possibilities were then objectively identified as alternative ways of filling "the same" instant. Thus in judging whether or not Autumn Jane saw to it that she was clean at 4:00 P.M., we considered what else might have happened at 4:00 P.M. It seems to us likely that there should be other interesting objective ways of picking out alternative possibilities. Here is one, the "transition stit," that we think has potential for illumination. The first of two fundamental thoughts is that we think about the transition aspect of agency, and that it seems to make intuitive sense to say that an agent sees to a transition from a concretely given initial situation or event to a propositionally expressed outcome (see §7A.4). (Both dstit and astit, in contrast, speak solely of seeing to a propositionally expressed outcome, leaving the initial situation to be brought into play via the moment of evaluation.) The second key thought is that we build into the stit construction itself an explicit reference to a chain of choices or actions witnessing the outcome of the transition. These two thoughts lead us to take as our target, instead of a two-place construction such as dstit or astit, a four-place construction relating agent, initial event, a given chain of choices, and a propositional outcome. The linguistic form we carry as [a tstit: mo ==4> A], the "transition stit," which we shall think of as being evaluated at a moment m. Because of English tenses, it is difficult to give the transition stit a reading in ordinary language. Permit us to sidestep some of this problem with English by using direct discourse, putting "A" in place of the name of a sentence, instead of in place of the sentence itself. We may then read [a tstit: m0 =^> A] as follows. Given that A was not settled true at mo, and considering that the fact that A is settled true (now, at m), it turns out that a is entirely responsible for this transition, which he or she secured by means of a certain chain of choices or actions, namely, c.

^ '

The idea of "double time reference" (Definition 5-12 and §8F) hovers in the background. The earlier "initial" moment mo is where we evaluate A. But the current moment, m, is where we are evaluating the stit sentence that contains A as a proper part. This is admittedly complicated, but we think that it needs to be so. We should think of A as moment-independent. The reason for this is that we wish to say, in the spirit of double time reference, that at m it is settled true that A, whereas at m0 it was not settled true that A. In this way we distinguish "causal transition" from simple change of state. For mere change of state, we evaluate A at different moments, as time moves on. On Monday the pond is full, then on Tuesday it is dry. But that sequence is not in its description objectively "causal" since it allows that the pond being dry on Tuesday may have been predetermined from all eternity. In line with §7B.3, the real causal transition needing an explanation is from (something like)

252

Foundations of indeterminism "The pond is dry on Tuesday" is not settled true at TOO

to

"The pond is dry on Tuesday" is settled true at TOI. As is typical in the use of double time references, we do evaluate the interior sentence, "The pond is dry," at different moments. In contrast, the entire sentence, "The pond is dry on Tuesday," is moment independent. By concentrating on moment-independent sentences, we zero in on the causal transition without getting confused by any change of state. In any event, here is our semantic account of the suggested four-argument "transition stit" representation of seeing to a transition. We include moment independence as a necessary condition. [ti tstit: t-2 =^ A}. READING: See (13). SEMANTICS: 371, mc, a, m/h N [ti tstit: £2 ==> A] iff, letting a = Val^^i), a 6 Agent; letting TOO = Val^fe), TOO € Tree; letting c = Valv(t3), c is a chain of moments; and — Moment independence of outcome. - /i/ter arid before. — Positive condition. — Essentiality condition. Recall, for the positive condition, that hi =™ h means that h\ is inseparable from h for a in c, Def. 13. The essentiality condition says that no member of c can be omitted. This condition is what gives bite to the idea of the "by means of" phrase in (13). Observe that this is an entirely objective, if limited, concept of "by means of." The essentiality condition, which says that each member of c is essential, evidently implies a negative condition to the effect that the witnessing is not complete until c is complete. Even though we do not have the space to develop the idea of stitting a transition by means of a series of choices, we append what seems to us the most promising feature of the idea: that [a tstit: TOO => A] encodes agent responsibility for an outcome based on an entire campaign of actions or choices. We are in a position to express, for example, that Mary accomplished the transition from dirty dishes at 7:00 to clean dishes, not by a single action, but by doing one thing after another until the job was done. This concludes our organized semantic account of some constructions whose exact nature we think important for the understanding of indeterminism. We have also finished the foundational discussions that have interrupted, for the space of three chapters, our consideration of stit and its uses in helping us to understand how agency fits into the causal structure of our world. Next we turn to some applications. Immediately following are two chapters that apply the achievement stit, and then come two involving the deliberative stit.

Part III

Applications of the achievement stit

This page intentionally left blank

9

Could have done otherwise The complexity of the connections among actions, moral responsibility, and the alternatives open to an agent have long tormented philosophers.* Hume, for example, the most famous of all compatibilists, claims that universal determinism is not only consistent with human freedom but necessary for morality (An Enquiry Concerning Human Knowledge, section 8, part 1). His view is that while agents are able to choose among alternatives, there is the liberty of voluntary action. We think that is right. On the other hand, Hume asserts, morality is without foundation if actions are not fully determined. We think that is wrong. Our intent, however, is neither to joust with Hume nor directly to engage more modern entrants in the "free will" tourneys. Instead we recount a tale that begins with the following proposition, one that might have puzzled even such as Don Quixote of La Mancha and his squire, Sancho Panza. If an agent is morally responsible for doing something, then the agent could have done otherwise.

, .

Modern replies to Moore 1912 by Austin 1961 (pp. 153-167) and to Frankfurt 1969 by Van Inwagen 1978 put the issue in just these terms. In an effort to unravel the complexities of (1), we here cleave it in twain, each conjunct seeming essential to its meaning. (See also the discussion in §5A.I, which relates the matter to promising.) If an agent is morally responsible for doing something, then the agent did it.

, .

If an agent did something, then the agent could have done otherwise.

,^\

The middle term, doing something, is thereby revealed. We vouchsafe that for the route from moral responsibility to "could have done otherwise" to be "With the permission of Kluwer Academic Publishers, this chapter draws on Belnap and Perloff 1992. You will note that here, as in that article, our diction sometimes quixotically imitates the source of our examples.

255

256

Applications of the achievement stit

accurately charted, heed must be paid this idea, formerly hidden, that stands at the crossroads. We shall not in this chapter further consider the claim of the first conjunct (2). We remark, however, that anyone who wishes to argue for or against the claim that moral responsibility implies "free will" should note at least the following: There is a deep difference between (2) as here displayed and its sound-alike, If an agent is morally responsible for an outcome, then the agent saw to that outcome.

/,..

This sound-alike, (4), is almost certainly false, a fact that in no way impeaches a claim that (2) holds. Take the following examples. Sancho Panza was morally responsible for having seen to it that Quixote's lance was unpolished. Therefore, Sancho Panza did see to it that Quixote's lance was unpolished.

(2x)

Sancho Panza was morally responsible for the fact that Quixote's lance was unpolished. Therefore, Sancho Panza did see to it that Quixote's lance was unpolished.

(4x)

The example (2x) of (2) rings true. The question of moral or legal responsibility of an agent for doing something (such as seeing to it that a lance is unpolished) seems to presuppose that the agent did that something. On the other hand, it sometimes makes good sense for a moral or legal code to assign responsibility for a fact (such as the fact that a lance is unpolished) without determining whether the agent saw to that fact. (If it was Panza's moral job to keep the lance polished, then it is plausible that we may hold him responsible for its lack of polish without inquiring into his agency in the matter.) So the example (4x) of (4) goes wrong. Stit theory helps to avoid confusing (4), which is false, with its true sound-alike cousin, (2). (See also §5A.l.) Leaving the first conjunct (2) aside, however, we press attention on the morality-free claim of the second conjunct, (3), that doing something implies having been able to do otherwise. There has been little effort to clarify the second conjunct in isolation from moral considerations; we deem worthwhile the enterprise of examining the relation between agentive doings and what it means to say that "an agent could, or might, have done otherwise." Hume's own account, "if we choose to remain at rest, we may; if we choose to move, we also may," seems, for example, to suggest a tie between "could" and "might": An agent could have done otherwise iff there is something else he or she might have done. Is that right? Our view is that the armory of ordinary language is inadequate to the task of deciding such questions. We need the weaponry provided by something like stit theory, a theory that is careful in placing agents and their doings in relation to the causal structure of our world. Our preparations having been completed in earlier chapters, we venture forth to battle a variety of questions in the topic of agency and "could have done otherwise" as embodied in (3). Our assembled conceptual apparatus, though weighty, is neither more nor less than is needed to complete the task. On the

9. Could have done otherwise

257

one side we avoid the perils induced by the vagaries of ordinary language, for example the differences between "might" and "could" in this context, and the exact target of the anaphoric reference of "otherwise." With stit normal forms to guide us, we sharpen the contrasts and bring each topic into clear relief. On the other side, we first render a question or conjecture in the language of stit, meaning in this chapter the achievement stit, unless we give explicit notice to the contrary. Then we employ stit theory to render formal judgment (an upshot). That, in turn, requires clarity about alternatives, about matters temporal and historical, and about possibilities in branching time, such as we have studied in earlier chapters.

9A

Could have been and might have been

First a simple question with a simple answer. QUESTION. Is what could have been the same as what might have been? UPSHOT. We think there is no difference—which stands in emphatic contrast to our treatment of the conjecture immediately following. Although could-have-been and might-have-been come to the same thing, branching time encourages one to see that both are ofttimes context-dependent, since the question of when something was possible is generally in order but hard to deal with. For both "could have been" and "might have been" we use a (tensed but) rigorous context-independent reading that works well for appropriately idealized cases: STIT VERSION.

Might-have-been:Q.

We hope it is obvious that our suggestion is not empty: "Might have been" and "could have been" are English, while "Might-have-been:" is formally characterized in §8F.6.

9B

Could have done and might have done

Now a conjecture not quite so simple that, in relation to the previous conjecture, emphasizes the importance of distinguishing agentive from non-agentive complements. CONJECTURE. What an agent might have done is different from what he or she could have done. By "the agent might have done it" we intend an impersonal modality, expressed perchance less ambiguously (and less idiomatically) by "it might have been that the agent did it."

258

Applications of the achievement stit

We confess that in spite of the importance of the distinction, it is easy to run together "could have done" and "might have done." As a first advance in distinguishing them, replace "do" by "stit," yielding "could have stit" versus "might have stit." As a second advance, interpret "could have stit" as "it was (simple past) in the agent's power to stit," and interpret "might have stit" as "there was a way that things could have gone such that it would have been in the agent's power to stit." These ordinary language statements remain, however, obscure, and discourage further advance. Stit theory suggests that the critical difference concerns the witness to the stitting. When we say that "the agent could have done," we demand that the witness to the stitting be in the past of the moment m, at which Q is evaluated. On the other hand, when we say that "the agent might have done," we allow the witness to be in the future of possibilities of a moment that is in the past of m. These differences suggest the following (using notation from §8F.6 and §8G.3). STIT VERSION, stit: Q}.

[a could-have-stit: Q] is not equivalent to Might-have-been:[a

UPSHOT. The conjecture is correct; [a could-have-stit: Q] implies Might-havebeen:[a stit: Q], but not conversely. As may be inferred by comparing the explanations of the two locutions given in §8F.6 and §8G.3, their semantics are different: Although the truth at m0 of each of [a could-have-stit: Q} and Might-have-been:[a stit: Q] requires the truth of [a stit: Q} at some moment co-instantial with mo, the former alone requires that the witness be in the direct past of m,Q. Figure 9.1 has a picture. The abstract situation is this. The rectangles are choices for a, whereas what happens at Wi is not up to the agent. The salient feature is that while [a stit: Q} at ma has a witness, namely, u>2, that witness does not stand in the past of mo. So at mo it is true that a might have seen to it that Q, but it is false that a could have seen to it that Q. In contrast, at mo it is true that a could have seen to it that R, since w0 witnesses [a stit: R] at each of mi-m3. Here is an example that fleshes out this abstract description of Figure 9.1. EXAMPLE. Don Quixote attacks the windmill. Commending himself most devoutly to his lady, Dulcinea, whom he begged to help him in this peril, he covered himself with his buckler, couched his lance, charged at Rozinante's full gallop, and rammed the first mill in his way. At the moment, WQ, that ends his commending, the Knight of the Mournful Countenance had the choice either to stand down or ride on. Once having begun his charge, however, there was a slightly later moment, Wi, at which Rozinante might by chance have collapsed. In the case of no collapse, there

9. Could have done otherwise

259

Figure 9.1: Don Quixote attacks the windmill

was then a later moment, W2, at which Quixote had the choice either to swerve toward or to swerve away from the disastrous confrontation. Let my, as shown in Figure 9.1 be the moment at which he rammed the windmill. Consider the history on which, as he finished his commending, the Knight regained his wits and stood down, and follow that history to a moment, mo, that is co-instantial with the moment, 7713, at which he rammed the windmill. At that moment, mo, it would be true to say that he might have attacked the windmill, but false to say that he could have. What decides the matter is that there is nothing he could have chosen at the end of his commending (WQ—which is the only choice point in the past of the moment of non-attack under consideration) that would guarantee his attack. Both chance (at wi) and the uncertainty of the outcome of a future choice (at w%) stand in the way of such a guarantee.1

9C

Might have been otherwise

To appreciate this next conjecture, consider a stit with a non-agentive state of affairs as complement, and let the anaphor, "otherwise," refer to just that non-agentive complement rather than to the entire stit sentence. CONJECTURE. If yon fellow sees to some state of affairs, then it might have been that the state of affairs not obtain—at that very instant. The final phrase accomplishes a task more easily than idiomatic English: Make sure that the "might" means that the absence of the state of affairs obtains 1 Permit us yet another restatement of methodology: Although we think the distinctions we are drawing are important for, e.g., moral analysis, we by no means fancy that our chosen expressions have a perfect fit with ordinary English.

260

Applications of the achievement stit

as a co-instantial alternative to the very moment in question. In notation it is unambiguous: STIT VERSION 1. [Q stit: Q] implies Might-have-been:~Q. By the semantics given in §8F.6, this is evidently equivalent to STIT VERSION 2.

[a stit: Q] is inconsistent with

Was-always-inevitable:Q.

These stit versions say that if [a stit: Q], then it has not been inevitable (determined) from all eternity that Q should obtain at the instant in question. UPSHOT. The conjecture is, in its stit versions, true. It is an obvious consequence of the negative condition (§8G). EXAMPLE. If it was inevitable from all eternity that the hog gelder's reed flageolet sounded four times while Don Quixote was at his meal, then the hog gelder did not see to it that his reed flageolet sounded four times while Don Quixote was at his meal. A hard determinist valiantly endorses the consequent; a soft determinist becomes cross, changes the topic, and exits the lists.

9D

Might not have done it

Next an important conjecture with a straightforward disposition. CONJECTURE. If a does something, then it might have been otherwise; that is, a might not have done it. Here let the "otherwise" refer anaphorically to the entire stit sentence, not just to its complement. STIT VERSION,

[a stit: Q} implies Might-have-been:~[a stit: Q}.

UPSHOT. True. We belabor the obvious by offering two proofs. First, since [a stit: Q] implies Q (by the positive condition), so that ~Q implies ~[a stit: Q], this is an immediate consequence of Upshot §9C. The second and more important proof is this: The consequent is a truth of logic, so that the implication is vacuous! This is related to the Triponodo principle of Makinson 1986, except here instead of the "trivial (legal) power not to do," we have the "trivial possibility of not doing." The argument that it is logically true is an easy reductio. If [a stit: Q] were settled true throughout an instant, then by the positive condition, Q would be settled true throughout that same instant—which would leave no room for satisfaction of the negative condition.

9. Could have done otherwise

261

EXAMPLE. The literature contains cases that stand as putative counterexamples to "that a did it implies that a might not have done it." The following gives the flavor of how stit theory might respond to a portion of these "counterexamples." Consider the apparently agentive statement: Don Quixote "armed himself cap-a-pie, mounted Rozinante, placed his ill-constructed helmet on his head, braced on his buckler, grasped his lance, and through the door of his back yard sallied forth into the open country." If the statement is taken at face value as agentive, then it must possibly be false. For example, if one looks to the description of Don Quixote as "having lost his wits completely" in order to judge the agentive statement false (on the grounds that a man without wits cannot see to anything), one sees that its possible falsity is trivial. If, however, one uses the fact that Don Quixote's brains have dried up as an excuse to reinterpret the apparently agentive content of the statement as really non-agentive, perhaps something like a metaphor, so that Quixote mounted Rozinante as the storm mounts a distant hill, then the description is not agentive and we concede that stit theory does not pretend to have a strategem—which would doubtless involve intentional elements—for probing the statement in question. (At this point we might have included a discussion of "might not have refrained from preventing," but we refrained, noting only that even though "refrain from preventing," in the sense of [a stit: ~[a stit: ~Q]], is a distinct agentive modality, "might not have refrained from preventing" does not create distinct analytical problems).

9E

Could not have avoided doing

This conjecture, inserted here because its disposition involves an application of the preceding result, is a proposal for a sufficient condition of doing. CONJECTURE. "The fact that a person could not have avoided doing something is a sufficient condition of his having done it" (Frankfurt 1969, p. 150). This appears to be an instance of "necessity implies truth"; but analysis reveals that the conjecture is plausible and interesting only because it is ambiguous. Its status depends on whether "avoided doing it" means just "didn't do it," and so is non-agentive; or whether, agentively, it means "refrained from doing it," that is, "saw to it that he or she didn't do it." This ambiguity is difficult to detect in ordinary English; but when it is revealed by using stit normal forms, either the plausibility or the interest of the conjecture disappears, as we see by considering the following two versions. STIT VERSION 1. ~Might-have-been:-~[a stit: Q] implies [a stit: Q}. STIT VERSION 2. ~[a could-have-sht: ~[a stit: Q]} implies [a stit: Q}.

262

Applications of the achievement stit

UPSHOT. Recall that the antecedent of Stit version 1 comes to Was-alwaysmevitable:[a stit: Q], which makes that version of Frankfurt's conjecture sound like an interesting truth. Stit version 1 is indeed true, but only vacuously so, since by the trivial possibility of not doing principle of §9D, its antecedent is a logical falsehood. Stit version 2 is evidently false; for a counterexample, choose Q as any tautology. It is then past doubt that the antecedent of version 2 is trivially true and the consequent trivially false. Thus, in spite of the plausibility derived from thick and interesting stories, attention to austere form suggests that there is no reading of Frankfurt's conjecture on which it is both interesting and true. 2 EXAMPLE. Since the first version is an easy application of §9D, we illustrate only stit version 2. On the side of the antecedent, it is evident that not even the great Mameluke of Persia, either before or after his nine-hundred-year enchantment, could have refrained from seeing to it that if the golden helmet of Mambrino was made of brass, then it was made of brass; but on the side of the consequent, that dignitary certainly did not in fact see to that, nor to any tautology.

9F

Could have prevented

The following conjecture is confusing in ordinary language but easy to settle correctly when expressed symbolically. CONJECTURE. That we are responsible for some state of affairs implies that it must have been possible for us to have been responsible for its absence. STIT VERSION. [a stit: Q] implies [a could-have-stit: ~Q}. Examples of this conjecture sound plausible in English: It appears to follow from the fact that if Sancho Panza remained at rest beneath the cork tree, then he could have seen to it that he moved (Hume). UPSHOT. But as all contemporary logicians of action know, the most elementary story tells us that the conjecture is false. In stit theory, the relevant point emerges through the negative condition, which requires only that the falsity of Q be risked, not that its falsity be guaranteed. EXAMPLE. La Tolosa, the fair cobbler's daughter from Toledo, saw to it that Don Quixote was girded with his sword; but given the rough company of carriers, to say nothing of La Molinera, that poor wench was evidently in no position to see to it that the knight failed to be girded. 2 Perhaps the evident plausibility of the Frankfurt examples derives from the fact that so many verbs give rise to both agentive and non-agentive readings, a matter that we have suggested can be at least partly resolved by applying the stit paraphrase thesis, Thesis 3.

9. Could have done otherwise

263

Figure 9.2: Journey to the ducal castle

9G

Could have refrained

A subtler question is the following: QUESTION. If a saw to something, could a have refrained from seeing to it? • Some think yes. Chisholm 1964a, for instance, says that if some varlet loosed his firelock, then "there was a moment at which it was true, both that he could have fired the shot and also that he could have refrained from firing it." (Observe that Chisholm's "also" is not "could have seen to it that the shot was not fired"; he does not make the superficial mistake of supposing that the Conjecture of §9F is true.) • Some think no. Frankfurt 1969 supposes it possible that there should be such a thing as "the fact that a person who has done something could not have done otherwise." On our view this question is not to be happily represented without taking into consideration the stit analysis of refraining, so that we are not surprised to find that on those few occasions that the literature notices the existence of the question, it seems to resort to sheer postulation. STIT VERSION. Does [a stit: Q] imply [a could-have-stit: ~[a stit: Q}}? We interpret the question as asking whether or not the fact that [a stit: Q] is true at m implies that there is a moment in the past of m that stands as witness to [a stit: ~[a stit: Q}} at a moment co-instantial with TO. That is, is there some single choice point in the past of TO that, had a different choice been made, would have guaranteed the agent's failure to stit Ql UPSHOT FOR STIT. The implication fails, with an easy example, though not quite so easy as the counterexample to the Conjecture of §9F. In Figure 9.2, Q is settled true at TOO and TOI, and settled false at 7712, all of which are coinstantial.

264

Applications of the achievement stit

Abstractly put, each of WA and WB picture a choice for a; you can see from the diagram that Choice0^,A(mn} — {TOO}, so that WA witnesses the truth of [a stit: Q] at mo (looking to m^ for satisfaction of the negative condition). On the other hand, Choice^,A(mi) = {mi, 7712}, so that since [a stit: Q] is true at mi, WA does not satisfy the positive condition for witnessing that [a stit: ~[a stit: Q}\ is true at m^. The choice point WA is "too soon." (The choice point, WB, however, does that job.) Therefore, [a could-have-stit: ~[a stit: Q}} fails at m0, and therewith the implication stated in the stit version. Here is a concrete example based on Figure 9.2. EXAMPLE. We take some literary license in the following idealization, recounted in such a way that Figure 9.2 serves as both spatial map and a model of choices in branching time. One afternoon the Knight of the Lions and his squire, Sancho Panza, journeyed to the castle of the duke and duchess.3 In order to process our English tenses, station yourself at some moment later than m0- At a certain point A the knight and his squire chose the north path, which (ideally) guaranteed their arrival at the castle by sunset. We suppose that their only other choice at point A was the northeast path, which itself split, after a few minutes, at point B. At point B they could have either elected the north path from B, which also would (ideally) have guaranteed their arrival by sunset, or they could have chosen the northeast path from B, which would have led them astray with no possibility of arrival by sunset. Thus, when Don Quixote and Sancho actually arrived at the castle by sunset, there was no choice point in the past of their entrance to the fortress at which they could have guaranteed refraining from arriving by sunset. That is, no choice in the past of their arrival could have positively prevented them from choosing to arrive by sunset. Take notice that the moment of departure from B, at which indeed they could have chosen to refrain from arriving by sunset, is not in the past of the moment of their actual arrival. Heed also that the example is purely structural—the desires, beliefs, and intentions of the agents are irrelevant. We add that for this conjecture, it makes a difference whether one considers stit or dstit. DSTIT VERSION. Does [a dstit: Q] imply Can:[a dstit: ~[a dstit: Q}}7 Here the dstit-suitable Can: is taken from §8F.4. UPSHOT FOR DSTIT. In contrast to the stit version, the implication holds. As in §8F.4, we take Can:[a dstit: Q] as simply Poss:[a dstit: Q], noting that the formal countenance of "can" for dstit can be less wrinkled than that of "can" with the achievement stit, because one need not worry about a double temporal reference. The implication then comes to this: [a dstit: Q] implies Poss:[a dstit: ~[a dstit: Q}}. A proof can be found in Horty 2001. 3 We follow Pellicer in identifying the unnamed duke and duchess with Don Carlos de Borja and Maria Luisa de Aragon, whose ducal descendant celebrated the third centenary of Qmxote in Pedrola in 1905.

9. Could have done otherwise

265

EXAMPLE. We redescribe the journey to the castle in terms of dstit, changing only the Q, which is now to the effect that ( WilLDon Quixote and Sancho Panza arrive at the castle by sunset), which we evaluate at WA and the history on which they choose to go left. On this construal the later moment WB is irrelevant: At WA, given the choice they did make, they dstit (WilLthey arrive by sunset); also at WA they could have chosen to dstit they did not dstit (Will:they arrive by sunset). Moment WB is relevant in considering whether or not at WA they can dstit (Will:they fail to arrive at sunset). They can't.

9H

Might have refrained

We pray your close attention to a question whose answer depends on whether an infinite number of choices is made in a finite time (busy choosers, Def. 14). QUESTION. Suppose that a sees to it that Q; does it follow that a might have refrained from seeing to it that Q in the sense that there is a co-instantial alternative at which a refrains from seeing to it that Q? STIT VERSION. Does [a stit: Q] imply Might-have-been:[a stit: ~[a stit: Q}}? This is a tricky question. Its answer depends, of all things, on whether or not there are busy choosers, §7C.5. UPSHOT WITHOUT BUSY CHOOSERS. If there are no busy choosers, the implication is valid. PROOF. Lettered steps are keyed to Figure 9.3. (a) Grant [a stit: Q] true at mo /ho, and let WQ be the witness in question, (b) Let mi be some moment in i(m0) a* which [a stit: Q] is settled true, and which has the further feature that it "has a closest witness" in the sense that there is a witness, w\, to [a stit: Q] at mi such that between wi and i(mo) there are no further witnesses to the settled truth of [a stit: Q} at any moment in i(mo)- Because there are no busy choosers, m1 must exist. By the negative condition, (c) there is a moment, m2, lying in i(mo) and above wi at which Q is not settled true. We claim that w\ is a witness to [a stit: ~ a stit: Q}} at 7712- The negative condition is easy: [a stit: Q] at mi is just what is required. Suppose, for reductio, that the positive condition failed; that is, (d) suppose that [a stit: Q] were true at some moment m3 £ Choice^im?.}, with witness at, say, w2. Where could w2 be? Because both wi and w2 precede ma, by no backward branching, either wi < w2 or w2 ^ wi. ( e i ) The first alternative is impossible, because w\ is a closest witness. (e2) The second alternative is equally impossible, because then the positive condition of w2 witnessing [a stit: Q] at ma would conflict with the failure of Q to be settled true at m2. (/) It cannot therefore be gainsaid that [a stit: ~[a stit: Q}} is settled true at m2, which is a co-instantial alternative to m0. Therefore, Might-have-been:[a stit: ~[a stit: Q}] holds at mo/h0.

266

Applications of the achievement stit

Figure 9.3: Implication without busy choosers

EXAMPLE WITHOUT BUSY CHOOSERS. The journey to the ducal couple's castle portrayed earlier in Figure 9.2 illustrates the subtle difference between "could have refrained" and "might have refrained." In that adventure the travelers might have refrained from arriving before sunset, though it was false that they could have refrained from doing so, since they were not busy choosers. And thus it is: The moment at which the wayfarers might have departed from point B is an excellent witness to the truth of [a stit: ~[a stit: Q}} at a moment co-instantial with the one in question, thus verifying "might have refrained." Since, however, that moment is not in the past of the moment of their actual arrival at the castle, it does not help verify "could have refrained." UPSHOT WITH BUSY CHOOSERS. In the presence of busy choosers and witness by chains, the implication fails. This is a near consequence of Theorem 18-8, which is based on a strong lemma, and in the vicinity of which there is more information. Here we offer a proof of the upshot with busy choosers that is based on a picture. PROOF. We turn to the busy picture of Figure 9.4 for a counterexample to the implication from [a stit: Q] to Might-have-been:[a stit: ~[a stit: Q}]. The abstract facts are these. We are ultimately interested only in a certain moment, mo, but we consider also (i) moments in i( mo ), which is represented by the dashed horizontal, and (ii) choice points of two sorts: those for a, represented by rectangles, and those for /3, represented by circles. Each choice point for a is binary, with the right possible choice containing a single history on which Q is assigned settled true where it intersects i( mo ). The left possible choice for a at that same choice point leads immediately to a choice point for (3, at which there

9. Could have done otherwise

267

Figure 9.4: The Mirror Game

are two possible choices, left and right, each of which leads immediately to a choice point for a of exactly the same kind as before. We suppose the temporal distance between choices for a is halving, and that each entire denumerable historical series of choice points for a and (3 approaches a unique member of i(mo)'i and we assign Q settled false at such members of i(ma)- The moment, mo, is the one lying above the right side of the first choice for a. We claim first for Figure 9.4 that each choice point, wi, for a witnesses the settled truth of [a stit: Q] at the moment, mi, in which the history belonging to the righthand possible choice for a at w1 intersects i(mo)- The positive condition is easy, since we assigned Q settled true at mi and since there is but a single history contained in that possible choice. The negative condition is satisfied by our having assigned Q settled false at those members of i(mo) approached by a denumerable historical series of choice points; one (and indeed many) of those members of i(mo) must be properly later than w\. As a special case, [a stit: Q] is true at mo- We claim next that nowhere in i(mo} is it settled true that [a stit: ~[a stit: Q}}. This is obvious for the members of i(mo), such as mo itself, that lie above some right-hand possible choice for a. Now suppose for reductio that [a stit: ~[a stit: Q}] is settled true at some moment m-2 in Z( m o ) that is approached by a denumerable series of choice points. There must then be a witness, and since we are allowing witness by chains, §8G.4, let the chain be 02, as indicated in Figure 9.4. The positive condition implies that ~[a stit: Q] be settled true at every moment in i(mo) that is choice equivalent to 7712 for a at C2. Choose some member, W2, of 02. Properly between W2 and m2 there will be a choice point, w3, for (3, and properly after w3, there will be a choice point, 104, for a that is not in the past of m,2 (w^ -f. m-i). Let 777.4 be the member of i(mo) lying above the right-hand possible choice for a at 104. The critical point is that 771,4 is choice equivalent to 7712 for a at 02, since—by the no choice between undivided histories principle—no choice for a in c? distinguishes 7774 from 7712So, since [a stit: Q] is certainly settled true at 7714, we have a contradiction.

268

Applications of the achievement stit

Thus [a stit: Q] is settled true at mo but Might-have-been:[a stit: ~[a stit: Q}} is not, so that the implication fails. 9-1 EXAMPLE WITH BUSY CHOOSERS. (The mirror game.) The Knight of the Mournful Countenance and the Knight of the Mirrors engage in serious combat. (We thank S. Sterrett for supplying a basic idea of this game.) They play a busy game that begins at noon and ends at sunset—at which time the vanquished is to remain entirely at the mercy of the victor. (Thomsen 1990 reminds us that "there is nothing more serious than play," p. 171.) Some plays of the game consist of infinitely many moves, which our knights-errant manage by halving the time spent on each successive move. Though busy, it is still a simple game. Don Quixote has the first move. On his turn the Manchegan has the following choice: Press on or retire. If he retires then at sunset he is the vanquished. If he decides to press on, it is the turn of the Knight of the Mirrors, whose move always consists in selecting a phantasmic replica of one of two giants for the Manchegan to face: either Pandafilando of the Malignant Eye, or Briareus with many arms, each later phantasm being, in appearance, half as tall as its predecessor. Whatever he of the Mirrors selects, the next turn belongs again to the Knight of the Mournful Countenance. At sunset there are but two relevant possibilities: Either Don Quixote has retired, in which case he is the vanquished, or he has succeeded in facing some denumerable sequence of phantasms, in which case he is the victor. The curious fact to be illustrated is this: It is possible for Don Quixote to retire from the contest, but it is not possible for him to refrain from retiring. Contrary to our untutored intuitions, not even an entire chain of choices to press on, right up to sunset, can witness that Don Quixote refrains from retiring; for such a chain of choices does not establish that it was entirely up to him that he persevere. The choices that in fact were made by the Knight of the Mournful Countenance bestow no hard information about "what he would have chosen" had the Knight of the Mirrors confronted him with an unrelenting sequence of replicas of Pandafilando of the Malignant Eye. Quixote's famous victory, however, does allow him to wrest from the fallen Knight of the Mirrors the confession that "the torn and dirty shoe of Lady Dulcinea of El Toboso is better than the ill-combed though clean beard of Casildea." Trust the concreteness of this fable, we beseech you, only to the extent that you understand its structural properties. Symmetrically, if you think our chronicle is wrong, please try to find an alternative rigorous account of witnessing, refraining, and so on, and not just another picturesque story without a precisely described structure.

91

Had available a strategy for not doing

In §9D we observed that by the negative condition, doing something implies that it might have been that it was not done ([a stit: Q] implies Might-have-been:~[a stit: Q}). In §9H we showed, in contrast, that in the presence of busy choosers, that a does something does not imply that it might have been that a refrained

9. Could have done otherwise

269

from doing it ([a stit: Q] does not imply Might-have-been:[a stit: ~[a stit: Q}]}If you look at any picture, however, it certainly seems as if whenever there is a witness, w, for [a stit: Q], at that witness a has available a strategy he could follow that, provided a never deviated from that strategy, would guarantee his not-doing. That is, [a stit: Q] plausibly implies that there was (in the past) available to a a strategy guaranteeing ~[a stit: Q}. The intuitive idea is that a could avoid seeing to it that Q by shooting for a "counter," as called for by the negative condition, at which Q will not be settled true. This intuition works; but rigorously establishing the fact (Theorem 13-28) turns out to be less trivial than one might suppose, requiring as it does much of chapter 13. In the end we are led to see the sharp difference between (i) something coming about for which there was a guaranteeing choice by a, and (ii) something coming about for which there was (in the past) available to a a guaranteeing strategy. Since past choices are a matter of fact, in case (i) we are entitled to a "because": The thing came about because of a choice by a. In case (iz), however, we are not entitled to a "because." since we have said that the strategy was available to a and that a chose in accord with the strategy, without saying anything about the agent following the strategy. Even though this book sometimes lapses into the language of "following," one must recognize that the concepts of the theory of agents and choices in branching time are too austere to support such loose talk. But more of this in chapter 13.

9J

Summary

The proposition If an agent is morally responsible for doing something, then the agent could have done otherwise

, ,

conceals complex connections among actions, moral responsibility, and alternatives open to an agent. We simplify by dividing the proposition in two: If an agent is morally responsible for doing something, then the agent did it.

,_,

If an agent did something, then the agent could have done otherwise.

/o\

This division isolates the idea of doing something, validating the use of a logic of agency. We use stit theory to clarify proposition (3). CONJECTURE. Proposition (3) can be disambiguated by means of a logic of agents who make choices against a background of branching time. UPSHOT. The conjecture is true. If we interpret "could have done otherwise" as "might have been otherwise," the implication holds; if we interpret it as "might not have done it," the implication still holds, but vacuously. If we read it as "could have prevented," the implication fails. If "could have done otherwise"

270

Applications of the achievement stit

means "could have refrained" then it fails for the achievement stit, but holds for dstit. If "could have done otherwise" is taken as "might have refrained," then without busy choosers the implication holds, but with busy choosers it fails. If "could have done otherwise" is taken as "had available a strategy for not doing," then busy choosers or not, the implication holds.

10

Multiple and joint agency The goal of this chapter is to inspect some structural aspects of multiple and joint agency, a task sufficiently complex to give pause to the three inseparables, Aramis, Athos, and Porthos.* First, in §10B, we treat complex nestings of stits involving distinct agents. The discussion is driven by the logical impossibility of "a sees to it that 0 sees to it that Q" in the technical sense, even though that makes sense in everyday language. Of special utility are the concepts of "forced choice," of the creation of deontic states, and of probabilities. Second, in §10C, joint agency, both plain and strict (every participant is essential), is given a rigorous treatment in BT + I + AC theory. A central theorem is that strict joint agency is itself agentive. In the final section, §10D, we combine these two perspectives, looking briefly at other-agent joint agentives. Throughout this chapter we use "stit" for the achievement stit, §8G.3. As elsewhere in this book, even minimal progress toward the goal of this chapter has required a variety of simplifications, (i) As in stit theory generally, we totally avoid the reification of actions, and (ii) we minimize reference to intention in order to concentrate on causal structure, (in) Of relevant notions, we omit stits that are based on "witness by chains," §8G.4, and (iv) the deliberative stit, §8G.l. (v) Also, we omit concepts requiring the chapter 13 notion of strategies, and (vi) we do not consider the evident fact that agents interact in space-time, a topic yet to be studied. Finally, (vii) for the scope of this chapter, for the sake of expository simplicity we assume no busy choosers in the sense of Def. 14. In exchange, although we directly employ only the achievement stit, we have in this chapter limited ourselves to ideas and applications that, we think, work equally well for either the achievement or the deliberative dstit. We refer to Tuomela 1989a and 1989b for an alternative methodology that, in contrast to stit theory, freely permits one to (i) reify actions and (ii) refer *With the kind permission of Baltzer Science Publishers, this chapter draws on Belnap and Perloff 1993 As in the case of chapter 9, you will observe that we sometimes let the source of our examples influence our mode of speech.

271

272

Applications of the achievement stit

to intentions. Those articles also provide access to some earlier studies of joint agency.

10A

Preliminary facts

We will need the following stit facts. (See Def. 12 for the notation mi =%, m?.) 10-1 FACT. (Some facts about the achievement stit) i. Backward monotony,

and

m2 imply

ii. Witness-identity lemma (Chellas 1992). Suppose that Qi implies Q-2, that m, Wi, and io2 are moments, and that a.\ and a2 are (possibly identical, possibly distinct) agents. Suppose further that w\ is a witness to [a\ stit: Qi] at m, and that w
, .

Although it might seem that the content of the imperative construction of (1) is a non-agentive describing d'Artagnan's whereabouts, according to the imperative content thesis, that is mere appearance. In truth the content of (1) is wellregimented by the explicit agentive, [d'Artagnan stit: d'Artagnan is in Saint-Cloud ...]. 1 We remark that the locution "Q &: ~[a stit: Q]" plays an important role in the completeness proof of chapter 16.

10. Multiple and joint agency

273

And when d'Artagnan's father says of the old yellow horse "Never sell him," the content of his imperative for each moment may appear to be a non-agentive that merely denies agency to d'Artagnan, ~[d'Artagnan stit: d'Artagnan sells the yellow horse]. The imperative content thesis, however, drives us to take as agentive the content of the imperative "Never sell him": [d'Artagnan stit: ^[d'Artagnan stit: d'Artagnan sells the yellow horse]].

(2)

That is, the father charges d'Artagnan to deny himself the agency—to refrain, where according to stit theory, refraining from seeing to it that Q is always definable as [a silt: ~[a stit: Q]]. We shall be dealing also with some deontic statements, in connection with which we remind the reader of the restricted complement thesis, Thesis 5. That thesis requires (or suggests) that deontic statements have one of the following forms. Oblg;[ stit: Q]; is obligated to see to it that Q is forbidden to see to it that Q Frbn:[ stit: Q]: is permitted to see to it that Q. Perm:[ stit: Q]: See §1B.2 and §2B.9, as well as chapter 11 and chapter 12.

10B

Other-agent nested stits

The stit construction encourages nesting, which is made explicit in Slogan 56. Talja 1980, extending Lindahl 1977, has examined situations describable by truth-functional combinations of distinct-agent stits. Included are clauses involving their deontic modalizations, but without regard to the restricted complement thesis. In other places we have discussed same-agent nesting. In this section we work with other-agent nested agentives and some of their cousins, where, by definition, each other-agent nested agentive is an expression (i) agentive for some agent a and (ii) whose complement is or contains a sentence agentive for a distinct agent (3. We consider the following examples of apparent other-agent nested agentives. Queen Anne sees to it that d'Artagnan retrieves her diamond tags.

(3)

Jussac sees to it that Biscarat surrenders.

(4)

Cardinal Richelieu sees to it that M. Bonacieux agrees to spy on his wife.

/j.\

Kitty sees to it that d'Artagnan seduces her.

(6)

Count de Wardes sees to it that Mme Bonacieux doesn't keep her rendezvous.

/~\

(6)

274

Applications of the achievement stit

Figure 10.1: Impossibility of [a stit: [f3 stit: Q}]

Mme Bonacieux sees to it that d'Artagnan doesn't follow her.

(8)

The examples fall into two groups. The difference lies in their complements. In (3)-(6) the complements appear to be agentives, whereas in (7)-(8) the complements appear to be negations of agentives. We treat the groups separately, taking first those with apparently agentive complements. It might appear that [a stit: [/3 stit: Q]] is the appropriate form to represent (3)-(6). (Obviously when a = 0, [a stit: [/3 stit: Q]} does not involve multiple agents, and indeed is equivalent to [a stit: Q}. Henceforth we ignore that case.) Appearances, however, can be misleading. When a ^ 0, then as Chellas 1992 shows, the witness-identity lemma, Fact 10-1, and the independence of agents, Post. 9, together imply 10-2 FACT. (Impossibility of one agent seeing to the action of another) [a stit: [/3 stit: Q}} is impossible. Since this fact is central to our present concerns, we give a proof. PROOF. Assume the following for reductio: (a) [a stit: [0 stit: Q}} is settled true at mi with w as witness, and with m2 a "counter" as required for the negative condition, so that (b) [/3 stit: Q} is not settled true at m^- By independence of agents, there must be an m^ such that both (c) m-i =° 7713 and (d) 7713 =^ m?.. By (a), (c), and the second witness-identity lemma (Fact 10-1), it must be that (e) w is witness for [a stit: [/3 stit: Q}} at m^. By (a) and (c) we must, by the positive condition, have (/) [0 stit: Q] settled true at 7713—let u>i be the witness for this. From (e), (/), and the witness-identity lemma, we infer (g) w\ < w. So (d) and (g) imply, by backward monotony (Fact 10-1), that (h) ms =£t mg. But then the second witness-identity lemma with (/) and (h) gives that [0 stit: Q] must be settled true at m^; which contradicts (b) and completes the proof. For a picture, think of Figure 10.1 as a diagram of a witnessing moment, mo, in which columns represent possible choices for a, and rows picture the choices for 0 (see §10C.2 for the attribution of this picture to von Neumann). If [a stit: [0 stit: Q}}, then, where A = [0 stit: Q], A must fill some "choice-column" for a in

10. Multiple and joint agency

275

mo. But then, because A represents a stit by /?, whenever A appears anywhere in a "choice-row" for /? inTOO,it must fill that row. So A must fill the entire diagram ofTOO-If so, then A is by definition settled true atTOQ.This contradicts the principle that, by the negative condition, stit statements are never settled true at their witness. So [a stit: [(3 stit: Q}] won't do as a representation of anything consistent. Before further considering (3)-(6), we turn to the examples (7)-(8), whose complements appear to be negations of agentives. We represent (7) straightforwardly with the form [a stit: ~[/3 stit: Q]], inasmuch as the count makes it true by kidnapping Mme Bonacieux. We observe that although kidnapping is unusual, the form [a stit: ~[/3 stit: Q}}, in contrast to the impossible [a stit: [j3 stit: Q}}, depicts a common enough occurrence. Indeed, everything we do tends to limit i] p choices of others, and much that we do guarantees such limitations. Although (8) appears to be similar to (7), appearances can be deceiving. We cannot represent (8) with [a stit: ~[/3 stit: Q]}: Mme Bonacieux is in no position to guarantee that d'Artagnan fails to follow her. Observe that among (3)-(8), we have so far provided a representation only for (7). In particular, we have pointed out that neither [a stit: [(3 stit: Q]] nor [a stit: ~[/3 stit: Q}] is appropriate to represent any of the others. Does this indicate a weakness of stit theory? We think not; we think it indicates a strength. The failures of [a stit: [0 stit: Q}} and [a stit: ~[/3 stit: Q}} encourage us to find more adequate stit-involving interpretations of (3)-(6) and (8). Chellas 1992 says that it would be "bizarre to deny that an agent should be able to see to it that another agent sees to something." Our acceptance of this view for everyday language is exactly what drives our search. We will look at four interpretations involving other-agent nested stits: deontic, disjunctive, probabilistic, and strategic.

10B.1

Deontic reading of other-agent nested stits

Let us suppose that the facts of (3) are as follows: The queen calls d'Artagnan to her chambers and says, Retrieve my diamond tags.

(9)

D'Artagnan retrieves the tags for the queen. He succeeds in the task she assigned. Shall we represent the situation by [a stit: \/3 stit: Q}} (with a = Queen Anne, /3 = d'Artagnan, and Q •*-> d'Artagnan retrieves the diamond tags)? No, that would be a mistake. It would be a mistake because, as we have seen, [a stit: \f) stit: Q}} claims that the queen guaranteed the truth of [/? stit: Q]. Since her issuance of an imperative does not guarantee that d'Artagnan retrieves the diamond tags, we must be more realistic in our analysis. We begin by reminding ourselves of three things, (i) As Austin 1975 taught us, issuing an order is doing something with words; it is, as everyone now says, a speech act; (ii) speech acts are acts; and (Hi) stit encourages us to understand action by asking what it is that an agent sees to. Let us therefore ask, what did the queen see to when she uttered (9)? Our answer is that she created an obligation. She saw to it,

276

Applications of the achievement stit

by her pronouncement, that an obligation existed where there was none before. Specifically, she saw to it that d'Artagnan was obligated to retrieve the diamond tags. We mean, incidentally, that really giving an order really does create an obligation, not just that the speaker so intends. In making this distinction with absolute sharpness, we separate ourselves from Searle 1965, who appears indifferently to use the language of intention in his "essential condition" but not to use it in his "essential rule" for promises. Having read Hamblin 1987, we also reject the claim of Searle and Vanderveken 1985 that "the point of orders and commands is to try to get people to do things" (p. 14). Instead the point of an order is to create an obligation. Nor does advice have the "causing of action" as its point. Aramis is plainly right that "as a general rule, people ask for advice only in order not to follow it; or, if they do follow it, in order to have someone to blame for giving it." What needs telling is a better story of what deontic states agents really see to when they use not only orders and commands, but also advice, requests, invitations, promises, ..., and indeed assertions and questions. Structural features of stit theory accordingly lead us to the following as a preliminary interpretation of (3).

That, however, won't do, since (3) is a "success" locution. It implies its complement, "d'Artagnan retrieves her diamond tags," whereas the form (10) does not imply [/3 stit: Q}. In a way that is precisely the point: Anne can stit the obligation, but not that d'Artagnan carries it out. We need to add that as a separate conjunct:

Now (11) shows on its face that it involves agency by Anne (as well as agency by d'Artagnan) without being agentive for Anne. 2 Such cases are many and important. When in (4) Jussac orders Biscarat to surrender and Biscarat replies, "You're my commander and I must obey you," he recognizes that his commander has seen to an obligation, [a stit: [/3 stit: Q}} is not the appropriate reading here because Jussac did not guarantee Biscarat's surrender. What did Jussac accomplish with his order? Jussac saw to the creation of an obligation: [Jussac stit: Oblg:[Biscarat stit: Biscarat surrenders]].

(12)

Jussac created the obligation to surrender, but it was Biscarat who surrendered. Regarding (8), [a stit: ~[/3 stit: Q]\ is not appropriate because when Mme Bonacieux sees to it that d'Artagnan does not follow her she does not prevent him from following her; rather she sees to it that a prohibition exists where none existed previously. The form 2

If it were possible for us to use the theory of agents and choices in branching time to represent that d'Artagnan retrieved the tags because of Anne's order, we would do so. We cannot, since the "because" in question seems to be an intentional matter, and so falls among the many aspects of agency for which we offer no theory.

10. Multiple and joint agency [Mme Bonacieux stit: Fr&n: [d'Artagnan stit: d'Artagnan follows]]

277 (13)

is therefore preferable as a representation of Mme Bonacieux' agency in the matter. When d'Artagnan obeys, he is an agent: He refrains from following her. It therefore takes the conjunction of the two agentives in order to reflect these features of the situation. To summarize the results of this section, we make a conceptual advance when we represent the character of the narrow-scope agentive in examples like (3) and (4) with the deontic form

and (8) with

One final point before leaving the topic. Notice that if the equivalence, Frbn:[a stit: Q] <-> Oblg:[a stit: ~[a stit: Q]], is correct, then by substitution in the complement, (13) is equivalent to [Mme Bonacieux stit: 06/(?:[d'Artagnan stit: ~[d'Artagnan stit: d'Artagnan follows]]].

(16)

We think this is right. If so, there is confirmation of the deontic equivalences worked out in §2B.9.

10B.2

Disjunctive readings of other-agent nested stits

Not all apparent other-agent nested agentives can be illuminated by using deontic-complemented stits. Some require use of a disjunctive complement. For example, while we cannot use the form (14) to represent (5), we can use an other-agent nested agentive provided that it has a disjunctive complement. That is, though the cardinal does not obligate M. Bonacieux to agree to spy on his wife, the cardinal does see to it that Either M. Bonacieux agrees to spy on his wife or M. Bonacieux returns to his cell.

/,--,

The normal form [The cardinal stit: ([M. Bonacieux stit: M. Bonacieux agrees to spy on his wife] V [M. Bonacieux stit: M. Bonacieux returns to his cell])],

(18)

helpfully articulates the situation in which a principal agent imposes a "forced choice" on another agent. Figure 10.2 provides a picture: Open to a at w? is a choice that forces /3 into the position at w\ of having to choose between Qi and Q2- This observation, that a form such as

278

Applications of the achievement stit

Figure 10.2: Simple forced choice

can successfully represent the wide-scope agentive of certain apparent otheragent nested agentives whereas [a stit: [(3 stit: Q]} always fails, seems surprising until one reflects that, if one considers future choices, it is common coin that every choice opens up some future possibilities and closes off others. As Calvin once said to his tiger, Hobbes, "each decision we make determines the range of choices we'll face next." Subtly different and somewhat more complicated is the following, for which Figure 10.3 gives a picture. Suppose that Cardinal Richelieu is determined to entangle M. Bonacieux more deeply in his plot. For this purpose, the cardinal sees to it that the unfortunate draper is given a "forced choice." By the act of the cardinal, the draper is compelled to choose whether or not to put himself into the situation of forced choice already pictured in Figure 10.2. For example, Cardinal Richelieu might see to it that M. Bonacieux has a way in which the latter can avoid the forced choice between (i) earning his freedom by agreeing to spy on his wife and (ii) returning to his cell. He can do so by instead choosing to face the executioner. That is, the cardinal is sufficiently powerful to arrange matters so that M. Bonacieux must himself choose between an awful alternative—facing the executioner—and putting himself in a position of forced choice—spying or returning to his cell. When looking at such complicated interactions, it is easy to lose sight of the fact that the principal agent is responsible for seeing to it that the other agent is forced into this terrible predicament. Stit theory has the resources to describe the dimensions of the situation in order to help us understand the relations between the choices of different agents.3 Even so, we hasten to advertise that reliance on stit as one's only linguistic tool makes it awkward to articulate what is obvious from Figure 10.3: The cardinal and the 3 Figure 10.2 approximates the structure of the young man's predicament in Frank Stockton's The Lady or the Tiger. The more complex Figure 10.3 approximates the structure of Sophie's situation in William Styron's Sophie's Choice.

10. Multiple and joint agency

279

Figure 10.3: Complex forced choice

draper both had a role in the transition to the outcome, l 'M. Bonacieux spies for the cardinal."

10B.3

Probabilistic reading of other-agent nested stits

The next reading that we shall consider for other-agent nested stits is probabilistic. As background we observe the following: In this world we seldom guarantee the outcomes that everyday language expresses, even when these outcomes are not agentive; this is one of the principal reasons that we think of stit as only an approximation to "sees to it that." Observe that we do not intend this as a "concession." In analogy, it is no "concession" to remark that the law of the lever is only an approximation to what happens on a playground teeter-totter occupied by a couple of screaming, bouncing children whose parents are all too likely to intervene. The importance of the sociology of the teeter-tooter is no excuse for casting aspersions on the law of the lever in its teeter-totter application. The importance of the way in which outcomes are expressed in everyday language is no excuse for failing to appreciate the idealization of agency embodied in stit theory. In any event, our "nonconcession" is not meant to suggest that we never guarantee anything expressible in ordinary language! One of the things we can guarantee (we think) is the high probability of some outcome expressed in ordinary language. When Aramis sees to it that Lord de Winter learns of Milady's iniquity,

(19)

his choice by no means guarantees that de Winter is informed, but the following is fine:

280

Applications of the achievement stit [Aramis stit: it is highly probable that de Winter learns of Milady's iniquity.]

In designing a language to help us understand this matter, there are two choices: (i) permanently build the probabilistic element into the stit construction, so that stit itself indicates high probability instead of guarantee; or instead (if) represent the idea of probability as a separate linguistic element to be combined with stit as wanted. We think (i) should be avoided. In practice it makes it more difficult, not easier, to analyze problems. We recommend (ii). There is, however, a difficulty: The concealed double time reference of the achievement stit makes it at the least confusing to think through the interactions of probabilities and stit. We could reduce confusion by using dstit (see (iv) at the beginning of the chapter), but not having dstit available in this chapter, we cannot now profitably carry out the analytical work required. Furthermore there really isn't much sense in localizing probabilities in outcomes of moments. Moments are just too big: One ought to be suspicious of the intelligibility of saying that moments have "outcomes" that might or might not be probable. (We mean to refer to objective probabilities. If the probabilities are "epistemic," then the history of analytic philosophy testifies that anything goes.) Instead, outcomes and therefore probabilities of outcomes should attach to small, local events. (A relativistic foundation for this notion is given in Belnap 1992.) For these reasons we only indicate by an example how we think ( i i ) applies to other-agent nested agentives. Consider (6) on p. 273. Surely it is not literally true that Kitty guarantees that d'Artagnan seduces her by the provocative course of behavior she chooses. [Kitty stit: [d'Artagnan stit: d'Artagnan seduces Kitty]] is false. But the following, which introduces the required element of high probability, is true: [Kitty stit: it is highly probable that [d'Artagnan stit: d'Artagnan seduces Kitty]].

, ,

It is also true that d'Artagnan seduces Kitty.

(21)

So we agree with Chellas and common sense that (6) can sometimes be true, and we add a suggestion that in such cases a conjunction of (20) with (21) represents the matter correctly. The underlying point is that if in fact Kitty guarantees that (21), then it cannot be that d'Artagnan is agentive in the matter, and if he is agentive, then Kitty can guarantee the probability of the seduction, but not the seduction itself.

10B.4

Strategic reading of other-agent nested stits

In §5C we suggested at some length that the restricted complement thesis could and should be modified so that a strategy (instead of a stit) could be taken as

10. Multiple and joint agency

281

the content of a promise. We mean "strategy" in the austere sense of chapter 13. It seems equally plausible to invoke a strategy as the content of a widescope operation of an apparent other-agent nested stit, perhaps along with, for example, the probabilistic reading. Consider (5). What the cardinal sees to may well be best expressed as the carrying out of a strategy whose desired issue is that M. Bonacieux spies on his wife. Perhaps, however, this thought is too superficial. Whether or not it proves helpful remains an open question.

10C Joint agency: Plain and strict In the previous section we explored some other-agent nested constructions with singular agents as subjects. In this section we will study agentives with jointagent subjects.

10C.1

Preliminaries

We start with English grammar. Constituent imperatives (see §1C) are embedded imperatives, analogous to embedded declaratives or embedded interrogatives. Their content, like that of agentive declaratives, can always be represented by stit sentences. An imperative, whether stand-alone or constituent, can have a collective term in subject position, as can an agentive declarative: M. de Treville announces: "I won't have my musketeers going to low taverns." The four friends scraped together nine or ten pistoles.

f^\ (23)

Example (22) might well be taken "distributively," and as analyzable in terms of stit sentences with subjects taken to denote a single agent (we call these "singular stits"), perhaps the subjects being individual variables bound by a quantifier. On a plausible reading, M. de Treville requires each musketeer to see to it that he does not go to low taverns. Examples like (23), however, drive us to widen the grammar of the language of agency. Here it is evidently the four friends "taken collectively" who succeeded in raising nine or ten pistoles; it is not something that each of them does. We cannot usefully represent (23) with only singular stits. We need to add to our formal grammar of singular stits the category of a "joint stit." Collectives can be represented by mereological sums as in Massey 1976; here we choose to represent collectives by sets. This choice limits applicability; the proposed apparatus cannot treat cases in which collectives change their membership over time (see Parks 1972), nor cases in which their membership is history-dependent. The limitation is for expository convenience only, and could be removed by using the language ML" of Bressan 1972 that we mentioned in §7C.l. We would realize Bressan's "cases" as moment-history pairs. In that language we would first represent Agent as an absolute concept, so that Agente would by definition be the extensionalization of Agent. Collectives of agents would be represented as properties F (possibly extensional, possibly not; possibly contingent, possibly not) such that F C

282

Applications of the achievement stit

Agente (this makes being a collective of agents a contingent property). Then for such F, [F stit: Q] would be denned as equivalent to the following: [Agent OF6 stit: Q}. The point is that at every moment-history m/h there is a unique subproperty T of Agent whose bearers are precisely those agents contingently identical at m/h to some individual concept falling under F. We can then use F in order to trace the same group of agents through the vicissitudes of time and history, since it must be a "substance concept"; see Gupta 1980 for a discussion of the use of substance concepts as principles of identity. The formal clothing of our decision is this, (a) We let F range over nonempty subsets of Agent and (&) we count [F stit: Q] as grammatical. Thus, we propose to represent (23) by [The four friends stit: the four friends have nine or ten pistoles],

(24)

where we let the four friends = {Athos, Porthos, Aramis, d'Artagnan} C Agent. (Note: We give this form for simplicity of illustration. The form [F stit: P]V[F stit: Q] seems more apt for (23) than the form of (24), namely, [F stit: P V Q]. They are certainly not equivalent, neither intuitively nor in stit theory.) There is more than one thing that one might mean by [F stit: Q}. First, one might mean that the bearers of F, without any outside help, guarantee that Q, on the basis of a prior simultaneous real choice by each of them. There is also a second, stronger, account. In this version the bearers of F, without any outside help, and with the essential input of each of them, guarantee that Q. Each account is useful and is worth a notation of its own. Since, however, we can give only one meaning to [F stit: Q], we choose the first. Later we introduce [F sstit: Q] as notation for the second account. We postpone to another occasion treatment of cases where Q is best seen as due to sequential efforts of the members of F. Our thought is that one must first be clear on sequential choices by a single agent, a topic on which we lightly touch in §8G.5 and in chapter 13. A consequence is that in this chapter we will often treat cases that in reality represent sequential choices as if they were simultaneous, provided the sequencing seems not important and the reconstrual as simultaneous seems enlightening.

10C.2

Plain joint stits

The key concept is the extension of choice equivalence to sets of agents. 10-3 DEFINITION. (Choice equivalence for sets of agents) For F a nonempty set of agents, we let m-i =£, m^ be defined as Va(a € F —> m\ =£, 7712). Choice equivalence for a set F of agents at a moment w is technically easy, but it is conceptually so important that we offer some further words. Let us go back to the idea that at bottom we are representing "possible choices" at a moment. The deepest idea of a possible choice for a single agent a at w is contained in its representation as a set of histories. The deepest idea of a possible choice for a set F of agents is also contained in its representation as a set of histories. We obtain

10. Multiple and joint agency

283

one from the other as follows: Given a possible choice for each member of F, we define a possible choice for F as a whole to be the intersection or "combination" of all the individual possible choices. The "independence of agents" condition guarantees that such a combination always exists. The image we have in mind is due to von Neumann. Let an outcome be dependent on the choices of two agents a and (3. Von Neumann represents this graphically as follows. All the outcomes are arranged in a rectangular grid. Agent a can pick the row and agent (3 can pick the column. What happens is indicated in the intersection of the row picked by a and the column picked by B. "Independence of agents" just says that some outcome is indicated at each intersection of a row and a column. For example, if there are three rows (choices for a) and four columns (choices for /?), then there are twelve possible outcomes for their combined choice. With the help of the concept of choice equivalence for sets of agents, we can state the truth conditions for [F stit: Q}. We say that 10-4 DEFINITION. (Joint stit) [F stit: Q] is true at m/h iff there is a choice point w—a "witness to [F stit: Q] at TO"—satisfying the following conditions (compare the definition of the achievement stit in §8G.3): Agency: Agent. Priority: w < m. Positive: Q is settled true at each m\ such that m Negative: Q is not settled true at some moment—a "counter"—on the horizon from w at i(m). If we apply this definition to (24), it tells us that the raising of the pistoles was due to a simultaneous antecedent choice of the four friends. By so much the definition makes (24) a good approximation to (23). Furthermore, Definition 10-4 is good logic: 10-5 FACT. (Carryover from singular to joint stits) Results or analyses concerning singular agents established without the use of the postulate of the independence of agents also hold for joint agents. •Results not transferring include those expressed with the help of "a ^ /3" when these rely on the independence of agents. The point is that the possible choices for a and {3 at w will be independent if a ^ /?, but this is by no means true of the possible choices for FI and F2 when FI ^ F2. The obvious reason is that non-identity between the two collectives does not prohibit their having members in common. By Fact 10-5 we mean for example that any implication or non-implication that holds between singular stit sentences with just a also holds between the joint stit sentences that result when F is substituted throughout for a. For instance, [F stit: [F stit: Q}\ is equivalent to [F stit: Q], and ~[F stit: Q] is not in general agentive for F (i.e., it is not in general equivalent to [F stit: ~[F stit:

Q}})-

We can use joint agentives to express the independence of agents, provided we have the help of the following version of "ability" (this generalizes to a set of agents the ability concept of §8G.3).

284

Applications of the achievement stit

10-6 DEFINITION. (Can stit) Let [T can i -stit: Q] be true at a moment-history pair w/h iff there is a moment m lying on the horizon from w at i such that w witnesses the settled truth of [F stit: Q] at m. So [T carn -stit: Q] says that T can see to it that Q is true at i. Then we have the following. 10-7 FACT. (Joint ability) Provided cant-stit: P& Q]. That is, if at w a can see to it that P at z and ß can see to it that Q at i, then at w they can jointly see to the conjunction P& Q at i. Someone might think that the following is a counterexample. Porthos can see to it that the pistoles are used to repay a debt. Athos can see to it that the pistoles are used to purchase meals. But the pistoles being so few, the conjunction is impossible even with their best joint effort.

(25)

If you take the situation described seriously, however, especially with regard to fixing the time references, you will find that it is impossible. Of course if Porthos chooses first, then what Athos can see to is not independent of Porthos's choice, and vice versa. But fix their choices as absolutely simultaneous, as required for our principle of the independence of agents. And fix the "can" not sloppily, but as Austin's all-in, no-holds-barred "can" (Austin 1961, p. 177). Suppose that there are only a few pistoles. Award Porthos the ability to see to it that the debt is repaid. You have by so much restricted the power to be ascribed to Athos; there is in this situation nothing Athos can do by his choice alone that guarantees that the pistoles are used to purchase meals. For unless you either supply more pistoles or weaken Porthos's ability, you must allow that no matter what choice Athos makes, it is not enough by itself to guarantee the availability of the pistoles. Since you have given Porthos the ability to use the pistoles to repay the debt, you have described a situation in which for Athos to use the pistoles for meals requires the de facto cooperation of Porthos. The second sentence of (25) is therefore not satisfied, so that (25) is not a counterexample to the principle of independence of agents as expressed in Fact 10-7. We believe that any conceivable counterexample to the principle of Fact 107 will be equi-peculiar with the quantum-mechanical phenomenon discovered by Einstein-Podolsky-Rosen, for in fact it would need to have the same form; namely, spatially separated events that are each absolutely indeterministic and perfectly correlated. (This formulation comes from Belnap 1992.) Ordinary language can easily fool us about this by permitting (normally useful) waffly readings of "can"; here is a place where theory helps.

10C.3

Strict joint stits

But [T stit: Q] still does not tell us all that we may wish to know. For instance, (24) does not imply that each of the four friends was involved. It might have

10. Multiple and joint agency

285

been, for example, that d' Artagnan was not essential in raising the pistoles in the sense that [The three musketeers stit: the four friends have nine or ten pistoles],

(26)

where, as everyone knows, d' Artagnan E the three musketeers = {Athos, Porthos, Aramis}. Here is the easy fact about [F stit: Q] that informs us of this possibility: 10-8 FACT. (Weakening of ]oint stits) Given F1 C F2 C Agent: if [F1 stit: Q] then [F2 stit: Q}. That is, joint stits are closed under "weakening" by the addition of further agents. We need to define some related properties of agents in two versions before we can go further. The first relativizes the concepts to F and Q. The second drops the F, relativizing only to Q. The point is to be careful as to which concept is at stake. (We remark that although the terminology to be introduced seems apt in context, one needs to be sensitive to the considerations mentioned in §10C.l concerning sequential choices.) 10-9 DEFINITION. (Essential and inessential for stits) a is essential for [F a is inessential for 10-10 DEFINITION. (Essential and bystander for outcomes) a is essential [inessential, a mere bystander, not a mere bystander] for Q <-> 3F[F stit: Q] & for every [not all, no, some] F such that [F stit: Q], a is essential for [F stit: Q]. Thus (26) says that d'Artagnan is inessential for (24), but (26) does not say that d'Artagnan is a mere bystander for the four friends having nine or ten pistoles. What then about the idea that F sees to it that Q, with the added provision that each of its members is essential? We think the intuitive concept thus described is rigorously definable (up to an approximation) just by saying that every member of F is essential for [F stit: Q] (but without requiring that every member is essential for Q). For preference we adopt an equivalent way of adding that there are no inessential members: F sees to it that Q, but no proper subset of F does so:

We will soon define "strictly stit" (an expression we introduce for joint agency when each of the agents is essential) by just this sentence, but first we must face a difficulty: Only the first part of (27) has an agentive form; the second conjunct is instead a denial of agency. So the whole may not itself be agentive! The difficulty is, however, easily overcome. In fact (27) is equivalent to each of the following.

286

Applications of the achievement stit

The equivalence of (27) and (29) establishes that (27) is agentive for F in spite of the fact that a conjunct of (27) is a denial of agency. We state this as a 10-11 THEOREM. (No inessential members) The sentences (27), (28), and (29) are mutually equivalent. In other words, where we let

the following are equivalent:

"NIM" is an acronym for "no inessential members." The proof of Theorem 10-11 goes in a circle as follows. (27) —> (28). Suppose (27) true at m1/h with prior witness w, so that in particular (a1) [F stit: Q] and (a 2 ) NIM are each true at m1/h. From (01) we have that (b) Q is settled true at all m2 such that m1 =£, m2, and there is a counter m3 on the horizon from w at i(m1) at which (c) Q is not settled true. We show that the same witness and counter will serve for (28). The part about the counter is evident, since if (c) Q is not settled true at m3 then neither is its conjunction with NIM. What we need to show for the positive condition, since we already have (6), is that supposing (d) m1 =£, m2, we have (x) NIM is settled true at rn 2 . Suppose for reductio that (x) fails, i.e., that (e) and (/) [F1 stit: Q] true at m 2 with witness w1. By (a1) and (d) and the second witness-identity lemma we know that [F stit: Q] is true at m2 with witness w, and therefore by (/) and the witness-identity lemma we know that (g) w = w\ (any two stits to Q at m2 must have the same witness). But m1 = F1 m2 from (d) and (e), and hence (A) m1 = F1 m2 by (g). Now (h) with (/) and the second witness-identity lemma puts [F1 stit: Q] true at m1. But (a 2 ) and (e) imply that [F1 stit: Q] is false at m1, contradiction. (28) —> (29). Suppose (a) (28) true at m1 with prior witness w. By the positive condition, each of (b) Q and (c) NIM is settled true at every m2 such that m1 =F, m2, and by the negative condition there is a counter m3 such that at m3/h3 for some h3 to which m3 belongs, (d) (Q & NIM) is false, that is, either (e1) Q is false or (e 2 ) [F1 stit: Q] is true, for some nonempty proper subset F1 of F. We need to establish the positive and negative conditions for (29). The part about the negative condition is easy; since [F stit: Q] implies Q, (d) implies that ([F stit: Q] & NIM) is false at m3/h3. Choose m2 such that m1 = F m2. We need to show that ([F stit: Q] & NIM) is settled true there. Now (c) already tells us that NIM is settled true at m2, and indeed from (6) we know that Q is settled true at every moment m2/ such that m2

287

10. Multiple and joint agency

which gives us the positive condition for w to witness [F stit: Q] at m2. We therefore are missing only the negative condition for w to witness [F stit: Q] at m2, namely (x) there is a moment on the horizon from w at i ( m 2 ) at which Q is not settled true. In case (e1) we obviously have that (x); we need to show that case ( e 2 ) also implies (x). But1 this is a consequence of ( e 2 ) with (c) and the sufficient condition for unsettledness of Fact 10-1. (29) -> (27). This is trivial, having the form that [F stit: P] implies P. We therefore enter the following definition, where "sstit" is to be read "strictly sees to it that," and connotes the absence of inessential members. 10-12 DEFINITION. (Joint strict stit) (T sstit: Q] «-> ([r stit: Q] & VTi[0 ^

10-13 DEFINITION. (Strict agency) sstit: Q}.

Q is strictly agentive for r iff Q <-» [r

In order to show that [F sstit: Q] is itself strictly agentive for F, which one would certainly expect, we enter the following 10-14 LEMMA. (Agentiveness of joint strict stit) PROOF. Suppose, for reductio, that O / = F1 C F and [F1 stit: [F sstit: Q]] at m1/h with witness w and counter m3. Choose any m2 such that m1 ={F1 m2; then both [F stit: Q] and NIM are settled true at m2- So Q is settled true at all such m2, and we have the positive condition for [F1 stit: Q] at m1 to be witnessed by w. If, then, Q is not settled true everywhere on the horizon from w at i(m1)i we shall have the negative condition as well, and w will witness the settled truth of [F1 stit: Q] at m1 contrary to the settled truth of NIM there. We obtain the desired unsettledness of Q from the counter at m3 as follows. We know that [F sstit: Q] is not settled true at m3, so that either ~[F stit: Q] or ~NIM is settled true at m3. Since both [F stit: Q] and NIM are supposed settled true at m1, in either case we can use the sufficient condition for unsettledness of Fact 10-1 to infer that there is a moment on the horizon from w at i ( m 1 ) on which Q is not settled true, as required. The following is then an easy calculation. 10-15 FACT. (S4 property for strict stits) [F sstit: Q] is strictly agentive for F, that is, is equivalent to [F sstit: [F sstit: Q]}. One direction comes from the fact that quite generally [F sstit: Q] implies Q. The other direction is a consequence of Theorem 10-11 and Lemma 10-14. The "S4" property that we just proved of sstit does not give us copious information about the behavior of strict seeing to it that; although it is doubtless a beginning, there is much that we do not know. 10-16 QUESTION. (Modal properties of strict stits) Suppose we treat sstit as a modal operator. What illuminating properties does it have? What about its modal interactions with plain stit? And so on.

288

Applications of the achievement stit

1OC.4

Applications of [F stit: Q] and [F sstit: Q]

We turn now to applications of the distinction between [F stit: Q] and [F sstit: Q]. A plausible hypothesis is that [F stit: Q] is not of much use, and that only [F sstit: Q] has application. We have come to think that this misses the mark. Although [F sstit: Q] is sometimes exactly right, often [F stit: Q] is or should be intended. One should keep in mind two quite different contexts: stand-alone agentive declaratives used descriptively, and agentives in their role as complements. First the stand-alone agentive. The queen's ladies in waiting brought the queen's diamond tags from the Louvre to the ball,

(30)

where the queen's ladies in waiting = {Mme de Guitaut, Mme de Sable, Mme de Montbazon, and Mme de Guemenee} C Agent. Does (30) in its normal use imply that all of the ladies were involved, so that (30) should be awarded the form [F sstit: Q], or does (30) report only something having the plain form [F stit: Q]? We really have no fixed opinion, and we recognize that "conversational implicature" might be at work, but we do think it worth entering our own "intuition" : It is inconsistent with (30) that Mme de Sable was an inessential lady. If this "intuition" is correct, then only the stronger [F sstit: Q] is appropriate. If not, then the weaker [F sstit: Q] might be wanted. In either case, the statement (30) is agentive. Embedding a construction with the same content as (30), however, changes our "intuitions." The weaker reading is then much more plausible. Consider even the truth-functional case of negation: The queen's ladies in waiting failed to bring the queen's diamond tags from the Louvre to the ball.

(31)

It would seem to us at least misleading to use (31) to describe the situation in which Mme de Sable alone was inessential. It is, however, when agentives are complements of deontics or imperatives that we are most struck with the appropriateness of using the plain [F stit: Q] form: The queen sent her ladies in waiting to bring the diamond tags from the Louvre to the ball. Presumably this royal order lays a joint obligation on the ladies in waiting. What is the content of that joint obligation? Which of the following seems right? [The queen stit: Oblg: [the ladies stit: the ladies bring the diamond tags from the Louvre to the ball]], or [The queen stit: Oblg: [the ladies sstit: the ladies bring the diamond tags from the Louvre to the ball]].

10. Multiple and joint agency

289

Though the matter is uncertain, surely it is plausible that the content of the queen's order has only the plain form [F stit: Q], so that it is quite consistent with the content of that order that Mme de Sable should be inessential. If Anne of Austria really wants all her ladies to be involved, she should explicitly say so, using something with the content [F sstit: Q}. By Theorem 10-11 she will have the satisfaction of knowing that the content of her order is indeed agentive. We think that the outcome is the same for other deontics with joint subjects such as permissions and prohibitions: Although there is no logical reason not to permit or forbid a collective to see to it that Q in the strict sense, often the plain sense is more likely to catch what is wanted. For example, Although de Treville does not forbid the four friends to spend a total in excess of 6,000 livres on their equipment for the siege of La Rochelle, he advises them not to do so.

(32)

This example describes a prohibition and some advice. By the restricted complement thesis, each should have an agentive complement. The content of the advice is that the friends should refrain from spending more that 6,000 livres, that is, that The four friends see to it that it is false that the four friends see to it that the four friends spend in excess of 6,000 livres.

(33)

It would follow from the stit deontic equivalences, Ax. Cone. 2, that the content of the prohibition (the one never issued by Treville), when reconstrued as an obligation, is exactly the same as the content of the advice, that is, (33). Perhaps this logical parallelism is why (32) sounds so eminently intelligible. So now the question is, with what sort of stit should we approximate the see-to-it-thats that occur in (33)? Let F = the four friends, and let Q <-> the four friends spend in excess of 6,000 livres. Do we want (i) [F stit: ~[F stit: Q]], or (ii) [F sstit: ~[F stit: Q}}, or (iii) [F stit: ~[F sstit: Q]], or (iv) [F sstit: ~[F sstit: Q]]? One can use Lemma 10-14 to show that (iv) is equivalent to (in); and neither is tolerable. Suppose Porthos in his vanity chooses to spend over 6,000 livres, and thus alone guarantees the truth of the complement, so that he alone guarantees that the other three friends are inessential. It seems clear that this behavior counts as not following Treville's advice to refrain, so that (33) is false on that story; but the candidate (in) is true and so cannot be an accurate representation of (33). We are left with (i) and (ii). The latter of course mixes plain and strict stits, but in the absence of a more thorough investigation, both logical and conceptual, we ought not say more. Our last example concerns a permission. The four friends allowed their servants Planchet, Grimaud, Mousqueton, and Bazin to finish the Beaugency wine. It seems implausible that the content of this permission should be represented by a strict stit. Instead [The four friends stit: Perm: [The servants stit: the servants finish the Beaugency]]

290

Applications of the achievement stit

seems a more likely normal form. On this version the four friends permit that the Beaugency is finished by the choice of Planchet, Grimaud, and Mousqueton alone, Bazin having antecedently gone off to study his theology.

10C..5

Overdetermination and "free riders"

Definition 10-9 and Definition 10-10 distinguished relativized and unrelativized notions of essentiality. Because of Overdetermination, they do not come to the same thing. The simplest example involves singular stits with a /= (3: [a stit: Q] and [(3 stit: Q] can both be true. By the witness-identity lemma of Fact 10-1, the witness for the two stits at m will have to be the same moment w, but there is no reason that Q, while satisfying the negative condition, cannot be true both at all TOI such that m =a m1 and at all m1 such that m = B m1. It may be that at a certain moment Bois-Robert makes a real choice that guarantees that Richelieu knows of Buckingham's meeting with the queen, and that the Marquis de Beautru quite independently makes an equally real choice guaranteeing the same thing. So this case can be represented by the truth of [a stit: Q], [(3 stit: Q], and [{a, (3} stit: Q], and the failure of [{a, B} sstit: Q}. Observe that each of a and B are inessential for [{a, /3} sstit: Q], and therefore inessential for Q, but that neither is a mere bystander for Q. For this reason we think it would be wrong to describe either Bois-Robert or de Beautru as a "free rider" even though each is inessential. Only mere bystanders should be called "free riders." The following point to the need for further work, (i) There is the statement NMB that F contains no mere bystanders for Q: . Evidently NIM -> NMB, but not conversely. Thus the proposition that ([F stit: Q] & NMB) stands as follows: [F sstit: Q] -> ([F stit: Q] & NMB) -> [F stit: Q}. (ii) There is the statement OMB that outside of F there are only mere bystanders for Q. The proposition ([F sstit: Q] & OMB) says that F is the one and only joint agent for Q; it is evidently not agentive (in the sense of stit). It is, however, something that could be seen to.

10D

Other-agent nested joint stits

It is evident that the investigations of §10B on other-agent nested stits and §10C on joint stits need combining. In this area there is much to be considered. Here we offer only a single illustration, which is that the apparatus developed can distinguish the content of the following in an illuminating way: The four friends required of one of Planchet and Fourreau that he see to it that Brisement has a proper burial.

(34)

The four friends required of Planchet and Fourreau that they see to it that Brisement has a proper burial.

(35)

The four friends required of Planchet and Fourreau that one of them see to it that Brisement has a proper burial.

(36)

10. Multiple and joint agency

291

Letting a = Planchet, B = Fourreau, and Q <-> Brisement has a proper burial, the normal forms are respectively [The four friends stit: [The four friends stit: and

[The four friends stit:

(37)

In (36) the obligation is jointly on Planchet and Fourreau as a pair, but the execution is supposed to be by one of them as an individual. This complex content, so subtly different from that of (34) and (35), can be clearly expressed by an other-agent nested joint stit as in (37). The lesson, easy to miss if you take these examples as little puzzles or tricks, is that seriously applicable deontic logic needs other-agent nested joint agentives; and it therefore needs to include a theory with at least the expressive power of joint stits.

This page intentionally left blank

Part IV

Applications of the deliberative stit

This page intentionally left blank

11 Conditional obligation, deontic paradoxes, and stit Consider the following argument, which is adapted from Castaneda 1981.* (a) Alchourron is obligated to do the following: If Bulygin sends him the draft of their latest joint paper, revise it. (b) Bulygin has sent Alchourron the draft of their latest joint paper. Therefore,

(1)

(c) Alchourron is obligated to revise the draft. Castaneda points out that this straightforward reasoning cannot be accommodated by most existing deontic calculi. To see why, let A stand for "Alchourron revises the draft" and B for "Bulygin sends Alchourron the draft." Then we can represent the argument (1) symbolically:

Here, Oblg: stands for the obligation operator. The proposition which would allow detachment of the obligation Oblg:A in (2), does not belong to the standard system of deontic logic. Consequently, the argument is invalid.1 *Paul Bartha is the author of this chapter. Publishers, it is based on Bartha 1993. 1 F011esdal and Hilpinen 1971 sets out the logic," known as KD, in which Oblg:(B>A) D is usually rejected in any system which is based Hintikka 1971.

With the kind permission of Baltzer Science axioms for the "standard system of deontic (BD Oblg:A) does not hold. This proposition on "deontically perfect world" semantics; see

295

296

Applications of the deliberative stit

The argument becomes valid if we replace Oblg:(BDA) by BD Oblg:A in (2)(a). No corresponding change is required in English. The English version of the argument seems acceptable as stated. This suggests that English is less sensitive than standard deontic logics to the difference between the two forms Oblg:(BD A) and BD Oblg:A. These two forms will be referred to as OD-statements and D 0-statements respectively. Castaneda has developed one approach that brings the reasoning of deontic logic closer to that of English. He suggested in Castaneda 1981 that the two forms of obligation are equivalent under one assumption. Specifically, he proposed that the equivalence

is valid whenever B is a "circumstance" or "condition" and A is "an action considered deontically as a deontic focus" (p. 46). He develops a calculus which takes the circumstance/action-as-focus distinction as primitive. This chapter takes a different approach, starting from the equally important distinction between agentive and non-agentive sentences. Earlier chapters formalize this distinction by providing two varieties of semantics for the agentive construction "a sees to it that A" (written as [a stit: A]]: the achievement stit, §2A.2, and the deliberative stit, §2A.3. In this chapter, a simple semantics of obligation is developed as an extension of the logic of the deliberative stit, written [a dstit: A]. The basic idea is inspired by the reduction of deontic logic to alethic modal logic in Anderson 1956. After the technical preliminaries of §11A, in §11B we explain our adaptation of Anderson's semantics of obligation. The resulting concept is compared to other systems of deontic logic. A partial completeness result is described in §11C. The semantical system is then used to analyze conditional obligation. In particular, as §11D and §11E show, it provides a precise way to define a "circumstance" such that the argument (2) becomes valid under the assumption that B is a circumstance. In the remainder of the chapter, the semantics is used to shed light on two paradoxes of deontic logic. The point is not to give a final solution to any paradox or problem. Rather, we hope to show that many of the problems of deontic logic are essentially problems about agency rather than obligation. We also hope that readers will see that stit theory can be a useful tool in thinking about such problems, a theme that we further develop in chapter 12.

11A

Technical preliminaries

For this chapter we limit our target grammar as follows. Sentences in our language are constructed from propositional variables by truth-functional connectives ~ and &, as well as modal operators Universally: and Sett:, deontic operators Frbn: and Perm:, and the agentive operator, [a dstit: ]. There will also be tense operators Will: and Was:. As usual, V, D, =, T and _L are introduced as abbreviations, and we will also introduce by abbreviation the Chellas

11. Conditional obligation, deontic paradoxes, and stit

297

agentive operator [a cstit: A]. We use A, B, and so on to range over sentences. We recall that the semantics of [a dstit: A], where a is an agent and A a sentence, is based on the fundamental notions moment, history, agent, choice set, and possible choice, as reflected in a BT + AC model, m, which is a pair consisting of a BT + AC structure © = (Tree, \<, Agent, Choice) as in §2, and an interpretation 3. The fundamental properties of 6 are described in detail in chapter 7 and reviewed in §3. The fundamental semantic ideas are presented in chapter 8, with the semantics of dstit being given in §8G.l and the other operators being characterized in §8F. We extract the following for purposes of this chapter:

Positive Condition. m, m/h 2 /= A for all h2 with h2 =a h. (By making the possible choice containing history h, a guarantees that A is true, since A holds on all histories consistent with a's choice.) Negative Condition. m, m/h3 \= ~A for some h3 with m E h3; that is, it is not the case that m, m/h \= Sett:A. The moment-history pair m/h3 is called a counter. (Thus, a has a real choice about A, since it's not the case that A is settled true regardless of what a does.) Validity of A for , written m \= A, and validity of A for G, written & \= A, are defined in the standard way, as in §6 of the appendix. It is important for our upcoming analysis to observe that Universally: involves quantification over all moments (and histories) in Tree, and must be distinguished explicitly from Sett:, that is, settledness at a single moment. To facilitate the presentation of a partial completeness result (§11C), it is useful to introduce the modal operator of Chellas 1992, which we call "the Chellas stit." 2 As we noted in §4C, Chellas writes AaA; in this book, in order 2 Note that Chellas in fact proposed two different operators for stit. One was proposed in Chellas 1969 and discussed in Chellas 1992, and the other was proposed in Chellas 1992. We are here talking about the latter, the operator A of Chellas 1992 (for convenience we use A). Note also that the background of Chellas's semantics for stit actually falls into the category of Tx W accounts, but his account can be easily translated into a semantics against the background of "tree-like frames."

298

Applications of the deliberative stit

to emphasize analogies with dstit, we instead write [a cstit: A] for that operator. We could add it as a primitive to our language and, as in §8G.2, provide the following truth definition:

Evidently, [a cstit: A] corresponds precisely to the positive condition for [a dstit: A]. In fact, if m, m/h \= [a cstit: A] holds on m, then either A is settled true at m, or the negative condition is satisfied and [a dstit: A] must be true at m/h. We make use of these observations to introduce [a cstit: A] instead as an abbreviation:

So defined, [a cstit: A] agrees with the semantic condition (3). One may also verify that any one of [a cstit: A], Sett:A, and [a dstit: A] could be defined in terms of the other two operators, since we have as valid sentences:

(Chellas 1992 has emphasized this interdefinability. As he notes, the same disentanglement is not possible for the achievement stit.) It should also be noted that the three operators Universally:, Sett:, and [a cstit: ], by their truth definitions, are each just like the necessity operator in S5 modal logic.

11B

Semantics of obligation

Earlier chapters suggest that deontic logic should be treated as an extension of a modal logic of agency. The restricted complement thesis, Thesis 5, requires that deontic constructions take agentive sentences as complements: In a sentence Oblg:A, A must be (or be equivalent to) a stit sentence. A justification for this claim is that each practical obligation (each "ought-to-do") should be connected to a specific action by a specific agent. Regardless of whether the restricted complement thesis is correct in the end, we think that ought-to-do's are a good place to start. This section develops a semantics for deliberative obligations—obligations binding on an agent at the moment when he or she makes a choice. (Thomason 1981b makes the contrast between deliberative and judgmental obligation.)

11B.1

Definition of obligation operator

Anderson 1956 suggests the following reduction of deontic to alethic modal logic. Let S be a constant proposition, which we call a sanction. Anderson's suggestion is that we exploit the connection between obligation and sanction by defining, for any sentence,

11. Conditional obligation, deontic paradoxes, and stit

299

The Universally: ensures that the implication is strict. We will use something similar to define the obligation operator. As indicated by the preceding discussion, however, we will restrict our attention to cases in which A is a dstit sentence. Three questions arise: i. What is the appropriate interpretation of the propositional constant S that is to be added to our language? ii. What should be the quantificational range of the modal operator? Hi. How should we deal with the negation of A, where A is a stit sentence? Ad (i). The intended meaning of S is unambitious—something such as "there is wrongdoing," or "there is a violation of the rules." The connection between obligation and S should be unproblematic. Even though the chapter sometimes speaks of S as a sanction, we are not entitled to interpret it as punishment or censure, which has no logical connection to obligation. So as to avoid confusing the obligations of different agents, S should be indexed by an agent. Then we can interpret Sa as "a does something wrong." We suppress the subscript, however, since throughout this chapter we will only be concerned with one agent's obligations at a time. (We might also have added a second index for the authority (individuals or perhaps institutions) whose rules are violated. Again, this added complexity is not required at present, but in §11H we briefly discuss the suggestion that obligations should always be thought of as seen to (as "stitted") by an authority.) Ad ( i i ) . The scope of the modal operator will not be all of Tree, but only all histories through a given moment; that is, we will replace Universally: with Sett:. There are two reasons for this. First, "failing to see to it that A" may be a case of wrongdoing at some moments, but not at others. We do not want to limit ourselves to obligations that remain constant for all time. Universally: is unsuitable. Second, the truth of [a dstit: A] involves consideration of all histories through a moment (by the negative condition), so it is not unreasonable to suppose that the truth of Oblg:[a dstit: A] does so as well. (In §11H, we will discuss some difficulties that arise from the choice of Sett:. Another alternative is to use the Chellas stit, [a cstit: ]. It turns out that this is not a good choice, since the resulting definition of obligation makes the sentence [a dstit: A] D Oblg:[a dstit: A] valid in all models.) Ad (iii). With regard to the negation of [a dstit: A], there are three evident possibilities, leading to three alternative definitions of Oblg:[a dstit: A]:

300

Applications of the deliberative stit

Figure 11.1: Basic picture for Oblg: [a dstit: A]

(6) and (7) are harsher definitions than (5), in the sense that if Oblg:[a dstit: A] holds under (5), then the obligation also holds for both (6) and (7). In fact, (6) and (7) must be rejected as too harsh. On either definition, it turns out that Oblg:[a dstit: A] holds for all a if A is any tautology or contradiction. In what follows, we work with (5): a is obligated to see to it that A just in case it is settled that if a doesn't see to it that A then there is wrongdoing. The basic picture for Oblg:[a dstit: A], then, is given in Figure 11.1. Although its interpretation is much like that of the figures for the achievement stit in §2B, for this and later figures featuring the deliberative stit, note the following conventions, (i) Here, in contrast to, for example, §2B, each sentence in a diagram represents its truth at a moment-history pair rather than its settled truth at a moment, ( i i ) Here we allow that different histories coming out of the same choice box need not necessarily split at m. They may split later. Also observe that in this chapter we sometimes refer to a possible choice as a "choice box."

11B.2

Comparison with other systems

We call the just-described system SA, noting that SA combines Stit theory with Andersonian devices; or SA combines the Sanction with a logic of Agency. It is interesting to compare SA to other systems of deontic logic—in particular, to the standard system KD and to Anderson's own work. It turns out that many of the axioms and rules in these other systems can be reformulated as valid principles in SA. Yet we have made no assumptions about S, or about the obligation operator, beyond the definition (5). To the extent that validities in SA are reasonable, the chapter's claim that a logic of obligation can be constructed as an extension to a logic of agency is strengthened. First, there are the standard equivalences between permission, forbidding,

11. Conditional obligation, deontic paradoxes, and stit

301

and obligation that we find in these systems:

As §2B.9 suggests, when we restrict the complements of the deontic operators to be stit sentences, the most reasonable equivalences, as listed in Ax. Conc. 2, are

and

Note that by nesting the agentive modality, we avoid the problem of "negated actions." Since we also have, symmetrically,

(8) and (10) imply, by transitivity, that

In fact, as indicated by T14 and 15 in §17A, it can be verified directly that [a dstit: A] = [a dstit: ~[a dstit: ~[a dstit: A]]] is valid. It says that refraining from refraining from seeing to it that A is equivalent to seeing to it that A.3 The standard system KD (the version cited here is based on Meyer and Wieringa 1991) contains the tautologies of prepositional calculus, the rule modus ponens, the equivalences between permission, forbidding, and obligation, and three additional axioms and rules:

(KD1)

(the K-axiom or principle of deontic

detachment)

(KD2) Oblg:A D Perm:A (the D-axiom, obligatory implies permitted) (KD3) Rule: if A is a theorem, so is Oblg:A (OWg.'-necessitation) The analog of (KD1) for agentive sentences is

which is a valid principle in SA. Indeed, (11) is a direct consequence of

3 Recall from §2B.6 that the "refref" equivalence, Ax. Conc. 1, between seeing to it that and refraining from refraining, unproblematic for dstit, is a more delicate matter for the achievement stit. See the proofs of Lemma 15-15 and Lemma 15-16.

302

Applications of the deliberative stit

Figure 11.2: Proof ol (12)

and it is easy to see that (12) is a fact about agency qua dstit (but see §2B.4 for the opposing verdict for the achievement stit). PROOF. Consult Figure 11.2. To see (12), suppose [a dstit: (ADB)} and [a dstit: A] hold at m/h on model m. Then ADB and A, and hence B, hold at all m/h2 for h2 =a h. Further, m, m/h \= [a dstit: A D B ] requires a counter m/h3 where ADB is false, and thus where B is false. The positive and negative conditions for [a dstit: B] at m/h are both satisfied. From (12), (11) follows easily. PROOF. Suppose Oblg:(a dstit: (ADB)} and Oblg:[a dstit: A] hold at m/h. If ~[a dstit: B] holds at any m/h2, then either ~[a dstit: A] or ~[a dstit: (AD B)] holds there, by (12). But by definition (5), S must then hold at m/h 2 . This proves Sett:(~[a dstit: B]DS), that is, Oblg:[a dstit: Corresponding to (KD2), we have

This seems a perfectly reasonable principle. If one is obligated to drive under the speed limit, then one is permitted to do so. In fact, (13) is valid in SA under the mild assumption that the sanction S is not settled true. Let us call a moment at which S is settled true a no-good-choice moment. In the no-goodchoice case, it is settled that everything a does or does not do leads to the sanction. In this pathological situation, everything is obligatory, everything is forbidden, and nothing is permitted for a, as a consequence of definitions (5), (8), and (9). (See §12D for more discussion of the no-good-choice situation, and a picture.) Clearly, (13) is false in the no-good-choice case.

11. Conditional obligation, deontic paradoxes, and stit

303

Anderson 1956 adopted the axiom that the sanction should be contingent. We do not believe we can accept the corresponding axiom, ~Sett:S. We are not entitled to assume that no-good-choice moments can be ruled out a priori. Genuinely conflicting obligations seem at least possible, especially if the authority issuing obligations is unreasonable. In any case, the valid analog of (KD2), stated without proof, is

There is nothing in SA that corresponds to the rule of 0%:-necessitation, (KD3), a rule that can also be derived in Anderson's system. The rule

would be vacuous, since [a dstit: A] is never a logical validity (because of the counter). Furthermore, it follows from definition (5) that Oblg:[a dstit: T] is always false, except in the no-good-choice case. Perhaps it will come as a relief to learn that we are under no obligation to see to it that 2 + 2 = 4.

11C

Completeness

For this section only, we reduce our language (and our definition of structure, model, truth in a model, etc.) by eliminating the operators Was: (past) and Will: (future), and by restricting Agent to contain only one agent, a. The language still contains the constant S. A completeness result is stated without proof. Our axiomatization is based on that of §17A; structures and models are of kind BT + AC. Recalling abbreviation (4), [a cstit: A] <=^ Sett:A V [a dstit: A], we take as axioms for a system, SAo, all instances of truth-functional tautologies as well as the following schemata:

304 All

Applications of the deliberative stit Sett:A D ~[a dstit: A]

As rules of inference we take modus ponens and the rule of necessitation RN Rule: if A is a theorem, so is Universally:A. It is easy to see from definition (4) and All that Tl and T2 hold (see §11A), and it is also clear that the following rules are admissible: Rl Rule: if A is a theorem, so is Sett:A R2 Rule: if A is a theorem, so is [a cstit: A]. Axioms A1-A9 reflect the fact that Universally:, Sett:, and [a cstit: ] are like the S5 modality. Al0 and All state the relationships between these different modalities. SAo contains no axioms about the Oblg: operator specifically, although the sanction S does occur in substitution instances of its axioms. Using the definition (5) of Oblg:, results such as (11) and (14) can be proven as theorems of SAo- This is a consequence of the completeness property stated in Theorem 11-2. Without S, the system is simply an axiomatization of dstit. 11-1 THEOREM. (Soundness) For every sentence A, A is provable in SAo only if m \= A for every BT + AC model m. PROOF. By induction on sentences. 11-2 THEOREM. (Completeness) For every sentence A, A is provable in SA0 if m \= A for every BT + AC model m. PROOF. This is a consequence of Theorem 17-10. That result is stated only for the connectives [a cstit: ] and Sett:, but is easily generalized to include Universally: as well. The proof for the simple fragment of this chapter is similar to completeness proofs for S5.

11D

Conditional obligation

Using the semantics to which we referred in §11 A, a precise condition can be given under which "detachment" of obligation is acceptable, so that argument (2) goes through.4 That argument is reformulated as follows:

In English: 4

The condition is intended only as sufficient. On the side of necessity, Davey 1999 substantially refines the analysis of this section.

11. Conditional obligation, deontic paradoxes, and stit

305

(a) Alchourron is obligated to see to it that if Bulygin sends him the draft of their paper, he sees to it that he (Alchourron) revises it. (b)

Bulygin sends him the draft.

Therefore, (c)

(16)

Alchourron is obligated to see to it that he revises the draft.

Version (15) is obtained from (2) in two stages. First, we replace A by [a dstit: A] in (2) (a) and (2) (c), since "revising the draft" is agentive. Second, we place the conditional B D [a dstit: A] inside the dstit sentence [a dstit: (B D [a dstit: A ] ) ] , since the obligation in (2) (a) is that Alchourron should see to it that the conditional is true. Argument (15) is not valid without an added assumption:

This says that it's settled that a doesn't see to it that B is false, that is, that a can't see to it that B is false. Condition (C) is one way to formalize the assumption that B is a circumstance, for it captures the idea that a cannot prevent B from being true. (Castaneda's idea of a circumstance is different, since in his system circumstances sometimes are within the agent's control.) Assuming (C) makes the argument (15) go through. On the other hand, if (C) is false, the argument tends in general to fail. (See Davey 1999 for the more detailed account.) PROOF. Consult Figure 11.3. We prove (15) with the help of the added assumption (C). We assume (15)(a) Oblg:[a dstit: (B D [a dstit: A ] ) ] , (15) (b) B, and (C) Sett:(~[a dstit: ~B]) are true at m/h in model m. By the first assumption, if h2 is any history through m, then by definition (5),

We want to show that for any h2 through m,

that is, that (15)(c) Oblg:[a dstit: A] is true at m/h. The crucial thing to notice is that from B and Sett:(~[a dstit: ~B]) at m/h, it follows that for each history h2 through TO, there is at least one choice equivalent history h3, such that B is true at m/h3. Less formally, out of each choice box at TO comes at least one history h3 on which B is true. For if this were not so, then [a dstit: ~B] would be true at m/h2, with the counter at m/h, where B is true. This would violate the fact that ~[a dstit: ~B] is settled true. Now if ~[a dstit: A] were true at m/h2, it would also be true at m/h3. Then the conditional

306

Applications of the deliberative stit

Figure 11.3: Proof of detachment given (C)

would be false at m/h 3 , so that [a dstit: (B D [a dstit: A])] would be false at m / h 2 - By (17), S would follow at m/h2, and we have shown (18). Thus, Oblg:[a dstit: (B D [a dstit: A])] D (B D Oblg:[a dstit: A] ) is valid if B is a circumstance in the sense of (C).

HE

OD-statements versus D O-statements

This result is a possible explanation of the fact that English is largely indifferent to the distinction between OD-statements and D O-statements. For the purposes of detaching the obligation Oblg:[a dstit: A], the two forms

and

are equivalent, provided that the antecedent B is a circumstance as defined by (C). The following examples show that many common OD-statements do satisfy the "circumstance condition."

11E.1

Duty to apologize

Consider the following illustration of (15)(a). It's your duty to apologize, if you behaved badly at the party.

(19)

If we want to represent this obligation as an OD-statement, we suggest that it has the following form:

11. Conditional obligation, deontic paradoxes, and stit

307

In English, "You are obligated to see to it that if you saw to it that you behaved badly, you see to it that you apologize." Now notice that the antecedent Was: a dstit: B] satisfies the "circumstance condition,"

Agent a cannot see to the falsity of a statement about his past seeing-to's. The reason is that either Sett:(Was:[a dstit: B]) or Sett:(~Was:[a dstit: B]) must hold at any moment-history pair. In the first case the positive condition for [a dstit: ~Was:[a dstit: B]] fails, and in the second case, the negative condition fails. PROOF. Suppose that Was:[a dstit: B] is true at m/h. Then for some m2 < m, [a dstit: B] is true at m2/h. But then [a dstit: B] is true at m2/h2 for every h2 with h2 =am2 h. Since every history h2 through m passes through m2 and satisfies h2 = am2 h by the no choice between undivided histories condition, Post. 8, it follows that Was:[a dstit: B] is true at m/h2 for all h2 through m. Thus, Was:[a dstit: B] at m/h implies Sett:(Was:[a dstit: B] ) at m/h. So either Was:[a dstit: B] holds at all histories or at none, which is precisely the "either-or" condition.5 Since Was: [a. dstit: B] is a circumstance, whenever it is true we can detach the obligation Oblg:[a dstit: A]. It seems to us that many conditional obligations have the form of (20), even when the antecedent seems to be present tensed. Consider the obligation: "It's your duty to apologize if you behave badly at the party." What is the tense of "behaving badly" relative to "apologizing"? It must be future, present (contemporaneous), or past. Taking the tense as either future or present is not a reasonable interpretation of the duty, since any apology given before or at the moment of behaving badly will hardly be convincing. This leaves us with the same obligation as before: It's your duty to see to it that you (see to it that you) apologize, if you have (seen to it that you) behaved badly at the party. The form (20) of this restatement of (19) is then the same as the form of (19) itself. Many conditional obligations whose antecedents express definite actions (aorists in ancient Greek) have the form of (20). An important exception will be discussed in S11H.. 5

As noted in §8F.5, on the Prior-Thomason history-relative semantics for branching time, it is not generally true that the past is settled in the semantic sense, which requires that for an arbitrary sentence A, Sett: Was:A or Sett:~ Was:A must hold at each moment-history pair. For instance, it may be that neither Was: Will:A nor ~ Was: Will:A is settled true. Further, [a dstit: A] is never settled true, because of the counter. It is only the combination of the past operator, Was:, with dstit that leads to a form that is bound to be settled true or settled false. In this special case, the history-relative semantics agrees with the intuition that it is always true that either Was:A or ~ Was:A is settled.

308

Applications of the deliberative stit

11E.2

Duty to admonish

Consider another illustration of (15)(a). It's your duty to admonish Bob if he behaves badly (has behaved badly).

(21)

This could be expressed as

Translated back into English, this reads, "It's your duty to see to it that if Bob saw to it that he behaved badly, you see to it that you admonish him." The only difference between (22) and (20) is that a different agent is involved. The same argument as before shows that Was:[B dstit: B] is a circumstance. Consequently, it does not really matter if we express the obligation as an OD-statement or as a D 0-statement. Expression (22) is also a plausible way to formalize Alchourron's conditional obligation to "revise the draft." In (15), we symbolized that obligation as

where a is Alchourron, B is "Bulygin sends Alchourron the draft," and A is "Alchourron revises the draft." We then had to make the extra assumption that B was a circumstance. Since Bulygin's sending the draft is a definite action which takes place prior to the revising, Alchourron's obligation is better formalized in accord with (22): "Alchourron is obligated to see to it that if Bulygin has seen to it that he sent Alchourron the draft, then he (Alchourron) sees to it that he revises the draft." The circumstance condition is now redundant, since Was:[B dstit: B] automatically satisfies it.

HE.3

Present-tensed circumstances

We might be tempted by examples (19) and (21) to suppose that circumstantiality is somehow bound up with the past tense. The supposition would be false, as we indicate by considering present-tensed circumstances. In fact, most present-tensed statements are circumstantial. As a typical example, consider the sentence "it is raining." If you have promised to bring an umbrella if it rains, then your obligation can be represented in either of the following forms:

Clearly, "it is raining" is circumstantial: Sett:(~[a dstit: ~R]), or, in English, it is settled that you don't see to it that it is not raining. You may chant or dance, but in the end it is up to nature to stop the rain. In fact, at a given moment, it is probably fair to assume either Sett:R or Sett:~R holds.

11. Conditional obligation, deontic paradoxes, and stit

309

Since "it is raining" is a circumstance, we can detach from (23) the obligation to bring an umbrella, Oblg:[a dstit: U], if it actually does rain. Once again, the system S A is indifferent to whether we represent the obligation as (23) or (24), just as English is indifferent between the corresponding English sentences. "It is raining" is a present-tensed non-agentive sentence. Can present-tensed agentive sentences also be circumstantial? It is an interesting fact about dstit that the answer depends entirely on the agent. It turns out that the following are true:

The first, (25), says that other agents' present doings are always circumstantial for agent a. (The proof of Fact 10-2, which relies on the independence of agents, Post. 9, can easily be adapted to dstit.) Expression (26) says that a's own doings are never circumstantial for a, except when it is settled that a doesn't see to something. This becomes important in the discussion of contrary-to-duty obligations in §11G. Momentarily confining our attention to the case of different agents, we have the result that SA is indifferent between

and

as ways of putting the obligation to see to it that you (a) show up to the meeting if your boss (B) shows up to the meeting. In either case, we can detach Oblg:[a dstit: A] provided [B dstit: B] holds. We could also provide examples of future-tensed sentences satisfying the circumstance condition, such as "It will rain." The point is that circumstantiality is not derivative of temporal ordering. It depends only on what it is possible for a to see to at a given moment.

11F

The Good Samaritan

The paradox of the Good Samaritan—the version cited here is that of Castaneda 1981, with minor changes—relies on the following principle: If that a performs A entails that a performs B, then that a is obligated to do A entails that a is obligated to do B.

(27)

The paradox now proceeds: (a) Arthur is obligated to perform the act, call it C, of bandaging the man he will murder a week from now.

310

Applications of the deliberative stit

Figure 11.4: Invalidity of (28)-version of Good Samaritan

(b) But Arthur's doing C entails his doing the act of murdering a man a week hence.

So, by (a), (b), and (27), (c) Arthur is obligated to murder a man a week hence. We want to reject (c); but (27), (a), and (b) all seem acceptable. People usually attempt to resolve the paradox through analysis of tense, agency, and the sense of entailment in (27) and (b). We can bring all these considerations to bear in a precise way by using the dstit semantics. Let M stand for "Arthur murders a man." Then "Arthur murders a man a week hence" can be approximately translated as Will:[a dstit: M]. Therefore we can represent (b) as [a dstit: C] D Will:[a dstit: M]. ("Arthur sees to it that C" entails that in the future, Arthur sees to it that he murders a man.) As a first stab at (27), we can try

Then (28) does not apply to (b), since Will:[a dstit: M] is not an agentive sentence. Instead it is a future tensing of an agentive sentence. So (c) does not follow. Furthermore, we should note that (28) is, quite properly, invalid, as Figure 11.4 illustrates. This figure shows a moment m at which [a dstit: A] D [a dstit: B] at h1, but not at h2 or h3. In such a situation, Oblg:[a dstit: A] D Oblg:[a dstit: B] fails generally, and so at h1. Indeed, if it is possible to see to it that A (bandaging a man) without seeing to it that B (killing a man), the conditional Oblg:[a dstit: A] D Oblg:[a dstit: B] should be false. When we strengthen the antecedent, we obtain a valid form of (27):

11. Conditional obligation, deontic paradoxes, and stit

311

The proof is straightforward. (Alternatively, we could put Universally: in place of Sett:.) But (29) still does not apply to (b). Two possibilities remain for preserving the paradox. First, we can suppose that Arthur can see to it right now that "a man is dead a week hence." We restate premiss (b) as "Arthur's doing C entails his seeing to it (now) that a man is dead a week hence": where D stands for "a man is dead." Premiss (a) is translated as Oblg:[a dstit: C]. Then Oblg:[a dstit: Will:D] does follow. But to assume (b') is to assume that Arthur cannot bandage the injured man without seeing to it that he is dead a week hence. (Perhaps the bandages are coated with poison.) It is doubtful that Oblg:[a dstit: C] holds under these conditions. The paradox has lost its sting, since Oblg:[a dstit: C] and

Oblg:[a dstit: WillD] are equally objectionable. A more reasonable approach is to represent (b) as Sett:([a dstit: C] D Will:[a dstit: M]) (it's settled that seeing to it that C entails, in future, murdering a man.)

(b")

and to propose yet another version of (27):

(If seeing to it that A entails in future seeing to it that B, then a present obligation to see to it that A entails a future obligation to see to it that B.) Of all the formulations of (b) and (27), these seem most natural. Together, (a), (b"), and (30) do imply that Arthur has a future obligation to murder the man he bandages—the paradoxical result. The argument fails, however, because (b") is always false (except in the trivial case where [a dstit: C\ is settled false, i.e., when Arthur cannot bandage the man). It cannot be settled that a present agentive sentence entails a future agentive sentence. The reason is the negative condition (genuine choice) required for the truth of the future agentive sentence. In Figure 11.5, the counter for [a dstit: M] at m2/h is at history h2. So (a dstit: M] fails at m 2 /h2- Since h2 =am h, [a dstit: C] D Will:[a dstit: M] is false at m/h2, proving that (b") is false. The Good Samaritan paradox rests on ambiguities of tense, entailment, and agency. The semantic system SA makes it possible to unravel these elements in a graphic fashion, so that the argument is either invalid, or one or more premisses are manifestly false.

312

Applications of the deliberative stit

Figure 11.5: Invalidity of (30)-version of Good Samaritan

11G

Contrary-to-duty obligations

One of the most commonly discussed problems of deontic logic is the paradox of contrary-to-duty obligations (or imperatives), so named in Chisholm 1963. Such paradoxes arise in cases in which something forbidden is done. Consider the "gentle murder" example: (a) A certain man is not obligated to murder his neighbor; in fact, he is obligated not to murder his neighbor. (b) If he does murder his neighbor, he is obligated to murder gently. (c) Murdering gently entails murdering. (d) He murders his neighbor. Chisholm calls the sort of obligation in (b) a contrary-to-duty imperative. Such an imperative says what a person ought to do if the person has violated his duties. It is widely recognized that deontic logic should be able to accommodate contrary-to-duty obligations, but they pose a difficulty for the standard system. Statements (a)-(d) seem perfectly consistent, but cannot be consistently represented in the standard system. Let "M" stand for the sentence "a (the man) murders his neighbor," and "C?" for the sentence "a murders his neighbor gently." Then the most reasonable way to represent the four statements in standard deontic logic is as follows: (KDa) (KDb)

11. Conditional obligation, deontic paradoxes, and stit

313

(KDc) (KDd)

From (KDb), (KDc) and (KDd) together with the valid schema A D B / Oblg:A D Oblg:B, we can infer Oblg:M, contradicting (KDa). A possible response to this problem is to try representing the contrary-to-duty obligation (b) as (KDb')

This avoids inconsistency, but (KDb') then becomes a redundant premiss. For Oblg:(M D G), and in fact Oblg:(M D A) for any sentence A, follows from Oblg:~M. We could equally well add Oblg:(M D ~G). This suggests that (KDb') is a bad way to represent the contrary-to-duty obligation (b). The semantics of SA provides a way to represent the statements as a consistent set without redundant obligations: (SAa) (SAb) (SAc) (SAd)

The first statement expresses a's obligation to refrain from murdering as well as the fact that a has no obligation to murder. The second statement expresses the contrary-to-duty obligation. Unlike the situation in the standard system, (SAb) does not follow from (SAa). Furthermore, we avoid a contradiction between (SAb) and (SAa) because we cannot detach the obligation to murder gently, Oblg:[a dstit: G]. The reason is that [a dstit: M] is not a circumstance in the sense of condition (C). Looking again at (26), we see that it is never the case, given (SAd), that Sett:(~[a dstit: ~[a dstit: M]] ). It is not settled that a cannot refrain from murdering; the counter to (SAd) guarantees a the choice to refrain from murdering. The obligation to murder gently always remains just a conditional obligation, as it should. Figure 11.6 illustrates the situation. On history h1, a violates both obligations (SAa) and (SAb). On history h2, a violates only the obligation not to murder. Finally, on h3,(where (SAd) is false) a satisfies both obligations. Clearly, Oblg:[a dstit: G] does not hold, since it is possible to have ~[a dstit: G] without S (as at h3). Chisholm's own example in Chisholm 1963 is the following: (a) It ought to be that a certain man go to the assistance of his neighbors. (b) It ought to be that if he does go he tell them he is coming. (c) If he does not go then he ought not to tell them he is coming.

314

Applications of the deliberative stit

Figure 11.6: Illustration of S A-version of gentle murder

(d) He does not go. It can be handled in the same way as the gentle murder case. Let A stand for the sentence "a (the man) goes to the assistance of his neighbors," and T for the sentence "a tells them he is coming." Then we symbolize the paradox as follows: (SAa) (SAb) (SAc) (SAd)

Provided [a dstit: A] is possible (he can assist his neighbors), ~[a dstit: A] is not a circumstance. So we may not detach the obligation to refrain from telling his neighbors he will come, and there is no contradiction. There is only a conditional, not a categorical, obligation not to tell. (If it is impossible for him to assist his neighbors, ~[a dstit: A] is a circumstance and we can detach a categorical obligation not to tell. Oblg:[a dstit: T] and Oblg:[a dstit: ~[a dstit: T]] will conflict in this special case, but it can plausibly be argued that there is a genuine conflict of obligations.)

11H

Problems with the proposed semantics of obligation

The semantics of obligation suggested here faces numerous difficulties. We discuss two of the most obtrusive.

11. Conditional obligation, deontic paradoxes, and stit

315

Figure 11.7: Promising to call

(i) The system does not permit different obligations to hold at different histories at the same moment. If an obligation holds at one moment-history pair m/h, it is settled at m. Yet there are situations in which an agent's choices lead to different obligations. The most obvious is making promises, as represented in Figure 11.7. P stands for "a promises to call"; C stands for "a calls." It seems that Oblg:[a dstit: C] should hold at m/h1, but not at m/h2. On the proposed semantics, there are two possible ways around this difficulty. One way is to argue that the obligation to call does not really come into force until a time after the promise is made, that is, once it is settled that a has made the promise. What really holds at h1 is the sentence P D Will:(Oblg:[a dstit: C]). At some later time, the obligation to call will hold over the entire moment. But it is not certainly satisfying to have to introduce extraneous temporal considerations to resolve the problem. On the other hand, §5C suggests that causal-temporal concerns may be essential to the idea of promising. A second approach, following the treatment of contrary-to-duty obligations, is to treat the obligation to call as the conditional obligation Oblg:[a dstit: ([a dstit: P] D [a dstit: C] )]. Since promising is not a circumstance, the obligation to call cannot be detached. The conditional obligation, which holds for the whole moment, is only violated on histories where the agent makes a promise to call but fails to call. This solution is also prima facie unconvincing, however, because we have to introduce a conditional structure into what seems an unconditional obligation created by the promise. The difficulty derives from the fact that our definition of obligation quantifies over all histories through a given moment. If we replace Sett: by [a cstit: ] in (5). and define

then we allow for obligations that vary depending on a's choices. The problem with this definition is that [a dstit: A] D Oblg:[a dstit: A] becomes valid because [a dstit: A] = [a cstit: [a dstit: A]]. Given [a dstit: A], [a cstit: (~[a dstit: A] D S)] follows classically. There might be some hope if we use a nonclassical logic (for example, relevance logic), but this suggestion falls outside our present focus.6 6

Bartha 1999 develops a refinement of the approach of this chapter that allows for obliga-

316

Applications of the deliberative stit

(ii) There are conditional obligations of the form

which the semantics proposed here seems unable to handle. A good example (due to Castaneda in correspondence) is the following: Mary is secretarially obligated to report to the manager by 8:45 that she won't open the office by 9:00 A.M., if she won't. This fits the given form if a is Mary, A is "Mary does not open the office by 9:00 A.M.," and B is "Mary reports to the manager by 8:45 A.M." Imagine that it is now 8:45, and Mary is staying home to care for a sick child. Since the conditional obligation is in force, it seems that we should be able to detach the obligation to call the manager: Oblg:[a dstit: B]. But we cannot do so on the given semantics. The antecedent is not a circumstance in the sense of (C), since Mary might still be able to make it to the office by 9:00. We mention two stit possibilities that might help with the problem of conditional obligations. The first is to take the content of an obligation to be more like a strategy, in the sense of chapter 13, than it is like a stit. This suggestion would be analogous to that of §5C for promising. The second is to consider the impact of the proposal of Wansing 1998. Just as the analysis of this chapter makes the obligated agent an intrinsic part of the obligation, Wansing suggests analyzing obligation even further by making the authority or creating agent an additional, inseparable part. Thus, two agents are needed for an obligation: the agent or authority who creates the obligation, and the agent who is obligated. The English normal forms of obligation and prohibition are then given as follows. obligates a2 to see to it that A forbids a2 to see to it that A. Then Wansing 1998 proposes that these be represented in terms of an agentindexed Andersonian sanction on a2, written Sa 2 , such as we mentioned in §11B.1. The representations would then be as follows.

We think that this proposal deserves thorough exploration, especially with regard to the light that it might shed on conditional obligations. The simple semantics of obligation developed here needs to be improved, as these difficulties illustrate. The point of this chapter was to show that a semantics based on the logic of agency provides a useful tool for thinking about problems tions which are not settled true or false, and that also provides a more sophisticated account of contrary-to-duty obligations. Horty 2001 addresses many of the issues discussed in this chapter.

11. Conditional obligation, deontic paradoxes, and stit

317

of conditional obligation and paradoxes of deontic logic. Considerations of tense and agency often lie at the heart of such problems, and the proposed semantics allows such considerations to be brought fully into play.

12

Marcus and the problem of nested deontic modalities Marcus 1966, "Iterated deontic modalities," is re-readable, and re-compelling.* Marcus argues that standard deontic logic makes too little sense of putting one deontic modality inside another. Even now deontic logicians shy away from nesting their modalities one within the scope of another. One of the reasons apparent from her argument is this: Direct iteration is of little use. If, however, one takes the trouble to insert stit-like modal agentive constructions after the deontic operators, then embedding suddenly becomes a powerful analytical tool. By "modal" we mean, as always, a construction that maps sentences into sentences. We think that the modal move is doubtless not essential but so useful as to be critical. The plan of this chapter is to explore this idea by intertwining certain themes from Marcus's work on "general purpose" modal logic with themes from stit theory in its deontic application. In fact the nesting of deontic modalities is taken up only quite late in this chapter. Before then we shall be considering—all too briefly—the representation of obligations and prohibitions, moral dilemmas, the serious use of quantifiers in modal logic, Barcan formulas, and problems of extensionality or its absence.

12A

The parking problem

As a device for the remainder of this chapter, we shall concentrate on an unpretentious example from Marcus 1966, p. 580. There is much to be said for unpretentious examples. After all, "7 + 5 = 12" was good enough for Kant. The Marcus example is, however, more serious than mere arithmetic: Parking on highways ought to be forbidden.

(1)

*Paul Bartha is co-author of this chapter. With the kind permission of Cambridge University Press, it draws on Belnap and Bartha 1995.

318

12. Marcus and the problem of nested deontic modalities

319

With this example Marcus is challenging all deontic logicians. Deontic logic, she says, could be a "misleading technical exercise" (p. 582) if it treats (1) as a simple iteration of an impersonal "ought" and an impersonal "forbidden." That is, the following symbols (2) won't do for (1):

Marcus, with conclusive arguments that we shall not repeat here, pins the blame on a failure to keep separate the evaluative modal family headed by "ought to be" and the prescriptive modal family headed by "ought to do" or "obligatory." With reference to the prohibition expressed by "forbidden," let us leave aside the "ought to be," turning instead to the "ought to do." Marcus endorses this turn. She says It seems to me that the least problematic reading of the deontic operators is the one which fits the use of 'obligatory', 'forbidden', 'permitted', as they occur in connection with rule governed conduct. Such a restriction to prescriptive use has the advantage of allowing us to be clearer about the semantical interpretation of deontic statements. (Marcus 1966, p. 581) She is suggesting that we can best use deontic logic if we focus on the "ought to do." In advancing our understanding of the "ought to do," Marcus, citing K. Baier, suggests the following: ... we may take 'It is obligatory that A' (where 'A' is an appropriate statement of the sort 'x does w at t') as meaning that A is entailed by the set of rules and standards. (Marcus 1966, p. 582) This is a rich suggestion. Much of the ensuing discussion can be construed as following the suggestion with the help of the modal logic of agency. In doing so we are going to concentrate on (1) to the exclusion of any more gripping examples. If you start to fall asleep, please substitute an illustration filled with dangerous and dreadful actions of the most excitingly electrifying sort. You know, something like "buttering the toast in the bathroom."

12B

The form of obligations

The analysis of (1) proceeds from the inside out. We shall construe (1) as containing the following as a grammatically proper part: Parking on highways is forbidden.

(3)

Most of the work of this chapter will consist of articulating the part (3) of (1), using rather simple deontic concepts in combination with the logic of agency. Marcus would hold that even before we reach semantics, the underlying grammar of a typical prescription is like this: It is obligatory that (x does w at t).

(4)

320

Applications of the deliberative stit

This suggestion, itself unpretentious, is worth taking seriously and has in practice guided our work. In fact the suggestion is so helpful that it is easy to overlook the following: When some philosopher says "x does w at t" we have no idea over what the variable "w" is supposed to range. In other, more pretentious, words, we don't—even after decades of the most arduous first-order work—have an "ontology of actions" that is at once useful and rigorous. Nor, of course, does Marcus—the consummate modal logician—suppose that we do. That stit theory is a modal logic is precisely why it can provide an illuminating framework in which one isn't constantly having to worry about turning actions into things to name or to quantify over. Stit theory suggests instead that the grammar of a prescription is this: It is obligatory that a see to it that Q, or, in more idiomatic English, a is obligated to see to it that Q.

(5)

Three ideas lie behind this formulation, ideas that we re-mention here in order to relate them to Marcus. The first is that the place of Q can meaningfully be filled by any sentence whatsoever (stit complement thesis, Thesis 2). This is the power of exactly that modal point of view championed and defended by Marcus against various forms of "first orderism." The second idea is that the place following "it is obligatory that" can be occupied only by an agentive, of which the stit sentence is both paradigm and normal form. This is the restricted complement thesis, Thesis 5. This thesis gives technical bite to Marcus's distinction between the evaluative and the prescriptive. That is, it forces us to keep the prescriptive "ought-to-do" separate from the evaluative modal family. The third idea is this. In contrast to the way everyone waffles when it comes to an ontology of "actions," we are in firm possession of a clear and useful semantic account of "a sees to it that Q" (§8G.l). This account also shares convictions with Marcus 1980. Two are paramount. The first is that obligation and agency are bound up in complex ways with possibility. The second is that you cannot get away from occasions and times. This is witnessed by the "t" in (4), and made explicit in passages such as the following. 'Ought' is indexical in the sense that applications of principles on given occasions project into the future. They concern bringing something about. (Marcus 1980, p. 135) Together these ideas help suggest the employment of an independent account of seeing to it that such as is given by the stit theory of this book. We do not review these concepts, relying on chapter 8 and the summary in §6. We do note that our use of branching time as a foundation presupposes answers to difficult and important questions about actuality and possibility, answers that are roundly challenged in Marcus 1977 and Marcus 1985/86. Even to begin discussing any of these matters here, however, would mire us without hope of

12. Marcus and the problem of nested deontic modalities

321

progress. Having had our say in chapters 6, 7 and 8, as well as elsewhere, we presume to call on your charity—recognizing at all times that what we certainly consider to be plausible, others with equal certainty consider to be a doubtful dream. Recall from the stit paraphrase thesis, Thesis 3, that a sentence Q is "agentive for a" iff Q can be paraphrased as [a stit: Q]. When Bartha applied stit theory to deontic concerns, in chapter 11, he chose to idealize seeing-to-it-that with the deliberative stit, dstit, which is in some respects simpler than the achievement stit. Here, partly because we are building on chapter 11, we also use dstit; see §8G.l. So, returning to the parking problem, let Pax <-> a parks on highway x.

(6)

This rendition leaves the "t" in Marcus' "x does w at t" to be indicated parametrically rather than explicitly, but the t is nevertheless there. That is, (6) speaks of a particular occasion of parking on highway x, not a habit or practice. Also we are using a as ranging over agents, as always, and for convenience we are using x as ranging over highways. The chief point, however, is that by the stit paraphrase thesis, Thesis 3, Pax is agentive because—or, better, to the extent that—it comes to the same thing as "a sees to it that Pax": 12-1 POSTULATE. (Parking is agentive) Pax <-> [a dstit: Pax]. In accord with our understanding of the stit normal form thesis, Thesis 6, we do not propose that the longer sentence is an "analysis" of the shorter one, but only a useful "normal form."

12C

The Anderson/dstit simplification

So much for the "bringing it about" part of the Marcus account of prescriptions. What about the normative part? As noted, her thought is to spell this out as entailment by a set of "rules and standards." Chapter 11 suggests that we take our bearings in this extremely complex area by use of the Andersonian "sanction," the bad thing, represented by a prepositional constant, S, indexed by agents, with a different "sanction" Sa for each agent a.1 Think of Sa as true at just those moment-history pairs at which the particular agent a has not lived up to the rules and standards. Then on the chapter 11 proposal, we can analyze (5) as follows: 12-2 DEFINITION. (Oblg:, obligation) Oblg:[a dstit: Q] <=df Sett:(~[a dstit: Q] Sa ) . 1

Because chapter 11 considered only a single agent, it dropped explicit reference to the index on the sanction. Horty 2001 shows that for certain logical properties—logical validity is paramount—it makes no difference whether one keeps the index or not. Here, however, we keep the personal index explicit, since we are more interested in reminding ourselves that obligations are never impersonal.

322

Applications of the deliberative stit

That is, at a particular concrete moment, a is obligated to see to it that Q iff in every history through that moment, either a sees to it that Q or else the bad thing happens. Observe the following analytically decisive point. If you take Definition 12-2 as a definition, then it should not be understood as a definition of "Oblg:" in isolation. We intend it only as a definition of the compound Oblg:[a dstit: Q}. Definition 12-2 gives no meaning whatsoever to Oblg:Q for non-agentive Q. That this should be so conforms perfectly to the restricted complement thesis, Thesis 5. It is precisely what makes this account of obligation prescriptive rather than evaluative. Note that there is no talk of globally and impersonally ideal worlds or histories. It's just that at that moment you are supposed to do what you are supposed to do, and if you don't, you are in trouble. §11H touches on more than one inadequacy of this proposal, but these we pass over instead of pause over. We go on at once to relate Definition 12-2 more closely to Marcus motifs.

12D

The form of prohibitions

A first and interesting problem is the representation of prohibitions. Marcus 1966 notes that "'forbidden' is more clearly prescriptive of action" (p. 590) than it is evaluative. If the rules forbid you to park on highway x, they are prescribing an action. But which action? The trouble is that ~Pax, that is, "a is not parking on highway x," does not describe an action (unless, of course, it is used to mean that a is refraining from parking). It describes a state. For instance, a is in that state when sleeping. As Marcus 1966 says in another case (p. 580), "How can it be forbidden that such a state of affairs prevail unless it is part of a blueprint for creation?" It's confusing. "Forbidden" is supposed to prescribe, and to prescribe is to prescribe action, but since "not parking" is not an action, it looks as if there is no action to be prescribed. At this point one might think of taking prohibition by a set of rules to be a second primitive notion, to be added to prescription. Dstit theory has, however, an absolutely satisfying solution to this puzzle. It goes like this. We know that what is forbidden by the rules is that a park on highway x. The problem is to say what is prescribed in this case, if we can. We already know it won't do to say that it is the state of not parking that is prescribed. Dstit theory solves the problem as follows: It says that what is prescribed is that a see to it that a does not park on highway x. It can do this because although you cannot "negate an action" (and keep your sanity), you can certainly negate any sentence, including any agentive sentence. And any sentence can be the complement of the dstit construction. More generally, as discussed in §3A, the proposal is this. STIT DEONTIC EQUIVALENCE. (Axiomatics concept. Reference: Ax. Conc. 2) Frbn:[a dstit: Q] «-» Oblg:[a dstit: ~[a dstit: Q]]. That is, a is forbidden (by the rules) to see to it that Q if and only if a is obligated (by the rules) to see to it that it is false that a sees to it that Q.

12. Marcus and the problem of nested deontic modalities

323

The key point here is that this obvious and elementary truth cannot be said, or said clearly, without the grammar of dstit. To be able to formulate this striking definition of a prohibition as an obligation to refrain, you need to be able to embed a negated dstit inside of a dstit. It is precisely this capability that seems to be absent from an ontology of "actions." This power of embedding, which belongs to the very idea of modality, needs to be exploited. You cannot exploit it if you are ashamed of modalities. That is one of the great themes of Marcus's work: Instead of treating modalities as an embarrassment in the fashion repeatedly preached by Quine, you should use them for what they are worth, which is plenty. Use them by themselves, with quantification, with identity, or with whatever helps. Combining Ax. Conc. 2 with Definition 12-2, we have the following symbolizations and equivalents. Frbn:[a dstit: Pax]: a is forbidden to park on highway x. «-> Oblg:[a dstit: ~[a dstit: Pax]]: a is obligated to see to it that a does not park on highway x. <-> Sett:(~[a dstit: ~[a dstit: Pax]] D Sa): It is settled true that if a does not see to it that a does not park on highway x, then Sa. That's a lot of words. One can see further into the proposal by a combination of cases and pictures. Take a particular concrete moment, TO, of concern to a and to the rules. There are three major cases, the first two of which are throwaways. No-good-choice case. Since §11B.1, in contrast to Anderson, has put no constraints on the sanction, there is the case in which Sa is settled true at m. In the no-good-choice case all is obligatory for a, but also all is forbidden. There is no available choice for a that avoids the sanction. Nothing is permitted, where, as in §3A, Perm:[a dstit: Q] <-> ~Frbn:[a dstit: Q]. Figure 12.1 provides a picture. What to make of the different decisions, Anderson's requiring that the sanction not be universal and ours making no such requirement? We think that both decisions are plausible, and that the point of view of Marcus 1980 explains why. Anderson's sanction applies to entire worlds or perhaps histories. It would be strange indeed if every world or every history were deemed unacceptable. In this case there would clearly be nothing left for deontic logic to worry over. As Marcus puts what we take to be the same point, if a set of rules is inconsistent in this strong sense, then it "provides no guide to action under any circumstances" (p. 129). This makes reasonable Anderson's postulate that the sanction is not inevitable. On the other hand, when we mark as "sanctioned for a" every history through a particular moment, that sad situation is an extremely local affair. It represents only that in that particular circumstance, there is nothing acceptable to do.

324

Applications of the deliberative stit

Figure 12.1: No good choices for a at m

Figure 12.2: Two proper no-can-do cases

This is what Marcus urges to be quite another thing. To deny that such a circumstance could obtain is, as she says, to deny something real, namely, that a moral dilemma can arise in a particular unfortunate circumstance. So, we conclude, the two choices are both in accord with Marcus's account, and they are both right. When all histories are at issue, it is right to assume with Anderson that the sanction is avoidable, but when we consider only the histories that are possible historical continuations of a particular moment, then it is right to allow with Marcus that in some circumstances there may be no way to avoid the sanction. No-can-do case. In the other throwaway case, at m, a cannot see to it that Q: It is settled true that a does not. Nothing a can choose at m counts as choosing to see to it that Q. No such choice is in fact open to a at m. There are two subcases. In one of them the previously discussed no-good-choice case obtains, and there is nothing more to say: All is obligatory and all is forbidden for a, regardless of what the agent can do. In the other subcase, which is new, Sa is false at m/h for at least one history h through m. We will call such a case a proper no-can-do case. Figure 12.2 conjures up two of them.

12. Marcus and the problem of nested deontic modalities

325

Figure 12.3: The normal case

You can easily calculate that in a proper no-can-do case, Definition 12-2 neither obligates nor forbids the agent to see to it that Q, which sounds right when the agent no can do. Of course it follows that in the proper no-can-do case the agent is permitted to see to it that Q, and also permitted to refrain from seeing to it that Q, which sounds peculiar given that the agent is powerless with respect to Q. But it sounds peculiar, we think, only because in ordinary English we confuse permission and power. (The effort to straighten us out on this goes back at least to Hohfeld 1919; see especially Makinson 1986.) For example, we often use the same modal verb, "may," for both. When we reflect, however, we easily see that in fact there are numerous situations in which an agent is permitted or authorized to do what in fact is not in their power to do. The agent a might be permitted to park at any meter in the entire state of Connecticut, even when a is enjoying Oregon. The proper no-can-do case highlights this. The normal case. In the central case of interest, the sanction Sa is false on at least one history through m, and there is at least one choice on which a sees to it that Q. We will call this the normal case, even though it certainly does not happen with much frequency. In the normal case, a can see to it that Q, and in the normal case, the sanction is not inevitable; see Figure 12.3. It is a calculation of dstit theory (as opposed to the achievement stit) that in the normal case, though not at all in general, "seeing to it that" and "refraining from seeing to it that" are contradictory. (The sanction is irrelevant to this calculation; one needs only can-do.)

12E

Generalized prohibitions

Let us pass on to the first of the two generalities ingredient in (1). It is not just one highway on which a is forbidden to park. In fact,

326

Applications of the deliberative stit Each highway is one on which a is forbidden to park. (Or "each highway within a given jurisdiction." We shall suppress this subtlety.)

(7)

In other words, we might say, a is forbidden to park on a highway. Or, put more ponderously in order to nail down a narrower scope for the quantifier: It is forbidden that a park on a highway.

(8)

So, obviously, to express that parking is forbidden on highways we need a quantifier. But which quantifier, and where should it go? The universally quantified prohibition (7) and the prohibition of an existential (8) sound more or less equivalent, but certainly the last hundred years of logic suggest that it is not very plausible that they should be. It will be instructive to observe a proposed train of reasoning that would show them to be so. This train contains some routine transitions and some interesting ones. Here is what will strike you: None of the interesting transitions concerns the deontic modalities themselves. All are concerned with either the background theory of branching time, or the theory of agency. In the end we shall find out that (7) and (8) are not equivalent, though nearly so. And coming to see that sort of thing is part of what logic is for. Marcus, in the shadow of some intolerant and irrational decades that had foolishly put modal considerations on the defensive, said that "establishing the foundations of mathematics is not the only purpose of logic" (Marcus 1960, p. 58). This should count as a tiny example of what she had in mind. Let us begin with the henscratches for (7) and (8), respectively. Symbols for (7): VxFrbn:[a dstit: Pax].

(9)

Symbols for (8): Frbn:[a dstit: 3xPax].

(10)

It would be all right to shorten [a dstit: Pax] in (9) to Pax, just because, according to Postulate 12-1, the latter is agentive, that is, [a dstit: Pax] and Pax are equivalent. But it is probably better to use the longer "normal form" in order to remind us that the complement of Frbn: must always be a dstit or something equivalent. This is in accord with the restricted complement thesis, Thesis 5. Let us add here that by the restricted complement thesis a third candidate, Frbn:Ex[a dstit: Pax], is not on the face of it grammatical. We should suspect that this third proposed complement of Frbn: might, in the language of Marcus 1966, describe a mere state of affairs that could not meaningfully be on a's list of prohibitions. And in fact the coming analysis of the 3x/dstit transition confirms this suspicion. Here is a list of plausible equivalences, which we call the "Main Calculation," that would lead us between (9) and (10). We have indicated on the right how each entry is tied to the one above it. In gently moving quantifiers from the outside to the inside we will run into a number of particular transitions: "QTF" means "quantifier-and-truth-function equivalence." Other transitions

12. Marcus and the problem of nested deontic modalities

327

are applications of previous postulates, as indicated. And the three key transitions "Vx/Sett," "Vx/dstit," and "Ex/dstit" are based on hypothesized quantifier/modality relationships. These three we explicitly discuss, with an eye to either confirmatopn or disconfirmation.

Main Calculation

We take up the undiscussed quantifier/modality transitions Vx/Sett, Vx/dstit, and Ex/dstit in turn. Let us first underline that they have nothing to do with deontic logic per se. They are all either about branching time or else about pure agency. They are about the world and our doings, independently of whether we are getting ready to make some rules or set some standards.

12E.1

Barcan formula for Sett:

The first transition does not even concern agency, much less obligation. It is a matter for the general theory of branching time, or perhaps the general theory of highways. What are we to say of the principle Vx/Sett that permutes the Sett: modality with a universal quantifier over highways? In symbols: 12-3 POSTULATE. (Barcan formula for Sett:) VxSett:Qx <-> Sett:VxQx. Not to put too fine a point on it, we think Postulate 12-3 is good. Here is why. Sett: at a moment means, close enough, "true in all histories" through that moment. You can therefore plainly see that Sett: has the intent of an S5 necessity. That means that the permutation of the universal quantifier with Sett: is precisely the Barcan formula in one of its guises. (From Barcan 1946. So named in Prior 1957.) The Barcan formula is not only time honored and of

328

Applications of the deliberative stit

exceedingly noble parentage, but of tremendous logical utility. Proof systems without the Barcan formula are awkward and difficult for us to use. Adding the Barcan formula simplifies and thereby increases our logical acumen. It makes us smarter. That's the benefit. But, you may ask, at what cost? If we use the Barcan formula we must suppose that the range of the quantifier does not vary with the history; so much is clear from the analysis of Kripke 1963. Here is a general point, and we mean to be echoing one made by Marcus, for example in Marcus 1972: Although English means what it means independently of what we say, our technical quantifiers mean exactly what we say they mean. Their meaning is up to us, and it is, as Carnap emphasized, a practical matter. In applying Carnap's advice, it makes a difference whether the quantifiers we are considering are for general use, or for a case at hand. For general use it seems clearly best to simplify our logical life by interpreting our technical quantifiers—the ones with which we calculate—so that their ranges do not depend on the particular historical continuation.2 As Marcus argues in Marcus 1961, for example, we shall still be able to find appropriate and indeed illuminating symbolic formulations for every one of those English sentences that might have led us to abandon the Barcan formula, had we been so weak on our philosophical pins as to be bowled over by derisive counterexamples hurled down the alley in an effort to obtain a crowd-pleasing strike. Need we add that such presumed counterexamples will be found on inspection to be made from a nonmetallic composite material, and to contain a hole the size of your thumb? Proceeding to the local case at hand, observe that the particular example involves quantification over highways. That makes it sound as if in this locality the Barcan formula should fail, since surely what highways there will be depends on what happens and thus on particular historical continuations. Well, yes, the "since" clause is true, but the argument is not. The argument derives from the paradigm of Kripke or Lewis total worlds, each with its worldwide quota of highways. That paradigm, however, is not wanted here. We are looking at a particular momentary event of forbidden parking. Whether or not something is a highway at a particular moment does not depend on historical continuations of that moment. The domain of highways at a moment (as we might say) is indeed independent of what historically happens next. It is, in the phrase of Castaneda, a "circumstance" of the parking (see the beginning of chapter 11). Thus we should accept the permutation labeled "Vx/Sett" in general, and therefore in the particular case. 2 This still leaves numerous options, of which the following are salient: (i) Marcus quantifiers ranging over actualia; (ii) quantifiers ranging over possibilia; (iii) Bressan "individual concept" quantifiers with intensional predication; and (iv) substitutional quantifiers. Arguments that some alternative is better or worse than one of these have tended to be based on mockery involving the double modal, "How could you possibly say such a thing?" That's amusing, but also sad. For many practical purposes of serious formalization, it is appropriate to use the scandalously neglected Bressan individual concept semantics with intensional predication, of which we say just a little more in §12F.

12. Marcus and the problem of nested deontic modalities

12E.2

329

Universal quantifier over dstit

The next transition we discuss, namely Vx/dstit, is also Barcan-like:

Here, however, appearances deceive: Dstit is not really a straightforward S5like necessity modality. As you can see from §8G.l, the semantics for [a dstit: Q] involves not only a "positive condition" saying that the (perhaps vacuous) choice that a makes necessitates Q, but also a "negative condition" saying that "it might have been otherwise than Q" (from which it follows that the choice was not after all vacuous). As indicated in §11A, we can separate the positive from the negative. For the positive we can adapt the Chellas stit, [a cstit: Q], defined in §8G.2, just as we did in §11A. For the negative we can take from §8F.4 the standard branching-time dual of Sett:, namely, Poss:, so Poss:Q <-» that Q is historically possible, and sometimes Can: (with the same meaning). We review the definitions. 12-4 DEFINITION, ([a cstit: Q], the Chellas stit; Poss:; Can:) • [a cstit: Q] is true at a moment-history pair m/h iff Q is true at m/h1 for every history h1 through m that lies with h in the same possible choice for a at m. (Note the absence of a "negative condition.")

The rule is that Poss: is a general-purpose modality whose complements are unrestricted, but that Can:, although it means quite the same thing, is grammatically restricted to agentives. (The parallel analysis for the achievement stit must go differently since for it there is a much more marked contrast between the "can-be" and the "can-do"; see §9B.) Then we can read "Can:[a dstit: Q]" as "a can see to it that Q." After the separation that the Chellas modality makes possible, as Chellas 1992 emphasized and as we reported in §11A and elsewhere, we can express [a dstit: Q] as a conjunction of separate positive and negative conditions: 12-5 FACT. (Definability of dstit) [a dstit: Q]

df

([a cstit: Q] & Poss:~Q).

The advantage here is this: The Chellas modality, having only a positive and not a negative condition, is an approximation to agency that is easier for certain analytic purposes than dstit. For example, we can transparently see that a Barcan formula is appropriate: 12-6 POSTULATE. (Barcan formula for [a cstit: Q]) Vx[a cstit: Qx] <-> [a cstit: VxQx].

330

Applications of the deliberative stit

The Barcan formula of Postulate 12-6 is indeed a principle distinctive of the modal logic of agency, but the explaining idea remains the same as for the permutation of the universal quantifier with the Sett: modality; namely, the domain of quantification—or anyway the domain of highways—at a moment is independent of what happens next, whether by choice or by chance. If we now chase through the adventures of the universal quantifier in the "Special Case" (11), it becomes clear that the left and right sides of (11) can, with the important help of both the Barcan formula Postulate 12-3 and the Barcan formula Postulate 12-6, be reduced to the following:

Anyone can now see that it is not plausible, even in the Special Case, that the Right Side (13) should imply the Left Side (12). Nothing is going to make it plausible that whenever some highway is such that a can park on it, it is also true that each highway is such that a can park on it. There are just too many highways, many of them remaining in Connecticut even when agent a is about to park in Oregon. As a consequence of the negative condition Poss:~Q, then, we lose the Barcan formula for Vx/dstit. It is natural to ask whether we should keep the negative condition at such a high price. We could re-define [a dstit: Q] as [a cstit: Q}. Then (11) would be valid; indeed, the entire Main Calculation would go through quite easily. There are, nevertheless, compelling reasons for retaining the negative condition. For one thing, the simplified semantics would make Oblg:[a dstit: Q] valid for every tautology Q. More generally, dropping the negative condition would blur the distinction we have made between the evaluative and prescriptive modal families. We intended to limit the discussion to the "ought to do." The negative condition is intended to reflect a basic fact about choice: An agent can only see to Q if it is possible (at that very moment) for Q to be false. What is the upshot? We return to this after discussing the third and last transition of the Main Calculation.

12E.3

Existential quantifier over dstit

The transition Ex/dstit of the Main Calculation is, interestingly, the one that is most bound up with agency itself. It will turn out to be valid, but delicately so: We write a General Form (invalid), a Special Case (also invalid), and then a Special Case in Context (valid).

12. Marcus and the problem of nested deontic modalities

331

You should see Ex/dstit as a motion of the existential quantifier between the inside and outside of the dstit modality. The General Form is certainly to be rejected in passing from right to left. When in a game of chance one rolls a couple of ordinary dice, one sees to it that they come up some number or other (two through twelve), but it is false that there is a number such that one sees to it that they come up that number. The former is taking one's turn. The latter is cheating. We must still, however, look at the Special Case (14). What is most obvious about the Special Case is that for each value of x, the complement of the dstit on the Left Side is itself agentive (by Postulate 12-1). Even with this extra information, however, the equivalence fails. Briefly put, what holds instead is this. Provided Pax is agentive for every value of x,

We might not have noticed this because the extra conjunct seems so overwhelmingly plausible in the highway-parking example; it seems difficult to imagine a moment at which it is absolutely settled that there is a highway on which a parks. It is important to realize, however, that the difficulty of imagination arises from the example, parking on highways, not from agency itself. We would belabor this subtlety except for the following: Even though the General and the Special Cases fail, the Special Case in Context (15) of the 3x/dstit transition is in fact logically valid, given Postulate 12-1. The surrounding context makes the difference. We skip the detailed calculation, which indeed depends heavily on the logic of dstit (the analysis for the achievement stit would have to be substantially different) in favor of a mere record: 12-7 FACT. (Special case in context of Ex/dstit) lf Vx[Qx <-> [a dstit: Qx]] then (a dstit: ~3z[a dstit: Qx]] <-» [a dstit: ~[a dstit: E x Q x ] ] .

12E.4

Upshot for the Main Calculation

If we put all these considerations together we find the following. The transition from top to bottom of the Main Calculation is secure at every link (given of course the postulates). In the bottom-to-top direction, however, the transition Vx/Sett is weak, requiring as it does an added and implausible premiss that amounts to this: If a can park on some highway or other, then a can park on every highway; that is, The upshot of the Main Calculation is that to forbid a to park on each highway (as in (9)) is implicitly to forbid a to park on a highway (as in (10)), but not conversely. The difference arises only in proper no-can-do cases such as those involving parking on highways in Connecticut when a is in Oregon. This suggests the following: If it is forbidden that a park on a highway (as in (10)), then a is forbidden to park on each highway on which a can park. The suggestion can be verified; (10) implies the following:

332

Applications of the deliberative stit

This implication is not reversible. What happens, however, if we adopt a rule according to which

Here as it turns out we have an even weaker statement, implied by but not implying (17). As a visual help in sorting this out, here is a definition, intended only for restricted complements, that highlights a form common to (17) and (18). 12-8 DEFINITION. (Frbn-if-can-do:) dstit: Q] D Frbn:[a dstit: Q]].

Frbn-if-can-do:[a

dstit: Q] <-> (Can:[a

Then (17) and (18) are more transparently written in terms of weak prohibition respectively as follows:

So now we have four prohibitions, all subtly different, with irreversible implications as follows:

Inspection of that array seems enough both to explain and to correct our feeling of equivalence between (7) and (8). Which of these is wanted when we are writing up the traffic regulations for the state of Oregon? The following equivalences should guide us in the task:

These equivalences suggest that (9) and (10) are unacceptable interpretations of the parking prohibition (3). By (21), (9) entails that if there is any highway that a can't park on, then a has no good choice. Given the abundance of highways, (9) would perpetually put all agents (including the lawmakers themselves!) in the no-good-choice situation. Similarly, as (22) shows, (10) entails that if there is no highway that a can park on (for example, when a is sitting at home), then a has no good choice. Neither of these objectionable consequences follows from (19) or (20). The only difference between these two versions of the parking

12. Marcus and the problem of nested deontic modalities

333

prohibition occurs in the (rare) case of a forced choice. For example, suppose you are moving toward a Y-split between two highways when your engine stops. You are forced to choose between parking on one or the other highway in the strongest possible sense: If you don't park on one you cannot escape parking on the other. In this case, since it is settled true that there is a highway on which you park, we can see from (23) that (19) implies that you suffer the sanction no matter what you do. You are in a "legal dilemma." In contrast, (20) does not provide a sanction in this case of forced choice—even if you yourself purposely stopped your engine by turning the key. It is plausible that the lawmakers in Oregon would want one of the weaker prohibitions, (19) or (20), but it seems less plausible that they would want the weakest. For the rest of this essay we will assume that (19) is the most natural choice.

12F

Generalization on agents

So far there has been talk only of a single agent, a. There is very little that is problematic in adding a generalization to all agents, so that we may say that "parking on highways is forbidden" comes to

Although this is straightforward, while our attention is concentrated on the agent position, it will be good to raise the question as to its "extensionality." In addressing this question, we want to consider two theses and one piece of advice from Marcus. Marcus has chiefly discussed when a system is extensional, but here we are asking whether a particular context [a dstit: Q] is extensional in the position a. The questions are of course related, and the first thesis of Marcus is exactly on target, namely, that extensionality always has to do with a plurality of relations "eq" of equality, some of which have stronger replacement principles than others. In the theory of branching-time-with-choices there is enormous room for such equivalence relations, for we may consider with profit at least equality relations indexed to a single moment-history pair, or indexed to a single moment with the histories quantified out, or indexed to a single history with the moments quantified out, or de-indexed by double quantification to give the strongest equality relation of all. This seems to be exactly the sort of situation that Marcus is picturing. The second thesis is this: Marcus says that "extensionality principles are absent on the level of individuals" (Marcus 1961, p. 305). In context her meaning is that plural equalities are available only for sentential or predicative parts of speech. For individuals there is only a single equality predicate, the strongest, that she identifies with identity. She says that weaker (e.g., contingent) equivalences are "misleadingly called 'true identities'" (Marcus 1961, p. 312; our emphasis). If her second thesis is right, we have no sensible question. And

334

Applications of the deliberative stit

indeed it surely is correct for the powerful and interesting languages that Marcus has investigated. There are, however, the at least equally powerful and equally interesting languages championed by Carnap and fully developed by Bressan and Montague. In these languages singular terms carry "individual concepts" as their intensional semantic value. These are good and useful languages. Quine has made delicious fun of them over the years, and Kripke and others have cooked up their own pungent mockeries, but surely it is high time to stop dining on proofs by spoof when these are pressed into the service of intolerance. As Pooh would say, such proofs are not sustaining. Here we are led to serve up a particularly satisfying portion of advice from Marcus: ... the polemics of modal logic are perhaps best carried out in terms of some explicit semantical construction. (Marcus 1961, p. 319) Without fully digesting this advice, however, let us say that such constructions, namely those of Bressan and Montague, are standing ready in the kitchen. Then let us abandon the metaphor and just wing it for a while. Our point is that there is something fundamentally non-extensional about the agent position of dstit statements. Even for the single evaluation of a single statement [a dstit: Q] at a single moment-history pair, we need to fix the agent at that moment independently of histories. We need to be able to say what choices are available to the agent regardless of what happens next. Furthermore, when we proceed, as we must, to speak of strategies or other concepts that take us from moment to moment, we shall need to fix the concept of the agent in an even more severe sense. All of this is carried by saying that the underlying concept of "agent" must be "absolute" in Bressan's sense. This is a great deal like saying, in the language of Marcus 1961 (p. 304) that agents as we need them are "constant objects of reference." There are, however, two differences. First, the Marcus phrase is informal and metalinguistic, whereas Bressan's account is carried by a rigorous object-language but second-order definition of the absoluteness of concepts. Second, the Marcus approach, which in this respect is the same as Kripke's, renders this kind of idea a presupposition of applying quantified modal logic. In contrast Bressan leaves it open for us to make it a matter of extra-logical theory—for example, a matter of serious empirical physics in the theory of general relativity—whether a concept is or is not absolute. There remains another side to the point. As we can learn from Bressan, even the most non-extensional of concepts have extensional cousins that arise by "extensionalization" with suitably weak equality relations. Given any Qx, however fierce, there is always Ex1[(x eq x1) & Qx1]. The predicate of parking on a highway, and dstit in general, are like that. All we want to add to these unfortunately cryptic remarks is this: Postulate 12-1 and the various definitions are intended to hold only for a that fall under the absolute concept of agent and not for extensional equivalents thereof. This leaves much that is problematic for us to treat on some other occasion, perhaps the same occasion on which we can discuss what is needed to cater to the natural wish to restrict the range of "a" in (24) to some sane set of agents.

335

12. Marcus and the problem of nested deontic modalities

12G

Temporal generalization

As we remarked in connection with (6), our development has suppressed explicit reference to "time" in favor of locutions that support "truth at a moment-history pair," where the moment-history is supplied externally (either indexically or by some implicit binding). This linguistic policy simplifies some considerations, but if we wish to frame prohibitions that generalize on moments in some carefully controlled way, then we should need to enrich the language. The easiest and most transparent thing to do would be to add variables explicitly ranging over moments or histories or moment-history pairs, depending on what is convenient.3 Since our purpose here is merely illustrative, however, we refrain from such additions. We suppose instead that we only require the language to describe a prohibition good throughout the entire future, regardless of what happens. For that all we need is the standard tense-logical "will-always" construction, §8F.5. We recall from §8F.4 and §8F.5 that Sett: Will-always: Q is true at m/h iff, come what may, Q is settled true at each moment that is properly later than m. We are now in a position to offer a "final" account of (3) as a generalized prohibition of parking on highways binding at all future moments no matter what happens: Sett:Will-always:VaVxFrbn-if-can-do:[a ways is henceforth forbidden.

dstit: Pax]:

Parking on high-

We hope our insertion of "henceforth" is self-explanatory.

12H

The outer ought

What, finally, of the outer deontic modality of (1)? Marcus 1966 says the "ought" in (1), "Parking on highways ought to be forbidden," should be taken to be evaluative. That is plausible, but we nevertheless want to run with the prescriptive possibility, just to indicate what is not so obvious, namely, that it makes perfectly good sense. Permit us therefore to take as a temporary target Parking on highways ought to be forbidden by someone, or, in the active voice, Someone ought to forbid parking on highways. The critical and much-overlooked point is this. To forbid is itself very naturally construed as an agentive verb.4 If we were pursuing ontology we would have to 3

Some people think doing so would be to import "the metalanguage" into "the object language." That opinion, which should by now be a relic, is demonstrably false; as Bressan 1972 has in effect shown, this linguistic power is not only useful, but already definable. 4 As we noted in §11H, Wansing 1998 forcefully argues that every obligation and prohibition should be personalized by indication of the agentive source, and he explores the consequences. It would take us too far afield to examine and profit from his suggestion, and so we again let a mere mention suffice.

336

Applications of the deliberative stit

force a search for the "action" of forbidding each agent to park on a highway. Fortunately, however, no such ontological chase is called for. The thing is that we already have what is needed to make entirely good modal sense out of one agent B forbidding another agent a, or as in the present example, forbidding all agents. To forbid an agent a to park is to see to it that a is forbidden to park. To forbid all agents to park is to see to it that all agents are forbidden to park. No tricks, just an appropriate dstit.

Now to say that b ought to forbid parking on highways is just to use language already at our disposal:

What goes wrong at this point is that we don't much feel like existentially generalizing this to say that there is some particular agent that ought to do the forbidding. And the restricted complement thesis prevents us from putting that existential quantifier inside the Oblg: modality (though we could perfectly well do that if the modality were impersonal). There is, however, a natural and proper way in which to express the generality of the not-very-personal "ought" while still treating the "ought" as prescriptive and while still hewing to the restricted complement thesis. Instead of "someone" in the logicians' sense conveying quantification over agents, it would be natural to say Here the subject "they" of the English sentence is intended not as a singular quantifier phrase, but as the name of a class or aggregate in the sense elaborated by Marcus in Marcus 1963 and Marcus 1974. The obligation is collective, not distributive. If there were no theory about such constructions this would be just throwing sand in your eyes, but in fact there is. Take F as some suitable aggregate of agents, and use the theory of joint agency from §10C. Then (25) is well symbolized by the following.

This says that there is a joint obligation on F taken collectively to see to it that parking on highways is henceforth forbidden. If we apply the Anderson/dstit simplification of §12C to this extended form of obligation, we shall find that we need to index sanctions not only by single agents a, as in Sa, but also by groups of agents: SF. That is of course just a formal remark. The concrete content stands in need of exploration. Our progress here is limited to seeing the difference between (26) on the one hand and, on the other, either

which doesn't mean what we wish, or

12. Marcus and the problem of nested deontic modalities

337

which is not only unwanted but is also a disagreeable violation of the grammar required by the restricted complement thesis. One last complication. The point of view of modality makes us see very clearly that by one more degree of nesting we could express that whereas (i) the obligation is joint on the group, nevertheless (ii) the particular dstit is to be carried out by some member of the group. In this case the dstit is of a forbidding, so that we have the following:

Again the thing to notice is that the new subtlety requires no tricks. Nothing is made up to order. All is transparent. We are just using the power of modalities combined with quantifiers to which we have become entitled by the philosophical pioneering of Barcan Marcus.

This page intentionally left blank

Part V

Strategies

This page intentionally left blank

13

An austere theory of strategies Suppose an agent a can see to a certain result Q in the achievement sense. It may even be, as illustrated in Figure 2.8, that a is unable actively to refrain from seeing to it that Q. Still, a always has available a strategy guaranteeing that a does not see to that Q (a strategy for inaction), a fact that we foreshadowed in §9I. Finding a strategy for inaction is trivial for the example of Figure 2.8: Choose right at w0, and then, if "nature" makes it necessary, choose right again at w1, thus guaranteeing inaction. Indeed, that there is always a strategy for inaction sounds quite generally trivial, partly because some confusingly soundalike facts, such as those described in chapter 9, are indeed trivial. When, however, one considers spelling out the exact content, and especially if the possibility of busy choosers is taken into account, it becomes clear that there is more complexity than one might have expected. One has to begin by saying what one means by a "strategy," and how strategies are related to stits. Since we are after a mathematical fact, rough ideas will not do. In analogy with stit theory, we sharpen focus by holding fast to an austere and purely causal idea of strategy: a system of choices. We rely on the concepts of choices in branching histories that earlier chapters have developed for the logic of "seeing to it that," or "stit." Restricting ourselves to this foundation leads to a theory of strategies that is ontologically, conceptually, and postulationally austere. Our only primitives will be the BT + AC structure primitives, (Tree, \<, Agent, Choice), derived from the theory of choices in branching histories: (i) concretely possible moments, (ii) a branching causal-temporal ordering on moments, (iii) agents, and (iv) for each agent and moment, a "choice function" encoding what the agent can choose at that moment. It follows that we shall be able to ask about the causal content of strategies, but it also follows that the mental idea of having a strategy "in mind" falls outside the scope of our inquiry. There will, however, be the following disanalogy with stit theory: In stit theory we discussed at some length the language of "seeing to it that," especially its grammar and 341

342

Strategies

semantics. There is in this chapter no parallel discussion of the grammar and semantics of the language of strategies. We note that most of the chapter is devoted to a self-sufficient discussion of strategies themselves, partly because the initial problem requires a detailed development, and partly because of the independent interest of the austere theory of strategies. This interest arises out of the thought that the restricted complement thesis, Thesis 5, which says that promises, obligations, and so on take stits as their complements, should be substantially modified. The modification that we in effect propose in §5C and §11H is that the contents of promises, obligations, and so on, can be not only stits, but strategies. The elaboration of the idea of a strategy is, then, in part meant to give some rigorous content to this suggestion. Only near the end of the chapter do we stop considering strategies in general. At that point we return to the problem with which we started: the existence of a strategy for inaction. The chapter is organized section-by-section as follows. §13A begins the chapter by comparing the theory of austere strategies with von Neumann's theory of strategies in extended form, and by indicating the proper role of the former. In §13B we reference the concepts that we shall need from the theory of agents and choices in branching time. Then §13C presents the elements of the austere theory of strategies, developing some fundamental concepts and proving central facts and lemmas. The concept of "favoring" is introduced in §13D, permitting the statement and proof of the "favoring extension lemma," whose corollaries are fundamental. In §13E we explain and establish the "strategy for inaction theorem," which, as promised, finally relates the austere theory of strategies to the logic of "seeing to it that."

13A

Nature of austere strategics

The present theory is very treelike. In this and other respects, it can and should be seen as a generalization of the theory of strategies for games in extended form of von Neumann and Morgenstern 1944. The following are several ways in which the present theory is more austere than von Neumann strategies for games in extended form. • Von Neumann games in extended form are discrete: Every node of the tree (except the first) is immediately preceded by a unique node, and every node (except a terminal node) is immediately followed by some accessible nodes. It is as if the structure of time were discrete. The austere theory refrains from any such assumption. Nor does it substitute any other assumption about the structure of time, for example that it is continuous (as in theories of infinite games). The austere theory is truly austere. • The von Neumann theory postulates that every branch on every tree is not only discrete, but finite in length. The austere theory refrains. Nor does it substitute an assumption that branches are infinite.

13. An austere theory of strategies

343

• The von Neumann theory postulates that at any node there are a finite number of available choices. The austere theory refrains. Nor does it substitute an assumption that there are infinitely many available choices. • The von Neumann theory postulates that at each node there is only one agent that can make a choice. The austere theory refrains. • The von Neumann theory adds as a primitive the notion of "information sets" to mitigate the "one node—one agent" restriction just mentioned. Since the austere theory doesn't start with such a restriction, it refrains from any such addition.1 • The von Neumann theory of games rests on the theory of utility, which provides a value for each outcome. It is as if sane consideration of strategies had to treat "strategies" as "strategies for" some outcome (e.g., winning in chess). The austere theory refrains. Instead it investigates the part of the theory of strategies that makes sense without values; this part is "pre-utilitarian," and interesting. One should think of the present work as more like geometry than anything else. The proposed concept of "strategy" is analogous to Euclid's concept of "triangle." Strategies, like triangles, will be tokens, not types; they will be instances, not kinds. When Euclid proves the Pythagorean theorem about any right triangle, he proves it about each instance. His proof is only derivatively about "the right triangle" or "right triangularity." The proof certainly does not give us information about either the psychological idea of the triangle or about the word "triangle"—these battles have been fought and won since Frege. It is just so when we prove the existence of, for example, a complete strategy satisfying certain conditions. We do not mean to prove the existence of a kind, much less an idea of a kind, much less something with linguistic structure. Instead we prove the existence of a concrete instance that has as much "location" in our indeterministic world as does a particular triangle. Strategies will be objective. Keeping the analogy with triangles in mind might help one get strategies out of the head and into the world. Strategies, in the austere sense treated here, are neither linguistic nor mental. Of course one's philosophy might make everything an idea, triangles too. The philosophical road now forks. Path A: One who goes on to infer that Euclid's work was therefore somehow misguided can safely skip this chapter, and indeed this book. Path B: One who still makes room for Euclid should treat the forthcoming account 1 Information sets are useful in practical cases, including many found in economics, in which it seems convenient to run together causal and epistemological considerations. They provide, however, an unsatisfactory foundation. Because information sets conflate causality and ignorance, their use forever prohibits the game theorist from distinguishing between causal constraints and epistemic constraints. One who uses them cannot even articulate a theory about how the two might relate! We remark that this point is not intended as new, nor as suggesting that the game theoretical community does not worry about the issue. It is only to suggest that the clearest way to proceed is to throw out information sets and then to start over with explicit and separate representations of causality and of ignorance.

344

Strategies

of strategies in the same way as the Pythagorean theorem, whatever way that might be. What can we expect and what should we not expect? We should not expect an account of "what it is to have a strategy"; the concepts of the austere theory are not rich enough to enable such an account. We should expect an account of what a strategy is in a structural sense. To put the matter tautologically, a strategy is what one "has" when one "has" a strategy. We should, to use untrustworthy phrases from the ethical literature, expect an account of what it is to act "in accord" with a strategy. We should not expect an account of what it is to act "on" a strategy. This is analogous to the following: Euclid tells us what it is to be a triangle, but he does not provide an account of what it is to see a triangle. The present theory, like Euclid's, has nothing interesting to say about distinctively mentalistic concepts—except, perhaps, the following: Strategies have both psychological/linguistic/cultural and structural/geometrical/causal aspects, and there is hope of a benefit if we disentangle the two. The analogy goes a little further. Just as Euclid's definition of "triangle" is by itself almost without interest, so is our definition of "strategy." It is only when one passes from the most general concepts to (what Whitehead called) "happy particularities" that one should hope for enlightenment. We offer several modes of classification that, with the help of established conceptual connections (which we call facts, lemmas, or even theorems), aim to deepen our understanding of strategies.

13B

Review of choices in branching histories

The needed concepts from the theory of agents and choices in branching time are as follows. • We assume a BT + I + AC structure (Tree, ^, Instant, Agent, Choice}, §2, subject to the various postulates listed in §3. We let m range over Tree, i over Instant, a over Agent, and we let Choice^ be the set of possible choices for a at m. (The theory of strategies does not itself use the concept Instant, but we shall need it for our intended application of that theory to the achievement stit.) • We use the following additional moment-history notation and concepts from Def. 4: h, m/h, H(m), h1 =m h2 (h1 and h2 are undivided at m). • As for instants, we will need all of the various concepts defined in Def. 9—but only at the very end: i, i(m), m ( i , h ) • We will need choice equivalence for a at m: h1 =amh2 (choice equivalence between histories through m), Def. 12. • We need the idea of a busy chooser, Def. 14. • The theory of strategies requires the idea of choice inseparability for an agent of two histories by a set of moments, written h1 =amh2, Def. 13.

13. An austere theory of strategies

345

• Not for the theory of strategies itself, but for the application we give in §13E, we need the achievement stit, here written just [a stit: Q], with the chain witness semantics, §8G.4. We remind the reader that we take (Tree, \<, Instant, Agent, Choice) sometimes as a (more or less adequate) idealized representation of how agents and their choices fit into Our World, in which case the BT + I + AC postulates are "postulates" ; and sometimes as an abstract structure, in which case the postulates governing BT + I + AC are clauses of the definition of "BT + I + AC structure." The foregoing finishes the list of concepts and postulates concerning choices in branching time that are needed for an austere theory of strategies. No "surprise postulates" or "commonsense assumptions" are to be expected. Nor will we add a concept unless it is defined from these according to the standard of rigor preached and practiced by Frege.

13C

Elementary theory of strategies

This section develops the most elementary part of the theory of strategies. We begin with the easiest ideas, and restrict ourselves to those needed for the principal results, which are Lemma 13-22 (the favoring extension lemma) and its corollaries.

13C.1

First definitions and facts

There is not much to our actual definition of a strategy: A strategy is an array of choices. The "domain" of a strategy s is the set of moments at which it gives advice, and we write "Dom(s)" for the domain of s. Since the governing theory encodes a choice at a moment as a set of histories, there is hardly a substantive alternative to the following. 13-1 DEFINITION. (Strategy for a; consistency; strictness) • s is a strategy for a iff s is a (partial) function on moments such that for m E Dom(s), s(m) is a subset of H( m ) that is closed under choice equivalence for a at m: s(m) C H(m), and Vh[h € s(m) —» Choiceam(h) C s(m)]. • s is available at m iff m E Dom(s). • For m E Dom(s), s is consistent at m iff s(m) /= 0, and s is consistent iff s is consistent throughout its domain. • For m E Dom(s), s is strict for a at m iff s(m) 6 Choiceam, and s is strict for a iff s is strict for a throughout its domain.

346

Strategies

So s(m), when defined, is the union of a subset of Choiceam, and is nonempty when consistent. It would make no difference if s(m) had been defined as a set of possible choices for a; that is, one could have defined s(m) as a subset of Choiceam instead of as the union of a subset of Choiceam. Regardless of the coding employed, the underlying idea is that a strategy for a says which available choices are admitted (advised, required, whatever) at each moment m in its domain. A strategy is available at all and only the moments in its domain. A strict strategy admits (or advises or requires) a unique choice at each moment m in its domain rather than, as we might say, a disjunctive choice of choices. We might call "loose" those strategies that are not strict. In the foundational study von Kutschera 1993, "loose" strategies are called "strategies in the wider sense." Loose strategies are of great significance. Hamblin 1987 calls a concept with a similar point a "partial" strategy, noting that "in practice, no one ever chooses or is allocated a strategy in the minute detail that specifies every deed; and certainly not to the end of time" (p. 157). Hamblin goes on to observe that "masterful inactivity, or simply doing whatever you feel like or getting on with something else, may often be the best or only course for stretches of time while events unfold." This seems to us just right. We nevertheless concentrate here on strict strategies, since only these are needed for the "main result" toward which this study points. Though everywhere-inconsistent strategies are of little or no interest, in practice we often need to deal with locally inconsistent strategies (compare the parallel point for obligation in the §12D discussion of the no-good-choice case); however, in this study we usually consider only strategies that are consistent—that is, strategies that are everywhere-consistent. Strict strategies are consistent. We shall be "extending" strategies to obtain completeness. Extension is a form of "strengthening"—all of these words to be taken in an improper sense. 13-2 DEFINITION. (Strengthening; extending) • Let S0, s1 be strategies for a. We say that S0 is (perhaps improperly) weaker than s1, or that s1 is a strengthening of S0, iff and for each

• We say that s1 extends so when the second clause can be improved to say that s1(m) = s0(m) for each Note the difference in directions of the subset relation. Adding a moment to the domain strengthens a strategy (we call such an addition an "extension"), while cutting down the number of options also strengthens it. In this study we shall be almost exclusively interested in those strengthenings that are extensions. The reason for this is that in this section we are interested primarily in strict strategies, and of course a strict strategy can be consistently strengthened only by extension—since to cut the number of options below one is to cut them to zero.

13. An austere theory of strategies

347

13-3 FACT. (Lattice of strategies) • The set of all strategies for a, ordered by the weaker-than relation, is a complete lattice. • Let E be a set of strategies, sE—the join of E in the lattice of strategies— is characterized by the following.

One should keep in mind that the technical idea of the "join" of some strategies represents their common strengthening; so it is more likely to be expressed in English with "and" than with "or." 13-4 DEFINITION. (Admission, exclusion) Suppose that s is a strategy for a. • s admits h iff Vmo[m0 € Dom(s)nh —> h € s(m0)]. • Admh(s) = {h: s admits h}. • s admits m1 iff Vmo[(m0 6 Dom(s) & m0 < m1) —> m1 E U s ( m0 )]• Admm(s) = {m: s admits m}. • s excludes a history or a moment if it does not admit it. The ideas of exclusion and admission are central to the theory of strategies. As for exclusion, one may start out along an excluded history, but if one chooses in accord with s, then one cannot forever remain on an excluded history. Eventually one reaches a moment at which the strategy goes one way and the history goes another. Nor can one ever reach an excluded moment if one follows a strategy. Admission is in one sense a weak notion; for example, any history that contains no member of Dom(s) is vacuously admitted. Apart from this trace of vacuity, which we remove on a case-by-case basis when we apply the notion—for example, in the next definition—an admitted history is one that might happen given that the strategy is faithfully followed. An admitted moment is one that might be reached provided that the strategy is followed without exception. Admissibility represents what is possible, given that the strategy is followed. It is therefore natural to define what a strategy "guarantees" in terms of admissibility: 13-5 DEFINITION. (Guarantee)

• s guarantees HO iff • s really guarantees HQ iff s guarantees HQ, and furthermor

348

Strategies

Think of H0 as a proposition expressed by a moment-independent sentence Q, that is, as the set of all histories in which such a Q is true. Then for SQ to guarantee H0 is for s to guarantee Q in the sense that Q is true in all those histories not forbidden by s. The reference to H(D 0 m(s)) insures that we don't have to worry, however, about histories that branch off who-knows-where long before the arrival of the domain of s. The definition of "real guarantee" adds a "negative condition" in accord with the general tenor of stit theory, §8G. For a strategy to be said to really guarantee H0, its choices must do at least some work of excluding non-H0 histories. The real sense of "guarantee" is exactly what we want for our main result, Corollary 13-29, to the effect that a certain strategy "guarantees" inaction. The following is obvious, but worth recording. 13-6 FACT. (Admissibility and extension) Extending or strengthening a strategy can only reduce admissibility, never increase it. Contrapositively, extending or strengthening can increase exclusion, but never reduce it. And what is guaranteed persists under extension or strengthening. Among strategies there is a sharp difference between those that do and those that do not tell you what to do after you have failed to follow their earlier advice. We call the former "secondary" and the latter "primary." (Our earlier work on strategies used the pair of words "redundant" and "irredundant." Horty 2001 uses "lean" where we say "primary." Although the ideas are stable, the best choice of vocabulary is unsettled.) The thought is that advice given by a strategy after you have already violated the strategy may be useful, but it is surely "secondary" to the primary portion of the strategy. So we call the strategy itself "secondary" or "primary" depending on whether it does or does not contain any secondary advice. 13-7 DEFINITION. (Secondary versus primary) s is secondary iff some moment that it excludes is anyhow in Dom(s); and s is otherwise primary. The strategies (for games in extended form) of von Neumann and Morgenstern 1944 are primary, or mostly so. A typical game theoretical strategy does not tell you what to do if you do not follow the strategy. Suppose a strategy for chess begins with advising king's pawn to KP4 as first move. That very strategy would typically not tell you what to do on your second move if instead you moved your king's rook pawn on your first move. Why should it? Such advice would be redundant (it would be secondary advice). One finds a contrast in the "deontic kinematics" of Thomason 1984, p. 155, which we discuss in chapter 14. In that scheme, obligation is laid on at every moment. If we think of these obligations as together making up a strategy (Thomason speaks of plans), the strategy would be secondary. And for good reason; as Thomason notes, secondary strategies enable consideration of "reparational" obligations, telling us what to do after we have done wrong. Secondariness can be important.

13. An austere theory of strategies

349

Secondariness of a strategy should be distinguished from a form of strategy that at a certain moment gives you a "second-best" choice besides its first choice. One might in English describe both a secondary strategy and a strategy that gives a second-best choice in the same language: Each tells you what to do "if" you don't do what it primarily advises. But the cases are really different. If a strategy offers a second-best choice, then the second-best must be labeled so: "PQ4 is best, but PK4 is second best." See Horty 2001 for ideas that involve such grading of the choices on which stits are based. In this chapter, however, we do not deal at all with strategic concepts that reach outside the BT +1 + AC structure, (Tree, ^, Instant, Agent, Choice), in order to label some choices as second-best: The "second-best" of a secondary strategy needs no extra labeling. What makes it second-best is already definable. Namely (and roughly), it is the choice (and the only choice) that the strategy advises at a moment that is excluded by an earlier choice dictated by the strategy. It is the exclusion of a moment that renders advice at that moment automatically secondary. Note, incidentally, that when there are no busy choosers, we can always count our way down from a forbidden moment through a finite series of "fresh mistakes" to the main line of the strategy. This gives an already-definable account of "n-ary strategy" as well as "secondary strategy." On the other hand, if there are busy choosers, it is not so clear that such gradations make sense. 13C.2

Simple strategies

We shall be dealing largely with strategies that are both strict and primary, partly because they are in some respects the simplest. In fact we shall cause these two properties to contribute to the upcoming definition of "simplicity." There is, however, another ingredient. To introduce it, we need to think ahead a little ways to the completeness of strategies. When we turn to "completeness," we need to have in mind some population of moments M that is typically larger than the domain of the strategy but much smaller than the entire universe. These are the moments that we care about. Typically they will not extend backward all the way to the Bang, nor forward past the heat death of the sun.2 When M is given, we may call it a "field" for the strategy. Always a field M will include the domain of a strategy, that is, the set of moments at which the strategy gives advice. We use the phrase "in M" to carry this information: 13-8 DEFINITION, (strategy in M; field for s) • s is a strategy for a in M iff s is a strategy for a such that Dom(s) C M. • When s is a strategy for a in M, we call M a field for s. 2

Less pictorially, a "field" M for a strategy might have the property that in each history h with which M has nonempty intersection, there are upper and lower bounds for h n M. Or it might be that all of the nonempty hdM have a common past or even contain a common lower bound. Or it might be that each nonempty h n M is a maximal chain in M. But even though typically interesting strategies will be bounded or even have one of the stronger properties, our austere refusal to impose such conditions seems to clarify the concepts and results that this chapter considers.

350

Strategies

It could be that a strategy gives advice for every moment in M, or even for every moment in all of Tree. Such a strategy is said to be "total": 13-9 DEFINITION. (Total strategy) A strategy s is total [in M\ iff Dom(s) = Tree [= M\. We shall not be interested in total strategies until chapter 14. We are instead presently interested in notions of "completeness," which are altogether different from totality. A strategy complete in a set of moments M will tell us what to do at appropriate moments within a field M—but by no means all. For instance, a "complete" strategy need not tell us what to do at moments in the field that the strategy itself forbids. That is precisely why a field for a strategy must be distinguished from its domain. Nor is a field uniquely determined by the domain; that is why we say "field for s" instead of "field of s." EXAMPLE. (Domain versus field) Picture Tree as representing a game of chess in extended form (von Neumann), that is, a tree, rooted in some concrete occasion, of all plays in accord with the rules of chess. That would be an appropriate "field" for a strategy. A particular strategy itself would be represented by a small subtree of Tree; for example, a tree that starts with "P to Q4." This strategy can be altogether "complete" in Tree, while giving no advice at all to the player who ignores it in favor of starting with KKt to KB3. As a small step toward completeness, we consider backward closure. Note that we confine ourselves to M partly so as not to have to go back to the beginning of the universe. 13-10 DEFINITION. (Backward closure) Suppose that s is a strategy for a in M. s is backward closed in M iff Vmo,mi[(mi € Dom(s) & mo < mi & mo 6 M) —> m0 G Dom(s)}. A "conditional" strategy might well not be backward closed: "If you reach Station A, get off the subway; but for earlier stations, there is no advice." So strategies that are not backward closed are important. But if a strategy is not backward closed, then that seems enough to give it a kind of not-yet-available flavor. Such a strategy gives us advice at a certain moment without giving us any advice whatsoever as to how to get there. It would in that respect be similar to an end-game strategy for chess as given in a book on end games, provided we chose an odd field for it. Namely, suppose we took as field not just the subtree where the book starts giving us advice. Suppose instead we oddly took as field the entire tree for chess that starts with the first move. Then that strategy would not be backward closed in that field. If we want to confine ourselves to the simple case in which a strategy is available from the very beginning of its field, we must think about backward closed strategies. This notion is therefore the third and final part of the concept of simplicity: 13-11 DEFINITION. (Simplicity) s is simple for a in M iff s is a strategy for a in M that is strict for a, primary, and backward closed in M.

13. An austere theory of strategies

351

So a strategy that is not simple might fail of simplicity in one of three ways, (i) It might be loose (sometimes giving disjunctive advice), (ii) It might be secondary (sometimes giving advice at moments it forbids). (Hi) It might not be backward closed (some parts of its domain might be isolated high up in M). Simple strategies are not like that.

13C.3

Pre-simplicity

Permit us to say something parenthetical about strategies that are not simple in virtue of being not backward closed in M—parenthetical in the sense that later sections do not build on this one. Suppose such a strategy is in fact strict and primary, and fix the set M of relevant moments. The strategy might have a simple extension in M, or it might not. There is only one case in which it does not: There are two moments TOI and m^ in its domain, and they have a lowerbounding moment m0 in M (but not in Dom(s)) such that every choice for a atTOOexcludes one of m1 and m2. Were this case to occur, we should not know how to specify s(mo) while retaining both strictness (at TOO) and primariness (with respect to each of m1 and m 2 ). This observation points to an alternative concept of "simplicity" for use when we wish to consider strategies that are not backward closed. To remind us that the alternative is helpful only when we are looking forward to simplicity, let us call it "pre-simplicity." 13-12 DEFINITION. (Pre-simplicity) s is pre-simple for a in M iff s is a strategy for a in M that is strict for a, primary, and such that each two members of its domain are inseparable for a in M. EXAMPLE. (Pre-simplicity) Let s be a strategy for you that might be described in this way: "If you reach Station A or Station B, get off the subway; but for the earlier moments at which you have a choice about reaching Station A or Station B, there is no advice." You may not care about completeness at all. If you do, you may not care about it with respect to a field M for s that includes those earlier moments. In these cases you will not care about pre-simplicity in M (though you may still care about it with respect to a smaller field). Suppose, however, that a field M for s includes at least one of these earlier moments, say TOO, at which you can choose between reaching Station A and Station B; and suppose that you care about strengthenings of s that are complete in M. You may or may not be interested in primariness. If you are not, you can think about extending the strategy to mo in any way you like. You could then think of some or all of the "if you reach Station A or Station B" advice as "reparational." You could think of it as "redundantly" advising you what to do after you have fallen off the strategy. In this case—see chapter 14—pre-simplicity would be of no interest. Suppose, however, that you are interested in primariness. Even then it makes sense to extend the strategy toTOO,provided the advice there is not strict but loose: Head either for Station A or for Station B, whichever you choose. That would be a primary (though loose) extension. Nothing wrong with that. But if you are interested in a strategy only if it can be backward closed to a strict and primary extension, you need to think about pre-simplicity.

352

Strategies

Here are key facts about pre-simplicity. They are put here for help in keeping our bearings, and are not otherwise used in this chapter. 13-13 FACT. (Pre-simplicity) i. s is pre-simple for a in M iff s has a simple extension for a in M. ii. s is pre-simple for a in M iff s is a strategy for a in M, and

iii. In application to a strategy that is backward closed in M, there is no difference between simplicity and pre-simplicity. The smallest simple extension as guaranteed by (i) might be called the "backward closure" of s. But we won't need the idea, and in any event that phrase could well be used in other ways when we are interested in loose or inconsistent or separable strategies. Part ( i i ) gives an equivalent account of pre-simplicity so as to exhibit that pre-simplicity is a unified idea, not a mere conjunction: Strictness, primariness, and inseparability are all part of the same package. And (Hi) tells us that when we know a strategy is backward closed in M, we just don't have to worry about simplicity versus pre-simplicity. In any event, we shall nearly always be dealing with simple strategies.

PROOF. Ad Fact 13-13 (i). Suppose SQ is pre-simple for a in M. Define si so that Dom(si) = Dom(s0)U{m0: mo £ M & 3mi[m0 < mi & mi 6 Dom(s0)}}; and for m define Evidently si is backward closed in M, and si is primary if SQ is primary. Also pre-simplicity of S0 guarantees that s1 is strict. So s1 is a simple extension of s0. Conversely, if s0 is not pre-simple, then if it is either not strict or secondary, no extension can remove that defect. And if its failure of pre-simplicity comes from having two members m1 and m2 of its domain that are not choice equivalent at some moment mo in M that lies in the past of each of m1 and m2, then there is no way to extend s0 backward to m0 without violating either strictness (at mo) or primariness: If Si(mo) is a single member of Choicean0, then either m1 or m2 will be excluded, causing s1 to be secondary. Ad Fact 13-13 (ii), left-to-right. Argue by contraposition. Suppose that s is a strategy for a in M, but that the right-hand side is false: mi, m2 G Dom(s), hi S s (mi), hi e s (mi), m0 e M, m0 ^ mi, mo ^ m2, hi _L^ 0 hi. Argue by cases. If mo = mi = mi, then there is nonstrictness at mo since non-choice equivalent histories belong to s(m 0 ). If mo = mi and mo < mi, then mo excludes m2, witnessing that s is secondary. Similarly if m0 < mi and mo = m2. And if m0 < mi and m0 <mi, then there is a violation of inseparability. So in either case s is not pre-simple. Ad Fact 13-13 (ii), right-to-left. Argue by contraposition. If s is not presimple, it is either not strict, or secondary, or separable. If it is not strict at mo then we can find a counterexample to the right-hand side with mo = mi

13, An austere theory of strategies

353

= m-2- If it is secondary in virtue of the exclusion of m\ at mo, we can find a counterexample to the right-hand side with mo < m\ andTOO= m^. And if s fails inseparability in virtue of mi,m,2 E Dom(s) being separable at mo & M, then we can find a counterexample to the right-hand side with mo < mi and m0 < TO2. Ad Fact 13-13 (Hi). This is a way of saying that simplicity implies inseparability. The argument is that a witness to separability that by backward closure lay in the domain of s would also witness either nonstrictness or secondariness.

13C.4

Ideas of completeness

We can now define "completeness." For convenience, we shall do this under the presupposition that we are dealing only with simple strategies; otherwise the concepts might need refining.3 13-14 DEFINITION. (Completeness) Suppose that s is a simple strategy for a in M. • s is complete along h in M iff M(~}h C Dom(s). • s completely admits h in M iff s admits h and is complete along h in M. • s is H-complete in M iff s is complete in M along every history in H that s admits. • s is complete in M iff s is complete in M along every history that s admits. • s is simply complete for a in M iff s is simple for a in M and s is complete in M. We offer the fourth of these five concepts as a conceptual analysis of the essential idea of strategic completeness. In contrast, the first three concepts are introduced as technically useful, while the last gives us a rhetorical variant of "simple and complete." (Strictly speaking, "simply" here is overdefined, but we nevertheless try to be consistent about including it.) Completeness is an easy idea: A complete strategy in M provides advice at each moment in the field M at which you can arrive without violating the dictates of the strategy itself. No matter which advised choice you make, no matter the outcomes of decisions by other agents or by nature, as long as you are within the field M, a complete strategy always tells you what to do. Not even a complete strategy in M, however, either gives or pretends to give any advice outside the confines of the specified field M. A strategy that is incomplete in a field M gives you some 3 For instance, the deontic trees defined in Definition 14-1 are "complete" in a sense almost unrelated to Definition 13-14: Deontic trees obligate at each and every moment. Strategies derived from that variety of deontic kinematics are not part of our present topic because they are not simple.

354

Strategies

advice, and then sometimes leaves you adviceless even when you follow it, even when you are still in M. Complete strategies are analogous to the complete theories known to modeltheoretic logic in the following respect: Although interesting specimens of either are few and far between, complete strategies, like complete theories, are important objects of study because of the limiting role they play. One can see that—and be a little surprised by the fact that—for strategies in M, completeness is converse to primariness: 13-15 FACT. (Completeness and primariness) Suppose that s is a simple strategy for a in M. Then the following give good accounts of completeness in M and primariness. • Completeness: (Admm(s)nM) • Primariness: Dom(s) C

C Dom(s).

(Admm(s)r\M).

Let us record that extending a strategy can never make it less complete than it was. 13-16 FACT. (Persistence of completeness) If s0 is a simple strategy for a in M, then if SQ has one of the following properties, so does any simple strategy si for a in M that extends SQ: (i) is complete along h in M; (ii) completely admits h in M; (Hi) is //-complete in M; (iv) is complete in M; (v) is simply complete for a in M. PROOF. Because extensions only add to the domain of a strategy, completeness evidently begets completeness. We need, however, to check the "admission" part of ( i i ) . This part says that if SQ and s\ are simple strategies for a in M such that si extends s0, and if s0 completely admits h, then sj admits h. So let mo € Dom(si)C\h; we need to show that h € SI(TOO). Because si is a strategy in M, mo € Mfl/i. So, since SQ is complete along h in M, mo 6 Dom(so), and so mo € (Dom(so)C\h). Therefore, since SQ admits h, h € So( m o)- But since mo £ (Dom(sQ)(~}Dom(si)), the definition of extension says that SI(TOO) = so( m o)So /i 6 si(mo), as required. The next definition and fact articulate the technically useful idea of extending a strategy by an entire admitted history (more accurately, by that portion of it in M). 13-17 DEFINITION. (Extension along h) Let s0 be a strategy for a in M. Define "si is the extension of S0 along h for a in ' as follows:

13. An austere theory of strategies

355

The purpose of this construction is the following. In general admission of a history is not preserved under extension. We can get around this by a wholesale extension that always points along h, and that totally uses up the portion of h in M. Then there will be no leftover moments at which a further extension could exclude h. The following small fact is thus critical. 13-18 FACT. (Extension along h) If (i) s0 is a simple strategy for a in M and (M) s0 admits h, then, where sj is the extension of s0 along h for a in M, • S1 is itself a simple strategy for a in M; • s1icompletely admits h for a in M; and • any extension of S1 in M continues to admit h. PROOF. If s1 is the extension of s0 along h for a in M in the sense of Definition 13-17, then s1 is on the face of it strict and backward closed for a in M. Suppose that s1 were secondary, with mo, m1 S Dom(s1) and mo < m1 but m1 If m1 did not belong to h it would belong to 0), which would make SQ secondary, contrary to hypothesis. So mi € h, and since histories are closed backward (by the postulate of no backward branching, Post. 3), mo h. But then, since mi ^ U s i( m o)i h £ si(mo). Definition 13-17 then implies that mo G Dom(so) and that «o( m o) = SI(TTIQ). So after all SQ does not admit h, contrary to hypothesis. So s\ is simple for a in M. The "complete" part of "completely admits" is on the surface of Definition 13-17, and the "admits" part is not far below. Were h excluded by si, one would have mo with h £ si(mo). But a glance at Definition 13-17 suffices for this case implying that mo £ Dom(so), whence s0 would itself exclude h, contrary to hypothesis. That extensions of Si continue to admit h comes from Fact 13-16. We shall also need some near-the-surface facts about joins of chains of strategies in the sense of the "weaker than" relation of Fact 13-3. 13-19 FACT. (Joins of chains) Let S be a nonempty chain of strategies for a in M. Then the join sg (Fact 13-3) is itself a strategy for a in M. Furthermore, if every member of S has one of the following properties, then so does s-^: (i) strict for a, (ii) primary, (m) backward closed in M, (iv) simple for a in M, (v] admits h, and (m) completely admits h. PROOF. Use Zorn's lemma. A violation of any of (i)-(vi) for SE would be finite, and hence would be a violation of the same property for some member of This concludes our introduction of the most basic strategy concepts. We go on to introduce "favoring," an important but somewhat less obvious idea.

356

Strategies

13D

Favoring

Let us return to the problem mentioned at the beginning of the chapter: As promised in §91, find a strategy the following of which will guarantee not seeing to it that Q. The core of an intuitive solution is easy: Shoot for ~Q whenever given a choice. To make clear sense out of this suggestion, we introduce the concept of a strategy "favoring" a set of histories. The root idea comes from Hamblin 1987, p. 157, where the topic is carrying out (or "extensionally" satisfying) an imperative: ... the addressee of an imperative would be expected, at least, to act in such a way as to keep extensional satisfaction within the bounds of possibility. This means that he must not do any deed d that would infringe—that is, that would ensure dissatisfaction of—the imperative. The next definition aims to capture exactly this idea. In reading it, think of HO as the set of histories on which Hamblin's iniDerative is satisfied. 13-20 DEFINITION. (Favoring)

Let s be a simple strategy for a in M.

• s favors H0 iff • s completely favors HQ in M iff s favors HQ and is flo-complete in M. If s favors HO, then whenever any moment in the domain of s enables a choice that keeps HQ as a live possibility, then s advises such a choice. Furthermore— and critically—s does so by means of a history such that the same is true of any further moments in the domain of s that lie on that history. Let us relate this critical conceptual point to the notation. At a key point in the definiens of "s favors H" we used "Admh(s)," which refers to the set of histories that can be reached if the strategy is followed everywhere. If instead we had used "s(m)," we would have represented a weaker notion of "favoring." For example, consult Figure 14.1. Let HQ be the set of all the histories, {/iw, hi, h^, ...}. Then the depicted strategy, s, would favor H0 in the weaker sense, since each sm is nonempty. It would not, however, favor HQ in the stronger sense, since, as declared by Fact 14-9, Admh(s) = 0. The property of favoring need not persist under extension; but the property of completely favoring does: 13-21 FACT. (Persistence of completely favoring) Any simple extension si of a strategy s0 for Q in M that completely favors HQ also completely favors HQ. PROOF. Suppose h € HO and si admits h. Then s0 admits h by Fact 13-6, so SQ is complete along h in M by hypothesis, so si is complete along h in M by Fact 13-16. So si is Ho-complete. For the "favoring" part, remark that HoH Admh(so) C Hor\Admh(si) by Fact 13-16. Now suppose mi € Dom(s\) and

13. An austere theory of strategies

357

and By the remark, r\Admh(so) = 0, so that to avoid violating the hypothesis that SQ completely favors HQ, mi ^ Dom(s0). Choose h 6 H(mi)r\H0; evidently /i ^ ^4dmft(s 0 ). Let the excluding occur at m0: m0 £ Dom(so)nh, h £ s0(m0). Since m0, TOI both belong to h, they are comparable; but mi ^ mo would put mi in Dom(so) by backward closure, so that mo < mj. So SQ excludes mi according to Definition 13-4, so that si excludes mi by Fact 13-6. So the supposal that mi € Dom(si) leads to the secondariness of si, contrary to hypothesis. We have developed just enough of the theory of strategies to prove a useful lemma: A favoring strategy can always be extended to a completely favoring strategy. This lemma will have important corollaries. 13-22 LEMMA. (Favoring extension lemma) Suppose s is simple for a in M, and favors H0. Then s has an extension s' that is simple for a in M and that completely favors HO. If furthermore m0 is such that m0 €. M and H^^CiHo then s' can be chosen to be available at m0. PROOF. Well-order H0 with the ordinals less than a limit ordinal A'. (We omit adapting the argument for finite H0.) For 7 < A', let /i7 be the history in H0 that is indexed by ordinal 7. Furthermore, when the "furthermore" clause of the lemma is wanted, in the special case 7 = 0, as the "furthermore" hypothesis we let h-y G H(mQ)
If s7 admits /i7, then s7 + i is the extension of s7 along /i7 in M. Otherwise, for limit ordinal We argue by ordinal induction up to (and including) A' that each s7 is simple for a in M and favors HQ. CASE. 7 = 0. Use the hypothesis of the lemma. CASE. 7 = 6+1, assuming for the induction that sg is simple for a in M and favors HQ. We need to show that 5,5+1 is simple for a in M and favors HQ. If sg+ i = sg, the inductive hypothesis carries us, so suppose that sg admits hg. That s,5 + i is simple comes from Fact 13-18; and we remark that hg 6 Admh(sg + i) does so as well. For favoring, suppose mo 6 Dom(sg + i) and If mo e hg then by the remark, hg itself witnesses Suppose then that mo ^ hg, so that mo G Dom(sg). By inductive hypothesis, let h e H(mo)nH0r}Admh(ss). Suppose for reductio that sg + i excludes h in virtue of mi € Dom(sg + i}r\h and h $ sg(mi). It cannot be that mi 6 Dom(sg), since sg admits h, so mi 6 hg. Since mo, mi G h, they are comparable, which gives us an absurd disjunction: Either (i) mo ^ mi or (ii) mi < mo- Fork (i) contradicts that the history hg is backward closed (which must be so by the postulate of no backward branching); and fork (ii) contradicts

358

Strategies

that the strategy sg is backward closed (given by inductive hypothesis). So sg+1 admits h, and we have the needed member of H^mo)r\HonAdmh(se + i). CASE. 7 = A is a limit. We have as inductive hypothesis that each predecessor strategy of s\ is simple for a in M and favors H0. Fact 13-19 lets us conclude that s\ is simple for a in M. For favoring, suppose mi € Dom(s\) and #(mi) flHo 7^ 0- We need to find a history in H(mi)r\H0r\Admh(s\). Let mj e Dom(s^), where 7 < A. Since s7 favors #Q; let h 6 s 7 (mi)n.Hon.Adm/i(s 7 ). Consider {7': 7' < A & sy excludes /i}. If this set is empty, h & Admh(s\) by Fact 13-19, and we are finished. Otherwise, choose the least such 7', noting that since exclusion is maintained upward (Fact 13-6), 7 < 7'. Also by Fact 13-19, 7' cannot be a limit. So, where £ + 1 = 7' (so that 7 ^ 6), we must have that sg admits h while s$ + i does not. Keep in mind that mi € Dom(sg) since sg extends s7. As witness to the exclusion of h by sg +1, let mg € .Dom(s6+i)n/i and /i ^ s« +1(7712). Since 5,5 + 1 agrees with sg on their common domain, the fact that sg admits h implies that m^ £ Dom(sg). Hence, by the way that 5,5 + 1 is defined, m? 6 kg. Since mi,m 2 € /i, they are comparable. It cannot be that m2 ^ mi, else by backward closure of sg we should have m2 G Dom(sg). So mi < m.2- But then hg € -f/( mi ) by backward closure of histories. Furthermore, since sg+i completely admits hg by Fact 13-18, hg is also admitted by the extension s\ (Fact 13-16). So hg G H^mi^r\Hor\Admh(s\), which is therefore not empty. This finishes the inductive argument that each s7 is simple for a in M and favors HQ. Penultimately, we argue that s\/ is .f/o-complete in M. Suppose it is not, so that s\i admits h € HQ but is not complete along h in M. Let h = h~{ in the well-ordering. Since h is admitted by s\', h must by Fact 13-6 be admitted by s7. But then by Fact 13-18 s7 + i is certainly complete along h in M, and so all its extensions in M, including s\>. In short, every history in HO admitted by s\> was "completed" in its turn. Furthermore, suppose the "furthermore" clause is operative. We know by the "furthermore" hypothesis that the privileged moment mo belongs to ho(~\M, where ho is the history in HQ that is indexed by the ordinal 0. Since ho is admitted by SQ = s, we know by definition that «i is complete along /i0 in M. This guarantees that mo S Dom(si). Since s\ extends si, it is sure that mo € Dom(s\). That is, the strategy s\' is available at mo, as desired. 13-23 COROLLARY. (Complete and favoring extensions) • If s0 is a simple strategy for a in M then s0 can be extended to a simply complete strategy s\ for a in M. • Suppose m G M and H^m^r\H0 ^ 0. Then there is available at m a simply complete strategy for a in M that favors HOPROOF. For the first part, supposing SQ to be a simple strategy for a in M, apply Lemma 13-22, choosing H0 as Admh(so)- Evidently SQ vacuously satisfies the requirement that it favor Admh(s0). Let si be the promised extension of SQ that, besides being simple for a in M, completely favors Admh(so)- We need to

13. An austere theory of strategies

359

show that Si is complete. Suppose si admits h. Then h € Admh(so) by Fact 136, so by complete favoring, si is complete along h, as required. For the second part, suppose m £ M and H(m)f~}H0 ^ 0. Start with the empty strategy, which vacuously is a simple strategy for a in M that favors HQ. This gives us the main hypothesis of Lemma 13-22, and we have also the "furthermore" hypothesis since the empty strategy admits all histories. So there is a strategy So available at m that is simple for a in M and completely favors HQ. This strategy, however, may not be complete. No matter, by the already-established first part of this corollary, SQ itself has an extension si that is simply complete for a in M. Clearly si remains available at m. Lastly, by Fact 13-21, si also completely favors HQ.

13E

Application to finding a strategy for inaction

As we said at the beginning of this chapter, we provide an elementary application of the theory of strategies to the logic of "seeing to it that." Previous work in stit theory, §9G, made it clear that the ability to see to some result, Q, does not imply the ability to refrain from seeing to Q. (Since two stits were put in play in §8G, we note that this failure applies, delicately, to the achievement stit, but not to the deliberative stit. We are here concerned only with the former.) In the course of developing stit theory, it became plausible to suggest, however, that whenever an agent, a, can see to it that Q, in spite of being unable to refrain from seeing to Q, there is always available to the agent a strategy guaranteeing that, if followed, the result would be that the agent not see to it that Q. Attempts to demonstrate this intuitively, especially in the presence of infinities, always ran up against the need for rigor, and therefore had to await the development of the austere theory of strategies. Chiefly because we now have the favoring extension lemma in hand, we are ready for the application of the theory of strategies to the problem of proving the existence of a "strategy for inaction." (We certainly hope, however, that the favoring extension lemma finds more surprising applications.)

13E.1

Semantics for stit

At this point we need an exact statement of the semantics of [a stit: Q] as the achievement stit, which we restate. Recall from §2A and elsewhere that indeterminism requires that truth with respect to a BT + I+AC model, m, must be relative to moment-history pairs m/h, so that we write "971, m/h t= Q" for "Q is true in the model m at the pair m/h." The semantics for [a stit: Q] relies on the apparatus of "instants" described in §13B, which was not needed or wanted for the theory of strategies. Reminder (Def. 4): H(M) 'ls the set °f histories having nonempty intersection with M; i(mi) is the instant of time on

360

Strategies

which mi lies; m( J(m ) t h ) is the moment in which the instant i(mi) intersects the history h. The following is the concept of the truth of [a stit: Q] based on witness by a chain (instead of witness by only a moment); it is a merely verbal variant of §8G.4, with which it may be compared. 13-24 DEFINITION. (Truth of [a stit: Q]) Given that mi £ hi, we define that 97t, mi/hi 1= [a stit: Q] iff there is a chain of moments CQ, called a witness, such that the following conditions are satisfied. • Priority-nonemptiness. All moments in CQ must be properly earlier than mi, and CQ must be nonempty: CQ ^ 0 and Vmo[mo G CQ —>TOO< mi]. • Positive condition. Q must be settled true at every moment in i( mi ) that lies on a history intersecting CQ that is inseparable from hi for a in CQ: • Negative condition. Every moment, mo, in CQ "risks" the falsity of Q in the sense that there is above m0 some moment in i(mi) at which Q is not settled true. In other words, for every moment, m0, in CQ, it is not settled at m0 that Q be true at

13E.2

Abstraction of stit witness by chains

The rigorous result we are after goes something like this. Suppose that [a stit: Q] is true at a moment-history pair mi/hi, witnessed by a chain CQ. Fix mi, hi, and CQ. Then at every moment m0 in the witnessing chain CQ there is available a strategy such that following it really guarantees that [a stit: Q] fails at m( i(m j./io)/^) f°r suitable /i0. To prove this result, we need to provide a suitable interface between stit and strategy. From the stit side, we need much less than what we have. We surely need the agent, a, and the moment-history pair mi/h\. But we do not need all of the syntactic information in Q, nor a general semantic story. We need only whether the particular Q holds at each m(i( ),h)lh. This we can represent by a set of histories depending only on mi and Q: We also need a field of choice points for a. On the "stit" side, it must be large enough to include all the chains that could possibly witness any of the stits considered in the desired result. On the "strategy" side, since the strategy is to be "available" at mo 6 CQ, the field must include mo. The field, however, must not include any moments earlier than mo, which are evidently no longer "available" at mo. Let us take the set of all moments from mo onward that are temporally prior to mi, that is, that precede some moment co-instantial with mi: MI — {m: mo ^ m & m < i( mi )}. We record these two abstractions for later reference.

13. An austere theory of strategies

361

13-25 STIPULATION. (Two critical sets) Suppose that TOO, mi, and Q have been fixed, and thatTOO< TOI. • Hi = {h: m, m ( l ( m i ) i h ) //i N Q}. • MI = {TO:TOO^ TO &TO< i(mi)}. Next notice that /ii doesn't really come into the Definition 13-24 account of truth for stit. Consequently we can take TOI to come in only as a peg on which to hang the definitions of H\ and M\. If we resolve the general nonemptiness condition on the chain witness c in favor of a particular member TOO of c, we can state the remainder of the truth condition for stit as a relation between TOO, HI, MI, a, and an arbitrary history ho. We codify this "stand-in" for stit as a general definition of a five-place relation, stit-stand-in(mo, HQ, M0, a, ho). 13-26 DEFINITION. (Stit-stand-in) For any moment TOO, set of histories HO, set of moments M0, agent a and history ho, stit-stand-in(m0, H0, M0, a, ho) iff there is a c such that the following all hold. • Priority-nonemptiness stand-in.TOO • Positive stand-in. • Negative stand-in. The positive stand-in says this. Let h be any history intersecting c such that c makes no choice distinction between ho and /i. Then h G HO- That is why we can say that the choices made in c by a are such as to "guarantee" being in HQ. Or to put it contrapositively, if h G (H(c) —Ho) then 3m[m G cD/ioPI/i & /IQ _L^j /i]. The negative stand-in says that every moment m in c keeps departure from HQ as a live possibility. That makes c nonvacuous as a series—though it does not and should not follow that a has a nonvacuous choice at each member of c. The condition says that at every moment TO in c there is a "live possibility" of finishing outside of HQ. It does not say that there is a currently available "live option." It is important to see that stit-stand-in is a concept entirely in the language of the theory of strategies. For the upcoming result about stit-stand-in to have its intended significance, however, we must verify that the concept is satisfactory on the "stit" side, as follows. 13-27 FACT. (Stit and stit-stand-in) Fix a and TOI, and h\ such that TOI € h\. Assume that 9JT, mi/h\ N [a stit: Q], and let CQ be a witness (Definition 13-24). Fix TOO G CQ. Define HI and MI as in Stipulation 13-25. Choose any history ho G H(mo). Then 971, m,(i(m },hQ)/ho ^ [<* stit: Q] iff stit-stand-in(m0, HI, MI, a, ho).

362

Strategies

PROOF. It is almost straightforward that given the stipulations for HI and MI, the stand-in conditions are restatements of the truth conditions for [a stit: Q] at the moment-history pairs in question. There is just one delicacy in the part of the argument from left to right: The hypothesis promises a witnessing c for [a stit: Q] at m,(,(m )tho)/ho> but it does not say, as required for stit-stand-in(mo, HI, MI, a, ho), that mo 6 c. Nevertheless, since mo is assumed to be part of a witness for [a stit: Q] at m\jhi, we can be sure that if c witnesses [a stit: Q] at TO(i ,,h 0 )//iO) then so does cUJmo}. The reason is this. We know by the negative condition for the witnessing of [a stit: Q] at mi jh\ that Q fails at m(j ( m /i)//i for some h 6 H(moy Therefore, the chain c cannot be entirely earlier than mo on pain of violating the positive condition for the witnessing of [a stit: Q] at m(i( }tho)/ho- So some member of c must come properly after mo. But then adding mo to c keeps the negative condition true, and does not disturb the positive condition, so that cU{mo} is also a witness to [a stit: Q] at m(j ( m ),h0)/ho- And of course that set is bound to contain mo, as required for stit-stand-in(mo, HI, MI, a, h0). (There is a discussion in Belnap 1996a of what can be added to or subtracted from a chain witness.)

13E.3

Main result

We use the stit-stand-in abstraction from the concept of seeing-to-it-that as follows. We prove generally that regardless of H0 and M0, whether as just defined or not, we can always find available at mo a strategy for a such that if a follows it faithfully at the choice points in MQ, then for any history /IQ that could possibly eventuate, stit-stand-in(mo, HO, MO, a, ho) must fail. This is a result in the pure theory of strategies. Then we use Fact 13-27 to argue that there is always a strategy the following of which guarantees that [a stit: Q] is false at the moment-history pairs in question. 13-28 THEOREM. (Strategy for inaction) Fix mo, MO, HO, and a such that mo Then there is available at mo a simply complete strategy s for a in MQ such that stit-stand-in(mo, HQ, MO, a, ho) fails for every history ho admitted by s. PROOF. By the second part of Corollary 13-23, let s be a simple strategy for a in MQ available at mo that is complete in MO and that favors -H(M 0 ) ~Ho m MQ. Suppose s admits ho (and is therefore complete along /i0 in MO), and, for contradiction, that stit-stand-in(m0, H0, MO, a, h0) holds (Definition 13-26). By priority-nonemptiness let mo G c C /i 0 nM 0 . By the negative condition, so that by favoring, we know that we may let So choose By the positive condition, hi (hence in h0r\Mo, therefore in Dom(s)) such that hi ±^ ll ho. Here is the contradiction: ho and hi are both admitted by s, so must both belong to s(mi). But then s is not after all strict. Our main result is a direct corollary.

13. An austere theory of strategies

363

13-29 COROLLARY. (Main result) Suppose that 9Jt, mi/hi 1= [a stit: Q], with witness CQ. Then at each moment mo G CQ, there is available a simply complete strategy s for a in that really guarantees PROOF. Start with the hypothesis, and choose m0 € CQ. Define MI, #1 in accord with Stipulation 13-25. Since m0 e MiHljHi, by Theorem 13-28 there is available atTOOa simply complete strategy s for a in MI such that stit-standOT(TOQ, -Hi, MI, a, ho) fails for every history /i0 admitted by s. By Fact 13-27, that amounts to saying that s guarantees And the hypothesis allows us to calculate that s is not vacuous for the set of histories

since we can promote "guarantee" to "really guarantee."

For this reason,

14

Deontic kinematics and austere strategics Deontic concepts are naturally linked with strategic concepts.* Surely any obligation can be viewed as an obligation to follow a certain strategy in the austere sense of chapter 13, always making such choices, depending on circumstances, as conduce to satisfying the obligation. Conversely, to follow a strategy can be viewed as very like living up to a set of deontic requirements, "doing what the strategy requires." We do not here directly discuss such common-speech linkages. Instead we report a specific and detailed theoretical linkage that allows reciprocal illumination between certain deontic concepts and certain strategic concepts. An unexpected result is the connection between the no choice between undivided histories condition from the austere theory of strategies, and a new deontic kinematic condition adduced by Thomason to help describe how oughts fit into branching time. Deontic kinematics I. On the deontic side we invoke Thomason 1984, a fundamental essay that describes a family of structures carrying information about (z) oughts-at-moments in (ii) branching time. The branching mentioned in (ii) represents that some things might happen that are not guaranteed to happen. In other words, there are inconsistent possibilities for the future. In particular, an existing obligation might be fulfilled, or might not. The purpose of relativizing oughts to moments as in (z) is simple and pervasive: One knows that oughts come to be and pass away. For example, an obligation can be created by a promise; for another example, a "reparational obligation" can be created by breaking a promise. One asks for general principles governing how such event-located oughts fit together in our world. This is the topic of "deontic kinematics." (Thomason 1984 calls the topic "ought-kinematics.") Deontic kinematics is an enrichment of the "static" or "timeless" versions of the deontic concepts that are prevalent in the literature. 'With the kind permission of Filosophia, this chapter draws on Belnap 1996b.

364

14. Deontic kinematics and austere strategics

365

Austere strategics I. The strategic concepts we use arise in the theory of agents making choices in branching time, BT + AC theory, §3. In earlier chapters we explored stit theory, which presupposes BT + AC or BT + I+AC theory; and then in chapter 13 we considered the austere theory of strategies. Austere strategics sits on the same foundation as stit theory, which is to say, upon the theory of choices in branching time; but this theory of strategies is not semantic and does not itself depend on the semantics of stit. We describe the theory as "austere" because it involves none of the psychological or value concepts typically taken as underpinnings for the concept of strategy. In fact, the austere theory of chapter 13 simply identifies a strategy as a pattern of choices in branching time, neglecting entirely the mental work that goes into adopting a strategy, and equally neglecting what a strategy is "for." It is surprising how rich and also clarifying such a minimal theory can be. Deontic kinematics and austere strategics both "take time seriously": As we see, each has a key postulate that in the former vertically integrates oughts across time, and in the latter vertically integrates choices. Our most striking result exhibits the interplay between these two apparently disconnected ideas.

14A

Basic concepts

First we review the underlying concepts.

14A.I

Branching time

We begin by noting with maximum brevity what is common to the two theories, namely, branching time. Deontic kinematics and austere strategics follow the lead of Prior 1957 and Thomason 1970 in representing the indeterminism of our world by means of a BT structure (Tree, ^}. That is, deontic kinematics and austere strategics have the Prior-Thomason theory of branching time as a common part. In this book the postulates and definitions of this theory are briefly described in §2A, more extensively discussed in chapter 7, and summarily presented in §3. See the latter for help with the notation.

14A.2

Deontic kinematics II

Thomason 1984 arrives at the theory of deontic kinematics by adding a deontic primitive "0" to the primitives Tree and ^ of branching time. He calls the resulting structures "treelike frames for deliberative deontic tense logic." Here, for our immediate purposes, we substitute the phrase "deontic tree." 14-1 DEFINITION. (Deontic tree) A deontic tree is a triple (Tree, ^, 0), where (Tree, <} is a branching time structure and where 0 is a function from moments to sets of histories (maximal chains) satisfying the following conditions for all m G Tree.1 1

Thomason 1984 takes < instead of < as primitive, and does not give names to the three conditions.

366

Strategies

i. Consistency. O(m) ii. Locality. 0(m) Hi. Kinematic. The idea is that 0(m), which we may think of as a proposition, represents what ought to be as of moment m—a proposition true in all and only those histories deontically acceptable at m. The locality condition (ii) says that deontically acceptable histories must be drawn from among those histories that remain possible at m. The contrast in Thomason 1984 is with histories that might have been but are no longer possible, and that therefore could support a nondeliberative (judgmental) use of "ought," for example, "Although it is now impossible, nevertheless I ought to pay back my debt." Thus (ii) signals that the present topic is limited to future-oriented "deliberative" oughts. In the context of (ii), the consistency condition (i) is a local consistency condition saying that at least locally, "ought to be implies can be." The kinematic condition (Hi), new to the literature in Thomason 1984, is offered as a condition explicating the coherence of plans (p. 155). We shall return to it. The most interesting result of this chapter concerns this kinematic condition. The principle says that if some history that is deontically acceptable at an earlier moment mo can take you to a later moment mi, then there is a simple relation between your later and your earlier obligations: Given that a history h passes through the later moment mi, h is acceptable at the later moment mi if and only if h is already acceptable at the earlier moment mo-

14A.3

Austere strategics II

Let us review the relevant portion of the austere theory of strategies with equal brevity. Chapter 13 built this theory on BT + AC theory, for which we refer to §3. We repeat the following series of definitions, which are lifted from chapter 13, since they constitute the key concepts from the austere theory of strategies on which we shall rely. 14-2 DEFINITION. (Strategic concepts: basic) • s is a strategy for a iff s is a (partial) function on moments such that for m 6 Dom(s), s(m) is a subset of H(m) that is closed under choice equivalence for a at m: s(m) C #(m), and Vh[h G s(m) —> Choice'^(h) C s(m}\. • s is available at m iff m € Dom(s). • For m S Dom(s), s is consistent at m iff s(m) ^ 0, and s is consistent iff s consistent throughout its domain. • For m 6 Dom(s), s is vacuous at m iff s(m) = H(my

14. Deontic kinematics and austere strategics

367

• A strategy s is total [in M] iff Dom(s) •=• Tree [= M]. These ideas are explained at more leisure in §13C. Recall that we represent a choice as a set of histories. The key idea is that a strategy tells you which of your available choices count as following the strategy—and which do not. A strategy that is inconsistent at m explicitly tells you that no history available at m counts as following the strategy (there is nothing good to do). At the other end of the spectrum, a strategy that is vacuous at m says that everything goes at m. A strategy is total if it gives you advice at every moment in the entire tree. 14-3 DEFINITION. (Strategic concepts: admission and exclusion) • s admits h iff Vrn0[m0 G Dom(s)r\h —> h G s(m0)}. • Admh(s) — {h: s admits h}. • s admits mi iff Vm0[(m0 G Dom(s) & mo < mi) —> mi e U s ( TO o)]• Admm(s) = {m: s admits m}. • s excludes a history or a moment if it does not admit it. The admission and exclusion concepts on this list, drawn from Definition 13-4, are of critical importance for this chapter. Let us start with exclusion, which is defined (with inescapable confusion) for both moments and histories. A moment is excluded by a strategy if never violating the strategy implies that you will never reach that moment. A history is excluded by a strategy if never violating the strategy implies that eventually you will fall off that history. Since admission is simply the contradictory of exclusion, a moment is admitted by a strategy if it is possible to arrive there without violating the strategy, and a history is admitted by a strategy if staying on that history never requires that you violate the strategy. 14-4 DEFINITION. (Strategic concepts: simplicity, completeness, etc.) • For m G Dom(s), s is strict for a at m iff s(m) G Choice^, and s is strict for a iff s is strict for a throughout its domain. • s is secondary iff some moment that it excludes is nevertheless in Dom(s); and s is otherwise primary. (This dichotomy arises from the following: The dictates of s are primary if they concern moments that are admitted by s, but secondary when they concern moments that are excluded by s.) • s is a strategy for a in M iff s is a strategy for a such that Dom(s) C M. • When s is a strategy for a in M, we call M a field for s. • Suppose that s is a strategy for a in M. s is backward closed in M iff

368

Strategies • s is simple for a in M iff s is a strategy for a in M that is strict for a, primary, and backward closed in M.

• s is complete along h in M iff Mfl/i C Dom(s). • s completely admits h in M iff s admits /i and is complete along h in M. • s is H-complete in M iff s is complete in M along every history in H that s admits. • s is complete in M iff s is complete in M along every history that s admits. • s is simply complete for a in M iff s is simple for a in M and s is complete in M. This last group of definitions supports a rich theory of simple strategies. For example, a major fact proved in chapter 13 is that every strategy for a in M that is simple can be extended to a strategy that is simply complete for a in m (Corollary 13-23). We mention this here, however, only for contrast, since we will not be dealing with simple strategies at all. And for the same reason, we refer to chapter 13 for explanations of these concepts: In this chapter, we occasionally mention some of these concepts for contrast, but we do not really use them.

14B

From Thomason's deontic kinematics to austere strategics

To think about oughts and strategies together, zero in on a single agent, a, and think of Tree as both supporting 0(m] representing what ought to be at m and also supporting a choice-function Choice^ defining what a can do at m. What, if anything, would then be wrong with thinking of the deontic tree itself as a candidate account of a strategy for a? That is, what would be wrong with taking 0 as a strategy? A great deal. From the agency point of view, the deontic tree would have the defect that it omits consideration of what a can do at m. This defect seems to us so substantial that it calls into question the propriety of using the adjective "deliberative" in connection with a deontic tree. Such a tree could not be a strategy for a, since there is no requirement that 0(m) be closed under choice equivalence for a at m. 0(m) picks out histories containing m, but not choices available to a at m. Strategies ("deliberative" strategies) tell you what to do; 0 does not—at least not directly. Perhaps there is an indirect route from 0 to strategic advice. Let us then ask: How to make a strategic choice at m given a particular 0(m)? Consider a possible choice for a at m, say H (H will be a subset of all the histories H(m) to which m belongs). If H contains only histories lying outside of 0(m), certainly the strategy should steer away from H. And if H contains only histories acceptable to 0(m), then with equal certainty the strategy should

14. Deontic kinematics and austere strategics

369

permit the choice H. But what if H is "mixed," containing both some histories acceptable to 0(m) and others that are unacceptable to 0(m)l There are then evidently two salient strategic policies available, neither happy. The "weak" policy says that choosing H, even though mixed, is in accord with the strategy. The "strong" policy says that choosing a mixed H violates the strategy. There is also a third policy, itself not entirely happy. The "Hamblin" policy, suggested in Hamblin 1987, makes it depend upon what other options there are, as follows. Let H be mixed. If there is at least one possible choice that contains only histories acceptable to 0(m), then the Hamblin strategy forbids choosing such a mixed H (like the strong policy). But if there is no possible choice containing only histories acceptable to 0(m), then the Hamblin strategy permits choosing mixed H. There is no point moralizing about this here; we simply put the matter in terms of clear definitions about which something definite can be said.2 14-5 DEFINITION. (Deontic trees as strategies: sweak,a,o and sstrong,a,o and SHambhn,a, o) Suppose that we are given a deontic tree (Tree, ^, 0) together with a choice partition Choice1^ for a at every moment m. We define the weak, the strong, and the Hamblin strategies for a determined by this deontic tree, written respectively as sweak^a, o, sstTong,c>, o, and sHam,bhn,a, o, by specifying the domain of each function as the same as the domain of 0—namely, the entire tree—and continuing as follows. • sweak,a,o(fn)

=

• sstrong,a,o(m) = • SHambhn,a, o("i): If there is a possible choice H for a at m (that is, H such that then otherwise, sHambhn,a, o(m) We can now intelligibly speak of a deontic tree as in effect one of the strategies {sweak,a, o, sstrong,a, o, sHamblin,a, o}, and ask what properties it may have. 14-6 FACTS. (Deontic trees as strategies) • For S € {Sweak,a, O, Strong,a, O, SHamblin,a, o}, s(m)

is a Subset of H(m)

that is closed under choice equivalence for a at m. That is, each of {sweak,ce,o, strong,a, o, sHamUin,a, o} is a strategy for a. None of the conditions (i)-(iii) of Definition 14-1 play a role here: The form of Definition 14-5 suffices.

• Since 0 is total on Tree, so is each of the three strategies sweak,a, o, sstrong,a,o, and Sffambim,a, o- Each is available at every m in Tree. Each strategy is ferociously "complete." 2

Horty 2001 shows that there is much more to be said in the context of an apparatus involving utilitarian degrees-of-value comparisons. Here, however, we have only the deontic O.

370

Strategies • The consistency condition (z) of Definition 14-1 implies that sweak, a, o and SHambiin,a,o are consistent. In contrast, the strong cousin s s tr<mg,a,o may well be inconsistent even though the consistency condition on 0 is satisfied. Suppose, for example, that the consistency condition is satisfied at m by means of just a single history h through m that is acceptable to 0(m). If the possible choice for a at m that contains h also contains an unacceptable history, then according to sstr0ng,a, o, that choice is not open to a. The strategy s s t r ong,a,o tells a that at m every option is forbidden. That is, sstr0ng,a,o 1S inconsistent. • Any member of {sweak,a,o, sstrong,a,o, sHa.mbiin,a,o}, except in trivial cases, is bound to be secondary. This feature distinguishes any strategy based on a deontic tree from any simple strategy. The positive role of secondariness in the deontic-kinematic framework is to permit consideration of reparational obligations, obligations that come into force after some earlier obligation has been violated. We underline that the feature is neither good nor bad; but one should bear it in mind. Primariness/secondariness is the most significant difference between simple strategies and strategies that are derived from deontic trees. • If S G {sweak,a, O, Sstrong.a, O, SHamblin,a, o}, S is total, 3S W6 Said. So it

is natural to expect that at some moments m, s is close to being undefined; i.e., s lays on a at m only a vacuous obligation, the set of all histories containing m: s(m) = H(my This is not primarily a technical point: Any strategic advice that covers every moment in Tree must be largely vacuous.

• A strategy s € {sweak,a,o, sstrong,a,o, SHambiin,a, o} derived from a deontic tree is therefore never simple: Although s is bound to be backward closed (in any M), it will never be both strict for a and primary. Notably missing from this account is any reference to the kinematic condition (in) of Definition 14-1. What does the kinematic condition mean? Thomason 1984 connects the notion with the idea of a "coherent plan." But there seems nothing incoherent about a strategy derived from a deontic tree, whether or not condition (Hi) is satisfied. This suggests that we have not found the best way of relating austere strategics and deontic kinematics.

14C

From austere strategics to Thomason's deontic kinematics

We now start from the other end. We start with austere strategics as given, and ask what we can make of deontic kinematics in this context. It turns out— unexpectedly—that here is where we find the most interesting results. These are the results that seem to shed the most light both on austere strategics and on deontic kinematics.

14. Deontic kinematics and austere strategics

371

So suppose we start at the other end, with a strategy s. One might think of the histories S(TO) as what a "ought to do" at TO. Suppose we were then to turn things around and define 0(m) as s(m). We should learn that conditions (i) and (ii) of Definition 14-1 would be satisfied at least when s is consistent. We should also find out that condition (lii) is something that does not readily spring to mind. Mutual illumination seems again absent.

14C.1

Admission onward and Os

There is, however, another idea that begins with strategies. This idea pays attention to a recurring theme in the deontic literature: The histories defining obligation should in some sense be "ideal." At least they should be "ideal" to the extent that in these histories no one ever violates an obligation. If we take the given strategy s as the source of obligations, then we have already defined these histories. They are the admitted histories (Definition 14-3). In an admitted history a never departs from the strategy. This never-failing agreement with the strategy is evidently "ideal." Therefore admission, referring as it does to a generality of moments, would appear as a concept of deontic kinematics. We might therefore think about defining 0(m) as Admh(s)r\H(mj so as to include the element of ideality. Observe that since it involves quantifying over moments, admission is essentially a "global" idea, not bound to a particular moment. Admitted histories are acceptable to the strategy, and are "ideal" as far as the strategy goes. None of them can be improved upon, as far as strategy-following goes. So we can talk deontically about what is true in all "ideal" histories, that is, the admitted ones, those in which the strategy is never violated.3 The idea of defining obligation in terms of admission will not do, however, for every case of a deontic tree. And that is because some of these trees are designed to allow intelligible "reparational obligations." In such a tree there can be things that you should do at a certain moment m even though those very things were excluded by earlier obligations. (You only reached TO because you did what you ought not to have done.) The suggestion to define 0(m) as Admh(s)C\ H(m) might be all right if we were only concerned with simple strategies. In the reparational case at hand, however, we should have 0(m) = Admh(s)r\ H(m) = 0, which would represent an inconsistency: The consistency condition, Definition 14-1 (i), on 0 would not be satisfied. This definition cannot, then, offer advice to those who have ever, even once, made a strategic mistake. If this seems too abstract, any of the old tales will do to make the point. You ought atTOOto visit your mother—perhaps because of a promise—and you have adopted that as your strategy. But atTOOyou choose not to catch the plane to visit your mother. You find yourself at TOI, not having caught the plane. At TOI you must choose whether or not to call your mother to tell her that you will not be coming. Both of these histories through TOI, however, are such that on them you have not visited your mother. They are therefore both excluded 3 We say that admitted histories are "acceptable" rather than "permitted" because we think that permission is an essentially agentive concept and therefore not applicable to individual histories. Only choices for a can be permitted—or not—for a.

372

Strategies

(not admitted) by your strategy to visit your mother. Your strategy therefore excludes all alternatives through mi. Your strategy is inconsistent at mi. What has gone wrong? This: One wants for the reparational or secondary situation a concept that gives a a "fresh start." That is why taking admission as an absolute character of histories did not work. It is better to invoke the idea of a history that is "admitted by s from m onward." In other words, the history may have been excluded earlier, but don't look back: From m onward it forever after counts as ideal. In still other words, past failures don't count. As one might expect of a purely deliberative idea, only the future of possibilities is relevant. Here is the definition. 14-7 DEFINITION. (Admission onward; Admh(s, m)) let mo € Dom(s). • s admits h from mo onward iff mo

Let s be a strategy, and and

• For any strategy s and moment mo, Admh(s, mo) is the set of histories admitted by s from mo onward. In applying this definition, we run into the contrast that strategies can be partial while deontic trees are always total. For present purposes we therefore choose to concentrate on total strategies. It is easy to see that from the point of view of admissibility, nothing is lost. Given any strategy SQ we can always consider instead the total strategy si obtained by vacuously defining Si(m) = H(m) for every m & (Tree —Dom(so)). (It cannot be inferred, however, that nothing is changed tout court; obviously total strategies are not simple at all, so that exclusive concentration on them would put potholes in the road to inquiring about the properties of simple strategies.) The upshot is that in the context of reparational obligations, given a total strategy s, Admh(s, m) seems a good candidate for the concept 0(m) of deontic kinematics. These are the histories that, having reached m, are and will remain "ideal" as far as the strategy goes. They are the histories through m on which (although obligations may have been violated earlier) no more violations occur. Sounds "ideal," doesn't it? Let us record the suggestion in a definition. 14-8 DEFINITION. (Strategies as candidate deontic trees: Os) Suppose we have a choice tree (Tree, ^, Choice). Given a total strategy s for a, we define the candidate deontic tree determined by s as (Tree, ^, Os), where for m e Tree, Os(m) = Admh(s, m). Here is a decisive observation: Os is not a strategy and should not be confused with a strategy. A strategy s picks out choices at m, whereas Os can pick out some individual history within a possible choice for a at m, while excluding other histories in that same choice. Such ideal histories are important; but one must also keep in mind that no agent can choose the set of such histories. What can we say about the so-induced candidate deontic tree, especially with respect to conditions (i)-(iii) that Definition 14-1 laid on deontic trees?

14. Deontic kinematics and austere strategics

373

The following sections report that all of these conditions—(i) consistency, (ii) locality, and (Hi) kinematic—fall into place given Definition 14-8. There you have it: Think of deontic kinematics as generated by austere strategics, and each of the deontic-kinematic conditions is illuminated. This is especially helpful for appreciating the least obvious of these conditions, the kinematic condition (Hi). We start at the first of the deontic-kinematic conditions.

14C.2

The consistency condition on deontic trees

Of course if s is inconsistent, Admh(s, m) can be empty. That's trivial. The interesting question concerns the consistency (nonemptiness) of Admh(s, m) given that s is consistent. We shall find that the consistency of Admh(s, m) is by no means guaranteed. But we shall also locate an unexpected condition under which consistency of Admh(s, m) is restored: Namely, it makes a difference whether or not there are busy choosers, Def. 14. 14C.2.1

Inconsistency of Os given busy choosers

The first relevant fact is that Admh(s, m) can be empty even if 5 is consistent, if one allows a to be a busy chooser. 14-9 FACT. (Consistency condition on Os, Definition 14-1, with busy choosers) There is a total consistent strategy s for a such that Os is nevertheless inconsistent. PROOF. Here is a (peculiar) case in which a particular Admh(s, mu) is empty, even though each s(m) is nonempty. We need to look at an example of "feathering" of histories (see Figure 14.1). Suppose after some moment mw there is an infinite chain of moments mo > mi > m^ > ... > ml > ml + i > ... descending toward mw (mw is not part of the sequence, but mw is the greatest lower bound of the sequence). We suppose that s offers a vacuous choice for a at TOO and also at every other moment other than TOI, 7712, ..., mu. At mu and at each member ml of the chain, i > 0, there is, however, a binary choice for a. (Since there are infinitely many nonvacuous choices for a between m^ and mo, a is by Def. 14 a busy chooser.) Going left stays in the chain, going right leaves it. The strategy at mw says: Go left (stay in the chain), that is, s(mu) = the set of all histories passing through some mz, 0 < i. But after m^ and before TOO, s always prescribes going right ("leave the chain"). So Admh(s, mu) is empty even though s(m) is nonempty for every m. To verify this, pick any "right" history, hi, intersecting the chain atTOZ,i > 0. You will find that ht is excluded by some (indeed every) moment that is strictly between TOW and TO;. For example, you can see from Figure 14.1 that h% is excluded at ma. Furthermore, the "leftmost" history hu, which containsTOO,is excluded by every m-i that is earlier than mo and later than mu. So every history through mw is excluded at some point at or after mu. None of the histories through ma is admitted from mu onward. None of them is "ideal." So Admh(s, mu) = 0. Which is to say, Os is inconsistent at mw, even though s is everywhere consistent. D

374

Strategies

Figure 14.1: Consistent s does not imply consistent Os

It does not appear that the strategy pictured in Figure 14.1 is any more incoherent than any other more "finite" reparational strategy. The appearance is more that of a merely set-theoretic construction without intuitive content. But perhaps that appearance is only due to our reluctance to think about the possibility of busy choosers; it's hard to be sure. 14C.2.2

Consistency of Os given no busy choosers

The strange counterexample is not, however, the end of the story about how consistency of s connects with consistency of Os. We may add the following: Absence of busy choosers is quite enough to provide the link between consistency of s and consistency of Os. 14-10 FACT. (Consistency condition on Os (Definition 14-1 (i)), given no busy choosers) Let s be a total consistent strategy for a. Then Os (m) is nonempty— and the consistency condition (i) of Definition 14-1 is satisfied—provided a is not a busy chooser, Def. 14. PROOF. Suppose that we have a choice tree (Tree, <, Choice) with no busy choosers. Suppose that s is a total consistent strategy, that is, that s(m) is invariably nonempty. Choose any mo € Tree. We show how to find a history in Os(mo), that is, in Admh(s, mo). Define as follows a countable sequence of moment-history pairs (mi,ht), 0^1 The first moment m0 is given. At this and any stage, choose ht as any history in s(m;). It may be that /it € (Admh(s, ml)). In this case the sequence terminates in (mi, hi). Otherwise, {m: m € hi & ml < m & hi. £ s(m)} ^ 0. Because there are no busy choosers, this set cannot contain an infinitely descending sequence, that is, it must have a least member.

375

14. Deontic kinematics and austere strategics

Define ml + i as that least member. Now repeat the definition of (ml,hl) unless the sequence terminates. Either the sequence (mo, ho), (mi, hi), ... terminates or not. If it terminates in (ml,ht), then ht & Admh(s, mo) (argument omitted). Suppose it does not terminate. Consider the infinite sequence of moments mo < mi < . . . . It cannot be upper bounded because there are no busy choosers. So, since it has no upper bound, that sequence of moments must define a unique history hw that contains all its members. We argue that hw <E Admh(s, m0). To this end, suppose mo ^ m and m e hu; we need to show that hu € s(m). There must be an i such that ml ^ m < m,+1. We know by choice of ht that hl £ S(TOJ); and since mz + 1 was chosen as the earliest excluder for ht, it must also be that hl € s(m). Also we have that m t +i € (hur\h^)\ so, since m < m, + i, hu =m hl. So by no choice between undivided histories, Post. 8, hu =5^ ht. Therefore, since we have established that hl 6 s(m), we may conclude that hu € s(m) as well. D Where are we? Without busy choosers, Os is bound to satisfy the consistency condition if s does; but if there is a busy chooser, the consistency of s by no means guarantees the consistency of Os. So we now know all there is to know about the nonemptiness condition on deontic trees Os (Definition 14-l(z)) induced by a total consistent strategy s (Definition 14-2).

14C.3

The locality condition on deontic trees

Next we consider the second condition on 0. 14-11 FACT. (Locality condition on Os (Definition 14-l(ii))) strategy for a. Then Os(m) C H(m).

Let s be a total

We record this fact, but it brings no illumination. It is trivially provable of Admh(s, m), and is independent of the requirement on strategies s that s(m) C H(m).

14C.4

The kinematic condition on deontic trees

We now turn to the most interesting of the deontic conditions Definition 14-1, namely, the kinematic condition on deontic trees (Definition 14-l(iii)). What we now show is considerably less obvious: The difficult kinematic condition on deontic trees can also be proved for Os. 14-12 FACT. (Kinematic condition on Os (Definition 14-l(iii))) Let s be a total strategy for a. Then if mo < m\ and H^mi)!~}0s(mo) ^ 0 then Os(mi) = Os(m0)r\H(mi). That is, if ro0 < mi, and if H(mi)r\Admh(s, mo) ^ 0, then Admh(s, m\) = Admh(s, mo)nJf( m i ). What is interesting about the proof is that it depends essentially on the principle of no choice between undivided histories, Post. 8, that hi =mo h2 implies hi s^0 h2.

376

Strategies

PROOF. Given the hypotheses of Definition 14-1, it is trivial from the definition of Admh(s, m) that (Admh(s, m 0 )nH( m i )) C Admh(s, mi). This containment has nothing to do with anything except the way we explained "ideality": We quantified over bits of strategic advice about the future. The other direction is what is interesting: Admh(s, mi) C Admh(s, m 0 )nH( TOl ), under the hypotheses. We are given that mo,mi G ho, that mo < m1, and that ho G Admh(s, mo). To show the desired subset relation, we suppose hi in Admh(s, mi). We need to show hi G Admh(s, mo). To show that, pick m? such that mo ^ m2, with m,2 G (Dom(s)nhi). What is needed is that hi G 5(7712). It is given that all of mo, mi, m2 lie on hi, with mo < mi and mo ^ 7712- Where is m2 relative to mi? If mi ^ m2, then the hypothesis that hi G Admh(s, mi) guarantees that hi £ 5(7713). So take the other case, m2 < mi. Note the proper less-than. Now it all falls together. We know that ho (don't forget ho) and hi share a point, namely mi, that is properly later than m,2. That is to say, ho and hi are undivided at 7712: ho = TO2 ^1- Therefore, by the no choice between undivided histories condition, it must be that ho and hi are choice equivalent for a at 7712: ho = TO2 hi. But we know that ho € 5(7712), since ho G Admh(s, mo) and since mo ^ 77i2. So by the constraint of Definition 14-2 on s that 5(7712) be closed under choice equivalence for a, we have that hi G 5(7712) as well. D We may sum up most of these facts as follows. 14-13 THEOREM. (From choice trees to deontic trees) Consider any choice tree (Tree, ^, Agent, Choice}, and any agent a who is not a busy chooser, Def. 14. Let s be an arbitrary total consistent strategy for a (Definition 14-2). Define O as the total function on Tree such that for each m G Tree, 0(m) = Admh(s, m) (Definition 14-7). Then (Tree, ^, 0) is a deontic tree satisfying all of the conditions laid down by Thomason 1984 (Definition 14-1). The deepest point is that satisfaction of the kinematic condition, which appears altogether normative, arises here out of the entirely nonnormative fact that although one may scheme, calculate, reckon, or even intend tomorrow's choices, it is literally impossible to make those choices today (no choice between undivided histories; no choice before its time). We conclude with a series of remarks.

14D

Remarks

It is clear that Os(m) is not very similar to s(m). Each may well have a place, and they are closely connected, but they are different. The concept s(m) is strictly local, but Os(m) is'partly global. It is less clear how the kinematic condition, Definition 14-l(ra), on deontic trees, and perhaps equivalently the no choice between undivided histories condition on choices in branching histories (Post. 8) are related to the "coherence" of plans (Thomason 1984, p. 155). Perhaps they are; it's just the "how." Both, if they are indeed connected in the fashion indicated in §14C.4, have to do

14. Deontic kinematics and austere strategics

377

with plans that involve attending to "can do," which sounds like coherence or consistency. But not necessarily in the exact way that one might think. When we prove that Os satisfies the kinematic condition, we do so for an arbitrary consistent total strategy. Even for the weirdest such you can imagine. Therefore, satisfaction by Os of the kinematic condition can never give us a reason to judge one total consistent strategy better than another. On the other hand, it certainly sounds as if coherence is a proper ground on which to evaluate a strategy. This evident tension strongly suggests the existence of deeper issues that need clarification. Let us add that when one considers primary (nonreparational) strategies, then the obligation concepts probably go differently. For one thing, there is then no essential difference between Admh(s, m) and Admh(s). In primary strategies, ideality does not depend on where you are. In conclusion, the following evaluative remarks appear warranted. First, deontic kinematics as defined in Thomason 1984 earns additional respect by making useful connection with the austere theory of strategies. Second, and conversely, the austere theory of strategies merits our appreciation for its ability to relate in an interesting way to that very theory of deontic kinematics. And we should prize both theories, simple as they are, for helping us to get a handle on how normative ideas fit within the real-world causal order.

This page intentionally left blank

Part VI

Proofs and models

This page intentionally left blank

15

Decidability of one-agent achievement-stit theory with refref As presented in earlier chapters, stit theory starts with the stit sentence [a stit: A] (read "a sees to it that A"), where a is a term for agents and A is any sentence.* Based on BT + I + AG structures (§2), [a stit: A], as an achievement stit (§8G.3), is interpreted as saying that A is guaranteed true by a previous choice of a. This chapter considers a subset of the BT + I + AC structures, namely, those that contain no "busy choice sequences" in the sense of Def. 14: • No busy choice sequences. There is no sequence of infinitely many nonvacuous choices for any agent occurring within a finite time; that is, no such sequence is bounded by moments both above and below. We say that any BT + I + AC structure with no busy choice sequences is a BT + I + AC + nbc structure (§2). In BT + I + AC + nbc structures, doing is equivalent to refraining from refraining from doing. (For a discussion of whether doing is in fact equivalent to refraining from refraining from doing, see §2B.6.) That is to say, the refref equivalence (Ax. Cone. 1) (refref)

is valid in all BT + I + AC + nbc structures. Figure 2.12 and its accompanying text suggests part of the proof of this. We will give a full proof in §15C. The converse, adumbrated via Figure 2.13, that refref is valid for a BT + I + AC structure only if it contains no busy choice sequences, is proved in chapter 18. Accordingly, we prove the existence of a decision procedure for the set of all single-agent sentences that are valid for all BT + I + AC + nbc structures. 'With the permission of Kluwer Academic Publishers, this chapter draws on Xu 1994b.

381

382

Proofs and models

Although in this chapter we restrict ourselves to a single agent, the validity of refref for those structures holds even with many agents. The decidability of achievement-stit theory with a single agent and the refref equivalence will be achieved by presenting an axiomatic system, Lai + rr, in §15A, with soundness proved in §15C, and completeness and the finite model property proved in §15G.1 Our results are grounded on a notion of "companions to stit sentences," which will be introduced in §15B. The syntactic features of "companions," "companion sets," "alternatives," "counters," and "semi-ref-counters" will be discussed in §15D-§15F. Another result will be obtained simultaneously with the completeness of Lai + rr: Restrict attention to BT + I + AC + nbc structures. Then every satisfiable sentence is satisfiable in a model in which at every moment every agent has at most two choices.

15 A

Preliminaries

We are considering achievement-stit theory with only a single agent. So all sentences are constructed from propositional variables, using truth-functional connectives ~ (negation) and & (conjunction), and the connective [a stit: ], with a as the only term for agents in our language. A stit sentence is any sentence of the form [a stit: A}. As usual, V (disjunction), D (material implication), = (material equivalence), T (truth) and _L (falsity) can be introduced as abbreviations. In addition we introduce the following technically useful abbreviation: (A, but a does not see to it that A). We will use A, B, C, and so on to range over sentences, and F, S, II, and so on to range over sets of sentences. Although for current purposes we admit only one agent term, we present our semantic account in terms of arbitrary BT + I + AC + nbc structures & = (Tree, ^, Instant, Agent, Choice), §2, in order to harmonize with the rest of this book. We use the postulates of §3 and definitions from the rest of the appendix, adapted to BT + I + AC + nbc structures when necessary, for example, for the meaning of "model" and the range of the structure variable 6 and the model variable 9JI. Because of the mathematical emphasis of this chapter, we tighten up on rigor in the matter of agents: Whereas elsewhere we use "a" sometimes as ranging over Agent and sometimes as ranging over the special agent-denoting terms, here we always let a be an agent-denoting term, with a its semantic value, 3(a). Furthermore, in this chapter we let a (rather than a) range over Agent. Let us first present some facts to be used tacitly in subsequent discussions. 15-1 FACT. (Settled truth of achievement stit) m, m/h \= [a stit: A] iff 9JI, m N [a stit: A].

For any m and h with m € h,

1 "Lai + rr" is intended as mnemonic, as follows: Logic of the achievement stit with 1 agent and the refref equivalence. See Ax. Cone. 3 for the list of axiomatic systems for which we employ such mnemonics.

15. Decidability of one-agent achievement-stit theory with refref

383

We can therefore ignore the difference between truth at a moment-history pair and settled truth at a moment for a stit sentence, [a stit: A], considered in isolation—while keeping in mind that we cannot ignore that difference either for its part, A, or for certain compounds containing it, especially tense sentences, which we do not treat in this chapter. 15-2 FACT. (Uniqueness of witness) Let m, m N [a stit: A], and let w be a witness to [a stit: A] at m. Then w is the witness to [a stit: A] at m. This second fact is a corollary of Chellas's witness identity lemma, Fact 10-1. As a special case, this fact can also be proved by applying Lemma 15-4. 15-3 FACT. (Truth of stit sentences at choice-equivalent moments) Let 971 be any model in which w < m, and let i = i(m) and M — {m1: m' € i & m' =^ m}. OH, m N [a stit: A] with witness w iff 971, M N A and 971, i\>w F A iff 371, M 1= [a stit: A] and 371, i\>w ¥ A. Hence, when w witnesses [a stit: A] at m, w also witnesses [a stit: A] and [a stit: [a stit: A]] at any m' choice equivalent to m for a at w. Let & = (Tree, \<, Instant, Agent, Choice) be any BT + I + AC structure, and m any model on G. We say that © (or m) is finite if Tree is finite. Let a E Agent be any agent in 6. A busy a-choice sequence in 6 (or in m) is a nonempty chain BC of moments in 6 satisfying the conditions upper-lower boundedness, that is, 3w3mVw'[w' e BC —> w ^ w' < m], forward denseness, that is, Vw[w e BC —* 3w'[w' 6 5(7 & w < w'}}, and nonvacuous choices, that is, Vtu[w 6 BC —> Choice^ ^ {H(wj}}.2 A &MSJ/ choice sequence in & (or 971) is a busy a-choice sequence in & (or 971) for some agent a in ©. It is easy to see that & is finite only if there is no busy choice sequence in &, but the converse does not hold. Let K be the set of all sentences A such that & 1= A for each BT + I + AC + nbc structure ©, and let Lai + rr be the axiomatic system that takes as axioms all substitution instances of truth-functional tautologies, refref, and the following schemata:

2 The idea of a busy choice sequence is listed with a some somewhat more general definition in Def. 14. A formal study of busy choice sequences can be found in Xu 1995a. Note that the forward denseness condition is only a particular way in which a chain can be busy. We only consider forward denseness because when considering the truth values of sentences in our language, only forward denseness is relevant. One can find, however, other issues in which backward denseness is more crucial. See, for example, Fact 14-9.

384

Proofs and models

and takes as rules of inference modus ponens and RE From A = B to infer [a stit: A] = (a stit: B\. The definitions of theorems (I-), consistency and maximal consistent sets are as usual, keeping in mind that in this chapter all these concepts refer to Lai + rr. Let A be any sentence and E any set of sentences. A is a deductive consequence of E, written E h A, if h (A1 &...&;Ak D A) for some >4i, ..., .Afc e E. E is deductively closed if j4 € E whenever E h .A. The deductive closure of ,4 will be denoted by DC(A). Identifying Lai + rr with the set of all its theorems, we show in the following sections that K = Lai + rr, and that Lai + rr has the finite model property: For every sentence A, V- A only if SOT ¥ A for some finite model SDt. This property will be obtained simultaneously with the completeness of Lai + rr. For the convenience of our upcoming discussions, we present some theorems and derived rules in Lai + rr. Note that all the derived rules and all the theorems TT1-19 can be proved without applying refref.

15. Decidability of one-agent achievement-stit theory with refref

15B

385

Companions

In this section, we will consider some basic semantic features of so-called companions to stit formulas, which we will not define until we obtain the companion theorem (Theorem 15-11). But it will be helpful to present a description of it at the beginning so that the reader may have some idea of where our discussion is going. Let 9m be any BT + I + AC model such that m, m t= [a stit: A] with witness w, and let M = {m1: m' € i(m) & m' =° m}. Clearly, m, M N [a stit: A}. We will show in the companion theorem that for each sentence C, either m, M N [a stit: A&Ca] or m, M ~[a stit: A,Ca\. Whichever is true together with [a stit: A] in M will be called a "companion" to [a stit: A}. Note that in this section, "structures" and "models" always mean BT + I+AC structures and BT + I + AC models, and we do not require that these structures and models satisfy the property of no busy choice sequences. Let us start with some simple lemmas.

386

Proofs and models

15-4 LEMMA. (A consequence of no choice between undivided histories) Let & be any structure in which w < m and w' < m, and let a be any agent, i = i(m), M - {m1: m1 6 i & m' =awm} and M1 = {m'\ m! € i & m' = , m}. Then

PROOF. Suppose that w < w'. Consider any TO' 6 i\>w>, that is, m' £ i and w' < m'. Assume that m' ^ m. Then by the axiom of choice there are two histories h and h' such that {w, w', m} C h and {w, w', m'} C h', and hence h' =w h by definition. This implies, by no choice between undivided histories, h1 =aw,h and hence m' =£, m, that is, m' 6 M. It follows that i| >UJ ' C M. (ii) Suppose that w' ^ w. By (%), if w' < w, then i\>w C M', and hence M C M'. If to' = w, then m' =£,, m iff m' =£, m for every m' € i, and hence M = M'.

n

15-5 LEMMA. (Some sufficient conditions for stit) Let SOT be any model in which w < m, i = i( m j and M = {m": m" e z & m" = ° m}, and let >1 be any sentence. Then

PROOF, (i) Suppose that SOT, M N A and for some m' € M, SOT, m' N [a stit: .A] with witness w'. Setting M' = {m": m" 6 i & m" =£,/ m'}, we then have

Since m' € M, w < m', and hence by no backward branching, either w < w' or w' ^ u/. If w < w', then by Lemma 15-4(i), i\>w> C M, and hence SOT, i|>u/ N .A, contrary to (1). Hence it must be true that w' ^ w, and then by Lemma 15-4(ii), M C M'. It follows from (1) that 971, M N [a stit: A], (ii) Suppose that !Ut, i\>w t= yl and for some m' € i\>w, 3H, m' N [a stit: A] with witness w'. Similarly, setting M' = {m": m" € i & m" =^, m'}, we also have (1), and applying no backward branching, we have either w ^ w' or w' < w. By the transitivity of ^, w ^ w' implies i\>w' C i\>w, which in turn impliesm,i\>w' t= A since SOT; i\>w N A . It follows from (1) that w' < w, and hence by Lemma 15-4(%), i|>«, C M'. Then DOT, i|>tu N [a stit: A] by (1). D 15-6 LEMMA. (Concerning [a stit: A&B]) Let 9H,TON [a stzt: A] with witness w, and let i = i(m). Suppose that 9JI, M 1= 5, where M = {m': TO' 6 i & m' = ^ m}. Then SOT, m 1= [a stit: A&B] with the same witness. PROOF. Suppose that 271, M 1= B. Then by hypothesis, Jt, M *= a&.B and 9Jt, i|>UI ^ 4, and hence mOT, i|>m J^ ^&5. It follows that £01, m N [a stit: A&B] with witness w. D

15. Decidability of one-agent achievement-stit theory with refref

387

15-7 LEMMA. (Concerning Aa) Let m be any model in which w < m, i = i(m) and M = { m / : m ' E i & m ' s ^ m } . Suppose that A is any sentence such that m, M F= A and M, M F [a stit: A]. Then m, i >w F= Aa. In particular, if m, m 1= [a stit: A&B] with witness w, and if M, M F [a stit: B], then Wl, i>w F= Ba and M, M F= [a stit: /4&Ba]. PROOF. First, if m, i\>w F A, then, since M, M F= A, M,F 1= [a stit: A], contrary to our hypothesis. It follows that M, i|>w, F= A. Next, suppose for reductio that M, i|>w, F ~[a stit: A]. Then there is an m" € i|>«, such that M, m" F= [a stit: A]. Since we have shown that M, i\>w F=A, M, i|>w, F= [a stit: A] by Lemma 15-5 (ii), contrary to the hypothesis. From this reductio we conclude that M, i\>w 1= ~[a stit: A] and hence M, i|>w F= Aa. In particular, if M, m F= [a stit: A&B] with witness w, and if M, M F [a stit: 5], then M, i|>w F= Ba and hence M, M F [a stit: A&;Ba] by Lemma 15-6. 15-8 LEMMA. (01 refraining lemma) Let M be any model, and let A and B be any sentences such that M, m F= [a stit: A&~[a stit: B]] with witness w, and let i = i( m ). Suppose that for some m' £ i|> w , M, m' F= [a stit: B] with witness w'. Then w < w'. PROOF. Let M' = {m": m" 6 i & m" =£, m'}. Then M, M' F= [a stit: B]. Since m' E i|>w,, w < m' and w' < m'. Then by no backward branching, either w' < w or w < w'. Suppose for reductio that w' < w. Then i|>w C M' by Lemma I5-4.(i). It follows that M, m F [a stit: B] since m € i\>w. But by hypothesis we have M, m F= ~[a stit: B], a contradiction. From this reductio we conclude that w < w'. 15-9 LEMMA. (Concerning Aa and stit) Let M be any model, and let A and B be any sentences. Then the following hold: i. If M, m F [a stit: A&B a ] with witness w, then M, m F= [a stit: A] with the same witness; ii. If M, m F= [a stit: A&~[a stit: A&B 0 ]] with witness w, then M, m F [a stit: A] with the same witness. PROOF. (i) Suppose that M, m F= [a stit: A&.Ba] with witness w. Then, setting i = i(m) and M = {m': m' € i & m' =aw m}, we have M, M F A&5a and M, i|>w ¥ A&Ba. Hence by Lemma 15-7, M, i\>w F= Ba, and hence M, i|>w, F A. It follows that M, m F= [a stit: A] with witness w. (ii) Suppose that M, m F= [a stit: A&~[a stit: A&B a ]] with witness w, and let i = i(m) and M = {m': m' € i & m' = awm}. Then

and either M, i|>w F A, or M, i\>w ¥ ~[a stit: A&Ba}. Consider the latter case. There must be an m' € i\>w such that M, m' t= [a stit: A&Ba} with witness w'. By applying Lemma 15-8, we have w < w', and hence i\>w' C i\>w.

388

Proofs and models

Since by (i), M, m' F= [a stit: A] with witness w', it follows that M, i\>w ¥ A. So in both cases, we have M, i\>w ¥ A. It follows from (2) that M, m F= [a stit: A] with the same witness w. 15-10 COROLLARY. (Validity of A7 For any model M, M F= A7. PROOF. Suppose that M, m F [a stit: ~[a stit: A&.B}&.Ba] with witness w, and let i = i(m) and M = {m": m" € i & m" =aw m}. By Lemma 15-7, we have that

Then there must be an m' £ i\>w such that M, m' F= [a stit: A&B] with witness w'. By Lemma 15-8, w < w', and hence, i\>w> C i\>w. Let M' = {m": m" € i & m" = aw, m'}. Then M' C i|>u, and hence by Lemma 15-6, M, m' F [a stit: A&.Ba] with witness w', and hence by Lemma 15-9(i), M, m' F= [a stit: A] with witness w', it follows that

We show as follows that M, M F ~[a sizt: A]. Suppose for reductio that there is an m* C M such that M, m* F [a stit: A] with witness w*. Set M* = {m": m" C i & m" = aw. m*}. Then M, M* F [astit:A]. Since m* € M, w < m*, and hence either w < w* or w* < w by no backward branching. Now if w < w*, then by Lemma 15-4(ii), M* i\>w, and hence by (3) and Lemma 15-6, M, M* F= [a stit: A & B ] , contrary to (3). If w* < w, then we have i\>w C M* by Lemma 15-4(i), and hence, since M, M* F [a stit: A], M, i\>w F= [a stit: A], contrary to (4). This reductio shows that M, M F= ~[a stit: A]. It follows from (4), (3) and Lemma 15-6 that M, M F= [a stit: ~[a stit: A]&Ba}. Now we are ready to prove the companion theorem. 15-11 THEOREM. (Companion theorem) Let M, m F [a stit: A] with witness w, and let i = i(m) and M — {m': m' € i & m' =aw, m}. Then for every sentence 5, either M, M F= [a stit: A&Ba] or M, M F ~[a soit: A&Ba], and consequently, either M, M F= [a stit: A&Ba] or M, M F= [astit:A&~[a stit: A & B a ] ] . PROOF. Suppose that M, M ¥ ~[a stit: A&Ba}. Then there is an m' € M such that M, m' F [a stit: A&Ba] with witness w'. Hence, setting M' = {m": m" € i & m" =aw, m'}, we have

By Lemma 15-9(i), we have M, m' F [a stait: A] with the same witness w'. This and our hypothesis imply

15. Decidability of one-agent achievement-stit theory with refref

389

Since m' E M, w < m'. Hence by no backward branching, either w < w', or w' < w, or w = w'. If w < w', then by Lemma 15-4(i), i\>w> C M, and hence by (6), m, i\>w> F A, contrary to (7). Similarly, if w' < w, then by Lemma 15-4(i), i\>w C M', and hence by (6), M, i\>w F= A, contrary to (7). We then conclude that w = w'. Since m'' =aw, m (i.e., m' € M), it follows that M = M', and hence by (5), M, M F [a stit: A&B"]. By the foregoing proof, we know that if M, M F [a stit: A&Ba], then M, M F ~[a stit: A & B a ] , and hence m, M F= [a stit: a&~[a stit: A&Ba}} by Lemma 15-6. Let M, m 1= [a stit: .A] with witness tu. For each sentence C, if M, m F= [a stit: A&Ca], we call [a stit: A&Ca] a (semantic] positive companion (poscompanion) to [a stit: A] at m (w.r.t. M), and C a pos-companion root of [a stit: A] at m ( w . r . t . M ) . Similarly, if M, m F ~[a stit: A&Ca], we call ~[a stit: A&(Ca] a negative companion (neg-companion) to [a stit: A] at m, and we call C a neg-companion root of [a stit: A] at m. Pos-companions and negcompanions to [a stit: A] at m are called companions to [a stit: A] at m. Note that every sentence must be either a pos-companion root or a neg-companion root of [a stit: A] at m. Let M be any model in which M = {m': m' € i(m) & m' = aw m} and suppose that m, m F= [a stit: A] with witness w. The companion theorem guarantees that for each sentence C, if M, m F= [a stit: A&. Ca], then M, M F [a stit: A &ca]; and That is to say,[a stit; A] must be true together with all its companions through all of

M, not only with its consequences. It can be seen from our further discussions that it does not suffice to consider only consequences of [a stit: A}; the notion of companions is essential to our proof of the completeness theorem. Of particular interest is that companions help us to compare witnesses to stit sentences, which we show as follows. 15-12 LEMMA. (Companions and witness) Let M be any model in which w < m, i = i(m) and m' e i >w, and let M, m F= [a stit: A&Ca\ with witness w, and M, m' (= [a stit: B&~[a stit: B&(Ca]] with witness w'. Then w' < w. PROOF. Since w < m' and w' < m', either w' < w or w < w' by no backward branching. Suppose for reductio that w < w'. Then, setting M = {m": m" € i & TO" =aw, m'}, we know that M C i >w. Since m, i\>w F= Ca by hypothesis and Lemma 15-7, it follows that m, M F Ca. By Lemma 15-9 (ii) we have that M, m' F= [a stit: B] with witness w', and hence by Lemma 15-6, M, M F= [a stit: B&C"]. But we know by hypothesis that M, M F ~[a stit: B&Ca], a contradiction. It follows from this reductio that w' < w. Among other things, what Lemma 15-12 tells us is this: If a sentence is a pos-companion root of [a stit: A] at m but a neg-companion root of [a stit: B] at m, then the witness to [a stit: A] at m must be strictly later than the witness to [a stit: B] at m, as formulated in the following corollary.

390

Proofs and models

15-13 COROLLARY. (Companions and witness) Let 97T, m 1= [a stit: A&cCa] with witness w, and 971, m 1= [a stit; B&~[a stit; B&C"*]] with witness w'. Then w' < w. 15-14 COROLLARY. (Validity of A9)

For any model 971, 97T N A9.

PROOF. Suppose that

and let i = i (m) and M = {m': m' £ i & m' =* m}. Then m, M \= Ca, and hence by Lemma 15-7, 971, i >w 1= <7Q and 97t, i|>tu ^ ~[a stit; ^4&[a stit; 5& ~[a stit; 5&C"*]]]. Hence there must be an m' 6 i| >U) such that

and by Lemma 15-9 (ii), 971, m' t= [a stit; B] with witness w'. Set M' = {m": m" 6 i & m" =°, m'}. Then 971, M' 1= [a stit; 5]. Applying Lemma 15-12 to (8) and (9), we have w' < w, and hence by Lemma 15-4^, i\>w C M'. It follows that 971, m E [a stit; B]. D

15C

Soundness: Validity of refref equivalence

In the last section, we showed the validity of all axioms A6-A9 of Lai + rr for all BT + I + AC structures (Lemma 15-7, Corollary 15-10, Theorem 15-11, Corollary 15-14). It is easy to see that for all BT + I + AC structures, all axioms A1-A5 are valid (see §15A) and that the rules modus ponens and RE are "validity-preserving." Thus, in order to establish the soundness of La1 + rr, the only thing left to show is the validity of refref for all BT + I+AC + nbc structures. 15-15 LEMMA. (Doing implies refraining from refraining) Let M be any BT + I + AC + nbc model. Suppose that M, m F= [a stit; A}. Then M, m F [a stit: ~[a stit; ~[a stit; A}}}. PROOF. Let M, m \= [a stit: A] with witness w and counter m', and let i = Since

Suppose for reductio that M, m¥ [a stit: ~[a stit; ~[a stit; A]]]. Then we have by (10) and Lemma 15-7 that

15. Decidability of one-agent achievement-stit theory with refref

391

We claim that there must be an mo € M' such that M, mo F [a stit: A] with witness WQ and counter mo; for if M, M' F= ~[a stit: A], then by (11), M, M' N= [a stit: ~[a stai: A]], contrary to (12). Furthermore, since mo € M', M' = {m": m" € i & m" =° m0}; and since w < mo and w0 < m0, either m < wo or Wo < w by no backward branching. If WQ < w, then, setting MO = {m": m" E i & mO" =aw0 M}, M' C M0 by Lemma 15-4, and hence, since M, MO F= [a stit: A], M, M' F [a stit: A], contrary to (11). So it must be true that w < Wo. Let M'o = {m": m" E i & TO" =awo mo'}. In general, we can inductively define two sequences m0, mi, ... and w0, w1, ... such that for each k > 0, mk + 1 E M''k = {m": m" € i & m"=awkmk} and M, mk + 1 F [a stit: A] with witness Wk + 1 > wk > w and counter m'fe + 1. The proof of this is similar to the earlier one. But {w, WQ, w\, ...} is obviously a busy choice sequence, contrary to our hypothesis. 15-16 LEMMA. (Refraining from refraining implies doing) Let 971 be any BT + I+AC + nbc model. Suppose that 971, m t= [a stit: ~[a stit: ~[a stai: 4]]]. Then 971, m N [a sizi; 4]. PROOF. Let 971, m 1= [a stit: ~[a stai: ~[a stit: A]}] with witness w and counter m*, and let i = i (m ) and M = {m": m" € i & TO" =£ TO}. Then 97t, m* (= [a siit' ~[a stit: A]] with the witness w* ^ w by Lemma 15-8, and hence

Suppose for reductio that 971, m K [a siii: /!]. Since

there must be an mo G M such that 97i, mo N [a sfoi: /IJ with witness WQ and counter m0; for if 071, M 1= ~[a stit: A], then 971, M N (~[a stt .4])Q by (14), and hence by Lemma 15-7, 971, i\>w t= (~[a stit: A ] ) a , contrary to (13). Since m0 € M, M = {m": m" E i & m" =™TOO};and since w < mQ and w0 < TOO, either w < w0 or w0 ^ w by no backward branching. If w0 ^ w, then, setting M0 = {m": m" G i & m" =aw0 mo}, we have M C M0 by Lemma 15-4, and hence, since 971, Mo1= [a stit: A], M, M F= [a stit: A], contrary to the assumption of our reductio. It follows that w < WQ. Let M0 = {m": m" € i & m" =aw0 In general, we can inductively define two sequences mo, m1, ... and w0, mo}. w1 ... such that for each k > 0, mK + 1 € M'k = {m": m" E i & m" =awk mk} and M, mk + 1 F= [a stit: A] with witness wk + 1 > Wk > w and counter m'k+1. The proof of this is similar to the earlier proof. But {w, WQ, w\, ...} is a busy choice sequence, contrary to our hypothesis. D These two lemmas enable us to establish the soundness of Lai + rr. 15-17 THEOREM. (Soundness theorem) For every sentence A, and for every BT + I + AC + nbc structure 6, h- A in La1 + rr only if 6 f= A.

392

15D

Proofs and models

Companion sets

In §15B we studied some basic features of semantic companions to stit sentences. From now on we study the syntactic features of companions. Our proof of the finite model property and the completeness of Lai + rr needs to consider, for each consistent sentence A, a finite set F of sentences including all subsentences of A. From now on, whenever F appears, we presuppose that F is finite and nonempty. Corresponding to §15B, we define (syntactic) companions to stit sentences as follows. Let F be any finite set of sentences, and let $ be any maximal consistent set (MCS) containing [a stit: A}. For any C € F, if [a stit: A&Ca] E $, we call [a stit: A&Ca] a (syntactic) T-pos-companion to [a stit: A] in <$, and C a T-pos-companion root of [a stit: A] w.r.t. $; and if ~[a stit: A &C"*] e $, we call ~[Q stit: A&zCa} a T-neg-companion to [a stit: A] in $, and C a T-neg-companion root of [a stit: A] w.r.t. <E>. F-pos-companions and F-neg-companions to [a stit: A] in $ are called F- companions to [a stit: A] in $. We define the F-companion set for [a stit: A] in $ as the deductive closure of [a stit: A] and all its F-companions in $. We say that S is a F- companion set for [a stit: A] if E is the F-companion set for [a stit: A] in some $, and we say that E is a F- companion set if E is a F-companion set for some sentence [a stit: A}. In our completeness proof for Lal + rr, we construct a BT + I + AC + nbc model containing a particular instant«, and we associate each moment in i with an MCS. We use MCSs and F-companion sets as follows. Let m be associated with an MCS , and let [a stit: A}.& F. Fn$ will represent the sentences true at m (we ignore the truth values of sentences not in F). When we have [a stit: A] true at m, we know that the witness w to [a stit: A] at m is uniquely determined, so that the F-companion set for [a stit: A] in $ can represent the set of all sentences in F that are settled true at all m' with m' =£, m. So, in a sense, the F-companion set for [a stit: A] in <j> represents one possible choice available to a at w, while other possible choices available to a are also represented by some F-companion sets. Keeping this in mind will help the reader to appreciate the following discussion. In this section we discuss some basic properties of F-companion sets. Note that by definition, every F-companion set is consistent. Note also that we do not make any use of refref in our discussion about F-companion sets in this section, or in the discussion about F-counters to F-companion sets in the next section. 15-18 REMARK. (An equivalent description of a companion set) Suppose that E contains [a stit: A] and all its F-companions in some MCS $. Then for any MCS $' including E, E contains all the F-companions to [a stit: A] in $'. In particular, if E is the F-companion set for [a stit: A] in $, then E is the F-companion set for [a stit: A] in every $' including E. Note that in this remark, when E contains [a stit: A] and all its F-companions in $, E is not necessarily a F-companion set for [a stit: A]. This is simply

15. Decidability of one-agent achievement-stit theory with refref

393

because [a stit: A] and all its F-companions in O could be consequences of some other stit sentence and its F-companions in O. By Remark 15-18, once E is chosen, we can drop the phrase "in O," and use "F-companions to [a stit: A] in E" and "F-companion roots of [a stit: A] w.r.t. E," and so on. In the following lemma we will, for the first time, apply Remark 15-18 to use "in E," "w.r.t. E," and so on, and drop "in O." 15-19 LEMMA. (Characteristic sentence of a companion set) Let E be a Fcompanion set for [a stit: A] with C"i, ..., Ck and DI, ..., Dm as, respectively, all the F-pos-companion roots and all the F-neg-companion roots of [a stit: A] w.r.t. E. Then E = DC([a stit: B}), where B is the following sentence.

PROOF. It suffices to set

and show that F ([a stit: B] = E). On the one hand, we obtain F(E D [a stit: B]) by A4, A8, and Rl. On the other hand, we obtain F ([a stit: B] D [a stit: A}) by T7 and T15, and for each i with 1 < i < k, F ([a stit: B} D [a stit: A &C?]) by applying T15 m times and T7 k — l times; and for each j with 1 < j < m, F ([a stit: B] D ~[a stat/ A&Daj]) by A2. It follows that F [a stit: B] -D E. By this lemma, we have two different but equivalent descriptions of a Fcompanion set E for [a stit: A], and we are free to use whichever is convenient. We will call the sentence [a stit: B] indicated in Lemma 15-19 the characteristic sentence of E w.r.t. A. 15-20 LEMMA. (A property of characteristic sentences of companion sets) Let B be the following sentence, where k > 0 and m > 0:v

Then for any sentence C, PROOF. As in our proof in the previous lemma, we obtain by applying T7 k times and T15 m times. 15-21 LEMMA. (Properties of companion sets) Let E be any F-companion set for [a stit: A], and let [a stit: B] be the characteristic sentence of E w.r.t. A. Then the following hold: i. For every sentence

394

Proofs and models

and consequently E h [a stit: B&C"*] or E F ~[a stit: 5&C"]; ii. If C"i, ..., Ck, and DI, ..., Dm are, respectively, all the F-pos-companion roots and all the F-neg-companion roots of [a stit: A] w.r.t. E, then

and consequently

PROOF, (i) By Lemma 15-20 and A2, we need only to show one direction of each equivalence. Suppose that C £ F and £ I- [a stit: A&Ca]. Since F [a stit: B] D Ca by A2, it follows from R3 that E F [a stit: B&C"]. Suppose next that E E ~[a stit: A & C a ] . Then E T [a stit: A&~[a stit: A&Ca}] by A8 since E T [a stit: A]. We show as follows that E F [a stit: B&~[o; stit: B & C a ] ] . By Lemma 15-20, F [a stit: B & < C a ] D [a stit: A&C a] . So E T ~[a stit: B & C a ] , and hence, T [a stit: B] D ~[a stit: B&C"*]. It follows from R3 that E h [a stit: B&~[a stit: B & C a ] ] . (li) By (i), we have that F [a stit: B] D [a stit: B&Cai] for all i with 1 < i < K, and that F [a stit: 5] D [a stit: B &~[a stit: 5 &£>"]] for all j with 1 <j < m. Then we can complete the proof by applying T7, T15, and A4. A direct consequence of Lemma 15-19 and Lemma 15-21 is the following. Let E be the F-companion set for [a stit: A] in an MCS $ with C\, ..., Ck, and DI, ..., Dm to be, respectively, all the F-pos-companion roots and all the Fneg-companion roots of [a stit: A] w.r.t. $. Let BQ, BI, B%, ... be defined as follows:

and let £„ = DC([a stit: Bn + i\) for all n > Q. Then E = E0 = EI = .... That is to say, for each n > 0, E is the F-companion set for [a stit: Bn] in $ (with [a stit: Bn + i] to be the characteristic sentence of E w.r.t. Bn), and [a stit: Bn] has the same F-pos-companion roots and the same F-neg-companion roots w.r.t. E as [a stit: A] has. 15-22 LEMMA. (Properties of companion sets) Let E be a F-companion set for [a stit: A]. Then for every sentence C 6 F,

15. Decidability of one-agent achievement-stit theory with refref

395

PROOF, (i) By A2, it suffices to suppose that E h Ca and show that E h [a stit: A&Ca]. Let [a stit: B] be the characteristic sentence of E w.r.t. .A. Since h [a stit: B] D Ca*, it follows that E h [a stit: B & < C a ] by R3, and hence E 1- [a stit: A & C a } by Lemma 15-21 (i). (ii) Suppose that E \- C and E F [a stit: C]. We know that Eu{~[a stit: C}} is consistent, and hence it must be included in some MCS <£. By Remark 15-18, E is the F-companion set for [a stit: A] in $, and hence by A2, it suffices to show that [a stit: A&Ca] € $. Set [Q stit: B] to be the characteristic sentence of E w.r.t. A. Since E I- C, h [a stit: B] D C, and then I- [a stit: B} D [a stit: B& C] by R3. It follows from Lemma 15-20 that h [a stit: B] D [a stit: A&C], and consequently [a stit: A &cC]k~[a stit: C] 6 $. Hence [a stit: A&Ca] € $ by A6, and hence E h ~[a stit: C] by A2. (iii) Since E is a F-companion set for [a stit: A], we know by (i) that E T [a stit: A & Ca ] iff E F Ca. Since E is also a F-companion set for [a stit: A'], we also know by (i) that E h Ca iff E h [Q stit: A'& Ca] (note that in the proof, we can set [a stit: B] to be the characteristic sentence of E w.r.t. A'). It follows that E h [a stit: A&Ca] iff E h [a stit: A & C a ] . This completes the proof. Lemma 15-22 (ii) shows that F-companion sets have a certain exhaustive character. A consequence of Lemma 15-22 (iii) is the following: Suppose that E is both a F-companion set for [a stit: A] and a F-companion set for [a stit: A']. Then [a stit: A] and [a stit: A'} have the same F-pos-companion roots and Fneg-companion roots w.r.t. E, and hence, if [a stit: B} and [a stit: B'] are the characteristic sentences of E w.r.t. A and A' respectively, then for some k, m > 0,

Let E be any F-companion set. We say that [a stit: B] is a characteristic sentence of E if there is a sentence [a stit: A] such that E is a F-companion set for [a stit: A], and such that [a stit: B] is the characteristic sentence of E w.r.t. A. Let E be any F-companion set with [a stit: B] to be a characteristic sentence of it. We know by Lemma 15-21 (ii) and Lemma 15-22 (iii) that for every sentence [a stit: A'] for which E is a F-companion set, [a stit: B] and [a stit: A'} have the same F-pos-companion roots and the same F-neg-companion roots w.r.t. E. 15-23 LEMMA. (Neg-companion roots and stit sentences in companion sets) Let E be any F-companion set, and let [a stit: B] be any characteristic sentence

396

Proofs and models

of E with D as a F-neg-companion root of [a stit: B] w.r.t. E. Then for any sentence ,A, if E h [a stit: A], then £ I- [a stit: A&~[a siz't; A & D a ] ] . PROOF. Suppose that £ h [a stit: A]. Then F [a stit: B] = [a stit: B]&[a stit: A}. Since E h ~[a stit: B&Da}, E h ~[a stit: [a stit: B]&Da} by A5. It follows from A3 and Rl that

Hence E h ~[a s t i t H z [a stit: A ] & D a ] by A4, and hence E F [a size: A]&~[a stit: A&Da] by T10. It then follows from A8 that E h [a stit: A&~[a stit: A &D a ]]. D In our proof of the completeness and the finite model property of Lai + rr, we need to associate each nontheorem A of Lai + rr with a F satisfying the following property: F is equivalently closed under conjunction iff for every C, C' € F, there is a D e F such that h D = C& C'. 15-24 LEMMA. (Correspondence between deducibility among stit sentences and inclusion of companion sets) Let F be equivalently closed under conjunction, and let E be any F-companion set in any MCS $. Suppose that E h [a stit: A] with A 6 F. Then E contains all the F-companions to [a stit: A] in $. PROOF. Let [a stit: B] be any characteristic sentence of E. Suppose that E h [a stit: A}. Then by Lemma 15-23, S h ~[a stit: A&Da] for every F-negcompanion root D of [a stit: B] w.r.t. $. Let C be any F-pos-companion root of [a stit: B] w.r.t. <£. Then we have S h A& C by A2. Since F is equivalently closed under conjunction, there is a C' € F such that I- C' = A&C and E h C'. Now, if ~[a stit: C'} € $, then E h ~[a stit: C'} by Lemma 15-22(ii), and E I- ~[a stit: A&C] by Rl, and hence E I- ~[a stit: 4 & C a ] by T9. If [a si#: C"] € $, E h [a st#: C'} by Lemma 15-22(M), and hence E h [a stit: A&C]. Since E h ~[a sizi: C] by A2, it follows from A6 that E h [a stit: A& Ca}. D The next three lemmas report combinatorial facts that we use later. 15-25 LEMMA. (Preservation ofpos-companion roots when extending companion sets) Let F be equivalently closed under conjunction, and let S be any FThen companion set. Suppose that and for every PROOF. Suppose for reductio that E h Ca and E V- [a stit: A] D [a stit: A& Ca] for some C & F. Then there is an MCS $ including SU{[a stit: A], ~[a stit: A&(Ca]}. Let [a stit: B] be any characteristic sentence of E. Then E F[a stit: B&Ca} by R3. Since E h ~[a stit: C}, E h [a stit: A&C] D [a stit: A&Ca] by A6, and hence ~[a stit: A&C] e $. Because [a stai: B&C a ]&[a stit: A] E $, [a stit: B&A&Ca} 6 $ by A4. Hence [a sfcf: B&yl&C 1 ] € $ by T9, and then [a stit: B&i(A&C)a} € $ by A6. Since F is equivalently closed under conjunction, there is a D e F such that h D = A&c C, and hence [a stit:

15. Decidability of one-agent achievement-stit theory with refref

397

BkDa] e $. Since D e F, we know by Lemma 15-21 (i) and Lemma 15-22fra,) that £ h [a sizi: 5&D a ]. Then by A2, £ h D, and hence, £1-4. It follows from Lemma 15-22(ii) that either £ h [a stit: A] or £ h ~[a stit: A], contrary to the hypothesis of the lemma. From this reductio, we conclude that £ h Ca implies £ I- [a stit: A] D [a stit: A& Ca] for every C € T. H 15-26 LEMMA. (Preservation of pos-companion roots when extending companion sets) Let F be equivalently closed under conjunction, let $ be any MCS, and let £ be any F-companion set in $ with [a stit: B] to be any characteristic sentence of it. Suppose that [a stit: A] & $, A e F and £ V- [a stit: A}. Then for any C € F, [a stit: A&Ca] <E $ iff [a sfzt: 4&[a stai: Sj&C 10 ] e $. PROOF. Suppose for reductio that there is a sentence C G F such that one of the following holds:

CASE 1. (15) holds. Since £ C $, [a stit: 5]&[a stit: A&Ca] e $. It then follows from A3 and A4 that [a stit: A&L (a stit: B\ & Ca] € <£, a contradiction. CASE 2. (16) holds. Since

[a siit- [a sfti: 5]&C"*] e $ by T4. Then [a s^i: B&C"*] 6 $ by A5. It follows from Lemma 15-21 (%) that £ I- [a stit: B&Ca], and hence £ h (7" by A2. By hypothesis, £ Y- [a siit- 4] and £ Y- ~[a siit: A], and hence £ h [a stit: A] -D [a stit: A&Ca] by Lemma 15-25. Since £ C $ and [a stit: A] € $, it follows that [a stit: A&Ca] e $, a contradiction. 15-27 LEMMA. (Used for Lemma 15-34 and Lemma 15-35) Let £ be any set of sentences, and let B* = 4 & ( & i ^ . , ^ m ~[a siii: 4&.D"]), where A, DI, ..., Dm are any sentences. Suppose that £ h ~[a stit: B*]. Then PROOF. For every j with 1 ^ j ^ m,we have by A8 that It is then clear by A4 that h [a stit: It follows from hypothesis that

15E

Alternatives and counters

As we said in §15B, in our proof of the completeness of Lai + rr, we select some F-companion sets to represent possible choices available to a at a given moment. In this section we discuss some relations between F-companion sets that accord with the relation between different choices available to a at any

398

Proofs and models

moment. We say that II is a T-alternative to E if for some A and A', E is a F-companion set for [a stit: A], II is a F-companion set for [a stit: A'}, and for every C € F,

Note that by definition, whenever we say that Ft is a F-alternative to E, then both E and II are F-companion sets, and the following holds for every C €. T (where A and A' are as specified):

Note also that if II is a F-alternative to E, S is also a F-alternative to II. So two F-companion sets for stit sentences are F-alternatives to each other if the two stit sentences have the same F-pos-companion roots and the same F-negcompanion roots. The next lemma follows immediately from our discussion in the last section. 15-28 LEMMA. (Properties of alternatives) Let II be any F-alternative to E, and let [a stit: B] and [a stit: B'\ be any characteristic sentences of E and Ft respectively. Then the following hold for every C € F:

PROOF. We can simply obtain (i) and (ii) by applying the definition of Falternative, Lemma 15-21, and Lemma 15-22 (Hi), and obtain (Hi) by Lemma 15-22 (i). D 15-29 LEMMA. (Properties of alternatives when a stit sentence is deducible from each of them) Let F be equivalently closed under conjunction, and let II be a F-alternative to E. Suppose that A is any sentence in F such that E I- [a stit: A] and II t- [a stit: A}. Then

PROOF, (i) Let A be any sentence such that E h [a stit: A] and II h [a stit: A], and let C be any sentence in F. Suppose that E h [a stit: A&, Ca\. Then by A2 and T9, S h A&C&[a stit: A & C} and S h Ca. Thus there is a D e F such that h D = A&C and hence E h £>&[a stit: D]. Then by Lemma 15-28(Hi), n P Da and II h Ca. Since II h A& C by A2, II h- £>&[« stit: D] by Lemma 15-22(ii). It follows that II h [a stit: A&C}&~[a stit: C], and hence II h [a

15. Decidability of one-agent achievement-stit theory with refref

399

stit: j4& Ca] by A6. The other direction of the proof is similar, (ii) Let $ be any MCS including E, and let £' be the F-companion set for [a stit: A] in $. By Lemma 15-24, E contains all the F-companions to [a stit: A] in $. Setting ^ to be any MCS including II, we know for the same reason that Ii contains all F-companions to [a stit: A] in ty. According to (i), each F-companion to [a stit: A] in $ must be in \? and vice versa. It follows that E' C EnII. D 15-30 LEMMA. (An inclusion relation among companion sets concerning a particular stit sentence) Let E and E' be F-companion sets in an MCS <J>, let II be a F-alternative to E such that E' C EnII, E h [a stit: A], II h [a stit: A] and £' V- [a stit: A], and let [a stit: B] be any characteristic sentence of £'. Suppose that F is equivalently closed under conjunction, A € F, and E" is the F-companion set for [a stit: A&[a stit: B}} in $. Then £' C E" C Enll. PROOF. Since E" h [a stit: B] by A2, it is then clear that E' C E". To show that E" C Enll, we need only to show that E and II contain all the F-companions to [a stit: A&c[a stit: B]} in $. To that end, we first note that since A € F and E h [a stit: A] and II h [a stit: A], it follows from Lemma 15-24 and Lemma 15-29 that

Consider any F-pos-companion root C of [a stit: A] w.r.t. $. Since £' C Enll, E h [a stit: AkCa]k[a stit: B} and II h [a stit: A&Ca}&[a stit: B} by (17). It then follows from A3 and A4 that E h [a stit: A&[a stit: B}&Ca] and II h [a stit: A&[a stit: B}&Ca}. Hence by Lemma 15-26 (substituting E' for E in that lemma), E and II contain all the F-pos-companions for [a stit: A & [a stit: B]] in $. Consider any F-neg-companion root D of [a stit: A] w.r.t. $. By (17), we have

It follows from E' C Enll that E' P [a stit: A] D [a stit: A&Da] (otherwise we would have E h ~[a stit: A] and II h ~[a stit: A]), and hence by Lemma 15-25, E' y [a stit: B&Da}. Then £' h ~[a stit: B&Da] by Lemma 15-21 ft), and hence by A5, E' h ~[a stit: [a stit: B]&Da]. It follows from Rl and T4 (by contraposition) that

Hence by (18), we have E h ~[Q stit: A&[a stit: B]&Da] and H h ~[a stit: A &[a stit: 5]&Z) a ]. It follows from Lemma 15-26 that E and H contain all the F-neg-companions—and hence all the F-companions—to [a stit: A&[a stit: B]} in $, that is, E" C EnH. This completes the proof. D II is a F-counter to £ relative to [a stit: A] iff H is a F-alternative to E, E h [a stit: A], and IIV- A. Assume that F is equivalently closed under conjunction. H is a T-semi-ref-counter to E relative to [a stit: A] iff

400

Proofs and models

II is a T-ref-counter to E relative to [a stit: A] if II is a F-counter to E relative to [a stit: A] and II h [a stit: ~[a stit: A}]. It is easy to see that ref-counters are semi-ref-counters. Note that when II is a F-counter (F-semi-ref-counter, F-refcounter) to E relative to [a stit: A], E does not have to be a F-companion set for [a stit: A}; but when F is equivalently closed under conjunction, by Lemma 1524, S does contain all the F-companions to [a stit: A] w.r.t. any MCS including S. As we said in the last section, a F-companion set E may represent a possible choice for a at a moment. In the same way, II, as a F-semi-ref-counter to E, may represent another possible choice for a at that moment. In the structure that we will construct in the proof of completeness and the finite model property, if S represents one choice for a at a moment, then II will represent the other choice for a at that moment (i.e., a has only binary choice). The semi-refcounter relation between F-companion sets is so important for our proof that we introduce the following notation for it. Let F be equivalently closed under conjunction. R£ (S, FT) iff the following conditions hold.

Note that whenever we use R£(E, II) in a discussion, we presuppose that F is equivalently closed under conjunction. Comparing (19), (20), and (21) earlier and (22), (23), and (24), the definition of -Rp(E, II) may not appear to give us the semi-ref-counter relation. But Lemma 15-31 guarantees that RT (E, II) iff [a stit: A] 6 F and II is a F-semi-ref-counter to S relative to [a stit: A], provided that F satisfies the following condition: F is closed under negated stit sentences if for every sentence A, [a stit: A] £ F implies ~[a stit: A] e F.3 3

Note that the notion Rp (E, II) is sufficient for our proof of the completeness and the finite model property. flp(E, II) does not imply that II is a F-semi-ref-counter to E relative to [a stit: A] if F does not have the property of being closed under negated stit sentences. That is to say, F's property of being closed under negated stit sentences is not essential for our proof. We give F this property only because it makes fip (E, II) match the notion of F-semi-ref-counters, while the ideas of F-semi-ref-counters and F-ref-counters are easier to follow than the idea of fl£(E, n).

15. Decidability of one-agent achievement-stit theory with refref

401

15-31 LEMMA. (Serm-ref-counter relation and RA ) Let F be equivalently closed under conjunction. Then

and if for every and if also

is a

pos-companion root of

is closed under negated stit sentences, then and is a semi-ref-counter to relative to

PROOF, (i) Assume that RA(£, II). We first show that I I P [a stit: A], which, by definition, is true when there is no F-neg-companion to [a stit: A] in E. Assume that D\, ..., Dm (m ^ 1) be all the F-neg-companion roots to [a stit: A] w.r.t. E. Suppose for reductio that E h [a stit: A}. Then, on the one hand, we know by definition that II I- V i< »«s m [a stit: A&D?]. On the other hand, Lemma 15-29 provides H t- &i^ is^m ~[a stit: A&cD?}. Hence II is inconsistent, contrary to our assumption of consistency of H. It follows from this reductio that H Y- [a stit: A}. We next show that H Y- A. Suppose for reductio that H h- A. Then by Lemma 15-22(ii) and II Y- [a stit: A], Ft h v4&~[a stit: A], and hence E h ~[a stit: A] by Lemma 15-28 (in), contrary to our assumption of consistency on E. It follows from this reductio that E V- A. (ii) Suppose that RA(S, II), ~[a stit: A] £ F and that every C 6 F is a F-pos-companion root of [a stit: A] w.r.t. E. Then by RA(E, II), II h ~[a stit: A}. Suppose for reductio that II Y- [a stit: ~[a stit: A}}. Then II h ~[a stit: ~[a siii: 4]] by Lemma 15-22^, and hence by Lemma 15-28(Hi), E h ~[a stit: A], contrary to our assumption of consistency on E. It follows from this reductio that II h [a stit: ~[a stit: A]]. It is easy to see that (Hi) follows from (i), (ii) and related definitions.

15F

Semi-ref-counters

From our discussion in the last two sections, one can see that F's property of being equivalently closed under conjunction is important for our proof. We now show that for every sentence A, we can associate a finite set F that has this property. Let A be any sentence, and let I" be the set of all subsentences of A, and let F* = F'U{~[a stit: C]: [a stit: C] e F'}. We define FA = {5j&...&5 n : BI, ..., Bn are distinct sentences in F*}. In the following lemma, for any set X = {Bi, ..., Bn} of sentences, &X = Bi&...&B n . 15-32 LEMMA. (Finding a set that has the desired properties) Let A be any sentence. Then TA is a finite set that is closed under subsentences, closed under negated stit sentences, and equivalently closed under conjunction. PROOF. It is easy to see by definition that TA is a finite set that is both closed under subsentences and closed under negated stit sentences. To show that TA

402

Proofs and models

is equivalently closed under conjunction, let B = £?i&...&5 m and C = C"i& ...&C n be any sentences in YA such that I ^ m, n ^ |F*|, and BI, ..., Bm are distinct sentences in F*, and C\, ..., C n are distinct sentences in F*. We show as follows that there is a D e FA such that 1- BhC = D. Let Conj(5) = {#!, ..., 5m} and Conj(C) = {C1; ..., Cn}. Clearly, if Conj(5) = Conj(C7), we can set D = B. Suppose that Conj(5) 7^ Conj(C'). Let us set X = Conj(5) -Conj((7), Y = Coiy(C)-Coiy(5), Z = Conj(B)nConj(C r ), B' = &X, C" = &Y, and E = &Z. It is easy to see that X, Y, and Z are disjoint subsets of F* and h 5'& C'hE = £& C. We can thus set D = £'& C"&£ to complete the proof. In this section we focus on F-semi-ref-counters to F-companion sets. From now on, whenever F appears in our discussion, we presuppose that F is a finite set that is closed under subsentences, closed under negated stit sentences, and equivalently closed under conjunction.4 15-33 LEMMA. (A property of semi-ref-counters corresponding to the absence of busy choice sequences) Let S, II and E' be any F-companion sets such that ,Rr(S, II), n C E' and E' h [a stit: A}. Suppose that there are n + l F-negcompanions to [a stit: A] in E. Then there are at most n F-neg-companions to [a stit: A] in S'. PROOF. We first show that

Consider any C € F. Suppose that E h [a stit: A&, Ca}. Then E h Ca by A2, and hence H h Ca by Lemma 15-28(Hi). Because H C S' and E' h [a stit: A], it is then clear that H F ~[a stit: A}. Since -Rp(E, H), II Y- [a stit: A] by Lemma 15-31 (i). Hence H h [a stit: A] D [a stit: A&Ca} by Lemma 15-25, and hence E' h [a stit: A&Ca}. It follows that (25) holds, which implies by Lemma 15-24 that

Setting DI, ..., Dn + i to be all the F-neg-companion roots of [a stit: A] w.r.t. E, we have by definition of R£ (E, II) that II h [a stit: A] D V i«s z< n + i [a stit: AkD?}. Hence E' H V i < t $ n + i [a stit: AkD?]. Since, by (26), S' contains all the F-companions to [a stit: A] in any MCS including S', it follows that there is a k such that 1 ^ k ^ n + l and E' h [a stit: A8z.D%}. Hence by Lemma 15-24, there are at most n F-neg-companions to [a stit: A] in S'. This completes the proof. D 4 As we said in note 3, the property of F of being closed under negated stit sentences is not essential. This property has no effect in our upcoming discussions, except that it helps one to see some of the ideas of our proof (by referring to Lemma 15-31).

15. Decidability of one-agent achievement-stit theory with refref

403

A direct consequence of Lemma 15-33 is the following: Let n > 0, and let EQ, • • • > E n and HO, ..., IIn be F-companion sets such that for each i with 0 ^ i ^ n, .R r (£j, II,), and for each i with 0 ^ i ^ n —1, IIj C EI+ I. Suppose that there are n F-neg-companions to [a stit: A] in EQ. Then Hn must be a F-ref-counter to E n relative to [a stit: A], and there can be no E and II such that -Rp(E, LT) and nra C E. This feature of F-semi-ref-counters corresponds to a semantic feature of Lai + rr—there is no busy choice sequence in its models. The following three lemmas establish some sufficient conditions for F-semiref-counters. 15-34 LEMMA. (A sufficient condition that semi-ref-counters exist) Let E be a F-companion set for [a stit: A] with [a stit: B] to be the characteristic sentence of E w.r.t. A. Let Ci, ..., Ck and D I , ..., Dm be, respectively, all the Fpos-companion roots and all the F-neg-companion roots of [a stit: A] w.r.t. E. Suppose that Ft = DC([a stit: ~[a stit: J5J&C 1 ]) is consistent, where C = &i< z < k C™. Then II is a F-companion set for (a stit: ~[a stit: B}} and is a Falternative to S, and II h- fa stit: A] D V i ^ j < m la s^: A&D"]. In particular, then PROOF. We first show that H is a F-companion set for [a stit: ~[a stit: B}} and is a F-alternative to £. To that end, we first note that by T7, II h [a stit: ~[a stit: B]}. Consider any D = D} with 1 ^ j < m. We know by Lemma 15-21 (ii) and T16 that

Hence h ~[a stit: ~[a stit: B]&Da} by Rl. It follows from hypothesis and Rl that

Hence by Lemma 15-19, II is a F-companion set for [a stit: ~[a stit: B}}. It is easy to check that H is a F-alternative to E. Next, since II h [a stit: ~[a stit: B]bC], and since h ~[a stit: C} by T6, it follows from Rl that H h [a stit: ~[a stit: B&~[a stit: C]]&.Ca]. Because B is the sentence v4&(7&;(&i^j^ m ~[a stit: A k D f } ) , setting B* = ^ & ( & i < j < m ~[a stit: A&DJ*]), we have II h [a stti: ~[a siii: B'&^j&C 1 0 ], and hence by T14, U\-[a stit: ~[a stti: 5*] & (7°]. It follows by A2 that II h ~[a sfit: B*], and hence by Lemma 15-27, II h [a sttt: v4] D V i 5 S j = £ m ta s^; ^&-0"]- If there is no F-neg-companion to [a stit: A] in E, then B = A& C, and hence by T6, II h [a stit: ~[a stai: ^4& <7a] & C10]. It follows from T14 that U\-[a stit: ~[a sttf; ^.] & C"*], and hence III[a stit: ~[a stzi: ^4]] by T7. In particular, if [a stit: A] e F, #r(S, II) follows by definition. 15-35 LEMMA. (A sufficient condition for extending a companion set to another and its semi-ref-counter) Let E be a F-companion set in an MCS $ such that

404

Proofs and models

E ¥- [a stit: A], [a stit: A] £ $ and [a stit: A] € F, let [a stit: B] be any characteristic sentence of £, and let E' be the F-companion set for [a stit: A& [a stit: B]} in $ with [a stit: B'} to be the characteristic sentence of X' w.r.t. A&[a stit: B}. Suppose that for every C 6 F, and that and

is consistent. Then

PROOF. Let C"i, ..., CA and Z?i, . . . , D m be, respectively, all the F-poscornpanion roots and all the F-neg-companion roots of [a stit: A & [a stit: B}] w.r.t. $. We first prove that II' is a F-companion set for [a stit: ~[a stit: B']k [a stit: B}] and is a F-alternative to E'. By R3, it is sufficient to show that for each C = Ct with 1 ^ i ^ k, and each D = D m with 1 ^ j ^ m, the following hold:

Since h [a stit: 5] D [a stit: B&Ca] by hypothesis and II' r- [a siit: B] by A2, II' h Ca by A2, that is, (27) holds. Next we show (28). By hypothesis and Lemma 15-21 (ii),

Applying (29) and Rl to T16, we have I [a stit: ~[a stit: £']&£> Q ]; and applying (30) and Rl to Til, we have I- ~[a stit: [a stit: £]&£>"]. It follows from T4 and Rl that I- ~[a stit: ~[a stit: B']&[a stit: 5]&D a ]. This completes our proof of (28). Hence II' is a F-companion set for [a stit: ~[a stit: B'} &[a stit: B]] and a F-alternative to E'. It is easy to see that E C E'nll' since E' h [a stit: B} and II' h [a siii: 5]. We show next that #r( s / > n / )- Since A e F and E' h A by A2, £' h [a stai: A] by Lemma 15-22 (h). Let B* and C1* be the following sentences:

Then by Rl, h [a stit: B1} = [a stit: 5*&[a stit: 5]&C*], and hence by hypothesis and A2, H' h ~[o; stit: 5*&[a siii; 5]&C7*]&[o! siif; B]. On the one hand, by applying A4 (by contraposition), we have II' h ~[a stit: B*]V ~[a stit: [a stit: B]&C""]. On the other hand, by applying Lemma \5-2l (ii) and T10, we have II' h [a stit: [a stit: 5]& C1?] for all i with 1 < i ^ fc (since IT h [a stit: B } ) , and hence by A4, II' h [a stit: [a stit: B}&C*}. It follows that II' I- ~[a stit: B*} (when there is no F-neg-companion to [a stit: A] in E', the same argument here, replacing B* by A, yields II' I- ~[a stit: A}). Hence by Lemma 15-27 (substituting A&[Q stit: B] for A in the lemma), we know that

15. Decidability of one-agent achievement-stit theory with refref

405

Since II' h [a stit: B] and h [a siz'i: 4]&[a sizi: B\ D [a s£zi: A&[a stit: B}} by A3 and A4, it follows that

We also know by Rl, T4, and A5 that for each j with 1 ^ j ^ TO,

But for each such j, £ h ~[a s£z't: 5&.D"] by hypothesis and Lemma 15-21 (ii), and hence IT h ~[a siif: 5&D°] by £ C n'. It follows from (31), (32), and (33) that

We have shown that £' h [a s£z't: .4], and hence by Lemma 15-24, £' contains all the F-companions to [a stit: A] in $. Since £ .K [a stit: A], we know by Lemma 15-26 that D I , ..., Dm are all the F-neg-companion roots of [a stit: A] w.r.t. E'. Hence by definition, R$(Z', IT). D 15-36 LEMMA. (A sufficient condition for extending a companion set to another and its semi-ref-counter) Let £ be a F-companion set in an MCS $ such that £F [a stit: A] and [a stit: A] e 3>nF, let E' be the F-companion set for [a stit: j4&[a stit: B}] in $, where [a stit: B] is a characteristic sentence of E, and let [a stit: B'} be the characteristic sentence of £' w.r.t. A&[o; stit: B]. Suppose that there is a C 6 F such that

and II' = DC([a stit: ~[a stit: 5']&C"]) is consistent, where C' = fci^j^fc C? and Ci, ..., Ck are all the F-pos-companion roots of [a stit: A&[a stit: B\] w.r.t. $. Then #r(E', n') and E C E'nn'. PROOF. Since A e F and E' h A by A2, it follows from Lemma 15-22(ii) that £' h [a sizi: yl]. By Lemma 15-34 we know that II' is a F-companion set for [a stit: ~[a stit: B'}} and is a F-alternative to E', and also, setting D I , ..., Dm to be all the F-neg-companion roots of [a stit: A&.[a stit: B}] w.r.t. <&, we have

We first show that E C E'nn'. Since h [a stit: B'} D [a stit: B} by A2, E C E' and

by A2, A3, and A4. Consider the sentence C given in the hypothesis. By Lemma 15-21 (ii), we know that

406

Proofs and models

Since II' is a F-alternative to £', we have by hypothesis, Lemma 15-21(i) and Lemma 15-28(i) that IT h [a stit: ~[a stit: B']&(7 a ]. It follows from (35), (36), and Rl that

and hence IT h [a stit: 5] by A9. That is to say, E C II', and hence £ C E'n II'. Since H' h [a s£ii: 5], it follows from Lemma 15-23 and A2 that II' h ~[a stit: S&.D"] for all j with 1 ^ j < m. Hence (34) and the same argument in the proof of Lemma 15-35 will yield R£ (£', II'), which completes the proof. D 15-37 LEMMA. (Criterion of consistency) If [a stit: A] is consistent, so are each of A and ~yl. PROOF. Suppose that either A or ~.4 is inconsistent. Then either h ~A or h A, and hence by R2 h- ~[Q sizi: 4], that is, [a stit: A] is inconsistent. 15-38 LEMMA. (Existence of semi-ref-counters in a simple case) Let £ be a F-companion set for [a sizi: A] with [a sizi: A] e F. Then there is a II such that R$(E, H). PROOF. Let [a stit: B] be the characteristic sentence of £ w.r.t. A, and let Ci, ..., Ck be all the F-pos-companion roots of [a stit: A] w.r.t. £, and let II = DC([a sirt: ~[a stit: B]b.C]), where (7 = &ioo C7f. We show as follows the consistency of II. By Lemma 15-21 (ii) and T6, h [a stit: B] = [a stit: B&t Ca], and hence by T20 and T6,

It follows from Lemma 15-37 that [a stit: ~[a stit: 5]&(7], and hence II, is consistent. Hence we can complete our proof by applying Lemma 15-34. 15-39 LEMMA. (Existence of semi-ref-counters in a simple case) Let H be a F-alternative to S. Suppose that [a stit: A] £ F, S I- [a stit: A] and H h [a stit: A]. Then there are £' and H' such that £' is a F-companion set for [a stit: and PROOF. We know by Lemma 15-24 that E contains all the F-companions to [a stit: A] in any MCS including S. Set E' to be the F-companion set for [a stit: A] such that £' C E. Then Lemma 15-29 implies E' C EnII. Hence by Lemma 15-38, we obtain a H' with R$(E', H'). 15-40 LEMMA. (Existence of semi-ref-counters in a complicated case) Let E be a F-companion set in an MCS $ such that [a stit: A] e F, [a stit: A] £ $ and Then there are £' and IT such that and

15. Decidability of one-agent achievement-stit theory with refref

407

PROOF. Let [a stit: B] be any characteristic sentence of E. Since [a stit: B] € $ and [a stit: A] e $, [a stii: .4&[a siit: B}] e $ by A3 and A4. Set E' to be the F-companion set for [a siii: A&[Q; stit: B]} in $, and set [a stit: B'} to be the characteristic sentence of E' w.r.t. A&c[a stit: B]. We need only to show that there is a II' such that fl£(E', II') and E C E'nlT'. There are two cases. CASE 1. E h- [a stit: B&Ca] iff E' h [a stat: 4&[a s£#: 5]&(7 a ] for every C € F. We show as follows that [a stit: ~[a stit: B']&[a sizi: B}} is consistent. Suppose for reductio that h ~[a siz£: ~[a sizi: £']&[a siii: 5]]. Then by Rl,

Hence by T21, E h [a stit: B'\. Consequently, S h A by A2, and hence, since [a siit: A] € <5, E h [a s£i£.- .4] by Lemma 15-22 (ii), contrary to our hypothesis of the lemma. It follows from this reductio that [a stit: ~[a stit: J5']&[a stii: B]} is consistent. Hence, setting IT = DC([a siz'i: ~[a stit: B']&i\a stit: B]}), we obtain #r( s / > n/ ) and s £ E'nH' by Lemma 15-35. CASE 2. There is a sentence C €. T such that either (37) or (38) holds:

We first show that (38) holds. Suppose for reductio that (37) holds. By A2 we have h [a stit: B'} D [a stit: B] and h [a sta!: 5] D Ca. It follows from R3 that E' h [a stit: B'&C01}, and hence by Lemma 15-21(%), S' h [a stit: Ak[a stit: 5]&C a ], a contradiction. Hence (38) must hold. Let Ci, ..., Ck be all the F-pos-companion roots of [a stit: A&[a stit: B]} w.r.t. $. As we did in the proof of Lemma 15-38, we can easily show the consistency of Setting II' = DC(£>*), we have by Lemma 15-36, which completes the proof. 15-41 LEMMA. (Existence of semi-ref-counters in a simple case) Let E" and E be F-companion sets in an MCS $, and let H be a F-alternative to E such that E" C Snn, E 1- [a stit: A], U \-[a stit: A] and E" Y- [a stit: A}. Suppose that [a stit: A] € F. Then there are E' and H' such that and PROOF. Applying Lemma 15-40, there are E' and H' such that R£(E', II') and Let [a stit: B] be any characteristic sentence of E". From the proof of Lemma 15-40 we know that we can choose E' to be the F-companion set for [a stit: A&[a stit: B}] in $. It follows from Lemma 15-30 that

408

15G

Proofs and models

Completeness and finite model property

Let W be a fixed denumerable set. For each F, we define Kr as the set of all sequences (Tree, ^, I, /, gi, g2) satisfying the following conditions C1-C6, where w < m iff w ^ m and w ^ m, where w is a successor of w' in Tree iff w' < w and Vw"[w" <E Tree -> ~(w' < w" < w)}, and where Cl IVee is a nonempty finite subset of W, and I is a nonempty subset of Tree; C2 ^ is a partial order on Tree satisfying VmVw[m £ ! & « ; € Tree & m historical connection, no backward branching, and the condition that for every w e Tree — I, there are at most 2x |F| different successors of w; C3 / is a function from I to the set of all MCSs of sentences, and
15. Decidability of one-agent achievement-stit theory with refref

409

C3. That is to say, for each k with 1 ^ k ^ n(n + !) + !, there is a sentence [a stit: Ak] e T such that either R? ( g \ ( w k ) , gz(wk}} or R^ ( g t ( w k ) , g i ( w k ) ) . It follows that there must be a sub-chain w\ < ... < w n + 2 (for the sake of convenience, we use wi, ..., wn+ 2 rather than wkl, ..., Wfc n + 2 ) and a single [a stit: A] G F such that Rr (g\(wj), g2(w3)) or Rr (g^w./), gi(w})) for all j with 1 ^ j ^ n + 2. We may assume, without loss of generality, that R^(gi(wj), <72(wj)) for all j with 1 ^ j ^ n + 2. Then by definition and Lemma 15-31, the following hold for all 1 ^ j ^ n + 2:

In the following, we show that there is a k ^ n + 2 such that gi(wk) Y- [a stit: A], which contradicts (39). To that end, we first show that

Consider any such j. Since w} < wj + i < m, either gi(wj) C f ( m ) or 52(wj) C f ( m ) by C4. If g i ( w } ) C /(m), then by C5, ffi(wj) C gi(w1 + i)r\g2(w] + i ) , and hence #2(^ + 1) I- [a stit: A], contrary to (39) since j + l ^ n + 2. It follows that gi(w3} C /(m), and hence by C5, g-2.(w0) C p 1 (w j + i), that is, (41) holds. Next, since in any F-companion set, there are n distinct F-companions to [a stit: A] in gi(wi), there are at most n F-neg-companions to [a stit: A] in g\(w\). Applying Lemma 15-33 with gi(w\) = E, gi(w\) = II and g\(w-i) = E', we know that there are at most n — l F-neg-companions to [a stit: A] in 51(^2). A simple induction will show that for each j with 1 ^ j ^ ra + 1, there are at most n— j + l F-neg-companions to [a stit: A] in gi(w}). It follows that there must be an I such that 1 ^ I ^ n +1 and there is no F-neg-companion to [a stit: A] in g\(wi). Then g-2(w{) \- ~[a stit: A] by (40) (recall from Lemma 15-31 that g2(u>i) is in fact a F-ref-counter to g\(w{) relative to [a stit: A}). Hence by (41), <7i(wj + i) h ~[a stit: A}. Since ^i(w; + i) is consistent, it follows that
PROOF. Since by C2, no element of I has any successor and every element of Tree —I has at most 2n successors, the displayed inequality follows from Lemma 15-42. To prove the completeness theorem, we need a (Tree, ^, I, /, g\ g%} satisfying C1-C6 as well as the following condition for all m £ I and all [a stit: A] € F:

410

Proofs and models

C7 if [a stit: A] € /(m), then there is a w e Tree — I such that w < m, and either #i(w) C /(m) and #r(5i(w), 52(«0), or ^(w) C /(w) and

/tf (02M,
Let p = (Tree, <, I, /, 51, g2) £ K r . For any m e I, and any [a sizt: 4] e F, if they satisfy the antecedent but not the consequent of C7, we say that they constitute a counterexample to C7 in p. Let p = (Tree, ^, I, /, gi, g%) and p' = (Tree', <', I', /',
We define a proper extension p' = (Tree', ^', I', /', g\, g'2) e Kp of p as follows:

Since k ^ |F|-1 (S h [a sizi: 4]) and n ^ |F|, there are at most 2 x |F| different successors of w in Tree'. It can be easily shown that p' satisfies all other conditions to be an element of Kr; details are omitted. CASE 2. w < m for some w € Tree. Let there be n* elements w' of Tree such that w' < m. Thus by C2, we have w\ < ... < wn* < m and wi ^ w' for all w' € Tree. We may assume, without loss of generality, that gi(wk>) C f ( m ) for every k' with 1 ^ fc' ^ n*. Consider wi. For the first subcase, suppose that gi(wi) I- [a stit: A}. Then 52(^1) l~ ^4> for otherwise [a stit: A] and m would not constitute a counterexample to C7 in p. It follows from Lemma 15-22 (ii) and Lemma 15-28(iii) that gi(wi) h [a stit: A]. Applying Lemma 15-39 with

15. Decidability of one-agent achievement-stit theory with refref

411

ffi(wi) = E and g 2 ( w i ) = II, we obtain £' and II' such that #r(£', n/ ) and £' £ 5 i ( w i ) n # 2 ( w i ) . Since wi < w' for all «/ e TVee, C4 implies that £' C /(m') for all m' € I. Then we repeat step (42) (replacing E and II in (42) by E' and II' here respectively) and define a proper extension p' of p by setting Tree', I' and /' the same way as in (43), and setting

Suppose now for the second subcase that 51(101) Y- [a stit; A]. This subcase has its own two subcases. Consider wn*. CASE (a). g\(wn-) V- [a stit: A}. Applying Lemma 15-40 with gi(wn*) — E and /(m) = $, we obtain E' and II' such that E' C /(m), #r(£', n/ ) and 9i(wrf) C E'nll'. Then we repeat step (42) (replacing E and II by E' and II' respectively) and define a p' by setting Tree', I' and /' the same way as in (43), and g\ and g2 as in (44), and setting

CASE (b). g\(wn*} \- [a stit: A]. Then it is easy to see that there is a k* such that 1 ^ fc* < n*, gi(wk*) Y [a stit: A], g\(wk" + 1) l~ [ex stit: A] and g^(wk* + 1) h A (or else m and [a stit: A] would not constitute a counterexample to C7 in p). Then by Lemma 15-22(ii) and Lemma l5-28(iii), g2(w ' + 1) I- [a stit: A]. Then, applying Lemma 15-41 with g\(wk») = E",
It is easy to see that p', so defined, satisfies all the conditions C1-C6. This completes the proof. D 15-45 LEMMA. (Constructing a structure for a consistent sentence) Let A be any consistent sentence. Then there is a p = (Tree, ^, I, /, gi, g%) G KJM satisfying C7 and A £ /(^o) f°r some mo € I. PROOF. Let $ be any MCS containing A. Select an m0 € W and define po = (Tree0, < 0 , ID, /o, 9i0, 520) £ K r x such that Tree0 = I0 = {m0}, ^o = {{m0, "io)}, /o = {{^o, *)}, 5i0 = 520 = 0- By repeatedly applying Lemma 15-44 with F = TA, we can construct a sequence {pn} (pn = (Treen, <„,!„, / n , 5i n ,
412

Proofs and models

This process will eventually provide us with a pk at the end of the sequence, that is, a p& in which no counterexample to C7 can be found, for otherwise we would have that for each n ^ 0, pn G K r x and \Treen\ > n, which contradicts Corollary 15-43. So let p be this pk- This completes the proof. Now we are ready to prove the completeness and the finite model property of Lai + rr, both of which are included in the following theorem. 15-46 THEOREM. (Completeness and finite model property) Let A be any consistent sentence. Then there is a finite BT + I + AC structure (which is therefore a BT + I + AC + nbc structure) & = (Tree, <, Instant, Agent, Choice), and a model yjl = {&, 3} on 6 such that i. 9Jt, mo \= A for some mo G Tree; ii. for each w G Tree, Choice^ contains at most two elements. PROOF. By Lemma 15-45, there is a sequence (Tree1, <', I, /, gi, i satisfying C7 and A G /(mo) for some mo G I. In order to construct a structure 6, we may first need to extend (Tree1, ^'} to a (Tree, <} in which each maximal chain is as long as all the others. Since |7Vee'| is finite, we can fix n to be the length of the longest chain in Tree', that is, n = max{|ft|: h is a maximal chain in Tree'}. For each maximal chain h in Tree', we know by C2 that there is a unique element m G I such that h = {w: w G Tree' & w ^ m}. Let us use m^ to denote such m. If there is any h in Tree' such that \h\ < n, then we choose some wi, ..., w n _|ft| G W—3>ee' and add them into Tree', earlier than m^ and Zoter than all w G ft —{m/,}. Finally, we obtain (Tree, ^) such that Tree' C TVee, ^' C ^, and ^ is still a partial order on Tree subject to the historical connectedness and no backward branching conditions, and such that for each maximal chain ft in Tree, \h\ = n. Details are omitted. For any maximal chain ft in (Tree, ^}, we will still use m^ to denote the unique m G I such that ft = {w: w G Tree & w ^ m}. Since Tree is clearly still finite (in fact, the inequality in Lemma 15-44 still holds), there must be a WQ G Tree such that WQ ^ w for all w G Tree. Now, letting a have an interpretation denoted by a, we can form a BT + I + AC + nbc structure 6 = (Tree, ^, Instant, Agent, Choice) by setting the following: Instant = {i\, ..., in}, where i± — {WQ}, and for each j with 1 < j ^ n, i1+ i = {w: w G Tree & 3w'(w' G i} and w is a successor of w')}; Agent

Note that since all histories in (Tree, <) are equal in length, in — I, and it is easy to see that Instant, so defined, satisfies unique intersection and orderpreservation conditions. Let us set i to be the last instant in Instant, that is, i

15. Decidability of one-agent achievement-stit theory with refref

413

= in. Note also that for any w G iL)(Tree — Tree'), H^ is a singleton. Note, finally, that for each w G Tree, either Choice^ is a singleton (when w & iL)(Tree — Tree')), or it contains exactly two elements (when w G Tree' — i). We claim that Choice, so defined, satisfies the no choice between undivided histories condition, Post. 8. We justify this claim as follows. Let us define a function g from {(m, w): m G i & w G Tree.1 & w < m} to the set of all F-companion sets as follows: For each m G i and each w G Tree' such that w < m, g(m, w) = g i ( w ) if gi(w} C /(m), and g(m, w) = g2(w) if g2(w) C /(m). Suppose that w G Tree and /i and /i' are distinct histories in H^w) such that w' G ftnft' for some w' > u>. Then, since ft and ft' are distinct, H^w) and .H"^/) are not singletons, and hence by an earlier remark, w, w' G Tree'. By definition g(mh, w) C /(m/i), and hence g(rrih, w) C gi(w')(~\g2(w') by C5. It is also true by definition that
Hence by definition of Choice, if /i G Hg2^w-), then g(rrih, w) = g \ ( w ) , and hence by (45), ly from Def. 9 and Def. 12) that for every m, m' G z, and every w G Tree,

Let OH = (6, J) be a BT + I + AC model on S such that 3(a) = a, and for every prepositional variable p € F'4, every ft in Tree and every m G i with m € ft (i.e., m = m/i), (m, ft) G 3(p) iff p G f ( m ) . Applying induction on sentences, one can show that for every C G F A and every m G i, OK, m t= C1 iff C1 G /(m), and consequently, since .4 G /(mo), 371, m 0 N ^4. We omit the inductive steps for truth-functional connectives, but provide the step for the connective [a stit: ] as follows. Let m G i and [a stit: C] G F = F"4. Suppose first that a sizi: C} G f ( m ) . Then by C7, there is a w G Tree'-i such that w < m, and either 51 (w) C /(m) and R f ( g i ( w ) , g 2 ( w ) ) , or 52(1") Q /(w) and R f ( g 2 ( w ) , g i ( w ) ) . Suppose that g \ ( w ) C /(m) (the other case is similar). Then g \ ( w ) (- [a shi: C1] and g2(w) V- C by Lemma 15-31. Since g i ( w ) = g(m., w), [a stit: C} G f ( m ' ) for every m' G i with m' =£, m by (46), and hence by A2, C G f ( m ' ) for every m' G i with m' =£, m. And also, since g2(111} ^ C, there is, by C6, an m" G i\>w with ~C" G f ( m " ) . It follows from the induction hypothesis that 9Jt, m' 1= C1 for all m' G i with m' =™ TO, and 971, j| >u , >^ C1. Hence 371, m ^ [a siii: C]. Suppose next that there is a w < m such that 371, m' 1= C for every m' G z with TO' =£, m, and 371, i| >tu >^ C1. Then H(w) is not a singleton, and hence by an earlier remark, w G Tree' — i. Assume by C4 that g\(w} C /(m) and hence g(m, w) — g i ( w ) (the other case is similar). By our induction hypothesis, C G f ( m ' )

414

Proofs and models

for every TO' 6 i with TO' =£ m, and C1 ^ /(TO") for some m" € z|> w . Hence by C6, (46) and C4, gi(w) h (7 and 0 2 (w) F (7. Now, if 9l(w) h ~[Q sizf: C1], then by Lemma 15-28(Hi), gi(w] \~ C, & contradiction. It follows that g i ( w ) V~[a stit: C], and hence by Lemma 15-22(H), g\(w) \- [a stit: C]. Since g i ( w ) C /(m), [a stit: C] 6 /(m), which completes the proof. D Let us say that a BT + I + AC structure 6 = (Tree, <, Instant, Agent, Choice) is at-most-binary if for every a € Agent and every w G TVee, Choicewa contains at most two elements. Let KI be the set of all sentences valid for all at-most-binary BT + I + AC + nbc structures, let K2 be the set of all sentences valid for all finite BT + I+AC structures, and let KS be the set of all sentences valid for all finite at-most-binary BT + I + AC structures. Identifying Lai -f-rr as the set of all its theorems, we have the following as a consequence of Theorem 15-46 and Theorem 15-17. 15-47 THEOREM. (Decidability and characterizations) Lal + rr is decidable, and Lai + rr = K = KI = K 2 = K3. Hence, r- A iff BT +1 + AC + nbc \= A.

16

On the basic one-agent achievement-stit theory Recall the refref equivalence, Ax. Cone. 1, that is, doing is equivalent to refraining from refraining from doing.* (refref)

It turns out that this equivalence is valid for every BT + AC structure for dstit, and it is valid for every BT + I + AC structure for astit iff the structure contains no busy choice sequences, and is therefore a BT + I + AC + nbc structure; see chapter 15 and chapter 18 for details. (BT + AC, BT + I + AC, and BT + I + AC + nbc structures are defined in §2.) We showed in chapter 15 that the astit theory with a single agent and the refref equivalence (i.e., the set of all sentences, with a single agent term, valid for all BT + I + AC + nbc structures) is decidable. The purpose of this chapter is to remove the no-busy-choice-sequence restriction and give an axiomatization for the resulting basic logic of astit. We present an axiomatic system Lai in §16A that is sound and complete w.r.t. BT + I + AC structures. Soundness is proved in §16B, and completeness is proved in §16E. Our proof uses the notions of companions and companion sets introduced and extensively used in chapter 15, and uses facts established in §15A-§15E, which continue to apply even in the absence of refref. In §16E we also establish some other results, for example that every satisfiable astit sentence is satisfiable in an at-most-binary model.

16A

Preliminaries

We take the language and the basic semantics of Lai—except for dropping the requirement of no busy choice sequences—from §15A. In particular, we use the 'This chapter draws on Xu 1995b, with the permission of the Association for Symbolic Logic. All rights reserved. This reproduction is by special permission for this publication only. (An error in the proof of Xu 1995b has here been corrected.) 415

416

Proofs and models

abbreviation Aa <=±dy ,4&~[a stit: A], and take models 9Jt to be pairs {©,3} based on BT + I + AC structures & and interpretations 3. We take over the recursive definition of truth from §8F, the semantics for the achievement stit from §8G.3, and the various derivative semantic concepts from §6. Turning now to axiomatics, let Lai be the axiomatic system which takes, as axioms, all substitution instances of truth-functional tautologies as well as the following schemata:

and takes, as rules of inference, modus ponens and the following:

Theorems (h), consistency and maximal consistent sets (MCSs) are defined w.r.t. Lai as usual. We will use $, \?,and so on to range over MCSs. Deductive consequences deductive closures are as defined in §15A. Setting KQ to be the set of all valid sentences and identifying Lai with all its theorems, we show in the following sections that K0 = Lai. To that end, it is convenient to list the following theorems and rules in Lai.

lr

The only use of the rule RS in our proof of the completeness theorem is to find a counter for each F-companion set for [a stit: A}. Actually the following rule serves the same purpose:

The reason why we choose RS instead of RS' is that RS is known to be admissible in Lai + rr, while it is an open question whether RS' is admissible in Lai + rr.

16. On the basic one-agent achievement-stit theory

417

Note that A1-A6 and A8 listed in §15A are identical to our Axl-Ax6 and Ax8 here, whereas we extend the conditional A7 in §15A to the biconditional Ax7 here. Note also that each of T1-T19 and R1-R4 listed in §15A is either identical to some of Thl-Th9, DR1 and DR2 listed here or derivable from them (together with Axl-Ax8). Note, finally, that there is an axiom in §15A, that is, A9, which is not included in our list of axioms, nor in our list of theorems; but the only place in chapter 15 where we applied A9 is Lemma 15-36, which is not necessary for the proof we give in this chapter. Hence we can use here the results established in §15C-§15E, where we did not apply A9 or refref.

16B

Soundness

It is easy to see that Axl-Ax5 are valid for all BT +1 + A C structures and that modus ponens and RE are validity preserving. A proof of the validity of Ax6 and Ax8 can be found in §15B. In order to establish the soundness of Lai, it is therefore sufficient to show that Ax7 is valid for all BT + I + AC structures and that RS is validity preserving. (In this section, as in chapter 15, we let a = 3(a) be the agent denoted by a.) 16-1 LEMMA. (Validity of Ax7)

Ax7 is valid for all BT + I + AC structures.

PROOF. It has been shown in §15B that is valid for all BT + I + AC structures. It is therefore sufficient to suppose that 971, m 1= [a stit: ~[a stit: A}&Ba} for any model 971 with m in it and show that 971, m 1= [a stit: ~{a stit: A&B}&Ba}. Let w be the witness to [a stit: ~[a stit: A]hBQ] at m. Setting M = {m': m' e i & m' we know by Lemma 15-7 and Lemma 15-8 that

418

Proofs and models

It follows from Lemma 15-6 that 271, i >w ¥• ~[a siz'f: A&B]. Suppose for reductio that 971, m\ 1= [a s£z£: A&B] for some mi e M with witness u>i. Let . Since w < mi and MI < mi, we know by no backward branching, Post. 3, that either w ^ w\ or w\ < w. If u; ^ wi, we have by (1) and Lemma 15-6 that 971, mi 1= [a stit: A&Ba], and hence by Lemma 15-9, 971, mi t= [a stit: A ] , contrary to (1) since TOI e M. So it must be the case that w\ < w. Since 971, MI N [a sizi: A&B] and 971, by (1) it follows from mi 6 MI and Lemma 15-7 that 971, MI 1= [a stai: 4&5 Q ], and hence by Lemma 15-9, contrary to (1). We conclude from this reductio that and hence, since It follows from (1) and Lemma 15-6 that 16-2 LEMMA. (Same pos- and neg-companion roots of stit sentences having the same witness) Let 971 be any BT + I + AC model with mi, m.2 €E i. Suppose that 271, mi t= [a stit: A] and 271, m 2 N [a sirt; S] with the same witness. Then for any C, 97T, PROOF. Suppose that 971, mi t= [a stit: A&Ca] with witness w. Then, setting we know that 97t, M 1= Ca, and hence by Lemma 15-7, Applying Lemma 15-9 we know that 971, mi 1= [a stit: A] with witness w, and hence by hypothesis, 971, mg N [a siif: 5] with the same witness. Since {TO: TO € i & TO =£j m 2 } C z|>™, it follows from Lemma 15-6 that 971, m-2 N [a stii: S&C 10 ]. The other direction is symmetrical. 16-3 LEMMA. (Validity preservation of RS) any BT + I + AC structure &.

RS is 6-validity preserving for

PROOF. Let 6 be any BT + I + AC structure, and let A, C, £>i, . . . , Dm be any sentences in each of which q does not occur. Suppose that 6 J^ [a stit: Then there is a model 971 = (6, 3) such that 971, mi I- [a stit: A& C > Q ] & ( & i ^ J < m ~[a stit: A k D f ] ) for some ml. Let i = i( m i ), and let w be the witness to [a stit: A&Ca] at mi. We know by Lemma 15-9 that 971, mi t= [a stit: A] with the same witness, and hence there is an TO2 € i\>w such that 971, ra-2 ¥• A (m^ _L^ rn\). Now let M = {TO: TO e z & TO =2 TO2}, let 3' be the same as 3 except 3'(q) = {(m,h): m e M n /i}, and let 97t' = {(3,3'). A routine induction will yield that for every sentence B in which q does not occur, 971, m/h N 5 iff 271', m/h 1= B, and in particular,

16. On the basic one-agent achievement-stit theory

419

By definition of 3' and the fact that mi £ M, 271', m^ (= [a stit: q] with witness w and a counter TOI. Then by (2) and Lemma 16-2. Hence by (3), It follows that sfa't- ^4] D V K J < m[ a s^; q&D"]. Hence RS is validity preserving for ©. 16-4 THEOREM. (Soundness theorem) h A only if (5 1= A, for every sentence j4 and for every BT + I+AC structure 6. PROOF. By Lemma 16-1, Lemma 16-3, Lemma 15-7, Lemma 15-9, Corollary 15-10, and Theorem 15-11.

16C

Companion sets and their alternatives

Our strategy to prove the completeness theorem is similar to that described in §15D, except this time we do not aim at a finite structure. Let us fix a finite set F of sentences. In this section, we discuss F-companion sets and their alternatives and counters, as defined in §15D and 15E. As we mentioned earlier, discussions in 15D and §15E can be applied here because they do not assume refref. It is easy to verify the following by Ax8, Ax4, Ax2, Th6 and RS. 16-5 LEMMA. (A derived rule of inference) Let q be any prepositional variable not occurring in A,C,D\,..., Dm with m ^ 0. Suppose that h [a stit: 0) be, respectively, all the F-pos-companion roots and all the F-negcompanion roots of [a stit: A] w.r.t. S, and let H = DC([a stit: B'}), where B' = Suppose for reductio that [a stit: A]. Then, setting C = (&i< z $ k C°) (when fc = 0, C = T), we have by Th8 (Axl when fc = 0) that [a stit: A], and hence by Lemma 16-5, that is, h ~[a stit: B], contrary to our assumption of consistency on E. From this reductio, we conclude that II.F [a stit: A}. Hence H is consistent, and therefore is a F-alternative to S. Since A £ F, and since S h [a stit: A] and Lemma 15-22(ii) and Lemma 15-28(mj imply H F A. It follows that H is a F-counter to E relative to [a stit: A].

420

Proofs and models

16-7 LEMMA. (Existence of counters, a simple case) Let F be equivalently closed under conjunction, and let II be a F-alternative to £. Suppose that A e T, £ h [a stit: A] and II h [a stit: A}. Then there are £' and IT' such that II' is a F-counter to £' relative to [a siii: A], and £' C £ n II. PROOF. Let £' be the F-companion set for [a stit: A] in any MCS $ including £. Then Lemma 15-29 (ii) implies £' C £nll. Applying Lemma 16-6, we obtain a II' which is a F-counter to £' relative to [a stit: A}. D 16-8 LEMMA. (A derived rule of inference) Let q be any prepositional variable not occurring in A,B,C,DI, ...,Drn with m > 0. Suppose that

Then PROOF. We first show the following holds:

Since I- ~[a siz'i: [a siz'i: B]&G] D ~[a siz'i: B] V ~[a 5^z'i: G] by Ax4 and Ax3, we know that h ([a stit: B]&G) Q D [a stit: B]&Ga. Now by Ax5, Ax6, and (i), we have H [a sizi: B]&G a &[a stti: [a sizi: J5J&G] D [a sfei: 5&G a ] &~[a stit: B&G"]. It follows from classical logic that I- [a stit: B}&.Ga D ([a stit: .B]&G) Q , and hence, (4) holds. Now it is easy to see by (ii) and Ax2Ax4 that D^})\ D [a stit: A&[a stit: B}]. Then, setting E = [a stit: B]&G and F} = [a stit: B}&,D} for each j with 1 ^ j ^ m, we obtain by (4) that h [a stit: q&.Ea and hence by Lemma 16-5 (substituting E for G, Fj for £>.,, and It follows from (4) that 16-9 LEMMA. (Existence of counters, a complicated case) Let S be any Fcompanion set in $ with [a stit: B} to be its characteristic sentence, and let [a stit: A] G $, A € F and E V- [a stit: A]. Then there are E' and II' such that II' is a F-counter to E' relative to is the F-companion set for PROOF. Since [a stit: A], [a stit: B} e $, [a stit: A&[a stit: B}} 6 $ by Ax3 and Ax4. Let E' be the F-companion set for [a stit: A&[a stit: B}] in $. Since and since Y! \- A and [a stit: .A] 6 $, we have by Lemma 15-22 (ii) that

16. On the basic one-agent achievement-stit theory

421

First suppose that for every C e F. Let IT = E. By hypothesis and [a stit: A] € <&, and then by Lemma 15-22 (ii), II' Y- A, and hence by (5), 13' is a F-counter to S' relative to [a stit: A}. It is easy to see that Next suppose that there is a C € F such that either (6) or (7) holds:

Let [a stit: B'] be the characteristic sentence of E' w.r.t. A&[a stit: B\. By hypothesis and Ax2, we know that E' I- [a stit: B]. Thus if (6) holds, Ax2 implies that h [a stit: B'} D Ca, from which, together with DR2 and Lemma 15-21 (%), it follows that E' h [a stit: A&[a stit: B]&Ca], a contradiction. Hence (7) must hold. Let q be any prepositional variable not occurring in B', and let Ci,...,Ck and Z?i,..., Z?m be, respectively, all the F-pos-companion roots and all the F-neg-companion roots of [a stit: A&[a stit: B}} w.r.t. E', and let II' = DC([a stit: B*]), where B* = qk[a stit: B]&(& 1< z < fc Cf) &(&i< ^ m ~[a stit:
Consider any D3 with 1 ^ j ^ TO. Applying Lemma 15-21 (i) and the fact that Dj e F, we know that either I- [a stit: B] D [a stit: B&D"} or h [a stit: B} D ~[a stit: B&D™}. Suppose for reductio that h [a stit: B} D [a stit: £?& Df]. Since E' h [a stit: B], it follows from Lemma 15-23 that E' \~ [a stit: B &~[a stit: B&D™}}, and hence by Ax2, E' \- ~[a stit: B&D™}, and hence by the supposition of our reductio, E' h ~[a stit: B], contrary to the consistency of E'. From this reductio we conclude that follows that (8) holds. Next, setting E = &i^ t <; kC", we show that

By (7) and Lemma 15-21^, we know that

and C must be Cr for some i with 1 ^ i ^ fc. We thus obtain h [a stit: B& E] D [a stit: J5&C Q ] by applying Th4 k — 1 times, and hence I- [a stit: B] D ~[a stit: B&E] by (10). It follows from Th8 and DR1 that (9) holds. Finally we show that II' Y- A. Suppose for reductio that II h [a stit: A]. Then h [a stit: q &[a stit: J B]&E&(&i s ;^ T n ~[a stit: "])] 3 [a sM: ^]. It follows from (8), (9), and Lemma 16-8 that h ~[a and hence

422

Proofs and models

by Th8 again, h ~[a stit: B'], contrary to the consistency of E'. We conclude from this reductio that II' V- [a stit: A}. Applying (5), the same argument in Lemma 16-6 (replacing S and II by E' and II') shows that II' is a F-counter to E' relative to [a stit: A]. In the following and the rest of this chapter, we assume that F is finite, closed under subsentences and equivalently closed under conjunction. Applying our strategy used in §15D, at each step of our construction of the structure, each history intersects the last instant. Since our construction here could be infinite, the whole construction could leave us some history h that does not intersect the last instant i in the structure. In such cases, we have to extend those infinite chains to meet the condition of unique intersection, which will be accomplished by applying the following notion of "parameters." Let 9 C F, and let £ be the F-companion set for [a stit: A] in <E>. 0 is a parameter for [a stit: A] in $ if 0 is the set of all pos-companion roots of [a stit: A] w.r.t. $, 0 is a parameter for £ in $ if 0 is a parameter for [a stit: A] in <&, and 0 is a parameter for E if 0 is a parameter for E in some MCS $. By our discussion in §15D, we know that if E is a companion set in $ for both [a stit: A] and [a stit: B], they must have the same F-pos-companion roots and the same F-neg-companion roots w.r.t. E. Furthermore, if 0 is a parameter for E in an MCS <E>, it is the parameter for £. We will therefore speak of a F-companion set E with parameter 0 to indicate that 0 is the parameter for E. Applying the definition of F-alternatives (15E), we know that E and II are F-alternatives iff if they have the same parameter. 16-10 LEMMA. (Comparability of parameters) Let $ be any MCS, and let E and E' be any F-companion sets in $ with parameters 0 and 0' respectively. Then either PROOF. Suppose for reductio that neither Then and C i 0', D e 0' and D £ 0 for some C and D. Let E and E' be the F-companion sets for [a stit: A] and [a stit: B] in <3> respectively. By Ax8, and then by Ax4, contrary to Th8 and the consistency of 16-11 LEMMA. (Simple properties of parameters) Let 0 C 0' C F, and let E and E' be F-companion sets in <£ with parameters 0 and 0' respectively. Then

for each

with parameter and consequently

only

PROOF. Let E be a companion set for [a stit: A] in $. Since and C £ 0 for some C. Then E V- [a stit: A&C"*], and then by Lemma 15-22(i),

16. On the basic one-agent achievement-stit theory

423

(i) We know that E' h Ca. If £' C E, we would have E I- Ca, contrary to (11). (ii) Let (a stit: A'] 6 with parameter 0'. If E h [a stit: A'], we would have by Lemma 15-24 that E h Ca since C € &', contrary to (11). (Hi) Consider any B e E n T. Then by Lemma 15-22 (n), either by Lemma 15-22(i,), and then Suppose that E h- [a stit: B}. We know that ~[a stif: £?&C a ] 6 $, for otherwise by Lemma 15-24, and then E h Ca, contrary to (11). It follows from Ax6 that ~[a stit: B&C] e $. Now let [a stit: A*} be any characteristic sentence of E'. We know by Lemma 15-21 (i) that [a stit: A* and hence [a sirt: yl*&C1] e $ by Th4. Since by Ax4, and then Because Y is equivalently closed under conjunction, there is a D £ T such that h D <-> 5&C and [a stai: A*&D Q ] e $. Then D e 6', and hence E' h B. It follows that E n T C E' n T. To see that E n T C £' n T, consider C. We know that C e E' and ~[a stit: C] 6 $. If C 6 E, we would have S h C, and thus by Lemma 15-22(ii), E h Ca, contrary to (11). It follows that C Provided E is a F-companion set in $ with parameter ©, E is Q-maximal in w.r.t. F if for each A € F, [a stit: A] G $ with parameter G only if It is easy to see that if E is 0-maximal in $ w.r.t. F, and if and E' is a F-companion set in $ with parameter 0, then E' is also 0-maximal in $ w.r.t. F. 16-12 LEMMA. (A sufficient condition for the stability of maximal companion sets w.r.t. r) Let E be 0-maximal in $ w.r.t. F, let E' be any F-companion set in $ with parameter ©' such that 0 C 6' and E C E'. Then for each MCS fy including E', E is ©'-maximal in W w.r.t. F. PROOF. Let * be any MCS including E', and let A e F such tha with parameter 0. We show that E h [a stit: A]. Let E* be the F-companion set for [a stit: A] in *. Since 0 C ©', E* OF C E'nF by Lemma 16-11 (Hi), and hence S' h A. By Lemma 15-22(MJ, either E' h Aa or E' h [a stit: A}. Since it then follows that E' (- [a stit: A], and hence by Lemma 15-24, £* C S'. Consequently, since E' C $, E* C $, and hence [a stit: A] e $ with parameter 0. Because E is ©-maximal in $ w.r.t. F, we have by definition that 16-13 LEMMA. (Finding suitable T-maximal companion sets) Let E and E' be F-companion sets in $ with parameters 0 and ©' respectively such that Then there is a F-companion set E* in $ such that E U E' C S* and maximal in $ w.r.t. F. PROOF. Let [a stit: B] and [a stit: B'} be characteristic sentences of E and S' respectively, let AI ,..., An (n ^ 0) be all the sentences in F such that with parameter Q (I ^ k ^ n), and let E* be the F-companion set for

424

Proofs and models

[a stit: A] in $, where A = [a stit: AI]&. .. &[a stit: An]&,\a stit: B]&[a stit: B'] ([a stit: A] G $ by Ax3 and Ax4). Clearly, £ U £' C £*. We show that £* has the parameter 6. If C G 6, E h- Ca by Ax2, and then E* h C Q , and hence by Lemma 15-22(i), [a stit: A&Ca] G $. Suppose that ( 7 ^ 6 . Then C1 ^ 6', and then by Ax2 and Lemma 15-23, ~[a stit: Ai&cCa],... ,~[a sttt: A n &C Q ] € $, and ~[a stit: S&(7a],~[a stit: 5'&(7Q] G $. Now if [a stit: A&Ca] G $, [a stit: AI&. .. &A l &5&5'&C m ] e $ by Ax5, and then by Th7 and DR1, either [a stit: Ai&Ca] 6 $ or ... or [a stit: An&Ca] € $ or [a stit: S&C a ] G $ or [a stit: S'&C"*] e $, contrary to the consistency of $. It follows that ~[a stit: A&C1"] e $. Hence 0 is the parameter for £*, and then S* is 0-maximal in $ w.r.t. F. D

16D

Construction of preliminary structures

In the rest of this chapter, we fix 0 to be the function that maps each Fcompanion set to its unique parameter. Let W be a fixed denumerable set. For each finite set F of sentences, we define Kp as the set of all sequences (Tree, ^, ixree, f , 9i, 92, 9+, 9-) satisfying the following conditions CO-C6, where w < m iff w ^m and w ^ m: CO Tree is a finite subset of W. Cl ^ is a partial order on Tree satisfying historical connection and no backward branching; and 0 ^ irree C Tree satisfying VmVw[m e irree & w € Tree —> m •£ w\ and Vw[w G Tree —iTree —* 3m(m € i-rree & w < TO)]. C2 / is a function from ifree to the set of all MCSs; and g\ and #2 are functions from Tree — i jyee to the set of all F-companion sets such that for every w € Tree - 1^^, 0(gi(w)) = 6(g2(w)) and gi(w) ^ 52 WC3 g+ and
16. On the basic one-agent achievement-stit theory

425

We will use p,p', and so on to range over sequences (Tree, iTree, /, 9i, 92, g+,
We will eventually convert "preliminary structures" (Tree, ^,iTree, f,9i,g2, g+,9-) into BT + I + AC structures, but we need them to satisfy C1-C6 as well as the following conditions for all m e i-rree, w & Tree and A 6 F: C7 If [a stit: A] e /(TO), there is a w' 6 Tree — iTree such that w' < m, and g_(m,u/) is a F-counter to g+(m,w') relative to [a stit: A}. C8 If g i ( w ) Y- A (g2(w) V- A), there is an m' 6 iTree such that w < m', g+(m',w) = gi(w) (= g2(w)) and ~A <E /(TO'). When we convert a p into a BT +1 + AC structure, the first part of C4 will guarantee that the choice function to be defined satisfies no choice between undivided histories, and C5-C6 will enable us to find a p, satisfying all C1-C8, in which each history intersects the last instant. Let p = (Tree, ^,iTree, f,gi,g2,g+,g~) be any sequence satisfying C1-C6. If any TO 6 i-rree and A 6 F satisfy the antecedent but not the consequent of C7, we say that they constitute a counterexample to C7 in p; and if any w € Tree —i Tree and A £ F satisfy the antecedent but not the consequent of C8 (in either case), we say that they constitute a counterexample to C8 in p. Let p= (Tree, <, i^ee, f, 9i, §2, 9+, 9-) and p' = (Tree1, ^', iTree', f, g{, 92, 5+, g'-)- Provided that both p and p' satisfy C1-C6, p' is a (finite) extension of p iff (identifying relations and functions with sets of ordered pairs) Tree C Tree1, ^ = <' H (Tree x Tree), iTree C in-ee', f C /', g1 C g(, g2 C g'2, g+ C g'+ and 5_ C g'_ (and Tree' — Tree is finite). Next we deal with counterexamples to C7-C8. The strategy is similar to that used in §15G. In our proof, we will only provide the crucial information about how to define the corresponding extension, and leave the rest of the construction (usually straightforward) and the verification to the readers. 16-14 LEMMA. (Killing counterexamples to C8) Let A G F and p = (Tree, ^, i-Tree, /,9\, 92, 9+, 9-) satisfying C1-C6 with w e Tree - iTree, and let w and A constitute a counterexample to C8 in p. Then there is a finite extension p' of p, in which w and A no longer constitute a counterexample to C8. PROOF. We deal only with one case (the other is similar). Since gi(w) Y- A, 9i(w) U {~A} C $ for some MCS $. Selecting a new TO, we can extend p to the desired p' such that Tree' = TreeU {TO}, w <' m 6 iTree', /'( m ) = ®, g'+(m,w) = gi(w) and g'_(m,w) = gzfa). D The following Lemma 16-15™Lemma 16-18 deal with counterexamples to C7 in particular cases. They serve as preparations for Lemma 16-19, which deals with counterexamples to C7 in general.

426

Proofs and models

16-15 LEMMA. (Killing counterexamples to C7, a particular case) Let A S F and p = (Tree, ^,iTree,f,9i,92,9+,9i) 6 K r with TO € irree such that m and A constitute a counterexample to C7 in p. Suppose that there is no w 6 Tree with w < m. Then there is an extension // e Kp of p, in which m and ^4 no longer constitute a counterexample to C7. PROOF. By hypothesis, Tree = ixree = {m}- Let E be the F-companion set for [a stit: A] in /(m). By Lemma 16-6, we obtain a F-counter II to E relative to [a stit: A}. Let * be any MCS including IIU{~A}. By Lemma 16-13, there is a #(E)-maximal F-companion set II* in ^. Since ~A e *, II* Y- A, and hence II* is a F-counter to S relative to [a stit: A}. Then we can select WQ,TOO€ W — Tree and extend p to the desired p' 6 Kr such that Tree' = TreeU {WQ,TOO},WQ <' m and u;0 <'TOO€ iyree', /'(TOO) = *, ffi^o) = ^("^^o) ~ 9-(m0,w0) = E and 52(^0) = g+(m0,w0) = g'_(m,w0) = II*. 16-16 LEMMA. (Killing counterexamples to C7, a particular case) Let .A e F and p = (Tree, ^,iTree, f,9i,92,g+,9i) € Kr with TO € ijvee such that TO and A constitute a counterexample to C7 in p. Suppose that w ^ w' for all w' € Tree and <7-|_(m,u;) h [a stit: A]. Then there is an extension p' € Kr of p, in which TO and A no longer constitute a counterexample to C7. PROOF. Since <7 + (m,w) f- [a stit: A], g_(m,w) h A; for otherwise TO and A would not constitute a counterexample to C7 in p. It follows that g^(m,w) h [a stit: A] by Lemma 15-22 (ii) and Lemma 15-28fiii,). Applying Lemma 16-7 with g+(m,w) = £ and g-.(m,w) = II, we obtain £' and II' such that II' is a F-counter to £' relative to [a stit: A] and E' C g+(m, w) Cig-(m, w). Similar to our proof of Lemma 16-15, we extend II' to a 0(II')-maximal F-companion set II* in a MCS * w.r.t. F (where II' U {~A} C <]/), which is also a F-counter to E' relative to [a stit: A}. Since w ^ w' for all w' e Tree, C3 and C4 imply that ^' C /(TO') for all TO' S iTree- Selecting WQ,TOO6 W — Tree, we can extend p to the desired p' such that Tree' = Tree\J{WQ,TOO}, WQ <'TOO€ irree' and WQ <' w for all w G Tree, /'(m 0 ) = *, ffi(wo) = 5'_(m 0 ,wo) = E' = g'+(m',wQ') for all m' e iivee, 52(^0) = 9+(m0,w0) = II* = g'_(m',w0) for all m' £ i^ee16-17 LEMMA. (Killing counterexamples to C7, a particular case) Let A 6 F and p = (Tree, ^,iTree,f,9i,92,9+,9i) with TO e i^vee satisfying C1-C6, let TO and A constitute a counterexample to C7 in p, and let g+ (TO, w) F [a stit: A] and w < m with w < w/ < TO for no w' e Tree. Then there is a finite extension p' of p, in which TO and A no longer constitute a counterexample to C7. PROOF. Applying Lemma 16-9 with g+(m,w) — E and /(TO) = <3>, we obtain S' and IT such that E' C /(TO), g+(m,w) C E'DIT, and II' is a F-counter to S' relative to [a stit: A}. Similar to our proof of Lemma 16-15, we extend II' to a 0(n')-maximal F-companion set IT in an MCS * w.r.t. F (where n'U{~A} C f and g+(m,w) C $), which is a F-counter to E' relative to [a stit: A]. Selecting new WQ and TOO, we extend p to the desired p' the same way as in Lemma 1615, except w' <' WQ for all w' ^ w, g((w0) = g'+(m,w0) = g'_(m0,w0) = E', g'+(mo,w') = g+(m,w') and g'_(m0,w') = g-(m,w') for all w' ^ w. D

16. On the basic one-agent achievement-stit theory

427

16-18 LEMMA. (Killing counterexamples to C7, a particular case) Let A 6 F and p= (Tree, ^,iTree,f,9i,92,9+,9i) with m G irree such that p satisfies ClC6, and such that TO and A constitute a counterexample to C7 in p. Suppose that w < w' < m such that there is no w" £ Tree with w < w" < w', g+(m, w) V\a stit: A] and g+(m,w'} I- [a stit: A}. Then there is a finite extension p' of p, in which m and A no longer constitute a counterexample to C7. PROOF. By hypothesis, g-(m,w') h A (or m and A do not constitute a counterexample to C7 in p). Then by Lemma 15-22(ii) and Lemma 15-28(Hi), g-(m,w') \- [a stit: A}. Applying Lemma 16-9 with g+(m,w) = E and /(m) = $, we obtain E' and II' such that II' is a F-counter to E' relative to [a stat: A], g+(m,w) C E' nil', and E' C /(m) (in fact E' is the F-companion set for [a stit: Ak[a stit: B}] in $, where [a stit: B] is a characteristic sentence of g+(m, w)). By C4, g+(m,w) C 1} C * and II* is a F-counter to E' relative to [a stit: A]. Selecting new WQ and TOO, we extend p to the desired p' such that Tree' = TreeL) {w0,m0}, w <' w0 <' w' and w <' w0 <' m0 e zjvee', /'( TO O) = *, ffi(wo) = g'-(mo,w0) = E', 5 2 ( w o) = ffV(mo,wo) = n*, 0+(m',u;o) = E' and g'__(m',wo) = FI* for all TO' € ijvee with w' < m', and g'+(mo,w") = g+(m,w") and g'_(mo,w") = g-(m,w") for all w" ^w. 16-19 LEMMA. (Killing counterexamples to C7) Let A € F and p = (Tree,^, i-Tree, f,g\,g2,g+,gi) G Kr with TO € iTree- Suppose that TO and A constitute a counterexample to C7 in p. Then there is an extension p' e Kr of p, in which TO and A no longer constitute a counterexample to C7. PROOF. If there is no w e Tree with w < TO, we apply Lemma 16-15. Suppose that w < TO for some u; € Tree. Then by Cl, we have a maximal chain w\ < ... < «;„ < m in Tree. Consider w\. If g+(m,wi) \~ [a stit: A], we apply Lemma 16-16. Suppose that g+(m,wi) Y- [a stit: A}. We then first consider wn. If g+(m, wn) Y- [a stit: A], we apply Lemma 16-17. If g+(m,wn) \- [a stit: A], then there must be a k such that 1 ^ k < n, g+(m,Wk) Y~ [a stit: A] and g+(m,u>k+i) r- [a stit: A}. Then we apply Lemma 16-18. Let K be any cardinal, and let {p^: 0 ^ £ < K} be a set of sequences such that for each p^ is an extension of p$. Then where Tree Let A be any sentence, and let F^ be the set of all subsentences of A. We define TA = {£?!&;... &B n : 1 < n < |F^| and B\,... ,Bn are distinct sentences

428

Proofs and models

in Fg4}. It is easy to verify that for each A, TA is finite, and is both closed under subsentences and equivalently closed under conjunction (see Lemma 15-32). 16-20 LEMMA. (Constructing "preliminary structures") Let A be any consistent sentence. Then there is a sequence p = (Tree, <, iTree,f,9i,92,9+,9-) satisfying C1-C8, and A g /(TOO) for someTOOG irreePROOF.

Let $ be any MCS containing A. Select an m0 G W and define such that Treeo = {m0}, 0 = {m0}, fo(m0) = {{m0,$}}, gi0 = 92Q = g+0 = g-0 = 0. By repeatedly applying Lemma 16-14 and Lemma 16-19 with F = TA, we can construct a sequence {pn} (pn = (Treen,^n,iTreen,fn,gln,g2n,g+n,g_n}) of elements of KpA in such a way that for each k ^ 0, Pk+i is an extension of Pk, and whenever we have a counterexample to C7 or C8 in p, there will be a pn in the sequence with k < n, in which it is no longer a counterexample. Since this process strongly resembles a standard method applied in modal logic and tense logic, we omit the details. Finally we define p = (Jn>0pn- It is easy to see that p satisfies all the conditions C1-C8.

16E

Completeness

The "preliminary structures" p= (Tree, ^,ziy e e ,/,91,g%,g+,g-) constructed in the last section provide all conditions we need for the completeness theorem except the order preservation condition and the following: C9 For each maximal <-chain h in Tree, h n i Tree ^ &• Let p = (Tree, =%, iTree, f , 9i,92,9+,9-) satisfy C1-C8, and let h be any history (maximal <-chain) in Tree, h is a counterexample to C9 in p if /iPUjVee = 0. Let h be any counterexample to C9 in p. It is easy to see by Cl that h is endless, that is, for each w € h, there is a w* e h such that w < w*, and then by Cl, there must be an TO 6 irree such that w < w* < TO. It follows from C4 that for each TO', TO" G iTree, if w < w' < TO' and w < w" < TO" for some w',w" € h, g+(m',w) = g+(m",w). Thus for each counterexample h to C9 in p, we define Sh to be the function on h such that for each w € h, Sh(w] — g+(m,w), where TO is any moment in iTree such that w < w' < m for some w' £ h (and hence, Sh(w) = g+(m,w) for every TO € iTree such that w < w' < m for some w' £ h); and let S be the function on all counterexamples to C9 in p such that for each such counterexample h, S(h) = {sh(w): w 6 h}. It is easy to see by C4 that for each counterexample h to C9 in p, S(h} is a chain of F-companion sets. 16-21 LEMMA. (Existence of a biggest member o f S ( h ) ) Let p — (Tree, ^, ixree, /) <7ii 92, 9+1 9-) satisfy C1-C8, and let h be a counterexample to C9 in p. Then there is a w e h such that Sh(w) = Sh(w'} for every w' G h with w < w', and such that Sh(w) is 0(s/((iu))-maximal in an MCS w.r.t. F.

16. On the basic one-agent achievement-stit theory

429

PROOF. We first observe that by (12), C4 and Lemma 16-10, both S(h) and {6(sh(w)): w G h} are C-chains. Because F is finite, there is a w* G h such that 9(sh(w*)} = 6(s(w')) for every w1 £ h with w* < w'. For the same reason, there must be a w G h such that w* < w and Sh(w) flF = s/ l (w') for every w' £ h with w < w', and then by C6 and our definition of Sh, Sh(w) is 0(s/ l (iu*))-maximal in some MCS w.r.t. F and Sh(w) = SH(W') for every w' € h with w < w'. D Let p = (Tree, ^,irree, f,91,92,9+,9-} satisfy C1-C8, and let h be a, counterexample to C9 in p. By Lemma 16-21, we know that there is a w e h such that Sh(w~) = Sh(w') for every w' & h with w < w', and such that s^w) is ©-maximal in some MCS w.r.t. F, where 0 = 0(s/,(iu)). We call such an h a Q-counterexample to C9 in p. Clearly, each counterexample h to C9 in p is a ©-counterexample to C9 in p for a unique 6. 16-22 LEMMA. (Killing counterexamples to C9) Let p = (Tree, ^, iTree, f , 9i, 92, g+,g-) satisfy C1-C8, and let 6 be any subset of F. Suppose that there are ©-counterexamples to C9 in p. Then there is an extension p* of p satisfying C1-C8, in which there is no ©-counterexample to C9, and in which each new counterexample to C9 is a G'-counterexample to C9 with PROOF. Let H be the set of all 6-counterexamples to C9 in p, and assume that and h = ho. For each £ < re, let Wj be a denumerable set of new moments, and assume that all WjS are disjoint from Tree as well as from each other. We first construct as follows (eventually we will have Since h is a 0-counterexample to C9 in p, there is, by Lemma 16-21, a u>* G /i such that Sh(w*) = Sh(w') for every u/ 6 h with iy* < w', and such that s/ l (w*) is ©-maximal in some MCS $ w.r.t. F. Selecting an TO* € WQ, we define p and g_ 0 == g_ U {{(m*,ti>),(?(u>)): to G /i}, where for each w € h, g(w) = gi(w) if Sh(w) = g2.(w], and g(to) = gz(w) if s^(w ; ) = 9i(w}- Then we construct Po,pi,... the same way as we did in Lemma 16-20, except this time we require that the following hold for every n ^ 0:

It is easy to see that all TO and w in po satisfy (13). Note that by hypothesis, for each m G iTreen, w G Treen - iTreen and A G F, if TO and A constitute a counterexample to C7 in p n , TO ^ Tree; and if w and ^4 constitute a counterexample to C8 in pn, w £ Tree. We show as follows that at each step n, how to kill any counterexample to C7 or C8. It is easy to see that if w and A constitute a counterexample to C8 in pn, we apply Lemma 16-14 and kill the counterexample by extending pn. So let TO G irreen and A G F constitute a counterexample to

430

Proofs and models

C7. Since p satisfies C1-C8, m G i-rreen — irree- Since [a stit: A] G f ( m ) , there must be a 0' that is the parameter for [a stit: A] in /(m). We first show that

By (13), Sh(w*) is 0-maximal in f ( m ) w.r.t. F, and by Lemma 16-10, either 6' C 9 or 0 C 0'. Suppose for reducho that 0' C 0. Then sh(w*} h ^4 by Lemma 16-11 (lii) and the fact that Sh(w*) is 0-maximal in /(m) w.r.t. F, and hence s/l(w;*) h [a stit: A] by Lemma 15-22 (ii). We know that there is an m' G ijvee such that w* < m' and g+(m',w*) = Sh(w*), and then [a stit: A] G f ( m ' ) . Since p satisfies C7, there is & w £ Tree — iivee such that w ^ w* (by C4) and g-(m! ,w) is a F-counter to g+(m',w) relative to [a stit: A]. But it is easy to see that g+n(m,w) = g+(m',w), which contradicts our assumption that m and A constitute a counterexample to C7 in pn. Hence (14) holds. There are two main cases. CASE 1. There is no w G Treen such that w' *) for g+(m,w) there, and Treen for Tree, etc.), except w' ^' WQ <' mo for all w' G ft. It is easy to see, by applying (14) and Lemma 16-12, that p' satisfies (13). CASE 2. There is a w G Treen such that w' fc <„ m. There are three subcases. SUBCASE 2.a. g+n(m,u>i) h [a stit: A}. By (14) and Lemma 16-11 (ii), we know that g+n(m,w'} C sn(w*) V- [a stit: A] for every w' G ft. Then, getting suitable E',n* and \I/, and selecting U>O,TOO 6 WQ — Treen, we extend pn to a p' = (Tree',^',iTree', f ,g'i,g2,9+i9'-) the same way as we did in Lemma 1618, (substituting Sh(wf) for g+(m,w) there, and Treen for Tree, etc.), except w' ^' u)Q <' toi for all u>' G ft. Applying (14) and Lemma 16-12, one can easily verify that (13) holds for p1. SUBCASE 2.b. g+n(m,wk) V- [a stit: A}. Then we apply Lemma 16-17 (substituting Treen f°r Tree, etc.). SUBCASE 2.c. g+n(m,wi) Y- [a stit: A] and g+n(m,Wk) I- [a stit: A}. Then there must be a k' such that 0 ^ k' < k such that g+n(m,Wk>) Y- [a stit: A] and g+n(m,Wk' + 1) r- [a sizi: A]. Thus we apply Lemma 16-18. What we have shown completes the proof that for each n ^ 0, we can kill any counterexample to C7 or C8 by extending pn to a p', satisfying C1-C6 and (13). It follows that we can construct po,pi,..., each pk + i extending p/~, in such a way that whenever there is a counterexample to C7 or C8 in p/^, there will eventually be a pn with k < n in which it is no longer a counterexample. Then let Po = \Jn^0Pn- It is easy to see that ;>*, satisfies all C1-C8. Note that if ft' is a ©-counterexample to C9 in p^, ft.' must be such a counterexample in p, i.e., there is no new 0-counterexample to C9 in p$. This is because, first, (13) holds when Treen is replaced by Tree^ since it holds at each step in constructing PQ, and second, each new maximal chain in p$ is a proper super-chain of fto-

16. On the basic one-agent achievement-stit theory

431

Therefore, each new counterexample h! to C9 in p$ must contain some Sh>(w) with w € (Tree^ — Tree] — irree*, and then by (13), 0 C s/ l /(tu), and hence h' cannot be a ©-counterexample to C9 in p$. Note also that by the same token, each new maximal chain in p^, being a super-chain of ho, cannot be a ©'-counterexample to C9 in p$ with ©' C 0. It follows from Lemma 16-10 that each new counterexample to C9 in p^, introduced by the construction of PQ, must be a ©'-counterexample to C9 in pg with 0 C ©'. Now the same argument can be used to deal with h% for any 0 < £ < K, except, when £ is a successor ordinal, we substitute PC_I for p- and when £ is a limit ordinal, we need first let p' = \J0^ ^, p£, and substitute p1 for p. Details are omitted. Finally, we let p* = U o < f < « ^ | ' an<^ ^ *s easy to see that p* satisfies all C1-C8, in which there is no ©-counterexample to C9, and in which each new counterexample to C9 is a ©'-counterexample to C9 with 0 C ©'. D 16-23 LEMMA. (Final construction of "preliminary structures") Let A be any consistent sentence. Then there is a sequence p = (Tree, ^, iTree, f , 92, 9+, <7_) satisfying C1-C9, and A G /(TOO) f°r someTOO6 irreePROOF. By Lemma 16-20 there is a sequence p = (Tree, ^, iTree, /, 9i, 92, g+, g-) satisfying C1-C8, and A G /(TOO) for someTOOG i-rree- Let us arrange all subsets of F into a linear ordering 0 0 ,0i,...,0 r a (recall that F is finite) such that Qz C ©^ only if i < j. Applying Lemma 16-22 repeatedly to get p0 (po = p),pi,... ,pn,pn+i, each Pk + i extending pk, such that for each k with 0 ^ k ^ n, if there is no ©^-counterexample to C9 in pk, Pk + i = Pk', and if there are ©^-counterexamples to C9 in p, there is no such counterexample in any of pk + 1 , . . • , pn +1 • Thus in pn + i, there is no counterexample to C9. n A structure (Tree, ^,iTree, Agent, Choice} (or a BT + I + AC structure (Tree, ^, Instant, Agent, Choice)) is at-most-binary if for each a G Agent and each w G Tree, Choice1^ contains at most two elements. Now we proceed to convert the "preliminary structures" (Tree,^,iTree,f,9i,92,9+,9-) satisfying C1-C9 into at-most-binary BT + I + AC structures. Let (Tree, ^, i-rree, f,9i,92,9+,9-) be any sequence satisfying all C1-C9. For each history h in Tree, we set m^ to be the unique element of h n i-pree, and for each TO € iTree, we set hm to be the unique history h in Tree with TO 6 h (hence m = m/im and h = hmh). For each w e Tree-zjvee, 9i(w) and gz(w) correspond to two possible choices for a at w, and consequently, g+(m,w) corresponds to the possible choice for a at w to which the history hm belongs, and g- (TO, w) corresponds to the possible choice for a at w to which hm does not belong. 16-24 LEMMA. (Constructing at-most-bmary structures) Let (Tree, ^, iTree, f , 9i, 92, 9+, 9-) satisfy all the conditions C1-C9, let a have an interpretation denoted by a, and let Agent = {a} and Choice = {{(a,w),{Hi(w),H2(w)}): w 6 Tree-ijYee} I) {((a,m),H(m)): m e i-pree}', where HI(W) = {hm: m € irree & w < TO & g+(m,w) = gi(w)} and H2(w) = {hm: m 6 i-Tree & w < m & g+(m,w) = g2(w)}. Then (Tree, ^,iTree, Agent, Choice) is at-most-binary.

432

Proofs and models

PROOF. It is sufficient to show that for each w € Tree — iTree, Choice^ is a partition of H(m), and Choice satisfies no choice between undivided histories. Let w € Tree-iTree. It is easy to see by C3 that for each h € H(w), g+(mh,w) is either g\(w) or gz(w), but not both, and hence either h € HI(W) or h £ H2(w), but not both. We know by Cl, C3, and C4 that neither H\(w) nor H^iw) is empty. It follows that Choice1^, is a partition of H^. Let h, h' € #(«,) such that h ^ h' and w' 6 h fl h' for some w' € Tree with w < w'. Then by Cl, w,w' € Tree — iTree, w < w' < rrih and w < w' < rrih<. Hence by C4, g+(mh,w) = g+(mh>,w). This implies that g+ (mh, w) = gl(w) iff g+(mh>,w) = gi(w), and hence by definition of Choice, h € HI(W) iff h' € H\(w). It follows that Choice satisfies no choice between undivided histories. D By this lemma, each p = (Tree, <, i-rree, f,9i,92,9+,9-) satisfying C1-C9 can be converted to an at-most-binary structure (Tree, ^,iTree, Agent, Choice), and we are very close to getting a BT + I + AC structure. The last thing we need to do is to make all histories isomorphic. We will sketch how that can be done. But first let us observe the following fact: When given a consistent sentence A, we apply Lemma 16-20 to construct a p = (Tree,^,iTree,f,9i,92,9+,9-) satisfying C1-C8, A € f(m) for some m e i-n-ee, and it is easy to see that Tree is countable, and hence so is each history in it. In extending p to a p' = (Tree1, ^',iTree',/',9i,92,9+,9-} satisfying C1-C9, applying Lemma 1622, we extend each history in (Tree, <} only finitely many times (since there are only finitely many subsets of F); and at each time, even though we may have added uncountably many histories, each of these histories is obtained by adding countably many new moments to a countable chain (a counterexample to C9). It is then clear that all histories in (Tree1, <'} are still countable. Let p = (Tree, ^,iTree, f,9i,92,9+,9-) satisfy all C1-C9 and be converted to an at-most-binary structure (Tree, ^, iTree, Agent, Choice) (where Agent contains a single agent denoted by a), and let each history in {Tree, ^} be countable. For each history h in (Tree, ^), we can insert (duplicated) real numbers into h and make the extended history isomorphic to, say, the interval [0,1], and let a have a vacuous choice at each newly added moment.2 In such a process, one needs to take care of the common portion of every two histories, which can be done, for example, by fixing a well ordering of all histories in (Tree, <), say, /i0, hi,..., / i £ , . . . (£ < K, where K is the cardinal of the set of all histories), and extending them in the following way. We first extend ho to h'Q such that h'0 is isomorphic to [0,1]. For each ordinal £, assuming that every h^ with £ < £ has wnere been extended to fy, let h'^ ^ is the initial segment of h1^ up to and including x, and extend h^ — P to an F such that F is isomorphic to (0,1] or [0,1] depending on whether P has its last element, and finally let 2

It is easy to see from our construction that there must be a ^-smallest moment in Tree, for each time we extend a structure {Tree!,^',iTree'>/'i9i)92>S+>5-) by adding a new moment to the past of the earliest one in Tree, it is because we have some [a stit: A] with A G F such that g^w) \- [a stit: A] and g'2(w) h [a stit: A] for the ^'-smallest member w e Tree'; and clearly, there can be only finitely many such stit sentences.

16. On the basic one-agent achievement-stit theory

433

When all histories become isomorphic to [0,1], we can easily define (Tree1, <', Instant, Agent, Choice1): (Tree1,^') is the extended tree-like structure just described (i.e., Tree! is the union of all extended histories) with all histories isomorphic to each other. Instant is a set of instants in (Tree!', ^'), with iTree to be the "last one" in it, satisfying unique intersection and order preservation. Agent is the same as before. Choice! is the choice function such that for each w € Tree— i-rree (i-G., each "old" moment not in iTree), Choice'(a,w) = where and for each w e (Tree' - Tree)Uirree (i-e., each "new" moment and each ("old") moment in iTree), Choice'(a, w) = {H(w)}, where H(w) is the set of all histories in (Tree1, <'} passing through w. Because a has vacuous choice at every newly added moment, and because vacuous choice will not affect the truth values of sentences when we evaluate them at moments in iTree, we will, for convenience, use Lemma 16-24 in a way as if we get an at-most-binary BT + I + AC structure (Tree, ^, Instant, Agent, Choice) (instead of (Tree, ^,iTree, Agent, Choice)) with iTree to be the last instant in Instant. This being the case, we will keep using our notations m^ and hm as before. The completeness theorem is included in the following theorem. 16-25 THEOREM. (Completeness theorem) Let A0 be any consistent sentence. Then there is an at-most-binary structure 6 = (Tree, ^, Instant, Agent, Choice) with a model 9Jt on it such that SOT,mo 1= AQ for some mo G Tree. PROOF. By Lemma 16-23, there is a sequence p = (Tree, ^, i-pree, f, ffi, g2, g+, g~) satisfying C1-C9 and AQ e /(mo) for some mo € in-ee- Let us set i = ^Tree- Applying Lemma 16-24, we convert p into an at-most-binary structure (Tree, ^ b, Instant, Agent, Choice) with i to be the last instant in Instant, where Choice is defined as in Lemma 16-24. By C5-C4 and the definition of Choice, it is easy to see that for each m, m' e i and each w € Tree — i,

Let SOT = (&,3) be a model on & such that for every propositional variable p € FA°, every h in Tree, and every m € i with m € h (i.e., m = m^), (m,h) G 3(p) iff p 6 /(m). Applying ordinary induction, one can show that for every A e F^0 and every m € i, 9JT,m t= ^4 iff A 6 /(m), and consequently, since ylo € /(mo), 9Jt, mo 1= AQ. We omit the induction steps for truth-functional operators, but provide the step for [a stit: ] as follows. • Let m £ i and [a stit: A] 6 F^0. Suppose first that [a stit: A] £ f ( m ) . Then there is, by C7, & w £ Tree — i such that w < m, g+ (m, w) h [a stit: A] and g-(m,w) V- A. (15) implies that [a stit: A] £ /(m') for every m' £ i with m' =° m, and, since g-(m,w) ¥• A, there is by C8 an m" e z >^ with A £ f(m"). It follows from induction hypothesis that 971, m' 1= A for all m' € i with m' =^ m, and 9JI, i >w ¥ A. Hence 2Jt, m t= [a stit: A}. Suppose next that there is a w < m such that 9Jl,m' N A for every m' £ i with m' =£j m, and By induction hypothesis, A 6 /(m-') for every m' G i with with

434

Proofs and models

m, and A <£ /(TO") for some m" & i\>w. It follows from C3, C8, and (15) that g+(m,w) h A and g-(m,w) V- A. Now if g+(m,w) \- ~[a stit: A], we would have g-(m,w) \- A by C2 and Lemma 15-28(Hi), a contradiction. It follows that and hence by Lemma 15-22(ii), g+(m,w) h [a stit: A}. Since by The following remarks are worth adding to the main completeness result, which has been established with Theorem 16-25. REMARK. (At-most-binary BT +1+AC structures) Because the BT + I+AC structure mentioned in Theorem 16-25 is at most binary, Theorem 16-25 gives us more than the completeness of Lai. Let Lali be the set of all sentences valid for all at-most-binary BT + I + AC structures. Identifying Lai as the set of all its theorems, we have as a consequence of Theorem 16-25 and Theorem 16-4 that Lai = Lali. REMARK. (Two kinds of validity) Let 6 be any BT + I + AC structure. A settled interpretation on & is any interpretation 3 on 6 such that for every propositional variable p, and for everyTO,h in 6 with TO e h, (m,h) € 3(p) iff V/i'[m 6 h' -> (m',h) e 3(p)]. An BT + I + AC model OT = (6,3) on 6 is settled if 3 is a settled interpretation on 6. A settled interpretation is obviously an assignment of truth values relative only to moments rather than moment-history pairs. Since we used settled models in our proof of Theorem 16-25, it is easy to see from Theorem 16-4 and Theorem 16-25 that validity for all BT + I+AC models and validity for all settled BT + I+AC models are the same, as far as the object language used in this chapter is concerned. REMARK. (About the finite model property) Lai does not have the finite model property. A simple example is refref, Ax. Cone. 1. Chapter 15 showed that refref is valid for a BT + I+AC structure iff the structure contains no busy choice sequence. Thus if Lai had the finite model property, refref would be a theorem of Lai; but, in fact, it is not.

17

Decidability of many-agent deliberative-stit theories In this chapter we give an axiomatization, Ldm, for the basic dstit logic, and prove its completeness and decidability by way of the finite model property. It is surely very natural to combine dstit theory with indeterminist tense logics, especially when we consider deliberative seeing to something to be connected with what the future will be like. In carrying out some basic technical work in dstit theory, however, we will use a formal language without tense operators, though we will use the historical necessity operator Sett:, §8F.4, as a primitive. In our formal language, we will introduce Chellas's cstit operator, §8G.2, as an abbreviation, just as in §11 A. The reader familiar with modal logic can easily see that the Chellas 1992 theory of cstit is decidable, since Chellas did not propose any condition concerning the relation among different agents. Our proof actually shows that cstit theory is decidable even if one adds the independence of agents condition, Post. 9. How many possible choices does an agent have at a given moment? For a logic of agency, this is an interesting question. We will show some results about the expressibility of our formal language in this aspect, that is, for each number n ^ 1, we have a scheme that corresponds to the condition that every agent has at most n possible choices at every moment. We will also give, for each n ^ 1, an axiomatization for the dstit logic with this condition, and prove its decidability.

17A

Preliminaries

The language for Ldm contains as primitive symbols, propositional variables po, Pi, P2, ..., agent terms QO> c*ij a2, ••-, &n equation symbol =, truth-functional operators ~ and &, the historical necessity operator Sett:, and the dstit operator [ dstit: ]. Sentences are constructed in the usual way, except that a = /3 is a sentence whenever a and /3 are agent terms, and [a dstit: A] is a sentence 435

436

Proofs and models

whenever a is an agent term and A is a sentence. We will use A, B, C, etc. to range over sentences. As usual, V, D, =, T, _L, and Poss: (§8F.4) are introduced as abbreviations. In addition we use a ^ 0 for ~(a = (3) and introduce the following as abbreviations, where A is any sentence, and a, (3o, @i, ... are any agent terms: 17-1 DEFINITION. (Chellas's stit operator) 17-2 DEFINITION. (Distinct agents) diff(/3

and for any

We shall be working with BT + AC structures 6 (no instants), §2. The independence of agents postulate Post. 9 is of particular importance in this chapter. Given a BT + AC structure 6, a model 97t on 6 is a pair (©, 3} where & is a BT + AC structure and 3 is an interpretation such that for each prepositional variable p, 3(p) is a subset of {(m, h): m € h}, and for each agent term a, 3(a) = a £ Agent.1 Let A be any sentence. We define the truth of A at m/h in 97T, written 97t,m//iN A, recursively as indicated by the relevant clauses in §8F.l, §8F.2 (including the clause for identity), §8F.4, and, especially, the clause for dstit, §8G.l. It is easy to verify by Definition 17-1 and the recursive definition of 1= that the following hold, where a is any agent term: 17-3 FACT. (Consequences of truth definition) • 97T, m/h t= Poss:A iff 97T, m/h' \= A for some h' with m € h; and • 971, m/h 1= (a cstit: A] iff m, m/h' N A for all h' with h' =° for all h' with h' = ^ h; and hence

h iff m,

Recall that the choice equivalence relation, Def. 12, is an equivalence relation, and hence by the recursion clauses of the truth definition, the operators Sett:, and [a cstit: ], [/3 cstit: ], and so on are just like the necessity operator in the modal logic S5, as noted in §11A. As in Def. 20, we define the validity of A for a model 9JT, 9JT 1= A, as 9JI, m/h 1= A for every m and h in 971 with m e h, and the validity of A for a structure &, 6 \= A, as 9711= A for every model 97T on 6.

Our logic Ldm takes as axioms all substitution instances of truth-functional tautologies as well as the following schemata, where a, (3, 7, and /?o, ..., (3k are any agent terms:

lr

That is to say, for every agent term a, both 3(a) and a denote the agent in Agent that 3 assigns to a. In each situation, we will use whichever is convenient.

17. Decidability of many-agent deliberative-stit theories

437

is any sentence obtained from A by replacing some or all occurrences of with

and takes as rules of inference modus ponens and RN from A to infer Sett:A. AIA can be called an axiom scheme for independence of agents. For convenience we sometimes refer to Ldm as Ldmo; and for each n ^ 1, we define Ldmra to be the axiomatic system obtained by adding the following to Ldm as an extra axiom scheme, where a is any agent term:

APC can be called an axiom scheme for possible choices. In the next section we show that for every BT + AC structure 6 and for every n ^ 1, (3 1= APCn iff every agent (in 6) has at most n possible choices at every moment (in &). Other syntactic notions such as theorems in Ldmn (l~Ldin n )) Ldmn-consistency (consistency w.r.t. Ldm n ), maximal consistent sets w.r.t. Ldmra (Ldm n -MCS), and so on are defined as usual. It is easy to see that for each i and k with 1 ^ i < k, hLdm, APCfc, and hence, identifying Ldmn with all its theorems, we have Ldm C ... C Ldm3 C Ldni2 C Ldmi. In fact, as the soundness theorem shows, we have Ldm C ... C Ldm3 C Ldm2 C Ldmi. It is easy to see by Definition 17-1 that the following are derivable in Ldm (and hence in all Ldmn):

2 When taking Sett: and cstit as primitive, one needs to introduce A}&,~Sett:A as an abbreviation, and replace A3 by Sett:A D [et cstit: A}. When taking dstit and cstit as primitive, one needs to introduce Sett:A f±dj [a cstit: A]&~[a dstit: A] for some particular a, and replace A3 by

438

Proofs and models

Note that although the decidability of Ldm will also give us the decidability of the dstit theory with a single agent, it would be nice to have an axiomatization of that theory. It is easy to check, along with our proof, that Al, A2, A3, and RN will be sufficient for that axiomatization. Note also that A2 does not directly show the a priori plausibility of the dstit operator. It is easy to see, however, that the following are derivable in Ldm:

Let L be the axiomatic system obtained from Ldm by replacing A2 with T5 — Til, and replacing AIA^ with T12fc. Then L and Ldm are equivalent in the sense that they have the same set of theorems (the proof is omitted). There is interest to be found in the following facts, which echo other discussions of refraining in this book. In both astit and dstit theories, doing (w.r.t. the fact that ^4) is represented by [a stit: A] ("a sees to it that A"), refraining from doing is represented by [a stit: ~[a stit: A}}, and refraining from refraining from doing is represented by [a stit: ~[a stit: ~[a stit: A]}}. In dstit theory, being able to do is represented by Poss:[a dstit: A}. (See §2B.6 and §2B.8 for an analysis of refraining and refraining from refraining; and see chapter 9 for information on the ability to do.) It is easy to see that the following are all derivable in Ldm:

3

Recall from, e.g., §2B.8 that the refref equivalence does not in general hold for astit. T13 indicates that the stit analysis of refraining coincides with von Wright's analysis of refraining—not doing conjoined with the ability of doing (see von Wright 1963). As noted in Horty 2001, the left-hand side of T13 is only an approximation of von Wright's analysis of refraining.

17. Decidability of many-agent deliberative-stit theories

439

T14 and T15 give us that doing is equivalent to refraining from refraining. That is to say, the following refref equivalence holds for dstit: (refref

17B

Soundness

In this section we prove the validity of AIA^ for all k ^ 1 and some results about the expressibility of the language concerning possible choices, and finally we establish the soundness theorem for Ldm n with all n ^ 0. 17-4 LEMMA. (Validity of AIA^) BT + AC structure.

For each k ^ 1, AIA^ is valid for every

PROOF. Let 6 be any BT + AC structure, and let 2JI = (&, 3) be any model on 6, and let k ^ 1. Suppose that 9JI, m/h \= diff(/30, ..., /?fc)&Poss:[/? 0 cstit: B0] & ... & Poss:[/3k cstit: Bk\ for some m and h in 9Jt with m 6 ft. Then by truth definition and Definition 17-2, all 3(/?o)> • • • , 3(/?fc) are different agents in Agent, and there are fto, ..., ftjt such that for each i with 0 ^ i ^ k, OT, m//i» ^ [/?, cstit: Bt}. Hence by Fact 17-3,

Let choicem be any function on Agent such that choice ftz} for each z with 0 fli, •••}• For each /c ^ 0, we set /i^ = {m, mjt}. It is easy to see that #( m ) = {ho, h\, ...}. Let us define Choice to be a function on Agent* Tree such that for each k ^ 0, C/ioicem = {Hk,i, #fc,2} where /f^i = {ho, ••-, h^}, Hk,2 = {/ifc + l. hk + 2, •••}• Since f l k s o Hk,2 = 0, we know that Choice does not satisfy the independence of agents postulate. If, however, we treat & = (Tree, ^, Agent, Choice) as a BT + AC structure, and if we keep all other semantic notions the same, then we will still have & N AIA& for every k ^ 0. Examples of this kind suggest that there are no sentences corresponding to the independence of agents condition since we can only talk about finitely many agents in each sentence. What {AIA^: k ^ 1} corresponds to is the weaker condition that for each m e Tree, and for each function choicem on any finite subset Agentf of Agent such that

440

Proofs and models

17-5 LEMMA. (APCn and numbers of possible choices) Let S = (Tree, ^, Agent, Choice) be any BT + AC structure, and let n ^ 1. Then 6 N APCn iff Choice^ has at most n elements for all a £ Agent and all m £ Tree. PROOF. Suppose that \Choiceam\ ^ n for all a € Agent and all m 6 IVee. Let 37T be any model on 6. If n = 1, then it is easy to see that 97T 1= Poss:[a cstit: A] D A. Assume that n > 1 and set a to be any agent term, and A\, ..., An any sentences. We show as follows that 971 N APC n . Suppose for reductio that 271, m/h ¥ APC n for some moment m and some history h with m £ h. Then 971, m//i t= POSS:[OJ cstz't: AI], 371, m//i 1= Poss:(~^i&[a cstit: A 2 ]), ..., 37T, m//i t= Poss:(~4i & ... & ~,4 n _i&[a cstz't: A n ]), and 971, m//i 1= ~^i & ... & ~>l n . Then there are hi, ..., hn such that for each i with 1 ^ i ^ n, m £ /i, and

For each i with 1 < z < ra, let us set e% = {h': h' =^ ht}. Since 371, m/h t= ~>li & ... & ~>4 n , we know by (2) and Fact 17-3 that for each i with 1 ^ i ^ n, /i _L^ /i,. That is to say, h £ e, for all i with 1 ^ i ^ n. Since Choice3^ is a partition of -ff( m ), there must be an e € Choice^"' such that h & e. But obviously e ^ e, for all z with 1 ^ z ^ n. It follows that |(7/iozce^a'| ^ n + 1, contrary to our supposition that \Choice3Jf^ < n. We conclude from this reductio that 371 N APC n . Next suppose that [Choice^ ^ n + 1 for some a £ j4
Let e, ei, ..., en be n +1 different elements of Choice^, let h € e, and let hl £ e, for each i with 1 ^ z ^ n. We define 37T = {©, 3), where 3 is any interpretation such that 3(a) = o, and for each i with 1 ^ z < n, 3(p») = {(m, h'): h' £ e,}. Since e, ei, ..., en are all different, it is easy to verify that 371, m/h \= ~pi & ... & ~p n , 971) m//ii 1= [a cstit: pi], and 97T, m/ht t= ~pi & ... & ~p l _i&[a csiz'i: p,] for every z with 1 < i < n. It follows that 971, m/h N Poss:[a csiz'i: pj] &Poss.-(~pi&[a cstit; p 2 ]) & ... & Poss:(~pi & ... & ~ p n _ i & [ a csizt: p n ]) and 97T, m/h ¥ p i V . . . V p n . It follows that (3) holds. Since it is easy to verify that Al — A6 are valid for all BT + AC structures and RN is validity preserving, Lemma 17-4 and Lemma 17-5 give us the following. 17-6 THEOREM. (Soundness) For every sentence A, \~Ldmo A only if 6 N A for every BT + AC structure 6; and for each n > 1, ^i,dmn A only if 6 N A for all BT+AC structures 6 in which every agent has at most n possible choices at every moment.

17. Decidability of many-agent deliberative-stit theories

17C

441

Completeness and compactness

In this section we prove the strong completeness theorem for all Ldmn with n ^ 0, which implies compactness. Our proof is based on a technique from modal logic, using canonical models, as explained, for instance, in Segerberg 1971 or Chellas 1980. We will assume the reader's familiarity with this technology, and will mainly deal with the general independence of agents condition and the conditions about possible choices. For any L = Ldm n with n ^ 0, let WL be the set of all L-MCSs, and let #L be the relation on W\, such that for each w, w' e WL, wR-^w' iff {A: Sett:A G w} C w'. By Al and RN, we know from modal logic that RL is an equivalence relation. Let us use X, X', and so on to range over RI.-equivalence classes. Let X be any fiL-equivalence class. We know that the restriction of RL to X is a universal relation. We define a relation = among all agent terms such that for each a and 0, a = j3 iff a — (3 G w for some w € X. It is easy to see by T3 that a ^ j3 iff a = (3 e w for all w e X, and a £ /3 iff a / 0 e w for all w £ X. We know by A4 that = is an equivalence relation. For each agent term a, we set [a] to be the ^-equivalence class to which a belongs. Let us assume that PQ, /3i, ... is an enumeration of the representatives of all = -equivalence classes. The agent frame $ for L on X is the sequence (X, [Po], [Pi], •••)• Let us define, for each agent term a, a relation R[a] on X such that for every w, w' € X, wR{a]U>' iff {A: [a cstit: A] € w} C w'. By ordinary modal logic, we know that for each a, R[a] is an equivalence relation. We set E[a] to be the set of all U[Q]-equivalence classes. It is easy to see that if a = /3, R^ is the same relation as R[p\. We will call the sequence (X, [Po], [Pi], • • • , R[00], R[0i]i • • • } the canonical frame for L w.r.t. X. 17-7 LEMMA. (Properties of the canonical frame for L w.r.t. X) Let L = Ldmra for any n ^ 0, and let (X, [Po], [Pi], •••, R[p0], R[0\]i • • • ) be the canonical frame for L w.r.t. X. Then the following hold: i. For each w € X and each A, Sett:A 6 w iff A € w' for all w' € X iff Sett: A e w' for all w' € X; ii. for each w G X and each a and A, [a cstit: A] G w iff A € w' for all w' E X with w'R[a]W iff [a csizt- ^4] € w' for all w' £ X with «/.R[Q]w; iii. for each w & X and each a and A, [a dstit: A] e w iff ^4 6 w' for all w' € X with w'R[a]W, and ~^4 6 w" for some w" 6 X. PROOF, (i) and fiij are trivial (apply modal logic and T2), and (Hi) can be easily obtained by Tl, (i) and (ii). D 17-8 LEMMA. (Canonical frames and independence of agents) Let L = Ldmn for any n ^ 0, and let (X, [po], [Pi], ..., R[p0\, R^], ...} be the canonical frame for L w.r.t. X. Suppose that / is any function on {[Po], [Pi], ...} such that /([&]) e %,] for all i 2 0. Then f l o o /([/?,]) ^ 0-

442

Proofs and models

PROOF. Let / be any function on {[/30], [/?i], ...} with /([/?,]) € E[0j for all i ^ 0. We show as follows that f ) t > o /([/?«]) ^ 0- We know by modal logic and Lemma 17-7(i, ii) that there are AQ, AI, A%, ... such that

and that for each i ^ 0, there are Blto, B^i, S t ,2, ••• such that

Since [/?o], [/?i], [,$2], ••• are different =-equivalence classes, we have that

We first show that 6 = As e «:U(U t ^oA,) is L-consistent. To that end, it is sufficient to show that

It is easy to see by (5) that for each i with 0 ^ i ^ k, there is a wt € /([/? t ]) € E[pt] such that [/?, cstit: Bt] 6 «;,. Then, selecting any w £ X, we have by (4), (6), and Lemma 17-7 (i) that

It follows from AIA fc that Sett:A&Poss:( 6 w, and hence by modal logic, Poss:(Sett:A Bk}) € w, and consequently, Sett:A consistent. It follows that (7) holds, and hence © is L-consistent, and is included in some L-MCS w* € W. Since Asett: C w*, we know by (4) that w* € X; and since Az C w* for each i ^ 0, we know by (5) that w* e /([/?«]) for each i ^ 0. It follows that 17-9 LEMMA. (Canonical frames and numbers of possible choices) Let L = Ldm n for any n ^ 1, and let (X, [/?o], [/3j], ..., ^?[/30], R[PI], •••) be the canonical frame for L w.r.t. Jf. Then for each i ^ 0, there are at most n .R^j-equivalence classes, that is, PROOF. Suppose for reductio that there is an i ^ 0, such that Setting a to be /?,, we know that there are CQ, ..., en G E[a] such that eo, ..., en are all different. Let us select WQ E e0, ..., wn £ en. It follows from Lemma 17-7(ii) that there are AI, ..., An such that for each i with 1 ^ i < n, [a cstit: Ai] e Wi and ~ylt € w/t for all A; with 0 ^ k ^ n and i ^ k. That is to say, It follows from Lemma 17-7(i) that

17. Decidability of many-agent deliberative-stit theories

443

and hence by APC n , J_ € WQ, contrary to our assumption of L-consistency on WQ. We conclude from this reductio that |j5[/3,] ^ n for all i ^ 0. 17-10 THEOREM. (Strong completeness) Every Ldm-consistent set 0 of sentences is satisfiable in a BT + AC structure; and every Ldmn-consistent set 0 of sentences (n ^ 1) is satisfiable in a BT + AC structure in which every agent has at most n possible choices at every moment. PROOF. Let L = Ldmn for any n ^ 0, and let G be any L-consistent set of sentences. Then there is a L-MCS w € Wi, including Q. Suppose that X is the ^-equivalence class to which w belongs, and let (X, [/30], [/?i], • • • , R[p0], R[0i]> ...} be the canonical frame for L w.r.t. X. We want to convert (X, [/30], [/?i], ..., R[p0], Rtfi], •••) into a BT + AC structure 6 = (Tree, ^, Agent, Choice}. Let us first set Tree = { X } U X ;

Then we set hw = {X, w} all w € X. It is clear that hw is the unique history in (Tree, ^) to which w belongs, and that there is a one-one correspondence between all w G X and all hw in (Tree, ^). Finally let us set

Note that at each w, each agent [/3t] has a vacuous choice. It is easy to see that for each i ^ 0, Choice % is a partition of H(x), and that the no choice between undivided histories condition is vacuously satisfied. To see that Choice satisfies independence of agents, let choicex be any function on Agent such that choicex ([0z]) € Choice£ f°r each i ^ 0. Then by definition of Choice, for each i ^ 0, there is an e% 6 ^[/g,] such that choice x ([(31}) = {hw: w € et}. Let / be the function on Agent such that for each i ^ 0, /([/?,]) = e, € £[/?,]• Since there is a one-one correspondence between all w £ X and all hw in (Tree, <}, it is easy to see that for any w & X and any i ^ 0, w 6 /([/3Z]) iff hw € choicex([/?i]), and hence by Lemma 17-8, Hoo choicex([/3i}) ^ 0- It follows that Choice satisfies independence of agents, and thus & is shown to be a BT + AC structure. When L = Ldmn with n ^ 1, we need to show that each agent in © has at most n possible choices at every moment. Since \Choice^\ = I for every i ^ 0 and every w £ X, we only need to show that \Choicex ^ n f°r every i ^ 0. But we know by definition of Choice that for each i ^ 0 and each k ^ 1, It follows from Lemma 17-9 that n.

444

Proofs and models

Let us define a model 9Jt = (&, 3) such that for each agent term a, 3(a) = [a]; and for each propositional variable p and each history hw in &, (X, hw) 6 3(p) iff p e 10. It is easy to see by our definition that for each h in 6, h € #(x) iff 3w e X[/i = hw]; and that for each agent term a, and for each w, w' € X, hw = %' hwi iff 3e € £[Q][w, «;' € e] iff wR^w'. Then one can show by induction that 3JT, X/hw N ^4 iff A £ w for every ,4 and every w 6 X, applying Lemma 17-7(i) and Lemma 17-7'(Hi). 17-11 COROLLARY. (Compactness) For every set © of sentences, if every finite subset of 6 has a BT + AC model, then so does 9; and if every finite subset of Q has a model in which every agent has at most n possible choices at every moment, then 6 has a model of the same kind. Some general remarks are worth making. Let us define a dstit logic as a set of sentences that contains all truth-functional tautologies, all Al —A5 and all AIAfc for k > 1, and is closed under substitution, modus ponens, and RN. Let L be any dstit logic. A model for L is any BT + AC model 9Jt such that 271 t= A for all A € L. L is complete for its models if for every A £ L, there is a model for L such that 371¥• A. Going over Lemma 17-7, Lemma 17-8, and the first part of Theorem 17-10, (considering L there as an arbitrary dstit logic), one can easily see that the following holds. 17-12 REMARK. (Completeness for models) Every dstit logic is complete for its models. A general BT + AC structure is any triple 03 = (&, A, Z) such that (i) & = (Tree, <, Agent, Choice) is a BT + AC structure; (ii) A = {QO, ai, ...}, where ao, ai, ... are all the agent terms in our language and an € Agent for every n > 0; and (in) Z is a subset of the power set of {(m, h): m € h} and is closed under Boolean operations as well as the following, where z G Z and a € A:

Let 03 = (6, A, Z) be any general BT + AC structure. A model on 23 is any BT + AC model m = (&, 3) such that 3(a n ) = an for every n ^ 0 and J(p) 6 Z for every propositional variable p. We set 53 1= A iff 9Jt t= A for all models 9Jt on 03. Let L be any dstit logic. 03 is a general BT + AC structure for L if 03 N A for all A £ L. L is complete for its general BT + AC structures if for every A £ L, there is a general BT + AC structure 23 for L such that 23 J^ A. 17-13 REMARK. (Completeness for general BT + AC structures) Every dstit logic is complete for its general BT + AC structures.

17. Decidability of many-agent deliberative-stit theories

445

PROOF. Let L be any dstit logic, and let A £L. We know by Remark 17-12 that there is a model SDt = (6, 3) for L such that VJl F A. Let us set an = 3(a n ) for every n ^ 0, and A = {a0, a{, ...}; and set \\B\\m = {(m, h): 971, m/h 1= 5} for every sentence 5, and Z = {||B|| : 5 is a sentence}. It is easy to show by induction that 05 = (S, A, Z) is a general BT + AC structure. We show as follows that 03 is a general BT + AC structure for L. Suppose for reductio that B e L but m* ¥ B for some B and some model 9JT = (6, 3*} on 03. Let B = B(p0, ..., pk) (all prepositional variables occurring in B are among p0, ..., pk). By definition 3*(p 0 ) = I I C o H ™ , • • • > ?*(Pk) = {{Ckf* • Consider the sentence B' = B(C0/po, ..., Ck/Pk), that is, the sentence obtained from B by substituting Ct for pt for each i ^ k. We claim that 97t ¥• B'. To justify this claim, it is sufficient to observe that for every sentence D(po, ..., p k ) , 9JT, m/h t= D(p0, ..., pk) iff 9JI, m//i N D(Co/po, ..., Ck/Pk), which can be easily established by induction. Since B e L, and since L is closed under substitution, we know that B' € L, and hence, since OT is a model for L, OK t= B', which is a contradiction. From this reductio we conclude that B 6 L implies SOT* t= B for every sentence B and every model 971* on 03, that is, 03 is a general BT + AC structure for L. It is easy to see that 9JI itself is a model on 03, and hence, since 971 ¥• A, 03 ¥• A. 0

17D

Finite model property

We show in this section that all Ldmra with n ^ 0 have the finite model property, and hence they are all decidable.5 The basic idea in our proof of the finite model property for Ldm,,, is borrowed from the filtration theorem in modal logic. Let A be any sentence. We define several sets of sentences as follows.

5 Note that the decidability of dstit theories is not a direct consequence of Gurevich and Shelah 1985a and Gurevich and Shelah 1985b, since in their result, the theory on tree-like frames does not contain a quantifier over sets of choice-equivalent histories, which is needed because of the presence of Choice in BT + AC structures, together with the truth conditions for dstit.

446

Proofs and models

It is easy to verify that HA is finite. HA is particularly designed, in accordance with AIA, for the purpose of dealing with the independence of agents postulate, with the help of a property of AA stated in the following lemma. 17-14 LEMMA. (A property of AA) Let L = Ldmn for any n ^ 0, and let A be any sentence. Then for each j3 occurring in A and each sentence [(3 cstit: B] € A A, there is a sentence B' such that KL [ßcstit: B'} = [ßcstit: ~[ßcstit: B}], and [ft cstit: B'} e AA. PROOF. It is easy to see that if [(3 cstit: B] £ AA, then either [ß cstit: B] € S0 or [ßcstit: B] = [ßcstit: ~[ßcstit: C]] with [ß cstit: C] e E0. Suppose that [ßcstit: B] e AA. If [ßcstit: B} e E0, then, setting B' = ~[/S cstit: B] 6 AA, we know that [ßcstit: B'] € Si- If [ßcstit: B] e £i-E 0 , then [ßcstit: B} = [ßcstit: ~[/S cstit: C]] for some [ßcstit: C1] € SQ, and hence [ßcstit: ~[ßcstit: 5]] = [ßcstit: ~[ßcstit: ~[/S cstit: (?]]]. Setting B' = C and applying modal logic, we have [/S cstit: B1} € EI. D Let L = Ldmn with n ^ 0, let .AT be any ^-equivalence class, and let $ = (X, [/So], [/Si], ...} be the agent frame for L on ^. Consider any sentence A. We first define a relation « on X as follows: For each w, w' £ X, w ~ w' if for every 5 6 11^, B & w iff B £ w'. w is obviously an equivalence relation. Since UA is finite, there are only finitely many ^-equivalence classes. Selecting a representative for each of these w-equivalence classes, we set Y to be the set of all those representatives. Next, since there are only finitely many agent terms occurring in A, there must be finitely many =-equivalence classes, say [/So], • • - , [f3k], such that each ßoccurring in A belongs to [/S,] for some i with 0 ^ i ^ k, and each [/St] with 0 ^ i ^ k contains at least some J5 occurring in A. We may assume, without loss of generality, that all /So, ..., j3k occur in A. A filtration of $ through AA and II^ is a sequence (Y, [/So], ..., [/3/t], —[/3 0 ], ..., ~[0 fc ]), where F is the set of representatives for ^-equivalence classes, and for each i with 0 ^ i ^ k, —\p,] is the relation on Y such that

Clearly each ~ ^j with 0 ^ i ^ k is an equivalence relation. Note that the filtration defined here is a little different from those in the area of modal logic. Each filtration there is obtained through a single set of sentences, while the filtration here is defined through two different sets: The «-equivalence classes are determined by HA, but the relation ~ ^ is defined in terms of AA rather than H^.6 Note also that when we choose different representatives for «-equivalence classes, we will have different nitrations, but they must be isomorphic to each other. We will thus speak of "the filtration of # through A^ 6

It is, however, not essential to define our filtrations in terms of two sets of sentences. It is possible to design a single set of sentences such that when we define a filtration in terms of this single set, the filtration still has a property corresponding to the independence of agents postulate.

447

17. Decidability of many-agent deliberative-stit theories

and 11^" as if there is only one such filtration. Consider each i with 0 ^ i ^ k. Since Y is finite and — [/?,] is an equivalence relation on Y, there are finitely many —[/?,]-equivalence classes. We will use t/[/3,], for each i with 0 ^ i ^ k, to denote the set of all —[/?,]-equivalence classes, and use u, u', and so on to range over its elements. 17-15 LEMMA. (Properties of filtmtions) Let L = Ldmn for any n > 0, let X be any ^-equivalence class, and,let $ = (X, [/So], [/Si], • • • } be the agent frame for L on X. Suppose that A is any sentence and (Y, [/So], • • - , [/Sfc], — [/90], • • • , — [/3fc]) is the filtration of ^ through A. A and H^. Then for any w £ y, then Sett:B

i. if Sett:B ii. if

iii. if and

iff B

for all

then

iff B

for all

then

iff B

for all

with

with

for some

iv. for each PROOF. (%) Let 5ett:5 e UA. Suppose that Sett:B £ w. Then B £ w' for all w' e .AT, and hence, B £ w' for all w' e Y. Suppose that Sett:B £ w. Then there is a, w" £ X such that B £ w". Since B £ 11^, and since there is a w' £ Y such that w' « w", it follows from our definition of fa that 5 ^ «/. (ii) Let [ßcsttt- 5] e A^. Suppose that [ßcstit: B] £ w. Let 7 be the representative of [/?]. Consider any u/ e F with w' —\p\ w. Then w' ~[7] w and [7 csiit: B] £ w by A5. Since [7 csttf: B] £ A.A, it follows from the definition of — [7] that [7 cstit: B] G w', and hence by A2, B G w'. It follows that B € w' for all w' £ Y with w' ~[^j w. Suppose that [ßcsiii: 5] ^ w. Then by A5, [7 cstit: B] £ w. Since w £ Y C X, it follows from Lemma 17-7 (ii) that there is a, w" £ X such that B £ w" and

Since there is a w1 £ Y such that w' « w", it follows from (8) and the definition of ~[ 7 j that w ~[ 7 j w' (i.e., w c^j w') and B ^ w'. (Hi) Let [yS dsiii: 5] 6 ZA. By Tl we know that [(3 dstit: B} £ w iff [ßcstit: B}&~Sett:B £ w. Since, by definition of UA, [ßcstit: B] £ AA and ~Sett:B £ A A £ HA, it follows from (i) and fz'z^ that [ßd5
448

Proofs and models

By (9) we know that j + l + 2 ^ |A^|. Since [/?» cstit: £>0], ..., [ß, cstit: £>,] Ayi, we know by Lemma 17-14 that there are £o, • • - , EI such that

Selecting some wt e w,, [ß, cstit: Co] & ... & [/Si cstit: CJ&f/S, cstit: E0] & ... & [0i cstit: EI] e wt by (10) and (12). Now let us set B, = Co & ... & C", &Eo & ... & -E1. Then by modal logic, [ß, 0cstit: jB t ] 6 u;,. In general, we can define BQ, ..., Bk the same way. Selecting w0 € MO, ..., Wk E ujt, we have It follows from modal logic that Poss:[f3o cstit: BQ] & ... & Poss:[ßk cstit: Bk] € to for some (actually, all) w e Y. Since [ß0], ..., [Pk\ are different =-equivalence classes, it follows from the definition of = that diff(ß0, ..., ßk)&Poss:[(30 cstit: B0] & ... & Poss:[j3k cstit: Bk] € w, and hence by AIAfc, Recall that for each i with 0 ^ i < k, Bt = C0 & ... & Cj&E0 & ... & £; with [ßt cstit: C"0], ..., [/?t cstit: Cj], [ßt cstit: £0], ..., [/?» cstit: jBj] e AA and j + i + 2 < |AA • Hence Poss:([/?0 cstit: 50] & ... & [/?fc cstit: B fc ]) 6 HA by the definition of IIA. It follows from (14) and (i) that In the following, we show that w' € uoC\...r\Uk- Consider any i with 0 ^ i ^ k. Let us assume that the same situation described in (9) — (12) still holds here. We have by (15) that [/?, cstit: (C0 & ... & C.,&E0 & ... & £;)] e ™'> and hence by modal logic [0t cstit: C0] & ... & [/?* cstit: C^]&[/?, cstit: J50] & ... & [ßt cstit: Sj] € w'. It follows from (12) that

and hence by A2, [ß, cstit: CQ] &...&[ßcstit: C f J ]&~[ß i cstit: D0] & ... & It follows from (9)-(11) that w' € w t. Thus we can show in general that w' £ MO, • • • , w' £ M& the same way. It follows that 17-16 LEMMA. (A property of filtrations related to possible choices) Let L = Ldmn for any n ^ 1, let X be any 7?L-equivalence class, and let # = (X, [ß0], [/?i], ...) be the agent frame for L on X. Suppose that A is any sentence and (y, [ß0], ..., [ßk], -[ß 0 ]i • • • > -[i9t]) is tne filtration of 5 through A A and HAThen for each i with 0 < i < fc, there are at most n ~[0t]-equivalence classes, that is, | [/[£,] | ^ n.

449

17. Decidability of many-agent deliberative-stit theories

PROOF. Suppose for reductio that |^[/3,]| ^ n+1 for some i with 0 ^ z ^ fc. Let a = @t, let UQ, ..., un G C/[Q] such that UQ, ..., w n are all different, and let Wo € UQ, ..., wn 6 w n . It follows from Lemma 17-15(ii) that there are [a cstit: AI], ..., [a cstit: An] € A^ such that for each i with 1 < i < n, [a cstii: .Aj] € Wi and ^4j ^ lu^ for all j with 0 ^ j ^ n and i ^ j. That is to say, ~(^4i V ...VAn) e w0, [a cstit: AI] e wi, ~^i&[a cstit: A2] & io2, ..., ~^4i & ... & It follows from Lemma 17-7(i) that

and hence by APC n , _L G WQ, contrary to our assumption of L-consistency on WQ. We conclude from this reductio that l-Ej/3,]! ^ n for all z < k. 17-17 THEOREM. (Finite model property) Every Ldm-consistent sentence A is satisfiable in a finite BT + AC model; and every Ldmn-consistent sentence (n ^ 1) is satisfiable in a finite BT+AC model in which every agent has at most n possible choices at every moment. PROOF. Let L = Ldmra for any n ^ 0, and let A be any L-consistent sentence. Set X to be any ./^-equivalence class such that A £ w for some w G X, and J = (X, [/?o], [/3i], ...) to be the agent frame for L on X, and (F, [/30], • • • , [/?n], ^[/30], - . - , —[/3 t ]} to be the filtration of $ through A^ and 11^. We first construct a structure 6 = (Tree, ^, Agent, Choice} as follows. Let us set Tree = {Y}UY,

Since Y is finite, Tree is clearly finite. Similar to our proof of Theorem 17-10, for each w £ Tree, we set hw to be the unique history in (Tree, ^) to which w belongs, that is, hw = {Y, w}. Note that there is a one-one correspondence between all w £ Y and all hw in (Tree, ^}, that is, for every h in (Tree, <},

Finally we set for each i with for each i with

and each

The no choice between undivided histories condition has been trivially satisfied. We show as follows that Choice, so defined, satisfies independence of agents. Consider any function choice y such that for each i with 0 ^ i ^ k, choiceY([0i]) = H! 6 Choicey . It is easy to see by definition that for each i with 0 ^ i < k, there is a Ui 6 t/^,] such that H% = {hw: w 6 ut}. Since there

450

Proofs and models

is a one-one correspondence between all w G Y and all hw in (Tree, ^), it is then clear that for each w G Y and each i with We know by Lemma 17-15(ii) that there is a w' 6 UQ(~}...r\Uk, and hence, h that is, When L = Ldmn with n > 1, we need to show that each agent in S has at most n possible choices at every moment. Since \ Choice^ \ = I for every w G Y and every i with 0 ^ i ^ k, we only need to show that \ChoiceY \ ^ n for every j with 0 ^ i ^ k. But we know by definition of Choice that for each i with 0 ^ i ^ fc and each number n' ^ 1, follows from Lemma 17-16 that for every i with 0 < i < fc. It is easy to verify that for every [7] 6 Agent and every w, w' G Y,

Let us now define an ET + AC model OK = (6, 3} on & by denning 3 in such a way that for each agent term a occurring in A, 3(a) = [a]; and for each prepositional variable p occurring in A, and each (Y, hw) with w & Y, (Y, hw) G 3(p) iff p & w. We show by induction that for every sentence B G IU and every w G V,

It is easy to see by definition of 3 that (18) holds when B is a prepositional variable or is of the form a = /?. The inductive steps for truth-functional operators are straightforward. Let B = Sett:C for some C. We know that 271, Y/hw 1= Sett:C iff 971, Y/A^ 1= (7 for all A™/ G # ( y ) , and then by (16), 971, Y/A™ t= Sett:C iff SOT, Y//!„,' 1= C for all w' G Y, and by induction hypothesis, m, Y/hw \= Sett:C iff C & w' for all w' G F, and hence by Lemma 17-15ft), OT, F//iw ^ Sett:C iff ^ett.-C1 G w. Let B = [7 dstzi: C]. Since OK, y/A^, 1= [7 dstit: C] iff OT, Y/hw> 1= C* for all hw> G ^ ( y ) with A^ =[Y] hw and 9Jt, y/Au," P C for some A™» G H(Y), we know by (16) and (17) that 971, Y/hw \= [7 dstit: C] iff 371, Y/hw> 1= (7 for all w' G y with w' ~ w w and DOT, y/A w » J^ (7 for some w" G y, and then by induction hypothesis, 971, Y/hw 1= [7 dstit: C} iff C G w' for all w' G y with w' ~ H w and C £ w" for some w" G y, and hence by Lemma 17-15 (Hi), 971, Y/hw N [7 dsM: C] iff [7 dstit: C] G w. It follows that (18) holds for all B G S^. Since X G IU and .A G w for some w G X, it follows that A G w for some w € Y. Hence by (16) and (18), 971, Y/hw 1= A for some hw G H(y)- This completes the proof. By Theorem 17-17 and a routine argument, the following theorem can be established. 17-18 THEOREM. (Decidability) For every n ^ 0, Ldmn is decidable.

18

Doing and refraining from refraining The refref equivalence for the achievement stit, Ax. Cone. 1, was discussed in §2B.6, §2B.8 and chapter 15.* We repeat it here for convenience: (refref) With [a stit: A] and [a stit: ~[a stit: A}} representing doing and refraining from doing respectively, this equivalence asserts that doing is equivalent to refraining from refraining from doing. This assertion holds if the BT + I + AC structure contains no "busy choice sequences" (sequences of infinitely many nonvacuous choices for an agent within a finite time). That is to say,

ques

in every BT + I + AC Structure containing no busy choice sequences, doing implies, and is implied by, refraining from refraining from doing

Figure 2.12 suggests a partial justification of (1), and we provided a proof in §15C. We now raise the converse question. Let BT +1 + AC + be be the set of all B T +1 + A C structures containing at least one busy choice sequence. We ask whether or not the refref equivalence fails in every BT + I + AC+bc structure. We know that the refref equivalence fails in some particular BT + I+AC+bc structures (see Figure 2.13). Here we prove that in every BT + I + AC + bc structure, doing neither implies nor is implied by refraining from refraining from doing.

We can then establish the following as a consequence of (1) and (2). The refref equivalence holds in a BT + I + AC structure iff the structure contains no busy choice sequences. Our first section contains some preliminaries, and clarifies the concept of a busy choice sequence. The second section presents the proof of (2). *With the permission of Kluwer Academic Publishers, this chapter draws on Xu 1994c.

451

452

ISA

Proofs and models

Preliminaries

In this chapter we deal only with the achievement-stit, §8G.3, and BT + I + AC structures, §2. Semantic clauses for prepositional variables, ~, D, and the achievement stit are as listed in §8, while other semantic concepts such as an interpretation 3 and a model 97t on a BT + 1 + AC structure are listed in §6. A past is defined, as in Def. 6, as a nonempty subset p of Tree such that 3m[m € Tree & Vu>(u) S p —> w ^ m)] and VwVw'[w 6 p & w/ ^ w —» «/ £ p] (intuitively, any cftazn of moments in Tree that closed in both past and future). We compare moments and instants as in Def. 9 by defining w < i iff w < m for some m e i, p < m iff Vw[w € p —> w < m], and p < i iff p < m for some m £ i. As in Def. 9, we set i\>w = {m: m € i & w < m} (tfte horizon from w at i), and we also set i\>p = {m: m & i & p < m} (the horizon from p at i). It should be noted that p is used for pasts (as in Def. 6) and q is used for prepositional variables. 18-1 REMARK. (An obvious property of horizons) Let w, p and i be in a BT + 1 + AC structure &. Then w e p implies i\>p C i\>w. We use m =£, m' (m and m' are choice equivalent for a at w), and m m' (m and m' are choice separated for a at w), as listed in Def. 12. Note that both m =£, m' and m J_£, m' imply w < m and w < m'. The following lemma is useful in later discussions. 18-2 LEMMA. (A consequence of no choice between undivided histories) Let be any BT + I + AC structure in which w' < w, w < m, w < m' and m, m' i. Then for every agent a, m' =£,, m. And in particular, if «/ ^ u; and m' m, then PROOF. By the axiom of choice there are two histories h and h' such that {w', and {w', w, m'} C ft'. Since by definition, which implies h' =£,/ ft for every agent a by no choice between undivided histories Post. 8, and hence m' =£,, m for every agent a. In particular, if m' =£, m, then w < m and w < m'. Hence w' ^ w and m' =£, m implies We use choice equivalence between moments, m' =£, m, rather than choice equivalence between histories, h' =£, ft, for convenience. The two concepts, defined in Def. 12, do the same work since one can easily verify that for some

such that and

It is the fact that Lemma 18-2 implies the uniqueness of w in the displayed right-hand clause that entitles us to call call w the witness to [a stit: A] at m; see Fact 15-2. Similarly we will call any m" satisfying the listed conditions a

453

18. Doing and refraining from refraining

counter to [a stit: A] at m. This being the case, we will allow ourselves to speak of "971. m 1= [a stit: A] with witness w and counter m"." Semantic definitions of the various uses of "!=", for example 9H, M\= A (settled truth of A throughout the set of moments M), are listed in §6. As special cases of settled truth throughout a set, here we use 971, i 1= A, 971, i\>w N A and 971, i\>p 1= A, and so on. The following easily verified facts prove useful.

18-3 REMARK. (Semantic properties of stit) i. Let m

with

ii. iii. Let m such that

iff for some

the

and

m such that (a) A implies Then

with witness w iff for every with the same witness w

Recall that although for each h, 9Jt, m/h t= [a stit: A] iff 971, m N [a stit: A] (that is, settled truth and truth relative to a history are the same w.r.t. stit sentences), 971, m/h 1= A is in general different from 971, m t= A (that is, settled truth is not the same as truth relative to a history w.r.t. all sentences). As listed in Def. 14, a chain c is a busy choice sequence for a iff (i) c is upperand lower-bounded in Tree, and (M) c is an infinite sequence of (nontrivial) choice points for a. What turns out to be essential for the present chapter is this: For any agent a, a busy a-choice sequence in a BT + I + AC structure & (or in a BT + I + AC model 971) is an upper- and lower-bounded chain of moments that does not terminate in the forward direction, and at each of those moments a has nonvacuous choice. That is to say, a busy a-choice sequence is a nonempty chain BC of moments such that

We will fix on a single agent for our discussion, and therefore we will say "busy choice sequence" instead of "busy a-choice sequence." We let BC, BC', and so on range over busy choice sequences, and for any busy choice sequence BC, we will use pBC to denote the smallest past including BC, that is, PBC — {w: 3w/[u/ G BC & w < w'}}. We define BC < i iff A w [ w e BC -» w < i}. Whenever we speak of a model 371 = ( © , 3 ) and a busy choice sequence BC in it, we assume that 3(a) = a where BC is a busy a-choice sequence. 18-4 REMARK. (Simple properties of busy choice sequences) i. Suppose that p = pBC for some BC < i in 6. Then for all m, m' 6 i\>p, all w € p and all a 6 Agent, m =£> m'. (This is a consequence of Lemma 18-2.)

454

Proofs and models

ii. Suppose that BC < m € i in 6, then for all m. Suppose that BC1 is any busy choice sequence. Then there is a countable subset BC' = {WQ, Wi, W2, •••} of BC that is still a busy choice sequence.

18B

Main results

We prove in this section that in every BT + I + AC structure that contains a busy choice sequence, doing neither implies nor is implied by refraining from refraining from doing. We start with some simple lemmas. 18-5 LEMMA. (Sufficient condition for ~[a stit: A}) Let 97T be any BT + I + AC model in which p < i, and let A be any sentence. Suppose that and that for every w G p, there is a w' G p with w < w' and Then PROOF. Suppose for reductio that 371, m t= [a stit: A] for some m G i|> p with witness w and counter m'. Then by no backward branching, we have either p < w or w G p. But this is impossible. For if p < w, then m' G i|> p , contrary to hypothesis; and if w G p, then by Lemma 18-2, for each w' £ p with w < w', m =^, m' for all m" 6 i|>u/, and hence 971, i >w> t= 4, contrary to hypothesis. It follows that 18-6 LEMMA. (Sufficient condition for ~[a stit: .A],) Let 971 be any BT + I + AC model in which p < m* G z, and let . Suppose that 971, M 1= A and that for eve Then PROOF. Suppose for reductio that 971, m N [a stit: .A] for some m e M with witness w. Two cases. CASE, w £ p. Since m £ i >p, there is a w' € p such that w' ^ m. It follows that w < w', and hence by the hypothesis of this lemma, there is an m' € i such that w' < m' and 97T, m' ¥ A. But Lemma 18-2 and w' < m* imply that m' =2, m*, and so m' =£j m since m € M; hence 971, m' 1= ^4, a contradiction. CASE, w ^ p. Consider any counter m" to [a stit: B] at m. We show that m" € M, which contradicts our hypothesis. First, since w < m" and w £ p, we have the following by no backward branching:

Then by Lemma 18-2, for every w' £ p with w' < m", m" =^, m since w < m and w < m", and hence TO" =^, TO* since TO =^, TO* (m 6 M). Next, if m" G i|> P , we would have p < m", and hence by (3), p < w which implies m G i|> p , contrary to our assumption that m G M. Hence, m" ^ i[ >p . It follows that m" G M. From this reductio we conclude that 971, M \= ~[a stit: .A], which completes the proof. D

18. Doing and refraining from refraining

455

18-7 LEMMA. (Mam lemma) Suppose that 6 is any BT + I + AC structure in which BC < i. Then there is a model 97T on 6 such that 971. i ^ ~[a stit: ~[a stit: q}}, and 97T, i ¥ B for every subsentence B of ~[a stit: q}. PROOF. Let p = pBc and fix m" £ i\>p. We define a model 971 in such a way that for all m and all /i in 6 with m £ h, (m, /z) € ,3(g) iff m e M = {m: m € z - i >p & 3w[w e p & w < r a & T O _ L ° m*]}. Clearly, z = i| > p UMUM', where M' = {m: m e z —i|> p & Vw[w £ p & w < m —> TO =£, w*]}. By our definition of 3 and Remark 18-3 (ii), we have

Consider any m e M. By definition of M, there is a w 6 p with w < m and m _L" m*. We first show as follows that

Let m' be any moment in i with TO' =2, TO. Since m _L^ TO*, m' _L£j m*. We know by m ^ z| >p that there is a w' e p with w' •£ m, and hence w < w'. Since m' _L^, TO*, we know by Lemma 18-2 that w' •£ m', and hence m' £ i >p. Hence m' G M, and it follows that (5) holds. Hence we have by 97t, M t= q that 971, m' N 9 for all m' e i with m' =2, TO. Since w e p, 971, ?!>,„ ^ g by (4) and Remark 18-1, and hence 971, m 1= [a stit: q] by Remark 18-3 (^. It then follows from Remark 18-3 (ii) that

(4), (6) and Remark 18-4 (n) give us the following:

Hence by (4), (7), and Lemma 18-5, we have 97T, We know by the definition of 3 and Remark 18-3 (n) that 971, and hence 971, M' t= ~[a siii: ~[a stit: q}} by (7) and Lemma 18-6. It follows from (6) that 971, i 1= ~[a stit: ~[Q; stit: 5]]. We can then complete the proof by applying (4), (6), and the definition of 3. We are now ready to show that in every BT + I + AC structure that contains a busy choice sequence, doing does not imply refraining from refraining from doing, 18-8 THEOREM. (Doing does not imply refraining from refraining from doing) Let & be any BT + I + AC structure in which there is a busy choice sequence. Then PROOF. By Lemma 18-7, there is a model 971 on & such that for some i, 971, i and 971, i 1= ~[Q stit: ~[a stit: q}}. and hence 971, for some m e i and 97?, i t= ~[a stit: ~[a stit: ~[a stit: q}}}. It follows that 6

456

Proofs and models

We show in the rest of this section that in every BT + I+AC + bc structure, refraining from refraining from doing does not imply doing. We prove this claim by applying the following concepts. A substructure of a BT + I + AC structure 6 = (Tree, ^, Instant, Agent, Choice) is any BT + I + AC structure ©' = (Tree1, ^', Instant', Agent1', Choice') satisfying the conditions Agent, and for every a £ Agent' and w e Tree', Note that if S' is a substructure of 6, (Tree1, ^') must satisfy the historical connection condition Post. 4 and must be closed forward w.r.t. (Tree, ^}, and that every pair {Tree', <') (Tree' C Tree and <' = ^n (Tree' x Tree')) satisfying these two conditions determines a unique substructure. Note also that if 6' is a substructure of 6, then for any a 6 Agent' and any w e Tree', m' = £, m iff Tree, where m' ='£, m means that in 6', m' and m are choice equivalent for a at w. A model 931' = (6', 3') is a submodel of a BT + I + AC model SJt = 3) w.r.L an instant i' in fH' if 6' is a substructure of 6 and for each agent term a, 3'(a) = 3(a); and for each m 6 i', each A in 6 with m & h, and each prepositional variable 9, {m, /i) € 3(g) iff (m, /i'} e 3'(g), where /i' is the history in 6' such that h1 = hn Tree'. 18-9 LEMMA. (Submodel lemma) Let OT' be a submodel of 9Jt w.r.t. n instant i' in 9Jt', and let F be any set of sentences closed under subsentences. Suppose that for each agent term a, and each sentence A, [a stit: A] 6 F only if SOT', i' A . Then for every sentence A e F, every m g z', and every /i in 931 with m

PROOF. Our proof is by induction on the structure of A. We know immediately by definition that the displayed biconditional holds for all prepositional variables. The inductive steps for truth-functional connectives are straightforward. So it suffices to suppose that the displayed biconditional holds for A and to show that the following are equivalent for any a:

Given our induction hypothesis and an earlier note about the relation between m' ='w m and m' =£, m, we need only to assume (9) and show that w stated in (9) must also be in Tree'. Let us suppose for reductio that w £ Tree'. Consider any m' 6 i'. If m' = m, then 931, m" t= A by induction hypothesis. If m' j^ m, then, w' <' m' and w' <' m for some w' € Tree' by historical connection. It

18. Doing and refraining from refraining

457

follows from no backward branching that w < w'. and then m' = ™ m by Lemma 18-2. Hence OT, m' 1= A by (9); and so 9JI, m" t= ,4 by induction hypothesis. It follows that 9JI, «' 1= A, contrary to hypothesis 18-10 THEOREM. (Doing is not implied by refraining from refraining from doing) Let 6 be any BT + I + AC + bc structure. Then PROOF. By Remark 18-4(iii), let {w*, WQ, u>i, ...} be any busy choice sequence earlier than i in 6. Then BC = {w0, wi, ...} is still a busy choice sequence in 6 and BC < i. Set c = {w: w* < w ^ w0}, Tree' = G c & w' ^ w]} and <' = ^n(Tree' x Tree'). It is easy to see that (Tree1, ^'} satisfies historical connection and closed forward w.r.t. (Tree, ^}, and hence it determines a unique substructure 6' of (3. Since BC C Tree', 5(7 is a busy choice sequence in 6' and BC < i' = id Tree'. So by Lemma 18-7, there is a model 9JT = (6', 3') such that

We can thus define a BT + I+ AC model 5Jt on 6 in such a way that for every m £ i and every h with m £ h, if m e i', then (m, /z) £ 3(
Let us select an arbitrary m* 6 i'. Consider any m 6 /. We know by definition of c that there is a, w & c such that w* < w, w < m* and w < m, and hence, m =", m* by Lemma 18-2. It follows that

Let us set M = by (12). Since 9JI, M h ~g, we know by Remark 18-3 (ii) that

Suppose for reductio that for someTOO€ M, 971, mo N [a stit: ~[a stit: 9]] with witness w'. Since w* < mo and w' < mo, either u>' ^ w* or w* < w' by no backward branching. If w' ^ w*, then by (12) and Lemma 18-2, m0 =°, m for all m e i', and hence, 9Jt, i' N ~[a stit: 5], contrary to (11). Thus it must be the case that w* < w'. Consider any counter mi to [a stit: ~[a stit: q}} at mg. Since w* < w' < mo and w' < mi, m-i —^ mo by Lemma 18-2, and hence mi m* since mo € M. It follows that either mi € i' or mi € M. But mi e

458

Proofs and models

M contradicts (13) because OT, m\ t= [a stit: q}. Hence mi e i'. Since w* < w' < mi, it follows from no backward branching that w' e Tree', and hence mo 6 i', contrary to the assumption that mo 6 M. From this reductio we conclude that 9Jt, M t= ~[a siii: ~[a sM:
Consider M' = {m: m e i - i ' & w * < m & m X^,, m*}. By our definition of m and Remark IS-Sfnj, OT, M' 1= ~[a sh<: gj. It is easy to see by (12) that for each m' e M', m" e M' for all m" e i with m" =£,, m', and hence by (11) and Remark 18-3(i), SEJl, M' t= [a 5^ii: ~[a sizi:
Appendix: Lists for reference Here we list in one place many of the items to which we repeatedly refer. Crossreferences to this appendix, either to its sections or to one of its numbered statements, are indicated by the use of one of the boldface forms occurring in the following display. • §1. Stit theses Thesis 1-Thesis 6. • §2. Structures of various kinds. • §3. BT + I + AC postulates Post. 1-Post. 10. • §4. Definitions of important BT + I concepts Def. 1-Def. 9. • §5. Definitions of concepts involving choice Def. 10-Def. 14. • §6. Fundamental semantic concepts Def. 15-Def. 16. • §7. Derivative semantic concepts Def. 17-Def. 20. • §8. The grammar of the mini-language we introduce. • §9. A few concepts from the axiomatics of stit theory, Ax. Conc. 1, Ax. Conc. 2, and Ax. Conc. 3. We do not list recursive semantic clauses in this appendix; instead, we refer to the proper sections of chapter 8.

1

Stit theses: Thesis 1-Thesis 6

By means of various "stit theses" we try to make explicit some of our central claims. (There are also several "slogans" in §5A.) Thesis 1 AGENTIVENESS OF STIT THESIS. (Stit thesis) agentive for a. Introduced in §1A.

459

[a stit: Q] is always

460

Appendix: Lists for reference

Thesis 2 STIT COMPLEMENT THESIS. (Stit thesis) (a stit: Q\ is grammatical and meaningful for any arbitrary sentence Q. Introduced in §1A. Thesis 3 STIT PARAPHRASE THESIS. (Stit thesis) Q is agentive for a just in case Q may usefully be paraphrased as [a stit: Q]. Therefore, up to an approximation, Q is agentive for a whenever Q <-> [et stit: Q}. Introduced in §1A. Thesis 4 IMPERATIVE CONTENT THESIS. (Stit thesis) Regardless of its force on an occasion of use, the content of every imperative is agentive. Introduced in §1B.1. Thesis 5 RESTRICTED COMPLEMENT THESIS. (Stit thesis) A variety of constructions concerned with agents and agency—including deontic statements, imperatives, and statements of intention, among others—must take agentives as their complements. Introduced in §1B.2. Thesis 6 STIT NORMAL FORM THESIS. (Stit thesis) In investigations of those constructions that take agentives as complements, nothing but confusion is lost if the complements are taken to be all and only stit sentences. Introduced in §1C. We remind the reader that we offer these theses as worthwhile heuristics, but not as pronouncements to be taken strictly. In some cases, indeed, we call explicit attention to how one or more can usefully be modified, while nevertheless continuing to advance them as excellent rough approximations.

2

Structures

We list the most important classes of structures that we treat. In each case, if quantifiers are not being considered, Domain may be missing. (Note. We use U BT" for example, both as a singular term naming a class of structures and as an adjective modifying "structure" or "model.") • A BT structure is any tuple (Tree, ^, Domain) satisfying each postulate of §3 that governs one or more of its elements. Such a structure is a branching time structure. Introduced in §2A.I. • A BT + AC structure is any tuple (Tree, ^, Agent, Choice, Domain) satisfying each postulate of §3 that governs one or more of its elements. Such a structure is an agents and choices in branching time structure. Introduced in §2A.2. • A BT +1 structure is any tuple {Tree, ^, Instant, Domain) satisfying each postulate of §3 that governs one or more of its elements. Such a structure is a branching time with instants structure. Introduced in §2A.2.

Appendix: Lists for reference

461

• A BT +1 + AC structure is any tuple (Tree, ^, Instant, Agent, Choice, Domain) satisfying each postulate of §3 that governs one or more of its elements. Such a structure is an agents and choices in branching time with instants structure. Introduced in §2A.2. • A BT + I + AC + nbc structure is a BT + I + AC structure in which there are no busy choosers (no busy choice sequences) in the sense of Def. 14. Introduced at the beginning of chapter 15. • A BT +1 + AC +bc structure is a BT + 1 + AC structure in which there is at least one busy chooser (at least one busy choice sequence) in the sense of Def. 14. Introduced at the beginning of chapter 18.

3

BT + I + AC postulates: Post. 1-Post. 10

The postulates for agents and choices i-n branching time were introduced informally in chapter 2 and studied at length in chapter 7. These postulates play two roles, (i) They serve to define the kinds of structures of §2, considered abstractly, (ii) When a structure is considered as an idealized representation of our world, the postulates count as part of a broadly substantive theory. To emphasize this second aspect of our project, this book sometimes uses "Our World" in place of "Tree." Note that several postulates are cast in terms of phrases that are denned in §4 and §5. Post. 1 NONTRIVIALITY. (BT +1 + AC postulate) Tree ^ 0. Discussed in §7A.

Tree is a nonempty set:

Post. 2 CAUSAL ORDER. (BT +1 + AC postulate) bv sJ:

Tree is partially ordered

Reflexivity. Transitivity Antisymmerty

Discussed in §7A. Post. 3 No BACKWARD BRANCHING. (BT' +1 + AC postulate) Incomparable moments in Tree never have a common upper bound; or contrapositively, if two moments have a common upper bound, then they are comparable: (mi ^ ma & m2 < m3) —> (mi ^ m2 or m2 ^ mi). Discussed in §7A.2. Post. 4 HISTORICAL CONNECTION. (BT +1 + AC postulate) Every two moments have a lower bound: VmiVm^Bmofmo ^ m-i & mo ^ m 2 ]. In other words, every two histories intersect. Discussed in §7A.3. Post. 5 Instant AND INSTANTS. (BT +1 + AC postulate)

462

Appendix: Lists for reference

i. Partition. Instant is a partition of Tree into equivalence classes; that is, Instant is a set of nonempty sets of moments such that each moment in Tree belongs to exactly one member of Instant. ii. Unique intersection. Each instant intersects each history in a unique moment; that is, for each instant i and history h,ir\h has exactly one member. Hi. Order preservation. Instants never distort historical order: Given two instants i\ and 1% and two histories h and h', if the moment at which i\ intersects h precedes, or is the same as, or comes after the moment at which 12 intersects h, then the same relation holds between the moment at which i\ intersects h' and the moment at which i% intersects h'. Discussed in §7A.5. Post. 6 AGENTS. (BT +1 + AC postulate) cussed in §7C.l.

Agent is a nonempty set. Dis-

Post. 7 CHOICE PARTITION. (BT +1 + AC postulate) Choice is a function defined on agents and moments. Its value for agent a and moment m is written as Choice^. For each agent a and moment TO, Choice^ is a partition into equivalence classes of the set H(m) of all histories to whichTObelongs. Discussed in §7C.2. Post. 8 NO CHOICE BETWEEN UNDIVIDED HISTORIES. (BT +1 + AC postulate)

If two histories are undivided at m, then no possible choice for any agent at m distinguishes between the two histories. That is, one of two histories undivided at TO belongs to a certain choice possible for a at TO if and only if the other belongs to exactly the same possible choice. In symbols from Def. 4 and Def. 12: hi = mo /i2 -» hi =^0 /i 2 . Discussed in §7C.3. Post. 9 INDEPENDENCE OF AGENTS. (BT' +1 + AC postulate) If there are multiple agents: For each moment and for each way of selecting one possible choice for each agent, a, from among a's set of choices at that moment, the intersection of all the possible choices selected must contain at least one history. In symbols: for each m e Tree, and for each function fm on Agent such that / m (a) 6 Choice1^ for all a e Agent, C\{fm(ct)'- et G Agent] ^ 0. Discussed in §7C.4. Post. 10 RICHNESS OF Domain. (BT +1 + AC postulate) The domain of quantification, Domain, must include Tree, History, Instant, and Agent as subsets. Discussed in §7D. That concludes the list of postulates for BT + I + AC theory, that is, the theory of agents and choices in branching time with instants.

Appendix: Lists for reference

4

463

Branching-time-with-instants definitions: Def. 1-Def. 9

We list key definitions of BT +1 + AC concepts. Most are intended as revelatory. In each case we suppose that we are given a BT + I+AC structure, (Tree, ^, Instant, Agent, Choice). Def. 1 MOMENTS. (Definition) A moment is defined as a member of Tree. We let m and w range over moments; and we let M range over sets of moments. Discussed in §7A.

Def. 2 CAUSAL ORDER. (Definition) • TOI < m2 is the strict partial order of Tree associated with the partial order ^; that is, m\ < m^ iff (m\ ^ m,2 & mi ^ m^*). We refer to either of these as the causal ordering of Tree. • We read ^ and its converse with the plain words earlier/later, below/above, lower/upper, backward/forward, and so on, and insert proper when we intend < or its converse. When mi is properly earlier than 7712, we also say, in a fashion much more revealing of our intentions, that mi is in the (causal) past of m,2; and m^ is in the (causal) future of possibilities of mi. Discussed in §7A. Def. 3 CHAINS AND HISTORIES.

(Definition)

• Moments TOI and m-z are comparable if ^ goes one way or the other: (mi ^ m2) or (m2 ^ mi). • A chain, c, in Tree is a subset of Tree such that every pair of its members is comparable: c C Tree & VmiVm2[mi,m2 € c —> mi and m,2 are comparable]. We let c range over chains in Tree. • A history, h, of Tree is a maximal chain in Tree: ft, is a history of Tree iff ft, is a chain in Tree, and no proper superset of ft, is itself a chain in Tree. We let ft range over all histories. • History is the set of all histories of Tree. We let H range over subsets of History. Discussed in §7A.

Def. 4 MOMENTS AND HISTORIES.

(Definition)

i. -ff( m ) is the set of histories in which m lies (or the set of histories "passing through" m): h G ff(m) iff m € ft.

464

Appendix: Lists for reference

ii. m/h is a moment-history pair provided m is a moment and h € -ff( m ) (which is to say, m € h & h £ History). The notation "m//i" is an alternative way of naming the ordered pair {m, h), except that the use of "m//i" always presupposes that m G h. Hi. Moment-History is the set of all moment-history pairs m/h. iv. h\ = mo h-2 iff mo £ hi^h^ and there is an m\ such that mo < mi, and mi 6 /zifl/12 (unless there is no mi such that mo < mi). We say that hi and h% are undivided at m®. We adapt undividedness-at to pasts as well: Two histories extending a past, p, are undivided at p iff they share a moment properly later than p, so that they appear as a single line as p comes to a close. v. hi _L mo hi iff hi ^ h-2 and mo is the least upper bound of /iin/i 2 . We say that hi and h-2 split at mo. vi. Tree is deterministic at a moment, m [or at a past, p] iff every pair of histories through m [or p] is undivided at m [or p}. vii. A set, H, of histories is an immediate possibility at a moment, m0 iff H is a subset of H(mo) that is closed under undividedness at mo: (hi 6 H and hi =mo h-2) -> ht 6 H. viii. ff(M) = {^: (Mn/i) 7^ 0}, so that .ff(M) is the set of histories that pass through at least one member of the set of moments, M. For suitable M, this is the set of histories in which M comes to be. ix. H[M) = {h: M C h}, so that H[M] is the set of histories entirely containing M. For suitable M, this is the set of histories in which M passes away. Discussed in §7A and §7A.4. Def. 5 BOUNDS.

(Definition)

• A moment, m, is a [proper] lower bound of a set of moments, M, iff m is [properly] earlier than every member of M; and similarly for [proper] upper bounds of chains. • We let m < M iff m is a proper lower bound of M, and similarly in other cases. • We let MI < M-2 iff every member of MI is earlier than every member of MZ, and similarly in other cases. • Greatest lower bounds and least upper bounds of sets of moments are as usual in the theory of partial orders. Discussed in §7A.

Def. 6 CUTS, PASTS, AND FUTURES.

(Definition)

Appendix: Lists for reference

465

• A historical cut for a history, h, is a pair (p, f} of sets of moments such that (i) neither p nor / is empty, (ii) p < /, and (iii) (pU/) = h. • p is £/ie past history of / iff / is a future history of p iff (p, /} is a historical cut. • M is the causal past of a future history, /, iff M is the set of all proper lower bounds of /. • We often say just past because given no backward branching, a causal past is the same as a historical past. • M is a future of possibilities, or a causal future of a past, p, iff M is the set of all proper upper bounds of p. Given indeterminism, a future history must be distinguished from a future of possibilities, so that in this book we never say merely "future." • We say that (p, M) is a causal cut iff p is a past, and M is the future of possibilities of p. • All of these usages may straightforwardly be adapted to speak of pasts and futures of single moments. Discussed in §7A.2. Def. 7 SEMI-LATTICE CONDITION. (Definition) The semi-lattice condition says that for every two (distinct) histories there is a moment at which they split (Def. 4(u)). Discussed in §7A.3. Def. 8 INITIAL AND OUTCOME EVENTS, AND TRANSITIONS. (Definition) • I is an initial event iff I is a nonempty and upper-bounded chain. • O is an outcome event iff O is a nonempty and lower-bounded chain. • O is an immediate outcome of I iff O is a (possible) outcome of /, and if furthermore no moment lies properly between / and O. • (I, O) is an [immediate] transition iff O is an [immediate] outcome of /. • (/, O) is a contingent transition iff (/, O) is a transition, and if some history is dropped in passing from the completion of / to the beginning of O: H[I]-H{0}^0. Discussed in S7A.4. Def. 9 INSTANTS. (Definition) • The members of Instant are called instants, i ranges over instants.

466

Appendix: Lists for reference • i( m ) is the uniquely determined instant to which the moment m belongs, the instant at which m "occurs." • rri(lth) is the moment in which instant i cuts across (intersects with) history h; that is, ir\h = {m^ith)}. • Order preservation can conveniently be stated in the symbols just introduced: rn (ni/ll ) < rn( i2)/ll ) implies m( lli/l2 ) < m^2,h2)• Fact: «i(1(m j ,/n,), a function of mo and h0, is the moment on history h0 that occurs at the same instant as does mo: • i >m = {WQ: m < mo & mo e z}. We say that i|>m is i/ie horizon from moment m ai instant i. • Where i\ and 12 are instants, we may induce a linear time order (not a causal order!) by defining i\ ^ i^ iff m\ ^ m^ for some moment m\ in ii and some moment 7712 in ^2. Instants can also be temporally (not causally) compared with moments, m: i\ < m iff mi < m for some moment m\ in i\\ and m < 12 iff m < m? for some moment m2 in i% .

Discussed in 57A.5.

5

Agent-choice definitions: Def. 10—Def. 14

Def. 10 AGENTS. (Definition) To be an agent is to be a member of Agent. We let a and j3 range over agents, and sometimes over terms denoting agents. F ranges over sets of agents. Discussed in §7C.l. Def. 11 CHOICE NOTATION. (Definition) i. Choice represents all the choice information for the entire Tree. ii. Choice1^ gives all the choice information for the agent a and the moment m; Choice1^ should be thought of as a set of possible choices, and we call it "the set of choices possible for a at m." in. Choice^h) is defined only when h passes through m, and is then the unique possible choice (a set of histories) for a at m to which h belongs. The notation is justified by the fact that according to Post. 7, each member of ff(m) picks out a unique member of Choice1^ to which it belongs. iv. C7iozce^j(mi) is defined only when mi is in the proper future of the moment m of choice. Pick any history, h, containing mi, a history that will a fortiori contain m. Then C7ioice^(mi) is defined as Choice^^). This definition is justified by no choice between undivided histories, Post. 8.

467

Appendix: Lists for reference

v. Choice^ (mi) is defined only when instants are present, and when m\ is properly future to m. Recall that i( mi ) is the instant on which TOI lies. Then Choice1^ (mi) is defined as the set of all moments on the instant i(mi) that also lie on some history in CViozce^(mi). In symbols, Choice°^L(m\] = *(mi) n (JChoice^(mi). 1 Discussed in §7C.2. Def. 12 CHOICE EQUIVALENCE. (Definition) choice equivalent for a at m.

and we say that /ij and /i2 are

/ii and /i2 are choice separated for a at m.

We say that

m. mi =$^ m2 is defined only when instants are present, and when m < mi, m2. Then mi =^ m2 iff Choice^(mi) = Choice^m-i). We say that mi is choice equivalent to m2 /or a at m. Be warned that in contrast to Choice, there is no mnemonic in either the phrase "mi is choice equivalent to m2 for a at m," nor in the symbols "mi =^ m,2," that forces the recognition that mi and m2 belong to the same instant. (Other studies—but not this book—use the same notation for a concept defined via Choice instead of Choice.} Discussed in §7C.2. Def. 13 INSEPARABILITY/SEPARABILITY. (Definition)

Discussed in §7C.2. Def. 14 ADDITIONAL CHOICE CONCEPTS. (Definition) • A moment m is a choice point for a iff there is more than one possible choice for a at m. J

The underlining on Choice is intended as a mnemonic calling to mind the horizontal picture of an instant.

468

Appendix: Lists for reference • A possible choice for a at m is vacuous or trivial iff it is the only possible choice for a at m; and is otherwise nonvacuous. There can only be a vacuous choice for a at m when Choice^ = { H ( m j } , in which case m itself is said to be a trivial choice point for a. • A chain c is a busy choice sequence for a iff (z) c is both lower and upper bounded in Tree, and (n) c is an infinite chain of (nontrivial) choice points for a. • a is a busy chooser iff some c in Tree is a busy choice sequence for a.

Discussed in §7C.2 and §70.5. That concludes the list of major definitions of the BT +1 + AC theory of agents and choices in branching time with instants.

6

Basic semantic definitions: Def. 15—Def. 16

Sometimes we use K as ranging over the kinds of structures listed in §2. Rather than rehearse basic semantic concepts for each K, we run through the following system of definitions only for the case in which K = BT + 1 + AC, leaving other cases to be understood by analogy. Def. 15 INTERPRETATION AND MODEL. (Definition) Let & = (Tree, ^, Instant, Agent, Choice, Domain} be a BT + I + AC structure. • 3 is an &-interpretation for a BT + I + AC structure 6 = (Tree, ^, Instant, Agent, Choice, Domain) iff 3 is a function defined on prepositional variables p, individual constants u (some of which are agent terms a and one of which, f, is to denote "the non-existing object"), operator letters /, and predicate letters F, such that 3 assigns to each prepositional variable a function from Moment-History into {T, F}; assigns to each individual constant (including f) a member of Domain; assigns to each agent term a member of Agent; assigns to each n-ary operator letter a function from Moment-History into functions from Domain™ into Domain; and assigns to each n-ary predicate letter a function from Moment-History into functions from Domain71 into {T, F}.2 • 971 is a BT + I+AC model (based) on 6 iff 971 is a pair (6, 3), where 6 is a BT + I + AC structure, and where 3 is an ©-interpretation. Discussed in §8B and §8C.

Def. 16 POINT, DENOTATION, AND TRUTH. (Definition) 2 Some chapters use the equivalent procedure of assigning a subset of Moment-History to prepositional variables, and more generally using a subset of Moment-History wherever we have used a function from Moment-History into {T, F}.

Appendix: Lists for reference

469

• A BT + I + AC point is a tuple (9H, mc, a, m/h), such that 971 is a BT + I + AC model, mc e Tree, a is a function from the individual variables into Domain, m 6 h, and ft 6 History. Henceforth we assume that (971, TOC, a, m/h) is a BT + I + AC point. We speak of the various parameters in (m, mc, a, m/h) in the following way: 971 is the BT + I+AC model, mc is the moment of use, a is the assignment (of values to the variables), m is the moment of evaluation, and h is the history of evaluation. • We let TT be the point (9JI, mc, a, m/h), and adopt the following convention: In any context in which we write "TT," we will understand the expressions Tree, ^, Instant, Agent, Choice, Domain, &, 3, m, mc, a, m, and h just as if we had written "(971, mc, a, m/h)." • Other structures and missing parameters. Points are defined in the same way for the other structures listed in §2. The assignment parameter, a, however, may be missing if quantification is not at issue, and the momentof-context parameter, TOC, may be missing if the context of use is not under discussion. • We say that a BT + I + AC point, (m, mc, a, m/h), is context initialized iff the moment of use, mc, is identical to the moment of evaluation, m. That is, context-initialized BT + I+AC points have always the form (971, m c , a, mc/h). • Semantic value. For any categorematic expression E, be it term or sentence, V / a/OT,m c ,a,m//i(-E') is "the semantic value of E at the point (SOT, TOC, a, m/h}." Valgjiim<;taim/h(E)'is defined recursively by clauses given in §8Fand §8G. • Denotation. Where t is any term, Va,l<m,mc,a,m/h(t) 6 Domain. We read Va/OT,m c a m/h W as "the denotation of t at the point (971, mc, a, m//i)." • Truth. VaZgn,m c ,a,m//i(-'4) is "the truth value of A at (971, mc, a, m/h)." Where A is any sentence, Valyx,mc,a,m/h(-A) 6 {T, F}. Alternative notation for truth: 971,TOC,a, m//i N A iff ^a^OT,m c ,a,m//i(^.) = T. Either is read "A is true at point (m, mc, a, m/h)." Discussed in §8D and §8E.

7

Derivative semantic definitions: Def. 17-Def. 20

A number of semantic concepts are standardly defined by quantifying over various parameters. As in Def. 16, we sometimes drop the assignment and momentof-context parameters when not relevant.

470

Appendix: Lists for reference

Def. 17 SETTLED TRUTH.

(Definition)

• A is settled true at a moment m with respect to SOT, mc, and a iff SOT, mc, a, m/h t= A for all /i £ #( m ). We may drop h, writing SOT, mc, a, m 1= A. • .A is settled true throughout a set of moments, M, with respect to SOT, mc, and a iff SOT, mc, a, m N A for all m £ M. In addition to dropping h, we may replace m by M, writing SOT, mc, a, M 1= A. • We most often write just SOT, M 1= A and SOT, m \= A since we use these concepts most often when assignment and context are not relevant. Discussed in §8E.

Def. 18 EQUIVALENCE. (Definition) • Expressions EI and £2, either both terms or both sentences, are (semantically) equivalent iff for all BT + I + AC points (SOT, mc, a, m/h), VaZan,m c ,a,m/fc(-Ei) = Val<m,mc,a,m/h(E2) (very same semantic value at all BT + I + AC points). We write EI = E2. • Expressions EI and E2, both terms or both sentences, are in-context equivalent iff for all context-initialized BT + I + AC points, (SOT, mc, a, mc/h), ^OT,m c ,a,m c //i(£i) = Valm>mc,a,mc/h(E2) (the very same value at all context-initialized points). We write EI ^in-ctx E2. Discussed in §8E.

Def. 19 IMPLICATION. (Definition) • A set F of sentences implies a sentence A iff for all BT + I + AC points (SOT, mc, a, m/h), if SOT, mc, a, m/h t= A\ for every member A\ of F, then SOT, mc, a, m/h 1= A (truth preservation at all points). We write F 1= A. • A set F of sentences in-context implies a sentence A iff for all contextinitialized BT + I + AC points (SOT, mc, a, mc/h), if SOT, mc, a, mc/h 1= A\ for every member A\ of F, then SOT, mc, a, mc/h N A (truth preservation at all context-initialized points). We write F t=m"cfcr; A. Discussed in §8E.

Def. 20 VALIDITY CONCEPTS.

(Definition)

• For SOT a model, A is valid in SOT (or 9Jl-valid) iff SOT, mc, a, m N A for every mc 6 Tree and a over Domain and m € TVee. We write SOT 1= A. • For & a structure, A is valid in & (or &-valid) iff SOT t= A for every &model SOT. We write & t= A. When (3 is understood, especially when & is taken as a representation of Our World, we say that A is vaZzcL

Appendix: Lists for reference

471

• When K is a class of structures such as those listed in §2, A is valid in K (or K.-valid) iff 6 1= A for every structure 6 in K. • Each of these validity concepts has also an in-context version defined by restricting points to context-initialized points. When symbols are wanted, we write «\=in-ctx" instead of "N". Thus, A is (i) in-context Oft-valid iff m \=in-ctxA> (£) in-context 6-valid iff 6 \=in~ctx A, and (in) in-context K-valid iff KI= i n - c t e A Discussed in §8E.

8

Grammar

Part of our project involves speaking of a mini-language that we offer as illuminating. Here we indicate our ways of speaking of its grammar, and, within the general semantic framework outlined in §6 and §7, we point to chapter 8 for the semantics of each feature of the language. Base clauses. Typically any one of our discussions draws on only some of the following items, which we introduce by simultaneously describing how we speak of the language that is our target. See §8F.l for their semantics. • p (and sometimes q) ranges over prepositional variables. • u ranges over individual constants, including two sorts of special terms: (i) a ranges over agent terms (and frequently over the agents themselves), and (M) f is a term that artificially denotes "the non-existing object," to be available as a throwaway value of definite descriptions when existence or uniqueness fails. • Xj ranges over individual variables. • / ranges over operator letters. • F ranges over predicate letters. Terms and sentences. • t ranges over terms of any kind. f ( t i , ..., tn) is a term. • A ranges over sentences. We also sometimes use B, C, D, P, and especially Q as ranging over sentences. F(ti, ..., tn) is a sentence. For recursive semantic clauses, see §8F.l Truth functions and identity. ~, &, V, D, =, T (truth), and J_ (falsehood) are the usual truth-functional connectives, and = is the usual identity predicate. See S8F.2.

472

Appendix: Lists for reference

Stit-free functors. • Vx3A and tXj(A). See §8F.3. • Sett:A, Poss:A, and Can:A. See §8F.4. • Prior tenses Was:A, Will:A, Was-always:A, and Will-always:A, and temporal operators At-instt:A and At-momt:A. See §8F.5. Stit functors. • The achievement stit, [a astit: A}. See §8G.3 (witness by moments) and §8G.4 (witness by chains). • The Brown stit, [a bstit: Q], is mentioned in §1D and discussed in Horty 2001. • The deliberative stit, [a dstit: A]. See §8G.l. • The Chellas stit, [a cstit: A}. See §8G.2. • The strict joint stit, [F sstit: Q}. See jjlOC.3. • The transition stit, [a tstit: m =^4> A]. See §8G.5. • Plain stit. [a stit: A]. Used both for the achievement stit when there is no ambiguity, and for the general stit idea.

9

Axiomatics concepts: Ax. Cone. 1, Ax. Cone. 2, and Ax. Cone. 3

Ax. Cone. 1 REFREF EQUIVALENCE. (Axiomatics concept) alence, also called just refref, is the following:

The refref equiv-

See §2B.6, §2B.8, chapter 15, and chapter 18. Ax. Cone. 2 STIT DEONTIC EQUIVALENCES. (Axiomatics concept) deontic equivalences are the following.

The stit

See §2B.9 and §4E. Ax. Cone. 3 AXIOM SYSTEMS. (Axiomatics concept) the following axiom systems.

We study or refer to

Appendix: Lists for reference

473

Lai is the Logic for the basic achievement stit with 1 agent, without the refref equivalence (hence with the possibility of busy choice sequences). The language of Lai contains truth functions together with- stit sentences for just a single agent, a. See chapter 16. Lai + rr is the Logic for the achievement stit with 1 agent and the refref equivalence (hence with no busy choice sequences). The language of Lai + rr is the same as the language of Lai. See chapter 15. Ldl is the Logic for deliberative stit with 1 agent. In addition to truth functions and stit sentences, the language of Ldl includes Sett:. See chapter 17. Ldm is the Logic for the deliberative stit with many agents. In addition to truth functions and stit sentences, the language of Ldm includes identity, and hence can express multiplicity of agents. See chapter 17. Ldmn (for n ^ 1) is the Logic for the deliberative stit with many agents, where at each moment each agent is limited to at most n + 1 choices. The language of each Ldmn is the same as the language of Ldm. Also Ldmo is defined as Ldm. See chapter 17. SA is the chapter 11 combination of Stit theory with Andersonian devices; or the Sanction with a logic of Agency. SAo is the fragment of SA described in §11C. KD is the standard system of deontic logic from F011esdal and Hilpinen 1971. S4 and S5 are the standard modal logics of C. I. Lewis.

This page intentionally left blank

Bibliography Almog, J., Perry, J., and Wettstein, H., eds. 1989, Themes from Kaplan, Oxford University Press, Oxford. Anderson, A. R. 1956, The formal analysis of normative systems, Technical report 2, Contract no. SAR-NONR-609(16), Office of Naval Research, New Haven. Reprinted in Rescher 1967, pp. 147-213. Anderson, A. R. 1962, "Logic, norms, and roles," Ratio 4, 36-49. Anderson, A. R. 1970, "The logic of Hohfeldian propositions," Logique et analyse 49-50, 231-242. Reprinted in the University of Pittsburgh law review 33 1971, 29-38. Anscombe, G. E. M. 1957, Intention, Basil Blackwell, Oxford. Aqvist, L. 1972, Performatives and venfiabihty by the use of language, Philosophical Studies No. 14, Uppsala (Philosophical Society and the Department of Philosophy, University of Uppsala). Aqvist, L. 1974, "A new approach to the logical theory of actions and causality," in Stenlund 1974, pp. 73-91. Aqvist, L. 1978, "An analysis of action sentences based on a 'tree' system of modal tense logic," in C. Rohrer, ed., Papers on tense, aspect, and verb classification, TBL Verlag Gunter Narr, Tubingen, pp. 111-161. Aqvist, L. 1984, "Deontic logic," in Gabbay and Guenthner 1984, pp. 605-714. Aqvist, L., and Mullock, P. 1989, Causing harm: A logico-legal study, Walter de Gruyter, Berlin. Austin, J. L. 1961, Philosophical papers, J. Urmson and G. Warnock, eds., Oxford University Press, Oxford. Essays from the 1940s and 1950s. Austin, J. L. 1975, How to do things with words, J. Urmson and M. Sbisa, eds., Harvard University Press, Cambridge, MA. The William James Lectures delivered at Harvard University in 1955. Badecker, W. 1987, Formal grammars and the analysis of infinitives, Indiana University Linguistics Club, Bloomington. Barcan, R. 1946, "A functional calculus of first order based on strict implication," Journal of symbolic logic 11, 1-16. See also entries under Marcus. Barcellan, B., and Zanardo, A. 1999, Actual futures in Peircean branching-time logic. Unpublished manuscript. Bartha, P. 1993, "Conditional obligation, deontic paradoxes, and the logic of agency," Annals of mathematics and artificial intelligence 9, 1-23. Bartha, P. 1999, "Moral preference, contrary-to-duty obligation and defeasible oughts," in P. McNamara and H. Prakken, eds., Norms, logics, and information systems: New studies in deontic logic and computer science, IOS Press, Amsterdam, pp. 93-108. Belnap, N. 1991, "Backwards and forwards in the modal logic of agency," Philosophy and phenomenological research 51, 777-807. 475

476

Bibliography

Belnap, N. 1992, "Branching space-time," Synthese 92, 385-434. Belnap, N. 1995, Various notes on the theory of outcomes. Unpublished manuscript. Belnap, N. 1996a, "Agents in branching time," in B. J. Copeland, ed., Logic and reality: Essays m pure and applied logic, in memory of Arthur Prior, Oxford University Press, Oxford, pp. 239-271. This article draws heavily on "Before refraining: concepts for agency," Erkenntnis 34, 1991, 137-169. Belnap, N. 1996b, "Deontic kinematics and austere strategies," in P. K. T. Childers and V. Svoboda, eds., Logica '96: Proceedings of the 10th International Symposium, pp. 21-40. Belnap, N. 1996c, "The very idea of an outcome," Serbian scientific review 19—20, 15— 16. Belnap, N. 1999, "Concrete transitions," in G. Meggle, ed., Actions, norms, values: Discussions with Georg Henrik von Wright, Walter de Gruyter, Berlin, pp. 227236. Belnap, N., and Bartha, P. 1995, "Marcus and the problem of nested deontic modalities," in D. R. W. Sinnott-Armstrong and N. Asher, eds., Modality, morality, and belief: Essays in honor of Ruth Barcan Marcus, Cambridge University Press, Cambridge, pp. 174-197. Belnap, N., and Green, M. 1994, "Indeterminism and the thin red line," in J. Tomberlin, ed., Philosophical perspectives, vol. 8, Ridgeview Publishing Company, Atascadero, CA, pp. 365-388. Belnap, N., and Perloff, M. 1988, "Seeing to it that: A canonical form for agentives," Theoria 54, 175-199. Corrected version in H. E. Kyburg, Jr., R. P. Loui, and G. N. Carlson, eds., Knowledge representation and defeasible reasoning, Studies in cognitive systems, vol. 5, Kluwer Academic Publishers, Dordrecht, 1990, pp. 167-190. Belnap, N., and Perloff, M. 1992, "The way of the agent," Studia logica 51, 463-484. Belnap, N., and Perloff, M. 1993, "In the realm of agents," Annals of mathematics and artificial intelligence, pp. 25-48. Bennett, J. 1988, Events and their names, Hackett Publishing Company, Indianapolis. Brand, M. 1970, The nature of human action, Scott, Foresman and Company, Glenview, IL. Brandom, B. 1994, Making it explicit: Reasoning, representing, and discursive commitment, Harvard University Press, Cambridge, MA. Bratman, M. 1983, "Castaneda's theory of thought and action," in Tomberlin 1983b, pp. 149-169. Braiiner, T., Hasle, P., and 0hrstr0m, P. 1998, "Ockhamistic logics and true futures of counterfactual moments," in Proceedings of the Fifth International Workshop on Temporal Representation and Reasoning (TIME-98), Sanibel Island, FL, IEEE Press, pp. 132-139. Braiiner, T., Hasle, P., and 0hrstr0m, P. 2000, "Determinism and the origins of temporal logic," in H. Barringer, M. Fisher, D. Gabbay and G. Gough, eds., Advances in temporal logic, Kluwer Academic Publishers, pp. 185-206. Bressan, P. 1972, A general interpreted modal calculus, Yale University Press, New Haven. Bressan, P. 1974, "Comments on Suppes's paper: The essential but implicit role of modal concepts in science," in K. F. Schaffner and R. S. Cohen, eds., PSA 1972, D. Reidel Publishing Company, Dordrecht, pp. 315-321. Brown, M. 1988, "On the logic of ability," Journal of philosophical logic 17, 1-26. Brown, M. 1990, "Action and ability," Journal of philosophical logic 19, 95-114.

Bibliography

477

Brown, M. 1992, "Normal bimodal logics of ability and action," Studio, logica 51, 519532. Burgess, J. 1978, "The unreal future," Theoria 44, 157-174. Burgess, J. 1979, "Logic and time," Journal of symbolic logic 44, 556-582. Burgess, J. 1980, "Decidability for branching time," Studio, logica 39, 203-218. Burgess, J. 1984, "Basic tense logic," in Gabbay and Guenthner 1984, pp. 89-133. Castaiieda, H.-N. 1954, "La logica general de las normas y la etica," Universidad de San Carlos, Guatemala, vol. 30, pp. 129-196. Castaneda, H.-N. 1974, The structure of morality, Charles C. Thomas, Springfield, IL. Castafieda, H.-N. 1975, Thinking and doing, D. Reidel Publishing Company, Dordrecht. Castaneda, H.-N. 1976, "The twofold structure and the unity of practical thinking," in M. Brand and D. Walton, eds., Action theory, D. Reidel Publishing Company, Dordrecht, pp. 105-130. Castaneda, H.-N. 1981, "The paradoxes of deontic logic: The simplest solution to all of them in one fell swoop," in Hilpinen 1981, pp. 37-86. Chellas, B. F. 1969, The logical form of imperatives, Perry Lane Press, Stanford, CA. Chellas, B. F. 1980, Modal logic: An introduction, Cambridge University Press, Cambridge. Chellas, B. F. 1992, "Time and modality in the logic of agency," Studia logica 51, 485518. Chisholm, R. 1963, "Supererogation and offence: A conceptual scheme for ethics," Ratio 5, 1-14. Chisholm, R. 1964a, "Human freedom and the self," The Lindley Lecture. Reprinted in K. Lehrer, ed., Freedom and determinism, Random House, New York, 1966, pp. 11-44; and in R. Chisholm, ed., On metaphysics, University of Minnesota Press, Minneapolis, 1989, pp. 5-15. Chisholm, R. 1964b, "Symposium: Action, motive, and desire. The descriptive element in the concept of action," Journal of philosophy 90, 613-625. Chisholm, R. 1966, "Freedom and action," in K. Lehrer, ed., Freedom and determinism, Random House, New York, pp. 11-44. Chisholm, R. 1969, "Some puzzles about agency," in K. Lambert, ed., The logical way of doing things, Yale University Press, New Haven, pp. 199-217. Chisholm, R. 1971, "Reflections on human agency," Idealistic studies 1, 36-46. Danto, A. 1973, Analytical philosophy of action, Cambridge University Press, Cambridge. Davey, K. 1999, Obligation and the conditional in stit theory. Unpublished manuscript. Davidson, D. 1966, "The logical form of action sentences," in Rescher 1966, pp. 81-95. Davidson, D. 1980, Essays on actions and events, Oxford University Press, London. Dazeley, H., and Gombocz, W. 1979, "Interpreting Anselm as logician," Synthe.se. 40, 71-96. Di Maio, M. C., and Zanardo, A. 1998, "A Gabbay-rule-free axiomatization of T x W validity," Journal of philosophical logic 27, 435-487. Dummett, M. 1991, The logical basis of metaphysics, Harvard University Press, Cambridge, MA. Elgesem, D. 1989, Intentions, actions and routines: A problem in Krister Segerberg's theory of action. Unpublished manuscript. Fitch, F. B. 1963, "A logical analysis of some value concepts," Journal of symbolic logic 28, 135-142. F011esdal, D., and Hilpinen, R. 1971, "Deontic logic: An introduction," in Hilpinen 1971, pp. 1-36.

478

Bibliography

Frankfurt, H. G. 1969, "Alternate possibilities and moral responsibility," Journal of philosophy 66, 828-839. Reprinted in J. Fischer, ed., Moral responsibility, 1986, pp. 143-152. Gabbay, D., and Guenthner, G., eds. 1984, Handbook of philosophical logic, vol. 2: Extensions of classical logic, D. Reidel Publishing Company, Dordrecht. Gale, R. M., ed. 1968, The philosophy of time, Macmillan and Co., London. Goldman, A. 1970, A theory of human action, Princeton University Press, Princeton. Gould, S. J. 1999, Questioning the millennium: A rationalist's guide to a precisely arbitrary countdown, Harmony Books, New York. Rev. ed. Green, M. 1998, The inferential significance of Frege's assertion sign. Unpublished manuscript. Green, M. 1999, "Elocutions, implicata, and what a conversation requires," Journal of pragmatics and cognition 7, 65-91. Green, M. 2000, Elocutionary force and semantic content. Forthcoming in Linguistics and philosophy, vol. 23. Green, M., and Bicchieri, C. 1997, "Symmetry arguments for cooperation in the prisoners' dilemma," in G. Holmstrom-Hintikka and R. Toumela, eds., Contemporary action theory: The philosophy and logic of social action, Kluwer Academic Publishers, Dordrecht, pp. 229-250. Reprinted in R. Jeffrey, B. Skyrms, and C. Bicchieri, eds., The logic of strategy, Oxford University Press, Oxford, 1999. Gupta, A. 1980, The logic of common nouns: An investigation in quantified modal logic, Yale University Press, New Haven. Gupta, A., and Belnap, N. 1993, The revision theory of truth, MIT Press, Cambridge, MA. Gurevich, Y., and Shelah, S. 1985a, "The decision problem for branching time logic," Journal of symbolic logic 50, 668-681. Gurevich, Y., and Shelah, S. 1985b, "To the decision problem for branching time logic," in G. Dorn and P. Weingartner, eds., Foundations of logic and linguistics: Problems and their solutions, Plenum Press, New York, pp. 181-198. Haag, R. 1990, "Fundamental irreversibility and the concept of events," Communications in mathematical physics 132, 245-251. Hamblin, C. L. 1987, Imperatives, Basil Blackwell, Oxford. Henry, D. P. 1953, "Saint Anselm on the varieties of 'doing'," Theona 19, 178-183. Henry, D. P. 1960, "Saint Anselm's 'De Grammatico'," Philosophical quarterly 10, 115-126. Henry, D. P. 1967, The logic of Saint Anselm, Oxford University Press, Oxford. See especially §4, which presents among other things Anselm's proposal (in a work composed c. 1100) that "the analysis of 'doing' will in fact be an analysis of x's doing so that p." Hilpinen, R. 1973, "On the semantics of personal directives," Ajatus 35, 140-157. Hilpinen, R., ed. 1971, Deontic logic: Introductory and systematic readings, D. Reidel Publishing Company, Dordrecht. Hilpinen, R., ed. 1981, New studies in deontic logic: Norms, actions, and the foundations of ethics, D. Reidel Publishing Company, Dordrecht. Hintikka, J. 1971, "Some main problems of deontic logic," in Hilpinen 1971, pp. 59105. Reprinted 1981. Hofstadter, A., and McKinsey, J. 1939, "On the logic of imperatives," Philosophy of science 6, 446-455. Hohfeld, W. N. 1919, Fundamental legal conceptions as applied in judicial reasoning, and other essays, Yale University Press, New Haven. Horty, J. F. 1989, An alternative stit operator. Unpublished seminar notes.

Bibliography

479

Horty, J. F. 2001, Agency and deontic logic, Oxford University Press, Oxford. Horty, J. F., and Belnap, N. 1995, "The deliberative stit: A study of action, omission, ability and obligation," Journal of philosophical logic 24, 583-644. Reprinted in The philosopher's annual, vol. 18, ed. P. Grim, Ridgeview Publishing Company, Atascadero, CA, 1997. Humberstone, I. L. 1976, An alternative account of bringing about. Unpublished manuscript; read to a meeting of the Australasian Association for Logic in 1976. Humberstone, I. L. 1977, "An alternative account of bringing about (with a pinch of relevance)," Bulletin of the section of logic 6, 144-145. Reprinted in Relevance logic newsletter 2, 1977, 107-108. This is an abstract of the unpublished Humberstone 1976. Kane, R. 1998, The significance of free will, Oxford University Press, Oxford. Kanger, S. 1957, New foundations for ethical theory, Stockholm. Reprinted in R. Hilpinen, ed., Deontic logic: Introductory and systematic readings, D. Reidel Publishing Company, Dordrecht, 1971, pp. 36-58. Kanger, S. 1972, "Law and logic," Theoria 38, 105-132. Kanger, S., and Kanger, H. 1966, "Rights and parliamentarism," Theoria 6, 85-115. Kaplan, D. 1989, "Demonstratives: An essay on the semantics, logic, metaphysics, and epistemology of demonstratives and other indexicals"; and "Afterthoughts," in Almog, Perry,- and Wettstein 1989, pp. 481-563; 565-614. Kenny, A. 1963, Action, emotion, and will, Routledge and Kegan Paul, London. Kripke, S. 1959, "A completeness theorem in modal logic," Journal of symbolic logic 24, 1-15. Kripke, S. 1963, "Semantical considerations in modal logic," Acta philosophica fennica 16, 83-94. Lemmon, E. J. 1966, "Comments on D. Davidson's 'The logical form of action sentences' ," in Rescher 1966, pp. 96-103. Lewis, D. 1970, "Anselm and actuality," Nous 4, 75-88. Lewis, D. 1986, On the plurality of worlds, Basil Blackwell, Oxford. Lindahl, L. 1977, Position and change: A study in law and logic, D. Reidel Publishing Company, Dordrecht. Lukasiewicz, J. 1920, "Three-valued logic," Ruch Filozoficzny 5. Reprinted in S. McCall, ed., Polish logic, 1920-1939, Oxford University Press, Oxford, 1976. MacFarlane, J. 2000, What does it mean to say that logic is formal? Ph.D. diss., University of Pittsburgh. Makinson, D. 1986, "On the formal representation of rights relations," Journal of philosophical logic 15, 403-425. Marcus, R. B. 1960, "Extensionality," Mind 69, 55-62. Marcus, R. B. 1961, "Modalities and intensional languages," Synthese 13, 303-322. Marcus, R. B. 1963, "Classes and attributes in extended modal systems," Acta philosophica fennica 16, 123-136. Marcus, R. B. 1966, "Iterated deontic modalities," Mind 75, 580-582. Marcus, R. B. 1972, "Quantification and ontology," Nous 6, 240-250. Marcus, R. B. 1974, "Classes, collections, and individuals," American philosophical quarterly 11, 227-232. Marcus, R. B. 1977, "Dispensing with possibilia," in Proceedings of the American Philosophical Association, pp. 39-51. Presidential Address, Western Division, American Philosophical Association, delivered April 30, 1976. Marcus, R. B. 1980, "Moral dilemmas and consistency," Journal of philosophy 77, 121136.

480

Bibliography

Marcus, R. B. 1985/86, "Possibilia and possible worlds," Grazer philosophische Studien 25, 26, 107-133. Massey, G. J. 1976, "Tom, Dick, and Harry, and all the king's men," American philosophical quarterly 13, 89-107. McArthur, R. P. 1974, "Factuality and modality in the future tense," Nous 8, 283-288. McCall, S. 1976, "Objective time flow," Philosophy of science 43, 337-362. McCall, S. 1984, "A dynamic model of temporal becoming," Analysis 44, 172-176. McCall, S. 1994, A model of the universe: Space-time, probability, and decision, Oxford University Press, Oxford. McKim, V. R., and Davis, C. C. 1976, "Temporal modalities and the future," Notre Dame journal of formal logic 17, 233-238. McTaggart, J. E. 1908, "The unreality of time," Mind 18, 457-484. Meyer, J., and Wieringa, R. 1991, "Deontic logic: A concise overview," in J. Meyer and R. Wieringa, eds., DEON '91, Proceedings of the First International Workshop on Deontic Logic in Computer Science, Amsterdam, pp. 2-14. Moore, G. E. 1912, Ethics, Home University Library of Modern Knowledge, no. 52, H. Holt and Company, New York. Reprinted by Oxford University Press, Oxford, 1965. Mullock, P. 1988, "Causing harm: Criminal law," Law and philosophy 7, 67-105. Needham, P. 1971, A semantic approach to causal logic, Master's thesis, Birmingham University. 0hrstr0m, P. 1981, "Problems regarding the future operator in an indeterminate tense logic," Danish Yearbook of Philosophy 18, 81-95. 0hrstr0m, P., and Hasle, P. 1995, Temporal logic: From ancient ideas to artificial intelligence, Kluwer Academic Publishers, Dordrecht. Parks, Z. 1972, "Classes and change," Jdurnal of philosophical logic I, 162-169. Perloff, M. 1991, "Stit and the language of agency," Synthese 86, 379-408. Perloff, M. 1995, "Stit and the imperative," American philosophical quarterly 32, 7181. Placek, T. 2000, Is nature deterministic? Jagiellonian University Press, Krakow. Porn, I. 1970, The logic of power, Basil Blackwell, Oxford. Porn, I. 1971, Elements of social analysis, Philosophical Studies No. 10, Philosophical Society and the Department of Philosophy, University of Uppsala, Uppsala. Porn, I. 1974, "Some basic concepts of action," in Stenlund 1974, pp. 93-101. Porn, I. 1977, Action theory and social science: Some formal models, D. Reidel Publishing Company, Dordrecht. Pratt, V. R. 1980, "Application of modal logic to programming," Studia logicaSQ, 257274.

Prior, A. N. 1957, Time and modality, Oxford University Press, Oxford. Prior, A. N. 1967, Past, present, and future, Oxford University Press, Oxford. Prior, A. N. 1968, Papers on time and tense, Oxford University Press, Oxford. Rakic, N. 1997, Common-sense time and special relativity, Ph.D. diss., University of Amsterdam. Reichenbach, H. 1947, Elements of symbolic logic, Macmillan Company, New York. Reichenbach, H. 1980, From Copernicus to Einstein, Dover Books, Mineola, NY. Rescher, N., ed. 1966, The logic of decision and action, Pittsburgh University Press, Pittsburgh. Ross, A. 1941, "Imperatives and logic," Theoria 7, 53-71. Ryle, G. 1954, Dilemmas, Cambridge University Press, Cambridge. Salmon, N. 1989, "Tense and singular propositions," in Almog et al. 1989, pp. 331-392.

Bibliography

481

Searle, J. R. 1965, "What is a speech act?" in Philosophy in America, Allen and Unwin, London, pp. 221-239. Reprinted in J. R. Searle, ed., The philosophy of language, Oxford Readings in Philosophy, Oxford University Press, Oxford, 1971, pp. 39-53. Searle, J. R., and Vanderveken, D. 1985, Foundations of illocutionary logic, Cambridge University Press, Cambridge. Segerberg, K. 1971, An essay in classical modal logic, Philosophical Society and Department of Philosophy,University of Uppsala, Uppsala. Segerberg, K. 1980, "Applying modal logic," Studia logica 39, 275-295. Segerberg, K. 1981, "Action-games," Acta philosophica fennica 32, 220-231. Segerberg, K. 1982, "The logic of deliberate action," Journal of philosophical logic 11, 233-254. Segerberg, K. 1984, "Towards an exact philosophy of action," Topoi 3, 75-84. Segerberg, K. 1985a, "Models for actions," in B. K. Matilal and J. L. Shaw, eds., Analytical philosophy in comparative perspective, D. Reidel Publishing Company, Dordrecht, pp. 161-171. Segerberg, K. 1985b, "Routines," Synthese 65, 185-210. Segerberg, K. 1987, "Action, logic and social theory," Acta philosophica fennica 38, 282-298. Segerberg, K. 1988a, "Talking about actions," Studia logica 47, 347-352. Segerberg, K. 1988b, Validity and satisfaction in imperative logic, Auckland Philosophy Papers, University of Auckland. Segerberg, K. 1989, Getting started: Beginnings in the logic of action, Auckland Philosophy Papers, University of Auckland. Segerberg, K. 1990, "Validity and satisfaction in imperative logic," Notre Dame journal of formal logic 31, 203-221. Stenlund, S., ed. 1974, Logical theory and semantic analysis: Essays dedicated to Stig K anger on his fiftieth birthday, D. Reidel Publishing Company, Dordrecht. Szabo, L., and Belnap, N. 1996, "Branching space-time analysis of the GHZ theorem," Foundations of physics 26(8), 989-1002. Talja, J. 1980, "A technical note on Lars Lindahl's Position and change," Journal of philosophical logic 9, 167-183. Thalberg, I. 1972, Enigmas of agency, George Allen and Unwin, Limited/Humanities Press, New York. Thomason, R. H. 1970, "Indeterminist time and truth-value gaps," Theoria 36, 264281. Thomason, R. H. 1981a, "Deontic logic as founded on tense logic," in Hilpinen 1981, pp. 165-176. Thomason, R. H. 1981b, "Deontic logic and the role of freedom in moral deliberation," in Hilpinen 1981, pp. 177-186. Thomason, R. H. 1984, "Combinations of tense and modality," in Gabbay and Guenthner 1984, pp. 135-165. Thomason, R. H., and Gupta, A. 1980, "A theory of conditionals in the context of branching time," Philosophical review 89, 65-90. Thomsen, M. 1990, The saddest pleasure, Graywolf Press, Saint Paul, MN. Thomson, J. J. 1977, Acts and other events, Cornell University Press, Ithaca. Thomson, J. J. 1990, The realm of rights, Harvard University Press, Cambridge, MA. Tomberlin, J. 1983a, "Contrary-to-duty imperatives and Castaneda's system of deontic logic," in Tomberlin 1983b, pp. 231-249. Tomberlin, J., ed. 1983b, Agent, language, and the structure of the world, Hackett Publishing Company, Indianapolis.

482

Bibliography

Tuomela, R. 1989a, "Actions by collectives," Philosophical perspectives 3, 471-496. Tuomela, R. 1989b, "Collective action, supervenience, and constitution," Synthese 80, 243-266. van Bentham, J. 1988, Manual of intensional logic, CSLI Publications, Stanford. Van Inwagen, P. 1978, "Ability and responsibility," Philosophical review 87, 201-224. Reprinted in J. Fischer, ed., Moral responsibility, Cornell University Press, Ithaca, 1986, pp. 153-173. Vanderveken, D. 1990, Meaning and speech acts, vol. 1, Principles of language use, Cambridge University Press, New York. Vanderveken, D. 1991, Meaning and speech acts: vol. 2, Formal semantics of success and satisfaction, Cambridge University Press, New York. Vendler, Z. 1957, "Verbs and times," Philosophical review 66, 143-160. von Kutschera, F. 1980, "Grundbegriffe der Handlungslogik," in H. Lenk, ed., Handlungstheorien interdisziplindr: Handlungslogik, formale und sprachwissenschaftliche Handlungstheorien, vol. 1, Wilhelm Fink Verlag, Munich, pp. 67-106. von Kutschera, F. 1986, "Bewirken," Erkenntnis 24, 253-281. von Kutschera, F. 1993, "Causation," Journal of philosophical logic 22, 563-588. von Neumann, J., and Morgenstern, O. 1944, Theory of games and economic behavior, Princeton University Press, Princeton, von Wright, G. H. 1963, Norm and action: A logical inquiry, Routledge and Kegan Paul, London. von Wright, G. H. 1966, "The logic of action: A sketch," in Rescher 1966, pp. 121-136. von Wright, G. H. 1981, "On the logic of norms and actions," in Hilpinen 1981, pp. 335. Walton, D. 1975, "Modal logic and agency," Logique et analyse 69-70, 103-111. Walton, D. 1976a, "Logical form and agency," Philosophical studies 29, 75-89. Walton, D. 1976b, "St. Anselm and the logical syntax of agency," Franciscan studies 36, 298-312. Walton, D. 1980, "On the logical form of some commonplace action expressions," Grazer philosophische Studien 10, 141-148. Wansing, H. 1998, "Nested deontic modalities: Another view of parking on highways," Erkenntnis 49, 185-199. Xu, M. 1994a, "Decidability of deliberative stit theories with multiple agents," in D. M. Gabbay and H. J. Ohlbach, eds., Temporal logic, First International Conference, ICTL '94, International Conference of Temporal Logic, Springer-Verlag, pp. 332348. Xu, M. 1994b, "Decidability of stit theory with a single agent and refref equivalence," Studia logica 53, 259-298. Xu, M. 1994c, "Doing and refraining from refraining," Journal of philosophical logic 23, 621-632. Xu, M. 1995a, "Busy choice sequences, refraining formulas, and modalities," Studia logica 54, 267-301. Xu, M. 1995b, "On the basic logic of stit with a single agent," Journal of symbolic logic 60, 459-483. Xu, M. 1997, "Causation in branching time (I): Transitions, events, and causes," Synthese 112, 137-192. Zampieri, G. 1982, "Diffeomorphisms constructively associated with mutually diverging spacetimes which allow a natural identification of event points in general relativity," Atti delta Accademia Nazionale dei Lincei, ser. 8 73, 132-137, 221225. Zampieri, G. 1982-1983, "A choice of global 4-velocity field in general relativity," Atti Istituto Veneto Scienze, Lettere e Arti 141, 201-206. Zanardo, A. 1996, "Branching-time logic with quantification over branches: The point of view of modal logic," Journal of symbolic logic 61, 1-39.

Index / assignment shift, 239 in moment-history pair, 147 (imperative), 83 &; (conjunction), 31, 241 V (disjunction), 241 (material implication), 241 (negation), 241 (truth, sentential constant), 241 falsehood, 241 splitting of histories, 188 (choice-separated), 214, 467 (separable), 215, 467 (negation), 32 (universal quantifier), 152, 241 (identity), 241 (definite description), 241 (the non-existing object), 240, 471 causal order, 30, 139, 180, 461 linear time order on instants, 195, 466 between sets of moments, 182, 464 proper causal order, 30, 139, 180, 463 proper lower bound, 182, 464 choice-equivalent, 214, 467 inseparable, 215, 467 material equivalence, 241 agent-term equivalence, 441 semantic equivalence, 236, 470 (m.context equivalence), 236, 470 (theorem), 384 implication, 236, 470 settled truth, 235, 470 truth at, 31, 142

validity in a model, 236, 470 validity in a structure, 236, 470 in-ctx in-context implication, 236, 470 in-context validity, 237, 471 (in transition stit), 251 ' ' (quote-function), 171 [ ] (= equivalence class), 441 A (a sentence), 6, 240, 471 a an agent, 382 an assignment, 146 Aa, 382 a lemma concerning, 387 and stit, 387 A-series, 134 ability, 45, 242, 249 joint, 283, 284 above. See causal-order relation absolute concepts, 211, 334 functors, 241 time, 194 TRL (see Thin Red Line) absurdity in-context, 237 strong, 237 achievement stit, 29, 32-37 applications of, 39-58 not closed under n, 47-49 not closed under logical consequence, 40 compared with deliberative stit, 37-39 and could have, 45-47 decidability of, 414 and deontic contexts, 53-55 instants, role of, 38 negative requirement, 37 number of non-equivalent modalities, 51-52 483

484

overdetermination of, 290 pictures, 39-58 positive requirement, 36 and refraining, 40-45 and refraining from refraining, 49-53 semantics for, 36, 248, 250 sufficient conditions for, 386 and ten-minute mile, 49 action verb (Kenny), 69 action-nominals, 12 action-state semantics, 26 activity verb (Kenny), 69 actual history, 161 actuality, 246 and TRL, 163, 164, 168 Actually l:, 153, 246 Actually2:, 246 addressee-action-reduction principle, 89 Admh(s) (set of histories admitted by s), 347 Admh(s, m) (set of histories admitted by s from m onward), 372 admissibility, 347 and extension, 348 from a moment onward, 372 and strengthening, 348 admission, 347 Admm(s) (set of moments admitted by s), 347 advice. See imperatives agency condition (for stit), 247, 248, 250 with multiple agents, 283 Agent, 33, 211, 462 agent frame, 441 agent-action reduction program, 119 agent-terms, 240 agentive infinitive constructions, 17 agentive predicates, 87 agentiveness of stit thesis, 7, 59, 86, 459 agentives, 3-9 agent of, 5, 62 canonical form for, 5 as complements of other agentives, 67 declarative complements of, 5 and deontic statements, 13 and imperatives, 10, 460 joint, 281-291 literature on, 5n with multiple agents, 271-291 nesting of, 65, 273-281

Index stit sentences as, 7, 459 strict, 287 test for, 7, 79, 87-88, 99, 460 agents, 211, 462, 466 distinctness of, 436 fission or fusion of, 211 generalization on, 333-334 independence of, 26, 217-218, 283, 462 joint, 112 multiple, 26, 34, 76-77, 271-291 theory of, 211 aggregate (of agents), 336 alternative (to companion set), 398 Always:, 243 anchored connective, in parameters, 239 Anderson, 211, 323 1956, 296, 298, 303 1962, 5n, 20, 22 1970, 5n Anscombe 1957, 118 Anselm, 18, 22 antactualism, 159 antisymmetry, 180, 461 APCn (max. n choices), 437 Aqvist 1972, 23 1974, 24 1978, 24 1984, 5n, 11 and Mullock 1989, 25 Aristotle, on deliberation, 37 assertability, 157 assertion, 171-175 as a relation between a person and a content, 173 and betting, 158 direct and indirect discourse, 171 history-independence of, 171 intentionality of, 173 problem, 141, 158-160 a sentence, 157 thesis, 110 and truth, 158 versus uttering, 157 vindication/impugnment and, 174 assignment parameter, 146, 155 not initialized by context, 150, 152, 233234 mobility of, 146 shifting, 239

Index and Tarski, 221 assignment-open sentences, 158 assertability of, 157, 175 and history-open sentences, 153-155, 156n, 175 truth values of, 175 astit:. See achievement stit At-instt: (true at instant t), 243 at-most-binary, 414, 431 BT + I + AC structures, 434 At-momt: (true at moment t), 244 atomic sentences semantics for, 31, 240 truth relative to history, 31n Augustine, 163 Austin 1961, 4, 97, 242, 249, 255, 284 1975, 97, 275 authority, 299, 316 avoiding, 261 axiomatics concepts of, 472-473 of Lai, 416-417 of Ldm, 436-439 of SA0, 303-304 B (a sentence), 6 B-series, 134 backward. See also causal-order relation branching (see no backward branching) denseness, 383 monotony (of achievement stit), 272 versus downward, 30n Badecker 1987, 16 Baier, K., 319 Barcan 1946, 327 Barcan formula for cstit:, 329 for Sett:, 327 Barcellan and Zanardo 1999, 136 Barth, K., 163 Belnap. See also Gupta and Belnap, Szabo and Belnap 1992, 29n, 137, 140, 178, 183, 188, 190, 280, 284 1995, 194 1996a, 209, 250, 362 1996c, 194 1999, 61, 192 below. See causal-order relation

485

Bennett 1988, 5n, 18 betting, 158 history-independence of, 171 on the past, 120n Bicchieri. See Green and Bicchieri Big Bang, 189 bivalence, 156 branches (of a bundled tree), 198 branching histories, 29n branching space-time, 29n, 178, 190 branching time. See also BT, BT + I, BT + AC, BT + I + AC alternatives to, 196-203 arguments against, 205-209 forward and backward, 182-187 and indeterminism, 139 sources for, 224-225 stepping outside of, 207 terminology, 29n theory of, 29, 139-141, 177-203 branching + TRLab, , 162 branching + TRLC, 163 branching + TRLfcn, 165 Brand 1970, 41n Brandom 1994, 173 Bratman 1983, 74 Braiiner, Hasle, and 0hrstr0m 1998, 136, 166n, 169 2000, 163 Bressan, 185, 328n, 334 1972, 195, 211, 281, 335n 1974, 195 Brown 1988, 1990, 1992, 26 Brunner, E., 163 bstit, 26 BT (branching time) definitions concerning, 462-465 model, 31 postulates for, listed, 461 structure, 30, 31 BT + I (branching time with instants) definitions concerning BT, 462-465 definitions concerning instants, 465-466 postulates for BT, listed, 461 postulates for instants, listed, 461-462 structure, 36, 141, 144 BT + AC (agents and choices in branching time) compactness of, 444

Index

486

definitions w.r.t. agents and choices, 466-468 definitions concerning BT, 462-465 general structure, 444 postulates for agents and choices, listed, 462 postulates for BT, listed, 461 SAo complete w.r.t., 304 SAo sound w.r.t., 304 structure, 35 theory, 28 BT + AC with max. n choices Ldmn complete w.r.t., 440-445 Ldmn sound w.r.t., 439-440 BT + I + AC (agents and choices in branching time with instants) at-most-binary structure, 434 definitions w.r.t. agents and choices, 466-468 definitions concerning BT, 462-465 definitions concerning instants, 465-466 Lai sound w.r.t., 417-419 model, 35, 228, 468 point, 228, 469 postulates for agents and choices, listed, 462 postulates for BT, listed, 461 postulate for Domain, listed, 462 postulates for instants, listed, 461-462 structure, 35, 226 theory, 28 BT + I + AC+bc (BT + I + AC with busy choosers), 451 refref invalid in, 454-458 BT + I + AC+nbc (BT + I + AC with no busy choosers), 381 Lai + rr sound w.r.t., 390-391 Lai + rr complete w.r.t., 407-414 bundled trees, 198-199, 202 Burgess 1978, 159, 198, 205, 207 1979, 198 1980, 198 1984, 224 busy choice sequence, 218, 250, 383, 453, 468 busy chooser, 49, 219, 265-268, 468 and consistency condition for deontic trees, 373, 374 definition of, 50

and refraining from refraining, 50, 52, 451 and strategies, 349 bystander, mere, 285, 290 c

a chain, 181, 463 a context of use, 145 Calvin, J., 163 Can:, 242, 329 canonical frame, 441 cant (can see to it at i), 249 with multiple agents, 284 caring about the future, 209 Carnap, R., 328, 334 Castaneda, 22, 74-78, 316, 328 1954, 21 1974, 12, 74, 75 1975, 16, 74, 75 1976, 75, 77 1981, 21, 77, 295, 296, 309 categorematic expressions, 234 causal future of possibilities, 180, 463 causal past, 180, 182, 463, 465 causal-order relation, 139, 179, 180, 463 postulates for, 180 causality, 251 in branching time, 209-210 cusp of, 210 chains definition of, 181, 463 lower bound of, 182, 464 upper bound of, 182, 464 witness by, 49, 249-250 characteristic sentence, 393 Chellas, 5n, 85-87 1969, 22, 85, 197, 297n 1992, 86, 86n, 248, 274, 275, 297, 297, 298n, 329 Chellas stit, 23n, 86, 297, 436 definition of, 298 semantics for, 248 Chisholm, 5n, 22, 65-68 1963, 312, 313 1964a, 19, 21, 263 1969, 65, 68 Davidson's objections to, 80 choice box, 300 choice equivalence, 36, 214, 248, 467

Index equivalence, for sets of agents, 282 function, 33 moment of, 212 nonvacuous, 50 partition, 212, 462 point, 218, 467 separation, 214, 467 vacuous, 218 choices. See also BT + AC, BT + l + AC forced, 278 independence of, 35, 217 multiple, 116 sequential, 282 theory of, 211-219 Choice, 213, 466 postulates for, 214, 215 Choiceam (choice function), 33, 212, 213, 462, 466 Choice^(h), 34, 213, 466 Choiceam, (m), 213, 466 Choice^(mi), 213, 467 Choice° -equivalent, 36 choosing to choose, 217 circumstance, 296, 305, 328 condition, 306 future-tensed, 309 and other agents' doings, 309 present-tensed, 308-309 claims closure under entailment, 108 Thomson's locution for, 108 unrestricted complement thesis for, 108 closed sentence assertability of, 157 by independence, 153 by initialization, 153 in parameters, 158 (see also Z-closed) closed set, under negated stits, 400 coherence of plans, 376 collectives, 281 companion, 385-390 neg-, 389, 392 pos-, 389, 392 set, 392-397 set, characteristic sentence for, 393 syntactic, 392 theorem, 388 and witness, 389 comparability (of moments), 181, 463 compatibilism, 255, 260

487

completeness (of strategies), 353 and completeness of theories, 354 persistence under extensions, 354 and primariness, 354 completeness (semantic) of dstit logics for models, 444 of Lai w.r.t. BT + I + AC, 428-434 of Lal + rr w.r.t. BT + I + AC + nbc, 407-414 of Ldrrin w.r.t. BT + AC with max.n choices, 440-445 of SA0 w.r.t. BT + AC, 304 strong, 443 concrete possible event, 190 conflicting obligations, 303 conjunction, 241 connectives, v, 239 historical modality, 242 indexical, 246-247 mixed temporal-modal, 245 non-truth-functional, 146 tense and temporal, 242-245 truth-functional, 241 consequence, closure of stit under, 40 consistency of propositions and events, 193 syntactic, 384 consistency condition (on deontic trees), 366, 373-375 and busy choosers, 373, 374 contemplation and action, incompatibility of, 55-58 content, 173 in-a-context notions of, 231 context of use, 145, 147, 225 context parameter(s), 145, 147-149, 225 and context-dependent expressions, 148 as fact-of-the-matter parameters, 148 immobility of, 145 and initialization, 148-149 and moment-of-use parameter, 150, 226 terminology, 147 two functions of, 145, 147 context-dependent expression, 148 (see also context parameter(s)) context-dependent TRL. See Thin Red Line context-initialized (point), 230, 469 context-mobile pairing, 231 contingent (transition), 194, 465

488 contrary-to-duty obligations, paradox of, 312-314 could have, 45-47 complement of, 46 could have been, 257 could have done, 257-259 versus might have done, 257 could have done otherwise, 73, 255 Chisholm's account, 68 and moral responsibility, 99 not implied by open future, 47 picture, 46 and strategies, 47 could have prevented, 262 could have refrained, 263-265 could not have avoided doing, 261-262 could-have-stit:, 46, 249 counter (for achievement stit), 36, 247, 248 counter (syntactic), 399 ref-, 400 semi-ref-, 399, 401-407 counterexample lemma, 410 counterpart theory, 207, 209 cstit:. See Chellas stit Curry, H., 239 d operator (von Wright), 61 D-axiom, 301 Danto, 5n 1973, 19 Davey 1999, 304n, 305 Davidson, 5n 1966, 78-81, 97 1980, 18 his analysis of agentives, 78-81 Davis. See also McKim and Davis Dazeley and Gombocz 1979, 19 decidability of achievement stit, 414 of Lai + rr, 414 deductive consequence, 384 deductively closed set, 384 definite description operator, 241 deliberation, 212, 368 deliberative stit, 29, 37-39 compared with achievement stit, 37-39, 247 definability of, 329 definition of, 298

Index negative requirement, 37 never settled true, 38 positive requirement, 37 role of instants, 38, 194 semantics for, 37, 247 and von Kutschera, 26 and word-giving, 115 demands. See imperatives denotation, 235 deontic contexts, 11-13, 53-55, 64-65 detachment, principle of, 301 equivalences, 53-55, 64-65, 91, 277, 301, 322, 472 logic, 291, 295-317 paradoxes, 78, 309-314 readings, of other-agent nested stits, 275-277 deontic kinematics, 348, 364-366 and austere strategies, 368-377 deontic statements. See also restricted complement thesis canonical forms for, 13, 53 complements of, 11-13, 77 as quasi-agentive, 13 deontic trees, 365 determined by a strategy, 372 as strategies, 369-370 dependent (on parameters). See Zdependent determinism feature-relative, 138 meaning of, 203-205 and stit theory, 204 determinist, 204 deterministic Tree, 204 at a moment or past, 204, 464 strongly anti-, 205 Di Maio and Zanardo 1998, 196 Diodorus, 167 direct discourse, 171, 174n directives, 82 divergence theories. See TxW theories Dom(s) (domain of strategy s), 345 Domain, 141, 219, 226, 462 domain (of a strategy), 345 versus field, 350 double indexing, 231 double time references, 120-123, 175, 244, 248, 251

Index downward versus backward, 30n dstit logic, 444 dstit:. See deliberative stit dthat (Kaplan), 242 duties Thomson's locution, 108 to admonish, 308 to apologize, 306 unrestricted complement thesis for, 108 dynamic logic, 25 earlier. See causal-order relation Einstein, A., 179 Einstein-Podolsky-Rosen, 218, 284 Elgesem 1989, 25 embedded sentences, 148-149 and assignment parameter, 221 evaluation not restricted to contextinitialized points, 230 and propositions, 190 semantic concepts for, 235-238 embedding of agentives, 78 of declaratives, 14 of imperatives, 14, 94-96 of interrogatives, 14 of sentences, 148 and stit, 94-96, 107, 323 equivalence choice (see choice equivalence) in-context, 236, 470 semantic, 236, 470 equivalently closed under conjunction, 396 essential agent for outcomes, 285 for stit, 285, 286 essentiality condition (for transition stit), 252

Euclid, 179, 343, 344 evaluative versus prescriptive, 319, 320, 322, 330, 335 events, 190-192. See also concrete possible event actions as, 18, 70, 79, 97, 247 beginning of, 193 completion of, 193 consistency of, 193-194 existence of, 192, 193 have no negations, 79

489

initial, 191, 465 local, 179, 190 occurrence of, 192 outcome, 191, 465 super, 178 and transitions, 192 exclusion, 347 and extension, 348 and strengthening, 348 extensionality, 146, 241n, 333-334 extensionalization, 334 extensions, 346 along a history, 354, 355 complete and favoring, 358

F predicate letter, 240, 471 Prior's future tense (see Will:} F (false), 235 / future history, 182, 465 operator, 61, 240, 471 5 (agent frame), 441 falsity at a point, 235 fatalism, 167n favoring, 356-359 completely, 356 definition of, 356 extension lemma, 357 persistence of completely, 356 feature-independently (in)deterministic, 137 fiats, 82, 94 Ross's counterexample, 83 field (for a strategy), 116, 349, 349n versus domain, 350 finite model property, 384 none for Lai, 434 for Lai + rr, 407-414 fit between word and world, 118, 125 Fitch, 5n 1963, 20, 22 Floyd, 25 F011esdal and Hilpinen 1971, 295n forbearing, 62. See also refraining forbidden. See Frbn: forbidding, 335 forward. See also causal-order relation branching, 184

490 denseness, 383 versus upward, 30n Frankfurt 1969, 255, 261, 263 Prbn-if-can-do:, 332 Frbn: (forbidden), 11 equivalences, 53-55 free riders, 290 free will, 256 Frege, G., 179, 343, 345 functors, 239 absolute, 241 stit, 247-252 variable-binding, 241 future, 184. See also future history, causal future of possibilities, wondering, caring the actual, 208 of the context, 232 contingents, 31 history, 182, 183, 465 of possibilities, 140, 183 open, 30, 136 tense (see Will:) future-tensed statement, evaluation of, 31 G (Prior's strong future tense). See Willalways:

Gale 1968, 134 game theory, 342-343, 348 general BT + AC structure, 444 gentle murder, 312 gerundives, 67 Goldman 1970, 18 Gombocz. See Dazeley and Gombocz Good Samaritan paradox, 309-311 Goodman, N., 237 grammar. See also embedding compositional, 142 of English agentives, 14-18 formal, 142, 240, 471 of stit, 14 greatest lower bound, 182, 464 Green 1998, 230 2000, 148 and Bicchieri 1997, 218 Gupta, 47. See also Thomason and Gupta 1980, 211, 282 and Belnap 1993, 171

Index Gurevich and Shelah 1985, 445n 1985a, 445n

H

a set of histories, 181, 463 Prior's strong past tense (see Wasalways:} h:T-+S theory, 197-198 h a history, 181, 463 history of evaluation, 147 H(m) (the set of histories through m), 30, 192, 463 H(M) (the set of histories passing through at least one member of M), 192, 464 H[M] (the set of histories entirely containing M), 192, 464 Haag 1990, 187 had a strategy for not doing, 268-269 Hamblin, 5n, 87-90, 92-94, 124, 125 1987, 10, 25, 26, 87, 92, 119, 276, 346, 356, 369 Hare, R., 20 has no truth value. See truth value Hasle. See Braiiner, Hasle, and 0hrstr0m, 0hrstr0m and Hasle Henkin general models, 199 Henry, 20 1953, 19 1960, 19 1967, 18, 19 Hilpinen, 5n 1973, 23 Hintikka 1971, 295 historical connection, 187-189 cut, 182, 465 modalities, 32 histories, 140, 181, 463 choice-equivalent, 214, 467 definition of, 30 ideal, 371, 372 matching versus overlap of, 196, 203 missing, 200, 203 versus possible worlds, 151, 181, 190, 233 quantification over, 335 separable, 215, 467 splitting of, 188

Index History, 181, 463 history of the context, 164, 232 of evaluation (see history-of-evaluation parameter) of use, 151 history-of-evaluation parameter, 147, 155, 224 not initialized by context, 151, 152 history-open sentences, 158 assertability of, 157, 175 and assignment-open sentences, 153155, 156n, 175 truth values of, 175 Hitchcock, C., 171 Hoare, C., 25 Hofstadter and McKinsey 1939, 82 Hohfeld, 5n 1919, 19, 22, 107, 325 equivalences, 108 hoping, history-independence of, 171 horizon, 85, 195, 466 a property of, 452 Horty 1989, 29, 247 2000, 41n, 52, 55, 85, 264, 321n, 348, 349, 369n Humberstone, 5n 1976, 19 1977, 19, 23 Hume, D., 255 3 (an interpretation), 31, 144, 227 i (an instant), 195, 465 i( m ) (instant of m), 195, 466 i >m (horizon from m at i), 195, 466 identity, 241 immobile parameters. See parameters imperative content thesis, 10, 22n, 42, 60, 84, 89, 272, 460 imperatives, 10-11 accountable, 94 and agent, 16 Chellas's theory, 85-87 constituent, 281 content of, 10, 460 contrary-to-duty, 312 embedding of, 94-96 and fiats, 82-84 force of, 10

491 grammar of, 94-96 kinds of, 92-94 logic of, 17 negations of, 17, 89-92 negative, 42 and obligations, 86 and orders, 87 Ross's paradox, 83 as speech acts, 91 willful and nonwillful, 92 implication in-context, 236, 470 strong, 236, 470 impugnment, 127 in-context, 236. See also absurdity, equivalence, implication, validity inconsistency (of propositions and events), 193 independent (of a parameter), 153 indeterminism, 134. See also determinism, deterministic Tree concepts of, 136-139 and theology, 163 indeterministic semantics. See semantics of indeterminism indexicals, 238. See connectives, indexical indirect discourse, 157n, 171 individual concept, 328n, 334 individual constant, 240 individual variable, 240, 471 inessential agent for outcomes, 285 for stit, 285, 286 Inevitably:, 245 information sets, 343n initial. See events, initial event (see events, initial) of a transition, 137 initialization, 148-149 inseparability of histories, 215, 467 of moments, 215, 467 nontransitiveness of, 215 Instant, 35, 140 order type of, 196 postulates for, 194, 462 subset of Domain, 152 instants, 35, 140 definition of, 195, 465

492 names for, 243, 244 postulates for, 194, 462 role in stit theory, 194 theory of, 194-196 and times, 194 instructions. See imperatives intensional predication, 328n intention, history-dependence of, 56n intentionality and action, 33, 79 and promises, 119 interdefinability of stit operators, 298, 298n interpretation, 227, 468. See also 6interpretation settled, 434 interpretation parameter(s), 144, 227 immobility of, 144, 228 terminology, 144 intervals, 179 invitations, 93 inviting, 111 joint agency, 281-291 strict, alternative concept of, 113 joint stit applications of, 288-290 carry-over from singular stit, 283 essential and inessential agents for, 286 essential and inessential for, 285 grammar of, 282 other-agent nested, 290-291 plain, 282-284 semantics for, 283 strict, 284-287 strict, alternative account, 113 weakening of, 285 K (class of structures), 237, 471 -valid, 237, 471 K-axiom, 301 Kamp structures, 198, 202 Kamp, H., 246 Kane 1998, 28, 204 Kanger, H., 5n Kanger, S., 5n 1957, 20, 22 1972, 23 and Kanger 1966, 21, 22 Kaplan, 242, 246

Index 1989, 141, 145, 149, 151, 221, 225-226, 229-234 KD, 295n, 300, 301 Kenny, 5n, 20, 68-74 1963, 21, 22, 87 kinematic condition (on deontic trees), 366, 375-377 knowledge history-independence of, 56 implies truth, 56 Kremer, P., 216 Kripke, 139, 151, 195, 224, 334 1959, 179 1963, 328 L, 141, 227 Lai (astit with 1 agent) axiomatics of, 416-417 completeness of, 428-434 no finite model property for, 434 soundness w.r.t. BT + I + AC, 417-419 Lali, 434 Lai + rr (Lai +refref), 383-385 and at-most-binary structures, 414 completeness of w.r.t. BT + I + AC + nbc, 407-414 decidability of, 414 finite model property for, 407-414 soundness w.r.t. BT + I+AC + nbc, 390-391 later. See causal-order relation laws, 245 Ldrn (dstit with many agents) axiomatics of, 436-439 Ldm0 (= Ldm), 437 Ldmn (Ldm + APCn), 437 completeness of w.r.t. BT + AC with max. n choices, 440-445 decidability of, 450 finite model property for, 445-450 soundness of w.r.t. BT + AC with max. n choices, 439-440 least upper bound, 182, 464 Lee-Hamilton, E., 183 legal sequence (of parameters), 142 Leibniz, G. W., 163 Lemmon 1966, 81 Lewis, 151, 164, 195, 246 1970, 163 1986, 170, 179, 188, 196, 205-209

Index Lindahl, 5n 1977, 23, 273 local (in parameters), 223. See also translocal locality condition (on deontic trees), 366, 375 lower bound (of a set of moments), 182, 464 Lukasiewicz 1920, 156n Luther, M., 163 m (model), 31, 144 -valid, 236, 470 M (a set of moments), 178, 463 m a moment, 178, 463 moment of evaluation, 146 m( t , h) (moment at which i intersects /i), 195, 466 m/h (moment-history pair), 31, 147 mc (moment of use), 150 MacFarlane 2000, 227 Main Calculation, 326, 331 Makinson 1986, 5n, 25, 260, 325 Marcus 1960, 326 1961, 328, 333-334 1963, 336 1966, 318-319, 322, 326, 335 1972, 328 1974, 336 1977, 320 1980, 320, 323-324 1985/86, 197, 320 Massey 1976, 281 maximal consistent, 384 maximal in, 423 McArthur 1974, 159 McCall 1976, 135 1984, 135 1994, 29n, 135, 207 McCullagh, M., 50 McKim and Davis 1976, 166n, 167 McKinsey. See Hofstadter and McKinsey MCS (maximal consistent set), 392, 416 McTaggart 1908, 134 Melden, A., 5n Meyer and Wieringa 1991, 301 might have been, 257

493

might have been otherwise, 259-260 might have done, 257-259 versus could have done, 257 might have refrained, 265-268 might not have done, 260-261 Might-have-been:, 245 Minkowski, H., 179 missing-history structures, 199. See also moment-history structures mixed nectors, 239 mobile parameters. See parameters modal construction, vi, 18, 318 modal logic, 223, 240 modal logic of agency binary relations, use of, 23-24 Castaneda's practitions, 74-78 Chisholm's undefined locution, 65-68 history of, 18-26 Kenny's canonical form, 68-74 and ontology, 18, 25, 78-81, 320, 323 von Wright's program, 60-65 modality, 51 model, 228, 468 model for, 444 model parameter, 144, 228, 468 moment of choice, 250 of outcome, 250 Moment-History (the set of momenthistory pairs), 190 moment-history pairs, 224 quantification over, 335 moment-history structures, 199 complete, 203 descriptive adequacy of, 199-203 moment-of-evaluation parameter, 146, 224 initialized by context, 150, 152 mobility of, 147, 229 moment-of-evaluation-dependent TRL. See Thin Red Line moment-of-use parameter, 150, 226 immobility of, 229 and stand-alone sentences, 229 momentary witness. See witness, momentary moments, 139 choice-equivalent, 213, 248, 467 comparability of, 181, 463 concreteness or nonrepeatability of, 180

494 connectedness of (see historical connection) definition of, 178, 463 incomparable, 181 names for, 244 as possible events, 190 quantification over, 335 separable, 215, 467 versus states or times, 140, 180 monsters, 145, 174n Montague, R., 334 Moore 1912, 255 moral responsibility, 99-100, 255 for actions versus for facts, 256 Morgenstern. See von Neumann and Morgenstern Mullock. See also Aqvist and Mullock 1988, 25 Murphy's Law, 210 necessity, historical. See settledness Needham, 5n 1971, 23 neg-companion [root], 389, 392 negation and imperatives, 89-92 internal and external, 42, 44 semantics for, 241 and stit, 44, 72, 89-92 negative acts, 43, 301 negative condition for achievement stit, 248 for achievement stit, with witness by chains, 250 for deliberative stit, 247, 330 for joint stit, 283 Newton, I., 179 NIM (no inessential members), 286 NMB (no mere bystanders), 290 no backward branching, 30, 140, 182-187, 196 and agency, 185 fails for epistemic possibility, 184 fails for repeatable states, 184-185 no choice between undivided histories, 26, 216-217, 462, 466 a consequence of, 386, 452 and deontic trees, 375 no inessential members, 286 no-can-do case, 324-325

Index proper, 324 no-good-choice case, 323-324 no-good-choice moment, 302, 303 nonemptiness condition for achievement stit, with witness by chains, 250 nontriviality (of Tree), 140, 178, 461 normal case, 325 Now:, 153, 246 O (primitive of deontic kinematics), 365 CD-statements versus DO-statements, 296 in English, 306 equivalence of, 306 Oa (O generated by strategy s), 372 consistency condition on, with busy choosers, 373 consistency condition on, with no busy choosers, 374 kinematic condition on, 375 locality condition on, 375 Oblg: (obligated), 11 definition of, 299 equivalences, 53-55 -necessitation, 301, 303 picture for, 300 obligation. See also Oblg: agency intrinsic to, 92 collective versus distributive, 336 conditional, 304-306 conflicting, 303 contrary-to-duty, 312-314 deliberative, 298 deliberative versus judgmental, 366 detachment of, 304, 313 history-dependent, 315 joint, 288 relativized to moments, 364 reparational, 348, 364, 370, 371 semantics of, 298-303, 314-316 Ockhamist structures, 198, 202 0hrstr0m, 166. See also Braiiner, Hasle, and 0hrstr0m 1981, 136 and Hasle 1995, 139, 156n, 201 OMB (only mere bystanders), 290 ontology, 18 of actions, 320, 323 and refraining, 42

Index and semantics, 139 open (in parameters). See Z-open open future. See future, open open sentences. See Z-open operator letter, 240, 471 operators, 239 semantics for, 241 orders. See also imperatives normal form of, 92 and obligation, 276 ordinary language and agentives, 87 other-agent nested stits, 273-281 deontic reading of, 275-277 disjunctive readings of, 277-279 probabilistic reading of, 279-280 strategic reading of, 280-281 ought-kinematics, 364 Our World, 139, 140, 179, 188. See also Tree outcome essential agent for and bystander for, 285 of an initial, 192 of a transition, 137 P (Prior's past tense). See Was: P a past, 182, 465 a prepositional variable, 240, 471 paradoxes of contrary-to-duty obligations, 312-314 Good Samaritan, 309-311 parameter for (syntactic), 422 parameters (of truth), 142, 223 assignment (see assignment parameter) auxiliary, 147 closed in (see Z-closed) context (see context parameter(s)) dependent on (see Z-dependent) fact-of-the-matter, 145, 234 history-of-evaluation (see history-ofevaluation parameter) immobile, 143-145, 223 independent of (see Z-independent) interpretation (see interpretation parameter^)) local in (see local) mobile, 143, 145-147, 149-152, 223 in modal logic, 223

495 model (see model parameter) moment-of-evaluation (see moment-ofevaluation parameter) moment-of-use (see moment-of-use parameter) open in (see Z-open) structure (see structure parameters) in tense logic, 224 translocal in (see translocal) TRL (see Thin Red Line) two senses of, 142 paraphrase. See also stit paraphrase thesis clarification by, 7, 9, 45, 84, 257 Parks 1972, 281 partial order, 140 Pascal, B., 163 past, 182, 183, 465. See also causal past determinacy of, 30 past history, 182, 183, 465 past tense (see Was:) patiency, 20 Peirce, C. S., 159 Peircean future tense, 159 performance verb (Kenny), 69 Perm: (permitted), 11 equivalences, 53-55 permission. See Perm: and power, 325 personal identity, 211 Placek 2000, 29n Plantinga, 164 point (of evaluation), 142, 228, 469 context-initialized, 149, 150 Porn, 5n 1970, 23 1971, 23 1974, 23, 72 1977, 23, 24 pos-companion [root], 389, 392 positive condition for achievement stit, 248 for achievement stit, with witness by chains, 250 for deliberative stit, 247 for joint stit, 283 for transition stit, 252 Poss: (historically possible), 32 semantics for, 32, 152, 242, 436 Poss-true, 122

496

possibility concepts of, 179, 188 epistemic, 137 historical (see Pass:) immediate, 204, 464 real, objective, 179, 188 possible worlds versus histories, 151, 233 Possibly-will:, 161 practitions (Castaneda), 21, 74-75 canonical form for (X to A), 75 Pratt 1980, 25 predicates, 239 semantics for, 241 predication, grammar of, 240, 471 predictions, 141, 158 preliminary structures, 424-428 preventing, 262 Prior, 31, 224-225, 243 1957, 221, 224, 242, 327, 365 1967, 29, 139, 141, 221, 224 1968, 55 prior choice, principle of, 188 priority condition for achievement stit, 248 for achievement stit, with witness by chains, 250 for joint stit, 283 Prisoner's Dilemma, 218 privileges Thomson's locution, 108 unrestricted complement thesis for, 108 probability and possibility, 137 role of, in stit theory, 279 prohibitions. See also Prbn: form of, 322-325 generalized, 325-335 promises breaking, 104 complements of, 101, 116 infringed, 124 lying, 104 and obligation, 315 satisfied, 124 and speech acts, 97 strategic content of, 119, 123-126 Thomson's account, 113-116 to be, 102 to refrain, 115

Index and word-giving, 113, 116-129 promising, act of, 119 proper causal-order relation, 180, 463 prepositional variable, 240, 471 propositions, 114, 189-190 assignment-dependent, 190 consistency of, 193 and events, 192-193 intensional versus intentional notions of, 189 time-dependent, 190 timeless, 189 psychological attitudes, 56n Q (a sentence), 6 QTF (quantifier-and-truth-function equivalence), 326 quantifiers. See also truth for quantifiers not truth-functional, 146n, 221 range of, 328, 330 semantics for, 241 quasi-agentive, 13, 46 Quine, W. v. O., 323, 334 quote-function, 171 R$, 400 Rakic 1997, 29n recipes. See imperatives recursive semantics, 143 and relativized truth, 143 ref-counter, 400 reflexivity, 180, 461 refraining, 40-45 is agentive, 42 is both doing and not doing, 40, 42 and not doing, 41, 45 from refraining, 49-53 (see also refref equivalence) and seeing to it that, 63 stit paraphrase of, 42, 43 von Wright's account, 63 refref (refref equivalence), 301n, 451. See also refraining with busy choosers, 53, 451 and deliberative stit, 55 and deontic equivalences, 54 invalid in every BT + I + AC + nbc structure, 454-458 and number of stit modalities, 52

Index validity of in B T +1 + A C + nbc models, 390-391 Reichenbach, 80 1980, 81 relativity. See special relativity requests. See imperatives responsibility, moral, 99-100 restricted complement thesis, 13, 54, 60, 64, 87, 101, 115, 116, 249, 273, 298, 320, 322, 326, 460 and promises, 102, 116 and strategies, 342 Ross 1941, 83 his counterexample, 83 his paradox, 83-85 RR (The Realm of Rights). See Thomson 1990 Ryle 1954, 135 5 (sanction), 298 Sa (sanction for a), 299, 321 Sr (sanction for T), 336 6 (structure), 31, 144, 226 -interpretation, 227, 468 -valid, 236, 470 s (strategy), 345 SE (join of E in lattice of strategies), 347 Sweak, a, o (weak strategy for a determined by O), 369 s strong, a, o (strong strategy for a determined by O), 369 SHambiin, a, o (Hamblin strategy for a determined by O), 369 S4, S5, 298, 473 SA, 300 SA0 axiomatics of, 303-304 completeness of w.r.t. BT + AC, 304 soundness of w.r.t. BT + AC, 304 Salmon 1989, 232-233 sanction, 298 satisfaction of imperatives, 83 by sequence, 222 saying, 158n Searle, 4 1965, 87, 276 and Vanderveken 1985, 87, 97, 276 and Vanderveken 1990, 97

497

second witness-identity lemma, 272 sees to it that, 6 Segerberg 1980 ff., 25 1988, 84 1989, 19, 20 1990, 84 semantic value, 235 semantics. See also recursive semantics, semantics of indeterminism, truth for basic definitions for, 468-469 derivative definitions for, 469-471 of indeterminism, 220 of indeterminism, sources for, 221-226 of indexicals, 221 of quantifiers, 221-223 semi-lattice condition, 188 semi-ref-counter, 399, 401-407 sentences assignment-open (see assignment-open sentences) atomic (see atomic sentences) considered as embedded, 148 (see also embedded sentences) considered as stand-alone, 148 (see also stand-alone sentences) deontic (see deontic sentences) history-open (see history-open sentences) stit (see stit sentences) separability, 215, 467. See also choice equivalence Sett: (historically necessary), 32 definability of, 298 semantics for, 32, 152, 242 Sett-false, 122 Sett-true, 122 settled false, 32 interpretation, 434 true, 32, 36n, 242 true, at a moment, 235, 470 true, throughout a set of moments, 235, 470 Settled-will:, 161 settledness, 32, 242. See also settled, Sett: of moment-independent sentences, 210 Shelah. See Gurevich and Shelah Snow, C. P., 177

498 Sometime:, 243 soundness of Lai w.r.t. BT + I + AC, 417-419 of Lai + rr w.r.t. BT + I + AC + nbc, 390-391 of Ldmn w.r.t. BT + AC with max. n choices, 439-440 of SA0 w.r.t. BT + AC, 304 special relativity, 178, 205 speech act theory, 4, 97, 118 and double time references, 120 speech acts, 150, 275 splitting. See histories, splitting of sstit:. See strict stit stand-alone sentences, 148-149, 221 evaluated only at context-initialized points, 149, 230 no syntactic mark of, 230 and propositions, 190 semantic concepts for, 235-238 starting evaluation of, 229 state transformations, 61 states, 184 states of affairs, 198 static verb (Kenny), 69 statives, 87 Sterrett, S., 268 stit, 6. See also achievement stit, Chellas stit, deliberative stit, joint stit, strict stit, transition stit agentiveness of, thesis (see agentiveness of stit thesis) as complement (see restricted complement thesis) complement of, 7, 460 complement thesis, 7, 460 (see stit complement thesis) and could have, 256-270 and imperatives (see imperative content thesis) modalities, 51-52 negated, 454 normal form thesis (see stit normal form thesis) paraphrase thesis (see stit paraphrase thesis) properties of, 453 and Ross's paradox, 84-85 singular versus joint, 281 slogans, 98, 99, 101, 107

Index theses, 459-460 stit complement thesis, 59, 67, 69, 75, 89, 108, 117, 280-281, 320 stit deontic equivalences, 54-55, 91, 472 stit normal form thesis, 15, 60, 96, 321, 460 stit paraphrase thesis, 7, 8, 17, 42, 59, 79, 84, 99, 101, 321, 460 stit pararaphrase thesis, 262n stit sentences, 6 always agentive, 7, 459 both declarative and imperative, 16 can paraphrase agentives, 7, 460 as complements, 15, 460 embedding of, 16-17 place of agent in, 62, 66, 70, 98 stit theory, 28 idealizations of, 33, 279 role of probability in, 279 stit-stand-in, 361 stit:. See also astit:, dstit:, cstit:, sstit:, tstit meaning depends on context, 29 Stockton, F., 278 strategies, 26, 47 admitting histories/moments, 347 austere theory of, 116, 341-377 available at a moment, 345 backward closure of, 350 complete, 353 complete along a history, 353 completely admitting a history, 353 conditional, 350 consistent (at a moment), 345 as contents of obligations, 316 as contents of other-agent nested stits, 281 as contents of promises, 123-126 as contents of word-giving, 126-127 definition of, 345 and deontic kinematics, 368-377 domain of, 345 elementary theory of, 345-355 excluding histories/moments, 347 extensions along a history, 354, 355 extensions of, 346, 358 favoring a set of histories, 356 field for, 349, 349n, 350 for games in extended form, 342, 348 guaranteeing a set of histories, 347, 348

Index //-complete, 353 Hamblin, 369 having, 344 in field M, 349 joins of chains of, 355 lattice of, 347 linguistic expression of, 117, 342 mental aspects of, 341 n-ary, 349 for not doing, 268-269, 341, 359-363 objectivity of, 343 pre-simple, 351, 352 primary, 348, 354 really guaranteeing a set of histories, 347 relative weakness of, 346 secondary, 348, 349 secondary versus primary, 118 simple, 350 simply complete, 353 as stit complements, 117 strengthening of, 346 strict (at a moment), 345 strong, 369 tokens, not types, 343 total [in M], 350 tough, 125 von Neumann's theory of, 342-343 weak, 369 wholehearted, 125 strict stit, 112 applications of, 288-290 modal properties of, 287 S4 property for, 287 semantics for, 287 strictly agentive in, 287 strongly antideterministic Tree, 205 structure parameter(s), 144, 226-227 immobility of, 144, 228 language-independence of, 144 terminology, 226 structures, 226. See also BT, BT + I, BT + AC, BT + I + AC kinds of, 460-461 Styron, W., 278 submodel, 456 subnectors, 239 substance sort, 211, 282 substructure, 456 suggestions. See imperatives super event, 178

499

supervaluations, 156, 156n supervenience of Tree, 179 of TRL, 168 Szabo and Belnap 1996, 194 T (true), 235 TxW structures and theories, 196-203 and content, 231 demerits of, 197 T-sentences (von Wright), 60 Talja, 5n 1980, 23, 273 Tarski, A., 141, 143, 221-223 temporal connectives. See truth for tense and temporal connectives temporal generalization, 335 ten-minute mile, 49 picture, 49 Tennyson, A., 184 tense connectives. See also truth for tense and temporal connectives tense logic linear, 31, 32, 167, 224 Prior-Thomason, 31, 224-225 tense operators, 31 term, complex, 240, 471 Thalberg 1972, 5n theorems, 384 Thin Red Line (TRL), 135-136, 160-170, 233 absolute, 162-163 absolute, as mobile parameter, 169 context-dependent, 163-164 moment-of-evaluation-dependent, 165168 moment-of-evaluation-dependent, as mobile parameter, 169 Thomason, 31, 139, 194, 224-225 1970, 29, 139, 141, 156n, 196, 221, 224, 365 1981a, 37 1981b, 298 1984, 29, 32, 139, 163, 196, 198, 199, 201, 221, 348, 364-366, 370, 376, 377 and Gupta 1980, 166n, 196, 246 Thomsen 1990, 268 Thomson 1977, 18, 97, 178, 179 1990, 104, 105, 107-116

500 time and events, 197 same (in different worlds), 197 times. See instants Times x Worlds theories. See TxW theories Tomberlin 1983, 74 transition stit, 250-252 transitions, 137, 192, 465 and agency, 251 causal, 251 contingency of, 194, 465 immediate, 192, 465 transitivity, 180, 461 translocal (in parameters), 223. See also local Tree, 30, 179. See also Our World postulates for, 178, 461 tree structure, 25 Triponodo principle, 260 trivial choice, 218, 468 TRL. See also Thin Red Line immobility of parameter, 161 TRLabs (absolute Thin Red Line), 162 TRLC (context-dependent Thin Red Line), 163 TRLcfcn (Thin Red Line function of the context of use), 169 TRLfcn (Thin Red Line function), 165 truth absolute, 237 at a moment, 30 at a moment-history pair, 31, 121 at a point, 235 relativized to context, 225 relativized to history, 225, 238 relativized to parameters, 141, 142, 221, 224, 225, 237 truth for achievement stit, with witness by chains, 250 achievement stit, with witness by moments, 36, 248 atomic sentences, 31, 240 Chellas stit, 248 context-anchored connectives, 153 definite description operator, 241 deliberative stit, 37, 247 historical modalities, 32, 152, 242 identity, 241

Index indexical connectives, 246-247 individual constants, 240 mixed temporal-modal connectives, 245 operators, 241 predicates, 241 quantifiers, 152, 241 tense and temporal connectives, 31, 152, 242-245 truth functions, 31, 241 variables, 241 truth functions, 31, 241 truth value, 235 absolute versus relative, 221 gaps, 156, 156n has no, 155 of open sentences, 155 third, 156, 156n tstit:. See transition stit Tuomela 1989a, 271 1989b, 271 u (an individual constant), 240, 471 unanchored (in parameters), 240 undefined locution, Chisholm's, 66 undivided at, 203, 464. See no choice between undivided histories uniqueness of witness, 272, 383 Universally:, 245 unsettledness, sufficient condition for, 272 upper bound (of a chain of moments), 182, 464 upper-lower bounded, 383 upward versus forward, 30n vacuous choice, 218, 468 Val (the semantic-value function), 235 validity, 236, 470 definition of, 32 in a class of structures, 237, 471 in a model, 236, 470 in a structure, 236, 470 in-context notions of, 237, 471 van Benthem 1988, 241n van Praassen, B., 195 van Inwagen 1978, 255 Vanderveken, 4. See also Searle and Vanderveken 1991, 97 variable polyadicity, 71

Index stit and, 71 variables semantics for, 241 Vendler 1957, 87 verb, kinds of (Kenny), 69 vindication, 126 volition Kenny's theory of, 73 stit and, 73 von Kutschera 1980, 26 1986, 26, 29, 216, 247 1993, 209, 346 von Neumann, 283 and Morgenstern 1944, 25, 342-343, 348 von Wright, 5n, 60-65 1963, 20, 22, 41, 41n 1966, 60 1981, 20, 192 w (a moment), 178, 463 Walton 1975-1980, 19, 23 Wansing 1998, 92, 99, 316, 335n Was-always-mevitable:, 245 Was-always:, 243 Was: (past tense), 32, 152, 242 weakness of will, 217 Whitehead, A. N., 139, 344 Will-always:, 243 will-stit not agentive, 103 Will: (future tense) Peircean semantics for, 159 semantics for, 32, 152, 243 TRL semantics for, 161 witness, 36 by chains, 49, 249-250 momentary, 248 to Zi-dependence, 153 witness-identity lemma, 272 wondering, 176 about the future, 170, 207

501 history-independence of, 171, 207 word-giving, 104 with future-tensed agentive content, 127 impugned, 127 and promises, 113, 116-129 strategic content of, 126-127 thesis, 110 Thomson's account, 110-113 vindicated, 126 world, concepts of, 179, 233 Xj

an individual variable, 240, 241, 471 the assignment-to-Zj parameter, 239 Xu 1995a, 52 1997, 209 Zt (ith parameter position), 142 zl (ith parameter value), 142 Z-closed, 153 Z-dependent, 153 Z-independent, 153 Z-open, 153 sentences, truth values of, 155 Zampieri 1982, 195 1982-1983, 195 Zanardo, 196, 198, 202. See also Barcellan and Zanardo, Di Maio and Zanardo 1996, 32, 196, 198, 199 (agent or agent term), 211, 240, 382, 466, 471 (value of a), 382 23, 86. See also Chellas stit 297n (a nonempty subset of Agent), 282 (a set of sentences), 382 (a MCS), 392, 416 (point abbreviation), 229, 469