Truth and Consequence in Mediaeval Logic (Studies in Logic and the Foundations of Mathematics, 11)

STUDIES IN LOGIC AND THE

FOUNDATIONS OF MATHEMATICS L.E.J.BROUWER I E. W.BETH I A.HEYTING EDITORS

Truth and Consequence • 1ft

Mediaev·al Logic ERNEST A. MOODY ORTH·HOLLA D PUBLISHING COMPA AMSTERDAM

Y

STUDIES IN L"OGIC AND

THE FOUNDATIONS OF MATHEMATICS

L. E. J. BROUWER E. W. BETH A. HEYTING Editors

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m ~

1953 NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM

~RUTH

AND CONSEQUENCE IN

MEDIAEVAL LOGIC

ERNEST A. MOODY Associate Professor of Philosophy Columbia University

1953 NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM

'I I',

COPYRIGHT

1953

N. V. Noord-HollandBche Uitgeversmaatschappi} Amsterdam

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/

PRINTED IN THE NETHERLANDS DRUKKERIJ HOLLAND N.V.• AMSTERDAM:

PREFACE The study which follows, written with frequent interruptions over the course of nearly three years, represents the attempt of a mediaeval scholar, who is only an amateur in the field of contemporary formal logic, to interpret some of the rich content of fourteenth century logical literature in the logical language of the twentieth century. Both the vast extent of mediaeval writings in logic, and the difficulties of an exact understanding of its content, make the task of representing this logic in modern logical language one of great difficulty. Such attempts as can be made toward formalized representation of mediaeval logic must be viewed as tentative and partly conjectural translations of rules and arguments originally presented in a word language which, though perhaps quite precise for those who used it, is for us a foreign and only half understood language. The small space at my disposal has made it impossible to give adequate presentation of the textual foundations justifYing the interpretations which had to be chosen. References to the source materials, and occasional quotations of pertinent passages, are all that could be given. It is my hope that this introductory sketch, for all its shortcomings, may stimulate others to pursue the investigation of mediaeval logical literature with greater thoroughness and greater competence. Grateful acknowledgement is made to the many scholars who have recognized the interest of mediaeval logic, even for the twentieth century, and who have done arduous pioneer work in editing texts, analyzing their content, and grasping the significance of this material not merely for the history of logic, but for logical theory itself. E.A.M.

CONTENTS Page PREFACE

•••

Conspectus of Symbolic Notations. I.

3. 4. 5. 6. 7.

The Mediaeval Conception of Logic . . . . . . . The Formal and Material Constituents of Discourse Signification and Supposition . . . . . . . . . . Personal (Formal) and Material Supposition . . . Object Language, Metalanguage, and the Transcendental Terms ". . . . . . . . . . . . .

The Formal Classification of Propositions . The Logical Import of the Elementary Copula The Sentential Operators. Quantification . . . . . . Time Range and Modality

26

30 32 38 43 53

THE THEORY OF CONSEQUENCE

§ 13. The Meaning of ·Consequence' in Mediaeval Logic § 14. Formal and Material Consequences § 15. The Mediaeval Logic of Propositions V.

13 16 18 23

THE THEORY OF TRUTH CONDITIONS

§ 8. § 9. § 10. § 11. § 12.

IV.

1 10

LOGIC AND LANGUAGE

§ § § § §

III.

VII

INTRODUCTION

§ 1., Survey of the Development of Mediaeval Logic. . § 2. Aim and Scope of the Present Study . . . . . II.

V

64 70 80

TRUTH AND CONSEQUENCE

§ 16. The Aristotelian Definition of Truth . § 17. The Paradox of the Liar. . . . . .

101 103

Abbreviations for Works Cited, and Bibliography

III

CONSPECTUS OF SYMBOLIC NOTATIONS I.

VARIABLES:

Pronominal: x, y, z ,(Cf. § 5 and § 9) Nominal: F, G, H (Replaceable by a term; cf. § 4) Sentential: p, q, r (Replaceable by a sentence) When we wish to represent the name of whatever term, or whatever proposition, may replace the nominal or the sentential variables of a given formula, we enclose the variable in single quotation marks. When we wish to represent an ostensive use of whatever term may replace a given nominal variable of a formula, corresponding to the use of that term with a demonstrative pronoun prefixed to it, we add a subscript numeral to the nominal variable in question. II.

SENTENTIAL CONNECTIVES:

Negation: -p Definition: Conjunction: p.q (or pq) Possibility: Disjunction: p v q (Cf. "Simple" Implication: P-1q (Cf. "Simple" Equivalence: p.=.q (Cf. "As of now" Implication: p-:Jq (Cf. "As of now" Equivalence: p-:=q III.

§ 14 § 14 § 14 § 14

and and and and

§ 15) § 15) § 15)

§ 15)

SYMBOLS OF QUANTIFICATION- AND OF IDENTITY

Universal quantifier: Particular quantifier: Identity: IV.

p.=Df.q (Cf. § 14)

Op

(x), (y), (x,y) "For every " (Ex), (Ey), (Ex,y)"For some " .... = .. .. (E.g., "(x).x = x")

TEMPORAL OPERATORS:

These are required to indicate, in sentences whose copula is of past or future tense, the time (relative to the time the sentence is stated) for which a term of the sentence is indicated to stand for its values, or for those for which the other term stands. Where no temporal or modal operator is prefixed to a matrix expression in the formula of an analysed sentence, the supposition of the term represented by the variable in that matrix expression is

VIII

CONSPECTUS OF SYMBOLIC NOTATIONS

understood for the present time in which the sentence is stated. Of. § 12. ">Fx" means: "x was an F at some time prior to now" "
V.

LOGICAL (OR "SEMANTIC") PREDICATES:

means: "(The term) ~F' stands for x" means: "(Thesentencerp'standsforp" (On these uses, cf. § 9 and § 17) Truth and Falsity: "T~ p' " means "~p' is true" "F~p'" means "~p' is false" Supposition:

"~F'8X" "~p'sp"

Note that these logical predicates, as well as others which we state in words rather than symbols (e.g., ~implies'), are asserted of the names of whatever terms or sentences replace the variables F or p in the formula to which these statements refer or with which they are correlated.

VI. PARENTHESES, DOTS, AND QUOTATION MARKS: Parentheses are ordinarily used, as in Principia Mathematica, to enclose the quantifYing prefixes. They are also used occasionally, in addition to dots, for bracketing expressions in formulas, whereever this achieves greater visual clarity. Square brackets are also used, on a few occasions, to enclose long expressions, especially where these expressions contain parentheses. Dots are used according to the conventions of Principia M athematica~ though on occasion more dots are used than are strictly necessary, where this contributes to greater visual clarity. Quotation marks, single or double, are used technically for the purpose of indicating the "autonymous" use of an expression i.e., as name for the expression enclosed within the quotation marks, or as name for whatever constant might replace the variable enclosed in the quotations marks. Quotation marks are also used, in the text, in ordinary manner.

I

INTRODUCTION § 1.

SURVEY OF THE DEVELOPMENT OF MEDIAEVAL LOGIC

As one of the three arts of language known as the trivium (the other two being grammar and rhetoric), logic was basic to the educational progranl of the enture Middle Ages. In the pages which follow, however, we shall restrict the meaning of the expression ~mediaeval logic' to the body of doctrine taught during the 13th and 14th centuries in the courses on logic given in the Arts Faculties of the universities at Paris and at Oxford. This teaching is accessible in text books, treatises and commentaries written by the logicians of the period. As so conveyed, it forms a relatively stable and coherent discipline whose development and elaboration can be traced from foundations laid in the 12th century up to the highly articulated and systematic formulations achieved during the middle part of the 14th century. In restricting the meaning of ~mediaevallogic' to the discipline taught on the Arts Faculties of the universities, we simultaneously achieve a restriction of the subject to formal logic, inclusive of foundational problems as approached from the formal point of view. Although questions of mixed logical and epistemological character were treated by the fanlous scholastic doctors of the period, and on a vast scale indeed, this enterprise was chiefly confined to the Faculties of Theology, where it was undertaken in connection with nletaphysical debates and speculations. At Paris there were severe restrictions against treatment of the more lofty philosophical questions by members of the Arts Faculty, who were expected to confine their teaching to the disciplines of the trivium and quadrivium - the arts of language and the mathematical sciences. Logic, as taught on the Arts Faculties, remained a science of language (sermocinalis scientia). The period in which this body of logical teaching developed and flourished extends through four hundred years, from the early

2

INTRODUCTION

12th century to the end of the 15th century. At its beginning stands Peter Abelard (1079-1142), a man of analytic genius and originality, who worked with the corpus of Boethian logical treatises in one hand, and with the lnstitutiones grammaticae of Priscian in the other. The extant logical writings of Abelard consist of a Dialectica, whose first part is missing, and some commentaries on the Predicables of Porphyry and on the Categoriae and De interpretatione of Aristotle. * Two facts about the logical treatises of Abelard are of considerable historical importance. First, the key ideas, and the techniques and terminology of logical analysis which are found in the so-called "terminist" logic of the 13th century are already present in the work of Abelard. Secondly, Abelard's writings show no influence of any Arabian or Byzantine sources, which were just beginning to infiltrate into the Latin West toward the end of his life; nor does he seem to have had access to any of the logical treatises of Aristotle other than those conveyed in the Boethian writings. The historical significance of these facts is twofold. First of all, they dispose of Prantl's hypothesis, which has also been discredited on other grounds, that the "terminist" tradition of mediaeval logic was an alien importation stemming from Byzantine or Arabian sources. The second point of significance is that Peter Abelard, working exclusively on the basis of the Boethian treatises and of Priscian's grammar, was utilizing materials which contained, alongside of the Aristotelian and Neo-Platonist ingredients, a large body of logical ideas which had originally been derived from the Stoic and Megaric traditions of the later Greek period. The genius of Abelard consisted in his selection, from this hybrid mixture of diverse traditions, of those elmnents representing the Stoic-Megaric extension and interpretation of Aristotelian logic, and in his rejection of those elements which represented the NeoPlatonist extension and interpretation of Aristotle's logic. No doubt the application of Priscian's grammatical approach to

* These commentaries, to which we will refer by the general title "Introductiones", were edited by Dr B. Geyer in the Beitrage zur Geschichte der Philosophie des Mittelalters, Bd. XXI, Miinster i.W., 1919-1927. The Dialectica is included in the "Ouvrages inedits d'Abelard" edited by V. Cousin, Paris 1836.

INTRODUCTION

3

language, itself largely derived from the Stoic doctrine of 4>wv~, played the part of a catalytic agency in this process; Abelard's constant citation of Priscian throughout his logical writings is witness to this fact. But the result of Abelard's critical reconstruction of the Boethian logic was to set the direction of development of mediaeval logic, for the ensuing two centuries, along that path which had been opened up in antiquity by the Stoic and Megaric schools. * It is known, from the testimony of John of Salisbury and from other historical evidence brought to light by Grabmann, that the study and teaching of formal logic was pursued with tremendous intensity throughout the 12th and early 13th centuries, especially at Paris. Little in the way of logical literature from this period has survived, though such fragments as have been found are sufficient to show the continuity of the tradition of Abelard with that of the "terminist" logicians of the middle 13th century. Our first landmarks in the literature of this school are the treatises written by Lambert of Auxerre (fl. 1250n, William ShYreswood (died after 1267), and Peter of Spain (died 1277).**

* The striking resemblance of certain features of mediaeval logic, such as the propositional logic involved in the theory of the Oonsequentiae, with the formal logic of the Stoic-Megaric tradition, has been indicated by Lukasiewicz ("Zur Geschichte der Aussagenlogik", Erkenntnis 5, 1935, pp. 111-27) and by Bochenski ("De consequentiis scholasticorum earumque origine", Angelicum 15, 1938, pp. 92-109). While Bochenski (p. 108) considered that the scholastic theory of consequence was a rediscovery rather than a continuation of the Stoic logic of propositions, my own study of the materials - especially Boethius and Abelard - convinces me that there was a genuine continuity between the two traditions. This seems to have been the view of Lukasiewicz, who says (p. 127): "Die von den Stoikern begriindete, von den Scholastikern fortgefiihrte und von Frege axiomatisch aufgebaute zweiwertige Aussagenlogik steht nunmehr als ein fertiges System vor unseren Augen da". ** The treatise of Lambert of Auxerre is described by Prantl, Vol. III, XVII, pp. 25-32. Shyreswood's Introductiones were edited by M. Grabmann, Sitzungsberichte d. Bayerischen Akad. d. Wissenschaften, Phil.-Hist. Abt., 1937, Hft. 10; Miinchen 1937. Shyreswood's treatise on the Syncategoremata has been edited by J. R. O'Donnell, C.S.B., in Mediaeval Studies (Toronto, Canada), Vol. III, 1941, pp. 46-93. The famous Summulae logicales of Peter of Spain, printed in more than 150 editions in the early modern period, has been edited from a good early manuscript by 1. M. Bochenski,

4

INTRODUCTION

While these terminist logicians appear to have been familair with the Prior Analytics, Topics, and Sophistical Refutations of Aristotle, as well as with the traditional Boethian writings, their text books do not follow the order, nor cover the whole content, of Aristotle's Organon. The "traditional" sections treat only of the following subjects: (1) the proposition, both categorical and hypothetical; (2) the predicables; (3) the categories (omitted by Shyreswood); (4) the syllogism; (5) the topics or loci communes; and (6) the fallacies. What is new in these text books is a section treating of "the properties of terms", introducing the conception of the supposition of subject and predicate for a common undetermined subject, and using this notion in the analysis of quantification, temporal determinations of the copula or verb, adjectival determinations of subject or predicate, and logical functions of relative pronouns. A sharp distirrction is made between the property of "signification" and that of "supposition", such that the logical analysis of categorical propositions, and of their truth conditions, is carried out in a formal and extensional manner in terms of the property of "supposition". In addition to their manuals of logic, both Peter of Spain and William Shyreswood wrote special treatises on the logical operators, which they called "syncategorematic signs". Such signs were divided into two classes: those determining the composition of subject and predicate terms in atomic sentences, and those determining the composition of two or more such sentences in molecular sentences (called "hypothetical propositions"). The first class included the verb ~is' and the quantifiers ~every', ~both', ~neither', ~none', ~one only', etc. The second class included the sentential connectives ~ir, ~unless', ~it not', ~and', ~or', etc. The treatment of the sentential connectives is truth-functional, although distinctions are made between the ordinary ("material") truth-functions and other stronger ones. * and published at Torino, Italy (Marietti), 1947. A text of the part of Peter's Summulae dealing with the "properties of terms," with an English translation, was prepared by J. P. Mullally and published at Notre Dame, Indiana, in 1945. * Shyreswood's treatment of the syncategoremata may be studied in

INTRODUCTION

5

These new branches of logical investigation, represented by the treatises on the supposition of terms and by those on the sYncategorematic signs, were destined to receive an elaborate development during the century following on ShYreswood and Peter of Spain. The part played by Priscian's grammatical sYntax in the development of this approach to the logical analysis of language is strikingly evident from the constant citations of the Roman grammarian which occur in ShYreswood and the other terminist logicians. . While logic continued to be taught on this formal basis on the Faculties of Arts, the theologians of the later 13th century, influenced by new philosophical literature translated from the Greek and Arabic, engaged in epistemological and metaphysical debates and speculations which gave rise to a kind of "philosophical logic". Since the terminology of the traditional formal logic was regularly used in these philosophical discussions, it became infected with speculative connotations and ambiguities which have remained with it throughout the modern period. The influence of Avicenna promoted the logical realism which Gilson has called "essentialism", and which, by way of Duns Scotus, Aegidius of Rome, Suarez, and Christian Wolff, entered into modern traditional logic. * With this "philosophical logic", which properly belongs to metaphysics and epistemology, we are not here concerned. It did not destroy the tradition of formal logic which had been firmly established in the Arts Faculties and which maintained its integrity into the 15th century. It was this formal logic, on the contrary, which provided William of Ockham and other 14th century philosophers with a powerful instrument for their criticism of the metaphysical logic or logicized metaphysics that had grown up in the late 13th century. The primary significance of what the text edited by O'Donnell, already mentioned. Peter of Spain's treatise exists only in manuscript at the present writing. Our own study will be based on the more articulated formulations found in 14th century logical literature. * Cf. E. Gilson, "Being and Some Philosophers", N.Y. 1949. Prantl, who took modem traditional logic to be the heir of the native Western tradition, and who regarded the "terminist" logic as an alien importation from Byzantine and Arab sources, reversed the real historical relation between the two traditions.

6

INTRODUCTION

is called the "nominalism" of William of Ockham is its rejection of the confusion of logic with metaphysics, and its vigorous defense of the older conception of logic as a scientia sermocinalis whose function is to analyze the formal structure of language rather than to hypostatize this structure into a science of Reality or of Mind. * Ockham's logical works, consisting of an elaborate and systematic Summa logicae and of a group of commentaries on the "old logic" (Porphyry, and the Categories, De interpretatione and Sophistical Refutations of Aristotle), are distinguish~d for their painstaking rigor, rather than by any radical novelties or innovations. ** The logic presented by Ockham is a well organized and clearly articulated exposition of the common body of logical teaching which had developed continuously from the time of Abelard through the 13th century. Precisely because this logic was a formal logic, it could be accepted and utilized by the scholastics of all parties regardless of the metaphysical or epistemological oppositions dividing Scotists from Thomists, or realists from nominalists. *** The maturity of mediaeval logic, in its formal development, is represented by the treatises written at Oxford and at Paris during the fifty years between 1320 and 1370. Of these treatises a large portion is extant only in fragmentary form, or in manuscripts which have not yet been fully studied or ep.ited. Such is the case with the Oxford logicians associated with the "Mertonian" school

* Cf. my work, "The Logic of William of Ockham", N.Y. and London, 1935, which was largely devoted to this aspect of Ockham's thought rather than to the formal structure of the Logic he expounded. ** Ockham's Summa logicae was printed at Paris in 1488, at Bologna in 1498, at Venice in 1508, 1522, and 1591, and at Oxford in 1675. We have used the Venice edition of 1508. Ockham's commentaries on the "old logic" (but not including the one on the Sophistical Refutations) were printed, with Questions by Albert of Saxony, under the title Expositio aurea et admodum utilis super artem veterem, etc., at Bologna in 1496. *** Failure to grasp the distinction between formal logic, and the doctrines of epistemology and metaphysics on which scholastic doctors of the 14th century were in opposition to each other, has led many historians of mediaeval philosophy to the paradoxical conclusion that all the prominant 14th century scholastics were "Ockhamists", simply because they all used the same logic that Ockham employed. The otherwise valuable work of Gilson, Baudry, De Wulf, and Michalski suffers from this confusion.

INTRODUCTION

7

of mathematical philosophers - William of Heytesbury, Richard Swineshead (the "calculator" whose work was so highly praised by Leibniz), John Dumbleton, Ralph Strode, and Richard Ferabrich. More accessible and adequate sources for the study of 14th century formal logic are provided by the writings of two famous teachers of the Arts Faculty at the University of Paris - Jean Buridan (died 1358) and his pupil Albert of Saxony (died 1390). Their comprehensive and systematic text books and special treatises, extant in reasonably good early editions, afford an ample and accurate picture of the mature form of mediaeval logic. Our study will be based chiefly on the works of these two men, with supplementation from the writings of Ockham and from thp, treatises of the earlier "terminist" logicians. The logical works of Jean Buridan consist of a Summula de dialectica in eight treatises, a treatise on the Sophismata, and a treatise entitled Oonsequentiae. In addition, there are some Questions dealing with problems of logic in the fourth book of Buridan's Quaestiones in M etaphysicam Aristotelis. * The Summula de dialectica of Buridan is often described as a mere edition of the Summulae logicales of Peter of Spain. This is true as regards the first three of the eight parts of the work, but Parts 4 to 7 deviate very substantially from Peter's treatise, and the eighth part, on demonstration and definition, has no analogue in Peter's manual. The Sophismata of Buridan was apparently intended to constitute a ninth part of his Summula, according to statements made in its prologue. The work is divided into eight chapters, each dealing with a distinct type of "sophism" or logical puzzle, and giving

* Buridan's Summula de dialectia was printed in two early editions, one printed at Venice in 1499 under the title Perutile compendium totius logicae Joannis Buridani, and the other at Paris in 1504 under the title Oommentum Joannis Dorp super textu summularum J ohannis Buridani. Both editions contain Dorp's lengthy commentary as well as Buridan's text; we have used the 1499 edition. Buridan's Sophismata was printed at Paris several times between 1495 and 1500; we have used the edition published by Antoine Denidel and Nicolas de la Barre, Paris s.a. (1496/1500). The Oonsequentiae Buridani was also printed at Paris, between 1495 and 1500, without date, under the imprint of Felix Baligault. Buridan's Quaestiones in Metaphysicam were printed at Paris in 1518, edited by Iodocu8 Badius Ascensius.

8

INTRODUCTION

general principles and methods for solving puzzles of that type. Since a "sophism" is a statement which appears to be both true and false, the analysis of sophisms reduces in large measure to a study of the functions of syncategorematic signs (orlogical constants) in determining multiple truth-conditions of apparently simple sentences. The Sophismata, as treated by Buridan, is an advanced treatise on the topic of formal specification of truth conditions of sentences of categorical form, within the basic framework of the theory of supposition of terms. The last chapter of the work is devoted to an elaborate treatment of the so-called "insolubles", of the type represented by the paradox of the liar. The importance of this paradox, in connection with the enterprise of formulating the truth-conditions for sentences of such forms as the rules of syntax of the language permit, is very clearly recognized by Buridan and his contemporaries; the discussions of the "insoluble" are therefore of interest in view of contemporary treatments of the problem of defining truth, in a formal manner, for sentences constructible in a given language. Buridan's Consequentiae is one of the most interesting works of mediaeval logic, because it undertakes an axiomatic derivation of the laws of valid deduction, and in so doing takes the laws of propositional logic as the basic and elementary part of the theory of deduction. This is apparently the first attempt in the history of logic to give a deductive derivation of the laws of deduction. Buridan states in his preface that although others had treated the "consequences" in a posteriori manner, he proposes to investigate the "causes" of the validity of these laws of inference. The work is in four parts or books, as follows:

I. On consequence in general, and on the laws of consequences among assertoric propositions. II. On laws of consequence among modal propositions. III. On syllogistic consequences among assertoric propositions. IV. On syllogistic consequences among modal propositions. Albert of Saxony, who studied under Buridan and taught on the Parisian Faculty of Arts until 1362 or later, wrote a succinct and highly systematic Summa logicae which was printed at Venice in 1522. This work closely reflects the logical teachings of William of Ockham, but also shows marked influence by the work of

9

INTRODUCTION

Buridan, especially in the treatment of the theory of Consequence, and in that of the paradox of the liar. The structure of Albert's Summa logicae is as follows: I. Treatise on Terms: (a) On the terms ~term', ~sign', ~predicate', ~noun', and ~verb'. (b) On terms of second intention. (c) On terms of first intention.

~subject',

II. Treatise on Suppositions: (a) On supposition in general, and the rules of personal supposition. (b) On the supposition of relative pronouns. (c) On ampliation. (d) On appellation. III. Treatise on Propositions: (a) On assertoric propositions. (b) On modal propositions. (c) On hypothetical (or molecular) propositions. IV. Treatise on Consequences: (a) On the general laws of consequence. (b) On formal consequences ~ong assertoric propositions (c) On consequences among modal propositions. (d) On syllogistic consequences. (e) On syllogisms containing modal propositions. (f) On the dialectical places, or topical inferences. V. Treatise on the Fallacies. VI. Treatise on the Insoluble, and on the Obligationes. An interesting feature of the logical system of both Albert of Saxony and Buridan is the organization of all the traditional forms of argumentation, such as the syllogism and the enthymeme, under the single heading of "consequence", with the treatment of consequences between unanalyzed propositions placed at the beginning and used as foundation for the other types. This constitutes something of a reversal of the procedure of Aristotle, who sought to exhibit all non-syllogistic forms of inference as incomplete syllogisms. That this change was not merely accidental, and that it reflected a conscious recognition of the priority of the 2

10

INTRODUCTION

general theory of consequence over the special theory of syllogistic argument, is attested by the following statement, made in the preface to a commentary on the Oonsequentiae of Ralph Strode written in the 15th century by an Italian logician named Alexander Sermoneta: I say that this book is the most universal part of the Prior Analytics, or else is introductory to it; and therefore it should be placed immediately after the De interpretatione, and before the Topics, Sophistical Refutations, and Posterior Analytics. This order is evident, because this book is concerned with the consequence as its su~ject, and this is more universal than any special kind of argumentation, or than the syllogism, with which the Prior A nalytics is concerned. * The last period of mediaeval logic, stretching fronl the end of the 14th century into the early 16th century, seems to have been one of continuation and elaboration of the 14th century tradition. Text books such as the Logica of Paul of Venice contain abridged compilations based on the more ample works of Ockham, Hentisberus, Albert of Saxony, and other 14th century writers. Some lengthy commentaries on 14th century logical treatises, such as Dorp's commentary on Buridan's Summula, were produced in the 15th century, many of them in Italy. This late scholastic logical literature, which has received comparatively little study up to the present time, is undoubtedly valuable because of the light it can cast on the 14th century logical tradition, whose literature was more familiar to the later scholastics than to us. **

§ 2.

AIM AND SCOPE OF THE PRESENT STUDY

The purpose of our present study is to investigate two basic branches of logical analysis which were developed by the logicians of the 13th and 14th centuries, and which appear to be of more than antiquarian interest in relation to contemporary research in the field of logic. These two branches of logical theory were known, in the mediaeval terminology, as the theory of Supposition and the theory of Consequence. In contemporary language we would

* **

Oonsequentie Strodi, cum commento Alexandri Sermoneta. Venice 1493. For information on the extant (printed) literature of this later period, see Prantl, Vol. IV, p. 94 ff.

INTRODUCTION

11

describe them as the theory of truth-conditions and the theor of inference-conditions or of deduction: those logicians who difler~nt­ iate between syntactical and semantical inquiries and sYstems would consider that these two theories belong to semantics. * The mediaeval development of these two theories took place within the context of mediaeval formal logic and found its expression in the terminology of that logic. The logic was sUbstantiall Aristotelian, with accretions from later Greek logic translnitte~ through Boethiu~; as such, it attempted to for~~~te the logical structure of ordinary language as used by the SCIentists" (and philosophers) of the mediaeval universities. Distinctive of such "natural" language is the fact that statements are lnade with verbs of present, past, or future time, in subjunctive as 'VeIl as indicative mood, and with numerous syntactical constructions whose logical analysis is of extreme subtlety or complexity. Because mediaeval logic was a formulation of the logical structure of a very rich natural language, and not restricted to the comparativel simple expressions of the language of mathematics, its rules (an~ the corresponding formulas) are not easily represented in the symbolism of contemporary logic. This difficulty can be overcome in part, by introducing additional symbolism. But the lUor~ difficult problem is that of determining a correct lUanner of representing, in well defined modern expressions, the basic logical expressions and laws of the mediaeval system. Our present study is chiefly concerned to contribute to the solution of this problem of fundamental interpretation, in connection with investigation of the logical import of the copula ~is' in categorical sentences and of the logical import of.the connective ~if ... then' in compound or "hypothetical" sentences. These questions were discussed, in the mediaeval treatises, in connection with attempts to give formal definitions, for the scholastic "scientific" Latin of the time ,

* A. Tarski, "The Semantic Conception of Truth", in Philo80Phy and Phenomenological Research IV, pp. 341-375, holds that a formal Specification of truth-conditions for a given language can only be given in a semallticall richer metalanguage. R. Carnap, "Introduction to Semantics", Cambrid Y ge, Mass. 1946, p. 10, states that the theory of truth and the theory of logical deduction belong to semantics, because these concepts are based on the relation of designation.

12

INTRODUCTION

of the one-place logical predicate ~true' and of the two-place logical predicate ~implies'. The material on which our study is based is restricted to a few of the more accessible and complete logical treatises of the 13th and 14th centuries, and primarily to three 14th century works: Albert of Saxony's Summa logicae, and the Sophismata and Consequentiae of Jean Buridan. Ockham's Summa logicae is used in supplementary manner, and some use is made of the works of Abelard, William Shyreswood, and Peter of Spain in connection with the origin and evolution of certain distinctive and fundamental concepts. It is beyond the scope of this study to attempt to represent the mass of derivative laws and rules making up the lnain body of mediaeval formal logic; our aim is only to examine the simpler and more primitive rules and -theorems, and the logical signs involved in their expression, on which the mediaeval logic of terms and the mediaeval logic of propositions were based. We propose to utilize some of the symbolic techniques of contemporary logic for the clarification and simplification of our exposition - particularly in dealing with the rules of consequence. It is to be understood, however, that mediaeval logic was expressed and formulated as a set of "rules", and not as a set of formulas or theorems composed of variables and operators. Although mediaeval logicians frequently employed letters of the alphabet in their statements of rules, these letters were not normally used as variables which could be directly replaced by expressions of the object language: rather, they were used as variables to be replaced by the names of such expressions. Nevertheless, since the mediaeval rules determine corresponding theorems or formulas, we shall employ variables in both ways, expressing the mediaeval rules with variables representing the names of terms or propositions of the object language, and then giving the analogous theorems in the modern formalized manner. Our notation will be a modified form of that used in Principia M athematica, supplemented by some special symbols required for the expression of certain laws peculiar to mediaeval logic.

II

LOGIC AND LANGUAGE § 3.

THE MEDIAEVAL CONCEPTION OF LOGIC

In mediaeval classifications of the arts and sciences, logic was normally described as one of the "rational" or "linguistic" disciplines, the others being grammar and rhetoric. While grammar teaches how to speak correctly, and rhetoric how to speak elegantly, logic teaches how to speak truly (vere loqui) or to make valid inferences.* The description of logic given by Hugh of St. Victor, in the 12th century, probably represents the conception of logic tacitly accepted by the teachers on the Arts Faculties throughout the later mediaeval period. It is not to be supposed that logic is called the science of language for the reason that, before its invention, there had been no language, or as if men had not previously talked to one another. There existed spoken communication and writing, previously. But the formal structure (ratio) of the language, in its spoken and written usage, had not previously been made explicit, and until then no rules for correct speech and writing had been laid down. For every science existed as a practice before it existed as an art. But when men considered that usage could be transformed into art, and that what had previously been vague and subject to caprice could be regulated by explicit rules and precepts, they began, as we said, to reduce to art the habits which had originated partly by chance and partly by nature - correcting what had a bad usage, supplying what was lacking, eliminating what was superfluous, and prescribing, furthermore, exact rules and directions for each usage..... Hence there arose the skill of the logical art, which exhibits methods of argumentation and means of recognizing the arguments as such, so that it can be known which

* Cf. Shyreswood: I ntroductiones, p. 30; also Ockham : Expositio aurea I, fol. Ir (quoted in Moody: Ockham, p. 32, Note 2).

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LOGIC AND LANGUAGE

arguments are sometimes true and sometimes false, which are always true, and which always false. * Hugh's description of logic, as the science which formalizes the usage of language for the purpose of achieving exact methods of discrimination between formally valid and formally invalid inference-schemes or "argumentations", is representative of the entire mediaeval tradition. Logic was conceived as a science of language, in a prescriptive rather than a merely descriptive sense. In reducing the usage of language to art, as Hugh expresses it, logic creates an "artificial" language - a language for scientific discourse. The use of word-signs instead of symbolic notations (e.g., of "if. .. then" instead of ~:J') is not of itself decisive, so long as the usage of the word-sign is regulated by exact rules of logic. Aside from the prescriptive function of logical rules, there is a non-arbitrary factor involved in the generic definition of logic itself, whereby logic as a whole is differentiated from such other sciences of language as grammar, rhetoric, phonetics, etc. This factor is the purpose for which the rules of logic are formulated. As Hugh states in the final sentence quoted, this purpose is to exhibit valid inference schemes (argumentations), and to achieve means of discriminating between logically true premises (arguments) and those not logically true. This is the intrinsic end of logical inquiry, defining it as a science. But logic is also an instrument of scientific inquiry, an "art of arts and science of sciences," in that it has an extrinsic purpose of validating inference in the positive sciences. The statement of Boethius, that logic is both a science and an instrument of science, was understood in this sense.**

* Hugh of St. Victor, Didascalion I, ch. 12; in J. P. Migne, Patrologia Latina Vol. 176, cl. 749-50. Compare this with Carnap: Foundations, pp. 3-6, where the distinction between historically given languages, and formulations of syntactical and semantical systems for such languages, is drawn in much the same manner. ** Cf. Buridan: Summula, I, last part of Dorp's commentary on the first sentence of Buridan's treatise. Here Dorp asks whether logic can establish its own principles, or only those of the other sciences. He first states that logic cannot establish the principles of any science in the sense of demonstrating them; rather, logic teaches how conclusions may be validly derived

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15

The criteria of validity of arguments, employed in the mediaeval logical tradition, were neither psychological nor metaphysical, but formal. As early as the time of Abelard, the conception of forlnally valid arguments as those which hold by reason of the arrangement of the linguistic constituents of the argument, independently of the interpretations that may be given to the terms, is clearly grasped. Abelard calls those arguments which are formally valid "perfect" or "complexional" arguments, and he contrasts them with merely "topical" arguments based on habitual associations of meaning or on usual 0 bserved connections among things. "Complexional arguments", he says, "are those which derive their validity from their very construction, that is, from the arrangement of their terms". * As these statements show, the mediaeval criterion of logical truth was formal in the modern sense, determined by the way in which the constituents of the sentence or argument occur, and not by the meaning or content of the terms. The general endeavor of the later mediaeval logicians, expressed in their theories of supposition and of consequence, was to devise means of giving formal definitions of truth-conditions and of inference-conditions for the "scientific" Latin whose logical syntax they sought to formulate. The theory of consequence, taken as a whole, constituted a formal specification of inference-conditions for the formulated language; the theory of supposition was an attempt to specify truth-conditions, in a formal manner, for the atomic or "categorical" sentences constructible within the rules of syntax of the language. ** from premises within any science. "But in this sense", Dorp adds, "logic also provides a method in respect of its own principles, because it teaches how to form argumentations from these". Thus Dorp indicates an awareness of the use of logic for deductive development of its own theorems or rules from its own primitive postulates, analogous to its use for the deductive development of the positive sciences. * Abelard: lntroductiones, p. 508. Cf. also Abelard: Dialectica, p. 328: "Wherever there is not a perfect inference, a topical connection is of value; but where there is a perfect inference, never. ..... We call perfect the inference of a syllogism, which does not depend on any connection (i.e., in content) between the terms. For of whatever terms the inference may be composed, if it has the structure of a syllogism, it stands unshakeable". ** The recent work of Alfred Tarski, "Der Wahrheitsbegriff in den formalisierten Sprachen", Studia Philo8ophica I (1936), investigates this

16

LOGIC AND LANGUAGE

Preliminary to our study of these attempts, certain general distinctions of importance, between the formal and material constituents of discourse,. between signification and supposition as properties of terms, and between levels of language, will be briefly explained.

§ 4. THE FORMAL AND MATERIAL CONSTITUENTS OF DISCOURSE The signs and expressions from which propositions can be constructed were divided by the Inediaeval logicians into two fundamentally different classes: syncategorematic signs, such as have only a logical or sYntactical function in sentences, and categorematic signs (Le., "terms" in the strict sense) such as have independent meaning and can be subjects or predicates of categorical propositions. We may quote Albert of Saxony's definitions of these two classes of signs, or of "terms" in the broad sense. A categorematic term is said to be one which, taken significatively, can be a subject or a predicate, or a part of the subject or a part of the distributed predicate, of a categorical proposition. For example, these terms ~man', ~animar, ~stone', are called categorematic terms because they have a definite and determinate signification. A sYncategorematic term, however, is said to be one which, taken significatively, cannot be the subject or the predicate, nor a part of the subject nor a part of the distributed predicate, of a categorical proposition. Of this kind are these terms ~every', ~not any', some' etc.. which are called signs of universality or particularity; and similarly, signs of negation such as this negative ~not', and signs of composition such as this conjunction ~ and', and disjunctions such as this disjunctive ~ or', and exclusive or exceptive prepositions such as ~ other than', ~ only', and words of this sort. * f

In the 14th century it became customary to call the categorematic terms the matter of propositions, and the sYncategorematic signs (as well as the order and arrangement of the constituents of the same problem. The mediaeval treatment shows many parallels, and also interesting differences, in relation to Tarski's analysis. * Albert: Logica T, ch. 3. "Taken significatively" here means "not taken autonymously"; cf. infra § 6. The qualification, that a syncategorematic term cannot be a part of a distributed predicate, is made on account of such sentences as "Socrates is every man", in which the term 'every' is a part of the predicate, but not of the distributed part of the predicate.

LOGIC AND LANGUAGE

17

sentence) the form of propositions. This distinction was used in defining the formal consequence, as one valid through any transformations of the categormnatic terms, in contrast to the material consequence whose validity depends on the particular categorematic terms which occur in the constituent propositions. Buridan makes this distinction as follows: Since we have spoken of the form of a proposition, and of the distinction of consequences into formal and material, we must lay down ... what we take to belong to the form of a consequence or proposition, and what we take to belong to its matter. And I say .... that by the matter of a proposition or consequence we understand the purely categorematic terms - namely the subjects and predicates - , as distinguished from the sYncategorematic signs adjoined to them, by which they are connected or negated or distributed or determined to some particular manner of supposition. But we say that everything else belongs to the form, Thus we say that the connectives (copulas) of both categorical and hypothetical propositions pertain to their form; and the negations, and the quantifYing signs, and the number of the terms or of the propositions, and their order or arrangement. Also the relations of relative pronouns, and the modes of signification involved in the quantity of the proposition, such as singularity and generality; and many other things which the diligent can discern as they occur. * As these quotations reveal, the syncategoremata were held to comprise all those signs and expressions which we call logical constants, or operators; such words as ~is', ~not', ~every', ~some', ~and', ~or', ~if .. . then', etc. The word ~term', though used in a broad sense for any expression insofar as it is an element or constituent of a proposition, was defined in a strict sense as any expression which, when taken "significatively" or in its normal usage, can be a subject or a predicate of a proposition. As so defined, only categorematic signs, or expressions formed from them such as are not sentences, are to be called terms. And in this strict sense, the logical signs or syncategoremata are not terms. ** We shall hereafter employ the word ~term' in its strict usage, such that the basic distinction between the material and the formal constituents

* Buridan : Oonsequentiae I, ch. 7. Almost the same words are repeated by Albert of Saxony, Logica IV, ch. l. ** Albert: Logica I, ch. 4.

18

LOGIC AND LANGUAGE

of propositions is equivalent to the distinction between terms (subjects and predicates) and syncategorematic (or logical) signs. We may note that the term, in this strict sense, was described in two ways: (1) as a sign having independent meaning; and (2) as a sign which, taken significatively or in normal usage, can be a subject or predicate of a proposition. These two properties of the term were distinguished by the mediaeval logicians as the property of signification and the property of supposition. Since formal logic, in concerning itself with the form of propositions in abstraction from their matter, abstracts from the significations of the terms which enter into propositions, it follows that the property of the term with which formal logic is exclusively concerned is its property of supposition. It is important, therefore, to exhibit the relation and the distinction between these two properties of the term as clearly as possible.

§ 5. SIGNIFICATION AND SUPPOSITION A term is a language sign. But not all signs are language signs. Ockham and Albert of Saxony define a sign in general as "anything which, if apprehended, makes something come into the cognition of someone". In this most general sense a sign is anything that reminds a person of something already known to him. But a language sign is defined as anything which is a sign in the above sense, and which, in addition, "is fitted to be used to stand tor that thing in a proposition, or to be adjoined to such signs in a proposition, or which is constructed from such signs". * Whereas signs in general have the property of bringing to mind something previously known - Le., the property of being "meaningful" in the psychological sense -, the differentiating property of the language sign, or term, is that of being used to stand tor something in a proposition. Terms, so defined, may be sounds, marks, or mental states or images ("intentions of the soul"); but it is not what kinds of entities they are, that makes them to be language signs or terms, .but the use made of them in forming statements about things which they are not. **

* **

Albert: Logica I, ch. 1; Ockham : Summa J, ch. 1 ; Moody: Ockham, p. 40. This is the case when they are given "normal interpretation" as standing for things such as they have been instituted to designate in dis-

LOGIC AND LANGUAGE

19

The 14th century logicians, including Ockham, Albert of Saxony, and Buridan, distinguished between sounds or marks used as language signs, and "intentions of the mind" or "concepts" used as language signs, by saying that the usage of the former is established by convention (secundum placitum instituentis), whereas the usage of the "mental term" (intention, concept) is established by nature. What is here meant is that although the choice of a particular sound or mark, as symbol of such objects as occasion a certain kind of experience or perception, is arbitrary, the ways in which such objects are experienced or perceived by the natural sensory powers of man are not arbitrary. Thus, although there might be some language in which the word-design ~white' was used to identify or designate the condition which is designated in English by the word-design ~black', the very possibility of determining this fact, or of translating from one language into the other, presupposes that the empirical or sensory criteria for identifying the objects which we call black in English are approximately the same as those by which the objects that are called white in the other language are recognized and identified. So we assume that these empirical or sensory means of identifying objects function as a natural human language on which, as a non-arbitrary foundation, the conventional languages are based. * Once the natural or "mental" sign is mentioned and acknowledged as prerequisite for the conventional institution of sounds or marks as language signs, the logical analysis of language proceeds as a formulation of the usage of conventional signs in some definite language. To employ the terminology of C. W. Morris, formal logic abstracts from the particular relations of "sign-vehicles" to their "interpretants" and to their "designata", and deals only with the relations of the conventional language signs (or "signvehicles") to each other. ** Nevertheless, since logic is understood ~.~

course. But terms may also be used "autonymously" to stand for themselves or for language signs of like design. As so used, they are really distinct signs from the materially identical or similar expression-designs which they stand for - just as the name •Socrates' is distinct from Socrates. * Albert: Logica I, ch. 2; Ockham: Summa I, ch. 1; Moody: Ockham pp. 39-40. ** C. W. Morris, "Foundations of the Theory of Signs", International Encyclopedia of Unified Science, Vol. I, No.2. Cf. also R. Carnap, "Introduction to Semantics", Cambridge, Mass., 1946, pp. 5-11.

20

LOGIC AND LANGUAGE

to be a formalization of the usage of language, and not merely a set of rules for spatio-temporal displacements of sounds or marks, it is assumed that language symbols have some definite "interpretants "determining their designative relations to "objects", or, in the case of the logical signs, determining the relations between other language signs for objects. Only on this assumption can the relations of language symbols to each other, which constitute the "syntactical" dimension of language, be understood as relations between signs - Le., between sounds or marks as interpretable for some thing or things other than the sounds or marks which they are. This is the distinctive property of the language-sign, whereby the linguistic "sign-sign" relation is distinguished from the physical "sound-sound" relation. In mediaeval logic, this property of "being interpretable for something" was called the property of supposition. It was understood as the capacity of a term to be interpreted for one or more objects in a proposition. The general definition of "supposition", as given by Peter of Spain, ockham, Buridan, and Albert of Saxony, varies in mode of expression but not in essential meaning. In all cases it is conceived as the capacity of a categorematic language sign (Le., a term) to be "taken for something" in virtue of being combined with some other language sign in a sentence or proposition. We give below the definitions of "supposition" offered by these logicians. Peter of Spain: Supposition is the interpretation of a substantive term for something. Supposition differs from signification, because signification arises through imposing on a vocal sound the function of signifying something, whereas supposition is the interpretation of the already significant term, for something. Thus when we say, "A man runs", this term ~man' stands for Socrates or Plato, and so on. Hence signification is a property of a vocal sound, whereas supposition is a property of the term already constituted from a vocal sound and a signification. William of Ockham: It remains to speak of supposition, which is a property of the term, but only when it is in a proposition. . . .. It is called supposition, in the sense that it is a positing for other things, such that when a term in a proposition stands for something, we use the term for that of which (or of a demonstrative pronoun indicating it) that term is verified.

LOGIC AND LANGUAGE

21

Jean Buridan: Supposition, as here understood, is the interpretation of a term in a proposition for some thing or things such that, if it or they be indicated by the pronoun ~this' or ~these' or an equivalent, that term is truly affirmed of the pronoun by way of the copula of that proposition. Albert of Saxony: Supposition, as here understood, is the interpretation or usage of a categorematic term which is taken for some thing or things, in a proposition. And I say that a term of a proposition is interpreted for something, in this sense: that the predicate of that proposition is indicated to be verified affirmatively or negatively of a demonstrative pronoun denoting that thing. For example, if we say "A man is an animal", .... this term ~man' stands for Socrates or Plato, because the term ~animal' (which is the predicate of the aforesaid proposition) is indicated to be affirmatively verified of a demonstrative pronoun denoting Socrates or Plato. And in the same way, in the proposition mentioned, the term ~ animal' is interpreted for these same things, because it is indicated by the proposition that the term ~animal' is affirmatively verified of a pronoun denoting Socrates or Plato. *

We may note that Ockham, Buridan and Albert of Saxony clearly stipulate that a term has supposition only when it occurs in a proposition. Peter of Spain does not say this explicitly, and it has been claimed that his conception of supposition was on that account different from the later theory - presumably more "realistic" in taking supposition as a direct relation of the term to an extra-linguistic object rather than as a relation of the term to the other constituents of a proposition, for extra-linguistic objects. ** There are good reasons, however, for believing that Peter's conception of supposition was in this respect the same as that of the later authors. In the first place, he explicitly states that supposition is a property of the term, and not, like signification,

* The above quotations are from: Peter of Spain, 6. 03; Ockham : Summa I, 64, fo!. 24v; Buridan: Sophismata, ch. 3; and Albert: Logica II, ch. 1. ** Cf. J. P. Mullally, The Summulae logicales of Peter of Spain, Notre Dame, Indiana, 1945, p. xlvii: "For Peter of Spain, supposition was the property of any substitutive term in virtue of which the term could substitute for a thing or things, regardless of whether it was a component part of a statement or not. Later, William of Ockham attributed the property of supposition to a term but only when the term functioned as part of a statement".

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22

a property of vocal sounds. But a term, as distinct from a mere word, is defined by Peter as "that into which a proposition is resolved, as into its subject and predicate". * Secondly, Peter's entire discussion of suppositions shows that the kind of supposition a term has, depends on its occurrence in a proposition. Finally, to construe supposition as the direct relation of a significant term to what it signifies, is completely trivial and allows no basis for the sharp distinction drawn by Peter between supposition and signification. To say that a name, taken alone, stands for its objects, is equivalent to the trivial statement that a name is the name of whatever it is the name. But if supposition is taken as the relation which one term of a proposition has to the other, as positing some values in extension of the other term, it has genuine logical significance. This significance may be more directly suggested by reference to the use of the individual variable in the functional formulation of general propositions in contemporary logic. In saYing that a term has supposition relatively to another term, tor some thing or things, the mediaeval logicians were expressing an analysis of propositions of subject-predicate form shnilar to that which is new effected through the use of quantified variables - e.g., ~ (x). Fx":JGx' , or ~(Ex):Fx.Gx'. The word "thing", as used in the definition of supposition, functions as a pronominal identification of the terms for a common value in their extensional domain. ** The theory of supposition is not only important in constructing a functional interpretation of the categorical proposition, but also in clarifYing the distinction between the "syntactical" and "semantical" aspects of language. Supposition is a SYntactical relation of term to term, and not a semantical relation of the term to an extra-linguistic "object" or "designatum". This is evident from the following consideration: the metalanguage in

* **

Peter of Spain, 4. 01. Cf. Oarnap: Logical Syntax pp. 292-296, on "universal words" such as the word
LOGIC AND LANGUAGE

23

which the semantical relation of a term to its designatum can be expressed, must contain not only the name of that term, but also a name which directly designates the designata of that term. Thus we would say, in such a metalanguage, that "the term "'man' designates man". Now it is possible to express the relation of supposition without use of names for the objects for which the terms stand, as when we say "In the sentence "'Some man is an animal' the term "'man' stands for something for which the term "'animal' stands". Here we use names for the terms "'man' and '" animal', but we do not use names for their designata; the word something' does indeed refer to whatever the terms designate, but it does not, like those terms themselves, possess independent meaning. Its function is a sYntactical one of quantification, of determining a connection in extension for the two terms, just as relative pronouns determine an extensional relation of the subject of the dependent clause to an antecedent subject. The property of supposition is grounded, not in the semantical relation of designation, but in,the logical or syntactical relation of predication. The meaning relation, of a term to its designata, is neither true nor false, and involves no "hypothesis" or "supposition". But the predicative relation, which is between one term and some other term, does involve a "hypothesis" or "supposition" such as is either true or false. It is for this reason that the mediaeval logicians formulated their theory of truth-conditions on the basis of the property of supposition, and not on the basis of the property of significance or meaning. * t"

§ 6.

PERSONAL (FORMAL) AND MATERIAL SUPPOSITION

Supposition has been defined as the interpretation of a term, in a proposition, for some thing or things specified by another term in that proposition. We may now ask whether it is possible for a term to have supposition with respect to any other term, without involving an ambiguous usage of the word 'supposition'. Take these two sentences: "Socrates is white", and "White is an

* Cf. Buridan: Sophismata, Ch. 2, Cone!. 8: "And so it seems to me that in determining the conditions of the truth and falsity of propositions it is not sufficient to look to the significations of the terms, but to look to their suppositions".

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LOGIC AND LANGUAGE

adjective". It is clear that we cannot construe the supposition of the term 'white', in these two sentences, in the same way. In the first sentence the term 'white' stands for things such as it was instituted to designate, whereas in the second sentence it stands for the word 'white' and not for things which this word was instituted to designate. Nowadays it is customary to indicate this distinction between the significative and autonymous usage of a word-expression, by enclosing it in quotation marks whenever it is to be interpreted in autonymous usage, or as a name for itself. Thus we avoid ambiguity by writing "~White' is an adjective", in the case of autonymous usage, and "Socrates is white" in the case of normal or significative usage. The mediaeval logicians distinguished between these two radically different usages of word-signs by a general distinction of "modes of supposition". When a word was used significatively, to stand for things such as it was instituted to be a sign of, it was said to be used in personal supposition, or in formal supposition. When used autonymously, as a name for itself or for the kind of languagesign of which it is an instance, it was said to be used in material 8upposition. * As far as formal logic is concerned, this generic distinction between the "significative" and "autonymous" suppositions of terms was the important one. While the terminology varied to some extent among the authors, the distinction itself was accepted and used in the same manner by all. ** With the single exception of Buridan, all the logicians under consideration distinguished a third generic type of supposition, which they called "simple supposition" (suppositio simplex). In the earlier period, with its background of extreme realism, simple supposition was described as the interpretation of a general term for the "universal thing" of which the individuals, for which the term stands when taken

* Oarnap: Logical Syntax, pp. 237-240, makes a similar distinction between autonymous interpretation of word-signs, and their interpretation for objects, as basis for his thesis that alleged metaphyscial entities arise from the ambigouus interpretation of autonymous expressions occurring in "quasi-syntactical sentences in the material mode of speech". ** For the various classifications of types of supposition, see Prantl Vol. III, pp. 17-19, 51-52, 373-379, and Vol. IV, pp. 25-29 and 66-67.

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LOGIC AND LANGUAGE

in personal supposition, ,are instances. Ockham, and nearly all the other 14th century authors, rejected the subsistent universal of the earlier tradition, and held that the term ~universal' is a purely logical predicate whose range of significance is confined to the domain of language signs. But since they distinguished between conventional language signs (sounds or marks) and natural language signs (concepts or mental "intentions"), they retained the older distinction by saYing that when a conventional language sign is used to stand for itself as a sound or mark, it has material supposition, but when it is used to stand for the concept or mental sign which is its psychological correlate (or "interpretant"), then it is used in simple supposition. It should not be said that simple supposition is that whereby a term is interpreted for the universal nature, as some people used to believe; for such a universal nature is not to be posited - unless we wish to mean, by ~ universal' nature, the concept representative of many things of which it is a natural likeness, and to which the vocal or written terms are subordinated in their significative use. * While the diverse views concerning suppositio simplex reflect important metaphysical and epistemological issues, they have little bearing on formal logic. Here the one important distinction is that between personal (or formal) supposition, where the term is interpreted for the things it was instituted to designate, and material supposition, where the term is interpreted autonymously as name for itself. The metaphysical issues involved in the controversies over "simple supposition" bear ultimately on the question, extraneous to logic itself, of what brings significant language into being - whether it is a light reflecting an unchanging model language in the mind of God, or a capacity of men to use their natural ways of experiencing things as instruments of communication and control in their social and physical activities. Since logic presupposes language as its domain of analysis, the problem of how language as such is possible lies beyond the concern or the competence of logic. Buridan, who reduces the distinction of suppositions to the two types important for logic - to personal and material supposition - , appears to recognize this fact.

* Albert: Logica II, ch. 2. On the "subordination" of spoken and written signs to their conceptual interpretants, see Moody: Ockham, pp. 39-40. 3

26

§ 7.

LOGIC AND LANGUAGE

OBJECT LANGUAGE, METALANGUAGE, AND THE TRANSCENDENTAL TERMS

The normal interpretation of a term, which is its usage as sign of things such as it was instituted to designate, is its significative interpretation in personal supposition. Developed languages contain terms which have been instituted to signify kinds of language signs, as well as terms which have been instituted to signify things which are not language signs. In mediaeval logic, those terms which, in their significative usage or in personal supposition, stand for language signs, were called terms 01 second intention; words such as ~ term', ~ proposition', and (at least for the 14th century) ~ 'l,tniversal', ~ genus' , ~ species' , ~ property' , etc., are words whose domain of significance is that of language signs. Those terms which, in their normal or significative usage in personal supposition, cannot stand for things which are language signs (Le., as signs, though they may stand for them as instances of physical objects or mental states), were called terms 01 first intention; such terms are those like ~ stone', ~ tree', ~ blue', etc. * These distinctions apply, properly, only to terms in the strict sense, and not to syncategorematic or logical signs. These latter are not of themselves interpretable as standing for any "things", whether individuals or signs; hence they are neither of first nor of second intention. Any developed natural language will contain terms of second intention as well as terms of first intention, and

* Albert: Logica I, ch. 9; Ockham: Summa I, ch. 11, lol. 5v. Albert makes a distinction, of no great importance for logic, between terms of first or second intention, on the one hand, and terms of first or seeond imposition, on the other. Only mental terms, or "intentions" of the mind, were said to be of first or second intention, and spoken or written signs were said to be of first or second imposition. The technical expressions 'first imposition' and 'second imposition' seem to derive from the Greek tradition. See PrantZ, Vol. I, p. 632, on Porphyry's distinction between the neOYl'Yj (}i(Jt~ of words, which is their designation of objects (neaypaTa) , and their &meea (}i(Jt~, which is their syntactical function as part of a proposition. Aristotle's Oategoriae were said to treat of words in their first imposition, while the De interpretatione was said to treat of their second imposition. This distinction, perhaps of Stoic origin, was transmitted by Boethius, and is given considerable development in Abelard's lntroductiones.

LOGIC AND LANGUAGE

27

any language whatsoever will contain some sYncategorematic or logical signs, at least implicitly. A developed natural language, containing terms of first intention, terms of second intention, and an adequate group of sYncategorematic signs, is a language which contains, as a part of itself, a vocabulary with which to describe its own grammatical or logical structure. Such a sub-language is constituted of the terms of second intention contained in the language, together with sYncategorematic signs common to both parts of the language. We nowadays call such a sub-language a "metalanguage" or "syntax language" in relation to the remaining part, the latter being called the "object language". A crucial problem is that of whether a language can contain a vocabulary sufficient for formulating its logical structure in a complete manner, without ambiguous use of its terms of second intention. This problem becomes critical in respect of such terms of second intention as the predicate ~true', as is revealed through construction of the paradox of the liar. In the mediaeval tradition, the "metalanguage" composed of terms of second intention, together with the logical or sYncategorematic signs, was considered to be (or to contain) the language of logic. It was held that the propositions constituting the science of logic consist only of rules or metatheorems, whose terms stand for language expressions. Such terms of second intention abstract wholly from the meaning, or "material" content, of the terms of first intention for which they stand, characterizing the latter only by their forma.! properties as constituents of statement. Mediaeval logic was thus conceived as a formulation, in a metalanguage or "sYntax language", of the logical structure of scholastic Latin in its exact or "scientific" usage. * The conception of logic as a set of formalized sentences in the object language, with letters of the alphabet replacing expressions of the object language rather than naming them or "standing for"

* "Semantical" sentences, such as the sentence "The term ~man' means man", were held by Ockham not to belong properly to logic, but to have only illustrative use in teaching. The sentences belonging to logic were held to contain no terms of first intention, except such words as 'thing',. 'individual', etc., which were considered by some to be terms of first intention, and which were called "transcendental" terms by others.

28

LOGIC AND LANGUAGE

them significatively, was not consciously grasped or employed. Mediaeval logic consists of rules, and not of theorems in the modern sense. But since the rules or metatheorems which constitute mediaeval logic are concerned wholly with the logical form of expressions in the object language, it is legitimate and possible to represent them in the modern manner by formalized sentences in the object language - i.e., by formulas. In this way the logical structures which are described through the terms of second intention occurring in the mediaeval rules may be directly exhibited by the corresponding formulas. While the great body of logical rules constituting mediaeval formal logic contain only terms of second intention along with the syncategorematic signs, a number of the more fundamental metatheorems employ, in addition to these, the terms ~thing', ~ one' , ~ same' , and similar terms. These were considered to be "metaphysical" or "transcendental" terms, in virtue of the fact that they transcend the generic distinctions of significations by which the terms of first intention are differentiated with respect to ways of designating objects. They were called "transcendental" not because they designate a specid domain of entities, but because they do not designate any special domain of entities but stand for whatever can be signified or posited in any manner whatsoever. For this reason they cannot be said to possess "independent meaning", as categorematic terms (classified in Aristotle's Oategories) were said to possess. They might be said to have supposition without signification, or extension without intension. As used in the mediaeval statements of logical principles, they appear to be no more "metaphysical" than the individual variable ex') of contemporary logic; their function is that of expressing the extensional dimension of language, as indices of the suppositional function of terms. *

* The mediaeval conception of a purely metaphysical proposition, as a statement which is true for anything whatsoever insofar as it is anything at all, is the analogue of the modern conception of a logically true sentence, or "formula", of the object language. Thus Buridan, in his Questions on the Metaphysics, Book IV, Qu. 13, says that the "first principle" in metaphysics is not the logical rule called the law of excluded middle, since logical rules are metatheorems formulated in terms of second intention. Rather, it is the analogue of this metatheorem, formulated in terms of first in-

LOGIC AND LANGUAGE

29

This will complete our introductory exposition of the logical status of the categorematic term, constituting the "matter" of propositions. As far as the development of the theory of formal deduction is concerned, the term is treated, in mediaeval logic, only as an operand for the logical operators or sYllcategorematic signs. Its pragmatical and semantical relations, of empirical application or of conceptual content, play no essential part. This formalization of the subject matter of logic, already undertaken by Aristotle, was given precision and uniform application through the mediaeval theory of the supposition of terms. We may now turn to the logically essential constituents of language - the syncategorematic signs -, and consider their functions as determinants of the suppositional relations between terms in sentences, and of the truth-functional and consequential relations between sentences. tention - namely, "Quodlibet est vel non est". For Buridan, the "first principle" of logic is the metatheorem, "For any proposition 'p', either 'p' is true or 'p' is not true". But the "first principle" of metaphysics is the analogous theorem, "p v -p".

'\l I):: ,

III

THE THEORY OF TRUTH CONDITIONS § 8.

THE FORMAL CLASSIFICATION OF PROPOSITIONS

The word ~proposition', as used by the mediaeval logicians, had the meaning which is nowadays given to the term ~sentence'. A proposition was understood to be a certain kind of language expression, and not an objective "state of affairs" constituting the content or meaning of such a language expression. * Peter of Spain, following Boethius, defined the proposition as a sentence in the indicative mood which signifies the true or the false by way of judging. ** The 14th century logicians, who held that ~true' and ~false' are terms of second intention which stand for propositions, and whose meaning cannot be understood unless the meaning of the term ~proposition' is independently specified, adopted a recursive method of defining ~proposition', through enumerating and describing the forms of language expression to be denoted by the term. An initial division was made between atomic ("categorical") and molecular ("hypothetical") propositions. A categorical proposition is one which has a subject and predicate and copula, and which does not contain several such propositions. A hypothetical proposition is one which is composed of several categoricals; and it is divided into five kinds, according to the usual view, namely into the conjunctive, the disjunctive, the conditional, the causal, and the temporal. The conjunctive is that which is formed of several categoricals or hypotheticals. . .. by

*

On the distinction between 'sentence' and 'proposition', cf. R. Carnap, Semantics", Cambridge, Mass., 1946, pp. 235-236. Certain m~ql~l'wval.'"o/riters used the expression "complex signifiable" (complexum 8ignificabilr~) 'to designate the objective content or meaning of the sentence, but .they re~ained the term 'proposition' as name of the language expression (sentence) by which such an objective meaning is expressed or designated. Cf. H. Elie, "Le complexe significabile", Paris 1937. ** Peter of Spain, 1. 06: "sola indicative oratio est propositio"; and 1. 07, "Propositio est oratio verum vel falsum significans iudicando". ~'lntro4~ction to

THE THEORY OF TRUTH CONDITIONS

31

means of this connective ~ and'. .... A disjunctive proposition is that which is formed of several categoricals by means of this connective ~or'. A conditional is one which is formed of several propositions by means of this connective ~if'. .... A causal is one which is formed of several propositions by means of this connective ~because'. .... A temporal is one which is formed of two propositions conjoined by an adverb of time. * A similar procedure is used for defining modal propositions, and for distinguishing affirmative, negative, singular, particular, and universal propositions. In each case it is the presence of a definite syncategorematic sign in the expression that defines it as a kind of proposition. The formal analysis of propositions, consequently, reduces to a determination of the logical functions of these sYncategorematic signs. Of these signs, some are primary -namely, the verbal copula ~is' in the case of categorical propositions, and the connectives ~ and', ~ or', ~if', etc. in the case of hypothetical or molecular sentences. Other sYncategorematic signs, such as the quantifiers, modal terms, or tense variations of the verb, were considered to be secondary in the sense that they presuppose one of the primary logical signs, modifYing its force in one manner or another. Albert of Saxony called the primary logical constants the "formulas" of propositions; the sentential connectives (notae hypotheticae) were called the formulas of hypothetical (molecular) sentences, and the verbal copula ~is' was called the formula (formale) of the categorical proposition. ** Since mediaeval logic was a formalization of the sYntactically rich Latin language, the grammatically admissible forms of sentence were numerous, and their classification complex. Aside from the pure categorical sentences, and the pure hypothetical (molecular) forms, there was a class of propositions caUed exponibilia, composed of sentences which, though grammatically categorical in form, are logically equivalent to molecular sentences containing several categoricals. The treatment of the exponibilia is of interest as an application of logical analysis to ordinary language, but we shall

* Ockham: Summa, II, ch. 1, Cf. also Albert: Logica III, ch. 1, where a similar descriptive enumeration is given. The procedure is comparable to that used by Carnap: Foundations, pp. 8-9, in laying down "formation rules" of a semantical or syntactical system. ** Albert: Logica, III, ch. 1.

32

THE THEORY OF TRUTH CONDITIONS

restrict our consideration to the basic forms of sentence: the categorical (atomic) sentence, and the hypothetical (molecular) sentence. Both atomic and molecular sentences were divided into assertoric and nl0dal types, the modal sentences being defined as those in which a modal term explicitly occurs. Molecular sentences were subdivided on the basis of the sentential connectives, such as the signs of conjunction, disjunction, and implication. Atomic sentences were classified primarily on the basis of time range and of quantification. With respect to time range, the division was into sentences of present, past, or future time. With respect to quantification, the division was into singular, indefinite (or particular), and universal propositions. The analysis of the categorical proposition, as differentiated according to time range and quantity, was carried out in terms of the concept of supposition. The elementary form of the copula was conceived to be represented by the substantive verb ~is', as a verb of ·present time and undistributed subject. * We shall first consider the logical import of the elementary copula of affirmative categorical propositions, and then introduce the fundamental sentential connectives ~ not', ~ and', ~ or', and ~ it'. These were used in the mediaeval analysis of categorical propositions of extended ("ampliated") time range, and of quantification. This analysis, carried out in terms of the notion ofsupposition, constituted the mediaeval theory of truth conditions as determined by the form of categorical propositions.

§ 9.

THE LOGICAL IMPORT OF THE ELEMENTARY COPULA

A categorical proposition was defined as an expression which has, as its essential constituents, a subject term and a predicate term and a copula. ** Aristotle had said that the simplest sentence is one formed of two constituents, a noun and a verb. But Albert points out that this, while true in a grammatical sense, does not conflict with the definition, since insofar as the verb is a predicate it contains what is logically a noun, in addition to its verbal force as a substantive verb connoting time. Since, in certain types of

* **

Albert: Logica, III, ch. 1; Ockham: Summa, II, ch. 1, fol. 29v-30v. Albert: Logica, III, ch. 1; Ockham: Summa, II, ch. 1, fo1. 3Or.

THE THEORY OF TRUTH CONDITIONS

33

conversions, the subject and predicate terms can be interchanged, the logician considers them to be similar constituents of the sentence - Le., both are logically "nouns", and only the copula is the "verb".* When the verb ~is' occurs without any predicate term following it, it is said to occur secundo adiacens. In such use, it obviously does not "copulate" a subject with a predicate term. The recourse of analyzing it into a copula followed by the predicate ~being' was not considered to be of much significance, since the term ~being' does not specify any interpretation for the subject term beyond that which is specified by the subject ternl itself. Albert of Saxony says that the verb ~is', when occurring secundo adiacens, "signifies the existence of that for which the subject term stands". As so understood, it would seem to have the force of an existential quantifier, indicating that the subject term has supposition - i.e., that it is to be "taken for something". ** When the verb ~is' occurs as elementary copula connecting two terms as subject and predicate, it is said to occur tertio adiacens. In this use its function was described as that of positing that the terms stand for the same. When the verb ~is' occurs as a third constituent, it signifies a certain composition of the predicate with respect to the subject - that is, it signifies that the subject and the predicate stand for the same. ***

* Albert: Logica I, ch. 5. Adjectives, and indeed all categorematic terms such as have independent meaning, are included under the class of "noun" in the logical sense. Shyreswood: Syncategoremata, p. 71, argues that the verb 'is' is not merely a syntactical sign of composition of subject with predicate, but is itself a predicate whose content is "specified" by the term following it. This view is not necessarily in contradiction to Albert's position, since for Albert the verb 'is' has more than the function of a sign of composition of the terms, in that it also posits the existence of something for which the terms, so conjoined, stand. ** Albert: Logica, I, ch. 6. It should be noted, however, that this is an existential quantifier of present time only, which does not extend to past, future, or merely possible instances. Existence statements of past time require the verb 'was' or its equivalent, those of future time the verb 'will be' or its equivalent, and those of possible instances (or of any instance without qualification) the verb 'can be' or an equivalent. *** Albert: Logica, I, ch. 6, Cf. also Buridan: Sophismata, ch. 2, cone!. 10.

34

THE THEORY OF TRUTH CONDITIONS

This is the so-called "identity theory" of the copula, frequently mentioned and criticized, but perhaps not fully understood in its mediaeval meaning. It involves the assumption that a categorical proposition is an expression which posits one or more instances, or values in extension, of the significative function formed of its categorematic terms. The meaning of the copula is the quasisYntactical one of indicating that this significative function is to be "taken for something", or interpreted in extension. The complex of the terms alone, without a copula, has meaning but not supposition; it constitutes a matrix expression with a free variable, which can become a sentence only when the variable is "bound" by the copula. The copula is essentially an existential quantifier. The meaning of the word ~true', as a predicate applicable to the term 'proposition', was expressed by the mediaeval logicians by the statement that a sentence is true if, howsoever it signifies things to be, so they are (qualitercumque significat esse, ita est). * Since Albert of Saxony's statement of the logical function of the copula, as signifying that subject and predicate stand for the same, constitutes a metatheoretical description of what an affirmative sentence signifies in virtue of its form, it corresponds exactly to a formal description of the truth condition of affirmative categorical sentences. Thus it is as a "rule of truth" for affirmative categoricals that Buridan expresses the requirement that subject and predicate stand for the same. For the truth of an affirmative categorical proposition it is required that the terms, namely the subject and the predicate, stand for the same thing or things. ** While conceding that this rule is indemonstrable, Buridan seeks to show that it accords with the common usage of language, as follows: It is certain that in this sentence "A is B", this term ~A' either stands for nothing at all, or else stands for A; and similarly with this term ~B' ..... Then it is evident that .... it is the same to say "A is B" and to say "A is the same thing that B is".....

* Buridan: Oonsequentiae, ch. 1. This statement closely reflects the general definition of 'true' given by Aristotle, Metaphysics IV, ch. 7, 1011 b 26 ff. But cf. infra, § 16, for Buridan's criticism of this definition. ** Buridan: Sophismata, Ch. 2, Concl. 10.

THE THEORY OF TRUTH CONDITIONS

35

And yet, if it is true that A is the same thing that B is, it follows that these terms ~A' and ~B' stand for the same thing, since ~A' stands for A and ~B' for B. * We may now attempt to devise a manner of representing this rule of truth in symbolic manner. If we employ variables, F and G, for any categorematic terms of the language, we may represent affirmative categorical sentences of indefinite or particular quantity by the word-formulas; Some F exists, Some F is a G. Then, to give symbolic representation of the descriptions of the truth conditions determined by the form of such sentences, we will use <: F' and ~G' (enclosed in single quotation marks) as variables for the names of whatever terms may replace F and G in the formulas. We shall also represent the name of any sentence which could result from substituting constants for F and G in these formulas, by enclosing the formula in single quotation marks; Le., for any sentence, p, if Some F is a G is the formula of p, then ~Some F is a G' is the name of p. We also require a symbol for the expression "stands for", and for this we shall use a subscript ~s' following on the symbols ~ F' or ~G' which represent the names of terms, and followed by the individual variable x. It is to be understood, however, that the expression "~F' 8X" is not the formula of a complete sentence, since the individual variable x is not a term, but only a pronominal link needed to express the identification of the suppositions of two terms. Even where the sentence has only one term, this remains true, since the expression, (Ex).~F'8x, has the force of this more adequate expression: (Ex): x = x. ~ F'8X. And this in turn may be rendered: (Ex).~F'8x.~F'8x. Now, using the letters ~T' and ~F' for the logical predicates "is true" and "is false" respectively, we may state two metatheorems expressing the formal truth conditions of indefinite affirmative sentences in which the copula occurs secundo adiacens and tertio adiacens respectively. 9.1 9.2 ~F

T T

~Some ~Some

F exists' :-:J.(Ex). ~ F'8X F is a G' .:J. (Ex). ~F'8X.

~G'8X

In the formulas of these sentences, the expressions ~ F exists' and is a G' represent the actual use of the terms of the sentences,

*

Buridan: Sophismata, eh. 2, Conel. 10.

36

THE. THEORY OF TRUTH CONDITIONS

for whatever things they can stand for in virtue of their imposition as terms. But the matrix expressions in the metatheoretical descriptions of the truth conditions of such sentences, i.e., "~F' 8 x" and "~G'8 x", represent parts of statements affirming the logical relation of "standing for the same" of the names of the terms occurring in the sentences whose truth conditions are being described. We may now add three corollaries to the above metatheorems: 9.21

-[(Ex). ~ F'8X. 'G',x ::): T

9.22 9.23

-(Ex). -(Ex).

~F'8X::):F ~Some ~F',x.

v .-(Ex).

~Some

F is a G']

F exists' F

~G'8X::):

~Some

F is a G'

Buridan states 9.21 as follows: "It does not follow that if the subject and predicate of an affirmative categorical stand for the same, the proposition is true". The reason given is that propositions which "reflect on themselves", as in the case of the Liar paradox, may have subjects and predicates which stand for the same, and yet be false. * The other two rnetatheorems, 9.22 and 9.23, express the mediaeval requirement of existential import represented by the statement: "If either the subject or the predicate of an affirmative categorical proposition stands for nothing, that proposition is false". ** The identity theory of the copula, adopted by practically all the logicians of the 14th century, was an attempt to express the meaning of affirmation in a formal and extensional manner. Both subject and predicate terms are taken in extension, the force of the copula being that of positing the identity of the extensional values of the terms, in the manner determined by the form of the proposition. In the period prior to the 14th century a different analysis of the copula was given, which may be called the "inherence theory". According to this theory, the copula determines that the subject term is to be taken in extension, for individuals for which it stands in personal supposition, but it determines that the predicate term (if a general term) is to be interpreted in intension, as standing for the "universal nature" which it was thought to denote in

* Buridan: Sophismata, eh. 2, Conel. 9. On the liar paradox and its solution, ef. infra, § 17. ** Buridan: Sophismata, eh. I, Conel. 5. Albert: Logica, I, eh. 6.

THE THEORY OF TRUTH CONDITIONS

37

simple supposition. On this theory, the copula could not be construed as a sign of identity of what the terms stand for, but it was taken as a sign of the "inherence" of the universal nature signified by the predicate term, in the individuals for which the subject term stands. Peter of Spain called this relation one of "adjectivation" (adiectivatio) , with the meaning we associate with the expression "characterizes" . * Ockham criticized the inherence theory on the ground that it involved a confusion of language levels analogous to what Carnap has called the "fallacy of the material mode of speech". ** According to Ockham, all such expressions as ~inheres in', ~participates in', <"belongs to', and the like, are relational predicates of second intention, of the same order as the expression ~is predicated of'. Hence the significant use of these relational predicates requires . that they take. as arguments, the names of terms which occur as subjects and predicates of sentences, and not the terms themselves in their significative use. Thus the sentence "Animal inheres in man", or "Animality belongs to man", was construed by Ockham as equivalent to this sentence: "The term ~ animal' is predicable of the term ~man' in a true proposition". The true proposition in question is, of course, the sentence "Man is an animal". But this sentence does not say anything about the terms ~man' and <" animal', and consequently it cannot be construed as asserting the relation of "inherence", since this is a relation between terms and not a relation between what they stand for when used significatively. By such propositions as "Socrates is a man" or "Socrates is an animal", it is not denoted that Socrates has humanity or animality. Nor is it denoted that humanity or animality is in Socrates, nor that man or animal is in Socrates, nor that man or animal is a part of the being or essence of Socrates, or a part of the concept of the essence of Socrates. But it is denoted that Socrates is in fact a man, and that he is in very truth an animal - not in the sense that Socrates is this predicate ~man' or this predicate <" animal', but in the sense that there is something for which this predicate ~ man' and this predicate ~ animal' stand, such that these predicates stand for Socrates. ***

*

** ***

Peter of Spain, 6. 02 and 6. 13. Cf. Shyreswood: Introductiones, pp. 77-78. Carnap: Logical Syntax, p. 384ff. Oclcham: Summa, II, ch. 2. Cf. Moody: Oclcham, p. 183, Note I; and

38

THE THEORY OF TRUTH CONDITIONS

While the inherence theory construed the function of the copula as that of associating an intension with an extension, the identity theory construed it as an identification of the extensions of the terms. Although terms have meanings or intensions, it is not because they have meanings that the sentences in which they occur are true rather than false; rather, it is the fact that there is (or can be) something for which the terms, in their meanings, are used, that determines the truth of sentences in which the terms occur as subject and predicate. A proposition which would be considered true, because tautologous, in intension, could well be false in extension; thus the sentence, "A chimera is a chimera", was held to be false by the 14th century logicians, because, even though the term ~ chimera' has meaning, there cannot be anything for which it can stand.

§ 10.

THE SENTENTIAL OPERATORS

Negative categorical propositions, according to the mediaeval analysis, are formed by application of the adverb 'not' to the affirmative copula 'is' (or 'was', 'will be', etc.). The function of the negation sign, in categorical propositions, is to destroy the force of the affirmative copula - i.e., to indicate that subject and predicate do not stand for the same. Silnilarly, when the negation sign is applied to molecular sentences formed by means of a sign of conjunction, or of disjunction, or of implication, its function is to destroy the force of these connectives. * Negation is clearly a "primitive idea" in the mediaeval logic, since the description of the function of the word 'not', as given in the logical metalanguage, involves use of that word. Buridan related the meaning of the negation sign to the opposition between truth and falsity, in these words: pp. 57-65 on abstract terms such as 'animality'. On the fallacy of confusing a logical sign of the object language with a relational predicate of the metalanguage, cf. Quine pp. 27-33. * Cf. Shyreswood: Syncategoremata, pp. 71-73. The use of 'not' in direct combination with a term, to form a so-called "infinite noun" such as 'not-man', was of course recognized; but sentences in which such terms occur were considered to be exponible, or reducible to compounds of sentences in which the word 'not' occurs only in its use as a sentential operator. Cf. Ockham: Summa, II, ch. 12, fol. 35r; Moody: Ockham, p. 201.

THE THEORY OF TRUTH CONDITIONS

39

The eleventh conclusion is, that for the truth of a negative categorical it is sufficient that subject and predicate do not stand for the same.... And thus also, for the falsity of the negative it is required that subject and predicate stand for the same. . ... This conclusion is evident from the preceding one. For contradictories are such that one is affirmative and the other negative, and such that it is necessary that one be true and the other false. . . .. and such that they are not both true, or both false. And this is for no other reason than this, that whatever be the conditions of one of the propositions being true, the same conditions are the causes of the other being false, and conversely. Therefore, whatever is required for the truth of the affirmative, is required for the falsity of the contradictory negative. And likewise, whatever suffices for the falsity of the affirmative, suffices for the truth of the contradictory negative. * The second part of this quotation Yields two metatheorems expressing the truth functional relationship between any sentence and the negation of that sentence. We here use ~ p' and ~ -p' as variables representing the names of any sentence and of its negation. 10.1 10.11

.==. F ~-p' T ~-p' .==. F ~p' T ~p'

These metatheorems are equivalent to the usual truth table for the negation sign: 'p'

T F

'-p'

F T

From the first part of the quotation frOln Buridan we may derive two further metatheorems stating, respectively, a sufficient condition of the truth of the negation of an indefinite affirmative, and a necessary condition of the falsity of such a negation. Since the negation of the particular affirmative is a universal negative, these metatheorems state truth conditions of sentences of universal negative form. 10.2

-(Ex). ~ F' aX. 'G' aX

10.21

F~No

*

::>. T ~N0 F is a G'

FisaG' .:>: (Ex).

~F'ax. ~G'ax

Buridan: Sophismata, ch. 2, Concl. 11.

40

THE THEORY OF TRUTH CONDITIONS

In virtue of 10.11 we may replace the expression "T ~N 0 F is a G'" by "F ~Some F is a G'" in 10.2, thereby obtaining a statement of the sufficient condition of the falsity of a particular or indefinite affirmative. Note that this condition is implied by 9.23. 10.22

-(Ex).~ F'sx. ~G'sx

:"J. F

~Some

F is a G'

It is to be observed that the converses of these rules, and of 9.23, are not asserted. The mediaeval logicians normally introduced the sentential connectives ~ and', ~ or', and ~ir, in their treatment of "hypothetical" (molecular) propositions, and after their analysis of the categorical forms. This order was a dictate of tradition rather than of logical priority, since the sentential connectives were used, as if already understood, in the analysis of the categorical proposition. On this account we shall give the mediaeval descriptions of these sentential operators before proceeding to the analysis of quantification and of temporal and modal "ampliation" of categorical sentences. The conjunctive sign ~ and', like the sign of negation, is a primitive operator in the mediaeval system, being required in the metatheoretical statement of the truth condition of molecular sentences formed by its use. This metatheorem was usually stated, in word language, as follows: "For the truth of the conjunctive it is required that both parts be true". * If we represent the sentences constituting the "parts" of a conjunctive by the variables p and q, we may then rapresent the names of whatever sentences may replace these variables by 'p' and 'q' (enclosed in single quotes). The metatheorem stating the truth condition of the conjunctive is then represented as follows: 10.3

~p.q'

is true :-:

~p'

is true . ~q' is true

This truth-functional description of the conjunctive sign would also be represented accurately by the usual truth table:

*

'p'

'q'

'p.q'

T T F F

T F T F

T F F F

Albert: Logica, II, ch. 5; Ockham: Summa, II, ch. 32, fo1. 43r.

THE THEORY OF TRUTH CONDITIONS

41

The disjunctive sign 'or', though given the meaning of exclusive disjunction in early mediaeval times, was normally understood in the inclusive sense (nowadays symbolized by ~v') in the later 13th and in the 14th century. The truth condition of a molecular sentence formed by the connective ~ or' was stated in the following metatheorem: "For the truth of an affirmative disjunctive it suffices that one of its parts be true". * As before, we may express this metatheorem with symbols Cp' and ~q') representing the names of the component sentences of the disjunctive, and the symbol ~p v q' (in single quotes) representing the name of the disjunctive: 10.4

~p

v q' is true:==: ~p' is true. v. ~q' is true

Again we may formulate this truth-functional description by means of the usual truth table of disjunction: 'p'

'q'

T

T

T

F

F F

T F

'p v q'

T T T F

Of interest is the use of the rule corresponding to the so-called "De Morgan theorem" to show the equivalence of the negation of a disjunctive to a conjunctive composed of the contradictories of the parts of the disjunctive. This rule is stated as follows by Albert of Saxony: "The contradictory of the affirmative disjunctive is a conjunctive composed of parts which are contradictories of the parts of the disjunctive". ** This rule Yields an implicit definition of the disjunctive operator in terms of the operators of negation and of conjunction. As stated, the rule governs the "De Morgan theorem" in this form: 10.5

-(p v q) :== :-p.-q

Both Ockham and Albert of Saxony, in their explanations of the truth rules for the conjunctive and disjunctive forms, state the

* Albert: Logica, II, ch. 5. Ockham: Summa, II, ch. 33, fo1. 43r, says that "for the truth of the disjunctive it is required that one or the other of its parts be true". The distinction between "affinnative" and "negative" disjunctives is nothing more than the distinction between a sentence of the form 'p v q' and a sentence of the form' -(p v q)'. ** Albert: Logica, III, ch. 5. 4

42

THE THEORY OF TRUTH CONDITIONS

rules of simplification for logical multiplication and for logical; addition, represented by these theorems: ~pq.-:J.p'; ~pq.-:J.q'; ~p.-:J.p v q'; and ~q.-:J.p v q'. They also state an alternative form of the "De Morgan" law, represented by this theorem: ~ -(p.q):=: -p v -q'. In all these cases, the statements are made in the form of rules or metatheorems, and not as formulas or theorems involving propositional variables, such as we have used to represent the rules. The metatheorems so far considered have stated the truth functional relation between conjunctive or disjunctive sentences and their components. Analogous rules are also given for the conditions under which conjunctive or disjunctive sentences are to be called possible, impossible, or necessary. As stated, these rules employ the modal terms as logical predicates applied to the names of sentences. Thus it is stated that the conjunctive requires, for its possibility, not merely that each component proposition be possible, but that they be "compossible". Similarly, for the impossibility of a conjunctive it suffices that the components be "incompossible", and hence it is not required that each component be impossible. If however the conjunctive is necessary, each of its components is necessary. * The mediaeval description of the import of the conditional operator ~if' involves a more difficult problem of interpretation, and it seems that the mediaeval authors themselves were not entirely in agreement concerning the way of formulating the truth condition of conditional sentences. A true conditional was generally considered to be equivalent to a "consequence", so that the problem of stating the truth condition of the conditional is the same as that of defining the term ~ consequence'. Most of the authors agreed, however, that a true conditional is a necessary proposition, a false conditional an impossible proposition, and that if a conditional is true, it is impossible that its antecedent be true and its consequent false. **

* Ockham: Summa, II, ch. 32, fo1. 43r; Albert: Logica, II, ch. 5. We shall consider these modal rules, and the meaning assigned to the term 'possible', in our examination of the theory of consequence. ** Albert: Logica, II, ch. 5: "omnis conditionalis vera est necessaria, et omnis falsa est impossibilis" .... "ad veritatem conditionalis requiritur quod impossibile est qualitercumque significat antecedens, esse, quin qualitercumque significat consequens, sit".

THE THEORY OF TRUTH CONDITIONS

43

While it seems correct to use the truth-functional symbols ~, and ~v' to represent the mediaeval use of the conjunctive and disjunctive connectives ~ and' and ~ or', the use of the modern symbol of "material implication", namely ~:Y, for the mediaeval conditional connective ~if', would be questionable. On this account we shall introduce the symbol ~ -I' as conditional connective, such that a sentence of the form ~If p, then q', if asserted or interpreted without further qualification, will be symbolized by the expression ~p-jq'.

*

The relation of the mediaeval conditional connective, symbolized by '-j', to the modern connectives of "material" and "strict" implication, will be left indeterminate at this point. But we may formulate the mediaeval truth rule for conditionals to the extent of stating the following metatheorem: 10.6

~p-jq'

is true :==:

~(p.-q)'

is impossible

If we should formulate a theorem corresponding to this rule, using the diamond symbol ~O' for the word "possible", it would presumably take this form: ~p-jq:==:-O(p.-q)'. While this same theorem serves to define the meaning of the connective of strict implication, in the system of Lewis and Langford, it would be hasty if we identified the mediaeval connective ~ -j' with that of strict implication, since the meaning of this connective depends on the meaning given to the word ~impossib1e', or to the expression ~-O(p.-q)', in the mediaeval system. We shall examine this problem in connection with the mediaeval theory of consequence. **

§ 11. QUANTIFICATION The basic formal constituent (formale) of categorical sentences has been said to be the verbal copula ~is'. So far we have considered the truth conditions of assertoric categoricals in which this sign occurs without further sYncategorematic determinations of time

* We have said expressly, "if asserted Or interpreted without further qualification", because of the fact that the mediaeval authors did recognize a use of the conditional connective
44

THE THEORY OF TRUTH CONDITIONS

range or quantity. The traditional classification of categorical sentences into those of present, past, or future time, and of particular or universal quantity, was based on the presence of secondary sYncategorematic signs in the sentences. We have now to consider these, and first ofall those which determine quantification. Ordinary language, and especially the Latin language, contains many words which operate as quantifYing prefixes. Since the logical analysis of the sYncategorematic signs developed, in the 13th century, under the inspiration of the grammatical tradition of Priscian, the treatment of the so-called "signa" (or quantifYing prefixes) constituted a lengthy and complex section of the mediaeval logical text-books. ShYreswood'sSyncategoremata analyze the terms ~every' (omnis) , ~the whole of' (totum) , ~both' (uterque) , ~none' (nullus), ~nothing' (nihil), ~neither' (neutrum), ~other than' (praeter) , ~only' (solus) , ~no more than' (tantum). * While the 14th century logicians continued this elaborate tradition, they tended to stress the derivative character of most of these signs, and to exhibit the two quantifYing prefixes ~some' (aliquis) and ~every' (omnis) as the fundamental ones. It is only with these that we shall here be concerned. Albert of Saxony defines a quantifYing prefix (signum reddens propositionem universalem vel particularem) as "a syncategorematic sign explicitly indicating the manner of supposition of the term following it". ** The primary division of such signs is into signs of universality, and signs of particularly; each kind is then subdivided into substantival and accidental quantifiers, the "accidental" ones being words such as "of some sort" (aliquale) and "wherever" (ubicumque) or "of any sort you please" (qualislibet) , etc. It is the substantival quantifiers that are fundamental in generating the traditional distinction of general propositions into particular and universal. Albert defines the particular sign (aliquis) and the universal sign (omnis) in these words:

* Shyreswood: Syncategoremata, pp. 48-70. Peter of Spain, Tractatus XII, "De distributionibus", treats the same signs, as well as the so-called "distributive signs of accidents" such as qualelibet ("of whatever sort"), quotienscumque, and infinitum in the syncategorematic sense of "no matter how much is assigned, more". ** Albert: Logica, III, ch. 2.

THE THEORY OF TRUTH CONDITIONS

45

A sign of universality is one through which a general term to which it is adjoined is denoted to stand, in a conjunctive manner, for everyone of its values (supposita). A sign of particularity is one through which a general term is denoted to stand, in a disjunctive manner, for everyone of its values. * Quantification was not thought to apply to terms taken in material or simple supposition, and hence the distinctions of "manners of supposition" determined by the quantifiers applied only to terms taken significatively or in personal supposition. A first distinction of modes of personal supposition, determined by the suppositional possibilities of terms themselves as established by their manner of institution as signs, was into "discrete" (singular) and "common" (general) supposition. Discrete or singular terms are those which stand for one and only one individual; they include proper nalnes, and they also include demonstrative pronouns and general terms to which a demonstrative pronoun is prefixed. vVhile these demonstrative expressions are not determined, by institution, to stand for one certain individual in the sense that a proper name is, they nevertheless stand, in any sentence in which they occur significantly, for one and only one individual. ** Common or general personal supposition (suppositio personalis communis) is had by a general term, such as has been instituted to stand for many, when it is not restricted to discrete supposition by apposition of a demonstrative pronoun. Albert of Saxony defines it as the interpretation of a general term, occurring in a proposition, for everyone of the things it signifies by its imposition. It is to be noted that the range of supposition (or the number of things for which the general term is interpreted) is the same in a particular proposition as in a universal one. If we say "Some man is white", for example, the term ~ man' stands for each and every man, just as it does in the universal statement "Every man

* **

Albert: Logica, III, ch. 2. Ockham: Summa, II, ch. 4, fo!. 32r. Albert: Logica, II, ch. 4: "Suppositio personalis discreta est acceptio termini discreti, vel communis cum pronomine demonstrativo, pro uno tantum". Ockham: Summa, I, ch. 70, fo!. 26r, says: "Suppositio discreta est in qua supponit nomen proprium alicuius, vel pronomen demonstrativum significative sumptum, et talis suppositio reddit propositionem singularem".

THE THEORY OF TRUTH CONDITIONS

46

is white". Although the particular proposition only affirms that at least one of the things that are men, is white, it will be true if any man whatsoever, out of the totality of all men, is white. Particular quantification does not reduce the range of supposition, but it only determines a disjunctive manner of supposition of the general term for its individual values. * General personal supposition was divided into three principal kinds, corresponding to three ways in which the individual values of a general term are posited, by reason of the form of the sentence in which it occurs, as identical with the individuals for which the other term of the proposition stands. These three kinds of general supposition were called "determinate supposition". (determinata) "confused and distributed supposition" (con/usa et distributiva) , and "merely confused supposition" (con/usa tantum). The first two kinds, determined respectively by the sign of particularity and the sign of universality prefixed to the subject of an affirmative sentence, are of primary interest; the third kind is reducible, essentially, to determinate supposition. Determinate supposition is described as follows by Albert of Saxony: Determinate supposition is the interpretation of a general term for each of the things it signifies by its imposition, ... in such manner that a reduction to its singulars may be effected, in virtue of this interpretation, through a disjunctive proposition. It is thus that the term ~man' stands, in this proposition "Man runs". This term ~man', in the said proposition, stands disjunctively for all the things which it signifies by its imposition. Now it suffices, for the truth of this proposition "Man runs", that this disjunctive proposition be true, "This man runs, or that man runs", and so on for each singular. ** Both the subject and the predicate of a particular or indefinite affirmative have determinate supposition. Since the particular and indefinite were considered equivalent, it seems that the sign of particularity (Some') merely makes explicit the existential quantification exercised by the copula. Thus the particular affirmative is represented by the same formula as the indefinite affirmative:

* **

Albert: Logica, II, ch. 4. Albert: Logica, II, ch. 4. Cf. Ockham: Summa, I, ch. 70, fo1. 26v: "Suppositio determinata est quando contingit descendere per aliquam disiunctivam ad singularia".

THE THEORY OF TRUTH CONDITION!'!

47

'(Ex).Fx.Gx'. It will likewise have the same truth condition, already stated in 9.2, and the same sufficient condition of falsity as was stated in 10.23. "Confused and distributive supposition" was described as the interpretation of a general term for each thing it signifies by its imposition, in such manner that a reduction to its singulars may be effected through a conjunctive proposition. Thus the sentence "Every man is an animal" will have the truth condition represented by the conjunctive set, "This man is an animal, and that man is an animal", extended for all men. A term receives distributed supposition only in virtue of the presence of a distributive sign in the sentence; this may be the universal quantifier ~Every', or it may be the negation sign, or some other syncategorematic operator. * The third mode of general supposition, "merely confused", was said to be that which is had by the predicate term of a universal affirmative; e.g., by ~animal' in the proposition "Every man is an animal". Albert of Saxony describes this mode of supposition as follows: Merely confused personal supposition is the interpretation of a term for each thing it signifies by its imposition, ... such that a reduction to its singulars may be effected, in virtue of this interpretation, through a proposition of disjunct predicate, but not through a disjunctive or a conjunctive proposition. . ... It is with this kind of supposition that the term ~ animal' stands, in the proposition "Every man is an animal"; for it follows that if every man is an animal, then every man is either this animal or that animal, in such manner that this whole disjunct ~this or that' is verifiable of this term ~man' taken significatively. ** The distinction between "merely confused" and "determinate supposition" seems to be required only insofar as we are considering the reduction of the predicate term of the universal affirmative to be effected without any corresponding reduction of the distributed subject to its singulars. Thus Ockham says that the term ~ animal', in the sentence "Every man is an animal", cannot be reduced to its singulars by a disjunctive proposition "without variation of

* **

Albert: Logica, II, ch. 5. Cf. Ockham: Summa, I, ch. 70, fo1. 26v. Albert: Logica, II, ch. 4. Cf. Ockham: Summa, I, ch. 70, fo1. 26v.

48

THE THEORY OF TRUTH CONDITIONS

the subject "(nulla variatione lacta ex parte 8ubiecti), because we would then be stating that "if every man is an animal, then every man is this animal, or every man is that animal", etc., which is obviously invalid. If however we reduce the distributed subject to its singular values, then the predicate will have determinate supposition with respect to each singular value of the subject. The scheme of reduction to singulars, for a universal affirmative and with respect to both terms, would be a conjunction of disjunctions, while that of a particular affirmative would be a disjunction of disjunctions. These 8chemata of "reduction to singulars" constitute a metatheoretic method of exhibiting the way in which particular and universal propositions determine, in virtue of their form, two distinct summations of an infinite set of truth conditions. The exhibition of these summations is analogous to the exhibition, effected by the so-called "truth tables", of the summations of the truth values of atomic sentences as determined by the various sentential connectives. The "reduction to singulars" is a method of explicating the function of the individual variable ~x' as index of identity of extensional values of subject and predicate, according to the diverse forms of cross reference determined by different modes of quantification. If we use the predicate variables F and G to represent the general terms occurring in any particular or universal affirmative, we might then use these same letters, with numerical subscripts, to represent singular ostensive uses of these terms for individuals for which they stand. The extension of the expression ~Some F', as occurring in a sentence of the form ~Some F is a G', would then be exhibited by the infinite disjunctive set, F 1 v F 2 V Fa v ..... We could then say that to assert a sentence of the form 'Some F is a G' is equivalent to asserting that at least one member of the disjunctive set, F 1 v F 2 V Fa v , and at least one member of the disjunctive set, G1 v G2 V Ga v , have discrete (i.e., singular) supposition for the same individual. The reduction to singulars, for an indefinite or particular affirmative, yields the following disjunction of disjunctions: (F1 =G1 .V. F 1 =G2 .V. F 1 =Ga.v. ... ) V (F2 =G1 • V. F 2 =G2 • v. F 2 =Ga •v. .... ) v .....

THE THEORY OF TRUTH CONDITIONS

49

It must be emphasized that the general proposition represented by the formula ~Some F is a G' does not itself make the singular statements represented by ~ F 1 = G1', ~ F1 = G2', etc., in disjunction. As occurring· in the general proposition the terms do not have "discrete supposition" and they are not used ostensively for their supposita. "That the general proposition asserts is that there is at least one case in which the ostensive use of the general terms 'F' and 'G', for individuals for which they stand, would Yield a true singular statement of the form x = y, where ~x' is replaced by one of the ostensive uses of ~ F' and ~ y' by one of the ostensive uses of ~ G'. To use the terminology of Principia M athematica, the singular sentences of which the disjunctive sets are composed have "elementary truth", while the general proposition has "second order truth" in the sense that it asserts that there is a condition sufficient to determine elementary truth for at least one sentence of the form x = y in which ~ x' is a singular ostensive use of ~F' and ~y' a singular ostensive use of ~G'.* The "reduction scheme" for the universal affirmative, in which the subject term has "confused and distributive supposition" and the predicate term "merely confused supposition", would be represented by a conjunction of sentences of disjunct predicate, as follows: (F 1 .=. G1 V G2 V G3 V ••• ) • (F2 .=. G1 V G2 V G3 V •••• ) ••••••• We may however achieve a further reduction of this scheme to a conjunction of disjunctive sentences, whereby the "merely confused" supposition of the predicate is reduced to determinate supposition with respect to each singular for which the subject term is distributed. We thus obtain (F1 =G1 .V. F 1 =G2 .v. F 1 =G3 .v.... ). (F 2 =G1 • V. F 2 =G2 • V. F 2 =G3 • V •••• ) ••••••• The function of the universal quantifying prefix ~every' (or ~(xY), in an affirmative proposition, is that of determining the cross references of identification between the individual values of the subject and predicate terms, in the nlanner exhibited by the above reduction scheme. This quantifying function of the universal sign ~ every' is expressible by the use of a conditional operator in the

* cr.

Principia Mathem..atica, Vol. I, p. 45.

50

THE THEORY OF TRUTH CONDITIONS

matrix expression, as is done in the modern formula, (x). Fx-:JGx. But the mediaeval analysis would require a more complex representation, such as would express the requirement of existential import which holds for all affirmative propositions regardless of quantification. An adequate formula for the universal affirmative, as interpreted in mediaeval logic, must conjoin the existential formula "Some F is a G' or "(Ex).Fx.Gx', with the distributive formula "(x).Fx-:JGx'. This is evident from the rule previously stated: "If either the subject or the predicate of an affirmative categorical proposition stands for nothing, that proposition is false". * Since a sentence of the form (x). Fx-:JGx, or its equivalent -(Ex).Fx.-Gx, would be true if its subject term (" F') stands for nothing, it is clear that the formula of the universal affirmative must include the expression (Ex).Fx.Gx, or at least (Ex).Fx. We may therefore express the mediaeval analysis of the universal affirmative by a conjunction of the existential formula with the distributive formula. Every F is a G .:=. (Ex).Fx:(x). Fx-:JGx The use of the sign of material implication ("-:J') in this formulation may be justified insofar as we are considering only propositions of present time. Where however a sentence of the form "Every F is a G' is interpreted as "simply" true, for all values of its terms at all possible times, the conditional connective "-/' would be required. ** In practice the mediaeval logicians assumed the rule that all affirmative categoricals have existential import, as an over-all postulate, without explicitly adding it to the special truth rules for sentences of the four traditional forms. Thus Buridan states: "For the truth of the universal affirmative it is required and it suffices that for each thing for which the subject term stands, the predicate stands". *** If however we add the requirement,

* **

Buridan: Sophismata, ch. 1, Concl. 5. Albert: Logica, I, ch. 6. Cf. infra, IV, § 14, on the distinction between "simple implication" and "implication as of now". *** Buridan: Sophismata, ch. 2, Concl. 13. Since this rule is immediately qualified by excepting the case of propositions which reflect on themselves, it obviously does not state the sufficient condition of the truth of the universal affirmative, even if we make the additional assumption that the

THE THEORY OF TRUTH CONDITIONS

51

common to all affirmative categoricals, that there exist something for which the terms stand, we obtain the following metatheorem expressing the formal condition of truth for the universal affirmative: 11.1

T ~Every F is a G'.:J:(Ex).~ F'8X:(X).~ F'8X.:J.~G'8X

The truth conditions of negative categorical sentences, of particular and universal form, are expressed by negating the truth conditions of the universal affirmative and of the particular affirmative respectively. Since existential import was considered to belong only to affirmative sentences, it is sufficient, for the falsity of an affirmative and hence for the truth of the contradictory negative, that one of the terms stands for nothing. Since -(Ex).~F'8X.:J.-(Ex).~F'8X.~G'8X, and likewise -(Ex). ~G'8x.:J.-(Ex). ~ F'8X.~G'8X,the truth condition of the universal negative is adequately expressed by negating the truth condition of the particular affirmative. 11.2

T

~No

F is a G'

.:J.-(Ex).~F'8X.~G'8X

It is evident that the particular negative, whose condition of truth is the same as the condition of the falsity of the universal affirmative, is not adequately represented by the existential formula, (Ex).Fx.-Gx, by which it is customarily represented in modern expositions. For this formula contradicts only the distributive part of the universal affirmative, and not the conjunction of the existential formula (Ex).Fx with the distributive formula (x). Fx:JGx. The particular negative must, as negation of the universal affirmative, be analyzed as a disjunction of the negations of the two parts of the universal affirmative. * Consequently the formula of the particular negative is not properly represented by terms stand for something. For this reason we state this rule, as well as the other rules of truth for categoricals, as implications and not as equivalences or definitions. * Buridan: Sophismata, ch. 2, ConcI. 14: "omnis particularis negativa vera, ex eo est vera ex quo univerdalis affirmativa sibi contradictoria est falsa". We find constant use, in arguments, of the principle that a universal affirmative has two conditions of its falsity - either the fact that its subject term stands for nothing, or the fact that the predicate term does not stand for everything for which the subject term stands. By the same token, there are two sufficient conditions of the truth of the particular negative.

52

THE THEORY OF TRUTH CONDITIONS

the word-formula ~Some F is not a G~, but only by the formula, ~Not every F is a G~, which is satisfied either because nothing is an F, or because something is an F which is not a G. The metatheoreIn stating the adequate truth condition of a sentence of the form ~Not every F is a G~ is therefore the following: 11.3

T

~Not

every F is a

G~:):-(Ex).~F~8X:V:(Ex) .~F~sx.-rG~sx)

The metatheorem stating the truth condition of the particular affirmative, ~Some F is a G~, has already been given in 9.2. In all these four cases, the mediaeval statements of truth conditions determine formulas analyzing the import of these forms of statement, which may be expressed symbolically by replacing the expressions "~F~sx" and "~G~sx". occurring in the metatheoretical descriptions of truth conditions, by the corresponding propositional functions ~Fx~ and ~Gx~. We may exhibit the "square of opposition", with the conditions regarding existential import explicitly expressed, by means of such formulas. A (Ex).Fx:(x).F£JGx I (Ex). Fx.Gx

E -(Ex). Fx.Gx

o -(Ex).Fx:v:(Ex).Fx.-Gx

It will be seen that with the square constructed from the above formulas, the traditional relations of contradiction, contrariety, subcontrariety, and subalternation all hold. That is, A and 0 are contradictories, as are E and I; A and E may both be false, but not both true; I and 0 may be both true, but not both false; and A implies I, and E implies 0, though the converse implications do not hold formally. Our consideration has so far been confined to assertoric propositions of present time, as if a sentence of the form (Ex).Fx.Gx should be read, "For some x, x is now an F and x is now a G". Such is the mediaeval meaning when the copula is the verb of present time, and the terms such that they do not "ampliate" supposition beyond present time by reason of implicit verbal force, as when the predicate is a past participle. But if the copulas of all the four forms, A, E, I and 0, are in the past tense, or in the future tense, or in the mode of possibility, the relations of opposition and subalternation will still hold. With these extensions

THE THEORY OF TRUTH CONDITIONS

53

of the existential import of the copula, to past and future time and to "that which can exist", we shall now be concerned. *

§ 12.

TIME RANGE AND :MODALITY

The apparently simple gramnlatical forms of ordinary language are notoriously complex in their logical structure. On this account mediaeval logic, which attempted to formulate the logical syntax of the Latin language, contained a tremendous array of rules and techniques for making explicit the complex truth conditions of sentences whose grammatical form is apparently simple. We shall here consider only one type of problem presented by the sentences of ordinary language, that of time range and modality as determined by the tense or modal connotation of the verbal copula. The usual statement, that a sentence is true if things are as the sentence states them to be, cannot be conceded in literal manner except for sentences of present time. If the sentence is of past time, it is true only if things were as it states them to have been; and if of future time, it is true only if things will be as it says they will be. Finally, if the sentence has, as copula, the verbal expression "can be" (or "possibly is"), the sentence is true only if things can be as it states that they can be. ** On account of the

* A. Becker, Die Aristotelische Theorie der Moglichkeitsschlil8se, Berlin 1933, p. 18, Note 4, holds that it is impossible to "save" the Aristotelian square of opposition except by making the "existence postulate" an overall assumption of the system. He seems to imply that this existential requirement cannot be represented in the formulas of the propositions, as we have done. Lewis: Langford, pp. 275-281, also argue at length to show that it is impossible, even on the assumption of existential import, to construct the traditional square so that all the relations will hold. But their arguments rest on the assumption, not made in mediaeval logic, that existential import is to be assigned to the negative forms as well as the affirmative ones, with the result that they represent 0 only by the existential formula (Ex).Fx. -Gx, which of course contradicts only the distributive part of the formula for A and fails to contradict its existential part. Yet it should beobviousthatifweexpressAasaconjunctiveof(Ex).Fxand-(Ex) .Fx.-Gx, we must express its contradictory 0 by a disjunctive composed of the negations of the two parts of A. And if this is done, the validity of the square is preserved. ** Buridan: Oonsequentiae, I, ch. 1: "aliqui ponunt ex eo omnem propositionem veram esse veram quia qualitercumque ipsa significat ita est in re significata vel in rebus significatis; et ego credo hec non esse vera

54

THE THEORY OF TRUTH CONDITIONS

multiple truth conditions of sentences containing explicitly or implicitly teInporal or modal operators, special rules for such sentences were formulated, whereby the general metatheorems expressing the truth conditions of affirmative and negative categorical propositions were applied to these special forms of sentence. When the verbal copula is of past or future tense, or when it is in the mode of possibility (equivalent to "can be"), it is said to extend or to "ampliate" the supposition of the subject term to things not existing in the present time of the statement. It does not however remove the supposition of the subject term for presently existing things; the extension of time range is by logical addition, or in disjunction with present time. The temporal import of the sentence "Some man will be dead", for example, would be expressed in this manner: "For some x, x is a man or x will be a man, and x will be dead". * A copula of past or future tense, or of possibility, does not operate on the supposition of the predicate term in this same way. It was said that the predicate has "appellation" according to the tense or mode of the verbal copula, whereby the condition signified by the predicate is posited as verifiable, in the time or mode connoted by the verbal copula, of that for which the subject term stands. ** What is meant may best be explained by an example. In the sentence "SOlnething white will be black", the predicate ~black' has appellation according to the tense of the verb in this de virtute sermonis, quia si equus Colini est mortuus qui bene ambulavit, hec est vera 'equus Colini bene ambulavit', et non est in re sicut ista propositio significat quia res corrupta est ..... sed bene hec propositio ideo est vera quia ita fuit in re sicut propositio significat fuisse. ... Similiter hec est vera, 'aliquid quod numquam erit potest esse', non quia sic est sicut propositio significat, sed quia sic potest esse sicut ipsa significat posse esse; et sic patet quod secundum diversa genera propositionum oportet diversimode causas (veritatis) earum assignare". * Albert: Logica, II, ch. 10. ** Albert: Logica, II, ch. 11: "Appellatio est proprietas predicati; solemus enim dicere predicatum appellare suam formam in ordine ad verbum quod est copula illius proposition:is. Unde 'predicatum appellare suam formam' est ipsum sub eadem forma, vel sub eadem voce si sit terminus vocalis, sub quo predicatur in propositione in qua appellat suam formam, esse verificabile in propositione de presenti de pronomine demonstrante illud pro quo supponit subiectum propositionis cuius est pars".

THE THEORY OF TRUTH CONDITIONS

55

sense: if the sentence is true, there will be some future time in which a demonstrative sentence of present time, "This is black" would, if stated at that future time, be true of something for which the subject term of the original sentence, ~white', stands. But it is not required that the subject term, ~white', be verifiable of the thing which will be black in the same future time in which the predicate ~black' will be verifiable of it; it is sufficient that something which is now white, or something which will be white, will be black. * Thus the copula of future time is said to "ampliate" or to extend the supposition of the subject term to the future as well as the present, whereas it transfers the supposition of the predicate term fronl the present to the future. Verbal copulas of past time, or of possibility, operate correspondingly. To represent the rules of ampliation, we require special operators to indicate past time, future time, and possible time (or supposition for things whose existence is possible). We shall use the symbol ~ 0' as indicator of supposition for what can exist, and we will introduce the two symbols, ~ >' and ~ <', as indicators of supposition for what has existed and for what will exist, respectively. In order to avoid separate quantification of subject and predicate terms, such as would involve use of two individual variables and a matrix expression asserting their identity, we shall prefix our temporal and modal operators to the matrix expressions ~ Fx' or ~Gx', rather than to the quantifying prefixes ~(Ex)' or ~(x)'. Thus, to represent the sentence "Something white will be black". we will write the formula: ~(Ex):Fx v
* 'Verifiable', in this use, has the same meaning as t stands for'. A term •F' is said to stand for all the individuals whose existence suffices, has sufficed, will suffice, or can suffice for the truth of a demonstrative sentence of the form "This is an F", accompanied by an act of pointing, and stated in whatever time (present, past, future, or possible) in which those individuals have existed, do exist, will exist, or can exist. Since such a demonstrative sentence has not been, is not, and will not be actually stated, as a general rule, the thorny problem of "contra-factual conditionals" is involved in this definition.

56

THE THEORY OF TRUTH CONDITIONS

sentences with copulas of past tense, future tense, or of possibility. * 12.1

Every term having supposition, as subject, with respect to a verb of past time, is ampliated to stand for that which exists or for that which has existed. Some F was a G::=:::(Ex):Fx v >Fx.>Gx Every F'wasaG: : : :(Ex) :Fx v> Fx :.(x) :Fx v> Fx.""J. >Gx

12.2

Every term having supposition, as subject, with respect to a verb of future time, is ampliated to stand for that which exists or for that which will exist. Some F will be a G::-::(Ex):Fx v
] 2.3

Every term having supposition, as subject, with respect to this verb ~ can be', is ampliated to stand for that which exists or for that which can exist. Some F can be a G: :-: :(Ex) :Fx v OFx.<:>Gx Every F can be a G: :=: : :(Ex) :Fx v OFx :.(x) :Fx v OFx.""J.OGx

It is evident that the quantifYing prefixes, ~(Ex)' and ~(xr, have no temporal import of themselves, and that they apply to every possible individual - i.e., to every x such that, for some F, OFx. Rules analogous to those given above are stated for sentences which, though having a copula of present time, have temporal or modal force by reason of the sYntactical form of the predicate. Thus, if the predicate is a past participle, or a future participle, or if it has an ending C.. . able') indicating potentiality, the sentence will be equivalent to one of past tense, future tense, or to one of possibility. We may note two rules, however, which are of interest because of their bearing on the mediaeval theory of existential import. 12.4

All verbs, even when they are in the present tense, which are such that they have the power of being transitive with respect to future, past, or possible things as well as present things, are ampliative of terms for every time present, past, future, or possible.

* The statements of the rules are paraphrased from Albert: Logica, II, ch. 10.

57

THE THEORY OF TRUTH CONDITIONS

The verbs which have this property include such verbs as ~to lIDderstand', ~to know', ~to be acquainted with' (cognoscere) , ~to :lignify', and ~to stand for' (supponere). The reason of this, Albert )f Saxony explains, is that a thing can be understood or thought )f (intelligi) without reference to one time rather than to any )ther time - Le., in abstraction from a time or a place. Hence, when a thing is thus understood or conceived, the act of understandlug terminates in that thing just as well, when the thing is something which has existed, or will exist, or can exist, as it does when the bhing exists at the same time as that act of understanding it. Albert contrasts verbs of this special kind with other transitive verbs which require that their objects exist in the same time as the action indicated by the verb. Thus it follows, "If I am now eating bread, bread now exists". But it does not follow, "If I am thinking of a rose, a rose now exists". Similarly, if these verbs are in the passive voice, they ampliate the subject term to stand for all possible instances, as when it is said, "A rose is being thought of". Of particular interest is the inclusion of the verbs ~ signify' and ~stand for' in this special class. Albert says: "When we say ~This term stands for something', the word ~ something' is ampliated for that which exists, has existed, will exist, or can exist - or for that which can be understood. Consequently this consequence is not valid: ~This term stands for something, therefore it stands for something which exists"'. * It seems evident, from this rule, that the rule expressed in the metatheorems 9.22 and 9.23, that "if either the subject or the predicate of an affirmative categorical proposition stands for nothing, that proposition is false", must be given an extremely broad interpretation. As long as the term stands for what can be understood to be possible, or for something whose existence as an individual is conceivable without contradiction, the term may be said to "stand for something". Thus "existential import" has as many meanings as there are temporal and modal senses of the affirmative copula; propositions of past, present, or future time have existential import in a factual, empirical, and historical sense, while modal propositions have "existential

*

Albert: Logica, II, ch. 10. 5

58

THE THEORY OF TRUTH CONDITIONS

import" in a sense which, though perhaps not intended to be merely logical, is at least commensurate with what we generally call the "logically possible". Thus Buridan states that a term may have supposition for things which never have existed, do not exist, and never will exist, as long as it would not be impossible for such a thing to exist. * One further rule of ampliation, stated by Albert of Saxony, makes it clear that the necessary propositions of the sciences, as understood by the logicians of the 14th century, have "existential import" only in the Pickwickian sense of requiring that the terms be interpretable for "possible things". 12.5

The subject term of any proposition whose copula is determined by the modal term ~necessary', is ampliated to stand for that which exists or for that which can be. **

Albert gives this example: "Every creating thing is necessarily God". This, he says, is equivalent to, "Everything which is creating or which can create, is necessarily God", in which the subject term ~ creating thing' is ampliated to stand for what is creating or for what can be creating. That the subject term is thus ampliated is shown, he says, by the fact that the sentence "Every creating thing is necessarily God" is equivalent to "No creating thing can not be God" (Nullum crean8 pote8t non e88e Deu8). Since this form has the verb ~ can be' as copula, its subject is ampliated to stand for that which exists or for that which can exist, by 12.3.*** There is a difference between a sentence of the form, "Every F can be a G", and a sentence of the form "Every F is necessarily a G". This difference does not lie in the supposition of the subject term, which in both cases is for that which exists or which can exist. It lies in the way in which the modal operator applies to the predicate. If we say, "Something white can be black", our

* Buridan : Oonsequentiae, I, ch. 1: "This proposition is true: 'Something which will not be, can be'''. ** Albert: Logica, II, ch. 10, Rule 9: "Cuiuslibet propositionis de necessario in sensu diviso, subiectum ampliatur ad supponendum pro eo quod est vel potest esse". A "propositio de necessario in sensu diviso" is a sentence whose copula is adverbially modified by the word 'necessarily', as in the sentence "Every man is necessarily an animal". *** Albert: Logica, II, ch. 10.

THE THEORY OF TRUTH CONDITIONS

59

[leaning is this: "Something is or can be white, and it is possible Jr that thing to be black". But if we say, "Something white is .ecessarily black", the meaning is, "Something is or can be white, nd it is impossible for that thing not to be black". The formula f the first sentence is the one given under 12.3; the formulas of ffirmative sentences, whose copula is determined by the modal Brm ~necessarily', are as follows: ome F is necessarily a G: :==: :(Ex) :Fx v OFx.-O-Gx :very F is necessarily a G: :==: :(Ex):Fx v OFx:.-(Ex):O(Fx.-Gx) The second part of the formula of the universal proposition of ecessity, ~ -(Ex) :O(Fx.-Gx)', is equivalent to the formula, x) :-O(Fx.-Gx)', which in turn may be represented by the lediaeval operator for a necessary or unqualified (simplex) conditinal, as follows: ~(x):Fx.-j.Gx'. Thus a "scientific" proposition, :ttisfYing the Aristotelian requirements of universality, necessity, nd essential truth, is analyzed by the 14th century logicians as guivalent to a "necessary proposition concerning that which ossibly exists", or as equivalent to a consequence or conditional alid for all possible values of the terms, but not requiring that lch values actually exist. Ockham is quite explicit on this point, 1 his discussion of the requirements of the premises of demonjration as set forth in the Posterior Analytics; after stating that sentence in which a definiens is predicated of its definiendum lch as "Man is a rational animal", is contingent and not necessary, , construed as an assertoric proposition of present time, he )ntinues: J

And therefore I say that no such proposition can be a principle r conclusion of demonstration. Nevertheless it is to be said that Lany propositions composed of such terms can be principles or )nclusions of demonstration, because conditional propositions, or ropositions equivalent to such, can be necessary. Thus this mtence is necessary: ~If a man exists, an animal exists'; and this, f a man laughs, an animal laughs'; and this, ~Every man can ,ugh', where the subject is taken for things which can exist. And ropositions equivalent to these are in the same sense necessary. * As we shall see in examining the mediaeval theory of consequence,

*

Ockham : Summa, III, 2, ch. 5; fo1. 64v. Cf. Moody :Ockham, pp. 228-233~

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THE THEORY OF TRUTH CONDITIONS

a distinction was drawn between two kinds of necessity and two kinds of impossibility. Both kinds are of the logical sort, being determined by rules of usage of language. One kind, determined by rules or laws of formal logic which hold for all sentences regardless of their "matter" (or terms), yields the rules of formal consequences, including the rules of syllogism. The other kind of necessity and impossibility is determined by "semantic" rules specifying fully determinate predicative relations between terms, such as are embodied in definitions, or in specifications of relations among given terms as <"genus', <"species', or <"property'. These specifications of determinate predicative relations among terms "essentially" related (in contrast to those "accidentally" related) constitute the basis for many of the rules of "simple" material consequences. * Such consequences are true and necessary by reason of the "matter" of the atomic sentences of which they are compos~d; that is, by reason of semantically determined (or "essential") relations of identity or diversity between what the terms can stand for in virtue of their imposition. The examples of necessary conditionals, given by Ockham, are material consequences whose necessity is determined intensionally by the "matter" of the component sentences. Albert of Saxony devotes a chapter of his Logica to a treatsmen of the "matter of propositions", which is of interest becaue itt throws some light on the question of how the mediaeval theory of "material consequences" is to be interpreted, and also because it reveals how the 14th century logicians understood the term <"necessary' as applied to the propositions of which the Aristotelian "demonstrative sciences" are composed. ** Since the four forms

* Cf. infra, § 14. Material consequences, or conditional sentences which are true, not by reason of their syntactical form alone, but by reason of their terms or "matter", were divided into "simple material consequences", valid for all possible values of the terms occurring in their component propositions, and "consequences as of now", valid for the values of the terms existing at the time the consequence is stated. While "simple material consequences" are necessary without qualification, "consequences as of now" are "contingently necessary" in that things happen to be such, at the time of statement, that for that time it is impossible for the antecedent to be true and the consequent false. ** Albert: Logica, III, ch. 10. The discussion of the "matter" of propositions, which follows, is all based on this chapter of Albert's work.

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61

of sentence constituting the "square of opposition" are generated only by sYntactical operators of quantification or negation, any such square will be composed of sentences whose "matter", or terms, are the same. But the relations of opposition or subalternation among the members of such a square are fully determinate when the sentences are of semantically determinate "matter", whereas some of these relations are indeterminate, if the sentences are of semantically indeterminate (or "contingent") matter. Those propositions are said to be of natural matter, which are such that the predicate signifies the same that the subject signifies, and cannot be truly denied of that subject; or they are propositions in which the more universal is predicated of a less universal term included under it, or a definition of its definiendum, or a part of the definition is predicated of the term defined, or in which a term is predicated of itself. Other propositions are said to be of contingent matter, whose predicate can be predicated of its subject either affirmatively or negatively, in contingent manner. But a proposition is said to be of remote matter, whose predicate cannot be (truly) predicated of its subject at all. An example of the first type is, 'Man is an animal'; an example of the second is, ~Man runs'; an example of the third is, ~Man is an ass'. * Given a sentence in which a term ~B' is affirmed of a term that sentence is of "natural matter" if the sentence ~-(Ex):O(Ax.-Bx)' is true; it is of "remote matter" if the sentence ~ -(Ex) :O(Ax.Bx)' is true; and it is of "contingent matter" if neither of the above sentences is true. While a sentence of natural matter need not itself be necessary, it will stand in a relation of subalternation, contradiction, or contrariety to a sentence which is necessary; and a sentence of remote matter will be thus related to a sentence which is impossible. The distinction between sentences of "natural" or "remote" matter, on the one hand, and those of "contingent matter", on the other, corresponds to the distinction made by Carnap between semantically determinate ("analytic") sentences, and semantically indeterminate or "factual" sentences. ** Having made these distinctions, Albert states several rules which may be summarized as follows: ~ A' ,

12.6

* **

Every necessary proposition, and every impossible proAlbert: Logica, III, ch. 10. Carnap: Foundations, pp. 12-13.

62

12.7

THE THEORY OF TRUTH CONDITIONS

position, is in either natural or remote matter; i.e., semantically determinate. Every proposition which is neither necessary nor impossible is in contingent matter; i.e., it is semantically indeterminate.

Albert points out that the question of whether a proposition is true or false is quite independent of the question of whether it is in natural or remote or contingent matter. "Some man is not an ass", which is true, is of the same matter as "Every lnan is an ass", which is false. It should also be noted that although every necessary categorical proposition is of "natural" or "remote" matter, the converse does not hold. "Every man is an animal", if stated as a proposition of present time, is not necessary, since it is not necessary that there exist any men at the present time; yet it is a sentence of "natural" matter, since the terms ~man' and ~animar are so related, by reason of their imposition as signs, that the sentence: ~-(Ex) :O(Ax.-Bx)' is true (where ~A' and ~B' are abbreviations for ~lnan' and ~animar respectively). On the other hand, any proposition of "natural" or "remote" matter, whose terms have supposition for "that which can be", will be either necessary or impossible, regardless of whether it is universal or particular in quantification. It follows from this that among sentences whose matter is semantically determinate, the particular affirmative implies the universal affirmative, and the particular negative implies the universal negative. Similarly, among such sentences contraries (A and E) are equivalent to contradictories, and can be neither both true nor both false. Again, among propositions of "natural" or "remote" matter, the subcontraries (1 and 0) can be neither both true nor both false, so that they also are equivalent to contradictories. These relations will hold, among sentences of semantically determinate matter, even if they have temporal existential import, so long as this existential import is of the same type in all four members of the square. For if -(Ex) :O(Ax.-Bx), then (Ex):Ax.Bx::::::>::(Ex):Ax:.(x):Ax.::::>.Bx; i.e., the distributive part of the universal a,ffirmative follows by reason of the semantically determinate "matter", and the existential part follows immediately from the particular affirmative in the same sense of "present existence".

THE THEORY OF TRUTH CONDITIONS

63

Since "scientific propositions", as understood in the Aristotelian tradition, are composed always of terms "essentially" related, the matter of such propositions is always semantically determinate - i.e., either "natural" or "remote" matter. And since such propositions were held to be necessary, their terms were construed to stand for "what can exist" in disjunction with what actually exists, so that the semantically determinate possibilities of supposition of the terms constitute a sufficient criterion for determining the necessity of such propositions and the impossibility of their contradictories or contraries. This does not imply, by any means, that the conventions of semantically determinate relations among the terms involved in the Aristotelian sciences were established in purely arbitrary manner and without guidance from empirical sources. But it does mean that the language of science, in this mediaeval Aristotelian tradition, was understood to be a constructed language in which the "principles" of each "subject matter" are, in a very definite sense, "laid down" or stipulated.

IV THE THEORY OF CONSEQUENCE § 13.

THE MEANING OF ~CONSEQUENCE' IN MEDIAEVAL LOGIC

The notion of ~ consequence', in its generic meaning, is the fundamental conception of formal logic. * This was as true of ancient and mediaeval logic as it is of contemporary systems. The laws of syllogism are laws of consequence, in the generic sense of logically valid discourse, quite as much as are the laws of sentential calculus expressed with propositional variables. The term ~conse­ quence', derived in Latin from the verb ~to follow' (sequi or consequi), occurs in Cicero and Boethius, and the corresponding Greek words are found fairly often in the writings of the later Greek logicians. The generic meaning of the term is sufficiently indicated, in a material sense, by the statement that discourse is "consequential" or logically valid, if it is such that it does not derive a false conclusion (or consequent) from a true premise (or antecedent). ** In the mediaeval Latin tradition this generic conception of consequence or of ~following' was of course retained; but the term ~ consequence' came to be used technically to designate sentences of conditional form, such as are true or necessary, or at least such as "claim" by their form to be true for necessary. *** The later mediaeval logicians tended to regard all forms of valid deduction,

*

Cf. Carnap: Logical Syntax, p. 168: "If for any language the term is established, then everything that is said concerning the logical connections within this language is thereby determined". ** Cf. Sextus Empiricus, Against the Logicians, II, 113 (Loeb Library ed., London 1935, p. 297); and Aristotle, Anal. Pro II, 2, 53a 6-8. *** Cf. Boethius, De syllogismo hypothetico, I (J. P. Migne, Patrologia Latina Vol. 64, cl. 843) : "Hypothetical propositions are not usually enunciated through necessity or contingency, but those which signify assertorically are chiefly used in ordinary speech. But all however claim (tenere volunt) a necessary consequence". Cf. Sextus Empiricus, loco cit., 111, who states -that the conditional "promises" that the second of two things follows consequentially on the first. ~consequence'

THE THEORY OF CONSEQUENCE

65

including the syllogism, as forms of ~ consequence' and therefore as equivalent to conditional propositions. In this way the entire theory of deduction was organized as a development of the rules governing the validity of conditional sentences. The mediaeval theory of consequence shows many resemblances to the doctrines of the Stoic and Megaric logicians of antiquity, as these have been described by Sextus Empiricus and Diogenes Laertius. While these Greek antecedents were not directly known to the scholastics, they were known to Boethius, whose discussions of the conditional proposition and of hypothetical syllogisms constituted the principal historical source of the mediaeval doctrines. We shall not here attempt to trace the historical development of the mediaeval theory, or to identify its sources. We shall, instead, expound and analyze the theory of consequence as presented in a few treatises devoted to this subject, written in the 14th century. The most complete and fully developed treatise of this nature, accessible to us, is the De consequentiis of Jean Buridan, the content of which is very faithfully echoed by Albert of Saxony in Part IV of his Logica. While other 14th century logicians did not on all points agree with Buridan and Albert, the points of disagreement were in most cases taken into account by these men, so that it will be convenient to base our own exposition on the work of Buridan, referring where necessary to other authors in order to clarify obscure questions or to exhibit issues which were in controversy. * We shall first examine the general definition of the term ~ consequence', and the distinctions of kinds of consequence made by the mediaeval logicians. Subsequently we shall state some of the principal rules of consequence and exhibit their relation

* In addition to the Logica of Albert of Saxony, and to the Consequentiae of Buridan, we have employed the following texts in supplementary manner: Ockham's Summa logicae, the Tractatus de consequentiis of Ralph Strodus, the Logica of Paul of Venice, the De puritate artis logicae of Walter Burleigh, and the discussions of •consequence' found in the Quaestiones in Librum Primum Priorum Analyticorum Aristotelis contained in the Opera omnia of Duns Scotus (Paris, 1891,) Vol. 2, which, being almost certainly not by Duns Scotus, will be cited as •Pseudo-Scotus'. The discussions of the "pseudoScotus" have been analyzed by I. M. Bochenski, "De consequentiis scholasticorum earumque origine", Angelicum 15, 1938, pp. 92-109. For full titles of the above works, see the Bibliography.

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THE THEORY OF CONSEQUENCE

to corresponding theorems of the modern sentential calculus. Only a few of our authors take pains to formulate adequate and exact definitions of 'consequence'. Ockham, for example, says that a consequence is a conditional proposition, and that it is true if and only if its antecedent implies (infert) its consequent. * Ralph Strodus and Paul of Venice briefly describe a consequence as "the illation of a consequent from an antecedent". ** The meaning given by these men to the word 'implies', in this context, is indicated by statements to the effect that a sentence 'p' implies a sentence 'q' if the contradictory opposite of 'q' is "repugnant" to 'p' , or if the conjunctive' p. -q' is impossible. Ockham distinguishes between absolute or "Wlqualified" consequences, valid only if 'p.-q' is impossible, and "consequences as of now" (ut nunc) which are valid if, at the time the consequence is stated, 'p.-q' is false, even though there could be some other time when this same conjunction of sentences might be true. *** For more exact definitions of the term 'consequence', we may turn to Buridan and to the pseudo-Scotus. A consequence is a hypothetical proposition, since it is formed from several propositions by means of the connective 'if', or by this word 'therefore', or an equivalent. These words indicate that, of the propositions connected by them, one follows on the other. They differ in this, that the word 'if' indicates that the proposition immediately following it is the antecedent, and the other the consequent, whereas this word 'therefore' indicates the contrary. t This is a formal description of the logical function of the connectives 'if', or 'therefore', in relation to the language expressions (sentences) which they cannect. It is analogous to the description given of the 10gica1 function of the copula of categorical propositions, as indicating that the language expressions (terms) preceding it

* Ockham: Summa, II, ch. 31, fo1. 43r. ** Strodus: Gonsequentiae, 2r; Paul of Venice: *** Ockham: Summa, III, 3, ch. 37, fo1. 92r;

Logica, III, ch. 1, 31v. Strodus: Oonsequentiae, 2r-v; Paul of Venice: Logica, I, ch. 12, 8v-9r. Strodus and Paul of Venice appear to recognize only the absolute or "simple" consequences, whereas Ockham, Buridan, Albert of Saxony, and the pseudo.Scotus also recognize the "as of now" consequences. t Buridan: Gonsequentiae, I, ch.3. Albert: Logica, IV, ch. 1, gives the same description.

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and following it "stand for the same". The predicates involved in these descriptions, namely ~implies~ (or ~is antecedent to~) and ~ stand for the same~, are binary predicates of second intention which take as arguments the names of the language expressions connected by the logical signs ~if~ and ~is~, respectively. If we use "~p~" and "~q~" (enclosed in single quotation marks) to represent the names of the component sentences of the conditional, with "~p. --1.q~" (enclosed in single quotation marks) representing the name of the conditional formed from these sentences by means of the connective ~if', we may restate Buridan's description as a metatheorem specifYing the meaning of the predicate ~true~ as applied to the name of any sentence of conditional form: 13.1

T

CP.--1.q'):==:~p~

implies ~q'

It should be enlphasized that the sentence "~p~ implies ~q~" is not stated to be equivalent, either in extension or intension, to the sentence ~ p. --1.q~; it is stated to be equivalent to a sentence whose subject term is the name of such a sentence, and whose predicate term is the word ~true'. The word ~implies~ cannot be substituted, significantly, for the word ~if~; it describes what ~if' does, but it does not do what ~if~ does. The next problem is to define, in a formal way, the meaning of the logical predicate ~implies~. Buridan does this as follows: r.nI2'o1.iC1""W'>~~i~

One proposition is antecedent to another, if it is so related to that other that, both propositions being stated, it is impossible that whatever the first signifies to be so, is so, and that whatever the second signifies to be so, is not so. * It will be noticed that this definition sets forth the requirement that the sentences exist, or that they are stated. This is one of the conditions which must be satisfied in order that the relation of implication holds; for implication is a relation between sentences, and consequently there must be sentences to be so related. The

* Buridan, loco cit. Practically the same definition is given by Albert: Logica, IV, ch. 1: "propositio illa dicitur antecedens ad aliam, quae sic se habet ad earn quod impossibile est, qualitercumque est significabile per earn, stante impositione terminorum, sic esse, quin qualitercumque alia significet, ita sit." The use of the term "whatever" (qualitercumque) in these definitions is of importance in connection with the resolution of the liar paradox, as will be seen infra, V.

68

THE THEORY OF CONSEQUENCE

rest of the definition sets forth the other condition which must be satisfied if a sentence ~ p' implies a sentence ~ q'. Buridan is careful not to say that this condition is satisfied if it is impossible that the sentence ~ p' is true and the sentence ~ q' false; he says, rather, that this condition is satisfied if it is impossible that what the sentence ~ p' states to be the case, is the case, without that which is stated to be the case by the sentence ~q' being the case. The impossiblity attaches to the conjunctive ~p.-q', and not to the conjunctive ~(T~p'.-T~q'r. It is in this latter manner, however, that the pseudo-Scotus defines the relation of consequence or implication: A consequence is a hypothetical proposition composed of an antecedent and a consequent, connected by the conditional or rational connective, which denotes that it is impossible, the antecedent and consequent being stated, that the antecedent is true and the consequent false. And then, if the case is as this connective denotes, the consequence is valid; and if not, then the consequence is invalid. * Buridan's objection to the formulation given by the pseudoScotus rests on a distinction which becomes evident in the case of sentences which "reflect on themselves". A sentence such as this, "No sentence is negative", states what can be the case, though such a sentence, if stated, cannot be true. Let the letter ~8' be an abbreviation for the sentence, "No sentence is negative". Then Buridan would concede this proposition: "~8' is possible", while denYing this proposition: "It is possible that ~8' is true". That ~8' is possible is evident, because it is not impossible that at some time no sentences· exist, either in books or as uttered by speaking beings; and hence the sentence ~8' states what might at some time be the case, even though it could not be the case at the time ~ l exists or is stated. But for ~ 8' to be true, ~ 8' must exist, and since ~ 8' is a negative sentence, its own existence determines that it cannot be true. ** Apart from this consideration of sentences which reflect on themselves, Buridan's formulation would appear to be equivalent

* **

Pseudo-Scotus, Qu. X, pp. 104-105. Buridan: Consequentiae I, ch. 3; Buridan: Sophismata, ch. 8, 2nd sophism; Albert: Logica, IV, ch. 1.

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69

to that of the pseudo-Scotus, since Buridan says, at the beginning of his treatise on Consequence, that a proposition is called true if what it states to be so, is so, and that it is called impossible if what it states to be so, cannot be so. If then the existence of the sentence itself does not interfere with the satisfaction of its truth condition, the statement "~p.-q' is impossible" is equivalent to the statement "'(T~p'.-T~q')' is impossible".* To formulate Buridan's definition of implication, we require a symbol expressing the relation which holds between the name of a sentence and the sentence of which it is the same. Since the name of a sentence is a term, it may be said to signify, 'or stand for, the sentence of which it is the name; we shall therefore use the subscript ~ s', as abbreviation for "stands for", to express this relation. If then ~ p' is the name of a sentence p, we shall write: ~ p' s p. The relation is between a sentence mentioned, and that same sentence used. Buridan's definition may then be formulated as follows: 13.2

~p'

implies

~q'.=Df::(Ex,y):~p'sx.x=p:~q'8y.y=q:-O(p.-q)

Since, by 13.1, ~p' implies ~q' .:=. TCp--1q'), the above definition of implication is equivalent to a formal definition of the truth condition of any sentence of conditional form. The existential quantification involved in the definition is required because the term defined, ~implies', is a logical predicate significantly affirmed only of the names of sentences. There must exist, then, sentences for which these names stand. If we assume that there are sentences, p and q, for which the names ~p' and ~q' stand, the definition may be abbreviated as follows: ~p' implies ~q' .:=. -O(p.-q). More properly, however, we should write the equivalence, P--1q·:=·- O(p.-q), in which the sentences ~ p' and ~ q' are used, without being mentioned, on both sides. In this second form, also, the equivalence is valid without the assumption of the existence of the sentences, since no reference is made to them.

* The pseudo-Scotus is not unconscious of the difficulty arising, for his definition of consequence, when one or both components have reflection on themselves; hence he states his definition as valid for all consequences, except where the existence of one of the propositions is repugnant to the truth condition denoted by the conditional operator. Cf. Pseudo-ScotUB, Qu. X, p. 104.

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THE THEORY OF CONSEQUENCE

Our next problem is that of determining the import of the modal term 'impossible', or of the expression '-O(p.-q)', included in this definition of implication. At first sight it would seem that Buridan is defining the relation of implication in exactly the sense of "strict implication" as understood in the system of Lewis and Langford. But this conclusion is only warranted if the term 'impossible' is understood in the same sense. Lewis and Langford interpret the symbol' 0' in the broadest sense, of "absolute" or "logical" possibility, so that its negation, '-0', is taken in the strictest sense of absolute or logical impossibility. This meaning is undoubtedly included in the mediaeval lneanings of the terms 'possible' and 'impossible', but the mediaeval logicians used these terms in a "material" sense as well as in a formal, or strictly logical, sense. To clarify these meanings we must examine the distinctions made in mediaeval logic among different kinds of consequences, and among correspondingly different senses of the expression '-O(p.-q)'.

§ 14. FORMAL AND MATERIAL CONSEQUENCES Consequences, or true conditionals, were distinguished into formal and material consequences. The latter were distinguished in turn into "simply valid" consequences, and consequences "valid as of now". We shall first consider the general distinction between formal and Inaterial consequences, and indicate the kinds of formal consequences recognized. Then we shall examine the notion of material consequence, and the distinction between those which are "simply" valid and those valid "as of now". A consequence is called formal if it holds in all terms while retaining the same form; or, if you wish to use exact language, a formal consequence is one such that every sentence of the same form, if stated, is a valid consequence. .... But a material consequence is one such that not every sentence of similar form is a valid consequence, or, as is commonly said, one which does not hold in all terms while retaining the same form. For example, 'If some man runs, then some animal runs' is not valid in these terms, 'If some horse walks, then some wood walks'. * Buridan's definition of formal consequence presents no difficulty; all conditional sentences which are "logically true" on sYntactical

*

Buridan: Oonsequentiae, I, ch. 4. Cf. Albert: Logica, IV, ch. 1.

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grounds, or which are true for any transformations of the categorematic terms occurring in the sentence, are valid by their form (bona de forma). Some of the English logicians, such as Ralph Strodus, introduced epistemic considerations into their definition of formal consequence, saYing that a consequence is formal if the consequent is known through the antecedent - if it is de intellectu antecedentis. * But this psychological emphasis is not found in most of the authors, such as Ockham, Buridan, Albert of Saxony, or the pseudo-Scotus. At the beginning of the section of his treatise devoted to syllogistic consequences, Buridan gives a brief review of the types of consequence valid on formal grounds alone. One class is composed of consequences from one categorical proposition to another, where the two propositions share the same terms; these include the traditional equipollences, conversions, subalternations, and 0 bversions. Like the syllogism, such consequences belong to the logic of terms, holding in virtue of the internal formal structure of the component sentences, as determining the suppositional relations among the terms. Four different types of formal consequence, belonging to the logic of propositions, are then described by Buridan. The first of these is determined by the sentential connectives of conjunction and disjunction, whereby a conjunctive sentence implies each of its components, and a disjunctive sentence is implied by each of its components. These consequences may be represented by the following theorems: 14.1

pq. -i.p

14.11

pq.-i.q

14.2

p. -i.p v q

14.21

q. -i.p v q

Another type of formal consequence, also pertaining to the logic of propositions, is said to contain an those consequences

* Strodus: Consequentiae, 2v. Cf. Paul of Venice: Logica, 31v-32r, who also gives a psychological definition, saying that a valid formal consequence is one whose antecedent "cannot be imagined" to be true without the consequent being true.

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generated by apposition of some sentence to a conditional sentence. Buridan mentions three such consequences, represented by the following laws: 14.3

p'P-1q: -1.q

14.31

-q'P-1q:-1'-P p-1q.q-1 r :-1'P-1r

14.32

This type of formal consequence, which includes all those forms of hypothetical syllogism in which one of the premises is a conditional sentence, presumably includes the whole array of formulas of this type found in the De syllogismo hypothetico of Boethius. A third type of formal consequence pertaining to the logic of propositions is said to contain those consequences which hold in virtue of the formal impossibility of the antecedent, or the formal necessity of the consequent. Buridan says that if the antecedent is a conjunction of contradictories, it is formally impossible, and if the consequent is a disjunction of a sentence and its contradictory, it is formally necessary. We may represent thus type of consequence by two theorems: 14.4

p.-p: -1.q

14.41

p. -1.q v -q

A fourth type of consequence belonging to the logic of propositions is said to be exemplified by the case where, from the negation of one member of a disjunctive sentence, the other member is inferred. We would represent this type by the following theorem: 14.5

p v q.-p: -1.q

These types of formal consequence, pertaining to the logic of propositions, contain three of the five indemonstrables of Chrysippos, and the important Theophrastian law of syllogism. In addition, the list contains the important laws of simplification for logical addition and multiplication. Buridan might well have added the so-called "De Morgan" theorem to his list, since he invokes it frequently in his logical treatises, as did the other 14th century logicians. He does mention, as another type of formal consequence,

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the case where, from an "exponible" proposition, one of its exponents is inferred; this is however a derivative type, holding in virtue of the law stated in 14.1, pq .-1.P'. Finally, and perhaps principally, the syllogism is a formal consequence, the two premises forming a conjunctive antecedent, and the conclusion being the consequent. * 'Ve may now turn to the notion of material consequence. These consequences comprise all conditionals such as are valid, but not in virtue of their syntactical form. To be valid consequences, however, they must satisfy the general requirement that the contradictory of the consequent is incompatible with the antecedent - or that p._q' is impossible. This condition is of course satisfied by formal consequences, which exemplify logical laws or "tautologies". In the case of valid material consequences, the impossibility of 'p.-q' is a function of the truth values of the component sentences •p' and •q', as determined by the "matter" or interpreted terms of these sentences. Among material consequences, two subdivisions were made by the mediaeval logicians. One of these is a distinction between consequences valid simpliciter or without temporal qualification, and consequences valid only "as of now" (ut nunc) or for the time in which the sentence is stated. The other distinction, which we will consider first, is between consequences valid by reason of some connection of supposition or meaning between a term occurring in the antecedent and another term occurring in the consequent, and consequences valid only because of the falsity or impossibility of the antecedent, or because of the truth or necessity of the consequent. Those material consequences whose validity rests on a connection of supposition or meaning among the terms of antecedent and consequent, are represented primarily by "enthymemes". The sentence, "If a man is running, an animal is running", would be a materially valid consequence of this type, its validity being established by means of the sentence "Every man is an animal", which is the "suppressed premise" of a valid syllogism in these terms. Ockham and Strodus considered this type of consequence t:

t:

* Buridan: Oonsequentiae, III, ch. 1. This enumeration of types of formal consequence is made by way of introduction to the main subject of this third book of Buridan's treatise, which is the syllogism. 6

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to be a kind of formal consequence - not in the sense that it would be valid in any terms, but in the sense that it is formally valid for those terms occurring in the antecedent and consequent. In any case, this type of consequence is proved by reduction to syllogistic form, and belongs to the logic of terms. * The kind of material consequence which involves no connection of meaning among the terms of antecedent and consequent, was called a "merely material" consequence by Strodus, who exemplified it with this example: "If some man is a stone, then a stick stands in the corner". He gives just two rules for material consequences: (1) An impossible proposition implies any proposition, and (2) a necessary proposition is implied by any proposition. ** Buridan also gives these two rules, as rules for material consequences "simply" valid. But he adds two further rules, for material consequences valid "as of now": (1) A false proposition implies any proposition, and (2) a true proposition is implied by any proposition. *** It thus appears that the truth functional interpretation of implication, which involves the so-called "paradoxical theorems of material implication", is included in the mediaeval system. But a distinction is made between two senses of "material" implication, such that some consequences are valid "simply" or without temporal restriction, while others are valid only "as of now" or for a restricted time range. Ockham makes this distinction as follows: A consequence is valid as of now, when its antecedent can be true and its consequent false at some time, though not at this time. Thus this is a valid consequence as of now, ~Every animal is running, therefore Socrates is running', because for that time in which Socrates is an animal the antecedent cannot be true and

* Cf. Ockham: Summa, III, 3, ch. 1, 74r, and Moody: Ockham, p. 285, Note 1. Ockham appears to reserve the term <material consequence' for those conditionals which are true merely because the antecedent is false (or impossible), or the consequent true (or necessary). Strodus classifies the enthymematic consequence as a comequentia formalis, but says that it is bona de materia and not bona de forma. ** Strodus: Consequentiae, 2v. Ockham: Summa, III, 3, ch. 37, 92r, gives these two rules, and then remarks, "But such consequences are not formal, so that those rules are not much used". *** Buridan: Comequentiae, I, ch. 8, Conc!. I.

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the consequent false. But a consequence is simply valid when for no time could its antecedent be true and its consequent not true. Thus this consequence is simply valid, ~No animal is running, therefore no man is running', because the antecedent never could be true without the consequent being true. * The distinction between these two senses of material implication, "simple" and "as of now", appears to correspond to the distinction between the Diodorean and the Philonian conceptions of implication as described by Sextus Empiricus. Philo declared that the ~hypothetical is true whenever it does not begin with what is true and end with what is false'. . .. But Diodorus asserts that ~the hypothetical proposition is true which neither admitted nor admits of beginning with truth and ending in falsehood'. ** As an example Sextus mentions the sentence, "If it is night, it is day". According to Philo this is true when stated in the daytime, and false when stated at night. But Diodorus says it is false, because there can be a time in which what is stated by its antecedent is so, and what is stated by its consequent is not so. The controversy rests, in part, on the false assumption that a sentence of present time, stated at two different times, is the same sentence; only on this assumption is it possible to speak of a sentence being true and becoming false. If the relativity of time range in the sentence, to the time of its utterance, were eliminated by introducing· explicit dates into the sentences composing the conditional, it would be unequivocally true, or false, regardless of when it might be stated. This same circumstance attaches to the mediaeval formulation, and accounts in part for the distinction between consequences "simply" valid and those valid "as of now". There is, however, another aspect not to be overlooked. On the assumption that a sentence of present time, stated at different times, is the same sentence, the only condition under which such a sentence can be said to be "simply" true, or true whenever it might be stated, is that its contradictory is impossible. This means that a "simply" valid conditional, if true on material grounds

* **

Ockham: Summa, III, 3, ch. 1, 74r. Sextus Empiricus, "Against the Logicians", II, 113 and 115; transl. by R. G. Bury, Loeb Library edition, London 1935, pp. 297 and 299.

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rather than on syntactical grounds, must be such that its necessity, or the impossibility of its contradictory, is sufficiently determined by the meanings of the terms which constitute its "matter". The impossibility of ~p.-q' is not determined by what the terms occurring in ~ p' and ~ q' stand for at some particular time, but it is determined by what they can stand for, in any possible time. In short, a material consequence which is valid in the "simple" sense is one whose truth is established in virtue of some proposition Le., "necessary" or which is semantically determinate "impossible". * Buridan states that material consequences are only evident, for purposes of inference, insofar as they are "reduced" to formal consequences. Such reduction is accomplished through the introduction of a proposition from which, as from a premise, the material consequence formally follows. If the material consequence in question is valid in the "simple" or temporally unrestricted sense, it is reduced to a formal consequence by means of a "necessary" proposition - i.e., a proposition which is true on intensional grounds, or for whatever its terms can stand for. If however the material consequence is valid only in the contingent or "as of now" sense, it is reduced to a formal consequence by means of a proposition which is contingent, or true "as of now". It seems to me that no material consequence is evident in inference except through its reduction to a formal consequence. It is however reduced to a formal consequence by the addition of some necessary proposition or propositions which, when added to the antecedent, Yield a formal consequence. Thus if I say, ~If some man runs, then some animal runs', I prove the consequence by this sentence, ~Every man is an animal'; for if every man is an animal and some man is running, then it follows by a formal consequence that some animal is running. And it is thus that all who argue by enthymemes seek to prove their consequences if they are not formal. Other material consequen.ces are called consequences as of now, such as are not in the unqualified sense valid, because it is possible that the antecedent be true without the consequent being true; but they are valid as of now, because it is impossible, if things

* Cf. ante, § 12, pp. 60-63, on the mediaeval theory of "the matter of propositions".

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are as they now are, that the antecedent is true and the consequent not true..... This kind of consequence is reduced to a formal consequence by the addition of some proposition which is true, but not necessary, or by some propositions which are true but not necessary. * To "reduce" a material consequence to a formal one is to derive it, by means of some formally valid inference scheme, from a proposition sharing in the "matter" or terms of the material consequence in question. In the case of enthymematic consequences, the reduction is effected by introducing a proposition which, if taken in conjunction with the antecedent sentence of the enthymematic consequence, implies the consequent as the conclusion of a valid syllogism. In the case of material consequences of the truth functional type, whose validity is grounded Inerely in the falsity (or impossibility) of the antecedent, or in the truth (or necessity) of the consequent, the reduction is effected by introducing a sentence which formally implies the nlaterial consequence in virtue of laws of the logic of propositions. Thus the conditional ~p-:Jq' is proved to be materially valid "as of now", by introducing ~ -p' as a premise, for the formula ~ p.-p: -1 :q' is itself a formal consequence, or logically true conditional (cf. ante, 14.4). The rules of consequence are themselves formal, so that if they are expressed as theorems or formulas of the object language, they constitute logically true sentences of conditional form. They are called rules of material consequences, because they determine formally valid transformations among conditional sentences which are themselves not formally true or false, but "materially" true or false in either the unqualified ("simple") sense, or in the "as of now" (or factual) sense. In stating theorems corresponding to the rules of consequence, the asserted implication is in all cases formal or logical, but the implications occurring in subordinate positions are in most cases "material". We may now consider the question of how the mediaeval theory of consequence is related to the modern systems of material implication and strict implication. It should be emphasized, at the outset, that the mediaeval theory of consequence is not a logical calculus, not a system of fornlulas expressed with variables

*

Buridan: Oonsequentiae, I, ch. 4.

cr.

Albert: Logica, IV, ch. 1.

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THE THEORY OF CONSEQUENCE

and logical constants. It is, rather, a system of rules governing logically valid inferences to or from sentences of conditional form. Our problem is to determine whether these rules govern the system of theorems constituting the calculus of strict implication, or whether they can be adequately represented by the system of theorems constituting the calculus of material implication. The difference between the two systems is primarily embodied in the two different specifications of the import of the conditional connective: the connective of strict implication is defined through the formula ~p-3q:= :-O(p.-q)', whereas the connective of material implication is defined through the formula ~p:Jq:= :-(p.-q),. Now we have seen that the mediaeval logicians use the formula ~ -O(p.-q)' in specifying the truth condition of a conditional proposition interpreted in the "simple" or unqualified sense. The rules for such consequences are developed on the basis of this definition of implication, and the rules so developed govern theorems which are distinctive of the system of strict implication. We conclude that the mediaeval rules of "simple" consequence determine a set of theorems formally similar to those of the system of strict implication. But whether the semantic interpretation of this formal system is the same, is a more obscure question. The symbol of possibility, ~ Op', is a primitive or undefined operator in the system. Lewis and Langford state that it is to be interpreted in the broadest sense, as the "logically" possible, and that the symbol of impossibility, ~ -OP', is to be understood in the narrowest sense of "logical inconceivability." The notion of logical impossibility is however susceptible of more than one interpretation. If restricted to sYntactical impossibility, as determined by the axioms of formal logic, it would be understood in a stricter sense than that in which it is understood in the mediaeval theory of "simple" material consequence. If however the notions of logical impossibility and necessity are understood in relation to a system of semantic rules governing the usage of descriptive terms of the language, such that the language contains a class of sentences whose truth or falsity is determinable by these semantic rules along with the SYntactical rules of the language, then we may perhaps say that the mediaeval conception of "materially" necessary or impossible propositions is equivalent to the conception of logical necessity

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or impossibility in this broader sense. In any event, the formal representation ofthe mediaeval rules for simple material consequences requires modal operators, and we shall use the symbol ~O', from the system of strict implication, as translation of the term ~ possible' in its modal use in the mediaeval system. The operators of impossibility and necessity will be constructed by adjoining the negation sign to the symbol ~ 0', such that ~ -OP' means "not possibly p", and ~ -O-p' means "necessarily p". For the operator of implication in the "simple" sense (whether the logical necessity connoted is "formal" or "material" - i.e., sYntactical or semantical) we shall use the symbol ~ -1'. In contrast to the mediaeval rules for "simple" consequences, those governing consequences "as of now" determine theorems which are distinctive of the modern system of material implication. Not all the mediaeval logicians accepted the consequence "as of now" as a valid consequence, and those who did recognize this sense of implication indicated that it was not of much importance for scientific purposes. * Ockham, Buridan, Albert of Saxony, and the pseudo-Scotus, recognize the "as of now" consequence, and define it in the manner expressed by the formula ~p")q:-=- :-(p.-q)'. Buridan develops a number of rules for consequences "as of now", contrasting them with analogous rules for simple consequences; the differences between these rules correspond to the differences between the relevant theorems of the two systems of material implication and strict implication, serving to distinguish them. In representing the rules for "as of now" consequences, by the theorems or laws which they determine, we shall employ the ordinary truth functional connective of "material implication", for the unasserted occurrences of the conditional operator in the theorem. The asserted implication, being a formally or logically necessary relation, will be represented by our symbol of "simple" or unqualified consequence, ~ -1'. We may now turn to the rules themselves, as developed in the systematic treatises of Buridan and Albert of Saxony. We shall

* Cf. Albert: Logica, IV, ch. 1: "Such consequences are often called consequentiae vulgares . .... and some people argue against the consequence as of now, saying that there is no such consequence because if there were, so they say, the impossible would be implied by the possible".

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THE THEORY OF CONgEQUENCE

state only the more elementary rules, which pertain to the logic of propositions. The interest of these treatises lies in the fact that they undertake something of an axiomatic derivation of the rules of consequence, and in the further fact that in this derivation the rules of consequence among unanalyzed propositions are the first to be stated and proved. It would appear that Buridan was the first to conceive of undertaking a deductive derivation of the rules of consequence from an initial set of definitions and postulates. While his execution of this project is extremely defective, by modem logistic standards, his work is of historical importance and interest, as being the first conscious attempt to axiomatize the logic of propositions.

§ 15.

THE MEDIAEVAL LOGIC OF PROPOSITIONS

The rules which were organized in the treatises De consequentiis produced in the fourteenth century normally included the whole body of rules governing valid inferences among analyzed propositions most of which were derived from Aristotle's Prior Analytics. But even as Aristotle himself found it necessary to invoke laws belonging to the logic of unanalyzed propositions, for proof of certain moods of the syllogism, so the mediaeval logicians found that the whole system of laws pertaining to the logic of terms had to be regarded as logically posterior to the laws of consequence among unanalyzed propositions. These laws, which we shall designate as the laws of propositional logic, came to be organized into a system in which, certain laws being assumed, the others could be proved from them. While this systematization was far from complete, and lacked the techniques of symbolization and of the "logistic" procedure, it did involve a conscious effort to derive many laws of propositional logic from a limited set of axioms, such that none of these laws was to be used in a proof until it had already been established in a previous proof. In the present section we shall bring together as many of the laws of propositional logic as we have been able to find, under the form of explicitly stated rules of consequence, in the various logical treatises which we have been able to examine. Many of these laws, including those which we regard as most elementary, are not explicitly integrated into the systematic treatisesDeconsequentiis

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but occur in connection with formal rules determining the truth conditions of conjunctive and disjunctive sentences, assertoric and modal, as functions of the truth values of their components. While these rules express "consequences" in the sense of determining valid implications to or from sentences of conjunctive or disjunctive form, they were not conceived to be rules concerning consequences (de consequentiis), but as rules concerning disjunctive or conjunctive sentences. The rules concerning consequences, embodied in the treatises De consequentiis, were chiefly those which, when expressed as formulas, contain the conditional connective in a subordinate position. Since this connective, in its subordinate occurrences, normally indicates a "material" implication (either simple or as ot now), the rules "concerning consequences" were usually described as rules concerning material consequences. On this account the important laws in which only conjunctive or disjunctive signs occur in the subordinate positions were not stated or proved in the treatises De consequentiis, though they were used in the proofs of the rules of consequence as laws already established. In bringing together all these rules, and formulating them as logical laws expressed with propositional variables and symbols for the connectives, we are constructing a system, for the purpose of representing mediaeval propositional logic, which was only partially constructed by the mediaeval logicians themselves. The materials of this system, nevertheless, are all found in rules explicitly stated, in one place or another, by these mediaeval authors. In expressing the mediaeval rules as formulas, in modern symbolic manner, we seek to give a more concise, accurate, and intuitive representation of the system than could be provided by direct translation of the rules in word language. For the same purpose we introduce the distinctions between primitive ideas, definitions, postulates, and theorems, which are customary today. We have not however attempted to reduce the postulate set to a minimum of axioms required for the system, but have grouped what seem to be the more fundamental theorems into a set from which an adequate group of postulates might be selected. Our presentation of the system will involve the following procedures: (1) The definitions, postulates and theorems will be preceded

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by a number with an asterisk prefixed to it; these numbers will commence with *.01, and will determine certain groupings of laws by the numbers preceding the decimal point. (2) The sources of the definitions and laws will be given in the foot-notes under the number (enclosed in parentheses) of the formula referred to; in each ~ase we shall quote one source verbatim, and then indicate other sources in which the same law is expressed. To facilitate comparisons with contemporary systems, we will also indicate theorems or definitions stated by Lewis and Langford, or by Principia Mathematica, corresponding to the mediaeval laws, these theorems being indicated by the numbers assigned to them in these modern works. For reference to these sources we shall employ a system of sigla, as follows: B(I,8,5) A(IV,2,6) 0(111,3,37) P(38v)

= Buridan: Consequentiae, (Book I, ch. 8, rule 5) =

=

S(6r) BU(p. 11)

=

PS(p. 104)

=

BO(22)

=

SA(15)

=

(Lewis 12.11)

=

(PM 2.07)

=

(A)

Albert: Logica (Treatise IV, ch. 2, rule 6)

= Ockham: Summa (Book III, Part 3, ch. 37)

=

Paul of Venice: Logica, (folio 38 verso of the 1544 edition) Strodus :Consequentiae (fol. 6 recto ofthe 1493edition) Walter Burleigh, De puritate artislogicae, edited by Philotheus Boehner, O.F.M., St. Bonaventure, N.Y. and Louvain, Belgium, 1951; (p. 11). Pseudo-Scotus (J. Duns Scoti Opera omnia, Paris 1891, Vol. II, p. 104). 1. M. Bochenski, "De consequentiis scholasticorum earumque origine", Angelicum 15, 1938, pp. 98-105; (No. 22 in the list of laws given on those pages). J. Salamucha, "Die Aussagenlogik bei Wilhelm Ockham", transl. by J. Bendiek, Franziskanische Studien, 1950, pp. 111-115; (No. 15 in the list of laws given on those pages). Lewis-Langford, "Symbolic Logic", N.Y. 1932; (No. 12.11 in the system oflogicallaws there stated). Principia Mathematica, Vol. I (2nd ed., Cambridge 1925), pp. 90-126, Section A; (asserted theorem No. *2.07).

The Mediaeval System of Simple Implication

Primitive Ideas: As in the system of Lewis and Langford, the mediaeval logic of propositions involves the four primitive ideas

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of proposition (p,q,r,s), negation (-p), conjunction or logical product (pq), and (for the system of "simple" consequences) possibility (0). In addition it requires the relation of equivalence by definition, which we will symbolize by the equality sign followed by the letters ~Df' (thus, p. = Df.q). The defining relation may be construed in the sense of the mediaeval "nominal definition" (definitio quid nominis), as determining that one expression may replace another in any sentence without altering the meaning or the truth value of that sentence. * The notions of negation and conjunction have already been introduced in our section § 10, in the discussion of the import of the sYncategorematic signs ~not' and ~ and'. The notion of Proposition, in its elementary sense of an atomic affirmation, was introduced in section § 9, through truth rules expressing the import of the affirmative copula. The idea of possibility is also introduced, in the mediaeval system, through semantic rules expressing the import of the term ~possible' when used as a logical predicate; thus Buridan says that "an assertoric proposition of present time is said to be possible because, howsoever it signifies the case to be, so it can be". ** In this semantic rule, as in the others, the idea being defined is used in the definition of it. The notions of ~is so', ~is not so', ~ are both so', and ~ can be so', are clearly primitive or irreducible in mediaeval logic, or in any logic adequate to ordinary language. A further idea which appears to be primitive in the mediaeval system is that of compossibility, or of ~ O(pq)'. The truth rule for a sentence of this form states that "for the possibility of the conjunctive it is required that each component be possible and that it be not incompossible with the other". *** The notion of compossibility corresponds to that of "consistency" in the system of Lewis and Langford, represented by ~ po q'. To define ~ O(pq)'

* **

Cf. Ockham: Summa I, ch. 26; and Moody: Ockham, p. 110, Note 1. Euridan: Consequentiae I, 1: "propositio de inesse et de praesenti possibilis ex eo dicitur possibilis quia qualitercumque significat esse ita potest esse". *** Ockham: Summa II, ch. 32: "Ad hoc quod (copulativa) sit possibilis, requiritur quod utraque pars sit possibilis, et alteri non incompossibilis' .

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through the expression ~ -(p -I-q r is however circular, since the latter expression is ultimately defined by the former. *

Definitions : We now introduce the connectives by the following definitions:

~v', ~

-1',

and

~ ='

*.01

p v q.=Df.-(-p.-q)

*.02

P

-I q.=Df·-O(p·-q)

*.03

p

=

q.=Df:p-lq.q-lP

(*.01): O(II,33): "opposita contradictoria disiunctivae est una copulativa. composita ex contradictoriis partium ipsius disiunctivae . Cf. P(38v), A(III,5), BU(p. 10), BO(14), SA(17), Lewis 11.01, and PM 4.57. (*.02): B(I,3): "illa propositio est antecedens ad aliam quae sic se habet ad illam quod impossibile est qualitercumque ipsa significat, sic esse, quin qualitercumque illa alia significat, sic sit". Cf. A (111,5), S(6v), O(III,3,1), PS(p. 104-105), P(31v-32r), Lewis 11.02. (*.03): We introduce this symbol of "simple co-implication" as a convenient notation for laws determined by rules which state that sentences of given forms "follow mutually on each other", or that one of them "implies the other and conversely", or that they have "the same conditions of truth or falsity". The kind of implication involved is, of course, "simple" implication. Cf. Lewis 11.03.

Fundamental Laws of the System of Simple Implication: The numbers *1.01 to *1.12, inclusive, comprise laws of fundamental character which are stated explicitly in mediaeval logical works, and which are assumed in the proofs given by Buridan and Albert of Saxony in their systematic development of the laws numbered *2.01 to *2.20. The numbers [*1.13] to [*1.17] contain additional postulates which are needed for the system and which were perhaps taken for granted by the mediaeval logicians; but since we have not found explicit statements of these postulates in the mediaeval writings, we indicate their merely "implicit" character by enclosing them in square brackets. * Cf. Lewis-Langford, p. 153, and No. 17.01. In their system, 'Op' is definable by the formula 'pop' or 'O(pp)', so that if '0 (pq)' were taken as primitive, 'Op' could be introduced by definition. A similar procedure could be applied to the mediaeval system, presumably.

THE THEORY OF CONSEQUENCE

*1.03

pq.-1.p p.-1.p V q p.-1.-(-p)

*1.04

p. v.-p

*1.01 *1.02

*1.05 *1.06 *1.07 *1.08 *1.09 *1.10

C)

85

-O(p·-p) p.p-1q:-1.q p-1q·q -1 r : -1·p-1 r p V q.-P:-1.q p·-1·0p p -1q: -1 :-O-p· -1·-0-q

(*1.01): A(III,6,I): "quaelibet pars copulativae sequitur ad ipsam copulativam cuius est pars". Cf. B(III,I), O(II,32), P(38r), BO(15), SA(18-19), and Lewis 11.2. (*1.02): A(III,5): "a qualibet parte disiunctivae affirmativae, ad disiunctivam affirmativam cuius est pars, est bona consequentia". Cf. B(III, 1), O(II,33), P(38r), BO(18), SA(22), and Lewis 13.2. (*1.03-*1.05): B(I,8): "appono haec esse supponenda praesentia: Omnis contradictionis unam contradictoriarum esse veram et aliam falsam, et impossibile esse ambas simul veras aut simul falsas. Item: Omnem propositionem esse veram aut falsam, et impossibile esse eandem simul veram et falsam". A(III,10,I): "de contradictoriis est regula, quod si una est vera, reliqua est falsa". These rules determine *1.05 and *1.06 directly, but also yield *1.03 and its transposed form • -p.-1. -(p)'. Note that *1.05, in virtue of the definition *.02, is logically equivalent to the law 'p.-1.p'. Cf. Lewis, 11.5, 13.5, and 18.8. (*1.06): P(38v): "a conditionali affirmativa cum suo antecedente, ad consequens eiusdem, est bona consequentia". Cf. O(III,I,68), S(6r), BO(I), SA(I), and Lewis 11.6. (*1.07): A(IV,2,5): "Si ad A sequitur B, et ad B sequitur C, tunc C sequitur ad A". Cf. B(III,I), P(33r), BO(3), SA(3), and Lewis 11.6. (*1.08): P(38v): "A disiunctiva affirmativa cum destructione unius partis, ad alteram partem, est bona consequentia". Cf. B(III, 1), and BO(19). (*1.09): O(III,3,1l): "illa de inesse semper infert illam de possibili". Cf. BO(33), and Lewis 18.4. (*1.10): S(6r): "Si antecedens est necessarium, ergo et consequens est necessarium". P(33r), BO(26). Cf. Lewis 18.53, which is in the form: p-1q. -0 -p :-1. -0 -q.

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THE THEORY OF CONSEQUENCE

*1.11

-OP· -1·-0(pq)

*1.12

-O-p·=·-O(-p)

(*1.11): A(III,5): "Ad impossibilitatem eius (scil. copulativae) sufficit unam partem eius esse impossibilem". Cf. O(II,32), and Lewis 19.16. (*1.12): B(II,5): "supponendum est illud quod Aristoteles supponit et communiter alii, quod equipollet 'necesse est esse' et 'impossible non esse' .... quoniam per se videtur esse manifestum quod omne illud quod necesse est esse, ipsum impossibile est non esse".

The additional postulates which seem to be "implicit" in the system as developed by the logicians, but which were perhaps too obvious or too subtle to gain explicit mention on their part, are five in number. The first three (*1.13-*1.15) are the familiar laws: [*1.13] [*1.14] [*1.15]

pq.-1.qp

(Lewis 11.1)

P'-1'PP (pq)r. -1.p(qr)

(Lewis 11.3) (Lewis 11.4)

The next postulate which we believe to be implicit in the system is that which is expressed in the "existence postulate" of the system of Lewis and Langford. It serves to distinguish "simply" valid conditionals from those valid only "as of now", so that the true is not reduced to the necessary, or the false to the impossible. We will state this postulate in the form given by Lewis 20.01: [*1.16]

(Ep,q) :-(P-1q).-(P-1-q)

The further postulate which is probably implicit in the system is one which is required for the reduction of multiple modal determinations of a single expression, to simple ones. We state the postulate designated as axiom Cll by Lewis and Langford, in their discussion (p. 497) of possible additional postulates of their system. [*1.17]

OP· -i·-O( -Op)

From this postulate the following logical equivalences may be easily derived, for simplification of modal determinations: [*1.171] [*1.172]

-O-p·=·-O-(-O-p) -O-p·=·O(-O-p)

THE THEORY OF CONSEQUENCE

[*1.173] [*1.174]

87

OP·=·O(Op) OP·=·-O-(Op)

Further corollaries of *1.17 could be derived ad infinitum, as is apparent. The "logistic rules" laid down for operations on symbols, in moderh calculi, were undoubtedly not differentiated by the mediaeval logicians from the rules of inference belonging to their system of consequences. The rule of substitution is perhaps implicitly contained as a consequence of the mediaeval distinction between the "form" and the "matter" of propositions, with the definition of a formal consequence as one which holds good through any transformations of the "matter" or terms of the propositions. Perhaps also some distinction was understood between the law *1.06, which asserts the formula •p. p -jq: -j.q' as a valid consequence, and the rule which is laid down by Strodus at the beginning of his treatise, and which is apparently distinguished by him from *1.06 since this law occurs later as one of his list of valid consequences. Strodus' rule of inference is expressed as follows: If any consequence is formally valid, and if its antecedent is true, then the consequent is true. * Derived Laws of the System of Simple Implication: The first group of these laws (*2.01 through *2.20) expresses the series of rules for simple consequences among unanalyzed propositions which are stated by Buridan and Albert of Saxony in the first parts of their treatises De consequentiis, and proved in sequence. We shall not attempt to reproduce their proofs, which are in most cases obvious. The definition *.02, together with the postulate *1.11, is important in *2.01 to *2.04; and the postulates *1.01, *1.02, and *1.04-*1.07, are used in proofs of other laws of this group. *2.01 *2.02 *2.03 *2.04

Group One (*2.01-*2.20) -OP. -j.p-jq -O-p· -j.q-jp -O(pq)· -j.p-j-q O(pq)· -j.-(p -j-q)

* S(6r): "Si aliqua consequentia est bona et formalis, et eius antecedens est verum, ergo et consequens est verum".

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("'2.01-*2.02): B(l,8,1): "Ad omnem propositionem impossibilem omnem aliam sequi, et omnem propositionem necessarium ad omnem aliam sequi". Cf. A(lV,2,1-2), O(IIl,3,37), PS(p. 106), P(32r), S(5r), BO(29-30). and Lewis 19.74 and 19.75. (*2.03-2.04): B(l,8,2): "Ad omnem propositionem sequi omnem aliam cuius contradictoria non potest stare simul cum ipsa; et ad nullam propositionem sequi aliam cuius contradictoria potest simul stare cum ipsa". Cf. A(lV,2,3), and Lewis 18.31 and 18.3.

The next six laws express the principles of transposition and of syllogism. vVith regard to the latter (*2.07-*2.10), it may be noted that Lewis and Langford do not admit these formulations as valid in their most strict system of strict implication, but admit them in the system which they designate as 83, and which contains a postulate equivalent to our postulate *1.10 (cf. Lewis-Langford, p. 500). *2.05 *2.06 *2.07 *2.08 *2.09 *2.10

p--1q· --1·-q--1-P -p--1-q· --1·q--1p P--1q. --1 :q --1 r . --1. P--1 r q--1 r. --1 :P--1q· --1·P--1 r P--1q. --1 :-(p --1 r ). --1.-(q --1 r ) q--1 r . --1 :-(p --1 r ). --1. -(p --1q)

(*2.05): B(l,8,3): "Omnis bonae consequentiae, ad contradictoriam consequentis sequitur contradictoria antecedentis". Cf. A(lV,2,4), O(lII,3,37), S(6r), BV(p. 8), BO(6), SA(6), and Lewis 12.44. (*2.06): B(l,8,3): "omnis propositio, per modum consequentiae formata, est bona consequentia si ad contradictoriam designatae consequentis sequitur contradictoria designatae antecedentis". Cf. A(lV,2,4), P(32v), and Lewis 12.41. (*2.07): B(l,8,4): "Omnis bonae consequentiae, quidquid sequitur ad consequens, sequitur ad antecedens". Cf. A(lV,2,5), O(IIl,3,37), S(6r), BV(p. 2), BO(5), SA(5). (*2.08): B(l,8,4): "ad quodcumque sequitur antecede:p.s, ad illud sequitur consequens". Cf. A(lV,2,5), O(IIl,3,37), S(6r), BV(p. 2), BO(5), SA(5). (*2.09): B(l,8,4): "quidquid non sequitur ad antecedens, non sequitur ad consequens". Cf. A(lV,2,5). (*2.10): B(l,8,4): "ad quodcumque non sequitur consequens, ad illud non sequitur antecedens'. Cf. A(lV,2,5).

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THE THEORY OF CONSEQUENCE

The next six laws of this group determine modal relations between the antecedent and consequent of a valid consequence; taking the antecedent ~ p' as representing the conjunction of the two premises of a syllogism, these laws were applied by Buridan and Albert to the analysis of modal relations of syllogistic premises to the conclusion, such as Aristotle discusses in detail in Book I of the f:rio~ Analytics. *2.11

p.-q: -i.-(p-iq)

*2.12

OP·-Oq:-i·-(p-iq) -O-p·O-q: -i.-(p -iq)

*2.13 *2.15

p-iq. -i·Op-iOq p -iq. -i. -Oq -i-Op

*2.16

p -iq. -i·O-q -iO-p

*2.14

(*2.11-*2.13): B(I,8,5) : "Impossibile est ex vero sequi falsum, vel ex possibili impossibile, vel ex necessario non necessarium". Cf. A(IV,2,6). Cf. Lewis 12.8' (for *2.11). (*2.14): A(IV,2,6): "si antecedens est possibile, etiam consequens". Cf. B(I,8,5), S(6r), BO(27), and Lewis 18.51 (variant form). (*2.15): A(IV,2,6): "si consequens alicuius consequentiae est impossibile, et antecedens est impossibile". Cf. B(I,8,5), S(6r), BO(25), and Lewis 18.5 (variant form). (*2.16): A(IV,2,6): "si consequens ... non est necessarium, nec antecedens eius est necessarium". Cf. B(I,8,5), and Lewis 18.52 (variant form).

The next two laws serve to validate the derivation of a "simply valid" enthymematic consequence from a formally valid syllogistic consequence, or series of such. *2.17 is stated by Lewis and Langford, with the comment that this law serves to distinguish the system of "strict implication" from that of "material implication", since the analogue, q.pq-:Jr: -i.p-:Jr, which is valid in material implication, does not hold in strict implication unless the first occurrence of ~q' is replaced by ~-O-q'. It is worth noting that Buridan, in his proof and discussion of *2.17, formulates the analogue as valid only in the "as of now" sense of implication. * *

Cf. LewiB-Langford, p. 165, and Buridan: Oonsequentiae, I, ch. 8, rule 6. 7

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90

*2.17

-O-q·pq --1 r : -I. p-lr

*2.18

-O-(q,s,t . .. ).p(q,s,t . .. ) -Ir: -I.p-lr

(*2.17-*2.18): B(I,8,6): "Ad quamlibet propositionem cum aliqua necessaria sibi apposita, vel cum aliquibus necessariis sibi appositis, sequitur aliqua conclusio, ad eandem propositionem sequitur eadem conclusio sine appositione illius necessariae vel illarum necessariarum". Cf. A(IV,2,7), and Lewis 18.61.

The last two laws of this first group are characterized by Buridan as laws of formal consequence, though they obviously hold as laws of simple material implication as well. Both laws are stated by Buridan in his enumeration of types of formal consequence (cf. ante, 14.4 and 14.41), but only the first, *2.19, is included in the sequence of laws of material consequences given in the first book of Buridan's De consequentiis. *2.19

p.-p:-I.q

*2.20

p.-I.q v-q

(*2.19): B(I,8, 7): "Ad omnem copulativam ex duabus contradictoriis constitutam sequi quamlibet aliam, etiam consequentia formali". Cf. A(IV,2,8), and BO(22).

(*2.20): B(!III,I): "Sunt autem aliae consequentiae formales propter .... formalem necessitatem consequentis. . .. et formaliter necessaria esset disiunctiva ex contradictoriis constituta".

This completes the sequence of laws which Buridan and Albert develop in systelnatic fashion, at the beginning of their treatises De consequentiis, as basis for their proofs of the laws pertaining to the logic of terms, to which the remainder of each treatise is devoted. vVe now turn to a group of laws, drawn fron1 diverse sources, which develop conjunctive and disjunctive forms, and some more complex conditional formulas related to the principle of syllogism. We shall first state a group of laws which do not involve modal determinations of the propositional variables (*3.01-*3.17), and then a group involving such modal determinations (*4.01-*4.20).

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91

Group Two (*3.01-*3.17) *3.01 *3.02 *3.03 *3.04 *3.05 *3.06 *3.07 *3.08 *3.09

C)

-(pq).=.-p v -q -(p v q).=.-p.-q p v q.-q:-i.p -po -i.-(pq) -q. -i.-(pq) p-i-p.-i.-p p.p-iq.q-ir: -i.r P-iq·q-i r.r -i8 . .... -iu: -i.p-iu p-iq. -i :q-i-r. -i.p-i-r

(*3.01): A (III, 5): "propositioni copulativae contradicit propositio disiunctiva composita ex partibus contradictoriis copulativae". Cf. O(II,32), P(38v), BU(p. 10), BO(14), SA(17). (*3.02): A(III,5): "contradictoria disiunctivae affirmativae est una copulativa ex partibus contradictoriis partium disiunctivae composita". Cf. O(II,33), P(38v), BU(p.1O), BO(17), SA(21). This law, and the preceding one, are forms of the so-called "De Morgan law", of which still another form was given in *.01, as definition of the disjunctive. (*3.03): P(38v): "A disiunctiva cum destructione unius partis, ad alteram partem, est bona consequentia". Cf. BO(19). This rule, which is that of the modus tollendo ponens of the disjunctive syllogism, also yields

the formula: pvq. -p:---1.q, which was stated as a postulate (*1.08). (*3.04-*3.05): A(III,5): "Ad falsitatem copulativae sufficit alteram partem esse falsam". Cf. O(II,32), and Lewis 12.72 and 12.73. (*3.06): BU(p.96): "omnis propositio includens oppositum infert suum oppositum". The example given (p. 97) is the sentence, "Si omnis propositio est vera, non omnis propositio est vera"; of this Burleigh says, "Hic denotatur oppositum sequi ad appositum suum, igitur prima est falsa". Cf. Lewis 12.87. (*3.07): O(III,I,68): "ex conditionali. .. cum aliquo antecedente ad antecedens illius conditionalis, sequitur consequens". Cf. SA(2), BO(2). (*3.08): P(33v): "A primo antecedente ad ultimum consequens, quando omnes consequentiae intermediae sunt bonae et formales, et non variatae, tunc consequentia est bona et formalis consequentiae intermediae non sunt variatae quando consequens primae consequentiae est antecedens posterioris consequentiae". Cf. Lewis 12.78. (*3.09): O(III,3,37) : "quidquid repugnat consequenti, repugnat antecedente". Cf. S(6r), P(33v), BO(II) and SA(12).

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92

*3.10 *3.11 *3.12

P-1q·-q: -1'-P p-1q· -1'P-1pq p-1q· -1.pr-1qr

(*3.10): P(39r): "A conditionali affirmativa cum contradictorio consequentis, ad contradictorium antecedentis, est bona consequentia". Cf. BO(8), where this formula is used to represent the rule, "Ex opposito consequentis sequitur oppositum antecedentis", which we have construed in the sense of *2.05. (*3.11): O(II,32): "si una pars copulativae inferat alteram partem, tunc ab illa parte ad totam copulativam est bona consequentia". Cf. BU(p.5) SA(20), and Lewis 16.33. BO(16) represents this rule by the law,

P-1q.P:-1·pq. (*3.12) : O(III,3,37) : "Quidquid stat cum antecedente, stat cum consequente". Cf. S(6r), P(33v), BU(p.5 and 9), BO(10), SA(l1), and Lewis 19.6. B(III,4,4) states: "Omnis copulativa est consequens ad copulativam constitutam ex una parte ipsius et consequente alterius partis"; but this text is surely corrupt, and we should read 'antecedens' instead of 'consequens'.

The law *3.12, whose analogue in material implication is called "the principle of the factor" (cf. PM 3,45), is of importance. The law *3.10 is an "imported" form of *2.05. Whether the mediaeval logicians recognized the law of "exportation" (pq -1r. -1 :P-1.q -1r) as valid in the system of simple consequence, is problematical; the law does not seem to have been stated, and yet most of the formulations of the principle of syllogism seem to have been given the "exported" form. The system 83 of strict implication (cf. LewisLangford p. 500) admits the exported form of these laws, in virtue of a postulate equivalent to our postulate *1.10, rather than by reason of the law of exportation itself. The mediaeval system appears to correspond to this form of the system of strict implication, in this respect. We now come to a group of laws involving three variables, which were probably intended to apply to the syllogism, interpreted as a "consequence" with a conjunctive antecedent. Walter Burleigh, alone among our authors, insists that the syllogism is not to be interpreted as a conditional of conjunctive antecedent. He denies that the law of transposition, *2.05, can be applied to the syllogism, because "the syllogistic antecedent is several propositions not

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93

conjoined", and "because such an antecedent has no opposite, not being a single proposition nor a conjunction of propositions". * He nevertheless states the laws of antilogism (*3.15 and *3.16 below), apparently basing their validity on the laws of syllogism, rather than on those of the logic of propositions.

*3.16

p --1q. --1 :pq --1 r. --1.q--1 r p --1q. --1 :qr --18. --1.pr--1 8 pq --1r. --1 :p.-r. --1.-q pq--1r. --1 :q.-r. --1.-p

*3.17

pq--1r. --1 :-r. --1.-p v -q

*3.13 *3.14 *3.l5

(*3.13): BU(p. 4): "Quidquid sequitur ex consequente et antecedente,

sequitur ex consequente per se". Cf. B(III,4,4). (*3.14): BU(p. 4): "Quidquid sequitur ad consequens cum aliquo addito,

sequitur ad antecedens cum eodem addito". Cf. Lewis 12.77, which is in the "imported" form, p--1q.qr--1s:--1.pr--1s. (*3.15-3.16): A(IV,8,4): "Omnis syllogismi boni, ex utraque praemissa cum contradictoria conclusionis, sequitur contradictorium alterius praemissae". Cf. B(III,3), BU(p. 9), O(III,3,37), BO(9), SA(9-10), and Lewis 12.61, 12.62. In contrast to Burleigh, Albert proves these laws by propositional logic, through *2.05, *3.01, *1.08, incidentally deriving our next law, *3.17. (*3.17): A(IV,84): "ex contradictoria conclusionis, quae dicitur consequens, sequitur .... una disiunctiva ex contradictoriis praernissarum composita" Cf. O(III,3,37): "si consequens sit falsum, oportet quod totum antecedens sit falsum vel quod aliqua propositio quae est pars antecedentis sit falsa". BO(7), SA(7).

We may here mention a rule given by Strodus, which seems literally to express the invalid law, ~pq--1r.--1:-r--1p.--1.-q', since he says: "If one argues from the opposite of the consequent to one of the premises, the opposite of the other premise follows" (S,6r). This may have been intended to mean that "the opposIte of the other premise follows from the opposite of the consequent", which would yield the valid law:

pq --1 r . --1 :-r--1p. --1.-r --1-q

*

BU(p. 9).

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THE THEORY OF CONSEQUENCE

Group Three (*4.01-*4.20) In our third group of derivate laws are included those containing modal determinations of the variables, other than the laws of such type (*2.12-*2.18) included in the sequence of theorems developed in the treatises De consequentiis of Buridan and Albert of Saxony. *4.01

-O-p·-j·p

*4.02

-OP·-j·-p

*4.03

-O-p· -j·Op

*4.04

-O-p·-j·-(-Op)

*4.05

OP·O-P: -j·Op

*4.06

OP·-j·-(-Op)

(*4.01): O(III,3,Il): "illa de necessaria semper infert illam de inesse". Cf. Lewis 18.42. (*4.02): O(III,3,Il): "illa de impossibili ... non infert suam de inesse, sed semper contradictorium illius". Cf. Lewis 18.41. (*4.03): O(III,3,12): "illa de necessario semper infert illam de possibili". Cf. Lewis 18.43. (*4.04): O(III,3,12): "illa de necessario infert contradictorium illius de impossibili". This is not to be confused with *1.12. (*4.05): O(III,3, 12): "illa de possibili non infert illam de contingenti sed econverso". The "contingent" (contingens ad utrumlibet) is defined as a sentence neither necessary nor impossible, so that it requires the conjunctive < OP.O -P' for its representation. (*4.06): O(III,3,12): "illa de possibili infert oppositum illius de impossibili".

The above six laws formulate the "sequence of modes", according to the postulate *1.09 (p. -{Op). None of these implications is reversible. Though we have only used one source, William of Ockham, for these laws, they are to be found in lllost of the mediaeval logical works, including those of the thirteenth century. Most of the remaining laws of this group are modal analogues of laws previously stated for conjunctive and disjunctive sentences of assertoric form.

THE THEORY OF CONSEQUENCE

*4.07 *4.08 *4.09 *4.10 *4.11 *4.12 *4.13 *4.14 *4.15 *4.16 *4.17 *4.18 *4.19 *4.20

95

O(pq)·-1·0p O(pq)· -1·0q O(pq)· -1·0P·Oq p-1-q· -1·-0(pq) -Oq· -1'-O(pq) -O(p v q). -j·-OP·-Oq OP· -j·O(p V q) Oq· -j·O(p V q) Op V Oq· -j.O(p V q) -O-q· -j·-O-(p V q) -O-p V -O-q· -j.-O-(p V q) -P-1q'-1'-O-(p V q) -O-(pq)· = ·-O-p·-O-q pq -jr. -j :.-O-p.-O-q. -j·-O-r :)

(*4.07-*4.09): O(II,32): "ad hoc quod (copulativa) sit possibilis, requiritur quod quaelibet pars sit possibilis". Cf. A(III,5), and Lewis 19.01, 19.13, 19.14. (*4.10-*4.11): A(III,5): "Ad hoc autem quod copulativa sit impossibilis, non requiritur utramque eius partem esse impossibilem, nee requiritur aliquam eius partem esse impossibilem, sed sufficit quod partes eius sint incompossibiles, seu etiam contradicentes. Tamen ad impossibilitatem eius sufficit unam partem eius esse impossibilem". Cf. O(II,32), and Lewis 18.31, 19.17. (*4.12-*4.15): A(III,5): "Ad impossibilitatem disiunctivae, requiritur quod utraque eius pars sit impossibilis Ad possibilitatem eius sufficit alteram partem esse possibilem". Cf. O(II,33), and Lewis 19.28, 19.3, 19.31, 19.32. (*4.16-*4.18): A(II,5): "Ad necessitatem disiunctivae .... requiritur quod altera eius pars sit necessaria .... vel potest esse necessaria si neutra pars sit necessaria, sed contingens, dum tamen eius partes contradicunt sibi". Cf. O(II,33), and Lewis 19.34, 19.19. (*4.19): O(II,32): "Ad necessitatem copulativae requiritur quod quaelibet pars sit necessaria". Cf. Lewis 19.81. (*4.20): BO(28): "Si praemissae sunt necessariae, conclusio est necessaria".

The foregoing laws represent the mediaeval system of simple consequence, based on the interpretation of the conditional connective represented by the formula ~ -O(p.-q)'. The formal

96

THE THEORY OF CONSEQUENCE

similarity of this system of laws to that of the modern system of "strict implication", in the form designated 83 by Lewis and Langford, is sufficiently apparent. The mediaeval development of this system is fairly ample; aside from the two laws stated as definitions in *.01 and *.02, a total of fifty-nine laws of this system are explicitly stated by our mediaeval authors, in the form of rules of consequence or of formally valid inference. We have now to consider the analogous system of propositional logic recognized by the mediaeval logicians as valid when the conditional connective is interpreted in a purely truth functional and non-modal sense, as indicating a "consequence as of now".

(B)

The Mediaeval System of Implication "As 01 Now"

The primitive ideas involved in the laws of consequence "as of now" are only four: namely, that of proposition (p,q,r) .. ), negation (-p), conjunction ((pq)), and definition (p.=Df.q). The modal ideas of possibility and compossibility are not required. The relations of disjunction (p v q), "as of now" implication (P":Jq) , and "as of now" equivalence (p==q), are introduced by definition. The definition of disjunction is the same as that already given in *.01, for the system of simple implication. The other two definitions are as follows: *.021 *.031

p-:Jq.=Df.-(p.-q) p==q. =Df.p-:Jq.q-:Jp

(*0.21): O(III,3,1): "consequentia ut nunc est quando antecedens pro aliquo tempore potest esse verum sine consequente, sed non pro isto tempore". Cf. A(IV,I), B(I,4), and ante, § 14. (*.031): This symbol of "as of now' , equivalence is introduced for convenience in expressing the bi-conditional relation interpreted in the "as of now" meaning of implication.

The postulates and laws of this system may be derived from those laws of simple implication which do not contain the modal sign ~ 0', through replacement of the signs ~ -1' and ~ =' by the signs ~-:J' and ~ ==' respectively. Such replacement is only significant, however, in the subordinate occurrences of these connectives, since the principal connective indicates a formally valid implication (or an asserted logical law) in both systems. We shall therefore

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retain the signs ~ -1' and ~ =' as principal connectives, indicating that the law itself is formally valid, or "simply" valid in the sYntactical sense. That the system of "as of now" consequence contains analogues of all the laws of simple consequence such as do not involve the modal sign ~ (y, is demonstrable within the system of simple consequence, though the mediaeval authors were content to state the obvious fact that what cannot be the case is not the case as of now and that what cannot fail to be the case (or what is necessarily the case) is the case as of now. A formal demonstration may nevertheless be given, as follows: *1.001 p-iq. -i.p:)q Proof: (*.02) (*1.12: p.-q/-p,-(p.-q)/p): (*.021) : (*4.01) :

P-iq·=·-O(p·-q) ·=·-O-[-(p·-q)] ·=·-O-(p:)q) . -i.p:)q

Q.E.D.

The entire system of simple consequences could be reduced to valid laws of "as of now" implication if, in addition to replacing ~ -f with ~:)', and ~ =' with ~ ==', we were to omit the affirmative modal determinations ,~-O-' and ~ 0', and if we were to replace the negative modal determinations, ~ -0' and ~ 0-', with the sign of negation ~ -'. Buridan indeed suggests this relationship between the two systems, when he describes the merely false as "impossible as of now". This means that although in simple implication the impossible implies the false and not conversely, in "as of now" implication the false is equivalent to the impossible. If, as Buridan states, the false is "impossible as of now", this theorem holds as a valid consequence "as of now": -P:)-OP. From this, however, we obtain converses of the whole "sequence of modes" of simple implication, which are valid in "as of now" implication. The comparison is interesting: Simple Modalities

As of Now Modalities

p·-i·Op -O-p·-i·p -OP·-j·-p -p·-j·O-p

OP·:)·p p·:)·-O-p -p·:)·-Op O-p·:)·-p

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Since, by *1.001 above, P-1q. -1.p:Jq, we may state analogues of the four laws for simple modalities on the left, by replacing the sign ~ -1' with the sign ~:J'. 'Ve then obtain, in the system of "as of now" consequence, the following equivalences: *1.002

OP·~·p

*1.003

-O-p·~·p

*1.004

-OPe

*1.005

O-p·~·-p

.-p

When we apply these laws of "as of now" consequence to the laws of "simple" consequence, we find that all laws of the latter system determine valid truth functional laws of what is nowadays called "material implication", although many of the laws containing modal functions reduce to simple tautologies, or to theorems already obtained from the laws of non-modal form through replacement of the sign ~ -1' with the sign ~:J'. It would be tedious to restate all the laws of the system of simple consequence, in the forms obtained by reduction to analogous laws of the system of consequence "as of now". The reader is invited to make these reductions for himself, and to consider how Buridan's identification of the "merely false" with the "impossible as of now", expressed in *1.004 above, may have been related to his reputation as a determinist. We shall content ourselves with the statement of three laws admitted as valid implications "as of now", which Buridan explicitly contrasts with their modal analogues in the system of simple consequence. We shall number these to correspond with their "simple" analogous, by adding the numeral ~1' to the previous numbers. *2.011 *2.021

-p·-1·p:Jq p.-i.q:Jp

(*2.011-*2.021): B(I,8,1): "Et est notandum quod de consequentia ut nunc modo proportionali ponenda est conclusio, scilicet quod ad omnem propositionem falsam omnis alia sequitur consequentia ut nunc, et omnis vera ad omnem aliam sequitur etiam consequentia ut nunc, quia impossible est, rebus se habentibus ut nunc, propositionem quae est vera non esse veram, ideo nec possibile est ipsam esse veram qualibet

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alia non existente vera; et si sit sermo de praeterito vel de futuro, tunc vocetur consequentia ut tunc aut qualiter voluerit nominare. Verbi gratia, sequitur consequentia ut nunc vel ut tunc vel ut nunc per tunc, 'Si Antichristus non generabitur, Aristoteles numquam fuit'; quia licet simpliciter sit verum 'Antichristum possibile est non fore', tamen impossibile est quod, rebus se habituris sicut se habebunt, ipse non erit. Ipse enim erit, et impossibile est quod ipse erit et non erit". Cf. PS(p. 107), Lewis 15.2, 15.22, and PM 2.21.

*2.171

q.p{Jr:-1.p"Jr

(*2.171): B(I,8,6): "ad quamcumque propositionem cum multis veris appositis, vel cum aliqua vera apposita, sequitur aliqua conc1usio, ad eamdem propositionem solam sequitur eadem conclusio consequentia ut nunc". Cf. ante, *2.17 and *2.18, and Lewis 18.61 and his discussion. PM 3.3 is a slightlX different form of the same law.

The first two of the above laws express the so-called "paradoxes of material implication", which follow from the definition of "p"Jq' as logically equivalent to "-(p.-q)'. Buridan's argument in support of *2.011 is somewhat curious, since he seeks to show that, in relation to conditions as they actually are at a given time, a false sentence is materially equivalent as 01 that time to an impossible proposition. In short, although "-p. -\.-Op' is false as a "simple" implication, "-p."J.-Op' is valid as a consequence "as of now", or for the time for which "-p' is asserted. The "merely false" is thus equated with the "impossible as of now", as was stated above in *1.004. If then the "as of now" consequence "-p."J.-Op' is valid, the consequence "-OP."J.-O(p.-q)' is valid "as of now", being implied (in virtue of *1.001) by the law of simple consequence *1.10. Hence "-p' may be said to imply "p -1q' as 01 now, or for the time for which "-p' is asserted. But since "p -\q' is logically equivalent to "-O-(p"Jq)' , as was shown in the proof of *1.001, and since "-O-(p"Jq)' is equivalent "as of now" to "p"Jq' (by *1.003), it turns out that "p-\q' is equivalent "as of now" to "p"Jq', and that" -O(p.-q)' is equivalent "as of now" to "-(p.-q)'. Thus Buridan's argument, though routed by way of the law of simple consequence *1.01, actually involves no other assumption than the law" -p."J.-(pq)' (which follows from *3.04 in virtue of *1.001), together with the definition *.021 of the "as of now" conditional. In support of his statement that" -p' implies" -OP' as of now,

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Buridan invokes the law of contradiction along with the principle of "exportation". The argument is again curious: If the discourse is of the past or of the future, then let the consequence be called ~as of then' or however it is desired that it be called. For example, this is a valid consequence ~ as of now' or ~as of then' or ~as of now in relation to then': "If Antichrist will not be born, then Aristotle never existed". For although this is true in the simple sense, "Antichrist possibly will not exist", nevertheless it is impossible, if things are going to be as they are going to be, that he will not exist. For he will exist, and it is impossible that he will exist and will not exist. * If we represent the conditional sentence given in Buridan's example by p"Jq, Buridan's argument takes this form: -p.p. --1.q: --1 :-p"J.p"Jq

The antecedent of the above formula is one of the laws asserted in the system of simple consequence as formally valid (*2.19); it would imply the "as of now" law ~-p.p."J.q' if we wish to state it in this truth functional form. The derivation of the consequent ~ -p"J.p"Jq' from this antecedent clearly involves the law of exportation: pq"Jr."J :p"J.q"Jr. Since this is one of the crucial laws which, while valid in the modern system of material implication, is not valid in the modern system of strict implication, Buridan's use of it here serves to show the formal correspondence of the mediaeval system of consequence "as of now" with the modern system of material implication. The same law of exportation is involved in the third theorem stated above, *2.171, so that this theorem also indicates the correspondence of the mediaeval "as of now" consequence to the modern truth functional systems. We conclude, therefore, that the mediaeval logicians recognized the distinction between the two systems, and developed them side by side. Most of them, as is obvious, gave precedence to the system of "simple" or necessary consequence. *

B(I,8,1). The Latin text was given in the footnote to *2.011-*2.021.

v TRUTH AND CONSEQUENCE § 16.

THE ARISTOTELIAN DEFINITION OF TRUTH

The terms ~true' and ~false' occur constantly in mediaeval logical works. For the most part this was merely due to the fact that mediaeval logic was formulated as a system of rules of inference, and not as a calculus of uninterpreted formulas developed in "logistic" manner through rules of operation on the symbols. But since the normal interpretation of a logical calculus is such that a formula and its negation represent contradictory propositions of which one is true and the other false, this usage of the terms ~true' and ~false' had no metaphysical implications but was simply a convenient means of describing logical relations among sentences. The problem of defining the term ~true', as a term connoting the relation between a sentence and things or events outside of language, is a metaphysical or philosophical problem. It was discussed primarily in works dealing with metaphysics, and in connection with the definitions of truth found in Aristotle's Metaphysics. Aristotle's definitions, however, were found to involve paradoxical consequences which seenled to undermine the funda('mental laws of logic. These consequences were made evident in the so-called "insolubles", or logical paradoxes, to which the mediaeval logicians devoted much attention. The investigation of the paradoxes, and particularly that of the Liar, led to a revision of the Aristotelian definition of truth which is of interest in connection with some recent discussions of the same problems. *

* Cf. A. Tarski, "Der Wahrheitsbegriff in den fonnalisierten Sprachen", Studia philosophica I, 1935, pp. 261-405; and "The Semantic Conception of Truth", Philosophy and Phenomenological Research, Vol. IV, March 1944, pp. 341-375. R. Carnap, in his "Introduction to Semantics" (Cambridge, 1946), and subsequent works, has adopted many of Tarski's ideas, giving them an extensive development of his own. These works have aroused almost continuous controversy during the past fifteen years.

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Aristotle states in Book IV of the Metaphysics: "To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, and of what is not that it is not, is true". In Book VI of the same work he says: "The true judgment affirms where the subject and predicate really are combined, and denies where they are separated". * From these statements the scholastics derived the two formulations: (1) A proposition is true if, howsoever it signifies to be, so it is (qualitercumque significat esse, ita est); and (2) An affirmative categorical proposition is true if its subject and predicate stand for the same. We have already seen that Buridan rejects this second statement; though he concedes that if an affirmative categorical is true, its subject and predicate stand for the same, he denies the universal validity of the converse implication. The rule ~For the truth of the affirmative it is sufficient that the terms stand for the same' does not hold universally. And this is made clear in the case of the so-called insolubles; as for example if, on a certain page of a book, there is written just this sentence, "The sentence written on this page is false". The terms stand for the same thing, because that subject, ~The sentence written on this page', stands for that sentence which is written on that page; and similarly this predicate ~false' stands for that sentence, because it is false and not true. Therefore it does not suffice, in order that an affirmative. proposition be true, that the terms stand for the same.** The other general rule of truth, embodied in the first quotation from Aristotle given above, seems also to assunle the validity of a consequence from any sentence to another sentence in which the ternl ~true' is affirmed of the name of the first sentence. Thus, if ~ p' is the name of whatever sentence replaces the variable p, the Aristotelian definition would be formulated as a logical equivalence in the following manner: 16.1

(p):

p.=.T~p'

This definition, if taken as a logical equivalence in the above manner, leads to paradoxical results in the case of sentences which "reflect on themselves", among which are included the so-called

* Aristotle, Metaphysics, IV, 7, 101lb 26; and VI, 4, 1027b 20. Oxford transl. ** Buridan: Metaphysics, VI, Qu. 7, fol. 38v. Cf. ante, 9.21.

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insolubilia or forms of the well known paradox of the liar. These paradoxes were the subject of extensive discussion in the 14th century, and several types of solution were developed. We shall examine Buridan's proposed resolution of the paradox, which appears to be one of the most advanced of the theories developed.

§ 17.

THE PARADOX OF THE LIAR

The sinlplest forms of the liar paradox were those represented by the sentence "What I am saying is false", and by the sentence "No proposition is true", the assumption being that Socrates ("Sortes") utters the sentence and says nothing else. :More subtle formulations involved indirect reflection, as when Socrates says "What Plato is saYing is false", while Plato is saying "What Socrates is saYing is true"; or again, where Socrates says that what Plato is saYing is false, while Plato says that what Socrates is saying is false. This last type does not involve a vicious circle between the two sentences, but since the sentences are related to each other in exactly the same way, there is no basis for determining whether to assign truth to the first and falsity to the second, or falsity to the first and truth to the second. * In one of his discussions of the paradox, Buridan mentions and criticizes several proposed solutions which had apparently been defended by his predecessors or contemporaries. The first of these he calls an evasion of the problem. Some, wishing to evade the problem, say that terms which are instituted to stand for propositions are not to be used in propositions as standing for those propositions in which they occur, but only for others. **

* Albert: Logica, VI, Insoluble 6, gives a classification of these types of indirect paradox. Buridan chooses the last of these types for his analysis of the paradox in Qu. Il, Book VI, of his Questions on the Metaphysics. In his Sophismata, Ch. 8, Buridan develops a great many varieties of the paradox, some of which are extremely subtle. ** Buridan: Sophismata, ch. 8, Soph. 7. This position seems to have been held by Ockham, Summa logicae III,3, ch. 45, fo!. 94r; "in ista propositione ·Sortes non dicit verum' predicatum non potest supponere pro tota ilIa propositione cuius est pars". On Ockham's theory, cf. Ph. Boehner, O.F.M., "Ockham's Theory of Supposition and the Notion of Truth", Franciscan Studies VI, Sept. 1946, pp. 261-292.

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TRUTH AND CONSEQUENCE

Buridan considers this solution inadequate. For if one sentence states that another is false, which in turn states that the first is true, the antinomy occurs, and yet the terms of the sentences do not stand for the sentences of which they are parts. Others have said that such a proposition is true and false at the same time. But this is inconvenient, as is apparent, because jf the contradictory of this statement were true, then both contradictories would be true; and if it were false, then both would be false. And both alternatives are impossible. * A third theory, which Buridan says he himself had formerly accepted, is the theory which Albert of Saxony defends in his Logica. Buridan summarizes it as follows, in his discussion of the sentence "Every sentence is false", supposedly uttered by Socrates. Some have said, and so it seemed to me at one time, that although this sentence, according to the signification of its terms, does not signify or affirm anything except that every proposition is false, nevertheless, because every proposition by reason of its form signifies or asserts itself to be true, therefore every proposition asserting itself to be false, either directly or indirectly, is false. For although the case is as it is signified to be, insofar as the sentence signifies itself to be false, nevertheless the case is not as it is signified to be, insofar as the sentence signifies itself to be true. It is therefore false and not true, because for its truth it is not only required that the case be as the sentence signifies it to be, but that how8oever the sentence signifies, so it is. ** This theory, though conceded to be near the truth, is criticized because of the assumption involved in it, that every proposition, by reason of its form, signifies or asserts itself to be true. In the first place, Buridan denies that propositions as such "signify" anything at all, except in the sense that their terms signify the things they stand for. Some people apparently held that a sentence, e.g. "Man is an animal", signifies an extralinguistic entity otherwise described by the substantive phrase ~man-being-animal'. Buridan says that no such entity is to be posited, as is more evident in

* Buridan: Sophismata, ch. 8, Soph. 7. Who upheld this view, I have not been able to determine. The reference may be to those mentioned by Aristotle, in his Metaphysics, as denying the laws of contradiction and excluded middle. ** Buridan: Sophismata, ch. 8, Soph. 7. Cf. Albert: Logica, VI, ch. 1.

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the case of negative or false sentences. For we should have to say that the sentence "No man is an animal", or the sentence "Man is a horse", signify respectively entities described as ~man­ not-being-animar and ~man-being-horse.' Unless those sentences are true or possible, as nobody admitted, their alleged designata cannot be anything at all. Sentences or propositions, which for Buridan are the same, do not signify or stand for corresponding entities; their terms do so, but not the sYncategorematic signs or the sentences whose form and unity are determined by the SYncategorematic or "logical" signs. The two sentences, "Socrates exists" and "Socrates does not exist" have exactly the same referent, which is Socrates; there are not any distinct or additional entities corresponding to the two sentences, as designata. * ApplYing these considerations to the theory, described above, that every sentence signifies itself to be true, by reason of its form, Buridan says that ~itself-being-true' is not any thing at all, unless it be the sentence in question; and in this case ~itself-being-false' would be that same sentence and same entity. ** Despite this criticism, Buridan considers the distinction between what a sentence implies, in virtue of its form, and what it implies in virtue of its matter or content, important. It is then said, in a manner closer to the truth, that every sentence implies virtually another sentence in which, of a subject term standing for that first sentence, this predicate ~true' is affirmed. I say ~implies virtually', in the sense in which an antecedent implies whatever is consequent to it. Therefore a sentence is not true, if in that implied consequent, which is affirmative, the subject and predicate do not stand for the same. ***

* Buridan concedes that the conventional signs, sounds or marks, which are sentences in a given language, "signify" corresponding "mental statements" in the thought of the speaker or listener, or of the writer or reader. But this sense of "signify" has nothing in common with the ordinary sense in which a sentence (written, spoken, or thought) is said to signify the things that its terms stand for. Many contemporary logicians, such as C. 1. Lewis and Rudolf Carnap, appear to accept the view criticized by Buridan, and to hold that sentences designate or stand for entities called "propositions", these latter being true or false in a manner corresponding to sentences. ** Buridan: Sophismata, ch. 8, Soph. 7. *** Burid,an: Sophismata, ch. 8, Soph. 7. 3

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TRUTH AND CONSEQUENCE

If this premise is granted, then, since a sentence which asserts that it is itself false implies, by reason of its matter, another sentence which states that that first sentence is not true, it follows that the sentence asserting itself to be false is one which implies contradictory consequents. But every sentence which implies contradictory consequents is false: p. -1.q.-q: -1.-p. This formulation is still imperfect, according to Buridan, because of its assumption that everyproposition implies another sentence in which the predicate ~true' is affirmed of the name of that original sentence. The Aristotelian definition of truth, 16.1, makes this assumption: (p) :p. -1. T~ p'. But this implication cannot be conceded, as Buridan shows by an example. Let this sentence, "A horse runs", be designated by the proper name ~B'. Then, says Buridan, this consequence is not valid: "If a horse runs, then B is true". For a horse could be running without there being any such sentence asserting that a horse is running; and in that case the consequent of the above conditional, "B is true", would be false, because it is affirmative and its subject would stand for nothing. Therefore, perfecting this solution, we ought to say that to every proposition, along with the addition of a sentence stating that that proposition exists, it follows that the proposition is true. Thus in the case mentioned, in which the sentence "A horse runs" is denoted by the proper name "B", this consequence is valid: "A horse runs, and B exists, therefore B is true". * This additional condition, that the proposition exists, validates the "virtual" consequence from the proposition to another sentence stating that the first is true. With this additional condition, every sentence whatsoever implies a sentence stating that it is true, so that to every sentence which is actually stated this implied consequent, which states that that sentence is true, can be added without altering the truth value of that original sentence. This addition can, permissibly, be made to any proposition. But it is not necessary to do so where the proposition does not have a reflection on itself - as for instance, "God exists", or "Socrates is running", etc. But where the proposition can have

*

Buridan: Sophismata, ch. 8, Soph. 7.

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107

such reflection, it is necessary to make this addition, on account of the possible implication of a contradiction. * In the case where a sentence reflects on itself, as when Socrates says "What I am saYing is false", the fact that Socrates says this fulfills the condition of the existence of this sentence; consequently it follows, from his sentence and the fact that he is saYing it, that it is true. But since he is saYing that the sentence he is uttering is false, his statement implies another sentence stating that Socrates' statement is not true. His statement, then, coupled with the fact that it exists (or that he is saYing it), implies contradictory consequents. But any sentence which implies contradictory consequents, is false. Therefore Socrates' statement, if it exists, is false. Let us now attempt to exhibit this argument, and the definition of truth on which it depends, in a formal manner. The condition that the sentence must exist, if it is to imply another sentence in which the term ~true' is affirmed of its name, fulfills the important office of determining the correlation between the sentence used and ' that same sentence mentioned. That is, if we are to say that p. ~.T~p', it must be understood that ~ p', occurring in the consequent, stands for the antecedent, p. Since it cannot stand for what does not exist, the assumption that ~p' stands for p is an assumption that what ~p' stands for (Le., p), exists. Thus we add, to the Aristotelian definition of truth, the existential sentence: (EX).~P'8X. x=p. For convenience we may abbreviate this to read: ~ p' 8 p, and then write the revised definition of truth as follows:

Let us now take one of the simplest forms of the direct liar paradox, and apply Buridan's argument to it. We suppose that on a certain page of a book nothing is written except this one sentence: "The sentence written on this page is not true". Let the letter s (without quotation marks) be an abbreviation for this sentence, and let ~s' (with quotation marks) be a proper name of that sentence. If, as is assumed, the sentence written on that page is identical with s, then the definite description, "The sentence

*

Buridan: Metaphysics, VI, Qu. 11.

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written on this page", which occurs as subject terln of that sentence, stands for s and for nothing else. We may therefore replace this definite description with the proper name ~ s', and state this identity: The sentence ~ s' is identical with the expression "~s' is not true". And we may state this logical equivalence: s.=.~s' is not true. This last statement, which may be written, s.=.-T~s', expresses the empirical ~ssumption, or "case" (casus), in the liar paradox. Let us see, then, if the substitution of -T~s' for p, and of ~s' for ~p', in our definition 17.1, leads to paradoxical results. We first argue from the assumption that the sentence ~s' is true, stating 17.1 as an implication: T~s'. -i:~s'ss.s, and then substituting -T~s' for s, as the "case" requires.

--1 :~s's-T~s' .-T~s'

(1)

T~s'.

(2) (3)

(By 17,1, with substitutions for the "case") ~s' 8-T~s' .-T~s': -i.-T~s' (By *1.01: pq--1q) T~s'. --1.-T~s' (By (1), (2), and *1.07: law of syll.)

(4)

:-i.-T~s'

(By *3.06: P--1-P'--1'-p)

This shows that if we assert the predicate ~true' of the name of a sentence in which the predicate ~true' is denied of that same name, we will be committed to asserting that that sentence is not true. Since our assumption formally implies it.s contradictory, that assumption was false. So now we will argue from the opposite assumption, namely, that ~s' is not true. Again we take 17.1, this time transforming it, by means of the law of transposition (*2.05) and the "De Morgan" law (*3.01), to a form expressing what is implied by the sentence "-T~s'''. (1) (2)

--1 :-(s's-T~s'). v .-(-T~s') (By 17.1, *2.05, *3.01, and substitutions for the "case") -T~s'. --1 :-(s's-T~s'). v .T~s' (Reducing -(-T~s') to T~', by law of double negation)

-T~s'.

On account of the "existential" component added by 17.1 to the Aristotelian definition of truth, it turns out that if we assume the falsity of the sentence ~s', we are not compelled to grant that it is true; we are only compelled to grant the disjunctive statement, that either ~s' is true or ~s' does not state that ~s' is not true. Now

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this disjunctive statement will hold through all the turns of the "vicious circle" in which the sentence ~s' is involved. For whenever "s' is interpreted as asserting that ~s' is not true, we assert the contradictory of what ~s' is asserting as so interpreted: and whenever ~s' is interpreted as asserting that "~s' is not true" is not true (or that ~s' is true), we are then asserting the contradictory of what ~s' is then asserting. To put it another way, whenever the subject term ~ s', in the sentence to which we refer in our statement that ~s' is not true, is replaced by "-T~s''', then the subject term "s' in our statement stands fot that interpretation of s, and so on ad infinitum. At each degree of substitution, providing it is uniformly made throughout the formula 17.1, we maintain our position of contradicting the sentence which would, together with the expression then being substituted for s in the component "~s'8S", imply that ~s' is true. This solution appears to eliminate the vicious circle, in assigning a truth value to the sentence which asserts its own falsity. The sentence itself is no doubt in a vicious circle, since it is a sentence which implies its own contradictory - on which account it is false, But when we say that ~s' is not true, our statement does not imply that ~ s' is true; for the adequate statement of the conditions under which a sentence is to be called true, given in 17.1, determines that it is not sufficient, for the truth of a sentence, that what it states to be the case, is the case. It is also required that the sentence does not imply, by reason of its form, that what it states to be the case is not the case. Since this only occurs when sentences "reflect on themselves", we tend to forget this requirement of "formal truth", and to treat the Aristotelian definition, T~p' .=.p, as if it were adequate and universally valid. Whether this solution of the liar paradox, or the definition of truth given in 17.1, involves the distinctions made by contemporary exponents of "the semantic conception oftruth" , between sYntactical and semantical systems, hierachies of metalanguages, and corresponding ambiguities in the word ~true', may be left to the reader to decide. The mediaeval treatment of the problem of truth, and of the logical paradoxes, does indeed involve a very basic distinction between statements made through language expressions, and statements made about language expressions. The theory of supposition,

IlO

TRUTH AND CONSEQUENCE

in general, is a formulation of rules by which the distinction is maintained, in any given case, between language signs and what they stand for. And though a language sign may stand for itself, as when we say "The term ~term' is a term", the distinction between that sign as an object, and as a vehicle used to refer to objects, is preserved. This basic distinction between "use" and "mention" permeates all parts of language, including those parts (e.g., the terminology of logic) used in expressing this distinction. But it was assumed, in mediaeval logic, that adequate laws governing the use of language could, without contradiction or paradox, be developed within language. It was the task of logic to develop these laws, to be applied not only in the positive sciences and in in philosophy but in logic itself.

BIBLIOGRAPHY (A)

Abbreviated Titles for Works Frequently Cited

Abelard: Dialectica V. Cousin, Ouvrages inedits d'Abelard, Paris 1836, pp. 173-503. Abelard: Introductiones B. Geyer, "Peter Abaelards Philosophische Schriften", Beitrage zur Geschichte der Philosophie des Mittelalters, Bd. XXI, Munster i.W., 1919-1927. Albert: Logica Perutilis Logica Magistri Alberti de Saxonia, ed. Petrus Aurelius Sanutus, Venice 1522. Buridan: Consequentiae Consequentiae Buridani, Paris: Felix Baligault, sans annee (ca. 1495). Buridan : Metaphysics J ohannis Buridani Quaestiones in M etaphysicam Aristotelis, ed. Iodocus Badius Ascensius, Paris 1518. Buridan: Sophismata Sophismata Buridani, Paris: Antoine Denidel and Nicolas de la Barre, sans annee (ca. 1496). Carnap: Foundations R. Carnap, "Foundations of Logic and Mathematics", International Encyclopedia of Unified Science, Vol. I, No.3, Chicago 1939. Carnap: Logical Syntax R. Carnap, The Logical Syntax of Language, N.Y. 1937. Lewis: Langford C. 1. Lewis and C. H. Langford, Symbolic Logic, N.Y. 1932. Moody: Ockham Ernest A. Moody, The Logic of William of Ockham, N.Y. and London, 1935. Ockham: Expositio aurea Gulielmi de Ockham, Expositio aurea et admodum utilis super artem veterem . ... cum quaestionibus Alberti parvi de Saxonia. Bologna 1496. Ockham : Summa Gulielmi Ockham, Summa totius logicae, Venice 1508. Paul of Venice: Logica Logica Pauli Venet'i : Summa totius dialecticae . . , Venice 1544. Peter of Spain Petri Hispani Summulae logicales, ed. 1. M. Bochenski, Torino 1947. Prantl C. Prantl, Geschichte der Logik im Abendlande, Vols. I-IV, Leipzig 1855-1870. Principia Mathematica A. N. Whitehead and B. Russell, Principia Mathematica Vol. I, 2nd ed., Cambridge 1925. Pseudo-Scotus "In librum Priorum Analyticorum Aristotelis Quaestiones", J. Duns Scoti Opera omnia, Vol. II, Paris (Vives) 1891. Quine W. V. Quine, Mathematical Logic, Cambridge, Mass. 1947.

112

BffiLIOGRAPHY

Shyreswood: Introductiones M. Grabmann (ed.), "Die Introductiones in logicam des Wilhelm von Shyreswood", Sitzungsberichte d. Bayerischen Akad. d. Wissenschajten, Phil.-Rist. Abt., 1937, Hft. 10; Miinchen 1937. Shyreswood: Syncategoremata J. R. O'Donnell (ed.), "The Syncategoremata of William of Sherwood," Mediaeval Studies Vol. III, Toronto 1941, pp. 46-93. Oonsequentiae Strodi, cum commento Alexandri Strodus: Oonsequentiae Sermoneta, Venice (BonetuB Locatellus), 1493. (B)

Other Works Used or Cited

~F

Aristotle, Analytica Priora, Oxford Translation, ed. by W. D. Ross, Vol. I, Oxford 1928. Aristotle, Metaphysica, Oxford Translation, ed. by W. D. Ross, Vol. VIII, Oxford 1928. Becker, A., Die Aristotelische Theorie der Moglichkeitsschusse, Berlin 1933. Bochenski, I. M., "De consequentiis scholasticorum earumque origine", Angelicum 15, 1938, pp. 92-109. Boehner, Ph., "Bemerkungen zur Geschichte der De Morganschen Gesetze in der Scholastik", Archiv fur Philosophie, 1951, pp. 113-146. Boehner, Ph., "Does Ockham Know of Material Implications?", Franciscan Studies, Vol. XI, 1951, pp.203-230. Boehner, Ph., "Ockham's Theory of Supposition and the Notion of Truth", Franciscan Studies, Vol. VI, 1946, pp. 261-292. Boethius, A. M. S., De syllogismo Hypothetico, J. P. Migne, Patrologia, Latina, Vol. 64, Paris 1891. Buridanus, Johannes, Perutile compendium totius logicae Joannis Buridani, Venice 1499 (Otherwise entitled "Summula de dialectica"). Burleigh, Walter, De Puritate Artis Logicae, ed. by Ph. Boehner, Franciscan Institute Publications, St. Bonaventure, N.Y., and E. Nauwelaerts, Louvain, Belgium, 1951. Carnap, R., Introduction to Semantics, Cambridge, Mass., 1946. Elie, H., Le Oomplexe Significabile, Paris 1937. Feys, R., "Les Logiques nouvelles des modalites," Revue N eoscolastique de Philosophie, 40, Nov. 1937, pp. 517-553. Gilson, E., Being and Some Philosophers, N.Y., 1949. Hugo de S. Victore, Didascalion, J. P. Migne, Patrologia Latina Vol. 176. Lukasiewicz, J., "Zur Geschichte der Aussagenlogik", Erkenntnis 5, 1935, pp. 111-127. Mates, Benson, "Diodorean Implication", The Philosophical Review, Ithaca, N.Y., May 1949, pp. 234-242. Morris, C. W., "Foundations of the Theory of Signs," International Encyclopedia of Unified Science, Vol. I, No.2, Chicago 1938. Mullally, J. P., The Summulae logicales of Peter of Spain, Notre Dame, Ind., 1945.

BIBLIOGRAPHY

113

Salamucha, J., "Die Aussagenlogik bei Wilhelm Ockham", transl. by J. Bendiek, Franziskanische Studien, 1950, pp. 97-134. Sextus Empiricus, Against the Log1:Cians, transl. by R. G. Bury, Loeb Library, London 1935. Tarski, A., "Der Wahrheitsbegriff in den formalisierten Sprachen", Studia Philosophica I, 1935, pp. 261-405. Tarski, A., "The Semantic Conception of Truth", Philosophy and Phenomenological Research, Vol. IV, March 1944, pp. 341-375.

STUDIES IN LOGIC AND THE FOUNDATIONS OF MATHEMATICS W. ACKERMANN

Solvable Cases of the Decision Problem . (1954) P. BERNAYS, A. A. FRAENKEL AND A. BORGERS

Axiomatization of Set Theory . (1954) J

I. M. BOCHENSKI

Ancient Formal Logic· (1951) P. BOEHNER, O. F. M.

Ein Beitrag zur mittelalterlichen Suppositionstheorie (to be translated) HASKELL B. CURRY

Outlines of a Formalist Philosophy of Mathematics· (195 1) K. DURR

The Propositional Logic

of Boethius

. (195 1)

R. FEYS

Modal Logics. (1954) A. A. FRAENKEL

Abstract Set Theory. (1953) E. A. MOODY

Truth and Consequence in Mediaeval Logic. (1953) A. MOSTOWSKI

Sentences Undecidable in Formalized Arithmetic (An Exposition of the Theory ~f Kurt Giidel) . (195 2) H. REICHENBACH

Nomological Statements and Admissible Operations . (1953) A. ROBINSON

On the Metamathematics of Algebra . (195 1) J.

B. ROSSER AND A. R. TURQUETTE

Many-'Jalued Logics . (1952) A. TARSKI, A. MOSTOWSKI AND R. M. ROBINSON

Undecidable Theories· (1953) G. H. VON WRIGHT

An Essay in Modal Logic· (1951)