Bressan and Suppes on Modality Bas C. van Fraassen PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association, Vol. 1972. (1972), pp. 323-330. Stable URL: http://links.jstor.org/sici?sici=0270-8647%281972%291972%3C323%3ABASOM%3E2.0.CO%3B2-7 PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association is currently published by The University of Chicago Press.
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BRESSAN A N D SUPPES O N MODALITY
Professors Bressan and Suppes have both argued that modality is of great importance to the philosophy of science. I shall not dispute this, but I shall raise some problems of interpretation, which arise if one considers the possibility of a philosophical retrenchment with respect to the use of modal concepts. I. A P R O B L E M C O N C E R N I N G T H E D E F I N I T I O N O F MASS
Professor Bressan uses the discussion initiated by Mach's attempted definition of mass as a paradigm example of the use of modal concepts in foundational work in physics. He mentions the modal terms used in informal axiomatizations of mechanics which followed Mach, and explains how mass is defined in his own formalized axiomatic mechanics. On the other hand, in his book Introduction to Logic, Professor Suppes had said unequivocally that he knew of no problem in philosophy of science to which modality is relevant.' This he said almost in the same breath with the assertion that mass is not definable in classical mechanics and that Mach's book was a mass of confusions. The argument to show that mass is not definable is simplicity itself: if a body were to be unaccelerated (throughout its existence), because the (total) force on it is zero, then the laws of mechanics tell us only that its mass multiplied by zero equals zero. Since that is no information at all, it follows that the term 'mass' cannot be eliminated by a definition in terms of the remaining concepts of mechanics. Could Mach possibly have missed this obvious point? It seems unlikely. There was some discussion in the literature (notably by Pendse) of the conditions under which information about a body (possibly concerning a number of distinct times) would allow one to calculate its mass.' The outcome of these discussions is most perspicuously formulated by Simon (in response to Suppes' view): Simon showed that for a natural choice of measure on the set of models of classical mechanics, Kenneth F. SchafSner and Robert S. Cohen (edr.), PSA 1972, 323-330. All Rights Reserved Copyright 0 1974 by D. Reidel Publishing Company, Dordrecht-Holland
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mass is definable almost e~erywhere.~ However, that is scant comfort for someone who wishes to eliminate mass as a primitive concept. The correct account of Mach's attempt must therefore, it seems to me, be that of Bressan: Mach made essential use of modal concepts. But I now wish to raise a problem concerning the specific formulation adopted by Bressan. Without a detailed examination it will still be clear from Bressan's exposition that for each body U he asserts a necessary conditional : (a certain experiment is performed on U 3 the outcome is real number p) for a unique number p. Upon analysis that means that in all physically possible cases, this experiment upon U yields p. But now we must ask: what are the physically possible cases? They cannot be those logically possible cases that are compatible with the laws of mechanics (plus factual information about U phrased in terms other than 'mass'.) For relative to all that (if U is always unaccelerated), it is as possible that U has one mass as that it has another. Because of this problem, I would naturally have sought for a formalization of Mach's ideas using (Stalnaker-Thomason) counter factual^.^ But alternatively one can deny that it is reasonable to try and construe physical modalities as logical modalities relativized along the above lines. This alternative response has behind it the tradition according to which essential properties and de re modalities are impervious to any philosophical reduction. However, if the mode of response of body U to a certain kind of experiment is introduced as an essential property of U, by postulate, then the manner in which mass has been eliminated from the primitive concepts of mechanics seems to me to signal no philosophical gain. I shall return to these issues below. 11. A P R O B L E M I N T H E P H I L O S O P H Y O F S P A C E A N D T I M E
In the section of Professor Suppes' paper which deals with geometry and space, he notes correctly that there is no exact correspondence between geometric and physical relations unless the latter are described in modal terms. This problem - for so it must seem to the ametaphysical philosopher -
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was discussed extensively in the development of the causal theory of time. While the issue occurs already in the earliest efforts of Reichenbach to formulate such a theory, it was not explicitly and adequately handled until the formulations by G r i i n b a ~ m .And ~ it is in Reichenbach's writings on the philosophy of space and geometry that we may find the paradigm effort at a solution. According to Reichenbach, a geometry is interpreted by giving coordinative definitions. The question which is the correct geometry (in our world, of our space) makes no sense until such coordinative definitions have been given; for example, until it has been stipulated that lightray paths are geodesics. This is a special - and, as I say, paradigm - case of correspondence rules linking a theory to physical facts. But equally clearly, the question which is the correct geometry has no answer unless the interpretation through coordinative definitions is a total interpretation: since there is quite possibly a certain paucity of actual lightrays and other physical connections, there would in principle be infinitely many geometries that fit the actual physical facts equally well. Hence it is necessary to say, for example, that the geodesics are the possible lightray paths. However, the problem of the interpretation of the modal notions used arises here in a manner similar to that in the above discussion of mass. To place the problem in its clearest light, let us choose as context the attempt to formulate a causal theory of space-time. Through the work of Reichenbach and Griinbaum especially, it is now clear that we must insist on two main points. The first is that, to avoid the morass of a general theory of causality, the 'causal connections' to be utilized in an account of spatio-temporal relations must be restricted to a definite class (light-signals, transport of material bodies). The second is that actual connectedness is not a relation sufficing to define spatio-temporal relations. For example, there is a time-like separation between events X and Y exactly if some possible signal emitted coincidently with Y were reflected or absorbed coincidently with X , or emitted coincidently with X and reflected or absorbed coincidently with Y. But could we explain this by saying that it is possible relative to physical law and antecedent information about X and Y that such a physical connection obtain? The antecedent information would have to be stated in terms which are logically independent of spatio-temporal terms, for otherwise
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the explication of spatio-temporal concepts would be circular. So the answer is no ; as far as the laws are concerned, the relative spatio-temporal locations of X and Y could be anything at all; if in addition, there is no actual physical connection between X and Y, the antecedent information about them can also not be utilized in connection with such (otherwise relevant) laws as those which place upper limits on the speed of causal propagation. The necessities depicted in physical law do not compel the placing of actually unconnected events in space-time, in any definite manner. But this was clearly recognized, and the recognition signalled by paraphrasing the 'possible connection' clause by means of a counterfactual conditional : if sufficientlyvaried signals were emitted coincidently with X and with Y, some would be absorbed or reflected coincidently with Y or X. The indefiniteness of the spatio-temporal relational structure disappears now, if these conditionals are read in the Stalnaker manner, for then only one possible world is considered in the evaluation: the world which would be the actual one, were we actually to carry out the experiment. Could we in principle, as in the Mach mass definition case, insist simply that all the relevant strict conditionals hold? This seems much less plausible here. It means that we would be insisting that in every physically possible world, in which these same events occur, they occur in exactly the same spatio-temporal arrangement. If I may suggest a thought-experiment: imagine a rocket leaving earth, all signal connections between rocket and earth ceasing for a while (call that while the 'silent period'), and the rocket returning one hundred years later. The people on earth cannot deduce what its pattern of flight was during the silent period. Could we really, in all conscience, insist that this pattern of flight would, in any physically possible case (in which the rocket has the same initial and final states) have been exactly what it actually was? 111. T H E P O S S I B I L I T Y O F A P H I L O S O P H I C A L R E T R E N C H M E N T
There are two dismissive reactions to the problems I have raised. The first is to insist resolutely that mass, force, field, space-time, and indeed all other theoretical entities of actual science be taken as undefined.
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These terms have significance: this is clear because they play an indispensible role in physics. What need is there for explication? Perhaps I have put this too uncharitably. When we look at the definitions or informal characterizations which physicists give them, we shall, if we take these literally, conclude that the entities they discuss are all mathematical. Do they not say that an observable is a linear operator, mass is a number, space-time is a differentiable manifold? A strict formalization, if read without attention to its informal commentary, would strengthen this impression: the 'particles' of Professor Suppes' systems of classical particle mechanics might be ordinals : the axioms allow it. Only under the pressure of philosophical questioning are we made to understand that the mathematical entities discussed model the physical phenomena, which are the theory's proximate topic of concern. And then the question is: what is this modelling relation? A naive scientific realist might say: for each of these mathematical entities, if the theory is true at all, there is a concrete entity with exactly the same structure. A scientist's ontology might be read from his theory, to paraphrase a critical article by Professor Griinbaum, like his telephone number from a d i r e ~ t o r y This . ~ would be the articulate counterpart of the dismissive reaction I describe in the preceding paragraph. Certainly, neither Bressan nor Suppes would join this first dismissive reaction; so let me consider the second. This second reaction is to insist that it would be perverse to require an explication of physical possibility as construable in terms of logical possibility. One should take what is physically possible as sui generis, not further explicable, and postulate that all the needed physical necessity governed conditionals (statements of form 'It is physically necessary that if A then B') happen to be true. I see this as the reaction of despair. Certainly there are cases where the modal qualifier 'It is logically necessary relative to physical law and relevant boundary conditions' will not give us the needed logcal apparatus for our rational reconstructions. But as I have suggested with reference to the Stalnaker-Thomason theory of conditionals, other logical apparatus is available, and more becomes available all the time. Whatever the scope of our present logical reach, I cannot accept the counsel of despair. This is a fortunate difference between philosophy and life: that we are not faced with the prospect that, by heeding our scruples, we may precipitate disaster. Not to have an explication, not to
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have as yet a totally intelligible, articulated world picture is a regrettable imperfection, but to accept a noxious metaphysics averts no disasters. What I called the second dismissive reaction asks us to accept unexplicated the whole machinery of possible worlds and access relations, miraculously tailored to the needs of the explicant. We are here in the train of the medieval nominalist-realist controversy. This was not, as commonly presented, mere philosophical foreplay to our seduction by Q ~ i n e .It~ concerned causal properties, causal modalities, and the justification of inductive inference; and when Newton, Locke and Boyle berated the Schools' occult qualities, they referred to the causal properties postulated by the realists to explain uniformity in behaviour. In this controversy, I cannot but take the antirealist side. Though I have worked and played with possibles, impossibles, non-existents, possible worlds, and functions thereon, for some years now, I have not believed for a moment in their reality. I was gladdened by Professor Stachel's remarks about the physicists' reception of Everett's multiple world interpretation of quantum theory; it shows a laudably robust sense of reality. This is not to say that I am not convinced of the benefits of modal logic. Were I to speak in my own persona, rather than as commentator, I would argue ardently for these benefits, both to philosophy of science in general and to the interpretation of quantum mechanics in particular. Specifically, the practice of modal logic has given us a much better idea of what to require by way of adequacy proofs when a formalism is putatively provided with an interpretation. In addition, I do not advocate that when making use of modal concepts, we foreswear the possible worlds/possible cases terminology. In practical situations, it is useful to allow oneself to be bewitched by a picture. But not without an eye to the safety of retreat. It would certainly be a marvellous dknouement if, at this point, I could demonstrate the existence of such a safe retreat. I have made a number of small explorations to this end, but cannot report complete success.' As I have indicated here, it seems to me that a judicious use of a certain theory of conditionals is a necessary preamble, and some remarks of Thomason's may show the way to clear this use of noxious metaphysical side effeck8 A rough characterization of my general approach is this: physical theory provides us with a mathematical model. It asserts that any empirical structures to be found can be embedded in
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this mathematical structure. The mood of this assertion is what Sellars calls an inductive 'neck sticking out', and this mood is signalled in the material mode by modal qualifiers. The actual physical phenomena fit in this structure in some way. We speak as if they fit it in one, in a certain definite way: we conceive of the actual world as precisely located in the general mathematical model. But the actual facts leave this fit underdetermined to some extent. The conception of the world outstrips its fundamentum in re. Let me close with some remarks about mass to illustrate this. When Mackey writes about the foundations of classical mechanics, he defines m a s 9 He observes, of course, that the remaining quantities may not determine a given body's mass. In that case, he says, what mass we assign it, is arbitrary. And he assigns such bodies a mass of his choice. So what happens is this. We think of each body as having a definite mass; we phrase our general statements, our putative laws of mechanics, in a way that clearly reflects this assumption. But when the facts underdetermine the mass of a given body, there is no human nor scientific purpose served by this body having one possible mass rather than another. Mackey makes a single wholesale choice of mass for all such bodies, like Russell, in his theory of descriptions, made a single wholesale choice of truth-value for all simple sentences containing nonreferring terms. But he might have chosen differently, and wholesale or piecemeal; the choice could not have contradicted the facts. The realist will say that one of these assignments of mass is correct, the others false. But the contrary position is not difficult to state: what is true is exactly what is true independent of this choice of mass for the factually indeterminate cases. And if I may close by tantalizing: this position is exactly the one reached if one adopts a definition of mass by Stalnaker conditionals, and then retreats philosophically to Thomason's view of the status of conditionals not compelled to a single truth-value by the facts. University of Toronto NOTES P. Suppes, Introduction to Logic, Van Nostrand, Princeton, 1957, p. 298.
For references and discussion, see M. Jammer, Concepts of Mass, Harper, New York,
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1964, pp. 92-95; Pendse's papers are in the Philosophical Magazine 24 (1937); 27 (1939); and 29 (1940). H. Simon, 'Discussion: The Axioms of Classical Mechanics', Philosophy of Science 21 (1954) 340-343; 'Definable Terms and Primitives in Axiom Systems', in The Axiomatic Method (ed. by L. Henkin et al.), North-Holland, Amsterdam 1954, pp. 443453; 'A Note on Almost-Everywhere Definability' (abstract), Journal of Symbolic Logic 31 (1966) 705-706. R. Stalnaker, 'A Theory of Conditionals', in Studies in Logical Theory (ed, by N. Rescher), Oxford 1968, pp. 98-112; R. Stalnaker and R. H. Thomason, 'A Semantic Analysis of Conditional Logic', Theoria 36 (1970) 2342. For references and discussion, see my Introduction to the Philosophy of Time and Space, Random House, New York, 1970 (henceforth IPTS), Chapter VI, Sections 2 and 3. A. Griinbaum, 'Why I Am Afraid of Absolute Space', Australasian Journal of Philosophy 49 (1971) 96. For one line of thought, see IPTS, pp. 97-107 and 195-198; for another, my 'Meaning Relations, Possible Objects, and Possible Worlds' (with K. Lambert) in Philosophical Problems in Logic (ed. by K. Lambert), Reidel, Dordrecht 1970, pp. 1-19. See my 'Theories and Counterfactuals' forthcoming in a Festschrift in honour of Wilfrid Sellars (ed. by H.-N. Castafieda) and R. H. Thomason, 'A Fitch-Style Formulation of Conditional Logic', Logique et Analyse 13 (1970) 397412; last paragraph. G. W. Mackey, The Mathematical Foundations of Quantum Mechanics, W. A. Benjamin, New York 1963, pp. 1 4 . A similar point could be made with reference to H. Simon's first approach in his 'The Axioms of Newtonian Mechanics', Philosophical Magazine 38 (1947) 888-905, and 'The Axioms of Classical Mechanics' (see note 3).
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