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Models and reality1

  • Hilary Putnam (a1)

In 1922 Skolem delivered an address before the Fifth Congress of Scandinavian Mathematicians in which he pointed out what he called a “relativity of set-theoretic notions”. This “relativity” has frequently been regarded as paradoxical; but today, although one hears the expression “the Löwenheim-Skolem Paradox”, it seems to be thought of as only an apparent paradox, something the cognoscenti enjoy but are not seriously troubled by. Thus van Heijenoort writes, “The existence of such a ‘relativity’ is sometimes referred to as the Löwenheim-Skolem Paradox. But, of course, it is not a paradox in the sense of an antinomy; it is a novel and unexpected feature of formal systems.” In this address I want to take up Skolem's arguments, not with the aim of refuting them but with the aim of extending them in somewhat the direction he seemed to be indicating. It is not my claim that the “Löwenheim-Skolem Paradox” is an antinomy in formal logic; but I shall argue that it is an antinomy, or something close to it, in philosophy of language. Moreover, I shall argue that the resolution of the antinomy—the only resolution that I myself can see as making sense—has profound implications for the great metaphysical dispute about realism which has always been the central dispute in the philosophy of language.

The structure of my argument will be as follows: I shall point out that in many different areas there are three main positions on reference and truth: there is the extreme Platonist position, which posits nonnatural mental powers of directly “grasping” forms (it is characteristic of this position that “understanding” or “grasping” is itself an irreducible and unexplicated notion); there is the verificationist position which replaces the classical notion of truth with the notion of verification or proof, at least when it comes to describing how the language is understood; and there is the moderate realist position which seeks to preserve the centrality of the classical notions of truth and reference without postulating nonnatural mental powers.

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Presidential Address delivered before the Winter Meeting of the Association for Symbolic Logic in Washington, D. C., December 29, 1977. I wish to thank Bas van Fraassen for valuable comments on and criticisms of an earlier version.

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2 Barwise has proved the much stronger theorem that every countable model of ZF has a proper end extension which is a model of ZF + V = L (in Infinitary methods in the model theory of set theory, published in Logic Colloquium '69). The theorem in the text was proved by me before 1963.

3 This is a very counterintuitive consequence of the axiom of choice. Call two objects A, B “congruent by finite decomposition” if they can be divided into finitely many disjoint point sets A 1, …, An, B 1, … Bn, such that A = A 1A 2 ⋃ … ⋃ An, B = B 1, ⋃ B 2 ⋃ … ⋃ Bn, and (for i = 1, 2, …, n) Ai is congruent to Bi. Then Tarski and Banach showed that all spheres are congruent by finite decomposition.

4 This axiom, first studied by Mycielski, J. (On the axiom of determinacy”, Fundamenta Mathematicae, 1963) asserts that infinite games with perfect information are determined, i.e. there is a winning strategy for either the first or second player. AD (the axiom of determinacy) implies the existence of a nontrivial countably additive two-valued measure on the real numbers, contradicting a well-known consequence of the axiom of choice.

5 Cf. Evans, Gareth, The causal theory of names, Aristotelian Society Supplementary Volume XLVII, pp. 187208, reprinted in Naming, necessity and natural kinds, (Schwartz, Stephen P., Editor), Cornell University Press, 1977.

6 Evans handles this case by saying that there are appropriateness conditions on the type of causal chain which must exist between the item referred to and the speaker's body of information.

7 For a discussion of this very point, cf. Wiggins, David, Truth, invention and the meaning of life, British Academy, 1978.

8 To the suggestion that we identify truth with being verified, or accepted, or accepted in the long run, it may be objected that a person could reasonably, and possibly truly, make the assertion:

A; but it could have been the case that A and our scientific development differ in such a way to make Ā part of the ideal theory accepted in the long run; in that circumstance, it would have been the case that A but it was not true that A.

This argument is fallacious, however, because the different “scientific development” means here the choice of a different version; we cannot assume the sentenceA⌉ has a fixed meaning independent of what version we accept.

More deeply, as Michael Dummett first pointed out, what is involved is not that we identify truth with acceptability in the long run (is there a fact of the matter about what would be accepted in the long run?), but that we distinguish two truth-related notions: the internal notion of truth (“snow is white” is true if and only if snow is white), which can be introduced into any theory at all, but which does not explain how the theory is understood (because “snow is white” is true is understood as meaning that snow is white and not vice versa, and the notion of verification, no longer thought of as a mere index of some theory-independent kind of truth, but as the very thing in terms of which we understand the language.

1 Presidential Address delivered before the Winter Meeting of the Association for Symbolic Logic in Washington, D. C., December 29, 1977. I wish to thank Bas van Fraassen for valuable comments on and criticisms of an earlier version.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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