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Published online by Cambridge University Press: 01 January 2020
Chomsky (in syntax) and Davidson (in semantics) have made much of the constraint that speakers’ competence must have a finite base. This base is often supposed to mean a finite axiomatization of beliefs. Section I shows why this is plausible. Section II shows why it is wrong. Section III shows why the semantic constraint is thereby trivialized.
'Finite minds cannot have infinitely many beliefs’ has been taken for a useful truism. A theory of meaning, say, for a language may, for each of its infinitely many sentences, attribute to competent speakers knowledge, in some sense, and so in a corresponding sense belief, about the meaning (or some substitute for it) of that sentence. Our truism seems to force this paradox to immediate and constructive resolution: as finite minds we have a finite view on our language that recognizably entails an infinity of propositions about it.
1 I am grateful to Peter Lipton and Christopher Peacocke for comments on an earlier draft of this paper.
2 This is the purest holism, which determines the number as well as the content of beliefs holistically: cp. ‘a truth characterization fits the underlying physical facts from the theorems upwards; not … from the axioms downward. The deductive apparatus used in deriving the theorems needs no anchoring in the physical facts, independently of the overall acceptability of the derived assingrnents of truthconditions’ (McDowell, J. ‘Physicalism and Primitive Denotation: Field on Tarski’ in Platts, M. ed., Reference, Truth and Reality [London: Routledge 1980], 123Google Scholar) - so the axioms need not be physically reali2ed. The passage from Davidson McDowell quotes in support seems weaker: it speaks of evidential, not factual, grounding (in Defense of Convention T,’ in Leblanc, H. ed., Truth, Syntax and Modality [London: North Holland 1973], 84Google Scholar). I am not quarrelling here with theories of holistic interpretation which insist that each belief (or other propositional attitude) have its own physical realization.
3 On these issues generally, see Field, H.H. ‘Mental Representation,’ Erkenntnis 13 (1978) 9–61.CrossRefGoogle Scholar
4 By Peacocke, C.A.B. at pp. 147-8 of ‘Finiteness and the Actual Language Relation.’ Proceedings of the Aristotelian Society, 75 (1974-75)Google Scholar. He informs me that he would not claim this possibility to be consistent with the actual laws of nature.
5 ‘What the tortoise said to Achilles,’ Mind 4 (1895) 278-80. See also W.V.O., Quine ‘Truth by Convention,’ repreinted as ch. 10 of Ways of Paradox (London: Random House 1966).Google Scholar
6 Cf. Cz. Ryll-Nardzewski, The Role of the Axiom of Induction in Elementary Arithmetic,’ Fundamenta Mathematicae, 39 (1952) 239-63; but cf. section II (here, all inference rules are assumed to belong to the underlying logic).
7 That all and only effectively computable functions, in the intuitive sense, are recursive. For evidence see Kleene, S.C. Introduction to Metamathematics (Toronto: Van Nostrand 1952), §§62, 70.Google Scholar
8 Kleene, S.C. ‘Finite Axiomatizability of Theories in the Predicate Calculus Using Additional Predicate Symbols,’ Memoirs of the American Mathematical Society, 10 (1952) 27–66.Google Scholar
9 Reprinted as Ch. 1 of Truth and other Enigmas (London: Duckworth 1978); cf. pp. 10-14.
10 The Compactness Theorem states that if every finite subset of a set of sentences has a model, so does the set itself.
11 Cf. Davidson, D. ‘Theories of Meaning and Learnable Languages,’ in BarHillel, Y. ed., Logic, Methodology and Philosophy of Science (Amsterdam: North Holland 1965), 383-94.Google Scholar For related criticisms, cf. Leeds, S. ‘Semantic Primitives and Learnability,’ Logique et Analyse, 85–86 (1979) 99-108.Google Scholar
12 Cp. Davies, M. Meaning, Quantification, Necessity (London: Routledge 1981), 57–62.Google Scholar The argument from a structural constraint to finite axiomatization seems to let in axiom schemata under the guise of infinite conjunctions.
13 ‘∼ is the concatenation sign: S⌒T means S followed by T ('grass’ ⌒ ‘is green’ = ‘grass is green’).
14 See Peacocke, C.A.B. ‘Necessity and Truth Theories,’ Journal of Philosophical Logic, 7 (1978) 473–500.CrossRefGoogle Scholar
15 For a sceptical view of a different rule of proof in semantics, cp. Davies, 32-3, 48-50.
