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The problem of interpreting modal logic

Published online by Cambridge University Press:  12 March 2014

W. V. Quine
W. V. Quine*
Affiliation:
Harvard University
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Extract

There are logicians, myself among them, to whom the ideas of modal logic (e. g. Lewis's) are not intuitively clear until explained in non-modal terms. But so long as modal logic stops short of quantification theory, it is possible (as I shall indicate in §2) to provide somewhat the type of explanation desired. When modal logic is extended (as by Miss Barcan1) to include quantification theory, on the other hand, serious obstarles to interpretation are encountered—particularly if one cares to avoid a curiously idealistic ontology which repudiates material objects. Such are the matters which it is the purpose of the present paper to set forth.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 12 , Issue 2 , 23 June 1947 , pp. 43 - 48
Copyright
Copyright © Association for Symbolic Logic 1947

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References

1 Barcan, Ruth C., A functional calculus of first order based on strict implication, this Journal, vol. 11 (1946), pp. 1–16.Google Scholar

2 See Tarski, Alfred, The semantic conception of truth and the foundation of semantics, Philosophy and phenomenological research, vol. 4 (1914), pp. 341–376.CrossRefGoogle Scholar

3 Lewis, C. I., The modes of meaning, Philosophy and phenomenological research, vol. 4 (1943), p. 245.CrossRefGoogle Scholar

4 Dr. Nelson Goodman has suggested (in conversation) the dismal possibility that what we think of as synonymy may be wholly a matter of degree, ranging from out-and-out orthographical sameness of expressions on the one hand to mere factual sameness of designatum (as in the case of ‘nine’ and ‘the number of the planets’) on the other. In this case analyticity in turn would become a matter of degree—a measure merely of our relative reluctance to give up one statement rather than another from among a set of statements whose conjunction has proved false. But if it does develop that the boundary between analytic and synthetic statements has thus to be rubbed out, no doubt it will be generally agreed that the logical modalities have to be abandoned as well. The explanation of modal logic in terms of analyticity remains of interest so long as there is interest in modal logic itself.

5 Cf. my Mathematical logic, pp. 27–33.

6 I use the word ‘matrix’ (as in Mathematical logic) for one of the meanings of the ambiguous phrase ‘propositional function.’ A matrix is an expression which is like a statement except for containing, at grammatically permissible places, some free occurrences of variables of the kind that are admissible in quantifiers. Briefly, a matrix is a non-statement which can be turned into a statement by applying quantifiers.

7 Op. cit., theorem 38.

8 See my Designation and existence, Journal of philosophy, vol. 36 (1939), pp. 701–709.

9 Such is Church's, procedure in A formulation of the logic of sense and denotation (abstract), this Journal, vol. 11 (1946), p. 31.Google Scholar I am indebted to Professor Church for several helpful letters in this connection.—I am also indebted, along more general lines, to Professor Rudolf Carnap; correspondence with him on modal logic over recent years has been very instrumental in clarifying my general position.