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On Theoretical Constructs and Ramsey Constants

Published online by Cambridge University Press:  14 March 2022

R. M. Martin
R. M. Martin*
Affiliation:
New York University
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Abstract

The method of Ramsey sentences has been proposed for handling theoretical constructs within a scientific system. Essentially it consists of constructing a certain “monolithic” sentence for an entire theory. In this present paper several improvements are suggested which help to overcome some of the awkward features of the method. In particular we have here many Ramsey sentences rather than just one, each erstwhile primitive theoretical term giving rise to a Ramsey sentence. Such a sentence in effect defines what we call a Ramsey constant. Using Ramsey constants, we attempt to improve the method in important logical and semantical respects. It is suggested also that such constants are of interest for the philosophy of mathematics.

Type
Research Article
Information
Philosophy of Science , Volume 33 , Issue 1 , March 1966 , pp. 1 - 13
Copyright
Copyright © 1966 by The Philosophy of Science Association

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References

1 R. Carnap, “On the Use of Hilbert's ∊-Operator in Scientific Theories,” in Essays in the Foundations of Mathematics, Dedicated to A. A. Fraenkel on His Seventieth Birthday (Jerusalem, The Magnes Press: 1961), pp. 156-164; C. G. Hempel, “The Theoretician's Dilemma,” in Minnesota Studies in the Philosophy of Science, Vol. II (Minneapolis, University of Minnesota Press: 1958), pp. 37-98; H. Bohnert, The Interpretation of Theory, to appear; and F. P. Ramsey, “Theories,” in The Foundations of Mathematics (New York, Harcourt, Brace and Co.: 1931), pp. 212-236.

2 We use this phrase in essentially the sense of Quine. See his Word and Object (The Technology Press of the Massachusetts Institute of Technology and New York and London, John Wiley and Sons: 1960), esp. pp. 119 f. and 241 ff.

3 See especially R. M. Martin, Truth and Denotation (Chicago, University of Chicago Press: 1958), pp. 166-169.

4 Op. cit., p. 99 ff.

5 See the author's “On Denotation and Ontic Commitment,” Philosophical Studies 13 (1962): 35-39.

6 Cf. D. Hilbert and P. Bernays, Grundlagen der Mathematik, Vol. I (Berlin, Springer: 1934), pp. 164 ff.

7 Cf. the author's “A Homogeneous System for Formal Logic,” The Journal of Symbolic Logic 8 (1943): 1-23, esp. the ancestral axioms.

8 Cf. esp. R. Carnap, “Meaning Postulates,” in Meaning and Necessity, Second Edition (Chicago, University of Chicago Press: 1956), pp. 222-229.

9 See the author's “On Logical, Analytic, and Postulational Truth,” to appear in Synthese.

10 See The Problem of Universals (Notre Dame, University of Notre Dame Press: 1956), pp. 35-54, esp. p. 42.

11 Cf. W. V. Quine, Set Theory and Its Logic (Cambridge, The Belknap Press of Harvard University Press: 1963), pp. 1-27, and Martin Davis, “First-order, Second-order, and Higher-order Systems,” Invited Address before the Association for Symbolic Logic, December 27, 1963.

12 See the author's “The Philosophic Import of Virtual Classes,” The Journal of Philosophy 61 (1964): 377-387.

13 N. Goodman, The Structure of Appearance (Cambridge, Harvard University Press: 1951).

14 “The Theoretician's Dilemma,” pp. 85-86.

15 William Craig, “On Axiomatizability within a System,” The Journal of Symbolic Logic 18 (1953): 30-32 and “Replacement of Auxiliary Expressions,” Philosophical Review 65 (1956): 38-55. For an interesting discussion of both the methods of Ramsey and Craig, see I. Scheffler, The Anatomy of Inquiry (New York, Knopf: 1963), pp. 193-222. The method of L″ above is perhaps comparable in some respects to that of Craig.