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Strict implication, deducibility and the deduction theorem

Published online by Cambridge University Press:  12 March 2014

Ruth Barcan Marcus
Ruth Barcan Marcus*
Affiliation:
Northwestern University
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Extract

Lewis and Langford state, “… it appears that the relation of strict implication expresses precisely that relation which holds when valid deduction is possible. It fails to hold when valid deduction is not possible. In that sense, the system of strict implication may be said to provide that canon and critique of deductive inference which is the desideratum of logical investigation.” Neglecting for the present other possible criticisms of this assertion, it is plausible to maintain that if strict implication is intended to systematize the familiar concept of deducibility or entailment, then some form of the deduction theorem should hold for it. The purpose of this paper is to analyze and extend some results previously established which bear on the problem.

We will begin with a rough statement of some relevent considerations. Let the system S contain among its connectives an implication connective ‘I’ and a conjunction connective ‘&’. Let A1, A2, …, An ⊦ B abbreviate that B is provable on the hypotheses A1, A2, …, An for a suitable definition of “proof on hypotheses”, where A1, A2, …, An, B are well-formed expressions of S.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 18 , Issue 3 , September 1953 , pp. 234 - 236
Copyright
Copyright © Association for Symbolic Logic 1953

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References

1 Lewis, C. I. and Langford, C. H., Symbolic logic, New York and London, 1932, p. 247Google Scholar.

2 The deduction theorem in a functional calculus of first order based on strict implication, this Journal, vol. 11 (1946), pp. 115–118Google Scholar. The results of this paper were obtained for functional extensions of the Lewis systems S2 and 54. The results of the present paper also obtain for the corresponding functional extensions of the Lewis systems under discussion.

3 See Lewis and Langford, op. cit., particularly Appendix II.

4 For this and subsequent references to what has been established with respect to the deduction theorem in S2 and S4, see my paper, op. cit., on the deduction theorem.

5 As suggested originally by the referee, this condition could have been weakened to

6 See Lewis and Langford, op. cit., the Group III matrix on p. 493.

7 Rosenbloom, Paul, The elements of mathematical logic, New York, 1950, p. 60Google Scholar.

8 A theorem corresponding to III for S5 was established by Carnap, R. in Modalities and quantification, this Journal, vol. 11 (1946), p. 56Google Scholar.

9 See McKinsey, J. C. C. and Tarski, Alfred, Some theorems about the sentential calculi of Lewis and Heyting, this Journal, vol. 13 (1948), p. 5Google Scholar.