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What is strict implication?

  • Ian Hacking (a1)
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C. I. Lewis intended his systems S1–S5 as contributions to the study of “strict implication”, but in his formulation, strict implication is so thoroughly intertwined with other notions, such as possibility and negation, that it remains a problem, to separate out the properties of strict implication itself. I shall solve this problem for S2–5 and von Wright's M. The results for S3–5 are given below, while the implicative parts of S2 and M, which are rather more complicated, are given in §5.

In this presentation, ‘⊃’ stands for strict implication, and missing brackets for ‘⊃’ are restored by association to the right. A strict formula is one of the form A ⊃ B. Axiom schemes are used throughout.

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References
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[1]Ackermann, Wilhelm, Begründung einer strengen Implikation, this Journal, vol. 22 (1956). pp. 113128.
[2]Anderson, A. R., and Belnap, N. J., The pure calculus of entailment, this Journal, vol. 27 (1962), pp. 1952.
[3]Curry, Haskell B., A theory of formal deducibility, Notre Dame Mathematical Lectures No. 6 (1950).
[4]Curry, Haskell B., The definition of negation by a fixed proposition in the inferential calculus, this Journal, vol. 17 (1951), pp. 98104.
[5]Curry, Haskell B., The permutability of rules in the classical inferential calculus, this Journal, vol. 17 (1951), pp. 245248.
[6]Gentzen, G., Untersuchungen über das logische Schliessen, Mathematische Zeitschrift, vol. 39 (19341935), pp. 176210 and 405–431.
[7]Kripke, Saul A., Distinguished constituents, this Journal, vol. 24 (1959), p. 323.
[8]Kripke, Saul A., The problem of entailment, this Journal, vol. 24 (1959), p. 324.
[9]Lewis, C. I., A survey of symbolic logic. University of California Press, 1918.
[10]Lewis, C. I. and Langford, C. H., Symbolic logic. Century Company, New York, 1932.
[11]McKinsey, J. C. C. and Tarski, A., Some theorems about the sentential calculi of Lewis and Heyting, this Journal, vol. 13 (1948), pp. 115.
[12]Wright, G. H. von, An Essay in modal logic, North Holland Publishing Co., Amsterdam, 1951.
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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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