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Two additions to positive implication

Published online by Cambridge University Press:  12 March 2014

A. N. Prior
A. N. Prior*
Affiliation:
Manchester University
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Extract

Let LIC be the implicational fragment of Dummett's LC, axiomatised by Bull in [1] by adding the axiom (L) CCCpqrCCCqprr to positive implication. The system which Bull in [2] calls OIC, i.e. positive implication plus (0) CCCCpqqrCCCpqrr, is contained in LIC (it is thesis 34 in [1], p. 193); but it does not contain it, as the following matrix (due to Bull)

verifies OIC but not (L) (CCC342CCC4322 = 2). Nor is (L) deducible in OIC supplemented by intuitionist axioms for negation, since the latter are all verified by the above matrix with Cp5 for Np.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 29 , Issue 1 , March 1964 , pp. 31 - 32
Copyright
Copyright © Association for Symbolic Logic 1964

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References

[1]Bull, R. A., The implicational fragment of Dummett's LC, this Journal, vol. 27 (1962), pp. 189–192.Google Scholar
[2]Bull, R. A., Some results for implicational calculi, this Journal, vol. 29 (1964), pp. 33–39.Google Scholar