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The system LD

Published online by Cambridge University Press:  12 March 2014

Haskell B. Curry
Haskell B. Curry*
Affiliation:
The Pennsylvania State College
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Extract

In lectures at Notre Dame University in 1948 I presented what was essentially a semantical analysis of the propositions formed from the elementary propositions of a formal system by compounding with propositional connectives and quantifiers. The system LD is one of the systems treated there in the chapter on negation. It is substantially the minimal calculus with excluded middle, and was first considered by Johansson. It corresponds semantically to the situation where we have refutability with completeness, and so it may be regarded as the natural system of strict implication. This paper gives the details of certain results concerning this system which have been reported elsewhere. Some related remarks concerning other systems will also be included. Acquaintance with the lectures cited is presupposed; they are referred to as TFD.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 17 , Issue 1 , March 1952 , pp. 35 - 42
Copyright
Copyright © Association for Symbolic Logic 1952

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References

1 Johansson, I. Der Minimalkalkül, ein reduzierter intuitionistscher Formalismus. Compositio mathematica, vol. 4 (1936), pp. 119–136Google Scholar, especially p. 129, end of §3.

2 Proceedings of the International Congress of Mathematicians, Cambridge, Mass., U.S.A., Aug. 30–Sept. 6, 1950.

3 This is true also for Yl and Yr.

4 Cited in TFD [37]. Johansson (I.c.) asserts that the Glivenko Theorem does not hold for LM, but he interprets it as a relation between LM and LK, not LD.

5 Cf. the proof for Λr below.