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On the definition of negation by a fixed proposition in inferential calculus

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

It is well known that negation can be defined by postulating a fixed proposition F and specifying that

where “≡” indicates identity by definition. I propose to discuss this way of introducing negation in relation to the inferential systems of my Notre Dame lectures of 1948. Acquaintance with these lectures is presupposed; they will be referred to as TFD. The definition (1) was used there, essentially, in connection with the T-systems; but it was not used in connection with the L-systems because of a certain difficulty which I was not able to overcome at that time. This paper contains proofs of results concerning L-systems which have been announced elsewhere.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 17 , Issue 2 , June 1952 , pp. 98 - 104
Copyright
Copyright © Association for Symbolic Logic 1952

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References

1 A theory of formal deducibility. Notre Dame mathematical lectures, no. 6. Published by the University of Notre Dame, 1950Google Scholar.

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

3 It would not be of much help if we had. The argument here depends on the fact that the elimination process shortens the proof, a fact which is not evident from a mere statement of the theorem.