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Extendible sentential calculus

Published online by Cambridge University Press:  12 March 2014

H. Hiz
H. Hiz*
Affiliation:
Pennsylvania State University and University of Pennsylvania
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Extract

The system of sentential calculus presented below has peculiarities which may be of some interest. It is complete in the sense that every classical (two-valued) tautology is provable. In spite of this it is not Post complete (nor is it absolutely complete), i.e., one can add to the system an unprovable formula without proving a sentential variable as a theorem (or without making every formula a theorem). Thus the system may be extended by adjoining an unprovable formula without making the system inconsistent in the Post (or in absolute) sense. There are many distinct extensions of this kind. For every formula α1 that is not a theorem and which adjoined to the system does not make the system Post (or absolutely) inconsistent there is a formula α2 such that adjunction of α2 to the original system leaves α1 unprovable.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 24 , Issue 3 , September 1959 , pp. 193 - 202
Copyright
Copyright © Association for Symbolic Logic 1959

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References

[1]Church, Alonzo, Introduction to mathematical logic, vol. I, 1956.Google Scholar
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[3]Harrop, R., On the existence of finite models and decision procedures for propositional calculi, Proceedings of the Cambridge Philosophical Society, vol. 54, 1958, pp. 1–13.CrossRefGoogle Scholar
[4]Hiż, Henry, Extendible sentential calculus. Abstract 538–34, Notices of the American Mathematical Society, vol. 5 (1958), p. 34.Google Scholar
[5]Łoś, J., An algebraic proof of completeness for the two-valued prepositional calculus, Colloquium mathematicum, vol. II, 1951, pp. 236–240.CrossRefGoogle Scholar