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A Gentzen-type calculus of sequents for single-operator propositional logic*

Published online by Cambridge University Press:  12 March 2014

John Riser
John Riser*
Affiliation:
State University of New York at Albany
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Extract

The following describes a calculus of sequents (SLK) patterned after the calculus LK of Gentzen in [1] but restricted to the formulas of propositional logic and modified so that the only connective is the Sheffer stroke. The ‘elimination theorem’ (Hauptsatz) is proved for SLK and a decision procedure is specified for determining whether a given formula in stroke notation is tautologous. In addition, SLK is proved consistent and complete. Subsequent remarks indicate briefly how the calculus can be modified so as to employ the dagger as the only connective.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 32 , Issue 1 , June 1967 , pp. 75 - 80
Copyright
Copyright © Association for Symbolic Logic 1967

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Footnotes

*

I wish to thank the referee for suggesting the general direction this paper should take and for commenting specifically upon several points.

References

[1]Gentzen, Gerhard, Investigations into logical deduction, American philosophical quarterly, vol. 1 (1964), pp. 288–306; vol. 2 (1965), pp. 204–218. Translated by M. E. Szabo. Originally published as Untersuchungen über das logische Schliessen in Mathematische Zeitschrift, vol. 39 (1935).Google Scholar
[2]Nicod, J. G. P., A reduction in the number of the primitive propositions of logic, Proceedings of the Cambridge Philosophical Society, vol. 19 (1916), pp. 32–41.Google Scholar
[3]Sheffer, H. M., A set of five independent postulates for Boolean algebras, with application to logical constants, Transactions of the American Mathematical Society, vol. 14 (1913), pp. 481–488.CrossRefGoogle Scholar