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The Shortest Possible Length of the Longest Implicational Axiom

Dolph Ulrich
Journal of Philosophical Logic
Vol. 25, No. 1 (Feb., 1996), pp. 101-108
Published by: Springer
Stable URL: http://www.jstor.org/stable/30227090
Page Count: 8
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The Shortest Possible Length of the Longest Implicational Axiom
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Abstract

A four-valued matrix is presented which validates all theorems of the implicational fragment, IF, of the classical sentential calculus in which at most two distinct sentence letters occur. The Wajsberg/Diamond-McKinsley Theorem for IF follows as a corollary: every complete set of axioms (with substitution and detachment as rules) must include at least one containing occurrences of three or more distinct sentence letters. Additionally, the matrix validates all IF theses built from nine or fewer occurrences of connectives and letters. So the classic result of Jagkovski for the full sentential calculus -that every complete axiom set must contain either two axioms of length at least nine or else one of length at least eleven-can be improved in the implicational case: every complete axiom set for IF must contain at least one axiom eleven or more characters long. Both results are "best possible", and both apply as well to most subsystems of IF, e.g., the implicational fragments of the standard relevance logics, modal logics, the relatives of implicational intutionism, and logics in the Lukasiewicz family.

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