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Independence of Two Nice Sets of Axioms for the Propositional Calculus

T. Thacher Robinson
The Journal of Symbolic Logic
Vol. 33, No. 2 (Jun., 1968), pp. 265-270
DOI: 10.2307/2269872
Stable URL: http://www.jstor.org/stable/2269872
Page Count: 6
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Independence of Two Nice Sets of Axioms for the Propositional Calculus
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Abstract

Kanger [4] gives a set of twelve axioms for the classical propositional Calculus which, together with modus ponens and substitution, have the following nice properties: (0.1) Each axiom contains $\supset$, and no axiom contains more than two different connectives. (0.2) Deletions of certain of the axioms yield the intuitionistic, minimal, and classical refutability1 subsystems of propositional calculus. (0.3) Each of these four systems of axioms has the separation property: that if a theorem is provable in such a system, then it is provable using only the axioms of that system for $\supset$, and for the other connectives, if any, actually occurring in that theorem. (0.4) All twelve axioms are independent. It is easily seen that two of Kanger's axioms can be shortened, and that two others can be replaced by a single axiom which is the same length as one of the two which it replaces, without disturbing properties (0.1)-(0.3). These alterations have advantages of simplicity and elegance, but bring property (0.4) into question, in that similarities among some of the axioms in the altered system make demonstrations of independence considerably more difficult. It is the purpose of this paper to show that independence is nonetheless provable for the simplified system, and in another system which also satisfies (0.1)-(0.3), in which f (falsehood) is taken to be primitive instead of ∼ (negation). Nonnormal truth-tables2 are used to obtain the independence of one of the axioms.

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