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Double-Negation Elimination in Some Propositional Logics

Michael Beeson, Robert Veroff and Larry Wos
Studia Logica: An International Journal for Symbolic Logic
Vol. 80, No. 2/3, Negation in Constructive Logic (Jul. - Aug., 2005), pp. 195-234
Published by: Springer
Stable URL: http://www.jstor.org/stable/20016716
Page Count: 40
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Double-Negation Elimination in Some Propositional Logics
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Abstract

This article answers two questions (posed in the literature), each concerning the guaranteed existence of proofs free of double negation. A proof is free of double negation if none of its deduced steps contains a term of the form n(n(t)) for some term t, where n denotes negation. The first question asks for conditions on the hypotheses that, if satisfied, guarantee the existence of a double-negation-free proof when the conclusion is free of double negation. The second question asks about the existence of an axiom system for classical propositional calculus whose use, for theorems with a conclusion free of double negation, guarantees the existence of a double-negation-free proof. After giving conditions that answer the first question, we answer the second question by focusing on the Łukasiewicz three-axiom system. We then extend our studies to infinite-valued sentential calculus and to intuitionistic logic and generalize the notion of being double-negation free. The double-negation proofs of interest rely exclusively on the inference rule condensed detachment, a rule that combines modus ponens with an appropriately general rule of substitution. The automated reasoning program Otter played an indispensable role in this study.

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