On permutation in simplified semantics

Journal of Philosophical Logic 38 (3):333 - 341 (2009)
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Abstract

This note explains an error in Restall’s ‘Simplified Semantics for Relevant Logics (and some of their rivals)’ (Restall, J Philos Logic 22(5):481–511, 1993 ) concerning the modelling conditions for the axioms of assertion A → (( A → B ) → B ) (there called c 6) and permutation ( A → ( B → C )) → ( B → ( A → C )) (there called c 7). We show that the modelling conditions for assertion and permutation proposed in ‘Simplified Semantics’ overgenerate. In fact, they overgenerate so badly that the proposed semantics for the relevant logic R validate the rule of disjunctive syllogism. The semantics provides for no models of R in which the “base point” is inconsistent. This problem is not restricted to ‘Simplified Semantics.’ The techniques of that paper are used in Graham Priest’s textbook An Introduction to Non-Classical Logic (Priest, 2001 ), which is in wide circulation: it is important to find a solution. In this article, we explain this result, diagnose the mistake in ‘Simplified Semantics’ and propose two different corrections.

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Author Profiles

Tony Roy
California State University, San Bernardino
Greg Restall
University of Melbourne

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References found in this work

Introduction to Non-Classical Logic.Graham Priest - 2001 - Cambridge and New York: Cambridge University Press.
An Introduction to Non-Classical Logic.Graham Priest - 2001 - Bulletin of Symbolic Logic 12 (2):294-295.
Simplified semantics for relevant logics (and some of their rivals).Greg Restall - 1993 - Journal of Philosophical Logic 22 (5):481 - 511.

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