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An Admissible Semantics for Propositionally Quantified Relevant Logics

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Abstract

The Routley-Meyer relational semantics for relevant logics is extended to give a sound and complete model theory for many propositionally quantified relevant logics (and some non-relevant ones). This involves a restriction on which sets of worlds are admissible as propositions, and an interpretation of propositional quantification that makes ∀ pA true when there is some true admissible proposition that entails all p-instantiations of A. It is also shown that without the admissibility qualification many of the systems considered are semantically incomplete, including all those that are sub-logics of the quantified version of Anderson and Belnap’s system E of entailment, extended by the mingle axiom and the Ackermann constant t. The incompleteness proof involves an algebraic semantics based on atomless complete Boolean algebras.

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Authors and Affiliations

  1. Centre for Logic, Language and Computation, Victoria University of Wellington, P.O. Box 600, Wellington, New Zealand

    Robert Goldblatt & Michael Kane

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  1. Robert Goldblatt
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  2. Michael Kane
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Correspondence to Robert Goldblatt.

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Goldblatt, R., Kane, M. An Admissible Semantics for Propositionally Quantified Relevant Logics. J Philos Logic 39, 73–100 (2010). https://doi.org/10.1007/s10992-009-9109-7

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  • DOI: https://doi.org/10.1007/s10992-009-9109-7

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