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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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