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An alternative semantics for quantified relevant logic. (English) Zbl 1100.03011

Summary: The quantified relevant logic RQ is given a new semantics in which a formula \(\forall xA\) it true when there is some true proposition that implies all \(x\)-instantiations of \(A\). Formulae are modelled as functions from variable-assignments to propositions, where a proposition is a set of worlds in a relevant model structure. A completeness proof is given for a basic quantificational system QR from which RQ is obtained by adding the axiom EC of ‘extensional confinement’: \(\forall x(A\vee B)\to(A\vee\forall xB)\), with \(x\) not free in \(A\). Validity of EC requires an additional model condition involving the Boolean difference of propositions. A QR-model falsifying EC is constructed by forming the disjoint union of two natural arithmetical structures in which negation is interpreted by the minus operation.

MSC:

03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)
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