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.


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