Identity in Mares-Goldblatt models for quantified relevant logic (Q2056221)

Relevant logics like Anderson and Belnap's \(\mathsf{R}\) described in [\textit{A. R. Anderson} et al., Entailment. The logic of relevance and necessity. Vol. I. Princeton, NJ: Princeton University Press (1975; Zbl 0323.02030)] are motivated by a thesis that valid entailments \(\varphi\rightarrow\psi\) should require some sort of relevance between the antecedent and consequent. In the propositional case, one necessary (but not necessarily sufficient) formal indicator of relevance is the \textit{variable sharing property} that an entailment's antecedent and consequent share a common propositional atom. Thus, a formula like \(p\rightarrow(q\rightarrow q)\) may be rejected on grounds of relevance despite the tautologous nature of its consequent. When moving to more complicated languages, formal characterizations of relevance are less obvious. However, some cases of irrelevance are nevertheless recognizable. In the case of \textit{identity}, the same intuitions that cause a relevant logician to reject \(p\rightarrow(q\rightarrow q)\) in general suggest that \(p\rightarrow (t=t)\) may be rejected on similar grounds. Having appropriate accounts of identity is particularly pressing for its importance in distinguishing \textit{relevant predication} from merely \textit{accidental} predication as discussed in [\textit{J. M. Dunn}, J. Philos. Log. 16, 347--381 (1987; Zbl 0638.03003)]. One can characterize the relevant predication of \(P(x)\) of \(t\) by the truth of the formula \(\forall x(x=t\rightarrow P(x))\). Thus, an account of \textit{e.g.} \(\mathsf{R}\) in which both quantification and identity receive appropriate treatment is important. The author focuses two frameworks for quantified relevant logic: the \textit{Mares-Goldblatt} models described in [\textit{E. D. Mares} and \textit{R. Goldblatt}, J. Symb. Log. 71, No. 1, 163--187 (2006; Zbl 1100.03011)] and the \textit{Fine-Mares} models emerging from the works [\textit{K. Fine}, J. Philos. Log. 3, 347--372 (1974; Zbl 0296.02013); \textit{K. Fine}, J. Philos. Log. 17, No. 1, 27--59 (1988; Zbl 0646.03013); \textit{E. D. Mares}, Stud. Log. 51, No. 1, 1--20 (1992; Zbl 0788.03028)]. Although the Fine-Mares model theory includes an account of identity, the Mares-Goldblatt semantics lacks identity. The author thus investigates equipping the Mares-Goldblatt model theory with a satisfactory mechanism to capture extensions of \(\mathsf{R}\) with quantification and identity. The author iteratively builds on the framework, first considering \(\mathsf{R}^{=}\), an extension of \(\mathsf{R}\) with an identity predicate that allows substitution of identicals in extensional contexts and proving soundness and completeness with respect to a modified model theory. Secondly, an extension \(\mathsf{R}^{=}_{sub}\) in which substitution of identicals is valid in all contexts (\textit{i.e.} in formulae in which relevant entailment \(\rightarrow\) appears) and shows soundness and completeness with respect to a modification of Mares-Goldblatt models with conditions placed on propositions. The development of the model theory concludes with an extension \(\mathsf{QR}^{=}_{sub}\) including both quantifiers and identity with substitution in general. The extension is shown to be sound and complete with respect to the author's modification of the Mares-Goldblatt model theory. \(\mathsf{QR}^{=}_{sub}\) is important as it provides sufficient expressivity for an appropriate account of relevant predication. The author concludes by demonstrating additional value for the framework by showing that the inclusion of identity in the Mares-Goldblatt semantics exposes subtle distinctions between the approach and the Fine-Mares semantics. In particular, the author uses this point of comparison to discuss a more nuanced view of the constant domains of the Mares-Goldblatt model theory.