Unifying quantified modal logic
Garson argues for the importance of non-possibilist first-order modal logical systems and intends to unify several of these systems in the same syntactic and semantic framework. Syntactically, many of the major formal systems for those logics are presented as extensions of a calculus that the author calls ``system G'', which is a Gentzen sort of calculus for modal free logic. Among such extensions of G there are some of the most important formal systems of free logic for the objectual interpretation of the quantifiers (and where terms are not necessarily rigid designators). It is important to note that Garson shows that certain rules for the non-possibilist first-order modal formal systems are simplified within the syntactic framework of G. The semantic unification is achieved by both characterizing a semantic system based on what Garson refers to as ``G-models'' and describing different conditions that can be imposed on such models together with the syntactic rules corresponding to such conditions. A G-model is a possible worlds semantic structure with an accessibility relation, containing in addition a non-empty set \(D\) of objects, a set \(I\) of individual concepts (that is, functions from possible worlds into \(D\)) over which will range the universal first-order quantifiers, additional sets of individual concepts that will serve as domains for the different possible worlds. Each one of those possible world domains are not necessarily subsets of \(I\) and the intensions of some terms may be chosen outside any of such domains and even outside \(I\), from the set of all individual concepts. For the extensions of G considered by author, he offers completeness and soundness proofs. He also shows several independence results and indicates the limitations of his unification
sequent calculus for first-order modal logic
conceptual interpretation of quantifiers
scientific article; zbMATH DE number 5000206