Modal ontology and generalized quantifiers
The paper focuses on the distinction between two modal metaphysical views, which \textit{T. Williamson} [``Necessitism, contingentism, and plural quantification'', Mind, New Ser. 119, No. 475, 657--748 (2010; \url{doi:10.1093/mind/fzq042})] has called ``necessitism'' and ``contingentism'' and that he has proposed as substitutes for the notions of possibilism and actualism, respectively. Necessitists will subscribe to the thesis that necessarily everything is necessarily something. Contingentists will deny this thesis. Williamson has shown that necessitists can draw distinctions that the contingentists cannot draw on the basis of certain first-order modal theory. The author wants to test Williamson's result in a logical context wider than that of Williamson's. In particular, the author considers the logical context resulting from the addition of generalized quantifiers to the logical syntax of Williamson's first-order modal theory. Reference of those quantifiers is to infinite classes of objects. The formal semantics proposed in the paper for the extended formal language is a possible world semantics with an outer domain and inner domains (i.e., domains associated to each possible world of the semantic models). Each inner domain is a subset of the outer domain. The outer domain is the union of the inner domains. The semantics will also satisfy the so called property actualism condition, i.e., the condition that, at any possible world \(w\), \(n\)-relational symbols are assigned sets of \(n\)-tuples of the domain associated to \(w\). Quantifiers whose range of quantification is based on the outer domain is referred by the author as outer quantifiers; those whose range of quantification is based on inner domains are called inner quantifiers. When the semantics interprets quantifiers on the basis of the outer domain, the author speaks of a logic of outer quantifiers; when it interprets them on the basis of inner domains, he speaks of a logic of inner quantifiers. The author provides a formal definition of what it is for necessitists to draw the distinctions contingentists can draw (and vice-versa) in terms of mappings of formulas into formulas that are neutral in the dispute between necessitism and contingentism. A model theoretic definition of neutrality is provided. The author proves that (1) contingentists can draw the distinctions necessitists can draw if and only if the logic with inner quantifiers is at least as expressive as the logic with outer quantifiers; and 2) necessitists can draw the distinction contingentists can draw if and only if the logic with outer quantifiers is at least as expressive as the logic with inner quantifiers. He also shows that necessitists can draw the distinctions in question if and only if the language with generalized quantifiers relativizes, and that contingentists can draw the above distinction if and only if generalized quantifiers are first-order definable. Finally, there is a discussion on the philosophical significance of the main technical results in the paper. This discussion includes the question of the support that those results would give to Williamson's claim in favor of necessiticism.
scientific article; zbMATH DE number 6217459
scientific article