Modal Quanti cation over Structured Domains

1 Quantiiers as Modal Operators 1.1 Motivations The semantics for quantiiers described in this paper can be viewed both as a new semantics for generalized quantiiers and as a new look at standard rst-order quantiication, bringing the latter closer to modal logic. The standard semantics for generalized quantiiers interprets a monadic generalized quantiier Q as a set of subsets of a domain. For example, the quantiier "there are precisely two" is interpreted by the set of all subsets of the domain which contain precisely two elements. A formula Qx' is true in a model if the set of elements satisfying ' belongs to the interpretation of the quantiier; in our example, if there are precisely two elements satisfying '. The existential quantiier can be treated as a generalized quantiier, too: it is interpreted as the set of all non-empty subsets of the domain. The universal quantiier is interpreted by the singleton set containing the whole domain. The quantiiers listed so far are rst-order deenable in the following sense: they can be deened using ordinary quantiiers and equality. Many interesting generalized quanti-ers are not rst-order deenable. The present study is motivated by the work of Michiel van Lambalgen (1991) on Gentzen-style proof theory for the quantiiers "for many" (its dual is interpreted as a non-principal lter), "for uncountably many" and "for almost all" (the latter contains all subsets of the domain which have Lebesgue measure 1). All those quantiiers are not rst-order deenable. They have Hilbert-style axiomatizations, but until lately no one believed that they can have a reasonable Gentzen-style proof theory. In order to devise such a proof theory, van Lambalgen used a translation of generalized quan-tiier formulas into a rst-order language enriched with a predicate R of indeenite arity. (all free variables displayed). Observe that this translation is reminiscent of the standard translation of modal formulas into rst-order logic, with the sequence of free variables playing the role of the "actual world" and the quantiier ranging over the variables "accessible" from the given sequence. The idea behind such a translation is as follows. When generalized quantiiers are viewed as rst-order operators (binding rst-order variables), it becomes clear that a variable bound by a generalized quan-tiier cannot in general take any possible value. Its range is restricted, and this restriction can be deened using an accessibility relation. Then the elimination rule for Q with a premise Qx'(x; y) would introduce …