Event Detail

In his 1950 dissertation, Leon Henkin showed how to provide higher- order quantifiers with non-standard, or “general” interpretations, on which, for instance, second-order quantifiers are taken to range over collections of subsets of the domain that may fall short of the full power-set. In contrast, first-order quantifiers are usually regarded as immune to this sort of non- standard interpretations, since their semantics is ordinarily taken to be completely determined once a first-order domain of objects is selected.

The asymmetry is particularly evident from the point of view of the modern theory of generalized quantifiers, according to which a first-order quantifier is construed as a predicate of subsets of the domain. But the generalized conception still views first-order quantifiers as predicates over the full power-set. Accordingly, the possibility that they, similarly to their second-order counterparts, might denote arbitrary collections of subsets has not been pursued in full generality.

This talk introduces a Henkin-style semantics for arbitrary first-order quantifiers, exploring some of the resulting properties, and emphasizing the effects of imposing various further closure conditions on the second-order component of the interpretation. Among other results, we show by a model- theoretic argument that in certain cases the notion of validity relative to models satisfying the closure conditions is axiomatizable.