Partially interpreted Henkin quantifiers

pp. 93-108

in: Jaakko Hintikka, Tadeusz Czarnecki, Tomasz Placek, Artur Rojszczak (eds), Philosophy and logic in search of the Polish tradition, Berlin, Springer, 2003

Abstract

The present paper defines the concept of a partially interpreted generalized quantifier and studies some of the basic model-theoretic properties of the resulting logic. As such, the paper continues and completes a line of research started by Kripke (1975) and continued by Feferman (1984), van Benthem (1985), Blarney (1986), and Langholm (1988) which studied the logic of partially interpreted predicates. Surprisingly enough, the logic studied here which combines the partial interpretation of predicates with that of quantifiers turns out to have very nice abstract properties: Compactness, the Lövenheim-Skolem Properties, an effective Interpolation Property, and the property that it defines its own truth-predicate.