Henkin and Function Quantifiers

Henkin [7] introduced quantifiers which arise when usual (universal and existential) quantifiers are arranged in non-linear order. These quantifiers bre known as partially ordered quantifiers or Henkin quantifiers. Is. was observed already in [7] that languages with these quantifiers are stronger than first-order logic. A result of [ll] shows that even the weakest enkin quantifier has essentially the same expressive power as second-order logic. In this paper we introduce a simplification of Henkin quantifiers called function quantifiers. We show that function quantifiers are intimately connected with partially ordered quantifiers and suggest that they provide a natural framework for the study of partially ordered quantifiers and second-order definability in general. This paper continues work started in [9]. We are indebted to Alistair Lachlan and Lauri Hella for helpful discussions concerning preliminary versions of Section 8. We are also grateful to the referee for his or her remarks and suggestions. For basic notions concerning extensions of first-order logic and generalized uantifiers, the reader is referred to [ 11. Our logics 5? will mostly be extensions of 5$,,. obtained by adding geiazralized quantifiers in the sense of [12]. We recall the definition of a generalized quantifier from [ 121. Let K be a class of models of a relational similarity type r = (no,. .. , &). K is assumed to be closed under isomorphisms. The generalized quantifier Q asociated with K is defined as fQllows: