On the General Interpretation of First-Order Quantifiers G. Aldo Antonelli University of California, Davis antonelli@ucdavis.edu 27 July 2013 Abstract While second-order quantifiers have long been known to admit non-standard, or “general” inter- pretations, first-order quantifiers (when properly viewed as predicates of predicates) also allow a kind of interpretation that does not presuppose the full power-set of that interpretation’s first-order domain. This paper explores some of the consequences of such “general” interpretations for (unary) first-order quantifiers in a general setting, emphasizing the effects of imposing various further con- straints that the interpretation is to satisfy. 1 Introduction It has been known since Henkin (1950) that second-order quantifiers admit of interpretations on which they are taken to range over some collection of subsets of the domain, a collection that possibly falls short of the full power set. To capture these interpretations formally, Henkin introduced the notion of a general model for second-order logic. A general model M supplies, besides an interpretation function I for the extra-logical constants, also a sequence ( D 1 , D 2 ,...), where D 1 is a non-empty domain of objects, and each D n+1 is a collection of n-place relations over D 1 (by identifying subsets of D 1 with the corresponding sets of 1-tuples, we can assume that D 2 ⊆ ( D 1 )). On Henkin’s account, D 1 is the range for the first-order variables, whereas each D n+1 provides the range for second-order relational variables whose values are n-place relations over D 1 . In order for the all the instances of the second- order comprehension schema to be satisfied, Henkin also assumed each domain D n+1 to be closed under Boolean operations (intersection and complement, say) as well to contain all relations R that can be obtained as projections of relations S ∈ D n+2 (see Enderton 2009, Henkin 1996). Many thanks to John Baldwin, Claude Laflamme, and Albert Visser for helpful comments. I am grateful to the referees for their useful criticisms, and especially for pointing me to the question of decidability, as well as to parts of the literature with which I was not previously familiar, notably Thomason and Johnson (1969), Bruce (1978), Väänänen (1978), López-Escobar (1991), and Andréka et al. (1998). 1 Penultimate version. Final version to appear in the Rev. Symb. Logic.