Beyond the frege boundary

Abstract

In sentences likeEvery teacher laughed we think ofevery teacher as aunary (=type <1>) quantifier — it expresses a property ofone place predicate denotations. In variable binding terms, unary quantifiers bind one variable. Two applications of unary quantifiers, as in the interpretation ofNo student likes every teacher, determine abinary (= (type <2>) quantifier; they express properties oftwo place predicate denotations. In variable binding terms they bind two variables. We call a binary quantifierFregean (orreducible) if it can in principle be expressed by the iterated application of unary quantifiers.

In this paper we present two mathematical properties which distinguish non-Fregean quantifiers from Fregean ones. Our results extend those of van Benthem (1989) and Keenan (1987a). We use them to show that English presents a large variety of non-Fregean quantifiers. Some are new here, others are familiar (though the proofs that they are non-Fregean are not).

The main point of our empirical work is to inform us regarding the types of quantification natural language presents — in particular (van Benthem, 1989) that it goes beyond the usual (Fregean) analysis which treats it as mere iterated application of unary quantifiers.

Secondarily, our results challenge linguistic approaches to “Logical Form” which constrain variable binding operators to “locally” bind just one occurrence of a variable, e.g., the Bijection Principle (BP) of Koopman and Sportiche (1983). The BP (correctly) blocks analyses likeFor which x, x's mother kissed x? forWho did his mother kiss? sinceFor which x would locally bind two occurrences ofx. But some of our irreducible binary quantifiers are naturally represented by operators which do locally bind two variables.

This paper is organized as follows: Section 1 provides an explicit formulation of our questions of concern. Section 2 classifies the English constructions which we show to be non-Fregean. Section 3 presents the mathematical properties which test for non-Fregean quantification and applies these tests to the constructions in Section 2. Proofs of the mathematical properties are given in the Appendix.