What a Linguist Might Want from a Logic of Most and Other Generalized Quantifiers

Abstract

When Dov and I received our logical education — Dov is quite a bit younger than I am, still we got our education at more or less the same time — the overall picture of what logic was, seemed comfortably clear. There were four main branches of mathematical logic — model theory, set theory, recursion theory and proof theory. Underlying this clear and simple picture were a number of widely shared assumptions, some of them to the effect that certain basic problems of logic had essentially been solved. Of central importance among these were: the belief that one had, through the work of Peano, Frege, Peirce, Russell, Hilbert, Gentzen and others, a definitive formal analysis of the notion of logical deduction (or logical proof); the belief that the conceptual problem of defining logical consequence and logical truth, and of explicating the relationship between those concepts and the concepts of truth, reference and satisfaction on one hand, and their relationship with the concept of a formal deduction on the other, had found a definitive solution in the work of Gödel and Tarski; and, finally, the conviction that with the characterizations of recursive functions proposed by Gödel, Turing and Church, one had uncovered what had to be the right concept of computability. With regard to set theory the situation was perhaps a little different; then as now, one could not help feeling that each of the available systems of set theory (the most popular ones, Z(ermelo-)F(raenkel) and G(ödel-)B(ernays), among them) embodied an element of arbitrariness. Nevertheless, for better or worse even in this domain a certain consensus had established itself which heavily favoured GB and ZF.