Quantifiers and Inference

Abstract

Quantifiers play a prominent role in logical inference. Not surprisingly, then, a general study of quantification shows many points of contact with the general theory of inference. In this brief paper, three encounters of this sort will be pointed out, providing a more systematic perspective behind ongoing work in the field of generalized quantifiers. First, we consider the role of quantifiers in a hierarchy of inference patterns, starting with purely “syllogistic” schemata such as ‘Q AB,Q BC/Q AC’ and ascending up to more complex schemata involving interaction with Boolean operators and standard quantifiers. We shall consider axiomatization (`direct logic’) as well as definability (‘inverse logic’). Next, we move from inference with quantifiers to inference as quantification. Starting from a well-known similarity between quantifiers and conditional operators, we develop an analogy between the study of general quantifiers and that of general inference relations, which have become prominent in recent years. In particular, the same technical questions often make sense in both fields. Finally, these matters are also placed in a proof-theoretic, rather than a model-theoretic setting, with generalized quantifiers arising from the combinatorics of deduction with variable binding. The aim of this paper is to put some current research in perspective, pointing out connections and further directions, rather than to make any profound technical contribution. Its style is that of generalized quantifier theory as it has developed in the newer ‘linguistic’ tradition (cf. [van Benthem 1986]), although we also seek further contacts with the older ‘mathematical’ tradition (see [Westerståhl 1989] for an extensive presentation merging both).