The Concept of Logical Consequence

Abstract

There is broad agreement among logicians about the technical definition of the notion of logical consequence, especially as it applies to extensional first-order languages. The definition most logicians accept is the familiar one framed in terms of assignments of truth values to sentence letters, denotations to individual constants, and extensions to predicates, these assignments being relative to some nonempty set of objects. There is less agreement, however, on the pre-theoretic notion this technical definition is supposed to represent, and very little discussion of whether it actually does represent it adequately.' This paper will be concerned with the latter two points, primarily as they arise in the context of extensional first-order languages. In the first section, I consider what several logicians say informally about the notion of logical consequence. There is significant variation among these accounts, they are sometimes poorly explained, and some of them are clearly at odds with the usual technical definition. In the second section, I first argue (subsection 2. 1) that a certain kind of informal account-one that includes elements of necessity, generality, and apriority-is approximately correct. Next (subsection 2.2) I refine this account and consider several important questions about it, including the appropriate characterization of necessity, the criterion for selecting logical constants, and the exact role of apriority. I argue, among other things, that there is no need to recognize a special logical sense of necessity and that the selection of terms to serve as logical constants is ultimately a pragmatic matter. (Subsection 2.2.1 contains my fully developed account of the informal notion of logical consequence; subsection 2.2.2 includes criticism of recent work by Sher and Etchemendy.) In the third section, I consider whether the informal