Logical Consequence Inside Out

Abstract

Tarski’s definition of logical consequence for an interpreted language rests on the distinction between extra-logical symbols, whose interpretation is allowed to vary across models, and logical symbols, aka logical constants, whose interpretation remains fixed. In this perspective, logicality come first, and consequence is a by-product of the division between logical and extra-logical symbols. The problem of finding a conceptually motivated account for this division is a long-standing issue in the philosophy of logic. Our aim here is to short-circuit this issue and lay the basis for a shift in perspective: let consequence come first, so that the demarcation of a set of constants can be viewed as the by-product of the analysis of a relation of logical consequence. The idea for extracting logical constants from a consequence relation is the following: they are the symbols which are essential to the validity of at least one inference, in the sense that replacing them or varying their interpretation would destroy the validity of that inference. Conversely, definitions of logical consequence can be construed as providing us with mappings from sets of symbols to consequence relations. Extraction of constants is then expected to be an ‘inverse’ to generation of consequence relations.

We thank Johan van Benthem and Frederik Engström for valuable comments. Our work was supported by IHPST and ANR (Logiscience program), ESF (the Eurocores LogICCC program), and the Swedish Council for Research.