Logical Consequence Inside Out

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 byproduct of the division between logical and extra-logical symbols. Our aim here is to 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 (the ones selected as logical constants) to consequence relations. Extraction of constants is expected to be an ‘inverse’ to generation of consequence relations. In Sections 1 and 2, we introduce a general substitutional framework for the abstract study of consequence relations. Extraction of constants is presented in Section 3. It is shown that extraction thus defined does not straightforwardly provide an inverse to generation of consequence relations. To circumvent this limitation, we consider in Section 4 ‘richness’ properties of languages that make things better. In Section 5, we prove that extraction and generation constitute a Galois connection, considering families of expansions of a language, instead of a single language at a time. This gives us the correspondence for non-substitutional Tarskian consequence as a limit case.