, Volume 41, Issue 4, pp 671-709
Date: 26 May 2012

Consequence Mining

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Abstract

The standard semantic definition of consequence with respect to a selected set X of symbols, in terms of truth preservation under replacement (Bolzano) or reinterpretation (Tarski) of symbols outside X, yields a function mapping X to a consequence relation \(\Rightarrow_X\) . We investigate a function going in the other direction, thus extracting the constants of a given consequence relation, and we show that this function (a) retrieves the usual logical constants from the usual logical consequence relations, and (b) is an inverse to—more precisely, forms a Galois connection with—the Bolzano–Tarski function.

We thank Johan van Benthem, Stephen Read, and Göran Sundholm for helpful remarks, and in particular Lloyd Humberstone, Dave Ripley, and an anonymous referee for very useful comments on an earlier version of this paper. We also thank the audiences at several seminars and workshops where we presented this material, in various stages, for inspiring discussion.