Logical constants: the variable fortunes of an elusive notion

Summary

Logic: inferential and expressive power Logic is usually taken to be the study of reasoning - and the highest rank of nobility for an assertion or an inference is ‘logical validity’. There are proof-theoretic methods for describing all validities for a given language, and completeness theorems assure us that we have found ‘all there is’ to a particular style of reasoning. But in addition to this inferential power, logic is also about expressive power. Reasoning needs a language supplying ‘logical forms’, and what determines the choice of these? Logical languages contain various forms of expression: propositional connectives, quantifiers, or modalities, whose meanings are analyzed in addition to their inferential behavior. But what makes these particular notions ‘logical’, as opposed to others? Perhaps the usual expressive completeness argument reassures us that the Boolean connectives capture all there is to two-valued propositional reasoning. But no similar result is known for first-order predicate logic, the major working system of modern logic.

This concern is not part of the basic ‘agenda’ in textbooks, partly because it is seldom raised, and partly because there is no consensus on an answer. Sol Feferman is one of the small group of authors who do think about this basic issue (cf. [9]), and I am happy to contribute a little piece of my own thinking on these matters. But before doing so, let me note that not all great logicianswould find the effort worthwhile. E.g., Bernard Bolzano's system, pioneering in so many ways, did not contain any privileged set of logical operations. For him, the distinction logical/non-logical is merely one of methodology. Forms of assertion for a style of reasoning arise by fixing the meanings of some expressions (these will have a ‘logic’ then), and letting others vary. But this distinction can go in more than one way. Granting this liberality, I still would like to find out what makes the usual ‘logical constants’ tick.

My aims in this are different from Feferman's excellent analysis, who mentions a ‘demarcation’ of logic as a major concern - and first-order logic as a preferred target. I myself feel no need for a principled separation of logic from other territories, mathematics, linguistics, computer science, psychology, or whatever.