Varieties of logic (Q2879057)

This book addresses the question, or questions, of pluralism in logic. Is there just one system of logic rightly considered correct (monism), or do more than one have equal claim (pluralism)? For that matter, what does it mean to say one, or more, logic is correct? By `logic', the author intends the sort of consequence relation at work in rigorous deductive enterprises, especially mathematics as it is practiced. Given that, what is one to make of the fact there seem to be serious mathematical theories that call on logics incompatible with classical logic, such as Heyting's intuitionistic arithmetic with Church's thesis, or infinitesimal analysis, or the inconsistent, but non-trivial set theories and arithmetics of some paraconsistent logicians? The author here adopts an `eclectic' attitude, inspired somewhat by Hilbert, that all of these may well be considered proper logics, consistency or even practical applicability being the only criterion of legitimacy. But the matter is complex.NEWLINENEWLINE According to the author, terms like `valid', `logical consequence', etc., in the natural language of mathematics, are polysemous, having variations of meaning, even extension, much like vague or open-textured predicates, and subject to sharpening in different ways, in different contexts. Thus, a claim that an inference is, or is not, valid may be understood in different, perhaps incompatible ways, and subject to relativization to context. If the logical terms of the language of different logics, e.g., `if/then', `not', `or', `and', `for all', `some', etc., have the same meaning in the different theories, then, the author finds, an `indexical contextualism', wherein the content expressed by a term may be different in different contexts, much like expressions of orientation, `left of', `right of', etc., is appropriate for judgements of validity. On the other hand, if those terms have different meanings in different systems, then a benign monism regarding validity is nearly automatic, while the logical terms themselves are subject to a non-indexical, contextual relativism. Whether the logical terms do, or do not, have the same meaning in different theories, is itself a context-sensitive question. In some conversational contexts, it is appropriate to say one thing, in others, the other.NEWLINENEWLINE Since metatheoretic investigations themselves demand logic, questions of pluralism arise at that level, too. Is there but one logic for foundational studies in logic and mathematics? Is there a problem applying one logic to study a different one? In particular, is there a problem in making judgements of validity, etc., for one logic, given that such judgements may presuppose a perspective from a different logic? The author finds that, for the most part, there are no special problems in these regards.NEWLINENEWLINE Throughout the development of these themes, the arguments, and lessons learned, extend well beyond philosophical logic and the philosophy of logic, to take in the philosophy of language quite broadly. The author does, however, keep his focus fixed on the questions of meaning, context, etc. as they pertain to the language of logic, or logics, more narrowly construed.