The Practical Turn in Logic

Abstract

The great advances in logic in the last century and a half saw a turn from its historical preoccupation with arguing and reasoning to a preoccupation with mathematics. It was a re-orientation that made possible important gains in both the foundations and the methodology of mathematics. The foundational contribution was largely of philosophical interest. It sought to establish a basis for logicism, for the reduction of mathematics to logic. 1 The methodological contribution also has its philosophical significance, but it threw its net more widely, capturing the interest of those who thought that mathematics could benefit deeply from the rigour and the standards of exact proof that the new logic was in process of articulating. It is difficult to overestimate the significance of the mathematical turn in logic. Not only did the new logic narrow logic’s former range of interests, it was able to do so only after determining that the traditional syllogistic approach to logic was inadequate for logic’s new ambitions. Ever since its inception 2500 years thence, logic had been in all essentials the logic of the syllogism. The mathematical turn brought an abrupt end to Aristotle’s hegemony. Given the venerability and persistence of that long-lived influence, it is perhaps not wholly inexplicable that mathematical logicians did not entirely break with the traditional line that logic is about reasoning and arguing. There are plenty of textbooks on mathematical logic, including some of the best and most senior, in which we find it said, without a shred of irony or embarrassment, that mathematical logic is the most general, or the basic, theory of reasoning. Those of greater circumspection would claim that the new symbolic logic is the root theory of mathematical reasoning. It would be a mistake to overlook the fact that mathematical logicians have been quick to recognize various respects in which even the claim of logic to be a theory of (mathematical) reasoning is implausible. To that end, various distinctions have been invoked: