Logic in the second half of the twentieth century

By the end of the first half of the twentieth century, logic had become a mature philosophical and mathematical discipline. As happens in mature disciplines, in the second half of the twentieth century logicians developed subspecialties and narrow research programmes and, as a result, extended both in depth and in subtlety the analyses of logic that began in the first half of the century (see Logic in the early twentieth century).

Topics that were of central importance during the first half of the twentieth century, including first-versus higher-order logic, the foundations of set theory, and Gödelian-style incompleteness results, continued to play an important role in the latter decades of the century (see Second and higher-order logics; Second-order logic, Philosophical issues in; Set theory; Gödel’s theorems). But important new strands of research emerged as well.

One such new approach was inspired by broadening the sorts of questions raised by the classic independence results proven by Gödel and others in the early twentieth century. While those results showed that certain principles were insufficient to prove this-or-that proposition, the new Reverse Mathematics focused on determining what principles are necessary in order to prove particular propositions.

Another topic of increasing importance was also inspired by Kurt Gödel’s diagonalisation lemma and the centuries-old Liar paradox. As a result, a resurgence of work on the Liar paradox emerged in the second half of the twentieth century (see Semantic paradoxes and theories of truth). This work included both theories of truth that retained classical logic, such as Alfred Tarski’s hierarchy (Tarski 1944) (see Tarski’s definition of truth), and theories of truth that eschewed classical logic in favour of non-classical systems, such as Saul Kripke’s fixed point approach (Kripke 1975).

Another area that was ripe for exploration during the later decades of the century involved various ways to extend these formal systems with new resources. The most important such extension was modal logic, which involved supplementing standard logical systems with additional operators for ‘necessarily’ ($\square$) and ‘possibly’ ($◇$) (see Modal logic; Modal logic, Philosophical issues in).

With Kripke’s work on truth demonstrating the potential for applying non-classical logic to puzzles regarding truth, a great deal of attention was also paid to potential applications of non-classical formalisms to other philosophical problems. Notable amongst these are the exploration of non-classical logics that lie between classical logic and intuitionistic logic (the intermediate logics), the use of various non-classical logics to formalise reasoning with vague notions (e.g. ‘is bald’), and the exploration of the idea that the meanings of logical operators might be given in terms of formal proof rules governing their use (which might not lead to a set of rules that recaptures traditional classical logic) (see Vagueness).

Finally, near the end of the twentieth century (and continuing into the twenty-first century), these examinations led to a predictable result: logicians have increasingly (though by no means universally) begun to embrace pluralism. This pluralism comes in a variety of forms – see Cook (2010) and Russell (2019) for useful taxonomies – but their central motivation is this: there are a lot of different things that seem to lay equal claim to the title ‘logic’. Mightn’t there be a framework that allows them to all do so?