Logic in the second half of the twentieth century
Authors: Cook Roy T. , Logan Shay Allen 
Published: November 2023

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’ (□) 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?