Editing Willard Van Orman Quine (section)

===Logic===

Over the course of his career, Quine published numerous technical and expository papers on formal logic, some of which are reprinted in his ''Selected Logic Papers'' and in ''The Ways of Paradox''. His most well-known collection of papers is ''From A Logical Point of View''. Quine confined logic to classical bivalent [[first-order logic]], hence to truth and falsity under any (nonempty) [[universe of discourse]]. Hence the following were not logic for Quine:

* Higher-order logic and set theory. He referred to [[higher-order logic]] as "set theory in disguise";
* Much of what ''[[Principia Mathematica]]'' included in logic was not logic for Quine.
* Formal systems involving [[intension]]al notions, especially [[modal logic|modality]]. Quine was especially hostile to modal logic with [[Quantification (logic)|quantification]], a battle he largely lost when [[Saul Kripke]]'s [[Kripke semantics|relational semantics]] became canonical for [[modal logic]]s.

Quine wrote three undergraduate texts on formal logic:
* ''Elementary Logic''. While teaching an introductory course in 1940, Quine discovered that extant texts for philosophy students did not do justice to [[quantification theory]] or [[first-order predicate logic]]. Quine wrote this book in 6 weeks as an ''[[ad hoc]]'' solution to his teaching needs.
* ''Methods of Logic''. The four editions of this book resulted from a more advanced undergraduate course in logic Quine taught from the end of World War II until his 1978 retirement.
* ''Philosophy of Logic''. A concise and witty undergraduate treatment of a number of Quinian themes, such as the prevalence of use-mention confusions, the dubiousness of [[modal logic|quantified modal logic]], and the non-logical character of higher-order logic.

''Mathematical Logic'' is based on Quine's graduate teaching during the 1930s and 1940s. It shows that much of what ''[[Principia Mathematica]]'' took more than 1000 pages to say can be said in 250 pages. The proofs are concise, even cryptic. The last chapter, on [[Gödel's incompleteness theorem]] and [[Tarski's indefinability theorem]], along with the article Quine (1946), became a launching point for [[Raymond Smullyan]]'s later lucid exposition of these and related results.

Quine's work in logic gradually became dated in some respects. Techniques he did not teach and discuss include [[analytic tableau]]x, [[recursion#Functional recursion|recursive function]]s, and [[model theory]]. His treatment of [[metalogic]] left something to be desired. For example, ''Mathematical Logic'' does not include any proofs of [[soundness]] and [[completeness (logic)|completeness]]. Early in his career, the notation of his writings on logic was often idiosyncratic. His later writings nearly always employed the now-dated notation of ''Principia Mathematica''. Set against all this are the simplicity of his preferred method (as exposited in his ''Methods of Logic'') for determining the satisfiability of quantified formulas, the richness of his philosophical and linguistic insights, and the fine prose in which he expressed them.

Most of Quine's original work in formal logic from 1960 onwards was on variants of his [[predicate functor logic]], one of several ways that have been proposed for doing logic without [[Quantification (logic)|quantifier]]s. For a comprehensive treatment of predicate functor logic and its history, see Quine (1976). For an introduction, see ch.&nbsp;45 of his ''Methods of Logic''.

Quine was very warm to the possibility that formal logic would eventually be applied outside of philosophy and mathematics. He wrote several papers on the sort of [[Boolean algebra (logic)|Boolean algebra]] employed in [[electrical engineering]], and with [[Edward J. McCluskey]], devised the [[Quine–McCluskey algorithm]] of reducing [[Boolean equation]]s to a minimum covering sum of [[prime implicant]]s.