Editing History of logic (section)

===Logic after WWII===
[[File:Kripke.JPG|alt=Man with a beard and straw hat on a beach|thumb|[[Saul Kripke]]]]

After World War II, [[mathematical logic]] branched into four inter-related but separate areas of research: [[model theory]], [[proof theory]], [[computability theory]], and [[set theory]].<ref>See e.g. Barwise, ''Handbook of Mathematical Logic''</ref>

In set theory, the method of [[Forcing (mathematics)|forcing]] revolutionized the field by providing a robust method for constructing models and obtaining independence results. <!-- Kunen p. 235ff, Kanamori p. 114ff --> [[Paul Cohen]] introduced this method in 1963 to prove the independence of the [[continuum hypothesis]] and the [[axiom of choice]] from [[Zermelo–Fraenkel set theory]].<ref>{{cite journal | jstor=72252 | last1=Cohen | first1=Paul J. | title=The Independence of the Continuum Hypothesis, II | journal=Proceedings of the National Academy of Sciences of the United States of America | date=1964 | volume=51 | issue=1 | pages=105–110 | doi=10.1073/pnas.51.1.105 | pmid=16591132 | pmc=300611 | bibcode=1964PNAS...51..105C | doi-access=free }}</ref> His technique, which was simplified and extended soon after its introduction, has since been applied to many other problems in all areas of mathematical logic.

Computability theory had its roots in the work of Turing, Church, Kleene, and Post in the 1930s and 40s. It developed into a study of abstract computability, which became known as [[recursion theory]].<ref>Many of the foundational papers are collected in ''The Undecidable'' (1965) edited by Martin Davis</ref> The [[Turing degree|priority method]], discovered independently by [[Albert Muchnik]] and [[Richard Friedberg]] in the 1950s, led to major advances in the understanding of the [[degrees of unsolvability]] and related structures. <!-- cooper 246 --> Research into higher-order computability theory demonstrated its connections to set theory. <!-- sacks "higher recursion theory" --> The fields of [[constructive analysis]] and [[computable analysis]] were developed to study the effective content of classical mathematical theorems; these in turn inspired the program of [[reverse mathematics]]. A separate branch of computability theory, [[computational complexity theory]], was also characterized in logical terms as a result of investigations into [[descriptive complexity]].

Model theory applies the methods of mathematical logic to study models of particular mathematical theories. Alfred Tarski published much pioneering work in the field, which is named after a series of papers he published under the title ''Contributions to the theory of models''. <!-- "alfred tarski's work in model theory", vaught, JSL, https://www.jstor.org/stable/2273900 --> In the 1960s, [[Abraham Robinson]] used model-theoretic techniques to develop calculus and analysis based on [[non-standard analysis|infinitesimals]], a problem that first had been proposed by Leibniz. <!-- Keisler, fundamentals of model theory, HML, p. 48 -->

In proof theory, the relationship between classical mathematics and intuitionistic mathematics was clarified via tools such as the [[realizability]] method invented by [[Georg Kreisel]] and Gödel's [[Dialectica interpretation|''Dialectica'' interpretation]]. This work inspired the contemporary area of [[proof mining]]. The [[Curry–Howard correspondence]] emerged as a deep analogy between logic and computation, including a correspondence between systems of natural deduction and [[typed lambda calculus|typed lambda calculi]] used in computer science. As a result, research into this class of formal systems began to address both logical and computational aspects; this area of research came to be known as modern type theory. Advances were also made in [[ordinal analysis]] and the study of independence results in arithmetic such as the [[Paris–Harrington theorem]].

This was also a period, particularly in the 1950s and afterwards, when the ideas of mathematical logic begin to influence philosophical thinking. For example, [[tense logic]] is a formalised system for representing, and reasoning about, propositions qualified in terms of time. The philosopher [[Arthur Prior]] played a significant role in its development in the 1960s. [[Modal logic]]s extend the scope of formal logic to include the elements of [[Linguistic modality|modality]] (for example, [[Logical possibility|possibility]] and [[Necessary and sufficient conditions#Necessary conditions|necessity]]). The ideas of [[Saul Kripke]], particularly about [[possible world]]s, and the formal system now called [[Kripke semantics]] have had a profound impact on [[analytic philosophy]].<ref>Jerry Fodor, "[http://www.lrb.co.uk/v26/n20/jerry-fodor/waters-water-everywhere Water's water everywhere]", ''London Review of Books'', 21 October 2004</ref> His best known and most influential work is ''[[Naming and Necessity]]'' (1980).<ref>See ''Philosophical Analysis in the Twentieth Century: Volume 2: The Age of Meaning'', Scott Soames: "''Naming and Necessity'' is among the most important works ever, ranking with the classical work of Frege in the late nineteenth century, and of Russell, Tarski and Wittgenstein in the first half of the twentieth century". Cited in Byrne, Alex and Hall, Ned. 2004. 'Necessary Truths'. ''Boston Review'' October/November 2004</ref> [[Deontic logic]]s are closely related to modal logics: they attempt to capture the logical features of [[obligation]], [[Permission (philosophy)|permission]] and related concepts. Although some basic novelties [[syncretism|syncretizing]] mathematical and philosophical logic were shown by [[Bernard Bolzano#Metaphysics|Bolzano]] in the early 1800s, it was [[Ernst Mally]], a pupil of [[Alexius Meinong]], who was to propose the first formal deontic system in his ''Grundgesetze des Sollens'', based on the syntax of Whitehead's and Russell's [[propositional calculus]].

Another logical system founded after World War II was [[fuzzy logic]] by Azerbaijani mathematician [[Lotfi Asker Zadeh]] in 1965.