In 1946, he became professor of logic and the foundations of mathematics in Amsterdam. Apart from two brief interruptions – a stint in 1951 as a research assistant to Alfred Tarski, and in 1957 as a visiting professor at Johns Hopkins University – he held the post in Amsterdam continuously until his death in 1964. His was the first academic post in his country in logic and the foundations of mathematics, and during this time he contributed actively to international cooperation in establishing logic as an academic discipline.
Beth's most famous contribution to formal logic is semantic tableaux, which are decision procedures for propositional logic and first-order logic. It is a semantic method—like Wittgenstein's truth tables or J. Alan Robinson's resolution—as opposed to the proof of theorems in a formal system, such as the axiomatic systems employed by Frege, Russell and Whitehead, and Hilbert, or even Gentzen's natural deduction. Semantic tableaux are an effective decision procedure for propositional logic, whereas they are only semi-effective for first-order logic, since first-order logic is undecidable, as showed by Church's theorem. This method is considered by many to be intuitively simple, particularly for students who are not acquainted with the study of logic, and it is faster than the truth-table method (which requires a table with 2n rows for a sentence with n propositional letters). For these reasons, Wilfrid Hodges for example presents semantic tableaux in his introductory textbook, Logic, and Melvin Fitting does the same in his presentation of first-order logic for computer scientists, First-order logic and automated theorem proving.
One starts out with the intention of proving that a certain set of formulae entail another formula , given a set of rules determined by the semantics of the formulae's connectives (and quantifiers, in first-order logic). The method is to assume the concurrent truth of every member of and of (the negation of ), and then to apply the rules to branch this list into a tree-like structure of (simpler) formulae until every possible branch contains a contradiction. At this point it will have been established that is inconsistent, and thus that the formulae of together entail .
Beth modelsedit
This section needs expansion. You can help by adding to it. (April 2014)
Evert W. Beth, The foundations of mathematics. A study in the philosophy of science. XXVΊ + 722 pp. Amsterdam, North-Holland 1959.
Evert W. Beth, Épistémologie mathématique et psychologie (with J. Piaget). 352 pp. Paris P.U.F. 1961.
Evert W. Beth, Formal Methods: An introduction to symbolic logic and to the study of effective operations in arithmetic and logic. D. Reidel Publishing Company / Dordecht-Holland, 1962.
Evert W. Beth, Aspects of Modern Logic. D. Reidel Publishing Company / Dordecht-Holland, 1971.
Mooij, J. J. A. "Beth, Evert Willem (1908–1964)". Biographical Dictionary of the Netherlands: 1880–2000. Retrieved 22 April 2013.
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This article includes a list of general references but it lacks sufficient corresponding inline citations Please help to improve this article by introducing more precise citations April 2013 Learn how and when to remove this template message Evert Willem Beth 7 July 1908 12 April 1964 was a Dutch philosopher and logician whose work principally concerned the foundations of mathematics He was a member of the Significs Group Part of Beth s publications Contents 1 Biography 2 Contributions to logic 2 1 Beth definability theorem 2 2 Semantic tableaux 2 3 Beth models 3 Books 4 See also 5 References 6 External linksBiography editBeth was born in Almelo a small town in the eastern Netherlands His father had studied mathematics and physics at the University of Amsterdam where he had been awarded a PhD Evert Beth studied the same subjects at Utrecht University but then also studied philosophy and psychology His 1935 PhD was in philosophy In 1946 he became professor of logic and the foundations of mathematics in Amsterdam Apart from two brief interruptions a stint in 1951 as a research assistant to Alfred Tarski and in 1957 as a visiting professor at Johns Hopkins University he held the post in Amsterdam continuously until his death in 1964 His was the first academic post in his country in logic and the foundations of mathematics and during this time he contributed actively to international cooperation in establishing logic as an academic discipline In 1953 he became member of the Royal Netherlands Academy of Arts and Sciences 1 He died in Amsterdam Contributions to logic editBeth definability theorem edit This section needs expansion You can help by adding to it April 2014 The Beth definability theorem states that for first order logic a property or function or constant is implicitly definable if and only if it is explicitly definable Further explanation is provided under Beth definability Semantic tableaux edit Beth s most famous contribution to formal logic is semantic tableaux which are decision procedures for propositional logic and first order logic It is a semantic method like Wittgenstein s truth tables or J Alan Robinson s resolution as opposed to the proof of theorems in a formal system such as the axiomatic systems employed by Frege Russell and Whitehead and Hilbert or even Gentzen s natural deduction Semantic tableaux are an effective decision procedure for propositional logic whereas they are only semi effective for first order logic since first order logic is undecidable as showed by Church s theorem This method is considered by many to be intuitively simple particularly for students who are not acquainted with the study of logic and it is faster than the truth table method which requires a table with 2n rows for a sentence with n propositional letters For these reasons Wilfrid Hodges for example presents semantic tableaux in his introductory textbook Logic and Melvin Fitting does the same in his presentation of first order logic for computer scientists First order logic and automated theorem proving One starts out with the intention of proving that a certain set G displaystyle Gamma nbsp of formulae entail another formula f displaystyle varphi nbsp given a set of rules determined by the semantics of the formulae s connectives and quantifiers in first order logic The method is to assume the concurrent truth of every member of G displaystyle Gamma nbsp and of f displaystyle neg varphi nbsp the negation of f displaystyle varphi nbsp and then to apply the rules to branch this list into a tree like structure of simpler formulae until every possible branch contains a contradiction At this point it will have been established that G f displaystyle Gamma cup neg varphi nbsp is inconsistent and thus that the formulae of G displaystyle Gamma nbsp together entail f displaystyle varphi nbsp Beth models edit This section needs expansion You can help by adding to it April 2014 These are a class of relational models for non classical logic cf Kripke semantics Books editEvert W Beth The foundations of mathematics A study in the philosophy of science XXVI 722 pp Amsterdam North Holland 1959 Evert W Beth Epistemologie mathematique et psychologie with J Piaget 352 pp Paris P U F 1961 Evert W Beth Formal Methods An introduction to symbolic logic and to the study of effective operations in arithmetic and logic D Reidel Publishing Company Dordecht Holland 1962 Evert W Beth Aspects of Modern Logic D Reidel Publishing Company Dordecht Holland 1971 See also editGerrit Mannoury Method of analytic tableauxReferences edit Evert Willem Beth 1908 1964 Royal Netherlands Academy of Arts and Sciences Retrieved 20 July 2015 Francella Miriam 1999 Evert Willem Beth s Scientific Philosophy Grazer Philosophische Studien 57 221 236 doi 10 5840 gps19995712 Heyting Arend 1966 In memoriam Evert Willem Beth 1909 1964 Notre Dame Journal of Formal Logic 7 4 289 295 doi 10 1305 ndjfl 1093958744 Mooij J J A Beth Evert Willem 1908 1964 Biographical Dictionary of the Netherlands 1880 2000 Retrieved 22 April 2013 External links edit nbsp Wikimedia Commons has media related to Evert Willem Beth Beth Prize 2013 Evert Willem Beth Foundation Retrieved from https en wikipedia org w index php title Evert Willem Beth amp oldid 1180245183, wikipedia, wiki, book, books, library,
