Evert Willem Beth
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Evert Willem Beth (7 July 1908 – 12 April 1964) was a Dutch philosopher and
logician Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
, whose work principally concerned the
foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathe ...
. He was a member of the Significs Group.


Biography

Beth was born in
Almelo Almelo () is a municipality and a city in the eastern Netherlands. The main population centres in the town are Aadorp, Almelo, Mariaparochie, and Bornerbroek. Almelo has about 72,000 inhabitants in the middle of the rolling countryside of Twente, ...
, a small town in the eastern
Netherlands ) , anthem = ( en, "William of Nassau") , image_map = , map_caption = , subdivision_type = Sovereign state , subdivision_name = Kingdom of the Netherlands , established_title = Before independence , established_date = Spanish Netherl ...
. His father had studied mathematics and
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
at the
University of Amsterdam The University of Amsterdam (abbreviated as UvA, nl, Universiteit van Amsterdam) is a public research university located in Amsterdam, Netherlands. The UvA is one of two large, publicly funded research universities in the city, the other being ...
, where he had been awarded a PhD. Evert Beth studied the same subjects at
Utrecht University Utrecht University (UU; nl, Universiteit Utrecht, formerly ''Rijksuniversiteit Utrecht'') is a public research university in Utrecht, Netherlands. Established , it is one of the oldest universities in the Netherlands. In 2018, it had an enrollme ...
, but then also studied philosophy and
psychology Psychology is the scientific study of mind and behavior. Psychology includes the study of conscious and unconscious phenomena, including feelings and thoughts. It is an academic discipline of immense scope, crossing the boundaries between ...
. His 1935 PhD was in philosophy. In 1946, he became professor of
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premise ...
and the foundations of mathematics in
Amsterdam Amsterdam ( , , , lit. ''The Dam on the River Amstel'') is the capital and most populous city of the Netherlands, with The Hague being the seat of government. It has a population of 907,976 within the city proper, 1,558,755 in the urban ar ...
. Apart from two brief interruptions – a stint in 1951 as a research assistant to
Alfred Tarski Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician a ...
, and in 1957 as a visiting professor at
Johns Hopkins University Johns Hopkins University (Johns Hopkins, Hopkins, or JHU) is a private research university in Baltimore, Maryland. Founded in 1876, Johns Hopkins is the oldest research university in the United States and in the western hemisphere. It consi ...
– 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. He died in
Amsterdam Amsterdam ( , , , lit. ''The Dam on the River Amstel'') is the capital and most populous city of the Netherlands, with The Hague being the seat of government. It has a population of 907,976 within the city proper, 1,558,755 in the urban ar ...
.


Contributions to logic


Definition theorem

The definition theorem states that a predicate (or function or constant) is implicitly definable if and only if it is explicitly definable. Further explanation is provided under Beth definability.


Semantic tableaux

Beth's most famous contribution to formal logic is
semantic tableaux In proof theory, the semantic tableau (; plural: tableaux, also called truth tree) is a decision procedure for sentential and related logics, and a proof procedure for formulae of first-order logic. An analytic tableau is a tree structure compu ...
, which are
decision procedure In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question of the input values. An example of a decision problem is deciding by means of an algorithm wheth ...
s for
propositional logic Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...
and
first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
. It is a semantic method—like
Wittgenstein Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 – 29 April 1951) was an Austrians, Austrian-British people, British philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy o ...
's
truth tables A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional argume ...
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 Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic ph ...
, Russell and Whitehead, and
Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...
, or even
Gentzen Gerhard Karl Erich Gentzen (24 November 1909 – 4 August 1945) was a German mathematician and logician. He made major contributions to the foundations of mathematics, proof theory, especially on natural deduction and sequent calculus. He died o ...
's
natural deduction In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning. This contrasts with Hilbert-style systems, which instead use ax ...
. 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 2''n'' 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 \Gamma \, of formulae imply another formula \varphi\, , given a set of rules determined by the semantics of the formulae's
connectives In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary c ...
(and quantifiers, in
first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
). The method is to assume the concurrent truth of every member of \Gamma \, and of \neg \varphi (the negation of \varphi\, ), 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 \Gamma \cup \ is inconsistent, and thus that the formulae of \Gamma\, together imply \varphi \,.


Beth models

These are a class of relational models for
non-classical logic Non-classical logics (and sometimes alternative logics) are formal systems that differ in a significant way from standard logical systems such as propositional and predicate logic. There are several ways in which this is done, including by way of ...
(cf.
Kripke semantics Kripke semantics (also known as relational semantics or frame semantics, and often confused with possible world semantics) is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke and André Jo ...
).


Books

* 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 In computer science, formal methods are mathematically rigorous techniques for the specification, development, and verification of software and hardware systems. The use of formal methods for software and hardware design is motivated by the exp ...
: 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

* Gerrit Mannoury *
Method of analytic tableaux In proof theory, the semantic tableau (; plural: tableaux, also called truth tree) is a decision procedure for sentential and related logics, and a proof procedure for formulae of first-order logic. An analytic tableau is a tree structure computed ...


References

* * *


External links


Beth Prize 2013Evert Willem Beth Foundation
{{DEFAULTSORT:Beth, Evert Willem 1908 births 1964 deaths Mathematical logicians Dutch logicians Formal methods people University of Amsterdam faculty Utrecht University alumni People from Almelo 20th-century Dutch mathematicians Members of the Royal Netherlands Academy of Arts and Sciences 20th-century Dutch philosophers