Evert Willem Beth
Encyclopedia
Evert Willem Beth was a Dutch
philosopher and logician, whose work principally concerned the foundations of mathematics
.
, 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 Ph.D
. Evert Beth studied the same subjects at Utrecht University
, but then also studied philosophy
and psychology
. His 1935 Ph.D.
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.
He died in Amsterdam
.
are a proof method for formal system
s. Cf. Gentzen's natural deduction
and sequent calculus
, or even J. Alan Robinson
's resolution and Hilbert's
axiomatic systems. It is considered by many to be intuitively simple, particularly for students not acquainted with the study of logic (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 imply 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 imply 
.
).
Netherlands
The Netherlands is a constituent country of the Kingdom of the Netherlands, located mainly in North-West Europe and with several islands in the Caribbean. Mainland Netherlands borders the North Sea to the north and west, Belgium to the south, and Germany to the east, and shares maritime borders...
philosopher and logician, whose work principally concerned the foundations of mathematics
Foundations of mathematics
Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...
.
Biography
Beth was born in AlmeloAlmelo
Almelo is a municipality and a city in the eastern Netherlands. The main population centres in the town are Aadorp, Almelo, Mariaparochie and Bornerbroek....
, a small town in the eastern Netherlands
Netherlands
The Netherlands is a constituent country of the Kingdom of the Netherlands, located mainly in North-West Europe and with several islands in the Caribbean. Mainland Netherlands borders the North Sea to the north and west, Belgium to the south, and Germany to the east, and shares maritime borders...
. His father had studied mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
and physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
at the University of Amsterdam
Universiteit van Amsterdam
The University of Amsterdam or the UvA is a public research university located in Amsterdam, the Netherlands. Founded in 1632 as the Athenaeum Illustre by the scholars Gerardus Vossius and Caspar Barlaeus, it is the third-oldest university in the Netherlands...
, where he had been awarded a Ph.D
Doctor of Philosophy
Doctor of Philosophy, abbreviated as Ph.D., PhD, D.Phil., or DPhil , in English-speaking countries, is a postgraduate academic degree awarded by universities...
. Evert Beth studied the same subjects at Utrecht University
Utrecht University
Utrecht University is a university in Utrecht, Netherlands. It is one of the oldest universities in the Netherlands and one of the largest in Europe. Established March 26, 1636, it had an enrollment of 29,082 students in 2008, and employed 8,614 faculty and staff, 570 of which are full professors....
, but then also studied philosophy
Philosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...
and psychology
Psychology
Psychology is the study of the mind and behavior. Its immediate goal is to understand individuals and groups by both establishing general principles and researching specific cases. For many, the ultimate goal of psychology is to benefit society...
. His 1935 Ph.D.
Ph.D.
A Ph.D. is a Doctor of Philosophy, an academic degree.Ph.D. may also refer to:* Ph.D. , a 1980s British group*Piled Higher and Deeper, a web comic strip*PhD: Phantasy Degree, a Korean comic series* PhD Docbook renderer, an XML renderer...
was in philosophy.
In 1946, he became professor of logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
and the foundations of mathematics in Amsterdam
Amsterdam
Amsterdam is the largest city and the capital of the Netherlands. The current position of Amsterdam as capital city of the Kingdom of the Netherlands is governed by the constitution of August 24, 1815 and its successors. Amsterdam has a population of 783,364 within city limits, an urban population...
. Apart from two brief interruptions – a stint in 1951 as a research assistant to Alfred Tarski
Alfred Tarski
Alfred Tarski was a Polish logician and mathematician. Educated at the University of Warsaw and a member of the Lwow-Warsaw School of Logic and the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and carried out research in mathematics at the University of...
, and in 1957 as a visiting professor at Johns Hopkins University
Johns Hopkins University
The Johns Hopkins University, commonly referred to as Johns Hopkins, JHU, or simply Hopkins, is a private research university based in Baltimore, Maryland, United States...
– 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.
He died in Amsterdam
Amsterdam
Amsterdam is the largest city and the capital of the Netherlands. The current position of Amsterdam as capital city of the Kingdom of the Netherlands is governed by the constitution of August 24, 1815 and its successors. Amsterdam has a population of 783,364 within city limits, an urban population...
.
Definition theorem
The definition theorem states that a predicate (or function or constant) is implicitly definable if and only if it is explicitly definable.Semantic tableaux
Semantic tableauxMethod of analytic tableaux
In proof theory, the semantic tableau is a decision procedure for sentential and related logics, and a proof procedure for formulas of first-order logic. The tableau method can also determine the satisfiability of finite sets of formulas of various logics. It is the most popular proof procedure...
are a proof method for formal system
Formal system
In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive an expression from one or more other premises that are antecedently supposed or derived . The axioms and rules may be called a deductive apparatus...
s. Cf. Gentzen's natural deduction
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...
and sequent calculus
Sequent calculus
In proof theory and mathematical logic, sequent calculus is a family of formal systems sharing a certain style of inference and certain formal properties. The first sequent calculi, systems LK and LJ, were introduced by Gerhard Gentzen in 1934 as a tool for studying natural deduction in...
, or even J. Alan Robinson
J. Alan Robinson
John Alan Robinson is a philosopher , mathematician and computer scientist. He is University Professor Emeritus at Syracuse University, United States....
's resolution and Hilbert's
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
axiomatic systems. It is considered by many to be intuitively simple, particularly for students not acquainted with the study of logic (Wilfrid Hodges
Wilfrid Hodges
Wilfrid Augustine Hodges is a British mathematician, known for his work in model theory. He was Professor of Mathematics at Queen Mary, University of London from 1987 to 2006, and is the author of numerous books on logic....
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
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
, First-order logic and automated theorem proving).
One starts out with the intention of proving that a certain set
of formulae imply another formula , given a set of rules determined by the semantics of the formulae's connectivesLogical connective
In logic, a logical connective is a symbol or word used to connect two or more sentences in a grammatically valid way, such that the compound sentence produced has a truth value dependent on the respective truth values of the original sentences.Each logical connective can be expressed as a...
(and quantifiers, in first-order logic
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate 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 imply .Beth models
These are a class of relational models for non-classical logic (cf. Kripke semanticsKripke semantics
Kripke semantics is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke. It was first made for modal logics, and later adapted to intuitionistic logic and other non-classical systems...
).
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, Epistemologie mathematique et psychologie (with J. Piaget). 352 pp. Paris P.U.F. 1961.
- Evert W. Beth, Formal MethodsFormal methodsIn computer science and software engineering, formal methods are a particular kind of mathematically-based techniques for the specification, development and verification of software and hardware systems...
: An introduction to symbolic logic and to the study of effective operations in arithmetic and logic. D. Reidel Publishing Company / Dordecht-Holland, 1970. ISBN 90-277-0069-9.