Evert Willem Beth was a Dutch philosopher and logician, whose work principally concerned the foundations of mathematics.
Background
Beth 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 Doctor of Philosophy Evert Beth studied the same subjects at Utrecht University, but then also studied philosophy and psychology.
Education
His 1935 Doctor of Philosophy was in philosophy.
Career
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.
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
Semantic tableaux are a proof method for formal systems
Compare Gentzen"s natural deduction and sequent calculus, or even J. Alan Robinson"s resolution and Hilbert"s axiomatic systems
lieutenant 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).
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 semantics).
Views
fAents in occupied Holland in the Second World 'Var generated in Beth an enduring concern to eounter the effects of irrational forms of thought. He worked intensively in logic, the foundations of oiathematics and theoretical philosophy, dividing ois attention between the history of these topics and contemporary theories. Beth's rejection of the logical positivist view that some statements are Slrr>ply meaningless led him to historical investigations and the identification of an invalid Principle, which he called the Principle of the Absolute, in Aristotle’s methodology of science and philosophy. He linked the Locke-Berkeley Problem concerning general terms to Pre-Fregean theories of quantification.
From his experiences during the war, and t rough contacts with Alfred Tarski and others, cth developed a completely new and fundamental orientation in which formal and mathematical ogic are supported by a semantical methodology. e now introduced into Dutch philosophy the
category Falsity, on a par with the category Truth, treating the search for counterexamples and countermodels as the core method ot logical thought. This method of Semantical Tableaux defines logical validity as the impossibility of ever finding or construing a countermodel to the pair given class of premises: expected or desired conclusion. For a two-valued semantics this means that one can speak of a deduction as being valid on logical grounds only if it is impossible to find a model with respect to which all the premises are true whereas the expected/desired conclusion is false. On this account the definitions of logical particles are given as rules for the systematic search for countermodels. Beth applied this semantical method widely: to modal logic, intuitionist mathematics, the theory of definition, the history of philosophy, and the foundations of the physical sciences. Beth's influence has been considerable. He wrote 26 books and over 160 articles and papers. His method of Semantical Tableaux is now standard. Sources: Personal acquaintance.
Membership
Royal Netherlands Academy of Arts and Sciences]
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.