%Nr: DS-2000-04
%Author: Paul van Ulsen
%Title: E.W. Beth als logicus
The subject of this dissertation is the logical work of E.W.~Beth. In
addition, there is a short biography and an introduction to some of
Beth's methodological and philosophical ideas.
Evert Willem Beth (1908--1964) was born in Almelo, a small town near
the Dutch-German border. He was the son of H.J.E.~Beth and H.~de
Groot. His father studied mathematics and physics at the University of
Amsterdam, where he received his Ph.D. in mathematics, thereafter
working as teacher in mathematics and physics in secondary
schools. E.W.~Beth studied mathematics and physics at the University
of Utrecht, followed by a study in philosophy and psychology. Evert
Beth's Ph.D. (1935) was in philosophy (faculty of arts), because the
borderland between philosophy and mathematics did not yet exist as an
academic discipline in the faculty of science at that time.
In 1946 Beth became in Amsterdam the first professor of logic and
foundations of mathematics in the Netherlands. He held this position
in Amsterdam until his death in 1964. He also held two positions
outside Holland: in 1951 as research assistent of A.~Tarski in Berkely
(UC) and in 1957 as professor of methodology at Johns Hopkins
University in Baltimore.
The aim of this study is to show the diversity of Beth's logical
systems and what binds them (both systematically and historically)
together. Beth's main contributions to logic were the definition
theorem, semantic tableaux and the Beth models. The foundation of his
work was Gentzen's extended Hauptsatz, the subformula theorem and an
extensive use of (Tarskian) model theory.
Beth's work was a combination of syntactical and semantical
components. The definition theorem (1953) is a counterpart of
deductive completeness. Beth's proof is primarily syntactic: he uses
the midsequent, topology and reduced logic.
With his Definition Theorem and his non-normal valuations, Beth
created the tools for the next stage in his development, the semantic
tableaux (1954--1955). With the semantic tableaux Beth explored
different areas: classical logic, modal logic and intuitionistic
logic. The semantic tableaux give a rapid decision procedure, their
basis is a binairy splitting tree. In combination with his semantic
tableaux Beth made a proof-theoretic variant: the deductive tableaux.
During his entire professional career Beth was interested in
intuitionistic logic, but he was himself not an intuitionist and
disliked intuitionistic philosphy. Beth combined his semantic tableaux
with trees and choice sequence, thus creating the Beth models
(1956). With the application (and in his case duplication) of
Brouwer's fundamental theorem he avoided non-intuitionistic
mathematics in his intuitionistic semantics and completeness proof.
Another area was the use of his earlier developed non-standard
valuations: in combination with his so-called help (subordinate)
tableaux and valuations it was possible to study intuitionistic and
modal logic as in Kripke world semantics.
During the last period of his life (1960--1964) Beth tried to make his
logical research subservient to a diverse range of applications: the
study of language, theorem proving, mathematical heuristics and
translation methods in natural languages.
Beth had considerable influence in international organisations. He
foresaw their importance as early as the 1940s. In Europe, directly
after World War II, there was no supportive climate for the studies in
formal methods: no money, no professors and a scientific community,
which disliked logic and the philosophy of science. So, he tried to
set up international networks and organised congresses to get the
recognition he needed for money and jobs. He understood that getting
recognition in the Netherlands was only possible with international
support. But there was also an idealistic component in the motives
underlying his efforts: he wanted not only to improve the Dutch
situation, but also to further the theories of logic on a world scale.
Nor did he limit himself to only pure logic. In Holland he worked
towards a combination of formal philosophy, philosophy of science,
pure logic and foundations of mathematics. He was the first to bring
these several studies together. Nowadays logic is an internationally
recognized scientific discipline. As I show in the biography, we owe a
lot of that to Beth.
Beth had ideas about philosophy, methodology and the philosophy of
science as well. His aim was to create what he called a scientific
philosophy, i.e. philosophy without speculation. He was furious at
those modern philosophical movements like existantialism and the
like. He thought of philosophy as not monolithic and static, but as
changing in time. In this dissertation there is a small part devoted
to this subject: Beth's wishes for pure philosophy and the philosophy
of mathematics. Finally, I give a short impression of Beth's
methodology (logic) for classical and quantum mechanics.