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Alfred Tarski (; January 14, 1901 – October 26, 1983), born Alfred Teitelbaum,School of Mathematics and Statistics, University of St Andrews

''School of Mathematics and Statistics, University of St Andrews''.
was a Polish-American
logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and ar ...

logic
ian and
mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...

mathematician
. A prolific author best known for his work on
model theory In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical ...
,
metamathematics Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories. Emphasis on metamathematics (and perhaps the creation of the term it ...
, and
algebraic logic In mathematical logic, algebraic logic is the reasoning obtained by manipulating equations with Free variables and bound variables, free variables. What is now usually called classical algebraic logic focuses on the identification and algebraic de ...
, he also contributed to
abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
,
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

topology
,
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

geometry
,
measure theory Measure is a fundamental concept of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contai ...
,
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal sys ...
,
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...
, and
analytic philosophy Analytic philosophy is a branch and tradition of philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about existence Existence is the ability of an entity to interact with physical reality ...
. Educated in Poland at the
University of Warsaw The University of Warsaw ( pl, Uniwersytet Warszawski, la, Universitas Varsoviensis), established in 1816, is the largest university A university () is an educational institution, institution of higher education, higher (or Tertiary education ...
, and a member of the Lwów–Warsaw school of logic and the Warsaw school of mathematics, he immigrated to the United States in 1939 where he became a naturalized citizen in 1945. Tarski taught and carried out research in mathematics at the
University of California, Berkeley The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California) is a public In public relations Public relations (PR) is the practice of managing and disseminating information from an individual or an organization ...

University of California, Berkeley
, from 1942 until his death in 1983. Feferman A. His biographers Anita Burdman Feferman and
Solomon Feferman Solomon Feferman (December 13, 1928 – July 26, 2016) was an American philosopher and mathematician who worked in mathematical logic. Life Solomon Feferman was born in The Bronx in New York City to working-class parents who had immigrated to the ...
state that, "Along with his contemporary,
Kurt Gödel Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician Logic is an interdisciplinary field which studies truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dict ...
, he changed the face of logic in the twentieth century, especially through his work on the concept of
truth Truth is the property of being in accord with fact A fact is something that is true True most commonly refers to truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In ...

truth
and the theory of models." Feferman & Feferman, p.1


Life

Alfred Tarski was born Alfred Teitelbaum (
Polish Polish may refer to: * Anything from or related to Poland Poland ( pl, Polska ), officially the Republic of Poland ( pl, Rzeczpospolita Polska, links=no ), is a country located in Central Europe. It is divided into 16 Voivodeships of Pol ...
spelling: "Tajtelbaum"), to parents who were
Polish Jews The history of the Jews in Poland dates back at least 1,000 years. For centuries, Poland Poland, officially the Republic of Poland, is a country located in Central Europe. It is divided into 16 Voivodeships of Poland, administrat ...
in comfortable circumstances. He first manifested his mathematical abilities while in secondary school, at Warsaw's ''Szkoła Mazowiecka''. Nevertheless, he entered the
University of Warsaw The University of Warsaw ( pl, Uniwersytet Warszawski, la, Universitas Varsoviensis), established in 1816, is the largest university A university () is an educational institution, institution of higher education, higher (or Tertiary education ...
in 1918 intending to study
biology Biology is the natural science that studies life and living organisms, including their anatomy, physical structure, Biochemistry, chemical processes, Molecular biology, molecular interactions, Physiology, physiological mechanisms, Development ...

biology
. Feferman & Feferman, p.26 After Poland regained independence in 1918, Warsaw University came under the leadership of
Jan Łukasiewicz Jan Łukasiewicz (; 21 December 1878 – 13 February 1956) was a Polish logician and philosopher who is best known for Polish notation and Łukasiewicz logic. He was born in Lemberg, a city in the Austrian Galicia, Galician Kingdom of Austria-Hungar ...

Jan Łukasiewicz
,
Stanisław Leśniewski
Stanisław Leśniewski
and
Wacław Sierpiński Wacław Franciszek Sierpiński (; 14 March 1882 – 21 October 1969) was a Polish people, Polish mathematician. He was known for contributions to set theory (research on the axiom of choice and the continuum hypothesis), number theory, theory of fu ...

Wacław Sierpiński
and quickly became a world-leading research institution in logic, foundational mathematics, and the philosophy of mathematics. Leśniewski recognized Tarski's potential as a mathematician and encouraged him to abandon biology. Henceforth Tarski attended courses taught by Łukasiewicz, Sierpiński,
Stefan Mazurkiewicz Stefan Mazurkiewicz (25 September 1888 – 19 June 1945) was a Polish mathematician who worked in mathematical analysis, topology, and probability. He was a student of Wacław Sierpiński and a member of the Polish Academy of Learning (''PAU''). ...

Stefan Mazurkiewicz
and
Tadeusz Kotarbiński Tadeusz Marian Kotarbiński (; 31 March 1886 – 3 October 1981) was a Poles, Polish philosopher, logician and ethicist. A pupil of Kazimierz Twardowski, he was one of the most representative figures of the Lwów–Warsaw school of logic, Lwów– ...

Tadeusz Kotarbiński
, and in 1924 became the only person ever to complete a doctorate under Leśniewski's supervision. His thesis was entitled ''O wyrazie pierwotnym logistyki'' (''On the Primitive Term of Logistic''; published 1923). Tarski and Leśniewski soon grew cool to each other. However, in later life, Tarski reserved his warmest praise for , which was reciprocated. In 1923, Alfred Teitelbaum and his brother Wacław changed their surname to "Tarski". The Tarski brothers also converted to
Roman Catholicism The Catholic Church, also known as the Roman Catholic Church, is the largest Christian church, with 1.3 billion baptised Baptism (from the Greek language, Greek noun βάπτισμα ''báptisma'') is a Christians, Christian r ...
, Poland's dominant religion. Alfred did so even though he was an avowed
atheist Atheism, in the broadest sense, is an absence of belief A belief is an attitude Attitude may refer to: Philosophy and psychology * Attitude (psychology) In psychology Psychology is the science of mind and behavior. Psy ...

atheist
. After becoming the youngest person ever to complete a doctorate at Warsaw University, Tarski taught logic at the Polish Pedagogical Institute, mathematics and logic at the University, and served as Łukasiewicz's assistant. Because these positions were poorly paid, Tarski also taught mathematics at a Warsaw secondary school; before World War II, it was not uncommon for European intellectuals of research caliber to teach high school. Hence between 1923 and his departure for the United States in 1939, Tarski not only wrote several textbooks and many papers, a number of them ground-breaking, but also did so while supporting himself primarily by teaching high-school mathematics. In 1929 Tarski married fellow teacher Maria Witkowska, a Pole of Catholic background. She had worked as a courier for the army in the
Polish–Soviet War The Polish–Soviet War (Polish–Bolshevik War, Polish–Soviet War, Polish–Russian War 1919–1921) * russian: Советско-польская война (''Sovetsko-polskaya voyna'', Soviet-Polish War), Польский фронт (' ...
. They had two children; a son Jan who became a physicist, and a daughter Ina who married the mathematician Andrzej Ehrenfeucht. Tarski applied for a chair of philosophy at Lwów University, but on
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell (18 May 1872 – 2 February 1970) was a British polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, "universal human") is an individual whose know ...
's recommendation it was awarded to
Leon Chwistek Leon Chwistek (Kraków Kraków (, also , , ), written in English as Krakow and traditionally known as Cracow, is the second-largest and one of the oldest cities in Poland Poland ( pl, Polska ), officially the Republic of Poland ( pl ...

Leon Chwistek
. In 1930, Tarski visited the
University of Vienna The University of Vienna (german: Universität Wien) is a public university, public research university located in Vienna, Austria. It was founded by Rudolf IV, Duke of Austria, Duke Rudolph IV in 1365 and is the oldest university in the Geograph ...
, lectured to
Karl Menger Karl Menger (January 13, 1902 – October 5, 1985) was an Austrian-American mathematician. He was the son of the economist Carl Menger. He is credited with Menger's theorem. He worked on mathematics of algebras, algebra of geometries, curve and ...
's colloquium, and met
Kurt Gödel Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician Logic is an interdisciplinary field which studies truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dict ...
. Thanks to a fellowship, he was able to return to Vienna during the first half of 1935 to work with Menger's research group. From Vienna he traveled to Paris to present his ideas on truth at the first meeting of the
Unity of Science The unity of science is a thesis in philosophy of science Philosophy of science is a branch of philosophy concerned with the foundations, methodology, methods, and implications of science. The central questions of this study concern Demarcatio ...
movement, an outgrowth of the
Vienna Circle The Vienna Circle (german: Wiener Kreis) of Logical Empiricism Logical positivism, later called logical empiricism, and both of which together are also known as neopositivism, was a movement in Western philosophy Western philosophy refers to ...
. In 1937, Tarski applied for a chair at Poznań University but the chair was abolished. Tarski's ties to the Unity of Science movement likely saved his life, because they resulted in his being invited to address the Unity of Science Congress held in September 1939 at
Harvard University Harvard University is a private Private or privates may refer to: Music * "In Private "In Private" was the third single in a row to be a charting success for United Kingdom, British singer Dusty Springfield, after an absence of nearly t ...

Harvard University
. Thus he left Poland in August 1939, on the last ship to sail from Poland for the United States before the German and Soviet
invasion of Poland The invasion of Poland (1 September – 6 October 1939), also known as the September campaign ( pl, Kampania wrześniowa), 1939 defensive war ( pl, Wojna obronna 1939 roku) and Poland campaign (german: Überfall auf Polen, Polenfeldzug), was an ...
and the outbreak of
World War II World War II or the Second World War, often abbreviated as WWII or WW2, was a global war A world war is "a war War is an intense armed conflict between states State may refer to: Arts, entertainment, and media Literatur ...
. Tarski left reluctantly, because Leśniewski had died a few months before, creating a vacancy which Tarski hoped to fill. Oblivious to the
Nazi Nazism ( ), officially National Socialism (german: Nationalsozialismus, ), is the ideology An ideology () is a set of belief A belief is an Attitude (psychology), attitude that something is the case, or that some proposition about th ...
threat, he left his wife and children in Warsaw. He did not see them again until 1946. During the war, nearly all his Jewish extended family were murdered at the hands of the German occupying authorities. Once in the United States, Tarski held a number of temporary teaching and research positions: Harvard University (1939),
City College of New York The City College of the City University of New York (City College of New York, or simply City College, CCNY, or City) is a public college#REDIRECT Public university#REDIRECT Public university A public university or public college is a univers ...

City College of New York
(1940), and thanks to a
Guggenheim Fellowship Guggenheim Fellowships are grants Grant or Grants may refer to: Places *Grant County (disambiguation)Grant County may refer to: Places ;Australia * County of Grant, Victoria ;United States *Grant County, Arkansas *Grant County, Indiana * ...
, the
Institute for Advanced Study The Institute for Advanced Study (IAS), located in Princeton, New Jersey Princeton is a municipality with a borough A borough is an administrative division in various English language, English-speaking countries. In principle, the term ...

Institute for Advanced Study
in
Princeton Princeton University is a private Private or privates may refer to: Music * "In Private "In Private" was the third single in a row to be a charting success for United Kingdom, British singer Dusty Springfield, after an absence of nearly tw ...
(1942), where he again met Gödel. In 1942, Tarski joined the Mathematics Department at the
University of California, Berkeley The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California) is a public In public relations Public relations (PR) is the practice of managing and disseminating information from an individual or an organization ...

University of California, Berkeley
, where he spent the rest of his career. Tarski became an American citizen in 1945. Although emeritus from 1968, he taught until 1973 and supervised Ph.D. candidates until his death. At Berkeley, Tarski acquired a reputation as an astounding and demanding teacher, a fact noted by many observers:
His seminars at Berkeley quickly became famous in the world of mathematical logic. His students, many of whom became distinguished mathematicians, noted the awesome energy with which he would coax and cajole their best work out of them, always demanding the highest standards of clarity and precision.
Tarski was extroverted, quick-witted, strong-willed, energetic, and sharp-tongued. He preferred his research to be collaborative — sometimes working all night with a colleague — and was very fastidious about priority.
A charismatic leader and teacher, known for his brilliantly precise yet suspenseful expository style, Tarski had intimidatingly high standards for students, but at the same time he could be very encouraging, and particularly so to women — in contrast to the general trend. Some students were frightened away, but a circle of disciples remained, many of whom became world-renowned leaders in the field.
Tarski supervised twenty-four Ph.D. dissertations including (in chronological order) those of
Andrzej Mostowski Andrzej Mostowski (1 November 1913 – 22 August 1975) was a Polish mathematician. He is perhaps best remembered for the Mostowski collapse lemma. Biography Born in Lemberg, Austria-Hungary, Mostowski entered University of Warsaw in 1931. He w ...

Andrzej Mostowski
, ,
Julia Robinson Julia Hall Bowman Robinson (December 8, 1919July 30, 1985) was an American American(s) may refer to: * American, something of, from, or related to the United States of America, commonly known as the United States The United States of Ame ...
, Robert Vaught,
Solomon Feferman Solomon Feferman (December 13, 1928 – July 26, 2016) was an American philosopher and mathematician who worked in mathematical logic. Life Solomon Feferman was born in The Bronx in New York City to working-class parents who had immigrated to the ...
,
Richard Montague Richard Merritt Montague (September 20, 1930 – March 7, 1971) was an American American(s) may refer to: * American, something of, from, or related to the United States of America, commonly known as the United States The United States ...
, James Donald Monk,
Haim Gaifman Haim Gaifman (born 1934) is a logician, probability theorist, and philosopher of language who is professor of philosophy at Columbia University. Education and career In 1958 he received his M.Sc. at Hebrew University. Then in 1962, he received h ...
, Donald Pigozzi and
Roger Maddux Roger Maddux (born 1948) is an United States, American mathematician specializing in algebraic logic. He completed his B.A. at Pomona College in 1969, and his Ph.D. in mathematics at the University of California, Berkeley in 1978, where he was one ...
, as well as
Chen Chung Chang Chen Chung Chang (Chinese: 张晨钟) was a mathematician who worked in model theory. He obtained his PhD from UC Berkeley, Berkeley in 1955 on "Cardinal and Ordinal Factorization of Relation Types" under Alfred Tarski. He wrote the standard text ...
and Jerome Keisler, authors of ''Model Theory'' (1973), a classic text in the field. Feferman & Feferman, pp. 385-386 He also strongly influenced the dissertations of Alfred Lindenbaum,
Dana Scott Dana Stewart Scott (born October 11, 1932) is an American logician who is the emeritus Hillman University Professor of Computer Science Computer science deals with the theoretical foundations of information, algorithms and the architecture ...
, and Steven Givant. Five of Tarski's students were women, a remarkable fact given that men represented an overwhelming majority of graduate students at the time. However, he had extra-marital affairs with at least two of these students. After he showed another of his female students' work to a male colleague, the colleague published it himself, leading her to leave the graduate study and later move to a different university and a different advisor. Tarski lectured at
University College, London University College London, which Trade name, operates as UCL, is a major public university , public research university located in London, United Kingdom. UCL is a Member institutions of the University of London, member institution of the Federa ...
(1950, 1966), the in Paris (1955), the Miller Institute for Basic Research in Science in Berkeley (1958–60), the
University of California at Los Angeles The University of California, Los Angeles (UCLA) is a public In public relations Public relations (PR) is the practice of managing and disseminating information from an individual or an organization An organization, or organ ...
(1967), and the
Pontifical Catholic University of Chile The Pontifical Catholic University of Chile (UC) ( es, Pontificia Universidad Católica de Chile) is one of the six Catholic Universities Catholic higher education includes university, universities, colleges, and other institutions of h ...
(1974–75). Among many distinctions garnered over the course of his career, Tarski was elected to the
United States National Academy of Sciences The National Academy of Sciences (NAS) is a United States nonprofit A nonprofit organization (NPO), also known as a non-business entity, not-for-profit organization, or nonprofit institution, is a legal entity organized and operated for a ...
, the
British Academy The British Academy is the United Kingdom's national academy#REDIRECT National academy A national academy is an organizational body, usually operating with state financial support and approval, that co-ordinates scholarly research Res ...
and the
Royal Netherlands Academy of Arts and Sciences The Royal Netherlands Academy of Arts and Sciences ( nl, Koninklijke Nederlandse Akademie van Wetenschappen, abbreviated: KNAW) is an organization An organization, or organisation (Commonwealth English The use of the English langu ...
in 1958, received
honorary degree An honorary degree is an academic degree for which a university (or other degree-awarding institution) has waived all of the usual requirements. It is also known by the Latin phrases ''honoris causa'' ("for the sake of the honour") or ''ad hono ...
s from the Pontifical Catholic University of Chile in 1975, from
Marseilles Marseille ( , , ; also spelled in English as Marseilles; oc, Marselha ) is the prefecture A prefecture (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languag ...
' Paul Cézanne University in 1977 and from the
University of Calgary The University of Calgary (U of C or UCalgary) is a public In public relations Public relations (PR) is the practice of managing and disseminating information from an individual or an organization An organization, or organisati ...
, as well as the Berkeley Citation in 1981. Tarski presided over the
Association for Symbolic Logic The Association for Symbolic Logic (ASL) is an international organization of specialists in mathematical logic and philosophical logic. The ASL was founded in 1936, and its first president was Alonzo Church. The current president of the ASL is Ju ...
, 1944–46, and the International Union for the History and Philosophy of Science, 1956–57. He was also an honorary editor of ''
Algebra Universalis ''Algebra Universalis'' is an international scientific journal In academic publishing Academic publishing is the subfield of publishing Publishing is the activity of making information, literature, music, software and other content avail ...
''.


Mathematician

Tarski's mathematical interests were exceptionally broad. His collected papers run to about 2,500 pages, most of them on mathematics, not logic. For a concise survey of Tarski's mathematical and logical accomplishments by his former student Solomon Feferman, see "Interludes I–VI" in Feferman and Feferman. Tarski's first paper, published when he was 19 years old, was on
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...
, a subject to which he returned throughout his life. In 1924, he and
Stefan Banach Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a Polish Polish may refer to: * Anything from or related to Poland Poland ( pl, Polska ), officially the Republic of Poland ( pl, Rzeczpospolita Polska, links=no ), is a country ...

Stefan Banach
proved that, if one accepts the
Axiom of Choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

Axiom of Choice
, a
ball A ball is a round object (usually spherical of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a geometrical object in three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional s ...
can be cut into a finite number of pieces, and then reassembled into a ball of larger size, or alternatively it can be reassembled into two balls whose sizes each equal that of the original one. This result is now called the
Banach–Tarski paradox The Banach–Tarski paradox is a theorem in set theory, set-theoretic geometry, which states the following: Given a solid ball (mathematics), ball in 3‑dimensional space, existence theorem, there exists a decomposition of the ball into a finite ...
. In ''A decision method for elementary algebra and geometry'', Tarski showed, by the method of
quantifier elimination Quantifier elimination is a concept of simplification (disambiguation), simplification used in mathematical logic, model theory, and theoretical computer science. Informally, a quantified statement "\exists x such that \ldots" can be viewed as a que ...
, that the
first-order theory First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal system A formal system is an abstract structure used for inferring theorems from axioms according to a set ...
of the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s under addition and multiplication is decidable. (While this result appeared only in 1948, it dates back to 1930 and was mentioned in Tarski (1931).) This is a very curious result, because
Alonzo Church Alonzo Church (June 14, 1903 – August 11, 1995) was an American American(s) may refer to: * American, something of, from, or related to the United States of America, commonly known as the United States The United States of America (US ...
proved in 1936 that
Peano arithmetic In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian people, Italian mathematician Giuseppe Peano. These axioms have been ...
(the theory of
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
s) is ''not'' decidable. Peano arithmetic is also incomplete by Gödel's incompleteness theorem. In his 1953 ''Undecidable theories'', Tarski et al. showed that many mathematical systems, including
lattice theory A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound ...
, abstract
projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, proj ...
, and closure algebras, are all undecidable. The theory of
Abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
s is decidable, but that of non-Abelian groups is not. In the 1920s and 30s, Tarski often taught high school
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

geometry
. Using some ideas of
Mario Pieri Mario Pieri (22 June 1860 – 1 March 1913) was an Italian Italian may refer to: * Anything of, from, or related to the country and nation of Italy ** Italians, an ethnic group or simply a citizen of the Italian Republic ** Italian language, a Ro ...
, in 1926 Tarski devised an original
axiomatization In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
for plane
Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic Coptic may refer to: Afro-Asia * Copts, an ethnoreligious group mainly in the area of modern ...
, one considerably more concise than Hilbert's.
Tarski's axioms Tarski's axioms, due to Alfred Tarski, are an axiom set for the substantial fragment of Euclidean geometry that is formulable in first-order logic with identity (mathematics), identity, and requiring no set theory (i.e., that part of Euclidean ge ...
form a first-order theory devoid of set theory, whose individuals are
points Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Scotland, Lismore, Inner Hebrides, ...
, and having only two primitive
relations Relation or relations may refer to: General uses *International relations, the study of interconnection of politics, economics, and law on a global level *Interpersonal relationship, association or acquaintance between two or more people *Public ...
. In 1930, he proved this theory decidable because it can be mapped into another theory he had already proved decidable, namely his first-order theory of the real numbers. In 1929 he showed that much of Euclidean
solid geometry In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
could be recast as a first-order theory whose individuals are ''spheres'' (a
primitive notion In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
), a single primitive binary relation "is contained in", and two axioms that, among other things, imply that containment partially orders the spheres. Relaxing the requirement that all individuals be spheres yields a formalization of
mereology In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statem ...
far easier to exposit than Lesniewski's variant. Near the end of his life, Tarski wrote a very long letter, published as Tarski and Givant (1999), summarizing his work on geometry. ''Cardinal Algebras'' studied algebras whose models include the arithmetic of cardinal numbers. ''Ordinal Algebras'' sets out an algebra for the additive theory of order types. Cardinal, but not ordinal, addition commutes. In 1941, Tarski published an important paper on binary relations, which began the work on relation algebra and its
metamathematics Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories. Emphasis on metamathematics (and perhaps the creation of the term it ...
that occupied Tarski and his students for much of the balance of his life. While that exploration (and the closely related work of Roger Lyndon) uncovered some important limitations of relation algebra, Tarski also showed (Tarski and Givant 1987) that relation algebra can express most axiomatic set theory and
Peano arithmetic In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian people, Italian mathematician Giuseppe Peano. These axioms have been ...
. For an introduction to relation algebra, see Maddux (2006). In the late 1940s, Tarski and his students devised cylindric algebras, which are to first-order logic what the two-element Boolean algebra is to classical sentential logic. This work culminated in the two monographs by Tarski, Henkin, and Monk (1971, 1985).


Logician

Tarski's student, Vaught, has ranked Tarski as one of the four greatest logicians of all time — along with Aristotle, Gottlob Frege, and Kurt Gödel. However, Tarski often expressed great admiration for Logic of relatives, Charles Sanders Peirce, particularly for his pioneering work in the Finitary relation, logic of relations. Tarski produced axioms for ''logical consequence'' and worked on deductive systems, the algebra of logic, and the theory of definability. His semantic methods, which culminated in the model theory he and a number of his Berkeley students developed in the 1950s and 60s, radically transformed Hilbert's proof-theoretic metamathematics.
In [Tarski's] view, metamathematics became similar to any mathematical discipline. Not only can its concepts and results be mathematized, but they actually can be integrated into mathematics. ... Tarski destroyed the borderline between metamathematics and mathematics. He objected to restricting the role of metamathematics to the foundations of mathematics.
Tarski's 1936 article "On the concept of logical consequence" argued that the conclusion of an argument will follow logically from its premises if and only if every model of the premises is a model of the conclusion. In 1937, he published a paper presenting clearly his views on the nature and purpose of the deductive method, and the role of logic in scientific studies. His high school and undergraduate teaching on logic and axiomatics culminated in a classic short text, published first in Polish, then in German translation, and finally in a 1941 English translation as ''Introduction to Logic and to the Methodology of Deductive Sciences''. Tarski's 1969 "Truth and proof" considered both Gödel's incompleteness theorems and Tarski's undefinability theorem, and mulled over their consequences for the axiomatic method in mathematics.


Truth in formalized languages

In 1933, Tarski published a very long paper in Polish, titled "Pojęcie prawdy w językach nauk dedukcyjnych", "Setting out a mathematical definition of truth for formal languages." The 1935 German translation was titled "Der Wahrheitsbegriff in den formalisierten Sprachen", "The concept of truth in formalized languages", sometimes shortened to "Wahrheitsbegriff". An English translation appeared in the 1956 first edition of the volume ''Logic, Semantics, Metamathematics''. This collection of papers from 1923 to 1938 is an event in 20th-century
analytic philosophy Analytic philosophy is a branch and tradition of philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about existence Existence is the ability of an entity to interact with physical reality ...
, a contribution to Mathematical logic, symbolic logic, semantics, and the philosophy of language. For a brief discussion of its content, see Convention T (and also T-schema). Some recent philosophical debate examines the extent to which Tarski's theory of truth for formalized languages can be seen as a correspondence theory of truth. The debate centers on how to read Tarski's condition of material adequacy for a true definition. That condition requires that the truth theory have the following as theorems for all sentences p of the language for which truth is being defined: : "p" is true if and only if p. (where p is the proposition expressed by "p") The debate amounts to whether to read sentences of this form, such as
"Snow is white" is true if and only if snow is white
as expressing merely a deflationary theory of truth or as embodying
truth Truth is the property of being in accord with fact A fact is something that is true True most commonly refers to truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In ...

truth
as a more substantial property (see Kirkham 1992). It is important to realize that Tarski's theory of truth is for formalized languages, so examples in natural language are not illustrations of the use of Tarski's theory of truth.


Logical consequence

In 1936, Tarski published Polish and German versions of a lecture he had given the preceding year at the International Congress of Scientific Philosophy in Paris. A new English translation of this paper, Tarski (2002), highlights the many differences between the German and Polish versions of the paper and corrects a number of mistranslations in Tarski (1983). This publication set out the modern model theory, model-theoretic definition of (semantic) logical consequence, or at least the basis for it. Whether Tarski's notion was entirely the modern one turns on whether he intended to admit models with varying domains (and in particular, models with domains of different cardinal number, cardinalities). This question is a matter of some debate in the current philosophical literature. John Etchemendy stimulated much of the recent discussion about Tarski's treatment of varying domains. Tarski ends by pointing out that his definition of logical consequence depends upon a division of terms into the logical and the extra-logical and he expresses some skepticism that any such objective division will be forthcoming. "What are Logical Notions?" can thus be viewed as continuing "On the Concept of Logical Consequence".


Work on logical notions

Another theory of Tarski's attracting attention in the recent philosophical literature is that outlined in his "What are Logical Notions?" (Tarski 1986). This is the published version of a talk that he gave originally in 1966 in London and later in 1973 in Buffalo, New York, Buffalo; it was edited without his direct involvement by John Corcoran (logician), John Corcoran. It became the most cited paper in the journal ''History and Philosophy of Logic''. In the talk, Tarski proposed demarcation of logical operations (which he calls "notions") from non-logical. The suggested criteria were derived from the Erlangen program of the 19th-century German mathematician Felix Klein. Mautner (in 1946), and possibly an article by the Portuguese mathematician Sebastiao e Silva, anticipated Tarski in applying the Erlangen Program to logic. That program classified the various types of geometry (
Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic Coptic may refer to: Afro-Asia * Copts, an ethnoreligious group mainly in the area of modern ...
, affine geometry,
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

topology
, etc.) by the type of one-one transformation of space onto itself that left the objects of that geometrical theory invariant. (A one-to-one transformation is a functional map of the space onto itself so that every point of the space is associated with or mapped to one other point of the space. So, "rotate 30 degrees" and "magnify by a factor of 2" are intuitive descriptions of simple uniform one-one transformations.) Continuous transformations give rise to the objects of topology, similarity transformations to those of Euclidean geometry, and so on. As the range of permissible transformations becomes broader, the range of objects one is able to distinguish as preserved by the application of the transformations becomes narrower. Similarity transformations are fairly narrow (they preserve the relative distance between points) and thus allow us to distinguish relatively many things (e.g., equilateral triangles from non-equilateral triangles). Continuous transformations (which can intuitively be thought of as transformations which allow non-uniform stretching, compression, bending, and twisting, but no ripping or glueing) allow us to distinguish a polygon from an annulus (mathematics), annulus (ring with a hole in the centre), but do not allow us to distinguish two polygons from each other. Tarski's proposal was to demarcate the logical notions by considering all possible one-to-one transformations (automorphisms) of a domain onto itself. By domain is meant the universe of discourse of a model for the semantic theory of logic. If one identifies the truth value True with the domain set and the truth-value False with the empty set, then the following operations are counted as logical under the proposal: # ''Truth-functions'': All truth-functions are admitted by the proposal. This includes, but is not limited to, all ''n''-ary truth-functions for finite ''n''. (It also admits of truth-functions with any infinite number of places.) # ''Individuals'': No individuals, provided the domain has at least two members. # ''Predicates'': #* the one-place total and null predicates, the former having all members of the domain in its extension and the latter having no members of the domain in its extension #* two-place total and null predicates, the former having the set of all ordered pairs of domain members as its extension and the latter with the empty set as extension #* the two-place identity predicate, with the set of all order-pairs <''a'',''a''> in its extension, where ''a'' is a member of the domain #* the two-place diversity predicate, with the set of all order pairs <''a'',''b''> where ''a'' and ''b'' are distinct members of the domain #* ''n''-ary predicates in general: all predicates definable from the identity predicate together with Logical conjunction, conjunction, disjunction and negation (up to any ordinality, finite or infinite) # ''Quantifier (logic), Quantifiers'': Tarski explicitly discusses only monadic quantifiers and points out that all such numerical quantifiers are admitted under his proposal. These include the standard universal and existential quantifiers as well as numerical quantifiers such as "Exactly four", "Finitely many", "Uncountably many", and "Between four and 9 million", for example. While Tarski does not enter into the issue, it is also clear that polyadic quantifiers are admitted under the proposal. These are quantifiers like, given two predicates ''Fx'' and ''Gy'', "More(''x, y'')", which says "More things have ''F'' than have ''G''." # ''Set-Theoretic relations'': Relations such as inclusion (set theory), inclusion, Intersection (set theory), intersection and Union (set theory), union applied to subsets of the domain are logical in the present sense. # ''Set membership'': Tarski ended his lecture with a discussion of whether the set membership relation counted as logical in his sense. (Given the reduction of (most of) mathematics to set theory, this was, in effect, the question of whether most or all of mathematics is a part of logic.) He pointed out that set membership is logical if set theory is developed along the lines of type theory, but is extralogical if set theory is set out axiomatically, as in the canonical Zermelo–Fraenkel set theory. # ''Logical notions of higher order'': While Tarski confined his discussion to operations of first-order logic, there is nothing about his proposal that necessarily restricts it to first-order logic. (Tarski likely restricted his attention to first-order notions as the talk was given to a non-technical audience.) So, higher-order quantifiers and predicates are admitted as well. In some ways the present proposal is the obverse of that of Lindenbaum and Tarski (1936), who proved that all the logical operations of Russell and Alfred North Whitehead, Whitehead's ''Principia Mathematica'' are invariant under one-to-one transformations of the domain onto itself. The present proposal is also employed in Tarski and Givant (1987). Solomon Feferman and Vann McGee further discussed Tarski's proposal in work published after his death. Feferman (1999) raises problems for the proposal and suggests a cure: replacing Tarski's preservation by automorphisms with preservation by arbitrary homomorphisms. In essence, this suggestion circumvents the difficulty Tarski's proposal has in dealing with a sameness of logical operation across distinct domains of a given cardinality and across domains of distinct cardinalities. Feferman's proposal results in a radical restriction of logical terms as compared to Tarski's original proposal. In particular, it ends up counting as logical only those operators of standard first-order logic without identity. McGee (1996) provides a precise account of what operations are logical in the sense of Tarski's proposal in terms of expressibility in a language that extends first-order logic by allowing arbitrarily long conjunctions and disjunctions, and quantification over arbitrarily many variables. "Arbitrarily" includes a countable infinity.


Works

;Anthologies and collections * 1986. ''The Collected Papers of Alfred Tarski'', 4 vols. Givant, S. R., and McKenzie, R. N., eds. Birkhäuser. * * 1983 (1956). ''Logic, Semantics, Metamathematics: Papers from 1923 to 1938 by Alfred Tarski'', Corcoran, J., ed. Hackett. 1st edition edited and translated by J. H. Woodger, Oxford Uni. Press. This collection contains translations from Polish of some of Tarski's most important papers of his early career, including ''The Concept of Truth in Formalized Languages'' and ''On the Concept of Logical Consequence'' discussed above. ;Original publications of Tarski: * 1930 Une contribution à la théorie de la mesure. Fund Math 15 (1930), 42–50. * 1930. (with
Jan Łukasiewicz Jan Łukasiewicz (; 21 December 1878 – 13 February 1956) was a Polish logician and philosopher who is best known for Polish notation and Łukasiewicz logic. He was born in Lemberg, a city in the Austrian Galicia, Galician Kingdom of Austria-Hungar ...

Jan Łukasiewicz
). "Untersuchungen uber den Aussagenkalkul" ["Investigations into the Sentential Calculus"], ''Comptes Rendus des seances de la Societe des Sciences et des Lettres de Varsovie'', Vol, 23 (1930) Cl. III, pp. 31–32 in Tarski (1983): 38–59. * 1931. "Sur les ensembles définissables de nombres réels I", ''Fundamenta Mathematicae 17'': 210–239 in Tarski (1983): 110–142. * 1936
"Grundlegung der wissenschaftlichen Semantik"
''Actes du Congrès international de philosophie scientifique, Sorbonne, Paris 1935'', vol. III, ''Language et pseudo-problèmes'', Paris, Hermann, 1936, pp. 1–8 in Tarski (1983): 401–408. * 1936
"Über den Begriff der logischen Folgerung"
''Actes du Congrès international de philosophie scientifique, Sorbonne, Paris 1935'', vol. VII, ''Logique'', Paris: Hermann, pp. 1–11 in Tarski (1983): 409–420. * 1936 (with Adolf Lindenbaum). "On the Limitations of Deductive Theories" in Tarski (1983): 384–92. * 1994 (1941). ''Introduction to Logic and to the Methodology of Deductive Sciences''. Dover. * 1941. "On the calculus of relations", ''Journal of Symbolic Logic 6'': 73–89. * 1944.

" ''Philosophy and Phenomenological Research 4'': 341–75. * 1948. ''A decision method for elementary algebra and geometry''. Santa Monica CA: RAND Corp. * 1949. ''Cardinal Algebras''. Oxford Univ. Press. * 1953 (with Mostowski and Raphael Robinson). ''Undecidable theories''. North Holland. * 1956. ''Ordinal algebras''. North-Holland. * 1965. "A simplified formalization of predicate logic with identity", ''Archiv für Mathematische Logik und Grundlagenforschung 7'': 61-79 * 1969.
Truth and Proof
, ''Scientific American 220'': 63–77. * 1971 (with Leon Henkin and Donald Monk). ''Cylindric Algebras: Part I''. North-Holland. * 1985 (with Leon Henkin and Donald Monk). ''Cylindric Algebras: Part II''. North-Holland. * 1986. "What are Logical Notions?", Corcoran, J., ed., ''History and Philosophy of Logic 7'': 143–54. * 1987 (with Steven Givant). ''A Formalization of Set Theory Without Variables''. Vol.41 of American Mathematical Society colloquium publications. Providence RI: American Mathematical Society.
Review
* 1999 (with Steven Givant)

''Bulletin of Symbolic Logic 5'': 175–214. * 2002. "On the Concept of Following Logically" (Magda Stroińska and David Hitchcock, trans.) ''History and Philosophy of Logic 23'': 155–196.


See also

* History of philosophy in Poland#20th century, History of philosophy in Poland * Cylindric algebra * Interpretability * Weak interpretability * List of things named after Alfred Tarski


References


Further reading

;Biographical references * * * * * Patterson, Douglas. ''Alfred Tarski: Philosophy of Language and Logic'' (Palgrave Macmillan; 2012) 262 pages; biography focused on his work from the late-1920s to the mid-1930s, with particular attention to influences from his teachers Stanislaw Lesniewski and Tadeusz Kotarbinski. ;Logic literature * The December 1986 issue of the ''Journal of Symbolic Logic'' surveys Tarski's work on
model theory In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical ...
( Robert Vaught), abstract algebra, algebra (Jonsson), decidability (logic), undecidable theories (McNulty),
algebraic logic In mathematical logic, algebraic logic is the reasoning obtained by manipulating equations with Free variables and bound variables, free variables. What is now usually called classical algebraic logic focuses on the identification and algebraic de ...
(Donald Monk), and
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

geometry
(Szczerba). The March 1988 issue of the same journal surveys his work on axiomatic set theory (Azriel Levy), real closed fields (Lou Van Den Dries), decidability (logic), decidable theory (Doner and Wilfrid Hodges),
metamathematics Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories. Emphasis on metamathematics (and perhaps the creation of the term it ...
(Blok and Pigozzi),
truth Truth is the property of being in accord with fact A fact is something that is true True most commonly refers to truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In ...

truth
and logical consequence (John Etchemendy), and general philosophy (Patrick Suppes). ** Blok, W. J.; Pigozzi, Don
"Alfred Tarski's Work on General Metamathematics"
''The Journal of Symbolic Logic'', Vol. 53, No. 1 (Mar., 1988), pp. 36–50 * Chen Chung Chang, Chang, C.C., and Howard Jerome Keisler, Keisler, H.J., 1973. ''Model Theory''. North-Holland, Amsterdam. American Elsevier, New York. * John Corcoran (logician), Corcoran, John, and José Miguel Sagüillo, Sagüillo, José Miguel, 2011. "The Absence of Multiple Universes of Discourse in the 1936 Tarski Consequence-Definition Paper", ''History and Philosophy of Logic 32'': 359–80

* Corcoran, John, and Weber, Leonardo, 2015. "Tarski’s convention T: condition beta", South American Journal of Logic. 1, 3–32. * John Etchemendy, Etchemendy, John, 1999. ''The Concept of Logical Consequence''. Stanford CA: CSLI Publications. * * Ivor Grattan-Guinness, Grattan-Guinness, Ivor, 2000. ''The Search for Mathematical Roots 1870-1940''. Princeton Uni. Press. * Kirkham, Richard, 1992. ''Theories of Truth''. MIT Press. * Roger Maddux, Maddux, Roger D., 2006. ''Relation Algebras'', vol. 150 in "Studies in Logic and the Foundations of Mathematics", Elsevier Science. * * * Karl Popper, Popper, Karl R., 1972, Rev. Ed. 1979, "Philosophical Comments on Tarski's Theory of Truth", with Addendum, ''Objective Knowledge'', Oxford: 319–340. * * Smith, James T., 2010. "Definitions and Nondefinability in Geometry", American Mathematical Monthly 117:475–89. * Jan Woleński, Wolenski, Jan, 1989. ''Logic and Philosophy in the Lvov–Warsaw School''. Reidel/Kluwer.


External links

* Stanford Encyclopedia of Philosophy: *
Tarski's Truth Definitions
by Wilfred Hodges. *
Alfred Tarski
by Mario Gómez-Torrente. *
Algebraic Propositional Logic
by Ramon Jansana. Includes a fairly detailed discussion of Tarski's work on these topics.
Tarski's Semantic Theory
on the Internet Encyclopedia of Philosophy. {{DEFAULTSORT:Tarski, Alfred 1901 births 1983 deaths 20th-century American mathematicians 20th-century American non-fiction writers 20th-century American philosophers 20th-century essayists 20th-century mathematicians 20th-century philosophers American atheists American logicians American male essayists American male non-fiction writers American people of Polish-Jewish descent American philosophers Analytic philosophers Converts to Roman Catholicism from Judaism Computability theorists Jewish atheists Jewish philosophers Linguistic turn Members of the Polish Academy of Sciences Members of the Royal Netherlands Academy of Arts and Sciences Members of the United States National Academy of Sciences Model theorists Philosophers of language Philosophers of logic Philosophers of mathematics Philosophers of science Polish atheists Polish emigrants to the United States Polish essayists Polish logicians Polish male non-fiction writers Polish people of Jewish descent Polish philosophers Polish set theorists Scientists from Warsaw University of California, Berkeley faculty University of California, Berkeley people University of California, Berkeley staff University of Warsaw alumni