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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
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 premi ...
ian and
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History ...
. A prolific author best known for his work on
model theory In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the ...
, metamathematics, and
algebraic logic In mathematical logic, algebraic logic is the reasoning obtained by manipulating equations with free variables. What is now usually called classical algebraic logic focuses on the identification and algebraic description of models appropriate fo ...
, he also contributed to
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...
,
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
,
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
,
measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simil ...
,
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 s ...
,
set theory Set theory is the branch of mathematical logic that studies 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, is mostly concer ...
, and
analytic philosophy Analytic philosophy is a branch and tradition of philosophy using analysis, popular in the Western world and particularly the Anglosphere, which began around the turn of the 20th century in the contemporary era in the United Kingdom, United ...
. Educated in Poland at the University of Warsaw, 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 land-grant research university in Berkeley, California. Established in 1868 as the University of California, it is the state's first land-grant u ...
, from 1942 until his death in 1983. Feferman A. His biographers Anita Burdman Feferman and Solomon Feferman state that, "Along with his contemporary, Kurt Gödel, 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 or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to represent reality or otherwise correspond to it, such as beliefs ...
and the theory of models." Feferman & Feferman, p.1

# Life

## Early life and education

Alfred Tarski was born Alfred Teitelbaum (
Polish Polish may refer to: * Anything from or related to Poland, a country in Europe * Polish language * Poles, people from Poland or of Polish descent * Polish chicken *Polish brothers (Mark Polish and Michael Polish, born 1970), American twin scree ...
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 was home to the largest and most significant Ashkenazi Jewish community in the world. Poland was a principal center of Jewish culture, because of the lo ...
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 in 1918 intending to study
biology Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary ...
. Feferman & Feferman, p.26 After Poland regained independence in 1918, Warsaw University came under the leadership of Jan Łukasiewicz,
Stanisław Leśniewski Stanisław Leśniewski (30 March 1886 – 13 May 1939) was a Polish mathematician, philosopher and logician. Life He was born on 28 March 1886 at Serpukhov, near Moscow, to father Izydor, an engineer working on the construction of the Trans-S ...
and
Wacław Sierpiński Wacław Franciszek Sierpiński (; 14 March 1882 – 21 October 1969) was a Polish mathematician. He was known for contributions to set theory (research on the axiom of choice and the continuum hypothesis), number theory, theory of functions, an ...
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 and
Tadeusz Kotarbiński Tadeusz Marian Kotarbiński (; 31 March 1886 – 3 October 1981) was a Polish philosopher, logician and ethicist. A pupil of Kazimierz Twardowski, he was one of the most representative figures of the Lwów–Warsaw School, and a member of the ...
, 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 Kotarbiński, 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 baptized Catholics worldwide . It is among the world's oldest and largest international institutions, and has played a ...
, Poland's dominant religion. Alfred did so even though he was an avowed
atheist Atheism, in the broadest sense, is an absence of belief in the existence of deities. Less broadly, atheism is a rejection of the belief that any deities exist. In an even narrower sense, atheism is specifically the position that there no d ...
.

## Career

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 mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ...
's recommendation it was awarded to
Leon Chwistek Leon Chwistek ( Kraków, Austria-Hungary, 13 June 1884 – Barvikha near Moscow, Russia, 20 August 1944) was a Polish avant-garde painter, theoretician of modern art, literary critic, logician, philosopher and mathematician. Career and philos ...
. 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 Geogra ...
, lectured to
Karl Menger Karl Menger (January 13, 1902 – October 5, 1985) was an Austrian-American mathematician, the son of the economist Carl Menger. In mathematics, Menger studied the theory of algebras and the dimension theory of low- regularity ("rough") curve ...
's colloquium, and met Kurt Gödel. 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 movement, an outgrowth of the Vienna Circle. 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 Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of highe ...
. 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) was a joint attack on the Republic of Poland by Nazi Germany and the Soviet Union which marked the beginning of World War II. The German invasion began on 1 September 1939, one week aft ...
and the outbreak of
World War II World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the vast majority of the world's countries—including all of the great powers—forming two opposing ...
. Tarski left reluctantly, because Leśniewski had died a few months before, creating a vacancy which Tarski hoped to fill. Oblivious to the
Nazi Nazism ( ; german: Nazismus), the common name in English for National Socialism (german: Nationalsozialismus, ), is the far-right totalitarian political ideology and practices associated with Adolf Hitler and the Nazi Party (NSDAP) in ...
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 (also known as the City College of New York, or simply City College or CCNY) is a public university within the City University of New York (CUNY) system in New York City. Founded in 1847, Cit ...
(1940), and thanks to a
Guggenheim Fellowship Guggenheim Fellowships are grants that have been awarded annually since by the John Simon Guggenheim Memorial Foundation to those "who have demonstrated exceptional capacity for productive scholarship or exceptional creative ability in the ar ...
, the
Institute for Advanced Study The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent schola ...
in Princeton (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 land-grant research university in Berkeley, California. Established in 1868 as the University of California, it is the state's first land-grant u ...
, 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, Bjarni Jónsson, Julia Robinson, Robert Vaught, Solomon Feferman, Richard Montague, James Donald Monk, Haim Gaifman, Donald Pigozzi and Roger Maddux, as well as Chen Chung Chang 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, 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 , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public research university , endowment = £143 million (2020) , budget = ...
(1950, 1966), the Institut Henri Poincaré 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 land-grant research university in Los Angeles, California. UCLA's academic roots were established in 1881 as a teachers college then known as the southern branch of the Californi ...
(1967), and the
Pontifical Catholic University of Chile The Pontifical Catholic University of Chile (''PUC or UC Chile'') ( es, Pontificia Universidad Católica de Chile) is one of the six Catholic Universities existing in the Chilean university system and one of the two pontifical universities i ...
(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, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the Nat ...
, the
British Academy The British Academy is the United Kingdom's national academy for the humanities and the social sciences. It was established in 1902 and received its royal charter in the same year. It is now a fellowship of more than 1,000 leading scholars s ...
and the Royal Netherlands Academy of Arts and Sciences 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 h ...
s from the Pontifical Catholic University of Chile in 1975, from Marseilles' Paul Cézanne University in 1977 and from the
University of Calgary The University of Calgary (U of C or UCalgary) is a public research university located in Calgary, Alberta, Canada. The University of Calgary started in 1944 as the Calgary branch of the University of Alberta, founded in 1908, prior to being ins ...
, as well as the Berkeley Citation in 1981. Tarski presided over the Association for Symbolic Logic, 1944–46, and the International Union for the History and Philosophy of Science, 1956–57. He was also an honorary editor of '' Algebra Universalis''.

# Work in mathematics

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 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, is mostly concer ...
, 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 mathematician who is generally considered one of the 20th century's most important and influential mathematicians. He was the founder of modern functional analysis, and an origi ...
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 of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
, a
ball A ball is a round object (usually spherical, but can sometimes be ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used fo ...
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-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can th ...
. In ''A decision method for elementary algebra and geometry'', Tarski showed, by the method of
quantifier elimination Quantifier elimination is a concept of 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 question "When is there an x such ...
, that the first-order theory of the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Ever ...
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 mathematician, computer scientist, logician, philosopher, professor and editor who made major contributions to mathematical logic and the foundations of theoretical computer sc ...
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 mathematician Giuseppe Peano. These axioms have been used nearly ...
(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 ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal ...
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, 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, pr ...
, and closure algebras, are all undecidable. The theory of
Abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s is decidable, but that of non-Abelian groups is not. In the 1920s and 30s, Tarski often taught high school
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
. Using some ideas of Mario Pieri, in 1926 Tarski devised an original axiomatization for plane
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axiom ...
, one considerably more concise than Hilbert's. Tarski's axioms form a first-order theory devoid of set theory, whose individuals are points, and having only two primitive relations. 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 could be recast as a second-order theory whose individuals are ''spheres'' (a
primitive notion In mathematics, logic, philosophy, and formal systems, a primitive notion is a concept that is not defined in terms of previously-defined concepts. It is often motivated informally, usually by an appeal to intuition and everyday experience. In a ...
), 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 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 number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. The ...
s. ''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 relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and i ...
s, which began the work on
relation algebra In mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation. The motivating example of a relation algebra is the algebra 2''X''² of all binary relations ...
and its metamathematics 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 Set theory is the branch of mathematical logic that studies 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, is mostly concern ...
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 mathematician Giuseppe Peano. These axioms have been used nearly ...
. For an introduction to
relation algebra In mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation. The motivating example of a relation algebra is the algebra 2''X''² of all binary relations ...
, see Maddux (2006). In the late 1940s, Tarski and his students devised cylindric algebras, which are to
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 quant ...
what the
two-element Boolean algebra In mathematics and abstract algebra, the two-element Boolean algebra is the Boolean algebra whose ''underlying set'' (or universe or ''carrier'') ''B'' is the Boolean domain. The elements of the Boolean domain are 1 and 0 by convention, so that '' ...
is to classical sentential logic. This work culminated in the two monographs by Tarski, Henkin, and Monk (1971, 1985).

# Work in logic

Tarski's student, Vaught, has ranked Tarski as one of the four greatest logicians of all time — along with
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ...
,
Gottlob 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 phil ...
, and Kurt Gödel. However, Tarski often expressed great admiration for
Charles Sanders Peirce Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American philosopher, logician, mathematician and scientist who is sometimes known as "the father of pragmatism". Educated as a chemist and employed as a scientist for ...
, particularly for his pioneering work in the logic of relations. Tarski produced axioms for ''logical consequence'' and worked on
deductive system A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A fo ...
s, 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. Around 1930, Tarski developed an abstract theory of logical deductions that models some properties of logical calculi. Mathematically, what he described is just a finitary closure operator on a set (the set of ''sentences''). In
abstract algebraic logic In mathematical logic, abstract algebraic logic is the study of the algebraization of deductive systems arising as an abstraction of the well-known Lindenbaum–Tarski algebra, and how the resulting algebras are related to logical systems.Font, 2 ...
, finitary closure operators are still studied under the name ''consequence operator'', which was coined by Tarski. The set ''S'' represents a set of sentences, a subset ''T'' of ''S'' a theory, and cl(''T'') is the set of all sentences that follow from the theory. This abstract approach was applied to fuzzy logic (see Gerla 2000).
In arski'sview, 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 using analysis, popular in the Western world and particularly the Anglosphere, which began around the turn of the 20th century in the contemporary era in the United Kingdom, United ...
, a contribution to
symbolic 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 ...
,
semantics Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference, meaning, or truth. The term can be used to refer to subfields of several distinct disciplines, including philosophy, linguistics and comp ...
, and the
philosophy of language In analytic philosophy, philosophy of language investigates the nature of language and the relations between language, language users, and the world. Investigations may include inquiry into the nature of meaning, intentionality, reference, t ...
. For a brief discussion of its content, see Convention T (and also
T-schema The T-schema ("truth schema", not to be confused with "Convention T") is used to check if an inductive definition of truth is valid, which lies at the heart of any realisation of Alfred Tarski's semantic theory of truth. Some authors refer to it a ...
). 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 In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...
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 or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to represent reality or otherwise correspond to it, such as beliefs ...
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-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 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".

## 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; it was edited without his direct involvement by 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 Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and grou ...
. 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 ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axiom ...
,
affine geometry In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting") the metric notions of distance and angle. As the notion of ''parallel lines'' is one of the main properties that is ind ...
,
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, 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 In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two t ...
from an 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 In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or '' false''). Computing In some pr ...
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-function In logic, a truth function is a function that accepts truth values as input and produces a unique truth value as output. In other words: The input and output of a truth function are all truth values; a truth function will always output exactly on ...
s'': 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
conjunction Conjunction may refer to: * Conjunction (grammar), a part of speech * Logical conjunction, a mathematical operator ** Conjunction introduction, a rule of inference of propositional logic * Conjunction (astronomy) In astronomy, a conjunction occ ...
,
disjunction In logic, disjunction is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is raining or it is snowing" can be represented in logic using the disjunctive formula R \lor ...
and
negation In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false ...
(up to any ordinality, finite or infinite) # '' 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, intersection and union applied to
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
s 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 In mathematics, logic, and computer science, a type theory is the formal presentation of a specific type system, and in general type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a founda ...
, but is extralogical if set theory is set out axiomatically, as in the canonical
Zermelo–Fraenkel set theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such a ...
. # ''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
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ...
Principia Mathematica The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. ...
'' 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.

# Selected publications

;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). "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. * 1937. ''Einführung in die Mathematische Logik und in die Methodologie der Mathematik''. Springer, Wien (Vienna). * 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.

*
History of philosophy in Poland The history of philosophy in Poland parallels the evolution of philosophy in Europe in general. Overview Polish philosophy drew upon the broader currents of European philosophy, and in turn contributed to their growth. Some of the most momentous ...
* Cylindric algebra * Interpretability * Weak interpretability * List of things named after Alfred Tarski

# References

;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, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the ...
( Robert Vaught),
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
(Jonsson), undecidable theories (McNulty),
algebraic logic In mathematical logic, algebraic logic is the reasoning obtained by manipulating equations with free variables. What is now usually called classical algebraic logic focuses on the identification and algebraic description of models appropriate fo ...
(Donald Monk), and
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
(Szczerba). The March 1988 issue of the same journal surveys his work on
axiomatic set theory Set theory is the branch of mathematical logic that studies 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, is mostly concern ...
(Azriel Levy), real closed fields (Lou Van Den Dries), decidable theory (Doner and Wilfrid Hodges), metamathematics (Blok and Pigozzi),
truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to represent reality or otherwise correspond to it, such as beliefs ...
and logical consequence ( John Etchemendy), and general
philosophy Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. S ...
(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 * Chang, C.C., and Keisler, H.J., 1973. ''Model Theory''. North-Holland, Amsterdam. American Elsevier, New York. * Corcoran, John, and 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. * * Gerla, G. (2000) ''Fuzzy Logic: Mathematical Tools for Approximate Reasoning''.
Kluwer Academic Publishers Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 i ...
. * Grattan-Guinness, Ivor, 2000. ''The Search for Mathematical Roots 1870-1940''. Princeton Uni. Press. * Kirkham, Richard, 1992. ''Theories of Truth''. MIT Press. * Maddux, Roger D., 2006. ''Relation Algebras'', vol. 150 in "Studies in Logic and the Foundations of Mathematics", Elsevier Science. * * * 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 ''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an ...
117:475–89. * Wolenski, Jan, 1989. ''Logic and Philosophy in the Lvov–Warsaw School''. Reidel/Kluwer.