Alfred Tarski
   HOME

TheInfoList



OR:

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 study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
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, mathematical structure, structure, space, Mathematica ...
. A prolific author best known for his work on
model theory In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mat ...
,
metamathematics Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheory, metatheories, which are Mathematical theory, mathematical theories about other mathematical theories. Emphasis on metamathematics (and ...
, 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 mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
,
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
,
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
,
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 magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...
,
mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
,
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 mathema ...
,
type theory In mathematics and theoretical computer science, a type theory is the formal presentation of a specific type system. Type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a foundation of ...
, and
analytic philosophy Analytic philosophy is a broad movement within Western philosophy, especially English-speaking world, anglophone philosophy, focused on analysis as a philosophical method; clarity of prose; rigor in arguments; and making use of formal logic, mat ...
. Educated in Poland at the
University of Warsaw The University of Warsaw (, ) is a public university, public research university in Warsaw, Poland. Established on November 19, 1816, it is the largest institution of higher learning in the country, offering 37 different fields of study as well ...
, 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 university, public Land-grant university, land-grant research university in Berkeley, California, United States. Founded in 1868 and named after t ...
, from 1942 until his death in 1983. Feferman A. His biographers Anita Burdman Feferman and
Solomon Feferman Solomon Feferman (December 13, 1928July 26, 2016) was an American philosopher and mathematician who worked in mathematical logic. In addition to his prolific technical work in proof theory, computability theory, and set theory, he was known for h ...
state that, "Along with his contemporary,
Kurt Gödel Kurt Friedrich Gödel ( ; ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel profoundly ...
, he changed the face of logic in the twentieth century, especially through his work on the concept of
truth Truth or verity is the Property (philosophy), property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth, 2005 In everyday language, it is typically ascribed to things that aim to represent reality or otherwise cor ...
and the theory of models." Feferman & Feferman, p.1


Life


Early life and education

Alfred Tarski was born Alfred Teitelbaum ( Polish 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 Jews, Jewish community in the world. Poland was a principal center of Jewish culture, because of the long pe ...
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 (, ) is a public university, public research university in Warsaw, Poland. Established on November 19, 1816, it is the largest institution of higher learning in the country, offering 37 different fields of study as well ...
in 1918 intending to study
biology Biology is the scientific study of life and living organisms. It is a broad natural science that encompasses a wide range of fields and unifying principles that explain the structure, function, growth, History of life, origin, evolution, and ...
. 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. His work centred on philosophical logic, mathematical logic and history of logi ...
, Stanisław Leśniewski 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 ...
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, 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, mainly due to the latter's increasing anti-semitism. 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 List of Christian denominations by number of members, largest Christian church, with 1.27 to 1.41 billion baptized Catholics Catholic Church by country, worldwid ...
, 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 ...
.


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 the Third Boys’ Gimnazjum of the Trade Union of Polish Secondary-School Teachers (later the Stefan Żeromski Gimnazjum), a Warsaw secondary school, beginning in 1925. Before World War II, it was not uncommon for European intellectuals of research caliber to teach high school. Hence until 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 (14 February 1919 – 18 March 1921) was fought primarily between the Second Polish Republic and the Russian Soviet Federative Socialist Republic, following World War I and the Russian Revolution. After the collapse ...
. They had two children; a son Jan Tarski, 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 philosopher, logician, mathematician, and public intellectual. He had influence on mathematics, logic, set theory, and various areas of analytic ...
's recommendation it was awarded to Leon Chwistek. In 1930, Tarski visited the
University of Vienna The University of Vienna (, ) is a public university, public research university in Vienna, Austria. Founded by Rudolf IV, Duke of Austria, Duke Rudolph IV in 1365, it is the oldest university in the German-speaking world and among the largest ...
, lectured to
Karl Menger Karl Menger (; January 13, 1902 – October 5, 1985) was an Austrian-born American mathematician, the son of the economist Carl Menger. In mathematics, Menger studied the theory of algebra over a field, algebras and the dimension theory of low-r ...
's colloquium, and met
Kurt Gödel Kurt Friedrich Gödel ( ; ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel profoundly ...
. 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 that says that all the sciences form a unified whole. The variants of the thesis can be classified as ontological (giving a unified account of the structure of reality) and/or as epistemic/p ...
movement, an outgrowth of the Vienna Circle. Tarski's academic career in Poland was strongly and repeatedly impacted by his heritage. For example, in 1937, Tarski applied for a chair at Poznań University but the chair was abolished to avoid assigning it to Tarski (who was undisputedly the strongest applicant) because he was a Jew. 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 university, private Ivy League research university in Cambridge, Massachusetts, United States. Founded in 1636 and named for its first benefactor, the History of the Puritans in North America, Puritan clergyma ...
. 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, also known as the September Campaign, Polish Campaign, and Polish Defensive War of 1939 (1 September – 6 October 1939), was a joint attack on the Second Polish Republic, Republic of Poland by Nazi Germany, the Slovak R ...
and the outbreak of
World War II World War II or the Second World War (1 September 1939 – 2 September 1945) was a World war, global conflict between two coalitions: the Allies of World War II, Allies and the Axis powers. World War II by country, Nearly all of the wo ...
. Tarski left reluctantly, because Leśniewski had died a few months before, creating a vacancy which Tarski hoped to fill. Oblivious to the
Nazi Nazism (), formally named National Socialism (NS; , ), is the far-right politics, far-right Totalitarianism, totalitarian socio-political ideology and practices associated with Adolf Hitler and the Nazi Party (NSDAP) in Germany. During H ...
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, public research university within the City University of New York (CUNY) system in New York ...
(1940), and thanks to a
Guggenheim Fellowship Guggenheim Fellowships are Grant (money), grants that have been awarded annually since by the John Simon Guggenheim Memorial Foundation, endowed by the late Simon Guggenheim, Simon and Olga Hirsh Guggenheim. These awards are bestowed upon indiv ...
, the
Institute for Advanced Study The Institute for Advanced Study (IAS) is an independent center for theoretical research and intellectual inquiry located in Princeton, New Jersey. It has served as the academic home of internationally preeminent scholars, including Albert Ein ...
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 university, public Land-grant university, land-grant research university in Berkeley, California, United States. Founded in 1868 and named after t ...
, 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: 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 worked primarily in logic and foundations of mathematics and is perhaps best remembered for the Mostowski collapse lemma. He was a member of the Polish Academy ...
, Bjarni Jónsson, Julia Robinson, Robert Vaught,
Solomon Feferman Solomon Feferman (December 13, 1928July 26, 2016) was an American philosopher and mathematician who worked in mathematical logic. In addition to his prolific technical work in proof theory, computability theory, and set theory, he was known for h ...
,
Richard Montague Richard Merritt Montague (September 20, 1930 – March 7, 1971) was an American mathematician and philosopher who made contributions to mathematical logic and the philosophy of language. He is known for proposing Montague grammar to formalize th ...
, 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
Adolf Lindenbaum Adolf Lindenbaum (12 June 1904 – August 1941) was a Polish-Jewish logician and mathematician best known for Lindenbaum's lemma and Lindenbaum–Tarski algebras. Life He was born and brought up in Warsaw. He earned a Ph.D. in 1928 un ...
,
Dana Scott Dana Stewart Scott (born October 11, 1932) is an American logician who is the emeritus Hillman University Professor of Computer Science, Philosophy, and Mathematical Logic at Carnegie Mellon University; he is now retired and lives in Berkeley, C ...
, 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 student's 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 (1950, 1966), the
Institut Henri Poincaré The Henri Poincaré Institute (or IHP for ''Institut Henri Poincaré'') is a mathematics research institute part of Sorbonne University, in association with the Centre national de la recherche scientifique (CNRS). It is located in the 5th arrondi ...
in Paris (1955), the Miller Institute for Basic Research in Science in Berkeley (1958–60), the University of California at Los Angeles (1967), and the Pontifical Catholic University of Chile (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 Nati ...
, the
British Academy The British Academy for the Promotion of Historical, Philosophical and Philological Studies 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 sa ...
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 hon ...
s from the Pontifical Catholic University of Chile in 1975, from
Marseille Marseille (; ; see #Name, below) is a city in southern France, the Prefectures in France, prefecture of the Departments of France, department of Bouches-du-Rhône and of the Provence-Alpes-Côte d'Azur Regions of France, region. Situated in the ...
's Paul Cézanne University in 1977 and from the
University of Calgary {{Infobox university , name = University of Calgary , image = University of Calgary coat of arms without motto scroll.svg , image_size = 150px , caption = Coat of arms , former ...
, 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 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 mathema ...
, 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 original ...
proved that, if one accepts the
Axiom of Choice In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...
, a
ball A ball is a round object (usually spherical, but sometimes 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 for 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-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 then ...
. 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 ..." can be viewed as a question "When is there an x such ...
, that the
first-order theory In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language. In most scenarios a deductive system is first understood from context, giving rise to a formal system that combines the language with deduct ...
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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
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 computer scientist, mathematician, logician, and philosopher who made major contributions to mathematical logic and the foundations of theoretical computer science. He is bes ...
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 nea ...
(the theory of
natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
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 bou ...
, 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 (''p ...
, and
closure algebra In abstract algebra, an interior algebra is a certain type of algebraic structure that encodes the idea of the topological interior of a set. Interior algebras are to topology and the modal logic S4 what Boolean algebras are to set theory and ord ...
s, 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 commu ...
s is decidable, but that of non-Abelian groups is not. While teaching at the Stefan Żeromski Gimnazjum in the 1920s and 30s, Tarski often taught
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
. Using some ideas of
Mario Pieri Mario Pieri (22 June 1860 – 1 March 1913) was an Italian mathematician who is known for his work on foundations of geometry. Biography Pieri was born in Lucca, Italy, the son of Pellegrino Pieri and Ermina Luporini. Pellegrino was a lawyer. Pie ...
, in 1926 Tarski devised an original axiomatization for plane
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
, 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 Solid geometry or stereometry is the geometry of Three-dimensional space, three-dimensional Euclidean space (3D space). A solid figure is the region (mathematics), region of 3D space bounded by a two-dimensional closed surface; for example, a ...
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 (knowledge), intuition or taken ...
), 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 Mereology (; from Greek μέρος 'part' (root: μερε-, ''mere-'') and the suffix ''-logy'', 'study, discussion, science') is the philosophical study of part-whole relationships, also called ''parthood relationships''. As a branch of metaphys ...
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, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the cas ...
s. ''Ordinal Algebras'' sets out an algebra for the additive theory of
order type In mathematics, especially in set theory, two ordered sets and are said to have the same order type if they are order isomorphic, that is, if there exists a bijection (each element pairs with exactly one in the other set) f\colon X \to Y su ...
s. Cardinal, but not ordinal, addition commutes. In 1941, Tarski published an important paper on
binary relation In mathematics, a binary relation associates some elements of one Set (mathematics), set called the ''domain'' with some elements of another set called the ''codomain''. Precisely, a binary relation over sets X and Y is a set of ordered pairs ...
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 re ...
and its
metamathematics Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheory, metatheories, which are Mathematical theory, mathematical theories about other mathematical theories. Emphasis on metamathematics (and ...
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 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 mathema ...
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 nea ...
. 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 re ...
, see Maddux (2006). In the late 1940s, Tarski and his students devised
cylindric algebra In mathematics, the notion of cylindric algebra, developed by Alfred Tarski, arises naturally in the Algebraic logic, algebraization of first-order logic with equality. This is comparable to the role Boolean algebra (structure), Boolean algebras pl ...
s, which are to
first-order logic First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over ...
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 ''B ...
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, Robert Lawson Vaught, has ranked Tarski as one of the four greatest logicians of all time — along with
Aristotle Aristotle (; 384–322 BC) was an Ancient Greek philosophy, Ancient Greek philosopher and polymath. His writings cover a broad range of subjects spanning the natural sciences, philosophy, linguistics, economics, politics, psychology, a ...
,
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 philos ...
, and
Kurt Gödel Kurt Friedrich Gödel ( ; ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel profoundly ...
. However, Tarski often expressed great admiration for
Charles Sanders Peirce Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American scientist, mathematician, logician, and philosopher who is sometimes known as "the father of pragmatism". According to philosopher Paul Weiss (philosopher), Paul ...
, 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 and formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms. In 1921, David Hilbert proposed to use formal systems as the foundation of knowledge in math ...
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, 200 ...
, 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). 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 Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the phi ...
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 broad movement within Western philosophy, especially English-speaking world, anglophone philosophy, focused on analysis as a philosophical method; clarity of prose; rigor in arguments; and making use of formal logic, mat ...
, a contribution to
symbolic logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
,
semantics Semantics is the study of linguistic Meaning (philosophy), meaning. It examines what meaning is, how words get their meaning, and how the meaning of a complex expression depends on its parts. Part of this process involves the distinction betwee ...
, and the
philosophy of language Philosophy of language refers to the philosophical study of the nature of language. It investigates the relationship between language, language users, and the world. Investigations may include inquiry into the nature of Meaning (philosophy), me ...
. 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 philosophical debate examines the extent to which Tarski's theory of truth for formalized languages can be seen as a
correspondence theory of truth In metaphysics and philosophy of language, the correspondence theory of truth states that the truth or falsity of a statement is determined only by how it relates to the world and whether it accurately describes (i.e., corresponds with) that worl ...
. 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" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
p. (where p is the proposition expressed by "p") The debate amounts to whether to read sentences of this form, such as as expressing merely a deflationary theory of truth or as embodying
truth Truth or verity is the Property (philosophy), property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth, 2005 In everyday language, it is typically ascribed to things that aim to represent reality or otherwise cor ...
as a more substantial property (see Kirkham 1992).


Logical consequence

In 1936, Tarski published Polish and German versions of a lecture, “On the Concept of Following Logically", 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 philosophical literature.
John Etchemendy John W. Etchemendy (born 1952) is an American logician and philosopher who served as Stanford University's twelfth Provost (education), Provost. He succeeded John L. Hennessy to the post on September 1, 2000 and stepped down on January 31, 2017 ...
stimulated much of the 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

Tarski's "What are Logical Notions?" (Tarski 1986) 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 In mathematics, the Erlangen program is a method of characterizing geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as ''Vergleichende Betrachtungen über neuere geometrische Forschungen.'' It is na ...
of the 19th-century German mathematician
Felix Klein Felix Christian Klein (; ; 25 April 1849 – 22 June 1925) was a German mathematician and Mathematics education, mathematics educator, known for his work in group theory, complex analysis, non-Euclidean geometry, and the associations betwe ...
. Mautner (in 1946), and possibly an article by the Portuguese mathematician José Sebastião e Silva, anticipated Tarski in applying the Erlangen Program to logic. The Erlangen program classified the various types of geometry (
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
,
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 i ...
,
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, 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 made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...
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 (
automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphism ...
s) of a domain onto itself. By domain is meant the
universe of discourse In the formal sciences, the domain of discourse or universe of discourse (borrowing from the mathematical concept of ''universe'') is the set of entities over which certain variables of interest in some formal treatment may range. It is also ...
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''). Truth values are used in ...
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 ...
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,
disjunction In logic, disjunction (also known as logical disjunction, logical or, logical addition, or inclusive disjunction) is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is ...
and
negation In logic, negation, also called the logical not or logical complement, is an operation (mathematics), operation that takes a Proposition (mathematics), proposition P to another proposition "not P", written \neg P, \mathord P, P^\prime or \over ...
(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 In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...
and union applied to
subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), 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 a ...
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 and theoretical computer science, a type theory is the formal presentation of a specific type system. Type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a foundation of ...
, 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 suc ...
. # ''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 philosopher, logician, mathematician, and public intellectual. He had influence on mathematics, logic, set theory, and various areas of analytic ...
's and Whitehead's ''
Principia Mathematica The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by the mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1 ...
'' 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 Solomon Feferman (December 13, 1928July 26, 2016) was an American philosopher and mathematician who worked in mathematical logic. In addition to his prolific technical work in proof theory, computability theory, and set theory, he was known for h ...
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
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
s. 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. Vann 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 Jan Łukasiewicz (; 21 December 1878 – 13 February 1956) was a Polish logician and philosopher who is best known for Polish notation and Łukasiewicz logic. His work centred on philosophical logic, mathematical logic and history of logi ...
). "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 Adolf Lindenbaum (12 June 1904 – August 1941) was a Polish-Jewish logician and mathematician best known for Lindenbaum's lemma and Lindenbaum–Tarski algebras. Life He was born and brought up in Warsaw. He earned a Ph.D. in 1928 un ...
). "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.
The Semantical Concept of Truth and the Foundations of Semantics
" ''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 Mostowski (feminine: Mostowska, plural: Mostowscy) is a surname. It may refer to: * Mostowski Palace (), an 18th-century palace in Warsaw * Andrzej Mostowski (1913 - 1975), a Polish mathematician ** Mostowski collapse lemma, in mathematical logi ...
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 Leon Albert Henkin (April 19, 1921, Brooklyn, New York – November 1, 2006, Oakland, California) was an American logician, whose works played a strong role in the development of logic, particularly in the Type theory, theory of types. He was an ...
and Donald Monk). ''Cylindric Algebras: Part I''. North-Holland. * 1985 (with
Leon Henkin Leon Albert Henkin (April 19, 1921, Brooklyn, New York – November 1, 2006, Oakland, California) was an American logician, whose works played a strong role in the development of logic, particularly in the Type theory, theory of types. He was an ...
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 *
Cylindric algebra In mathematics, the notion of cylindric algebra, developed by Alfred Tarski, arises naturally in the Algebraic logic, algebraization of first-order logic with equality. This is comparable to the role Boolean algebra (structure), Boolean algebras pl ...
* Interpretability * Weak interpretability * List of things named after Alfred Tarski * Timeline of Polish science and technology


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, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mat ...
( Robert Vaught),
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
(Jonsson), 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 (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
(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 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 mathema ...
(Azriel Levy), real closed fields (Lou Van Den Dries), decidable theory (Doner and
Wilfrid Hodges Wilfrid Augustine Hodges, Fellow of the British Academy, FBA (born 27 May 1941) is a British mathematician and logic, logician known for his work in model theory. Life Hodges attended New College, Oxford (1959–65), where he received degrees i ...
),
metamathematics Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheory, metatheories, which are Mathematical theory, mathematical theories about other mathematical theories. Emphasis on metamathematics (and ...
(Blok and Pigozzi),
truth Truth or verity is the Property (philosophy), property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth, 2005 In everyday language, it is typically ascribed to things that aim to represent reality or otherwise cor ...
and
logical consequence Logical consequence (also entailment or logical implication) is a fundamental concept in logic which describes the relationship between statement (logic), statements that hold true when one statement logically ''follows from'' one or more stat ...
(
John Etchemendy John W. Etchemendy (born 1952) is an American logician and philosopher who served as Stanford University's twelfth Provost (education), Provost. He succeeded John L. Hennessy to the post on September 1, 2000 and stepped down on January 31, 2017 ...
), and general
philosophy Philosophy ('love of wisdom' in Ancient Greek) is a systematic study of general and fundamental questions concerning topics like existence, reason, knowledge, Value (ethics and social sciences), value, mind, and language. It is a rational an ...
(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 in ...
. * 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 peer-reviewed scientific journal of mathematics. It was established by Benjamin Finkel in 1894 and is published by Taylor & Francis on behalf of the Mathematical Association of America. It is an exposi ...
117:475–89. * Wolenski, Jan, 1989. ''Logic and Philosophy in the Lvov–Warsaw School''. Reidel/Kluwer.


External links

*
Stanford Encyclopedia of Philosophy The ''Stanford Encyclopedia of Philosophy'' (''SEP'') is a freely available online philosophy resource published and maintained by Stanford University, encompassing both an online encyclopedia of philosophy and peer-reviewed original publication ...
: *
Tarski's Truth Definitions
by
Wilfred Hodges Wilfrid Augustine Hodges, FBA (born 27 May 1941) is a British mathematician and logician known for his work in model theory. Life Hodges attended New College, Oxford (1959–65), where he received degrees in both '' Literae Humaniores'' and ( ...
. *
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 philosophers 20th-century American essayists 20th-century Polish mathematicians Jewish American atheists American logicians American male essayists American male non-fiction writers Analytic philosophers Converts to Roman Catholicism from Judaism Computability theorists Jewish American academics 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 People from Warsaw Governorate 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 20th-century Polish philosophers Set theorists Scientists from Warsaw University of California, Berkeley faculty University of California, Berkeley people University of California, Berkeley staff University of Warsaw alumni 20th-century American male writers Corresponding fellows of the British Academy