Alfred Tarski (1901 - 1983)

Until the early 20th century, the classical concept of truth, originating in the writings of Aristotle, was understood rather intuitively, as the conformity (or correspondence) of judgments with facts: "To say that what is, is not, and what is not, is, is false; and to say that what is, is, and what is not, is not, is true" (Łukasiewicz 1910). The Aristotelian definition of truth has long posed interpretative difficulties for philosophers, with its most significant flaw being that when applied without restrictions to natural language, it generates paradoxes, such as the well-known liar paradox. Tarski was the first to demonstrate— in an extraordinarily elegant and mathematically precise manner— how one could formulate a definition of truth that is consistent with Aristotle's concept, substantively accurate, and formally correct. Tarski's concept of the semantic theory of truth was first presented in 1933 in his article – “Concept of Truth in Formalized Languages," written in Polish and later translated into many languages.

Tarski's goal was to construct a definition of truth for sentences in any formalized language (i.e., a language with a precisely defined formal structure and vocabulary) that would be consistent with the so-called Convention T ("A sentence 'A' is true if and only if A") and free from the paradoxes of the classical definition of truth. The foundation of Tarski's definition of truth rests on several fundamental assumptions. First, according to Tarski, the concept of truth always refers to expressions of a specific language, meaning it has a relative nature: the concept of truth cannot, therefore, belong to the language to which it applies. Natural languages (e.g., Polish or English) do not meet this requirement. For this reason, Tarski believed it was impossible to provide a definition of truth for all sentences of natural language, as any such attempt must lead to paradoxes. The starting point in Tarski's construction was, therefore, the rigorous separation of the language whose sentences are being examined—called the object language—from the metalanguage, that is, the language in which the object language is described and analyzed. Second, truth—understood as the correspondence between sentences and extra-linguistic reality—must also be relativized to an interpretation, that is, a structure (model) consisting of a domain of objects described by the language and distinguished relations, properties, etc. The central idea of Tarski's definition is to reduce the truth of complex sentences to the truth of simpler sentences, from which the complex sentences are composed. This kind of reduction is continued until the simplest irreducible sentences (formulas) are encountered, whose truth (satisfaction) depends on their interpretation in the domain (model) that the object language is meant to describe.

A key concept in Tarski's definition of truth (semantics) is the satisfaction relation, which holds between the formulas of the object language and models (interpretations of the object language) and valuations (sequences of objects from the domain of the interpretation). The satisfaction relation is defined by induction, meaning the conditions defining the satisfaction relation are first formulated for the simplest formulas and then progressively for more complex ones. Consider a simple example. Suppose our object language contains a formula of the form xRy, where the letters x and y represent objects from the domain of the model, and R represents the relation between these objects. Let M be a model whose domain is the set of people, and let the interpretation of R in this model be the "is taller than" relation. Each valuation a assigns specific elements from the domain of model M to the letters x and y, so in our example, a(x) and a(y) belong to the set of people. The definition of the satisfaction relation for the considered object language will particularly include the following inductive condition defining the satisfaction of formulas of the form xRy:

The formula xRy is satisfied in M under the valuation a if and only if a(x) is taller than a(y).

Thus, the formula xRy will be satisfied in M by all and only those valuations that assign to the letter x a person from the domain of model M who is taller than the person assigned by valuation a to the letter y. Inductive satisfaction conditions are defined for all possible forms of formulas in the object language. The concept of truth for sentences in the object language is then defined as follows:

A sentence S is true in model M if and only if every valuation in M satisfies S.

Tarski's definition of truth is not merely the definition of a single concept but a whole theory of truth—a theory that made Tarski famous as "the man who defined truth," as evidenced by an anecdotal letter in the Tarski Archive at Berkeley, which reads: "Dear Professor Tarski, You have defined truth. I am wondering how you arrived at this. I made a bet with a classmate that you would reply. Sincerely, Patti Rossi." At the top, Tarski wrote in pencil: "Patti lost the bet." Tarski's definition of truth is now the foundation upon which modern model theory and logical semantics are built. For this reason, Tarski is considered the father of model theory.

As a complement to his definition of truth, Tarski demonstrated that formal languages cannot define their own truth. This result is known as Tarski's undefinability theorem, which, in a more formal statement, asserts:

If a consistent formal system L contains the language and axioms of arithmetic for natural numbers, then the concept of truth for L cannot be defined within L itself.

The undefinability of truth theorem is considered one of the most important metalogical theorems, indicating significant limitations of formal systems: if we wish to define the concept of truth for a given language, we must resort to a language that is much richer.

In 1924, Tarski, together with Stefan Banach, published one of the most astonishing mathematical theorems in Fundamenta Mathematicae, which states that any sphere (a mathematical object) can be divided into a finite number of parts, from which a sphere of any other size can then be reassembled. This result contradicted ordinary intuition, and for this reason, the theorem is often referred to as the Banach-Tarski paradox. However, unlike other well-known paradoxes that do indeed lead to contradictions, the Banach-Tarski theorem was consistent with the commonly accepted axioms of set theory. The proof of the theorem regarding the paradoxical decomposition of a sphere critically relied on the axiom of choice, which posits the existence of a set without specifying how it is constructed. The Banach-Tarski result thus indicated surprising consequences of adopting the axiom of choice in the foundations of set theory, which, prior to this, had been considered fairly intuitive in mathematical proofs. The Banach-Tarski theorem had significant implications for the development of research into the foundations of set theory, particularly initiating the search for constructive methods of proof without using the axiom of choice. In the later period, Tarski formulated around 30 equivalent versions of the axiom of choice, mostly in terms of operations and relations on cardinal numbers.

Tarski achieved many fundamental results in the foundations of mathematical theories such as algebra, geometry, and topology. Specifically, using the method of quantifier elimination, he demonstrated the completeness and decidability of elementary real algebra and elementary Euclidean geometry, meaning he provided an algorithm to check in finitely many steps whether a given statement formulated in the language of these theories is true. Tarski introduced into the logical literature the concept of an essentially undecidable theory, defined as a theory whose all consistent extensions are also undecidable, and then proved a series of metalogical theorems establishing sufficient conditions for a theory to be undecidable. Thanks to these results, many interesting theorems regarding the (in)decidability of various mathematical theories were proven.

In the 1940s, along with his collaborators, Tarski achieved many important results concerning algebras of cardinal numbers, order numbers, and topological algebras. Using an abstract-algebraic approach, Tarski demonstrated, among other things, that many theorems regarding finite and infinite sums of cardinal numbers could be proven without using the axiom of choice.

At the beginning of the 1940s, Tarski began, and continued until his death, his research on relational algebras. Some consider Tarski's results in the field of relational algebras to be as groundbreaking as his definition of truth. In 1941, Tarski proposed an elegant and relatively simple formalization of the two-argument relational calculus in the form of abstract relational algebras and posed the question of whether every equality that is true under any interpretation in all two-argument relational algebras defined on non-empty domains can be derived from the axioms of this system. In other words, Tarski raised the problem of whether, similarly to Boolean algebras, the representation theorem holds for relational algebras, according to which every relational algebra has a representation as a two-argument relational algebra. Unexpectedly, the solution to the problem was negative: there exist equalities that are true in all two-argument relational algebras but are not derivable from Tarski's axioms. Despite the negative result, it turned out that the concept of relational algebras had immense potential, as many theories from various fields (e.g., computer science and logic) can be interpreted as theories of certain relational algebras. The significance of Tarski's results in relational algebras was realized much later when these algebras began to be applied in research on relational databases and methods for efficiently managing large databases.

Biography

Alfred Tarski was born on January 14, 1901, in Warsaw. He was the first of two sons of Róża (Rachela) – née Prussak – and Ignacy (Isaak) Teitelbaum. In 1910, he began his studies at the IV Government Gymnasium in Warsaw, and in 1915, he entered the sixth grade of the eight-year Mazovian School, where he studied until 1918, obtaining his school-leaving certificate. In October 1918, he started studying biology at the University of Warsaw, but in the following year, he abandoned biology and switched to mathematics. Between 1919 and 1924, Tarski devoted almost all his time to studying mathematics and logic. He participated in classes led by some of Poland's greatest mathematicians, logicians, and philosophers, including Zygmunt Janiszewski, Tadeusz Kotarbiński, Kazimierz Kuratowski, Stanisław Leśniewski, Jan Łukasiewicz, Stefan Mazurkiewicz, and Władysław Sierpiński. In 1920, he changed the spelling of his surname from "Teitelbaum" to "Tajtelbaum." In 1921, Tarski published his first article, "Przyczynek do aksjomatyki zbioru dobrze uporządkowanego" ("Contribution to the Axiomatization of the Well-Ordered Set"), in the Przegląd Filozoficzny (trans. The Philosophical Review) ******journal, where he presented a solution to a problem in set theory posed by Stanisław Leśniewski. In 1922, he began working as a mathematics teacher at the Girls' Gymnasium of Zofia Kalecka, where he taught mathematics. In the same year, he was baptized and applied for a surname change to "Tarski." In 1924, he received his doctorate based on his dissertation titled "O wyrazie pierwotnym sylogistyki" ("On the Primitive Term of Syllogistics"). His doctoral supervisor was Leśniewski, and the reviewers of his work were Łukasiewicz and Sierpiński. In April 1924, he officially received permission to change his surname. In 1925, he began working as a mathematics teacher at the Stefan Żeromski Gymnasium and became an associate professor at the University of Warsaw. During this time, he met mathematicians from the Lviv school and established a collaboration with Stefan Banach and Adolf Lindenbaum.

In June 1929, Tarski married Maria Witkowska. In the same year, he applied for the chair in mathematical logic in Lviv. His candidacy was supported by philosophers Kazimierz Twardowski, Kazimierz Ajdukiewicz, Jan Łukasiewicz, and Stanisław Leśniewski. Ultimately, the chair was awarded to Leon Chwistek, a philosopher and logician supported by Hugo Steinhaus, Stefan Banach, and Bertrand Russell. In 1930, at the invitation of Hans Hahn, Tarski spent a week at the University of Vienna, where he met Rudolf Carnap and Kurt Gödel. He delivered a series of lectures at the famous Vienna Circle seminar. In the same year, Mojżesz Presburger earned his master's degree under Tarski's supervision, presenting a proof of the completeness of the theory of natural numbers with only addition, obtained using quantifier elimination. Today, Presburger's theorem is regarded as one of the most important results of the Polish mathematical school, though in 1930, Tarski thought it did not merit a doctoral degree. In 1934, Tarski's son, Janusz Andrzej, was born. In 1935, Tarski took a leave of absence and, thanks to a Rockefeller Foundation scholarship, spent several months in Vienna and then Paris, where he met Kurt Gödel and Karl Popper. In 1937, he applied for a professorship at the University of Poznań, but once again unsuccessfully: the competition was not resolved, and the chair was abolished. In the early 1930s, Andrzej Mostowski, who would later become Tarski's first doctoral student, began attending Tarski's lectures. Mostowski earned his Ph.D. in 1938, based on a dissertation written under Tarski's supervision, which analyzed the concept of finiteness and the independence of various definitions of it. In 1937, Wanda Szmielew began her studies under Tarski, earning her doctorate after the war. In March 1938, Tarski's daughter Ina Krystyna was born. From the time of his doctoral degree until the outbreak of World War II, Tarski led an intensive academic and social life both in Warsaw and abroad. By 1939, he had published 3 books, 62 articles (including one of his greatest works on the concept of truth in formalized languages), and 16 abstracts; he participated in numerous domestic and international conferences and established collaborations with leading scholars such as Carnap, Gödel, Popper, and Quine. In the spring of 1939, Tarski received an invitation from Quine to give a lecture at the Fifth International Congress of Unity of Science, scheduled for September at Harvard University. However, in May 1939, Leśniewski passed away, and Tarski hoped to take over his chair, which delayed his response to Quine’s invitation. He decided to travel at the last moment. On August 11, 1939, with a passport, an American visa, and a suitcase of summer clothes, Tarski took a train to Gdynia and sailed on the M/S Piłsudski to America. He arrived in New York on August 21, 1939. His wife and children remained in Warsaw, where the war broke out, leaving Tarski in the United States without money or employment.

In September 1939, Tarski participated in the Unity of Science Conference at Harvard University, where he delivered a lecture. With the help of fellow mathematicians, he secured a guest professorship at City College in New York for the spring semester of 1940, followed by a one-year academic position at Harvard University for the 1940–1941 academic year. During this time, he met J.C.C. McKinsey, with whom he would later publish significant works on the algebraic aspects of topology and its applications in the semantics of intuitionistic logic and modal logics. Between 1940 and 1942, Tarski traveled extensively, visiting major universities on the East Coast and in the Midwest of the United States. He delivered numerous lectures on his recent results concerning undecidability and relational algebras. In January 1942, he went to Princeton, where he spent time at the Institute for Advanced Study as a Guggenheim Fellow. During this period, he collaborated with Paul Erdős. In July 1942, Tarski left Princeton and moved to the University of California, Berkeley, where he took up a one-year lecturer position with the possibility of an extension at least until the end of the war.

In 1942, Tarski took up a lecturer position at the University of California, Berkeley. Three years later, he was promoted to associate professor, and in 1948, he became a full professor. In 1945, he acquired American citizenship. In 1946, he succeeded in bringing his wife and children to Berkeley. Unfortunately, his parents and brother did not survive the war. Tarski remained at Berkeley for the rest of his life. He visited Poland several times (1956, 1959, 1961, 1964, 1966), always for scientific reasons. Throughout his time in California, Tarski maintained an incredibly intense research pace, actively promoting logic and his research programs. At Berkeley, he established a special doctoral program in logic and the foundations of mathematics. In 1958, he founded the Group for Logic and Methodology of Sciences, which integrated logicians, mathematicians, and philosophers. He also developed the conceptual and organizational framework for international congresses on logic, methodology, and the philosophy of science, which continue to be held regularly to this day. Tarski organized many other more specialized scientific events. He maintained contact with many scholars from Poland, actively bringing them to Berkeley for short or long-term stays. He also supported Polish intellectual émigrés, such as Leszek Kołakowski and Czesław Miłosz. Tarski closely followed political events in Poland, reading Polish journals, including the Paris-based "Kultura," one edition of which he financed.

He passed away in Berkeley on October 27, 1983.