Alfred Tarski

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Alfred Tarski

Born January 14, 1901(1901-01-14)
Warsaw, Poland (under Russian rule at the time)
Died October 26, 1983
Berkeley, California
Fields Mathematics, logic, philosophy of language
Institutions UC Berkeley
Alma mater Warsaw University
Doctoral advisor Stanisław Leśniewski
Doctoral students Andrzej Mostowski, Julia Robinson, Robert Vaught, Solomon Feferman, Richard Montague, J. Donald Monk, Howard Jerome Keisler
Known for work on the foundations of modern logic, formal notion of truth, development of model theory

Alfred Tarski (January 14, 1901, Warsaw, Russian-ruled Poland – October 26, 1983, Berkeley, California) was a Polish logician and mathematician. Educated in the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and did research in mathematics at the University of California, Berkeley, from 1942 until his death.[1]

A prolific author best known for his work on model theory, metamathematics, and algebraic logic, he also contributed to abstract algebra, topology, geometry, measure theory, mathematical logic, set theory, and analytic philosophy.

He is regarded as one of the four greatest logicians of all time, perhaps matched only by Aristotle, Kurt Gödel, and Gottlob Frege.[2][3][4]His biographers Anita 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 and the theory of models." [2]

Contents

[edit] Life

Alfred Tarski was born Alfred Teitelbaum (Polish spelling: "Tajtelbaum"), to parents who were Polish Jews in comfortable circumstances. He first manifested his mathematical abilities while in secondary school, at Warsaw's Szkoła Mazowiecka.[5] Nevertheless, he entered the University of Warsaw in 1918 intending to study biology.[6]

After Poland regained independence in 1918, Warsaw University came under the leadership of Jan Łukasiewicz, Stanisław Leśniewski and Wacław Sierpiński and quickly became a world leading research institution in logic, foundational mathematics, and the philosophy of mathematics. Leśniewski recognized Tarski's potential as a mathematician and persuaded him to abandon biology.[citation needed] Henceforth Tarski attended courses taught by Łukasiewicz, Sierpiński, Stefan Mazurkiewicz and Tadeusz Kotarbiński, and became the only person ever to complete a doctorate under Leśniewski's supervision. Tarski and Leśniewski soon grew cool to each other. However, in later life, Tarski reserved his warmest praise for Kotarbiński, as was mutual.

In 1923, Alfred Teitelbaum and his brother Wacław changed their surname to "Tarski," a name they invented because it sounded more Polish, was simple to spell and pronounce, and seemed unused. (Years later, Alfred met another Alfred Tarski in northern California.) The Tarski brothers also converted to Roman Catholicism, Poland's dominant religion. Alfred did so even though he was an avowed atheist. [7] Tarski was a Polish nationalist who saw himself as a Pole and wished to be fully accepted as such. In America, he spoke Polish at home. With a non-Jewish name and as a nominal Catholic he hoped to be more successful in future applications for a university position in Poland since anti-semitic resentments were strong in Polish academia at the time.

In 1929 Tarski married a fellow teacher Maria Witkowska, a Pole of Catholic ancestry. She had worked as a courier for the army during Poland's fight for independence. They had two children, a son Jan who became a physicist, and a daughter who married the mathematician Andrzej Ehrenfeucht.

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. Tarski applied for a chair of philosophy at Lwów University, but on Bertrand Russell's recommendation it was awarded to Leon Chwistek. In 1937 Tarski applied for a chair at Poznań University; but, the chair was abolished.[8]

In 1930, Tarski visited the University of Vienna, lectured to Karl Menger'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 travelled to Paris to present his ideas on truth at the first meeting of the Unity of Science movement, an outgrowth of the Vienna Circle. Tarski's ties to this movement saved his life, because they resulted in his being invited to address the Unity of Science Congress held in September 1939 at Harvard University. Thus he left Poland in August 1939, on the last ship to sail from Poland for the United States before the German invasion of Poland and the outbreak of World War II. Tarski left reluctantly, because Leśniewski had died a few months before, creating a vacancy which Tarski hoped to fill. He was so oblivious to the Nazi threat that he left his wife and children in Warsaw; he did not see them again until 1946. During the war, nearly all his extended family died 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 (1940), and thanks to a Guggenheim Fellowship, the Institute for Advanced Study at Princeton (1942), where he again met Gödel. Tarski became an American citizen in 1945.[9] In 1942, Tarski joined the Mathematics Department at the University of California, Berkeley, where he spent the rest of his career. Although emeritus from 1968, he taught until 1973 and supervised Ph.D. candidates until his death.[10] At Berkeley, Tarski acquired a reputation as an awesome and demanding teacher, a fact noted by many observers:

"His seminars at Berkeley fast became a power-house of logic. His students, many of them now distinguished mathematicians, recall 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." [11]
"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." [12]
"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."[13]

Indeed, 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-Chun Chang and Jerome Keisler, authors of Model Theory (1973)[14], a classic text in the field.[15][16] 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.[16]

Tarski lectured at University College, London (1950, 1966), the Institut Henri Poincaré in Paris (1955), the Miller Institute for Basic Research in Science in Berkeley (1958-1960), 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 British Academy and the Royal Netherlands Academy of Arts and Sciences, received honorary degrees from the Pontifical Catholic University of Chile in 1975, from Marseilles' Paul Cézanne University in 1977 and from the University of Calgary, 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.[17]

[edit] Mathematician

Tarski's mathematical interests were exceptionally broad for a mathematical logician. His collected papers run to about 2500 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.[18]

Tarski's first paper, published when he was 19 years old, was on set theory, a subject to which he returned throughout his life. In 1924, he and Stefan Banach proved that, if one accepts the Axiom of Choice, a ball 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.

In A decision method for elementary algebra and geometry, Tarski showed, by the method of quantifier elimination, that the first-order theory of the real numbers 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 proved in 1936 that Peano arithmetic (effectively the theory Tarski proved decidable, except that the natural numbers replace the reals) 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, and closure algebras, are all undecidable. The theory of Abelian groups is decidable, but that of non-Abelian groups is not.

In the 1920s and 30s, Tarski often taught high school geometry. In 1929, he showed that much of Euclidian solid geometry could be recast as a first order theory whose individuals are spheres, a primitive notion, 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 that Lesniewski's variant. Starting in 1926, Tarski devised an original axiomatization for plane Euclidian geometry, one considerably more concise than Hilbert's. Tarski's axiomatization is 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. Near the end of his life, Tarski wrote a very long letter, published as Tarski and Givant (1999), summarizing his work on geometry.

Cardinal Algebras studied algebras whose models include the arithmetic of cardinal numbers. Ordinal Algebras sets out an algebra for the additive theory of order types. Cardinal, but not ordinal, addition commutes.

In 1941, Tarski published an important paper on binary relations, which began the work on relation algebra and its metamathematics that occupied Tarski and his students for much of the balance of his life. While that exploration (and the closely related work of Roger Lyndon) uncovered some important limitations of relation algebra, Tarski also showed (Tarski and Givant 1987) that relation algebra can express most axiomatic set theory and Peano arithmetic. For an introduction to relation algebra, see Maddux (2006). In the late 1940s, Tarski and his students devised cylindric algebras, which are to first-order logic what the two-element Boolean algebra is to classical sentential logic. This work culminated in the two monographs by Tarski, Henkin, and Monk (1971, 1985).

[edit] Logician

Along with Aristotle, Gottlob Frege, and Kurt Gödel, Tarski is generally considered one of the four greatest logicians of all time.[2][3][19] Of these four, he was by far the most prolific author.

Tarski produced axioms for logical consequence, and worked on deductive systems, the algebra of logic, and the theory of definability. His semantic methods, which culminated in the model theory he and a number of his Berkeley students developed in the 1950s and 60s, radically transformed Hilbert's proof-theoretic metamathematics.

"In [Tarski's] view, metamathematics became similar to any mathematical discipline. Not only its concepts and results can 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."[20]

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 indefinability theorem, and mulled over their consequences for the axiomatic method in mathematics.

[edit] Truth in formalized languages

In 1933, Tarski published a very long (more than 100pp) 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 had to await the 1956 first edition of the volume Logic, Semantics, Metamathematics. This enormously cited paper is a landmark event in 20th century analytic philosophy, an important contribution to symbolic logic, semantics, and the philosophy of language. For a brief discussion of its content, see Truth for a brief description of the "Convention T" (see also T-schema) standard in Tarski's "inductive definition of truth".

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 truth definition. That condition requires that the truth theory have the following as theorems for all sentences p of the language for which truth is being defined:

'p' is True if and only if p.

(where p is the proposition expressed by "p")

The debate amounts to whether to read sentences of this form, such as

"Snow is white" is true if and only if snow is white

as expressing merely a deflationary theory of truth or as embodying truth as a more substantial property (see Kirkham 1992).

[edit] 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.[21]

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."

[edit] What are 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 in 1966; it was edited without his direct involvement.

In the talk, Tarski proposed a demarcation of the logical operations (which he calls "notions") from the non-logical. The suggested criteria were derived from the Erlangen programme of the German 19th century Mathematician, Felix Klein. (Mautner 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, affine geometry, topology, etc.) by the type of one-one transformation of space onto itself that left the objects of that geometrical theory invariant. (A one-to-one transformation is a functional map of the space onto itself so that every point of the space is associated with or mapped to one other point of the space. So, "rotate 30 degrees" and "magnify by a factor of 2" are intuitive descriptions of simple uniform one-one transformations.) Continuous transformations give rise to the objects of topology, similarity transformations to those of Euclidean geometry, and so on.

As the range of permissible transformations becomes broader, the range of objects one is able to distinguish as preserved by the application of the transformations becomes narrower. Similarity transformations are fairly narrow (they preserve the relative distance between points) and thus allow us to distinguish relatively many things (e.g., equilateral triangles from non-equilateral triangles). Continuous transformations (which can intuitively be thought of as transformations which allow non-uniform stretching, compression, bending, and twisting, but no ripping or glueing) allow us to distinguish a polygon from an annulus (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 a logic. If one identifies the truth value True with the domain set and the truth-value False with the empty set, then the following operations are counted as logical under the proposal:

1. Truth-functions: All truth-functions are admitted by the proposal. This includes, but is not limited to, all n-ary truth-functions for finite n. (It also admits of truth-functions with any infinite number of places.)

2. Individuals: No individuals, provided the domain has at least two members.

3. Predicates:

  • One-place total and null (the predicate that has all members of the domain in its extension and the predicate that has no members of the domain in its extension).
  • Two-place total and null, as well as the identity and diversity predicates (the predicate with the set of all ordered pairs of domain members as its extension, the predicate with the empty set as extension, the predicate with the set of all order-pairs <a,a> where a is a member of the domain and the predicate with the set of all order pairs <a,b> in its extension, 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 and negation (up to any ordinality, finite or infinite).

4. 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."

5. Set-Theoretic relations: Relations such as inclusion, intersection and union applied to subsets of the domain are logical in the present sense.

6. Set membership: Tarski ended his lecture with a discussion of whether the set membership relation counted as logical in his sense. (Given the reduction of (most of) mathematics to set theory, this was, in effect, the question of whether most or all of mathematics is a part of logic.) He pointed out that set membership is logical if set theory is developed along the lines of type theory, but is extralogical if set theory is set out axiomatically, as in the canonical Zermelo-Fraenkel set theory.

7. Logical notions of higher order: While Tarski confined his discussion to operations of first-order logic, there is nothing about his proposal that necessarily restricts it to first-order logic. (Tarski likely restricted his attention to first-order notions as the talk was given to a non-technical audience.) So, higher-order quantifiers and predicates are admitted as well.

In some ways the present proposal is the obverse of that of Lindenbaum and Tarski (1936), who proved that all the logical operations of Russell and Whitehead's Principia Mathematica are invariant under one-to-one transformations of the domain onto itself. The present proposal is also employed in Tarski and Givant (1987).

Solomon Feferman and Vann McGee further discussed Tarski's proposal in work published after his death. Feferman (1999) raises problems for the proposal and suggests a cure: replacing Tarski's preservation by automorphisms with preservation by arbitrary homomorphisms. In essence, this suggestion circumvents the difficulty Tarski's proposal has in dealing with 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.

[edit] See also

[edit] References

  1. ^ Feferman A.
  2. ^ a b c Feferman & Feferman, p.1
  3. ^ a b Vaught, Robert L. (Dec. 1986). "Alfred Tarski's Work in Model Theory". J. of Symbolic Logic (ASL) 51 (4): 869-882. 
  4. ^ Restall, Greg (2002-2006). "Great Moments in Logic". http://consequently.org/writing/logicians/. Retrieved on 2009-01-03. 
  5. ^ Feferman & Feferman, pp.17-18
  6. ^ Feferman & Feferman, p.26
  7. ^ Feferman & Feferman, p.294
  8. ^ Feferman & Feferman, pp.102-103
  9. ^ Feferman & Feferman, Chap. 5, pp.124-149
  10. ^ "Alfred Tarski, Mathematics: Berkeley". University of California (System) Academic Senate. 1985. http://content.cdlib.org/xtf/view?docId=hb4d5nb20m&doc.view=frames&chunk.id=div00159&toc.depth=1&toc.id=. Retrieved on 2008-12-26. 
  11. ^ Obituary in Times, reproduced here
  12. ^ Gregory Moore, "Alfred Tarski" in Dictionary of Scientific Biography
  13. ^ Feferman
  14. ^ Chang, C.C., and Keisler, H.J., 1973. Model Theory. North-Holland, Amsterdam. American Elsevier, New York.
  15. ^ Alfred Tarski at the Mathematics Genealogy Project
  16. ^ a b Feferman & Feferman, pp. 385-386
  17. ^ O'Connor, John J.; Robertson, Edmund F., "Alfred Tarski", MacTutor History of Mathematics archive 
  18. ^ Feferman & Feferman, pp.43-52, 69-75, 109-123, 189-195, 277-287, 334-342
  19. ^ Restall, Greg (2002-2006). "Great Moments in Logic". http://consequently.org/writing/logicians/. Retrieved on 2009-01-03. 
  20. ^ Sinaceur, Hourya. "Alfred Tarski: Semantic Shift, Heuristic Shift in Metamathematics". Synthese (Springer Verlag) 126 (1-2): 49-65. ISSN 00397857. 
  21. ^ Etchemendy, John (1999). The Concept of Logical Consequence. Stanford CA: CSLI Publications. ISBN 1575861941. 

[edit] Bibliography

[edit] Works of Tarski

Anthologies and collections
  • 1986. The Collected Papers of Alfred Tarski, 4 vols. Givant, S. R., and McKenzie, R. N., eds. Birkauser.
  • Givant, Steven, 1986. "Bibliography of Alfred Tarski", Journal of Symbolic Logic 51: 913-41.
  • 1983 (1956). Logic, Semantics, Metamathematics, Corcoran, J., ed. Hackett. 1st edition edited and translated by J. H. Woodger, Oxford Uni. Press.

The latter 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
  • 1931. "Sur les ensembles définissables de nombres réels I," Fundamenta Mathematica 17: 210-239.
  • 1936 (with Adolf Lindenbaum). "On the Limitations of Deductive Theories" in Tarski (1983): 384-92.
  • 1994 (1941). Introduction to Logic and to the Methodology of Deductive Sciences. Dover.
  • 1941. "On the calculus of relations," Journal of Symbolic Logic 6: 73-89.
  • 1944. "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 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. Providence RI: American Mathematical Society.
  • 1999 (with Steven Givant). "Tarski's system of geometry," 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: 155-96.

[edit] Biographical references

[edit] Logic literature

The December 1986 issue of the Journal of Symbolic Logic surveys Tarski's work on model theory (Robert Vaught), algebra (Jonsson), undecidable theories (McNulty), algebraic logic (Donald Monk), and geometry (Szczerba). The March 1988 issue of the same journal surveys his work on axiomatic set theory (Azriel Levy), real closed fields (Lou Van Den Dries), decidable theory (Doner and Wilfrid Hodges), metamathematics (Blok and Pigozzi), truth and logical consequence (John Etchemendy), and general philosophy (Patrick Suppes).

  • Ivor Grattan-Guinness, 2000. The Search for Mathematical Roots 1870-1940. Princeton Uni. Press.
  • Kirkham, Richard, 1992. Theories of Truth. MIT Press.
  • Karl R. Popper, 1972, Rev. Ed. 1979, "Philosophical Comments on Tarski's Theory of Truth," with Addendum, Objective Knowledge, Oxford: 319-340.
  • Sinaceur, H., 2001. "Alfred Tarski: Semantic shift, heuristic shift in metamathematics," Synthese 126: 49-65.
  • Wolenski, Jan, 1989. Logic and Philosophy in the Lvov–Warsaw School. Reidel/Kluwer.
  • Chang, C.C., and Keisler, H.J., 1973. Model Theory. North-Holland, Amsterdam. American Elsevier, New York.
  • Etchemendy, John, 1999. The Concept of Logical Consequence. Stanford CA: CSLI Publications. ISBN 1-57586-194-1
  • Solomon Feferman, 1999. "Logic, Logics, and Logicism," Notre Dame Journal of Formal Logic 40: 31-54.
  • Maddux, Roger D., 2006. Relation Algebras, vol. 150 in "Studies in Logic and the Foundations of Mathematics." Elsevier Science.
  • Mautner, F. I., 1946. "An Extension of Klein's Erlanger Program: Logic as Invariant-Theory," American Journal of Mathematics 68: 345-84.
  • McGee, Van, 1996. "Logical Operations", Journal of Philosophical Logic 25: 567-80.

[edit] External links

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