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Alfred Tarski's work on general metamathematics

  • W. J. Blok (a1) and Don Pigozzi (a2)


In this essay we discuss Tarski's work on what he called the methodology of the deductive sciences, or more briefly, borrowing the terminology of Hilbert, metamathematics, The clearest statement of Tarski's views on this subject can be found in his textbook Introduction to logic [41m].1 Here he describes the tasks of metamathematics as “the detailed analysis and critical evaluation of the fundamental principles that are applied in the construction of logic and mathematics”. He goes on to describe what these fundamental principles are: All the expressions of the discipline under consideration must be defined in terms of a small group of primitive expressions that seem immediately understandable. Furthermore, only those statements of the discipline are accepted as valid that can be deduced by precisely defined and universally accepted means from a small set of axioms whose validity seems evident. The method of constructing a discipline in strict accordance with these principles is known as the deductive method, and the disciplines constructed in this manner are called deductive systems. Since contemporary mathematical logic is one of those disciplines that are subject to these principles, it itself is a deductive science. Tarski then goes on to say:

“The view has become more and more common that the deductive method is the only essential feature by means of which the mathematical disciplines can be distinguished from all other sciences; not only is every mathematical discipline a deductive theory, but also, conversely, every deductive theory is a mathematical discipline”.

This identification of mathematics with the deductive sciences is in our view one of the distinctive aspects of Tarski's work. Another characteristic feature is his broad view of what constitutes the domain of metamathematical investigations. A clue to this aspect of his work can also be found in Chapter 6 of Introduction to logic. After a discussion of the notions of completeness and consistency, he remarks that the investigations concerning these topics were among the most important factors contributing to a considerable extension of the domain of methodological studies, and caused even a fundamental change in the whole character of the methodology of deductive sciences.



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Andréka, H., Németi, I., and Sain, I. [198-] Abstract model approach to algebraic logic (to appear).
Birkhoff, G. [1935] On the structure of abstract algebras, Proceedings of the Cambridge Philosophical Society, vol. 31, pp. 433454.
Blok, W. J. and Pigozzi, D. [198-] A characterization of algebraizable logics, Memoirs of the American Mathematical Society (to appear).
Blok, W. J. and Pigozzi, D. [1956] Introduction to mathematical logic, vol. I, Princeton University Press, Princeton, New Jersey.
Curry, H. B. and Feys, R. [1958] Combinatory logic, vol. I, North-Holland, Amsterdam.
Felscher, W. and Schulte-Mönting, J. [1984] Algebraic and deductive consequence operations, Universal algebra and its links with logic, algebra, combinatorics and computer science (proceedings of the 25th workshop on general algebra, Darmstadt, 1983; Burmeister, P. et al., editors) Research and Exposition in Mathematics, vol. 4, Heldermann Verlag, Berlin, pp. 4166.
Gödel, K. [1933] Eine Interpretation des intuitionistischen Aussagenkalküls, Ergebnisse eines Mathematischen Kolloquiums, vol. 4, pp. 3940.
Herbrand, J. [1928] Sur la théorie de la démonstration, Comptes Rendus Hebdomadaires des Séances de I’Académie des Sciences, vol. 186, pp. 12741276.
Heyting, A. [1930] Die formalen Regeln der intuitionistischen Logik, Sitzungsberichte der Preussischen Akademie der wissenschaften, Physikalisch-Mathematische Klasse, pp. 4256.
Hilbert, D. and Bernays, P. [1934] Grundlagen der Mathematik, vol. I, Springer-Verlag, Berlin.
JaŚkowski, S. [1936] Recherches sur le système de la logique intuitioniste, Actes du congrès international de philosophic scientifique, VI: Philosophie des mathématiques, Actualités Scientiflques et Industrielles, vol. 393, Hermann, Paris, pp. 5861.
Kalicki, J. [1955] The number of equationally complete families of equations, Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings, Series A: Mathematical Sciences, vol. 58 = Indagationes Mathematicae, vol. 17, pp. 660662.
Kalicki, J. and Scott, D. [1955] Equational completeness of abstract algebras, Koninklijke Nederlandse Akademie van Wetenschappen. Proceedings, Series A: Mathematical Sciences, vol. 58 = Indagationes Mathematicae, vol. 17, pp. 650659.
Kalish, D. and Montague, R. [1965] On Tarski's formalization of predicate logic with identity, Archiv für Mathematische Logik und Grundlagenforschung, vol. 7, pp. 81101.
Langford, C. H. [1927] Some theorems on deductibility, Annals of Mathematics, ser. 2, vol. 28, pp. 1640.
Lewis, C. I. [1918] A survey of symbolic logic, University of California Press, Berkeley, California.
Łukasiewicz, J. [1920] I: On the notion of possibility; II: On three-valued logic, Ruch Filozoficzny, vol. 5, pp. 169171. (Polish).
Łukasiewicz, J. [1930] Philosophische Bemerkungen zu mehrwertigen Systemen des Aussagenskalküls, Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, Classe III, vol. 23, pp. 5177.
Lyndon, R. C. [1951] Identities in two-valued calculi, Transactions of the American Mathematical Society, vol. 71, pp. 457465.
Lyndon, R. C. [1954] Identities in finite algebras, Proceedings of the American Mathematical Society, vol. 5, pp. 89.
McKenzie, R. [1970] Equational bases for lattice theories, Mathematica Scandinavica, vol. 27, pp. 2438.
Mckinsey, J. C. C. [1941] A solution of the decision problem for the Lewis systems S2 and S4, with an application to topology, this Journal, vol. 6, pp. 117134.
McNulty, G. F. [1986] Alfred Tarski and undecidable theories, this Journal, vol. 51, pp. 890898.
McNulty, G. F. and Shallon, C. [1983] Inherently nonfinitely based finite algebras, Universal algebra and lattice theory, Lecture Notes in Mathematics, vol. 1004, Springer-Verlag, Berlin, pp. 206231.
McNulty, G. F. and Taylor, W. [1975] Combinatorial interpolation theorems, Discrete Mathematics, vol. 12, pp. 193200.
Monk, J. D. [1965] Substitutionless predicate logic with identity, Archiv für Mathematische Logik und Grundlagenforschung, vol. 7, pp. 102121.
Monk, J. D. [1986] The contributions of Alfred Tarski to algebraic logic, this Journal, vol. 51, pp. 899906.
Mostowski, A. [1937] Abzählbare Boolesche Körper und ihre Anwendung auf die allgemeine Metamathematik, Fundamenta Mathematicae, vol. 29, pp. 3453.
Padmanabhan, R. and Quackenbush, R. W. [1973] Equational theories of algebras with distributive congruences, Proceedings of the American Mathematical Society, vol. 41, pp. 373377.
Peirce, C. S. [1885] On the algebra of logic: a contribution to the philosophy of notation, American Journal of Mathematics, vol. 7, pp. 180202.
Post, E. L. [1921] Introduction to a general theory of elementary propositions, American Journal of Mathematics, vol. 43, pp. 163185.
Rasiowa, H. [1974] An algebraic approach to nonclassical logics, North-Holland, Amsterdam.
Rasiowa, H. and Sikorski, R. [1963] The mathematics of metamathematics, PWN, Warsaw.
Resnikoff, I. [1965] Tout ensemble de formules de la logique est équivalent à un ensemble indépendant, Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences, vol. 260, pp. 23852388.
Scott, D. [1956] Equationally complete extensions of finite algebras, Koninklijke Nederlandse Akademie van Wetenschappen. Proceedings, Series A: Mathematical Sciences, vol. 59 = Indagationes Mathematicae, vol. 18, pp. 3538.
Skolem, T. [1919] Untersuchungen über die Axiome des Klassenkalküls und über Produktations- und Summationsprobleme, welche gewisse Klassen von Aussagen betreffen, Skrifter utgit av Videnskapsselskapet i Kristiania, I: Matematisk-Naturtidenskabelig Klasse, no. 3.
Stone, M. H. [1936] The theory of representations for Boolean algebras, Transactions of the American Mathematical Society, vol. 40, pp. 37111.
Stone, M. H. [1937a] Applications of the theory of Boolean rings to general topology, Transactions of the American Mathematical Society, vol. 41, pp. 375481.
Stone, M. H. [1937b] Algebraic characterizations of special Boolean rings, Fundamenta Mathematicae, vol. 29, pp. 223305.
Stone, M. H. [1937/1938] Topological representations of distributive lattices and Brouwerian logics, Časopis pro Pěstováni Matematiky a Fysiky, vol. 67, pp. 125.
Taylor, W. [1979] Equational logic, Houston Journal of Mathematics Survey.
Vaught, R. L. [1986] Alfred Tarski's work in model theory, this Journal, vol. 51, pp. 869882.
Wójcicki, R. [1969] Logical matrices strongly adequate for structural sentential calculi, Bulletin de l’Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 6, pp. 333335.
Wójcicki, R. [1973] Matrix approach in methodology of sentential calculi, Studia Logica, vol. 32, pp. 737.
Wójcicki, R. [198-] Theory of logical calculi. An introduction (to appear).
Wronski, A. [198-] Some problems and results concerning BCK-algebras (to appear).


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