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Alfred Tarski's work on general metamathematics

Published online by Cambridge University Press:  12 March 2014

W. J. Blok  and
Don Pigozzi
W. J. Blok
Affiliation:
Department of Mathematics, University of Illinois at Chicago, Chicago, Illinois 60607
Don Pigozzi
Affiliation:
Department of Mathematics, Iowa State University, AMES, Iowa 50011
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Extract

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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 53 , Issue 1 , March 1988 , pp. 36 - 50
Copyright
Copyright © Association for Symbolic Logic 1988

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References

REFERENCES

Andréka, H., Németi, I., and Sain, I. [198-] Abstract model approach to algebraic logic (to appear).Google Scholar
Birkhoff, G. [1935] On the structure of abstract algebras, Proceedings of the Cambridge Philosophical Society, vol. 31, pp. 433–454.CrossRefGoogle Scholar
Blok, W. J. and Pigozzi, D. [198-] A characterization of algebraizable logics, Memoirs of the American Mathematical Society (to appear).Google Scholar
Blok, W. J. and Pigozzi, D. [1956] Introduction to mathematical logic, vol. I, Princeton University Press, Princeton, New Jersey.Google Scholar
Curry, H. B. and Feys, R. [1958] Combinatory logic, vol. I, North-Holland, Amsterdam.Google Scholar
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. 41–66.Google Scholar
Gödel, K. [1933] Eine Interpretation des intuitionistischen Aussagenkalküls, Ergebnisse eines Mathematischen Kolloquiums, vol. 4, pp. 39–40.Google Scholar
Herbrand, J. [1928] Sur la théorie de la démonstration, Comptes Rendus Hebdomadaires des Séances de I’Académie des Sciences, vol. 186, pp. 1274–1276.Google Scholar
Heyting, A. [1930] Die formalen Regeln der intuitionistischen Logik, Sitzungsberichte der Preussischen Akademie der wissenschaften, Physikalisch-Mathematische Klasse, pp. 42–56.Google Scholar
Hilbert, D. and Bernays, P. [1934] Grundlagen der Mathematik, vol. I, Springer-Verlag, Berlin.Google Scholar
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. 58–61.Google Scholar
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. 660–662.Google Scholar
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. 650–659.Google Scholar
Kalish, D. and Montague, R. [1965] On Tarski's formalization of predicate logic with identity, Archiv für Mathematische Logik und Grundlagenforschung, vol. 7, pp. 81–101.CrossRefGoogle Scholar
Langford, C. H. [1927] Some theorems on deductibility, Annals of Mathematics, ser. 2, vol. 28, pp. 16–40.CrossRefGoogle Scholar
Lewis, C. I. [1918] A survey of symbolic logic, University of California Press, Berkeley, California.Google Scholar
Łukasiewicz, J. [1920] I: On the notion of possibility; II: On three-valued logic, Ruch Filozoficzny, vol. 5, pp. 169–171. (Polish).Google Scholar
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. 51–77.Google Scholar
Lyndon, R. C. [1951] Identities in two-valued calculi, Transactions of the American Mathematical Society, vol. 71, pp. 457–465.CrossRefGoogle Scholar
Lyndon, R. C. [1954] Identities in finite algebras, Proceedings of the American Mathematical Society, vol. 5, pp. 8–9.CrossRefGoogle Scholar
McKenzie, R. [1970] Equational bases for lattice theories, Mathematica Scandinavica, vol. 27, pp. 24–38.CrossRefGoogle Scholar
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. 117–134.Google Scholar
McNulty, G. F. [1986] Alfred Tarski and undecidable theories, this Journal, vol. 51, pp. 890–898.Google Scholar
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. 206–231.CrossRefGoogle Scholar
McNulty, G. F. and Taylor, W. [1975] Combinatorial interpolation theorems, Discrete Mathematics, vol. 12, pp. 193–200.Google Scholar
Monk, J. D. [1965] Substitutionless predicate logic with identity, Archiv für Mathematische Logik und Grundlagenforschung, vol. 7, pp. 102–121.CrossRefGoogle Scholar
Monk, J. D. [1986] The contributions of Alfred Tarski to algebraic logic, this Journal, vol. 51, pp. 899–906.Google Scholar
Mostowski, A. [1937] Abzählbare Boolesche Körper und ihre Anwendung auf die allgemeine Metamathematik, Fundamenta Mathematicae, vol. 29, pp. 34–53.CrossRefGoogle Scholar
Padmanabhan, R. and Quackenbush, R. W. [1973] Equational theories of algebras with distributive congruences, Proceedings of the American Mathematical Society, vol. 41, pp. 373–377.CrossRefGoogle Scholar
Peirce, C. S. [1885] On the algebra of logic: a contribution to the philosophy of notation, American Journal of Mathematics, vol. 7, pp. 180–202.CrossRefGoogle Scholar
Post, E. L. [1921] Introduction to a general theory of elementary propositions, American Journal of Mathematics, vol. 43, pp. 163–185.CrossRefGoogle Scholar
Rasiowa, H. [1974] An algebraic approach to nonclassical logics, North-Holland, Amsterdam.Google Scholar
Rasiowa, H. and Sikorski, R. [1963] The mathematics of metamathematics, PWN, Warsaw.Google Scholar
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. 2385–2388.Google Scholar
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. 35–38.Google Scholar
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.Google Scholar
Stone, M. H. [1936] The theory of representations for Boolean algebras, Transactions of the American Mathematical Society, vol. 40, pp. 37–111.Google Scholar
Stone, M. H. [1937a] Applications of the theory of Boolean rings to general topology, Transactions of the American Mathematical Society, vol. 41, pp. 375–481.CrossRefGoogle Scholar
Stone, M. H. [1937b] Algebraic characterizations of special Boolean rings, Fundamenta Mathematicae, vol. 29, pp. 223–305.CrossRefGoogle Scholar
Stone, M. H. [1937/1938] Topological representations of distributive lattices and Brouwerian logics, Časopis pro Pěstováni Matematiky a Fysiky, vol. 67, pp. 1–25.Google Scholar
Taylor, W. [1979] Equational logic, Houston Journal of Mathematics Survey.Google Scholar
Vaught, R. L. [1986] Alfred Tarski's work in model theory, this Journal, vol. 51, pp. 869–882.Google Scholar
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. 333–335.Google Scholar
Wójcicki, R. [1973] Matrix approach in methodology of sentential calculi, Studia Logica, vol. 32, pp. 7–37.CrossRefGoogle Scholar
Wójcicki, R. [198-] Theory of logical calculi. An introduction (to appear).Google Scholar
Wronski, A. [198-] Some problems and results concerning BCK-algebras (to appear).Google Scholar