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Correspondences between gentzen and hilbert systems

Published online by Cambridge University Press:  12 March 2014

J.G. Raftery
J.G. Raftery*
Affiliation:
School of Mathematical Sciences, University of Kwazulu-Natal, Durban 4001, South Africa.E-mail:raftery@ukzn.ac.za
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Extract

Most Gentzen systems arising in logic contain few axiom schemata and many rule schemata. Hilbert systems, on the other hand, usually contain few proper inference rules and possibly many axioms. Because of this, the two notions tend to serve different purposes. It is common for a logic to be specified in the first instance by means of a Gentzen calculus, whereupon a Hilbert-style presentation ‘for’ the logic may be sought—or vice versa. Where this has occurred, the word ‘for’ has taken on several different meanings, partly because the Gentzen separator ⇒ can be interpreted intuitively in a number of ways. Here ⇒ will be denoted less evocatively by ⊲.

In this paper we aim to discuss some of the useful ways in which Gentzen and Hilbert systems may correspond to each other. Actually, we shall be concerned with the deducibility relations of the formal systems, as it is these that are susceptible to transformation in useful ways. To avoid potential confusion, we shall speak of Hilbert and Gentzen relations. By a Hilbert relation we mean any substitution-invariant consequence relation on formulas—this comes to the same thing as the deducibility relation of a set of Hilbert-style axioms and rules. By a Gentzen relation we mean the fully fledged generalization of this notion in which sequents take the place of single formulas. In the literature, Hilbert relations are often referred to as sentential logics. Gentzen relations as defined here are their exact sequential counterparts.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 71 , Issue 3 , September 2006 , pp. 903 - 957
Copyright
Copyright © Association for Symbolic Logic 2006

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References

REFERENCES

[1]Adillon, R. J. and Verdú, V., On a contraction-less intuitionistic propositional logic with conjunction and fusion, Studia Logica, vol. 65 (2000), pp. 11–30.CrossRefGoogle Scholar
[2]Ardeshir, M. and Ruitenberg, W., Basic propositional calculus I, Mathematical Logic Quarterly, vol. 44 (1998), pp. 317–343.CrossRefGoogle Scholar
[3]Avron, A., The semantics and proof theory of linear logic, Theoretical Computer Science, vol. 57 (1988), pp. 161–184.CrossRefGoogle Scholar
[4]Babyonyshev, S. V., Metatheories of deductive systems, Doctoral Dissertation, Iowa State University, 2005.Google Scholar
[5]Blok, W. J. and Hoogland, E., The Beth property in algebraic logic, Studia Logica, vol. 83 (2006), in press.CrossRefGoogle Scholar
[6]Blok, W. J. and Jónsson, B., Algebraic structures for logic, a course given at the 23rd Holiday Mathematics Symposium, New Mexico State University, January 1999. Available at http://math.nmsu.edu/holysymp/.Google Scholar
[7]Blok, W. J. and Jónsson, B., Equivalence of consequence operations, Studia Logica, vol. 83 (2006), in press.CrossRefGoogle Scholar
[8]Blok, W. J., Köhler, R, and Pigozzi, D., The algebraization of logic, manuscript, 1983.Google Scholar
[9]Blok, W. J. and Pigozzi, D., Protoalgebraic logics, Studia Logica, vol. 45 (1986), pp. 337–369.CrossRefGoogle Scholar
[10]Blok, W. J. and Pigozzi, D., Algebraizable logics, Memoirs of the American Mathematical Society, (1989), no. 396.Google Scholar
[11]Blok, W. J. and Pigozzi, D., Local deduction theorems in algebraic logic, Algebraic logic (Proc. Conf., Budapest, 8–14 August 1988), Colloquia Mathematica Societatis JÂnos Bolyai (Andréka, H., Monk, J. D., and Nemeti, I., editors), vol. 54, North-Holland, Amsterdam, 1991, pp. 75–109.Google Scholar
[12]Blok, W. J. and Pigozzi, D., Algebraic semantics for universal Horn logic without equality, Universal algebra and quasigroup theory (Smith, J. D. H. and Romanowska, A., editors), Research and Exposition in Mathematics, vol. 19, Heldermann Verlag, Berlin, 1992, pp. 1–56.Google Scholar
[13]Blok, W. J. and Pigozzi, D., Abstract algebraic logic and the deduction theorem, manuscript, 1997, [See http://orion.math.iastate.edu/dpigozzi/ for updated version, 2001.].Google Scholar
[14]Blok, W. J. and Raftery, J. G., Assertionally equivalent quasivarieties, manuscript.Google Scholar
[15]Blok, W. J. and Raftery, J. G., Ideals in quasivarieties of algebras, Models, algebras and proofs (Caicedo, X. and Montenegro, C. H., editors), Lecture Notes in Pure and Applied Mathematics, no. 203, Marcel Dekker, New York, 1999, pp. 167–186.Google Scholar
[16]Blok, W. J. and Raftery, J. G., On congruence modularity in varieties of logic, Algebra Universalis, vol. 45 (2001), pp. 15–21.CrossRefGoogle Scholar
[17]Blok, W. J. and Rebagliato, J., Algebraic semantics for deductive systems, Studio Logica, vol. 74 (2003), pp. 153–180.CrossRefGoogle Scholar
[18]Bloom, S. L., Some theorems on structural consequence relations, Studia Logica, vol. 34 (1975), pp. 1–9.CrossRefGoogle Scholar
[19]Bou, F., García-Cerdaña, À., and Verdú, V., On two fragments with negation and without implication of the logic of residuated lattices, Archive for Mathematical Logic, to appear.Google Scholar
[20]Celani, S. and Jansana, R., A closer look at some subintuitionistic logics, Notre Dame Journal of Formal Logic, vol. 42 (2001), no. 4, pp. 225–255.CrossRefGoogle Scholar
[21]Czelakowski, J., Protoalgebraic logics, Kluwer, Dordrecht, 2001.CrossRefGoogle Scholar
[22]Czelakowski, J., The Suszko operator. Part I, Studia Logica, vol. 74 (2003), pp. 181–231.CrossRefGoogle Scholar
[23]Czelakowski, J. and Jansana, R., Weakly algebraizable logics, this Journal, vol. 65 (2000), pp. 641–668.Google Scholar
[24]Czelakowski, J. and Pigozzi, D., Amalgamation and interpolation in abstract algebraic logic, Models, algebras and proofs (Caicedo, X. and Montenegro, C. H., editors), Lecture Notes in Pure and Applied Mathematics, no. 203, Marcel Dekker, New York, 1999, pp. 187–265.Google Scholar
[25]Dellunde, P., A finitary 1 –equivalential logic not finitely equivalential, Bulletin of the Section of Logic, vol. 24 (1995), no. 3, pp. 120–122.Google Scholar
[26]Dellunde, P., Contributions to the model theory of equality-free logic, Doctoral Dissertation, University of Barcelona, 1996.Google Scholar
[27]Dellunde, P. and Jansana, R., Some characterization theorems for infinitary universal Horn logic without equality, this Journal, vol. 61 (1996), pp. 1242–1260.Google Scholar
[28]Dunn, J. M., Partial gaggles applied to logics with restricted structural rules, Substructural logics (Schroeder-Heister, P. and Došen, K., editors), Clarendon Press, Oxford, 1993, pp. 63–108.CrossRefGoogle Scholar
[29]Elgueta, R., Algebraic model theory for languages without equality. Doctoral Dissertation, University of Barcelona, 1994.Google Scholar
[30]Font, J. M. and Jansana, R., A general algebraic semantics for sentential logics, Lecture Notes in Logic, vol. 7, Springer-Verlag, 1996.CrossRefGoogle Scholar
[31]Font, J. M., Jansana, R., and Pigozzi, D., Fully adequate Gentzen systems and the deduction theorem, Reports on Mathematical Logic, vol. 35 (2001), pp. 115–165.Google Scholar
[32]Font, J. M., Jansana, R., and Pigozzi, D., A survey of abstract algebraic logic, Studia Logica, vol. 74 (2003), pp. 13–97.CrossRefGoogle Scholar
[33]Font, J. M., Jansana, R., and Pigozzi, D., On the closure properties of the class of full g-models of a deductive system, Studia Logica, vol. 83 (2006), in press.CrossRefGoogle Scholar
[34]Font, J. M. and Verdú, V., Algebraic logic for classical conjunction and disjunction, Studia Logica, vol. 50 (1991), pp. 391–419.CrossRefGoogle Scholar
[35]Galatos, N. and Ono, H., Cut elimination and strong separation for substructural logics: an algebraic approach, manuscript.Google Scholar
[36]Galatos, N. and Ono, H., Algebraization, parametrized local deduction theorem and interpolation for substructural logics over FL, Studia Logica, vol. 83 (2006), in press.CrossRefGoogle Scholar
[37]Galatos, N. and Tsinakis, C., Equivalence of consequence relations: An order-theoretic and categorical perspective, in preparation.Google Scholar
[38]Gil, A. J. and Rebagliato, J., Protoalgebraic Gentzen systems and the cut rule, Studia Logica, vol. 65 (2000), pp. 53–89.CrossRefGoogle Scholar
[39]Gil-Férez, J., Categorical abstract algebraic logic: the isomorphism theorem (abstract), Algebraic and Topological Methods in Non-Classical Logics II, Barcelona, 15–18 June, 2005.Google Scholar
[40]Gyuris, V., Variations of algebraizability, Doctoral Dissertation, University of Illinois at Chicago, 1999.Google Scholar
[41]Hart, J., Rafter, L., and Tsinakis, C., The structure of commutative residuated lattices, International Journal of Algebra and Computation, vol. 12 (2002), pp. 509–524.CrossRefGoogle Scholar
[42]Herrmann, B., Equivalential and algebraizable logics, Studia Logica, vol. 57 (1996), pp. 419–436.CrossRefGoogle Scholar
[43]Herrmann, B., Characterizing equivalential and algebraizable logics by the Leibniz operator, Studia Logica, vol. 58 (1997), pp. 305–323.CrossRefGoogle Scholar
[44]Kearnes, K. A. and Szendrei, A., The relationship between two commutators, International Journal of Algebra and Computation, vol. 8 (1998), pp. 497–531.CrossRefGoogle Scholar
[45]Lewin, R., Mikenberg, I. F., and Schwarze, M. G., On the algebraizability of annotated logics, Studio Logica, vol. 57 (1997), pp. 359–386.CrossRefGoogle Scholar
[46]Łoś, J. and Suszko, R., Remarks on sentential logics, Proceedings of Koninklijke Nederlandse Akademie van Wetenschappen, Series A, vol. 61 (1958), pp. 177–183.Google Scholar
[47]Ono, H., Proof-theoretic methods in nonclassical logic—an introduction, Theories of types and proofs, MSJ memoirs 2 (Takahashi, M., Okada, M., and Dezani-Ciancaglini, M., editors), Mathematical Society of Japan, 1998, pp. 207–254.CrossRefGoogle Scholar
[48]Ono, H., Substructural logics and residuated lattices—an introduction, 50 years of Studia Logica (Hendricks, V. F. and Malinowski, J., editors), Trends in Logic, vol. 20, Kluwer, 2003, pp. 177–212.Google Scholar
[49]Pałasińska, K., Deductive systems and finite axiomatization properties, Doctoral Dissertation, Iowa State University, 1994.Google Scholar
[50]Pałasińska, K., Finite basis theorem for filter-distributive protoalgebraic deductive systems and strict universal Horn classes, Studia Logica, vol. 74 (2003), pp. 233–273.CrossRefGoogle Scholar
[51]Pałasińska, K. and Pigozzi, D., Implication in abstract algebraic logic, manuscript, 1995.Google Scholar
[52]Pigozzi, D., A lattice-theoretic characterization of equivalent quasivarieties, Proceedings of the International Conference on Algebra, Part 3 (Novosibirsk, 1989), American Mathematical Society, Providence, RI, 1992, pp. 187–199.Google Scholar
[53]Pigozzi, D., Abstract algebraic logic, Encyclopaedia of Mathematics, supplement III(Hazewinkel, M., editor), Kluwer, Dordrecht, 2001, pp. 2–13.Google Scholar
[54]Pigozzi, D., Partially ordered varieties and quasivarieties, manuscript, 2004.Google Scholar
[55]Pynko, A. P., Definitional equivalence and algebraizability of generalized logical systems, Annals of Pure and Applied Logic, vol. 98 (1999), pp. 1–68.CrossRefGoogle Scholar
[56]Raftery, J. G., The equational definability of truth predicates, Special issue of Reports on Mathematical Logic in memory of W. J. Blok, (to appear).Google Scholar
[57]Rebagliato, J. and Verdú, V., Algebraizable Gentzen systems and the deduction theorem for Gentzen systems, Mathematics Preprint Series No. 175 (June 1995), University of Barcelona.Google Scholar
[58]Rebagliato, J. and Verdú, V., On the algebraization of some Gentzen systems, Fundamenta Informaticae, vol. 18 (1993), pp. 319–338.CrossRefGoogle Scholar
[59]Rybakov, V. V., Admissibility of logical inference rules, Studies in Logic and the Foundations of Mathematics, vol. 136, Elsevier, 1997.CrossRefGoogle Scholar
[60]Suzuki, Y., Wolter, F., and Zakharyaschev, M., Speaking about transitive frames in propositional languages, Journal of Logic, Language and Information, vol. 7 (1998), pp. 317–339.CrossRefGoogle Scholar
[61]Torrens, A., Multi-dimensional deductive systems, manuscript, 1991.Google Scholar
[62]Troelstra, A. S., Lectures on linear logic, Csli Lecture Notes, no. 29, 1992.Google Scholar
[63]Alten, C. J. Van and Raftery, J. G., Rule separation and embedding theorems for logics without weakening, Studia Logica, vol. 76 (2004), pp. 241–274.CrossRefGoogle Scholar
[64]Visser, A., A propositional logic with explicit fixed points, Studia Logica, vol. 40 (1981), pp. 155–175.CrossRefGoogle Scholar
[65]Voutsadakis, G., Categorical abstract algebraic logic: equivalent institutions, Studia Logica, vol. 74 (2003), pp. 275–311.CrossRefGoogle Scholar
[66]Wójcicki, R., Theory of logical calculi, Kluwer, Dordrecht, 1988.CrossRefGoogle Scholar