# On the Leibniz congruences

Banach Center Publications (1993)

- Volume: 28, Issue: 1, page 17-36
- ISSN: 0137-6934

## Access Full Article

top## Abstract

top## How to cite

topFont, Josep. "On the Leibniz congruences." Banach Center Publications 28.1 (1993): 17-36. <http://eudml.org/doc/262564>.

@article{Font1993,

abstract = {The aim of this paper is to discuss the motivation for a new general algebraic semantics for deductive systems, to introduce it, and to present an outline of its main features. Some tools from the theory of abstract logics are also introduced, and two classifications of deductive systems are analysed: one is based on the behaviour of the Leibniz congruence (the maximum congruence of a logical matrix) and the other on the behaviour of the Frege operator (which associates to every theory the interderivability relation modulo the theory). For protoalgebraic deductive systems the class of algebras associated in general turns out to be the class of algebra reducts of reduced matrices, which is the algebraic counterpart usually considered for this large class of deductive systems; but in the general case the new class of algebras shows a better behaviour.},

author = {Font, Josep},

journal = {Banach Center Publications},

keywords = {deductive system; protoalgebraic logic; Gentzen calculus; closure operator; abstract logic; algebraizable logic; Leibniz congruence; selfextensional logic; logical matrices; algebraic logic; self extensional logic; algebraic semantics; deductive systems; abstract logics; maximum congruence of a logical matrix; Frege operator; interderivability relation; protoalgebraic deductive systems; algebra reducts of reduced matrices},

language = {eng},

number = {1},

pages = {17-36},

title = {On the Leibniz congruences},

url = {http://eudml.org/doc/262564},

volume = {28},

year = {1993},

}

TY - JOUR

AU - Font, Josep

TI - On the Leibniz congruences

JO - Banach Center Publications

PY - 1993

VL - 28

IS - 1

SP - 17

EP - 36

AB - The aim of this paper is to discuss the motivation for a new general algebraic semantics for deductive systems, to introduce it, and to present an outline of its main features. Some tools from the theory of abstract logics are also introduced, and two classifications of deductive systems are analysed: one is based on the behaviour of the Leibniz congruence (the maximum congruence of a logical matrix) and the other on the behaviour of the Frege operator (which associates to every theory the interderivability relation modulo the theory). For protoalgebraic deductive systems the class of algebras associated in general turns out to be the class of algebra reducts of reduced matrices, which is the algebraic counterpart usually considered for this large class of deductive systems; but in the general case the new class of algebras shows a better behaviour.

LA - eng

KW - deductive system; protoalgebraic logic; Gentzen calculus; closure operator; abstract logic; algebraizable logic; Leibniz congruence; selfextensional logic; logical matrices; algebraic logic; self extensional logic; algebraic semantics; deductive systems; abstract logics; maximum congruence of a logical matrix; Frege operator; interderivability relation; protoalgebraic deductive systems; algebra reducts of reduced matrices

UR - http://eudml.org/doc/262564

ER -

## References

top- [1] N. D. Belnap, Jr., A useful four-valued logic, in: Modern Uses of Multiple-Valued Logic, J. M. Dunn and G. Epstein (eds.), Reidel, Dordrecht 1977, 8-37.
- [2] W. J. Blok and D. Pigozzi, Protoalgebraic logics, Studia Logica 45 (1986), 337-369. Zbl0622.03020
- [3] W. J. Blok and D. Pigozzi, Alfred Tarski's work on general metamathematics, J. Symbolic Logic 53 (1988), 36-50. Zbl0651.03002
- [4] W. J. Blok and D. Pigozzi, Algebraizable logics, Mem. Amer. Math. Soc. 396 (1989). Zbl0664.03042
- [5] W. J. Blok and D. Pigozzi, Local deduction theorems in algebraic logic, in: Algebraic Logic, H. Andréka, J. D. Monk and I. Németi (eds.), Colloq. Math. Soc. János Bolyai 54, North-Holland, Amsterdam 1991, 75-109. Zbl0751.03036
- [6] W. J. Blok and D. Pigozzi, The deduction theorem in algebraic logic, preprint, 1991, to appear. Zbl0755.03034
- [7] W. J. Blok and D. Pigozzi, Algebraic semantics for universal Horn logic without equality, in: Universal Algebra and Quasigroups, A. Romanowska and J. D. H. Smith (eds.), Heldermann, Berlin 1992, to appear. Zbl0768.03008
- [8] S. L. Bloom, A note on Ψ-consequences, Rep. Math. Logic 8 (1977), 3-9.
- [9] S. L. Bloom and D. J. Brown, Classical abstract logics, Dissertationes Math. 102 (1973), 43-51.
- [10] D. J. Brown and R. Suszko, Abstract logics, ibid., 9-42.
- [11] S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Springer, New York 1981.
- [12] J. Czelakowski, Equivalential logics, I, II, Studia Logica 40 (1981), 227-236 and 355-372. Zbl0476.03032
- [13] J. Czelakowski and W. Dziobiak, A deduction theorem schema for deductive systems of propositional logics, Studia Logica, Special Issue on Algebraic Logic, 50 (1991), 385-390. Zbl0755.03014
- [14] J. Czelakowski and G. Malinowski, Key notions of Tarski's methodology of deductive systems, Studia Logica 44 (1985), 321-351. Zbl0615.03014
- [15] B. A. Davey and H. A. Priestley, Introduction to Lattices and Order, Cambridge Univ. Press, Cambridge 1990. Zbl0701.06001
- [16] K. Dyrda and T. Prucnal, On finitely based consequence determined by a distributive lattice, Bull. Sec. Logic Polish Acad. Sci. 9 (1980), 60-66. Zbl0435.03006
- [17] J. M. Font, On some congruence lattices of a topological Heyting lattice, in: Contributions to General Algebra 5, J. Czermak et al. (eds.), Teubner, Stuttgart 1987, 129-137.
- [18] J. M. Font and J. L. García Lapresta, Logics and Algebras motivated by cardinality restrictions in the Deduction Theorem, manuscript, 1992.
- [19] J. M. Font, F. Guzmán and V. Verdú, Characterization of the reduced matrices for the {∧,∨}-fragment of classical logic, Bull. Sec. Logic Polish Acad. Sci. 20 (1991), 124-128.
- [20] J. M. Font and R. Jansana, A general algebraic semantics for deductive systems, preprint, 1992, to appear.
- [21] J. M. Font and M. Rius, A four-valued modal logic arising from Monteiro's last algebras, in: Proc. 20th Internat. Sympos. on Multiple-Valued Logic, Charlotte 1990, 85-92.
- [22] J. M. Font and G. Rodríguez, Note on algebraic models for relevance logic, Z. Math. Logik Grundlag. Math. 36 (1990), 535-540. Zbl0696.03004
- [23] J. M. Font and G. Rodríguez, Algebraic study of system R of relevance logic, manuscript, 1992.
- [24] J. M. Font and V. Verdú, Abstract characterization of a four-valued logic, in: Proc. 18th Internat. Sympos. on Multiple-Valued Logic, Palma de Mallorca 1988, 389-396.
- [25] J. M. Font and V. Verdú, A first approach to abstract modal logics, J. Symbolic Logic 54 (1989), 1042-1062. Zbl0687.03008
- [26] J. M. Font and V. Verdú, Completeness theorems for a four-valued logic related to De Morgan lattices, Fac. Math. Preprint Ser. 57, Barcelona 1989.
- [27] J. M. Font and V. Verdú, Algebraic logic for classical conjunction and disjunction, Studia Logica, Special Issue on Algebraic Logic, 50 (1991), 391-419. Zbl0753.03027
- [28] J. M. Font and V. Verdú, The lattice of distributive closure operators over an algebra, Studia Logica, to appear. Zbl0773.03039
- [29] J. M. Font and V. Verdú, Algebraic study of Belnap's four-valued logic, manuscript.
- [30] J. L. García Lapresta, Finitely deductive logics, Ph.D. dissertation, Univ. of Barcelona, 1991 (in Spanish).
- [31] A. Grzegorczyk, An approach to logical calculi, Studia Logica 30 (1972), 33-43 Zbl0286.02029
- [32] F. Guzmán and V. Verdú, On two Gentzen logifications of the variety of semilattices, manuscript, 1992.
- [33] R. Jansana, The Box fragments of modal logic K, in: Actas del VII Congreso de Lenguajes Naturales y Lenguajes Formales, Vic, Barcelona 1991, C. Martín-Vide (ed.), 409-413 (in Spanish).
- [34] R. Jansana, Abstract modal logics, preprint, 1992, to appear.
- [35] J. Łoś and R. Suszko, Remarks on sentential logics, Indag. Math. 20 (1958), 177-183. Zbl0092.24802
- [36] D. Pigozzi, Fregean algebraic logic, in: Algebraic Logic, H. Andréka, J. D. Monk and I. Németi (eds.), Colloq. Math. Soc. János Bolyai 54, North-Holland, Amsterdam 1991, 473-502. Zbl0749.03055
- [37] J. Pla and V. Verdú, Quasi-Hilbert algebras, Publ. Mat. 20 (1980), 97-99 (in Catalan).
- [38] W. A. Pogorzelski and J. Słupecki, Basic properties of deductive systems based on nonclassical logics, I, II, Studia Logica 9 (1960), 163-176 and 10 (1960), 77-95. Zbl0129.25602
- [39] M. Porębska and A. Wroński, A characterization of fragments of the intuitionistic propositional logic, Rep. Math. Logic 4 (1975), 39-42. Zbl0318.02027
- [40] H. Rasiowa, An Algebraic Approach to Non-Classical Logics, North-Holland, Amsterdam 1974. Zbl0299.02069
- [41] W. Rautenberg, Axiomatizing logics closely related to varieties, Studia Logica, Special Issue on Algebraic Logic, 50 (1991), 607-620. Zbl0749.03020
- [42] W. Rautenberg, On reduced matrices, preprint, 1992, to appear.
- [43] J. Rebagliato and V. Verdú, On the algebraization of some Gentzen systems, Fund. Inform., Special Issue on Algebra and Logic in Computer Science (1992), to appear. Zbl0788.03006
- [44] J. Rebagliato and V. Verdú, A Hilbert-style axiomatization of the {∧,∨,¬}-fragment of IPC, manuscript, 1992. Zbl0806.03010
- [45] M. Rius, Tetravalent modal logics, Ph.D. dissertation, Univ. of Barcelona, 1992 (in Catalan).
- [46] A. J. Rodríguez, A. Torrens and V. Verdú, Łukasiewicz logic and Wajsberg algebras, Bull. Sec. Logic Polish Acad. Sci. 19 (1990), 51-55. Zbl0717.03027
- [47] A. Tarski, Über einige fundamentale Begriffe der Metamathematik, C. R. Soc. Sci. Lettres Varsovie Cl. III 23 (1930), 22-29.
- [48] A. Torrens, Model theory for sequential deductive systems, preprint, 1991.
- [49] A. Torrens and V. Verdú, Abstract Łukasiewicz logics, preprint, 1990.
- [50] V. Verdú, Contribution to the study of some classes of abstract logics, Ph.D. dissertation, Univ. of Barcelona, 1978 (in Catalan).
- [51] V. Verdú, Distributive and Boolean logics, Stochastica 3 (1979), 97-108 (in Catalan). Zbl0419.03041
- [52] V. Verdú, Some algebraic structures determined by closure operators, Z. Math. Logik Grundlag. Math. 31 (1985), 275-278. Zbl0553.03041
- [53] V. Verdú, On some relations between closure operators and congruences, preprint, 1986.
- [54] V. Verdú, Logics projectively generated from [M] = (F₄,[{1}]) by a set of homomorphisms, Z. Math. Logik Grundlag. Math. 33 (1987), 235-241. Zbl0607.03020
- [55] R. Wójcicki, Lectures on Propositional Calculi, Ossolineum, Wrocław 1984. Zbl0647.03019
- [56] R. Wójcicki, Theory of Logical Calculi. Basic Theory of Consequence Operations, Synthese Library 199, Reidel, Dordrecht 1988. Zbl0682.03001

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.