Studia Logica

, Volume 78, Issue 1–2, pp 171–212 | Cite as

Fregean logics with the multiterm deduction theorem and their algebraization

  • J. Czelakowski
  • D. Pigozzi


A deductive system \(\mathcal{S}\) (in the sense of Tarski) is Fregean if the relation of interderivability, relative to any given theory T, i.e., the binary relation between formulas
$$\{ \left\langle {\alpha ,\beta } \right\rangle :T,\alpha \vdash s \beta and T,\beta \vdash s \alpha \} ,$$
is a congruence relation on the formula algebra. The multiterm deduction-detachment theorem is a natural generalization of the deduction theorem of the classical and intuitionistic propositional calculi (IPC) in which a finite system of possibly compound formulas collectively plays the role of the implication connective of IPC. We investigate the deductive structure of Fregean deductive systems with the multiterm deduction-detachment theorem within the framework of abstract algebraic logic. It is shown that each deductive system of this kind has a deductive structure very close to that of the implicational fragment of IPC. Moreover, it is algebraizable and the algebraic structure of its equivalent quasivariety is very close to that of the variety of Hilbert algebras. The equivalent quasivariety is however not in general a variety. This gives an example of a relatively point-regular, congruence-orderable, and congruence-distributive quasivariety that fails to be a variety, and provides what apparently is the first evidence of a significant difference between the multiterm deduction-detachment theorem and the more familiar form of the theorem where there is a single implication connective.


abstract algebraic logic protoalgebraic logic quasivariety equivalential logic algebraizable logic self-extensional logic Leibniz congruence deduction theorem 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Agliano, P., ‘Congruence quasi-orderability in subtractive varieties’, J. Austral. Math. Soc. Ser. A, 71:421–455, 2001.Google Scholar
  2. [2]
    Agliano, P., ‘Fregean subtractive varieties with definable congruences’, J. Austral. Math. Soc. Ser. A, 71:353–366, 2001.Google Scholar
  3. [3]
    Blok, W. J., P. Köhler, and D. Pigozzi, ‘On the structure of varieties with equationally definable principal congruences, II’, Algebra Universalis, 18:334–379, 1984.Google Scholar
  4. [4]
    Blok, W. J., and D. Pigozzi, Algebraizable logics, volume 396 of Mem. Amer. Math. Soc. Amer. Math. Soc., Providence, January 1989.Google Scholar
  5. [5]
    Blok, W. J., and D. Pigozzi, ‘Local deduction theorems in algebraic logic’, in J. D. Andréka, H. Monk and I. Németi, (eds.), Algebraic Logic (Proc. Conf. Budapest 1988), volume 54 of Colloq. Math. Soc. János Bolyai, North-Holland, Amsterdam, 1991, pp. 75–109.Google Scholar
  6. [6]
    Blok, W. J., and D. Pigozzi, ‘Algebraic semantics for universal Horn logic without equality’, in A. Romanowska and J. D. H. Smith, (eds.), Universal Algebra and Quasigroup Theory, Heldermann, Berlin, 1992, pp. 1–56.Google Scholar
  7. [7]
    Blok, W. J., and D. Pigozzi, ‘Abstract algebraic logic and the deduction theorem’, Bull. of Symbolic Logic, To appear.Google Scholar
  8. [8]
    Büchi, J. R., and M. T. Owens, ‘Skolem rings and their varieties’, in S. MacLane and D. Siefkes, (eds.), The Collected Works of J. Richard Büchi, Springer-Verlag, 1990, pp. 161–221.Google Scholar
  9. [9]
    Czelakowski, J., ‘Algebraic aspects of deduction theorems’, Studia Logica, 44: 369–387, 1985.Google Scholar
  10. [10]
    Czelakowski, J., ‘Local deductions theorems’, Studia Logica, 45:377–391, 1986.Google Scholar
  11. [11]
    Czelakowski, J., and D. Dziobiak, ‘The parameterized local deduction theorem for quasivarieties of algebras and its applications’, Algebra Universalis, 35:373–419, 1996.Google Scholar
  12. [12]
    Czelakowski, J., and D. Dziobiak, ‘A deduction theorem schema for deductive systems of propositional logics’, Studia Logica, Special Issue on Algebraic Logic, 50:385–390, 1991.Google Scholar
  13. [13]
    Czelakowski, J., and D. Pigozzi, ‘Fregean logics’, Ann. Pure Appl. Logic, 127:17–76, 2004.Google Scholar
  14. [14]
    Diego, A., Sur les algébres de Hilbert, Gauthier-Villars, Paris, 1966.Google Scholar
  15. [15]
    Font, J. M., ‘On the Leibniz congruences’, in C. Rauszer, (ed.), Algebraic Methods in Logic and in Computer Science, number 28 in Banach Center Publications, Polish Academy of Sciences, Warszawa, 1993, pp. 17–36.Google Scholar
  16. [16]
    Font, J. M., and R. Jansana, A general algebraic semantics for sentential logics, Number 7 in Lecture Notes in Logic. Springer-Verlag, 1996.Google Scholar
  17. [17]
    Font, J. M., R. Jansana, and D. Pigozzi, ‘Fully adequate Gentzen systems and the deduction theorem’, Rep. Math. Logic, 35:115–165, 2001.Google Scholar
  18. [18]
    Grätzer, G., ‘Two Mal’cev-type conditions in universal algebra’, J. Combinatorial Theory, 8:334–342, 1970.Google Scholar
  19. [19]
    Idziak, P. M., K. Słomczyńska, and A. Wroński, Equivalential algebras: a study of Fregean varieties, Preprint, ver. 1.1, June 1997.Google Scholar
  20. [20]
    McKenzie, G. F., R. N. McNulty and W. F. Taylor, Algebras, Lattices, Varieties, volume I, Wadsworth & Brooks/Cole, Monterey, California, 1987.Google Scholar
  21. [21]
    Pigozzi, D., ‘Fregean algebraic logic’, in J. D. Andréka, H. Monk and I. Németi, (eds.), Algebraic Logic, number 54 in Colloq. Math. Soc. J ános Bolyai, North-Holland, Amsterdam, 1991, pp 473–502.Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • J. Czelakowski
    • 1
  • D. Pigozzi
    • 2
  1. 1.Institute of MathematicsOpole UniversityOpolePoland
  2. 2.Department of MathematicsIowa State UniversityAmesUSA

Personalised recommendations