Skip to main content
SpringerLink
Log in

2-Element matrices

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

Sections 1, 2 and 3 contain the main result, the strong finite axiomatizability of all 2-valued matrices. Since non-strongly finitely axiomatizable 3-element matrices are easily constructed the result reveals once again the gap between 2-valued and multiple-valued logic. Sec. 2 deals with the basic cases which include the important F ∞i from Post's classification. The procedure in Sec. 3 reduces the general problem to these cases. Sec. 4 is a study of basic algebraic properties of 2-element algebras. In particular, we show that equational completeness is equivalent to the Stone-property and that each 2-element algebra generates a minimal quasivariety. The results of Sec. 4 will be applied in Sec. 5 to maximality questions and to a matrix free characterization of 2-valued consequences in the lattice of structural consequences in any language. Sec. 6 takes a look at related axiomatization. problems for finite algebras and matrices. We study the notion of a propositional consequence with equality and, among other things, present explicit axiomatizations of 2-valued consequences with equality.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. G. Asser and W. Rautenberg, Ein Verfahren zur Axiomatisierung gewisser zweiwertiger Aussagenkalküle, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 6 (1960), pp. 303–318.

    Google Scholar 

  2. K. A. Baker, Congruence-distributive polynomial reducts of lattices, Algebra Universalis 9 (1979), pp. 142–145.

    Google Scholar 

  3. J. Berman, A proof of Lyndon's finite base theorem, Discrete Mathematics 29 (1980), pp. 229–233.

    Article  Google Scholar 

  4. D. M. Clark and P. H. Krauss, Varieties generated by para-primal algebras, Algebra Universalis 7 (1977), pp. 93–114.

    Google Scholar 

  5. -, Plain para-primal algebras, to appear in: Algebra Universalis.

  6. G. Kreisel and J.-L. Krivine, Modelltheorie, Berlin 1972.

  7. G. Grätzer, Universal algebra, 2. edition, Berlin 1979.

  8. L. Henkin, Fragments of the propositional calculus, The Journal of Symbolic Logic 14 (1949), pp. 42–48.

    Google Scholar 

  9. B. Jónsson, Algebras whose congruence lattice are distributive, Mathematica Scandinavica 21 (1967), pp. 110–121.

    Google Scholar 

  10. J. Łoś and R. Suszko, Remarks on sentential logics, Indagationes Mathematicae 20 (1958), pp. 177–183.

    Google Scholar 

  11. R. C. Lyndon, Identities in 2-valued calculi, Transaction of the American Mathematical Society 71 (1951), pp. 457–465.

    Google Scholar 

  12. A. I. Malcev, Algebraic systems, Berlin 1973.

  13. R. McKenzie, A finite algebra A with SP {A} not elementary, Algebra Universalis 8 (1978), pp. 5–7.

    Google Scholar 

  14. A. J. Olshewski, Conditional identities in finite groups, Siberian Mathematical Journal1975, pp. 1000–1003.

  15. A. E. Pixley, Functionally complete algebras generating distributive and permutable classes, Mathematische Zeitschrift 114 (1970), pp. 361–372.

    Google Scholar 

  16. E. Post, Two-valued iterative systems of Mathematical Logic, Princeton 1941. Reprint New-York 1960.

  17. R. W. Quackenbush, Algebras with minimal spectrum, Algebra Universalis 10 (1980), pp. 117–129.

    Google Scholar 

  18. W. Rautenberg, Klassische und Nichtklassische Ausagenlogik, Vieweg, Wiesbaden 1979.

    Google Scholar 

  19. I. G. Rosenberg, Completeness properties of multiple-valued logic algebras, Computer Science and Multiple-valued logic (ed. D. C. Rine), North-Holland, Amsterdam 1977.

    Google Scholar 

  20. A. Selman, Completeness of calculi for axiomatically defined classes of algebras, Algebra Universalis 2 (1972), pp. 20–32.

    Google Scholar 

  21. S. Surma (editor), Studies in the history of mathematical logic, Ossolineum, Wrocław 1973.

    Google Scholar 

  22. W. Taylor, The fine spectrum of a variety, Algebra Universalis 5 (1975), pp. 263–303.

    Google Scholar 

  23. R. Wójcicki, Matrix approach in methodology of sentential calculi, Studia Logica 32 (1973), pp. 7–37.

    Google Scholar 

  24. P. Wojtylak, Matrix representation for structural strengthenings of a propositional logic, Studia Logica 38 (1979), pp. 263–266.

    Google Scholar 

  25. —, Strongly finite logics, Bulletin of the Section of Logic 8 (1979), pp. 99–111.

    Google Scholar 

  26. A. Wroński, A 3-valued matrix whose consequence is not finitely based, Bulletin of the Section of Logic 8 (1979), pp. 68–71.

    Google Scholar 

  27. -, Consequence operations of 2-element matrices (Manuscript in Polish).

Download references

Author information

Authors and Affiliations

  1. Fachbereich Mathematik, Freie Universität WestBerlin, Deutschland

    Wolfgang Rautenberg

Authors
  1. Wolfgang Rautenberg
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rautenberg, W. 2-Element matrices. Stud Logica 40, 315–353 (1981). https://doi.org/10.1007/BF00401653

Download citation

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00401653

Keywords

Search

Navigation