Skip to main content
SpringerLink
Log in

The existence of matrices strongly adequate for E, R and their fragments

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

A logic is a pair (P,Q) where P is a set of formulas of a fixed propositional language and Q is a set of rules. A formula α is deducible from X in the logic (P, Q) if it is deducible from X∪P via Q. A matrix \(\mathfrak{M}\) is strongly adequate to (P, Q) if for any α, X, α is deducible from X iff for every valuation in \(\mathfrak{M}\), α is designated whenever all the formulas in X are. It is proved in the present paper that if Q = {modus ponens, adjunction } and P ε {E, R, E +, R +, E I, R I } then there exists a matrix strongly adequate to (P, Q).

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

Access this article

Log in via an institution

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. A. R. Anderson and N. D. Belnap, Modalities in Ackermann's “rigorous implication”, Journal of Symbolic Logic 24 (1959), pp. 107–111.

    Google Scholar 

  2. A. R. Anderson and N. D. Belnap, The pure calculus of entailment, Journal of Symbolic Logic 27 (1962), pp. 19–52.

    Google Scholar 

  3. A. R. Anderson and N. D. Belnap, Entailment, Vol. I, Princeton University, Press., Princeton 1975.

    Google Scholar 

  4. N. D. Belnap, Intensional models for first degree formulas, Journal of Symbolic Logic 32 (1967), pp. 1–22.

    Google Scholar 

  5. A. Biela and P. Hallala, Studies on Church's calculus, Bulletin of the Section of Logic Polish Academy of Sciences 5 (1976), pp. 57–62.

    Google Scholar 

  6. J. M. Dunn, Algebraic completeness results for R-mingle and its extensions, Journal of Symbolic Logic 35 (1970), pp. 1–13.

    Google Scholar 

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

    Google Scholar 

  8. L. L. Maksimova, Princip razdélénija péréménnykh v propozicijonal' nykh logikakh (The principle of separating variables in sentential logics, in Russian), Algébra i Logika 15 (1976), pp. 168–184.

    Google Scholar 

  9. R. K. Meyer, J. M. Dunn and H. Leblanc, Completeness of relevant quantificational theories, Notre Dame Journal of Formal Logic 15 (1974), pp. 97–121.

    Google Scholar 

  10. R. Wójcicki, Some remarks on the consequence operation in sentential logics, Fundamenta Mathematicae 58 (1970), pp. 269–279.

    Google Scholar 

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

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. The Section of Logic, Polish Academy of Sciences Institute of Philosophy and Sociology, Poland

    Marek Tokarz

Authors
  1. Marek Tokarz
    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

Tokarz, M. The existence of matrices strongly adequate for E, R and their fragments. Stud Logica 38, 75–85 (1979). https://doi.org/10.1007/BF00493673

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00493673

Keywords

Search

Navigation