Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-05-04T18:27:53.740Z Has data issue: false hasContentIssue false
  • English
  • Français

Algebraic completeness results for R-mingle and its extensions

Published online by Cambridge University Press:  12 March 2014

J. Michael Dunn
J. Michael Dunn*
Affiliation:
Indiana University
Get access Rights & Permissions [Opens in a new window]

Extract

Schiller Joe Scroggs in [9] established remarkable facts concerning “normal” extensions of the modal sentential calculus S5, the most notable of these facts being that all such proper extensions have finite characteristic matrices. The major import of the present paper is that like facts hold for the relevant sentential calculus R-Mingle (RM). Robert K. Meyer in [6] has obtained an important completeness result for RM, which will play a central role in our results. However, in §2 we shall obtain a new proof of Meyer's result as a by-product of the algebraic logic that we develop in §1. Also in §2 we shall obtain the promised results for extensions of RM. In §3 we shall provide a strong completeness theorem for RM by generalizing the semantics of Meyer.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 35 , Issue 1 , March 1970 , pp. 1 - 13
Copyright
Copyright © Association for Symbolic Logic 1970

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Anderson, A. R., Some open problems concerning the system E of entailment, Acta Philosophica Fennica, vol. 16 (1963), pp. 7–18.Google Scholar
[2] Anderson, A. R. and Belnap, N. D. Jr., The pure calculus of entailment, this Journal, vol. 27 (1962), pp. 19–52.Google Scholar
[3] Belnap, N. D. Jr., Intensional models for first degree formulas, this Journal, vol. 32 (1967), pp. 1–22.Google Scholar
[4] Łoś, J., On logical matrices, Travaux de la Société des Sciences et des Lettres de Wroclaw, Seria B, no. 19, 1949. (Polish)Google Scholar
[5] McKinsey, J. C. C., A solution to the decision problem for the Lewis systems S2 and S4 with an application to topology, this Journal, vol. 6 (1941), pp. 117–134.Google Scholar
[6] Meyer, R. K., R-mingle and relevant disjunction, this Journal (to appear).Google Scholar
[7] Meyer, R. K. and Dunn, J. M., Entailment logics and material implication, Notices of the American Mathematical Society, vol. 15 (1968), pp. 1021–1022.Google Scholar
[8] Meyer, R. K. and Dunn, J. M., E, R, and γ, this Journal, vol. 34 (1969), pp. 460–474.Google Scholar
[9] Scroogs, S. J., Extensions of the Lewis system S5, this Journal, vol. 16 (1951), pp. 112–120.Google Scholar
[10] Sugihara, T., Strict implication free from implicational paradoxes, Memoirs of the Faculty of Liberal Arts, Fukui University, Series 1, no. 4 (1955), pp. 55–59.Google Scholar
[11] Stone, M. H., Topological representations of distributive lattices and Brouwerian logics, Časopis pro Pžstováni Mathematiky a Fysiky, vol. 67 (1937), pp. 1–25.Google Scholar
[12] Ulrich, D. E., Matrices for sentential calculi, Doctoral dissertation, Wayne State University, Detroit, Mich., 1967.Google Scholar