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Algebraic completeness results for R-mingle and its extensions

  • J. Michael Dunn (a1)

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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.

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[1]Anderson, A. R., Some open problems concerning the system E of entailment, Acta Philosophica Fennica, vol. 16 (1963), pp. 718.
[2]Anderson, A. R. and Belnap, N. D. Jr., The pure calculus of entailment, this Journal, vol. 27 (1962), pp. 1952.
[3]Belnap, N. D. Jr., Intensional models for first degree formulas, this Journal, vol. 32 (1967), pp. 122.
[4]Łoś, J., On logical matrices, Travaux de la Société des Sciences et des Lettres de Wroclaw, Seria B, no. 19, 1949. (Polish)
[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. 117134.
[6]Meyer, R. K., R-mingle and relevant disjunction, this Journal (to appear).
[7]Meyer, R. K. and Dunn, J. M., Entailment logics and material implication, Notices of the American Mathematical Society, vol. 15 (1968), pp. 10211022.
[8]Meyer, R. K. and Dunn, J. M., E, R, and γ, this Journal, vol. 34 (1969), pp. 460474.
[9]Scroogs, S. J., Extensions of the Lewis system S5, this Journal, vol. 16 (1951), pp. 112120.
[10]Sugihara, T., Strict implication free from implicational paradoxes, Memoirs of the Faculty of Liberal Arts, Fukui University, Series 1, no. 4 (1955), pp. 5559.
[11]Stone, M. H., Topological representations of distributive lattices and Brouwerian logics, Časopis pro Pžstováni Mathematiky a Fysiky, vol. 67 (1937), pp. 125.
[12]Ulrich, D. E., Matrices for sentential calculi, Doctoral dissertation, Wayne State University, Detroit, Mich., 1967.

Algebraic completeness results for R-mingle and its extensions

  • J. Michael Dunn (a1)

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