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Fragments of R-Mingle

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Abstract

The logic RM and its basic fragments (always with implication) are considered here as entire consequence relations, rather than as sets of theorems. A new observation made here is that the disjunction of RM is definable in terms of its other positive propositional connectives, unlike that of R. The basic fragments of RM therefore fall naturally into two classes, according to whether disjunction is or is not definable. In the equivalent quasivariety semantics of these fragments, which consist of subreducts of Sugihara algebras, this corresponds to a distinction between strong and weak congruence properties. The distinction is explored here. A result of Avron is used to provide a local deduction-detachment theorem for the fragments without disjunction. Together with results of Sobociński, Parks and Meyer (which concern theorems only), this leads to axiomatizations of these entire fragments — not merely their theorems. These axiomatizations then form the basis of a proof that all of the basic fragments of RM with implication are finitely axiomatized consequence relations.

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Authors and Affiliations

  1. Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851S Morgan Street, 60607-7045, Chicago, Illinois, USA

    W. J. Blok

  2. School of Mathematical Sciences, University of KwaZulu-Natal, Howard College Campus, 4041, Durban, South Africa

    J. G. Raftery

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  1. W. J. Blok
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Special issue of Studia Logica: “Algebraic Theory of Quasivarieties” Presented by M. E. Adams, K. V. Adaricheva, W. Dziobiak, and A. V. Kravchenko

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Blok, W.J., Raftery, J.G. Fragments of R-Mingle. Stud Logica 78, 59–106 (2004). https://doi.org/10.1007/s11225-005-0106-8

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