Semantic Decision Procedures for Some Relevant Logics

This paper proves decidability of a range of weak relevant logics using decision procedures based on the Routley-Meyer semantics. Logics are categorized as -logics, for those proved decidable using a filtration method, and -logics, for those proved decidable using a direct (unfiltered) method. Both of these methods are set out as reductio methods, in the style of Hughes and Cresswell. We also examine some extensions of the -logics where the method fails and infinite sequences of worlds can be generated. In Relevant Logics and their Rivals [12, pages 399–406], Richard Routley (as he was then known) purported to have established the decidability of a range of weak relevant logics. He first claimed to have proved decidability for the system , on pp. 401-2, using a filtration method upon the Routley–Meyer semantics, which yielded the finite model property for . After warning that the method would by no means extend to all the systems with semantic postulates given in Chapter 4 and 5 of [12], he extended the result to the postulate Raaa, for the axiom A & (A → B) → B, the postulate Raa∗a, for A → ∼A → ∼A, and also to the postulate, if Rabc then Rac∗b∗, for A → ∼B → .B → ∼A. There are also a couple of less significant postulates given. Routley then examined the use of a second filtration, but realized its shortcomings in establishing postulate 2 — if a ≤ b and Rbcd then Racd — for the finite model. He briefly examined a third filtration, realized that it would not yield a decidability result without deductive closures on the worlds of the finite models, and hence suggested using operational semantics instead of the Routley–Meyer semantics. However, Routley had erred in two places in his first filtration. Fortunately for those of us interested in weak relevant logics and Routley–Meyer-style semantics, his second filtration can still be used and his results can be revived and extended using the simplified semantics of Priest and Sylvan [10] and Restall “Semantic Decision Procedures for Some Relevant Logics”, Australasian Journal of Logic (1) 2003, 4–27 http://www.philosophy.unimelb.edu.au/ajl/2003 5 [11], which has subsequently become available. The object of §3 of this paper is to show how this is to be done. In the process, I do not want to detract from what has otherwise been an outstanding volume, of which all of us in the field make heavy use. It must also be said that Fine had first proved the decidability of weak relevant logics without A → B → .B → C → .A → C and A → B → .C → A → .C → B in [6], pp. 365–8, using his own semantics which embraces both theories and prime theories. The above postulate 2 simplifies to — (ii) if t ≤ u then (t ·v) ≤ (u ·v) [6, page 348] — which enables the decidability proof to go through without anything resembling the simplified semantics of Priest, Sylvan and Restall. Moreover, a different range of systems have been shown decidable using proof-theory in Brady [3,4,5], viz. the contraction-less logics, ,  and , by using Gentzen systems based on the work of Dunn [1, pages 381–391] and Giambrone [7]. This suggests that there may be a semantic method, based on a Routley-Meyer semantics, for establishing the same result, which might then extend to other systems. We will show this in §5 of the paper, indicating what goes wrong with some of these extensions.