Semantic Decision Procedures for Some Relevant Logics

Slogica, 1985

Journal of Applied Non-classical Logics, 2005

Some investigations into the algebraicconstructive aspects of a decision procedure for various fragments of Relevant Logics are presented. Decidability of these fragments relies on S. Kripke's gentzenizations and on his combinatorial lemma known as Kripke's lemma that B. Meyer has shown equivalent to Dickson's lemma in number theory and to his own infinite divisor lemma, henceforth, Meyer's lemma orIDP. These

"Relevant Logics and their rivals, Volume 2" ed. by Richard Sylvan and Ross Brady, 2003

In this section, we provide a sketch of the undecidability results for some of the stronger propositional relevant logics. Interest in the decision problem for these systems dates from the late 1950s. The earliest result was obtained by Anderson and Belnap who proved that the first degree fragment of all these logics is decidable. Kripke [8] proved that the pure implicational fragments R → and E → of R and E are decidable. His methods were extended by Belnap and Wallace to the implication-negation fragments of these systems [3]; Kripke's methods also extend easily to include the implication-conjunction fragments of R and E. Meyer in his thesis [11] extended the result for R to include a primitive necessity operator. He also proved decidable the system RM, and LR, the system obtained from R by omitting the distribution axiom. Various weak relevant logics are also known to be decidable by model-theoretic proofs of the finite model property (see Fine ). Finally, Giambrone [6] has solved the decision problem for various logics obtained by the omission of the contraction axiom, including RW+. Even where positive results were obtained, the decision methods were usually of a complexity considerably greater than in the case of other nonclassical logics such as intuitionistic logic or modal logic, a fact that already indicates the difficulty of the decision procedure. The following section contains a discussion of the complexity of the decision procedure for some relevant logics.

We develop an axiomatic theory of "generalized Routley-Meyer (GRM) logics." These are first-order logics which are can be characterized by model theories in a certain generalization of Routley-Meyer semantics. We show that all GRM logics are subclassical, have recursively enumerable consequence relations, satisfy the compactness theorem, and satisfy the standard structural rules and conjunction and disjunction introduction/elimination rules. So far as the author has explored, all major realworld examples of logics which have these properties are GRM logics.

Journal of Symbolic Logic, Vol. 37, 159-169, 1972

Introduction. In what follows there is presented a unified semantic treatment of certain "paradox-free" systems of entailment, including Church's weak theory of implication (Church [7D and logics akin to the systems E and R of Anderson and Belnap (Anderson [3], Belnap [6D.1 We shall refer to these systems generally as relevant logics.

1991

Recently, the well-founded semantics of a logic program P has been strengthened to the well-founded semantics-by-case (WF C ) and then again to the extended well-founded semantics (WF E ). An important concept used in both WF C and WF E is that of derived rules. We extend the notion of derived rules by adding a new type of derivation and define the strong semantics of P, which has the following important property, known as the GCWA-property: if an atom p = false in all minimal models of P, then p = false in the strong semantics of P. In general, the WF C -semantics and the WF E -semantics do not have the GCWA-property. If we first apply the WF E -semantics to P and then apply the strong semantics to a suitably simplified form of P based on its WF E -semantics, then the resulting semantics is stronger than the WF E -semantics and has the GCWA-property.

The Journal of Symbolic Logic, Vol. 49, 1059-1073, 1984

Introduction. In this paper we show that the propositional logics E of entailment, R of relevant implication and T of ticket entailment are undecidable. The decision problem is also shown to be unsolvable in an extensive class of related logics. The main tool used in establishing these results is an adaptation of the von Neumann coordinatization theorem for modular lattices.

2006

In the context of combinations of theories with disjoint signatures, we classify the component theories according to the decidability of constraint satisfiability problems in finite and infinite models, respectively. We exhibit a theory T 1 such that satisfiability is decidable, but satisfiability in infinite models is undecidable. It follows that satisfiability in T 1 ∪ T 2 is undecidable, whenever T2 has only infinite models, even if signatures are disjoint and satisfiability in T2 is decidable.

Bulletin of the Section of Logic, 45(2), 93-109, 2016

The logic BN4 can be considered as the 4-valued logic of the relevant conditional and the logic E4, as the 4-valued logic of (relevant) entailment. The aim of this paper is to endow E4 with a 2-set-up Routley-Meyer semantics. It is proved that E4 is strongly sound and complete w.r.t. this semantics.

Journal of Symbolic Logic Vol. 64, 1774-1802, 1999

Introduction. In this paper, we show that there is no primitive recursive decision procedure for the implication-conjunction fragments of the relevant logics R, E and T, as well as for a family of related logics. The lower bound on the complexity is proved by combining the techniques of an earlier paper on the same subject [20] with a method used by Lincoln, Mitchell, Scedrov and Shankar in proving that propositional linear logic is undecidable. The decision problem for the pure implicational fragments of E and R were solved by Saul Kripke in a tour deforce of combinatorial reasoning, published only as an abstract [9]. Belnap and Wallace extended Kripke's decision procedure to the implication-negation fragment of E in [3]; an account of their decision method is to be found in [1, pp. 124-139]. The decision method extends immediately to the implication/negation fragment of R. In fact, in the case of R we can go farther; Meyer in his thesis [13] showed how to translate the logic LR, which results from R by omitting the distribution axiom, into RA, so that the decision procedure can be extended to all of LR. This decision procedure has been implemented as a program KRIPKE by Thistlewaite, McRobbie and Meyer [17]. The program is not simply a straightforward implementation of the decision procedure; finite matrices are used extensively to prune invalid nodes from the search tree. The decision methods of Kripke, Belnap, Wallace and Meyer are of a truly marvelous complexity. In fact, they are so complex that it is not immediately clear how to compute an upper bound on the number of steps required by the procedures for an input formula of a given length. The key combinatorial lemma of Kripke that forms the basis for all these decision procedures (see [1, pp. 138-139] or Lemma 11.1 below) simply asserts the finiteness of the search tree without giving an explicit bound on its size. Here we provide a lower bound on the complexity of these decision problems by showing that there is no primitive recursive decision procedure for them. This confirms (in fact, goes beyond) a conjecture of Saul Kripke (in a letter of 1981 to Michael McRobbie [17, p. 40]) that the decidability proof for LR is unprovable in elementary recursive arithmetic.