Algebraic Study of Two Deductive Systems of Relevance Logic

The Review of Symbolic Logic

Relevance logics are known to be sound and complete for relational semantics with a ternary accessibility relation. This paper investigates the problem of adequacy with respect to special kinds of dynamic semantics (i.e., proper relation algebras and relevant families of relations). We prove several soundness results here. We also prove the completeness of a certain positive fragment of R as well as of the first-degree fragment of relevance logics. These results show that some core ideas are shared between relevance logics and relation algebras. Some details of certain incompleteness results, however, pinpoint where relevance logics and relation algebras diverge. To carry out these semantic investigations, we define a new tableaux formalization and new sequent calculi (with the single cut rule admissible) for various relevance logics.

Outstanding Contributions to Logic, 2021

The theory and methods of algorithmic correspondence theory for modal logics, developed over the past 20 years, have recently been extended to the language L R of relevance logics with respect to their standard Routley-Meyer relational semantics. As a result, the non-deterministic algorithmic procedure PEARL (acronym for 'Propositional variables Elimination Algorithm for Relevance Logic') has been developed for computing first-order equivalents of formulas of the language L R in terms of that semantics. PEARL is an adaptation of the previously developed algorithmic procedures SQEMA (for normal modal logics) and ALBA (for distributive and nondistributive modal logics). It succeeds on all inductive formulas in the language L R , in particular on all previously studied classes of Sahlqvist-van Benthem formulas for relevance logic. In the present work we re-interpret the algorithm PEARL from an algebraic perspective, with its rewrite rules seen as manipulating quasi-inequalities interpreted over Urquhart's relevant algebras. This enables us to complete the part of the Sahlqvist-van Benthem theorem still outstanding from the previous work, namely the fact that all inductive L R-formulas are canonical, i.e., are preserved under canonical extensions of relevant algebras. Via the discrete duality between perfect relevant algebras and Routley-Meyer frames, this establishes the fact that all inductive L Rformulas axiomatise logics which are complete with respect to first-order definable classes of Routley-Meyer frames. This generalizes the "canonicity via correspondence" result in [43] for (what we can now recognise as) a certain special subclass of Sahlqvist-van Benthem formulas in the "groupoid" sublanguage of L R where fusion is the only connective. By extending L R with a unary connective for converse and adding the necessary axioms, our results can also be applied to bunched implication algebras and relation algebras. We then present an optimised and deterministic version of PEARL, which we have recently implemented in Python and applied to verify the first-order equivalents of a number of important axioms for relevance logics known from the literature, as well as on several new types of formulas. In the paper we report on the implementation and on some testing results.

Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1990

NOTE ON ALGEBRAIC MODELS FOR RELEVANCE LOGIC by JOSEP M. FONT and GONZALO ROD~GUEZ in Barcelona (Spain)') We denote by 8 = (Fm, A , v, +, i) the algebra of sentential formulas; it is the absolutely free algebra of type (2, 2, 2, 1); the set of propositional variables (the generators) will be denoted by Var. As usual, we take p ++ y to be a shorthand for (p + y) A (y + p). The relevant entailment relation bR is defined in the standard way (finite proofs from hypotheses) with the axioms and inference rules given in [l], pp. 340-341. If we want to speak about its algebraic models, then we must refer to 628.2 of [l]. In page 359 we find the following 1. Definition (DuNN). Let ' ? I = (A, A , *, 1) be an algebra of type (2, 2, 1). We say that U is a de Morgan semigroup (DMS for short) iff the following conditions are satisfied: Al. The reduct (A , A , 1) is a de Morgan lattice, with ordering relation 5 and supremum operation v. A2. The reduct (A , *) is an Abelian semigroup. A3. x 5 x * x for all x E A. A4. x * (y v z) = (x * y) v (x * z) for all x, y, z E A. A5. x * y 5 zx * i z 5 i y for all x, y, z E A.

Archive for Mathematical Logic, 2002

Substructural logics are obtained from the sequent calculi for classical or intuitionistic logic by suitably restricting or deleting some or all of the structural rules . Recently, this field of research has come to encompass a number of logics -e.g. many fuzzy or paraconsistent logics -which had been originally introduced out of different, possibly semantical, motivations. A finer proof-theoretical analysis of such logics, in fact, revealed that it was possible to subsume them under the previous definition (see e.g. .

Mathematical Structures in Computer Science, 2008

This paper proposes a new relevant logic B + , which is obtained by adding two binary connectives, intensional conjunction and intensional disjunction , to Meyer-Routley minimal positive relevant logic B + , where and are weaker than fusion • and fission +, respectively. We give Kripke-style semantics for B + , with →, and modelled by ternary relations. We prove the soundness and completeness of the proposed semantics. A number of axiomatic extensions of B + , including negation-extensions, are also considered, together with the corresponding semantic conditions required for soundness and completeness to be maintained. † Dunn's general approach is algebraic, where each logical connective is characterised as an operation on distributive lattices, which 'distributes' in each of its places over at least one of ∧ and ∨, leaving ∧ or Y. Gao and J. Cheng 146 • † , and shares with + ‡. Then, additional axioms or rules can be added to make coincide with •, and with +. This qualifies and as weaker versions of intensional conjunction and disjunction, respectively. To give a semantics for B + , we apply Dunn's strategy (Dunn 1990), that is, we use n + 1-placed accessibility relations to model n-placed connectives. The semantics is defined by adapting and extending the traditional relational semantics for relevant logics. There are four ternary relations: R 1 and R 2 for →; S 1 for ; and S 2 for. To construct canonical models, as well as theories, we define dualtheories and antidualtheories such that R 1 , R 2 , S 1 , S 2 are canonically defined as derivatives of operations on theories and anti-dualtheories. The crucial tools for completeness are extensions or reductions of a given theory or anti-dualtheory to a prime theory. Then, by well-known standard techniques, together with our extra definitions, we can establish the soundness and completeness of the proposed semantics for B +. Furthermore, we consider a number of axiomatic extensions of B + (including negation-extensions with negation modelled by the Routley ' * '-operation), together with the corresponding semantic conditions to ensure that soundness and completeness are maintained. 2. The basic system B + 2.1. An axiom system for B + B + is expressed in a language L, which has the two-place connectives →, ∧, ∨, and , parentheses (and), and a stock of propositional variables p, q, r, ... Formulas are defined recursively in the usual manner. We use the following scope conventions: the connectives are ranked , , ∧, ∨, → in order of increasing scope (that is, binds more strongly than , binds more strongly than ∧, and so on), otherwise, association is to the left. A, B, C, ... will be used to range over arbitrary formulas. We begin by giving an axiom system for B + , which is defined in the same way as that of Priest and Sylvan (1992) and Restall (1993) § : Axioms

Logica Universalis, 2010

Free download. 68 pages. Ten recommendations and over 260 reads on ResearchGate as of January 16, 2025. What is logical relevance? Anderson and Belnap say that the “modern classical tradition[,] stemming from Frege and Whitehead-Russell, gave no consideration whatsoever to the classical notion of relevance.” But just what is this classical notion? I argue that the relevance tradition is implicitly most deeply concerned with the containment of truth-grounds, less deeply with the containment of classes, and least of all with variable sharing in the Anderson–Belnap manner. Thus modern classical logicians such as Peirce, Frege, Russell, Wittgenstein, and Quine are implicit relevantists on the deepest level. In showing this, I reunite two fields of logic which, strangely from the traditional point of view, have become basically separated from each other: relevance logic and diagram logic. I argue that there are two main concepts of relevance, intensional and extensional. The first is that of the relevantists, who overlook the presence of the second in modern classical logic. The second is the concept of truth-ground containment as following from in Wittgenstein’s Tractatus. I show that this second concept belongs to the diagram tradition of showing that the premisses contain the conclusion by the fact that the conclusion is diagrammed in the very act of diagramming the premisses. I argue that the extensional concept is primary, with at least five usable modern classical filters or constraints and indefinitely many secondary intensional filters or constraints. For the extensional concept is the genus of deductive relevance, and the filters define species. Also following the Tractatus, deductive relevance, or full truth-ground containment, is the limit of inductive relevance, or partial truth-ground containment. Purely extensional inductive or partial relevance has its filters or species too. Thus extensional relevance is more properly a universal concept of relevance or summum genus with modern classical deductive logic, relevantist deductive logic, and inductive logic as its three main domains. By kind permission of the journal publisher, this paper was superseded by the better book of the same name, which is also here and on ResearchGate.

Slogica, 1985

Studia Logica - An International Journal for Symbolic Logic, 2003

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.

This paper defines a Sahlqvist fragment for relevant logic and establishes that each class of frames in the Routley-Meyer semantics which is definable by a Sahlqvist formula is also elementary, that is, it coincides with the class of structures satisfying a given first order property calculable by a Sahlqvist-van Benthem algorithm. Furthermore, we show that some classes of Routley-Meyer frames definable by a relevant formula are not elementary.