Routley–Meyer semantics (RM-semantics) is defined for Gödel 3-valued logic G3 and some logics related to it among which a paraconsistent one differing only from G3 in the interpretation of negation is to be remarked. The logics are defined in the Hilbert-style way and also by means of proof-theoretical and semantical consequence relations. The RM-semantics is defined upon the models for Routley and Meyer’s basic positive logic B+, the weakest positive RM-semantics. In this (...) way, it is to be expected that the models defined can be adapted to other related many-valued logics. (shrink)

Łukasiewicz 3-valued logic Ł3 is often understood as the set of all valid formulas according to Łukasiewicz 3-valued matrices MŁ3. Following Wojcicki, in addition, we shall consider two alternative interpretations of Ł3: ‘truth-preserving’ Ł3a and ‘well-determined’ Ł3b defined by two different consequence relations on the 3-valued matrices MŁ3. The aim of this article is to provide a Routley–Meyer ternary semantics for each one of these three versions of Łukasiewicz 3-valued logic: Ł3, Ł3a and Ł3b.

We provide Routley-Meyer type semantics for relevant logics including Contractionless Ticket Entailment TW (without the truth constant t and o) plus reductio R and Ackermann’s rule γ (i.e., disjunctive syllogism). These logics have the following properties. (i) All have the variable sharing property; some of them have, in addition, the Ackermann Property. (ii) They are stable. (iii) Inconsistent theories built upon these logics are not necessarily trivial.

We develop an axiomatic theory of “generalized Routley-Meyer 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. We also show that the GRM logics include classical logic, intuitionistic logic, LP/K3/FDE, and the relevant logics.

One of the most dominant approaches to semantics for relevant (and many paraconsistent) logics is the Routley-Meyer semantics involving a ternary relation on points. To some (many?), this ternary relation has seemed like a technical trick devoid of an intuitively appealing philosophical story that connects it up with conditionality in general. In this paper, we respond to this worry by providing three different philosophical accounts of the ternary relation that correspond to three conceptions of conditionality. We (...) close by briefly discussing a general conception of conditionality that may unify the three given conceptions. (shrink)

Routley-Meyer ternary relational semantics was introduced in the early seventies of the past century. RM-semantics was intended to model classical relevant logics such as the logic of the relevant conditional R and the logic of Entailment E. But, ever since Routley and Meyer’s first papers on the topic, this essentially malleable semantics has been used for characterizing more general relevant logics or even non-relevant logics. The aim of this paper is to provide an (...) RM-semantics with respect to which Łukasiewicz 3-valued logic Ł3 is sound and complete. Ł3 is understood as the set of all valid formulas in Łukasiewicz 3-valued matrices MŁ3. In this sense, leaning on previous work by us, Ł3 is axiomatized as an extension of Routley and Meyer’s basic positive logic B+, labelled Ł3. And the RM-semantics for Ł3 is actually defined for this particular axiomatization of Ł3. The result presented in the paper is interesting from the Universal Logic perspective, in the sense that it connects Łukasiewicz many-valued logics and similar systems to relevant logics from the point of view of the latter, the 3-termed relational point of view, in particular. (shrink)

The aim of this paper is to provide a Routley-Meyer semantics for Ackermann’s logics of “strenge Implikation” Π ′ and Π ′′ . Besides the Disjunctive Syllogism, this semantics validates the rules Necessitation and Assertion. Strong completeness theorems for Π ′ and Π ′′ are proved. A brief discussion on Π ′ , Π ′′ and paraconsistency is included.

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 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.

Modus ponens provides the central theme. There are laws, of the form A→C. A logic L collects such laws. Any datum A provides input to the laws of L. The central ternary relation R relates theories L,T and U, where U consists of all of the outputs C got by applying modus ponens to major premises from L and minor premises from T. Underlying this relation is a modus ponens product operation on theories L and T, whence RLTU iff LTU. (...) These ideas have been expressed, especially with Routley, as worlds semantics for relevant and other substructural logics.Worlds are best demythologized as theories, subject to truth-functional and other constraints. The chief constraint is that theories are taken as closed under logical entailment, which clearly begs the question if we are using the semantics to determine which theory L is Logic itself. Instead we draw the modal logicians’ conclusion—there are many substructural logics, each with its appropriate ternary relational postulates.Each logic L gives rise to a Calculus of L-theories, on which particular candidate logical axioms have the combinatorial properties expected from the well-known Curry–Howard isomorphism . We apply their bubbling lemma, updating the Fools Model of Combinatory Logic at the pure → level for the system BT. We make fusion an explicit connective, proving a combinator correspondence theorem. Having taken relevant → as a left residual for , we explore its right residual mate →r. Finally we concentrate on and prove a finite model property for the classical minimal relevant logic CB, a conservative extension of the minimal positive relevant logic B+. (shrink)

Routley-Meyer type ternary relational semantics are defined for relevant logics including Routley and Meyer’s basic logic B plus the reductio rule and the disjunctive syllogism. Standard relevant logics such as E and R (plus γ ) and Ackermann’s logics of ‘strenge Implikation’ Π and Π ′ are among the logics considered.

Routley-Meyer Ternary Relational Semantics for Intuitionistic-type Negations examines how to introduce intuitionistic-type negations into RM-semantics. RM-semantics is highly malleable and capable of modeling families of logics which are very different from each other. This semantics was introduced in the early 1970s, and was devised for interpreting relevance logics. In RM-semantics, negation is interpreted by means of the Routley operator, which has been almost exclusively used for modeling De Morgan negations. This book provides (...) research on particular features of intuitionistic-type of negations in RM-semantics, while also defining the basic systems and many of their extensions by using models with or without a set of designated points. (shrink)

We provide a semantics for relevant logics with addition of Aristotle's Thesis, ∼(A→∼A) and also Boethius,(A→B)→∼(A→∼B). We adopt the Routley-Meyer affixing style of semantics but include in the model structures a regulatory structure for all interpretations of formulae, with a view to obtaining a lessad hoc semantics than those previously given for such logics. Soundness and completeness are proved, and in the completeness proof, a new corollary to the Priming Lemma is introduced (c.f.Relevant Logics and (...) their Rivals I, Ridgeview, 1982). (shrink)

In this paper, the positive fragment of the logic math formula of contraction-less relevant implication is extended with the addition of a paraconsistent negation connective similar to the strong negation connective in Nelson's paraconsistent four-valued logic math formula. This extended relevant logic is called math formula, and it has the property of constructible falsity which is known to be a characteristic property of math formula. A Gentzen-type sequent calculus math formula for math formula is introduced, and the cut-elimination and decidability (...) theorems for math formula are proved. Two extended Routley-Meyer semantics are introduced for math formula, and the completeness theorems with respect to these semantics are proved. (shrink)

A ternary relation is often used nowadays to interpret an implication connective of a logic, a practice that became dominant in the semantics of relevance logics. This paper examines two early manuscripts --- one by Routley, another by Meyer --- in which they were developing set-theoretic semantics for various relevance logics. A standard presentation of a ternary relational semantics for, let us say, the logic of relevant implication R is quite illuminating, yet the invention of (...) this semantics was fraught with false starts. Meyer's manuscript, in which he builds on some ideas from Routley's manuscript, essentially contains a relational semantics for which R^{ot} is sound and complete. (shrink)

Sylvan and Plumwood’s is the relevant De Morgan minimal logic in the Routley-Meyer semantics with a set of designated points. The aim of this paper is to define the logic and some of its extensions. The logic is the non-relevant De Morgan minimal logic in the Routley-Meyer semantics without a set of designated points.

In this paper we investigate some logics with weak Boolean negation , calling wB logics, obtained by dualizing intuitionistic negation . We first provide Routley-Meyer semantics for wB-IC , its neighbors wB-LC, wB-LC* ), and wB-S4, wB-S4c . We give completeness for each of them by using RM semantics. We next provide RM semantics for IC, the Dummett's LC, the wB-S4 with ¬ in place of − , and the pB-S4 with c , and give (...) completeness for each system. Finally, we give a translation of the classical propositional logic PC into wB-IC, and a translation of PC into wB-S4c.1. (shrink)

Philosophers of modern logic have cherished no project more dearly than that of extensional reduction. Despite occasional protests that this project was ill-conceived from the start, or that it fails to account for important areas of experience and thought, the extensionalist mills have been grinding away anyhow. Their grinding has brought with it a number of important technical successes, replete with philosophical claims that light has finally been shed on areas hitherto buried in incomprehensible darkness.

Six interesting variants of the logics BN4 and E4—which can be considered as the 4-valued logics of the relevant conditional and entailment, respectively—were previously developed in the literature. All these systems are related to the family of relevant logics and contain Routley and Meyer's basic logic B, which is well-known to be specifically associated with the ternary relational semantics. The aim of this paper is to develop reduced general Routley-Meyer semantics for them. Strong soundness (...) and completeness theorems are proved for each one of the logics. (shrink)

The logics BN4 and E4 can be considered as the 4-valued logics of the relevant conditional and (relevant) entailment, respectively. The logic BN4 was developed by Brady in 1982 and the logic E4 by Robles and Méndez in 2016. The aim of this paper is to investigate the implicative variants (of both systems) which contain Routley and Meyer’s logic B and endow them with a Belnap-Dunn type bivalent semantics.

An Adequate theory of meaning and truth is semantically important. Such a theory necessarily includes in its analysis nonentities, items that do not exist. So what is semantically, and hence logically, important is bound to include nonentities. In virtue of the modifier ‘semantically“, the first premiss is analytic, and it is comparatively uncontroversial. By contrast the second premise of the syllogism, which we want to stick to, is decidedly controversial. So too is the thesis – which implies the inadequacy of (...) classical logical theories – that there are a great many natural language statements, statements an adequate theory should be able to treat of, which cannot be analysed logically, and semantically, without the equivalent of an appeal to nonentities. Defence of the thesis has been somewhat piecemeal, taking the form that all the theories so far offered which try to dispense with nonentities break down or run into insuperable difficulties, difficulties which are readily surmounted given appropriate talk about nonentities. In what follows we shall outline more general sorts of argument for the thesis, designed to show that no theory which dispenses with nonentities as objects of discourse can do justice to the data. (shrink)