Ross T. Brady

On the Ternary Relation and Conditionality

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.

The gentzenization and decidability of RW

which corresponds in LR, to the axiom (W) of R, would, in an attempted decision procedure based on LR, , produce unbounded multiplicities of the above structures a. It then seemed likely that the system R without the axiom (W), called now RW, would be decidable. Giambrone, in [8], has shown, by using Dunn’s Gentzenization LR, without (WI t) (thus yielding LRW,), that RW, is decidable. Giambrone’s decidability argument involves a refinement of Gentzen’s original argument for the decidability of sentential calculi in [7], together with original insights appropriate to Dunn’s style of Gentzenization, which involves both intensional and extensional sequences in the antecedents of consecutions. In this paper, we apply Giambrone’s decidability argument to achieve decidability for RW, after establishing a Cut-free Gentzenization for the logic by extending the work of Dunn in [l]. It is the establishment that the rule Cut:

Relevant Restricted Quantification

The paper reviews a number of approaches for handling restricted quantification in relevant logic, and proposes a novel one. This proceeds by introducing a novel kind of enthymematic conditional.

Relevant implication and the case for a weaker logic

We collect together some misgivings about the logic R of relevant inplication, and then give support to a weak entailment logic DJd. The misgivings centre on some recent negative results concerning R, the conceptual vacuousness of relevant implication, and the treatment of classical logic. We then rectify this situation by introducing an entailment logic based on meaning containment, rather than meaning connection, which has a better relationship with classical logic. Soundness and completeness results are proved for DJd with respect to a content semantics, which embraces the concept of meaning containment.

Depth relevance of some paraconsistent logics

The paper essentially shows that the paraconsistent logicDR satisfies the depth relevance condition. The systemDR is an extension of the systemDK of [7] and the non-triviality of a dialectical set theory based onDR has been shown in [3]. The depth relevance condition is a strengthened relevance condition, taking the form: If ⊢DR- A→B thenA andB share a variable at the same depth, where the depth of an occurrence of a subformulaB in a formulaA is roughly the number of nested ‘→’s required to reach the occurrence ofB inA. The method of proof is to show that a model structureM consisting of {M0, M1, ..., Mω}, where theMi′s are all characterized by Meyers 6-valued matrices (c. f, [2]), satisfies the depth relevance condition. Then, it is shown thatM is a model structure for the systemDR.

Where gamma fails

A major question for the relevant logics has been, “Under what conditions is Ackermanns ruleγ from -A ∨B andA to inferB, admissible for one of these logics?” For a large number of logics and theories, the question has led to an affirmative answer to theγ problem itself, so that such an answer has almost come to be expected for relevant logics worth taking seriously. We exhibit here, however, another large and interesting class of logics-roughly, the Boolean extensions of theW — free relevant logics (and, precisely, the well-behaved subsystems of the 4-valued logicBN4) — for which γ fails.

The Simple Consistency of Naive Set Theory using Metavaluations

The main aim is to extend the range of logics which solve the set-theoretic paradoxes, over and above what was achieved by earlier work in the area. In doing this, the paper also provides a link between metacomplete logics and those that solve the paradoxes, by finally establishing that all M1-metacomplete logics can be used as a basis for naive set theory. In doing so, we manage to reach logics that are very close in their axiomatization to that of the logic R of relevant implication. A further aim is the use of metavaluations in a new context, expanding the range of application of this novel technique, already used in the context of negation and arithmetic, thus providing an alternative to traditional model theoretic approaches.

A content semantics for quantified relevant logics. II

In part I, we presented an algebraic-style of semantics, which we called “content semantics,” for quantified relevant logics based on the weak systemBBQ. We showed soundness and completeness with respect to theunreduced semantics ofBBQ. In part II, we proceed to show soundness and completeness for extensions ofBBQ with respect to this type of semantics. We introducereduced semantics which requires additional postulates for primeness and saturation. We then conclude by showing soundness and completeness forBBdQ and its extentions with respect to this reduced semantics.

Distribution in the Logic of Meaning Containment and in Quantum Mechanics

The paper examines the sentential and quantificational forms of distribution in classical, intuitionistic and relevant logics, and in relation to quantum mechanics. Particular attention is also paid to the first author’s logic MCQ of meaning containment. The ultimate aim is to determine whether distribution should be included and, if so, in what forms. We conclude that the sentential and existential rule-forms, both derivable from the two-premise meta-rules of MCQ, are the only two forms that generally hold.

Rules in relevant logic - I: Semantic classification

Primitive and derived rules of a formal system of logic, by their very nature, will preserve validity in any semantics with respect to which the logic is sound and complete. However, tighter semantic preservation properties, such as truth-preservation, will often apply and various concepts of the semantics can be used to express such properties. The number of preservation properties for the rules of a logic depends to some extent on the variety of rules but also on the complexity of the semantics.