Greg Restall

Negation in Relevant Logics (How I Stopped Worrying and Learned to Love the Routley Star)

Negation raises three thorny problems for anyone seeking to interpret relevant logics. the frame semantics for negation in relevant logics involves a ‘point shift’ operator *. Problem number one is the interpretation of this operator. Relevant logics commonly interpreted take the inference from A and ~ A ⋁ B to B to be invalid, because the corresponding relevant conditional A ⋀ (~A ⋁ B) → B is not a theorem. Yet we often make the inference from A and ~ A ⋁ B to B, and we seem to be reasoning validly when we do so. Problem number two is explaining what is really going on here. Finally, we can add an operation which Meyer has called Boolean negation to our logic, which is evaluated in the traditional way: x ⊨ −A if and only if x ⊭ A. Problem number three involves deciding which is the ‘real’ negation. How can we decide between orthodox negation and the new, ‘Boolean’ negation? In this paper, I present a new interpretation of the frame semantics for relevant logics which will allow us to give principled answers to each of these questions.

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.

Moral fictionalism versus the rest

In this paper we introduce a distinct metaethical position, fictionalism about morality. We clarify and defend the position, showing that it is a way to save the ‘moral phenomena’ while agreeing that there is no genuine objective prescriptivity to be described by moral terms. In particular, we distinguish moral fictionalism from moral quasi-realism, and we show that fictionalism possesses the virtues of quasi-realism about morality, but avoids its vices.

How to be really contraction free

A logic is said to becontraction free if the rule fromA → (A →B) toA →B is not truth preserving. It is well known that a logic has to be contraction free for it to support a non-trivial naïve theory of sets or of truth. What is not so well known is that if there isanother contracting implication expressible in the language, the logic still cannot support such a naïve theory. A logic is said to berobustly contraction free if there is no such operator expressible in its language. We show that a large class of finitely valued logics are each not robustly contraction free, and demonstrate that some other contraction free logics fail to be robustly contraction free. Finally, the sublogics of Łω (with the standard connectives) are shown to be robustly contraction free.

Simplified semantics for relevant logics (and some of their rivals)

This paper continues the work of Priest and Sylvan inSimplified Semantics for Basic Relevant Logics, a paper on the simplified semantics of relevant logics, such asB+ andB. We show that the simplified semantics can also be used for a large number of extensions of the positive base logicB+, and then add the dualising‘*’ operator to model negation. This semantics is then used to give conservative extension results for Boolean negation.

Displaying and deciding substructural logics 1 : Logics with contraposition

Many logics in the relevant family can be given a proof theory in the style of Belnaps display logic (Belnap, 1982). However, as originally given, the proof theory is essentially more expressive than the logics they seek to model. In this paper, we consider a modified proof theory which more closely models relevant logics. In addition, we use this proof theory to show decidability for a large range of substructural logics.

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.

A Note on Naive Set Theory in LP

Recently there has been much interest in naive set theory in nonstandard logics. This note continues this trend by considering a set theory with a general comprehension schema based on the paraconsistent logic LP. We demonstrate the nontriviality of the set theory so formulated, deduce some elementary properties of this system of sets, and also delineate some of the problems of this approach.

Relevant and Substructural Logics

Publisher Summary This chapter discusses proof theory of relevant and substructural logics, and the model theory of these logics. The discipline of relevant logic grew out of an attempt to understand notions of consequence and conditionality where the conclusion of a valid argument is relevant to the premises, and where the consequent of a true conditional is relevant to the antecedent. The structural rules dictate admissible forms of transformations of premises contained in proofs. The chapter discusses how relevant logics are naturally counted as substructural logics, as certain commonly admitted structural rules are responsible for introducing irrelevant consequences into proofs.

Assertion, Denial and Non-classical Theories

In this paper I urge friends of truth-value gaps and truth-value gluts—proponents of paracomplete and paraconsistent logics—to consider theories not merely as sets of sentences, but as pairs of sets of sentences, or what I call ‘bitheories,’ which keep track not only of what holds according to the theory, but also what fails to hold according to the theory. I explain the connection between bitheories, sequents, and the speech acts of assertion and denial. I illustrate the usefulness of bitheories by showing how they make available a technique for characterising different theories while abstracting away from logical vocabulary such as connectives or quantifiers, thereby making theoretical commitments independent of the choice of this or that particular non-classical logic. Examples discussed include theories of numbers, classes and truth. In the latter two cases, the bitheoretical perspective brings to light some heretofore unconsidered puzzles for friends of naive theories of classes and truth.