This paper deals with substructural nuclear (image-based) logics and their algebraic and Kripke-style semantics. More precisely, we first introduce a class of substructural logics with connective _N_ satisfying nucleus property, called here substructural _nuclear_ logics, and its subclass, called here substructural _nuclear image-based_ logics, where _N_ further satisfies homomorphic image property. We then consider their algebraic semantics together with algebraic characterizations of those logics. Finally, we introduce _operational Kripke-style_ semantics for those logics and provide two sorts of completeness results for (...) them, one of which is based on algebraic completeness for all the logics and the other of which is based on set-theoretic one for the nuclear image-based logics. (shrink)

Abstractionism in the philosophy of mathematics aims at deriving large fragments of mathematics by combining abstraction principles (i.e. the abstract objects $\S e_1, \S e_2$, are identical if, and only if, an equivalence relation $Eq_\S$ holds between the entities $e_1, e_2$) with logic. Still, as highlighted in work on the semantics for relevant logics, there are different ways theories might be combined. In exactly what ways must logic and abstraction be combined in order to get interesting mathematics? In this paper, (...) we investigate the matter by deriving the axioms of second-order Peano Arithmetic from Frege's Basic Law V (the extension of $F$ is identical with the extension of $G$ if, and only if, $F$ and $G$ are extensionally equivalent) in the presence of a relevant higher-order logic. The results are interesting. Not only must we take on logic as true, and not only must we apply our logic to abstraction principles, but also we have to apply our theory of abstraction back to the logic in order to arrive at arithmetic. Thus, what Abstractionism gives us is not simply what we get from abstraction via logic, but also what we get from logic via abstraction. (shrink)

Here's the thing: when you look at it from just the right angle, it's entirely obvious how semantics for second-order relevant logics ought to go. Or at least, if you've understood how semantics for first-order relevant logics ought to go, there are perspectives like this. What's more is that from any such angle, the metatheory that needs doing can be summed up in one line: everything is just as in the first-order case, but with more indices. Of course, it's no (...) small matter finding the magical angle from which everything becomes obvious. And even having found this perspective, one cannot assume one's audience will find things as obvious as oneself. All that to say this: if the results in the paper below strike you as obvious, pay attention to the perspective that makes that possible. And if they don't, feel free to ignore this preamble in its entirety. (shrink)

Jc Beall has made several contributions to the theory of restricted quantification in relevant logics. This paper examines these contributions and proposes an alternative account of restricted universals. The alternative is not, however, a theory of relevant restricted universals in any real sense. It is, however, a theory of restricted universals phrased in the most plausible general quantificational theory for relevant logics—Kit Fine’s stratified semantics. The motivation both for choosing this semantic framework and for choosing the particular theory of restricted (...) quantification we use is because they are the best way of dealing with these topics from Beall’s theory-building theory picture of logic, establishing a second point of contact with Beall’s work. (shrink)

There appears to be few, if any, limits on what sorts of logical connectives can be added to a given logic. One source of potential limitations is the motivating ideology associated with a logic. While extraneous to the logic, the motivating ideology is often important for the development of formal and philosophical work on that logic, as is the case with intuitionistic logic. One family of logics for which the philosophical ideology is important is the family of relevant logics. In (...) this paper, I explore the limits of what a relevant connective is, showing how some basic criteria motivated by the ideology of relevant logicians provide robust limits on potential connectives. These criteria provide some plausible necessary conditions on being a relevant connective. (shrink)

Among Dunn’s many important contributions to relevance logic was his work on the system RM (R-mingle). Although RM is an interesting system in its own right, it is widely considered to be too strong. In this chapter, I revisit a closely related system, RM0 (sometimes known as ‘constructive mingle’), which includes the mingle axiom while not degenerating in the way that RM itself does. My main interest will be in examining this logic from two related semantical perspectives. First, I give (...) a purely operational bisemilattice semantics for it by adapting previous work of Humberstone. Second, I examine a more conventional algebraic semantics for it and discuss how this relates to the operational semantics. A novel operational semantics for J (intuitionistic logic) as well as its conventional Heyting algebraic semantics emerge as special cases of the corresponding semantics for RM0. The results of this chapter suggest that RM0 is a more interesting logic than has been appreciated and that Humberstone’s operational semantic framework similarly deserves more attention than it has received. (shrink)

Relevant logics infamously have the property that they only validate a conditional when some propositional variable is shared between its antecedent and consequent. This property has been strengthened in a variety of ways over the last half-century. Two of the more famous of these strengthenings are the strong variable sharing property and the depth relevance property. In this paper I demonstrate that an appropriate class of relevant logics has a property that might naturally be characterized as the supremum of these (...) two properties. I also show how to use this fact to demonstrate that these logics seem to be constructive in previously unknown ways. (shrink)

I present a version of Kit Fine's stratified semantics for the logic RWQ and define a natural family of related structures called RW hyperdoctrines. After proving that RWQ is sound with respect to RW hyperdoctrines, we show how to construct, for each stratified model, a hyperdoctrine that verifies precisely the same sentences. Completeness of RWQ for hyperdoctrinal semantics then follows from completeness for stratified semantics, which is proved in an appendix. By examining the base category of RW hyperdoctrines, we find (...) reason to be worried about the ontology of stratified models. (shrink)

This paper presents a new semantics for the weak relevant logic DW that makes the role of the infamous Routley star more explicable. Central to this rewriting is combining aspects of both the American and Australian plan for understanding negations in relevance logics.

Fine (2017a) sets out a theory of content based on truthmaker semantics which distinguishes two kinds of consequence between contents. There is entailment, corresponding to the relationship between disjunct and disjunction, and there is containment, corresponding to the relationship between conjunctions and their conjuncts. Fine associates these with two notions of parthood: disjunctive and conjunctive. Conjunctive parthood is a very useful notion, allowing us to analyse partial content and partial truth. In this chapter, I extend the notion of disjunctive parthood (...) in terms of a structural relation of refinement, which stands to disjunctive parthood much as mereological parthood stands to conjunctive parthood. Philosophically, this relation may be modelled on the determinable- determinate relation, or on a fact-to-fact notion of grounding. I discuss its connection to two other Finean notions: vagueness (understood via precisification) and arbitrary objects. I then investigate what a logic of truthmaking with refinement might look like. I argue that (i) parthood naturally gives rise to a relevant conditional; (ii) refinement underlies a relevant notion of disjunction; and so (iii) truthmaker semantics with refinement is a natural home for relevant logic. The resulting formal models draw on Fine’s (1974) semantics for relevant logics. Finally, I use this understanding of relevant semantics to investigate the status of the mingle axiom. (shrink)

Ultralogic as Universal? is a seminal text in non-classcial logic. Richard Routley presents a hugely ambitious program: to use an 'ultramodal' logic as a universal key, which opens, if rightly operated, all locks. It provides a canon for reasoning in every situation, including illogical, inconsistent and paradoxical ones, realized or not, possible or not. A universal logic, Routley argues, enables us to go where no other logic—especially not classical logic—can. Routley provides an expansive and singular vision of how a universal (...) logic might one day solve major problems in set theory, arithmetic, linguistics, physics, and more. It circulated in typescript in the late 1970s before appearing as the Appendix to Exploring Meinong's Jungle and Beyond. With engaging, forceful prose, unsparing criticism of entrenched institutions, and many tantalizing proof sketches, Ultralogic? has had a major influence on the development of paraconsistent and relevant logic. This new edition makes this work available for a modern audience, newly typeset and corrected, along with extensive notes, and new commentary essays. (shrink)

This paper introduces the inquisitive extension of R, denoted as InqR, which is a relevant logic of questions based on the logic R as the background logic of declaratives. A semantics for InqR is developed, and it is shown that this semantics is, in a precisely defined sense, dual to Routley-Meyer semantics for R. Moreover, InqR is axiomatized and completeness of the axiomatic system is established. The philosophical interpretation of the duality between Routley-Meyer semantics and the semantics for InqR is (...) also discussed. (shrink)

There is a natural story about what logic is that sees it as tied up with two operations: a ‘throw things into a bag’ operation and a ‘closure’ operation. In a pair of recent papers, Jc Beall has fleshed out the account of logic this leaves us with in more detail. Using Beall’s exposition as a guide, this paper points out some problems with taking the second operation to be closure in the usual sense. After pointing out these problems, I (...) then turn to fixing them in a restricted case and modulo a few simplifying assumptions. In a followup paper, the simplifications and restrictions will be removed. (shrink)

The main object of this article is to give two novel proofs of the admissibility of Ackermann’s rule (γ) for the propositional relevant logic R. The results are established as corollaries of cut elimination for systems of tableaux for R. Cut elimination, in turn, is established both nonconstructively (as a corollary of completeness) and constructively (using Gentzen-like methods). The extensibility of the techniques is demonstrated by showing that (γ) is admissible for RQ* (R with constant domain quantifiers). The status of (...) the admissibility of (γ) for RQ* was, to the best of the author’s knowledge, an open problem. Further extensions of these results will be explored in the sequel(s). (shrink)

I develop and defend a truthmaker semantics for the relevant logic R. The approach begins with a simple philosophical idea and develops it in various directions, so as to build a technically adequate relevant semantics. The central philosophical idea is that truths are true in virtue of specific states. Developing the idea formally results in a semantics on which truthmakers are relevant to what they make true. A very natural notion of conditionality is added, giving us relevant implication. I then (...) investigate ways to add conjunction, disjunction, and negation; and I discuss how to justify contraposition and excluded middle within a truthmaker semantics. (shrink)

In 1988, Kit Fine published a semantic theory for quantified relevant logics. He referred to this theory as stratified semantics. While it has received some attention in the literature, 1–20, 1992; Mares & Goldblatt, Journal of Symbolic Logic 71, 163–187, 2006), stratified semantics has overall received much less attention than it deserves. There are two plausible reasons for this. First, the only two dedicated treatments of stratified semantics available are, 27–59, 1988; Mares, Studia Logica 51, 1–20, 1992), both of which (...) are quite dense and technically challenging. Second, there are a number of prima facie reasons to be worried about stratified semantics. The purpose of this paper is to revitalize research on stratified semantics. I will do so by giving a ‘user friendly’ presentation of the semantics, and by giving reasons to think that the prima facie reasons to be worried about it are too simplistic. (shrink)

This paper deals with a collection of concerns that, over a period of time, led the author away from the Routley–Meyer semantics, and towards proof- theoretic approaches to relevant logics, and indeed to the weak relevant logic MC of meaning containment.

This paper proves decidability of a range of weak relevant logics using decision procedures based on the Routley-Meyer semantics. Logics are categorized as F-logics, for those proved decidable using a filtration method, and U-logics, for those proved decidable using a direct (unfiltered) method. Both of these methods are set out as reductio methods, in the style of Hughes and Cresswell. We also examine some extensions of the U-logics where the method fails and infinite sequences of worlds can be generated.

A variety of modal logics based on the relevant logic R are presented. Models are given for each of these logics and completeness is shown. It is also shown that each of these logics admits Ackermann's rule γ and as a corollary of this it is proved that each logic is a conservative extension of its counterpart based on classical logic, hence we call them “classically complete”. MSC: 03B45, 03B46.

Relevant logics are a family of non-classical logics characterized by the behavior of their implication connectives. Unlike some other non-classical logics, such as intuitionistic logic, there are multiple philosophical views motivating relevant logics. Further, different views seem to motivate different logics. In this article, we survey five major views motivating the adoption of relevant logics: Use Criterion, sufficiency, meaning containment, theory construction, and truthmaking. We highlight the philosophical differences as well as the different logics they support. We end with some (...) questions for future research. (shrink)

Logic: the Basics is an accessible introduction to the core philosophy topic of standard logic. Focussing on traditional Classical Logic the book deals with topics such as mathematical preliminaries, propositional logic, monadic quantified logic, polyadic quantified logic, and English and standard ‘symbolic transitions’. With exercises and sample answers throughout this thoroughly revised new edition not only comprehensively covers the core topics at introductory level but also gives the reader an idea of how they can take their knowledge further and the (...) philosophical questions around logic. -/- Logic: the Basics is essential reading for first-year undergraduate philosophy students on standard introductory logic courses. (shrink)

Following a proposal of Humberstone, this paper studies a semantics for modal logic based on partial “possibilities” rather than total “worlds.” There are a number of reasons, philosophical and mathematical, to find this alternative semantics attractive. Here we focus on the construction of possibility models with a finitary flavor. Our main completeness result shows that for a number of standard modal logics, we can build a canonical possibility model, wherein every logically consistent formula is satisfied, by simply taking each individual (...) finite formula (modulo equivalence) to be a possibility, rather than each infinite maximally consistent set of formulas as in the usual canonical world models. Constructing these locally finite canonical models involves solving a problem in general modal logic of independent interest, related to the study of adjoint pairs of modal operators: for a given modal logic L, can we find for every formula phi a formula f(phi) such that for every formula psi, phi -> BOX psi is provable in L if and only if f(phi) -> psi is provable in L? We answer this question for a number of standard modal logics, using model-theoretic arguments with world semantics. This second main result allows us to build for each logic a canonical possibility model out of the lattice of formulas related by provable implication in the logic. (shrink)

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)

This paper deals with one kind of Kripke-style semantics, which we shall call algebraic Kripke-style semantics, for relevance logics. We first recall the logic R of relevant implication and some closely related systems, their corresponding algebraic structures, and algebraic completeness results. We provide simpler algebraic completeness proofs. We then introduce various types of algebraic Kripke-style semantics for these systems and connect them with algebraic semantics.

The Logic R4 is obtained by adding the axiom □(A v B) → (◇A v □B) to the modal relevant logic NR. We produce a model theory for this logic and show completeness. We also show that there is a natural embedding of a Kripke model for S4 in each R4 model structure.

In "The Logical Structure of Linguistic Commitment I" (The Journal of Philosophical Logic 23 (1994), 369-400), we sketch a linguistic theory (inspired by Brandom's Making it Explicit) which includes an "expressivist" account of the implication connective, →: the role of → is to "make explicit" the inferential proprieties among possible commitments which proprieties determine, in part, the significances of sentences. This motivates reading (A → B) as "commitment to A is, in part, commitment to B". Our project is to study (...) the logic of →. LSLC I approximates (A → B) as "anyone committed to A is committed to B", ignoring issues of whether A is relevant to B. The present paper includes considerations of relevance, motivating systems of relevant commitment entailment related to the systems of commitment entailment of LSLC I. We also consider the relevance logics that result from a commitment reading of Fine's semantics for relevance logics, a reading that Fine suggests. (shrink)

This paper reformulates and decides a certain conjecture in Dunn's 'Relevant Predication 1: The Formal Theory' (Journal of Philosophical Logic 16, 347-381, 1987). This conjecture of Dunn's relates his object-language characterisation of a property's being relevant in a variable x to certain grammatical characterisations of relevance, analogous to some given by Helman, in 'Relevant Implication and Relevant Functions' (to appear in Entailment: The Logic of Relevance and Necessity, vol. 2, by Alan Ross Anderson, Nuel Belnap, and J. Michael Dunn et (...) al.) In the course of the investigation this paper also investigates Kit Fine's semantics for quantified relevance logics, which appears in his appropriately titled 'Semantics for Quantified Relevance Logics'. (shrink)

This paper explores allowing truth value assignments to be undetermined or "partial" and overdetermined or "inconsistent", thus returning to an investigation of the four-valued semantics that I initiated in the sixties. I examine some natural consequence relations and show how they are related to existing logics, including ukasiewicz's three-valued logic, Kleene's three-valued logic, Anderson and Belnap's relevant entailments, Priest's "Logic of Paradox", and the first-degree fragment of the Dunn-McCall system "R-mingle". None of these systems have nested implications, and I investigate (...) twelve natural extensions containing nested implications, all of which can be viewed as coming from natural variations on Kripke's semantics for intuitionistic logic. Many of these logics exist antecedently in the literature, in particular Nelson 's "constructible falsity". (shrink)

Most of the logics of grounding that have so far been proposed contain grounding axioms, or grounding rules, for the connectives of conjunction, disjunction and negation, but little attention has been dedicated to the implication connective. The present paper aims at repairing this situation by proposing adequate grounding principles for relevant implication. Because of the interaction between negation and implication, new grounding principles concerning negation will also arise.

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)

Models for the Lambek calculus of syntactic categories surveyed here are based on frames that are in principle of the same type as Kripke frames for intuitionistic logic. These models are extracted from the literature on models for relevant logics, in particular the ternary relationed models introduced in the early seventies. The purpose of this brief survey is to locate some open completeness problems for variants of the Lambek calculus in the context of completeness results based on various types of (...) ternary relational models. (shrink)

This paper provides an interpretation of the Routley-Meyer semantics for a weak negation-free relevant logic using Israel and Perry's theory of information. In particular, Routley and Meyer's ternary accessibility relation is given an interpretation in information-theoretic terms.

This paper considers the question: what becomes of the notion of a logic as a way of codifying valid arguments when the customary assumption is dropped that the premisses and conclusions of these arguments are statements from some single language? An elegant treatment of the notion of a logic, when this assumption is in force, is that provided by Dana Scott's theory of consequence relations; this treatment is appropriately generalized in the present paper to the case where we do not (...) make this assumption of linguistic homogeneity. Several applications of the resulting concept of a heterogeneous logic are suggested, but the main emphasis is on the formal development. One topic touched on is a certain contrast between the boolean and the intensional sentence-connectives in this more general setting. (shrink)

Models are constructed for a variety of systems of quantified relevance logic with identity. Models are given for systems with different principles governing the transitivity of identity and substitution, and the relative merits of these principles are discussed. The models in this paper are all extensions of the semantics of Fine's Semantics for Quantified Relevance Logic (Journal of Philosophical Logic 17 (1988)).

In this paper we set out a semantics for relevant (counterfactual) conditionals. We combine the Routley-Meyer semantics for relevant logic with a semantics for conditionals based on selection functions. The resulting models characterize a family of conditional logics free from fallacies of relevance, in particular counternecessities and conditionals with necessary consequents receive a non-trivial treatment.

Consider the set of inferences that are acceptable to use in all our theory building endeavors. Call this set of inferences the universal theory building toolkit, or just ’the toolkit’ for short. It is clear that the toolkit is tightly connected to logic in a variety of ways. Beall, for example, has argued that logic just is the toolkit. This paper avoids making a stand on that issue and instead investigates reasons for thinking that, logic or not, the toolkit is (...) substructural. It is presented as a dialogue for the simple reason that it summarizes a range of dialogues on this subject that the author has had with various folks over the past few years. The method I use to investigate the toolkit is inspired in both philosophical and technical details by Alasdair Urquhart’s work on semantics for relevance logics from the early 1970s. (shrink)

Presenting the first comprehensive, in-depth study of hyperintensionality, this book equips readers with the basic tools needed to appreciate some of current and future debates in the philosophy of language, semantics, and metaphysics. After introducing and explaining the major approaches to hyperintensionality found in the literature, the book tackles its systematic connections to normativity and offers some contributions to the current debates. The book offers undergraduate and graduate students an essential introduction to the topic, while also helping professionals in related (...) fields get up to speed on open research-level problems. (shrink)

Mares and Goldblatt, 163–187, 2006) provided an alternative frame semantics for two quantified extensions of the relevant logic R. In this paper, I show how to extend the Mares-Goldblatt frames to accommodate identity. Simpler frames are provided for two zero-order logics en route to the full logic in order to clarify what is needed for identity and substitution, as opposed to quantification. I close with a comparison of this work with the Fine-Mares models for relevant logics with identity and a (...) discussion of constant and variable domains. (shrink)

In the early 1970s, Alasdair Urquhart proposed a semilattice semantics for relevance logic which he provided with an influential informational interpretation. In this article, I propose a BHK-inspired reinterpretation of the semantics which is related to Kit Fine’s truthmaker semantics. I discuss and compare Urquhart’s and Fine’s semantics and show how simple modifications of Urquhart’s semantics can be used to characterize both full propositional intuitionistic logic and Jankov’s logic. I then present (quasi-)relevant companions for both of these systems. Finally, I (...) provide sound and complete labelled sequent calculi for all of the systems discussed. (shrink)