Relevance Logic

Relevance and irrelevance, it is argued, is constitutive to our access to “information objects” on three interconnected levels: (1) access to the information object itself, (2) the information gained from it, (3) the use of that information. Relevance selectively shapes our experience and action, but the “irrelevance” of what is left out is not simply the opposite or absence of relevance. The complex relation between relevance and irrelevance expresses itself in different shades of knowledge and ignorance, and in a fuzzy border between information we do not want to access and information we cannot access. This implies both chances and risks for communication as a process of producing and exchanging information objects. In a second step, previous research on relevance and irrelevance is sketched with respect to different traditions and approaches: (1) Alfred Schutz and Aron Gurwitsch; (2) Paul Grice, Dan Sperber and Deirdre Wilson; (3) library and information science; (4) signs and language; and (5) epistemology and logic. Finally, the role of the word “relevance”, which is not found in all languages, is briefly considered after distinguishing the explicit reflection of relevance from the constitutive role of relevance, which often remains implicit.

How to say no less, no more about conditional than what is needed? From a logical analysis of necessary and sufficient conditions, we argue that a proper account of conditional can be obtained by extending the logical notation of Frege's ideography. From a dialogical redefinition of truth-values as moves in a game, it becomes possible to characterize the logical meaning of " If " , and only " If ". That is: by getting rid of the paradoxes of material implication, whilst showing the bivalent roots of conditional as a speech-act based on affirmations and rejections. Finally, the two main inference rules for conditional, viz. Modus Ponens and Modus Tollens, are reassessed in this algebraic and game-theoretical light.

This course expands on material learned in Logic I: classical and modal logic. Where the first course introduced classical logic along with its modal expansions, the logics learned in this course are more naturally thought of as rivals of classical logics: all of them deny some of the valid consequences of classical logic, and thus imply that classical logic gets it wrong in certain cases. The course aims to introduce the student to the philosophical issues involved in these logics, while instilling technical proficiency in each of them.

We need to understand the impossible. Francesco Berto and Mark Jago start by considering what the concepts of meaning, information, knowledge, belief, fiction, conditionality, and counterfactual supposition have in common. They are all concepts which divide the world up more finely than logic does. Logically equivalent sentences may carry different meanings and information and may differ in how they're believed. Fictions can be inconsistent yet meaningful. We can suppose impossible things without collapsing into total incoherence. Yet for the leading philosophical theories of meaning, these phenomena are an unfathomable mystery. To understand these concepts, we need a metaphysical, logical, and conceptual grasp of situations that could not possibly exist: Impossible Worlds. This book discusses the metaphysics of impossible worlds and applies the concept to a range of central topics and open issues in logic, semantics, and philosophy. It considers problems in the logic of knowledge, the meaning of alternative logics, models of imagination and mental simulation, the theory of information, truth in fiction, the meaning of conditional statements, and reasoning about the impossible. In all these cases, impossible worlds have an essential role to play.

A dialetheia is a sentence, A, such that both it and its negation, ¬A, are true (we shall talk of sentences throughout this entry; but one could run the definition in terms of propositions, statements, or whatever one takes as her favourite truth-bearer: this would make little difference in the context). Assuming the fairly uncontroversial view that falsity just is the truth of negation, it can equally be claimed that a dialetheia is a sentence which is both true and false...

This article introduces, studies, and applies a new system of logic which is called 'HYPE'. In HYPE, formulas are evaluated at states that may exhibit truth value gaps (partiality) and truth value gluts (overdeterminedness). Simple and natural semantic rules for negation and the conditional operator are formulated based on an incompatibility relation and a fusion operation on states. The semantics is worked out in formal and philosophical detail, and a sound and complete axiomatization is provided both for the propositional and the predicate logic of the system. The propositional logic of HYPE is shown to contain first-degree entailment, to have the Finite Model Property, to be decidable, to have the Disjunction Property, and to extend intuitionis-tic propositional logic conservatively when intuitionistic negation is defined appropriately by HYPE's logical connectives. Furthermore, HYPE's first-order logic is a conservative extension of intuitionistic logic with the Constant Domain Axiom, when intuitionistic negation is again defined appropriately. The system allows for simple model constructions and intuitive Euler-Venn-like diagrams, and its logical structure matches structures well-known from ordinary mathematics, such as from optimization theory, combinatorics, and graph theory. HYPE may also be used as a general logical framework in which different systems of logic can be studied, compared, and combined. In particular, HYPE is found to relate in interesting ways to classical logic and various systems of relevance and paraconsistent logic, many-valued logic, and truthmaker semantics. On the philosophical side, if used as a logic for theories of type-free truth, HYPE is shown to address semantic paradoxes such as the Liar Paradox by extending non-classical fixed-point interpretations of truth by a conditional as well-behaved as that of intuitionistic logic. Finally, HYPE may be used as a background system for modal operators that create hyperintensional contexts, though the details of this application need to be left to follow-up work.

Wittgenstein’s discussion of the colour exclusion problem in ‘Some Remarks on Logical Form’ led to a series of papers by Lewy, von Wright and others on Entailment. Central to this stream of thought was concern with C.I. Lewis’ discussion of the so-called ‘Paradoxes of Strict Implication’.  Lewy and the rest were concerned to avoid Lewis’ entailments to necessary consequents, and from impossible antecedents, for instance.  Later, Anderson and Belnap took up the issue of Entailment in a different way.  Indeed, it was not until Anderson and Belnap’s work that a thorough study of the distinctive grammar of remarks like ‘That something is red all over entails that it is not green’ was undertaken. Nevertheless followers of this second train of thought were happy to take ‘entails’ as a sort of propositional connective, although naturally everyone realised that the plain hook of standard propositional logic would not suffice to express the relation, because of the so-called ‘Paradoxes of Material Implication’.  The problem for both of these traditions, however, was that they had no way of formalising ‘that’-clauses as referential phrases, allowing the relation to be formalised in predicate logic terms.  Indeed that would have brought up for them difficult philosophical questions to do with Realism.  But now that we are moving out of the Empiricism of those times it is easier for us to see just what it was that Wittgenstein had in mind.

ABSTRACT: This paper contends that Stoic logic (i.e. Stoic analysis) deserves more attention from contemporary logicians. It sets out how, compared with contemporary propositional calculi, Stoic analysis is closest to methods of backward proof search for Gentzen-inspired substructural sequent logics, as they have been developed in logic programming and structural proof theory, and produces its proof search calculus in tree form. It shows how multiple similarities to Gentzen sequent systems combine with intriguing dissimilarities that may enrich contemporary discussion. Much of Stoic logic appears surprisingly modern: a recursively formulated syntax with some truth-functional propositional operators; analogues to cut rules, axiom schemata and Gentzen's negation-introduction rules; an implicit variable-sharing principle and deliberate rejection of Thinning and avoidance of paradoxes of implication. These latter features mark the system out as a relevance logic, where the absence of duals for its left and right introduction rules puts it in the vicinity of McCall's connexive logic. Methodologically, the choice of meticulously formulated meta-logical rules in lieu of axiom and inference schemata absorbs some structural rules and results in an economical, precise and elegant system that values decidability over completeness.

This paper provides a formal account of non-circular reasoning, i.e. of reasoning in which the conclusion of an argument is not somehow presupposed in its premises.  Martin (along with R. Meyer) had previously shown that the implicational system P-W does not contain as theorems any circularities of the form A -> A.  This result is then extended  to systems that also contain conjunction.

We examine the relationship between the logics of nonsense of Bochvar and Halldén and the containment logics in the neighborhood of William Parry’s AI. We detail two strategies for manufacturing containment logics from nonsense logics—taking either connexive and paraconsistent fragments of such systems—and show how systems determined by these techniques have appeared as Frederick Johnson’s RC and Carlos Oller’s AL. In particular, we prove that Johnson’s system is precisely the intersection of Bochvar’s B_3 and Graham Priest’s non-symmetrized connexive logic and that Oller’s system lies just beneath the intersection of B_3 and Priest’s paraconsistent LP. We conclude by examining Oller’s system in more depth, giving it a characterization in terms of LP and showing that it plays the same role to Harry Deutsch’s paraconsistent containment logic S that Aleksandr Zinov'ev’s S_1 plays with respect to A I.

The analogy between inference and mereological containment goes at least back to Aristotle, whose discussion in the Prior Analytics motivates the validity of the syllogism by way of talk of parts and wholes. On this picture, the application of syllogistic is merely the analysis of concepts, a term that presupposes—through the root ἀνά + λύω —a mereological background.

In the 1930s, such considerations led William T. Parry to attempt to codify this notion of logical containment in his system of analytic implication AI. Parry’s original system AI was later expanded to the system PAI. The hallmark of Parry’s systems—and of what may be thought of as containment logics or Parry systems in general—is a strong relevance property called the ‘Proscriptive Principle’ (PP) described by Parry as the thesis that:

No formula with analytic implication as main relation holds universally if it has a free variable occurring in the consequent but not the antecedent.

This type of proscription is on its face justified, as the presence of a novel parameter in the consequent corresponds to the introduction of new subject matter. The plausibility of the thesis that the content of a statement is related to its subject matter thus appears also to support the validity of the formal principle.

Primarily due to the perception that Parry’s formal systems were intended to accurately model Kant’s notion of an analytic judgment, Parry’s deductive systems—and the suitability of the Proscriptive Principle in general—were met with severe criticism. While Anderson and Belnap argued that Parry’s criterion failed to account for a number of prima facie analytic judgments, others—such as Sylvan and Brady—argued that the utility of the criterion was impeded by its reliance on a ‘syntactical’ device.

But these arguments are restricted to Parry’s work qua exegesis of Kant and fail to take into account the breadth of applications in which the Proscriptive Principle emerges. It is the goal of the present work to explore themes related to deductive systems satisfying one form of the Proscriptive Principle or other, with a special emphasis placed on the rehabilitation of their study to some degree. The structure of the dissertation is as follows:

In Chapter 2, we identify and develop the relationship between Parry-type deductive systems and the field of ‘logics of nonsense.’ Of particular importance is Dmitri Bochvar’s ‘internal’ nonsense logic Σ0, and we observe that two ⊢-Parry subsystems of Σ0 (Harry Deutsch’s Sfde and Frederick Johnson’s RC) can be considered to be the products of particular ‘strategies’ of eliminating problematic inferences from Bochvar’s system.

The material of Chapter 3 considers Kit Fine’s program of state space semantics in the context of Parry logics. Recently, Fine—who had already provided the first intuitive semantics for Parry’s PAI—has offered a formal model of truthmaking (and falsemaking) that provides one of the first natural semantics for Richard B. Angell’s logic of analytic containment AC, itself a ⊢-Parry system. After discussing the relationship between state space semantics and nonsense, we observe that Fabrice Correia’s weaker framework—introduced as a semantics for a containment logic weaker than AC—tacitly endorses an implausible feature of allowing hypernonsensical statements. By modelling Correia’s containment logic within the stronger setting of Fine’s semantics, we are able to retain Correia’s intuitions about factual equivalence without such a commitment. As a further application, we observe that Fine’s setting can resolve some ambiguities in Greg Restall’s own truthmaker semantics.

In Chapter 4, we consider interpretations of disjunction that accord with the characteristic failure of Addition in which the evaluation of a disjunction A ∨ B requires not only the truth of one disjunct, but also that both disjuncts satisfy some further property. In the setting of computation, such an analysis requires the existence of some procedure tasked with ensuring the satisfaction of this property by both disjuncts. This observation leads to a computational analysis of the relationship between Parry logics and logics of nonsense in which the semantic category of ‘nonsense’ is associated with catastrophic faults in computer programs. In this spirit, we examine semantics for several ⊢-Parry logics in terms of the successful execution of certain types of programs and the consequences of extending this analysis to dynamic logic and constructive logic.

Chapter 5 considers these faults in the particular case in which Nuel Belnap’s ‘artificial reasoner’ is unable to retrieve the value assigned to a variable. This leads not only to a natural interpretation of Graham Priest’s semantics for the ⊢-Parry system S⋆fde but also a novel, many-valued semantics for Angell’s AC, completeness of which is proven by establishing a correspondence with Correia’s semantics for AC. These many-valued semantics have the additional benefit of allowing us to apply the material in Chapter 2 to the case of AC to define intensional extensions of AC in the spirit of Parry’s PAI.

One particular instance of the type of disjunction central to Chapter 4 is Melvin Fitting’s cut-down disjunction. Chapter 6 examines cut-down operations in more detail and provides bilattice and trilattice semantics for the ⊢-Parry systems Sfde and AC in the style of Ofer Arieli and Arnon Avron’s logical bilattices. The elegant connection between these systems and logical multilattices supports the fundamentality and naturalness of these logics and, additionally, allows us to extend epistemic interpretation of bilattices in the tradition of artificial intelligence to these systems.

Finally, the correspondence between the present many-valued semantics for AC and those of Correia is revisited in Chapter 7. The technique that plays an essential role in Chapter 4 is used to characterize a wide class of first-degree calculi intermediate between AC and classical logic in Correia’s setting. This correspondence allows the correction of an incorrect characterization of classical logic given by Correia and leads to the question of how to characterize hybrid systems extending Angell’s AC∗. Finally, we consider whether this correspondence aids in providing an interpretation to Correia’s first semantics for AC.

Anderson and Belnap, in their otherwise very thorough grammatical appendix to volume 1 of Entailment, knew that they were treading on shaky ground.  For they wanted to make it philosophically respectable, they said, to ‘confuse’ two things.  Their attempt was to ‘make it philosophically respectable to “confuse” implication or entailment with the conditional’ (Anderson and Belnap 1975, 470).  But there are other cases they needed to consider besides the ones they did which show that they definitely created confusion, thereby failing to make the matter ‘philosophically respectable.’  Clarifying the matter has a number of significant consequences not only for their work on Entailment, but also for the Relevance Logic tradition (see, for example, Read 1988) and standard propositional logic.

Agents require a constant flow, and a high level of processing, of relevant semantic information, in order to interact successfully among themselves and with the environment in which they are embedded. Standard theories of information, however, are silent on the nature of epistemic relevance. In this paper, a subjectivist interpretation of epistemic relevance is developed and defended. It is based on a counterfactual and metatheoretical analysis of the degree of relevance of some semantic information i to an informee/agent a, as a function of the accuracy of i understood as an answer to a query q, given the probability that q might be asked by a. This interpretation of epistemic relevance vindicates a strongly semantic theory of information, according to which semantic information encapsulates truth. It accounts satisfactorily for several important applications and interpretations of the concept of relevant information in a variety of philosophical areas. And it interfaces successfully with current philosophical interpretations of causal and logical relevance.

In March 1615 His Majesty King James the VI of Scotland and I of England participated in a debate concerning the use of logic by dogs. The debate was one of several entertainments provided for the King during a visit to the University of Cambridge. At first glance, this event might seem wholly frivolous. Certainly the King’s enthusiasm for hunting played a part in the choice of topic. But King James was no idle, anti-intellectual prince. He had assembled a hothouse of Protestant theologians to produce the English Bible that still bears his name, and his own collected works were to appear the following year. Moreover, as we shall see, the arguments rehearsed before King James echo down the history of logic, from antiquity to the twenty-first century.

An exact truthmaker for A is a state which, as well as guaranteeing A’s truth, is wholly relevant to it. States with parts irrelevant to whether A is true do not count as exact truthmakers for A. Giving semantics in this way produces a very unusual consequence relation, on which conjunctions do not entail their conjuncts. This feature makes the resulting logic highly unusual. In this paper, we set out formal semantics for exact truthmaking and characterise the resulting notion of entailment, showing that it is compact and decidable. We then investigate the effect of various restrictions on the semantics. We also formulate a sequent-style proof system for exact entailment and give soundness and completeness results.

However broad or vague the notion of connexivity may be, it seems to be similar to the notion of relevance even when relevance and connexive logics have been shown to be incompatible to one another. Relevance logics can be examined by suggesting syntactic relevance principles and inspecting if the theorems of a logic abide to them. In this paper we want to suggest that a similar strategy can be employed with connexive logics. To do so, we will suggest some properties that seem to be hinted at in Nelson's work. Following this strategy will ideally shed some light over the notion of content and will also help make a clear comparison between relevance and connexive logics.

We provide a logical matrix semantics and a Gentzen-style sequent calculus for the first-degree entailments valid in W. T. Parry's logic of Analytic Implication. We achieve the former by introducing a logical matrix closely related to that inducing paracomplete weak Kleene logic, and the latter by presenting a calculus where the initial sequents and the left and right rules for negation are subject to linguistic constraints.

In this paper, a connexive extension of the relevance logic R→ is presented. It is defined by means of a natural deduction system, and a deductively equivalent axiomatic system is presented too. The goal of such an extension is to produce a logic with stronger connection between the antecedent and the consequent of an implication.

We introduce a number of logics to reason about collective propositional attitudes that are defined by means of the majority rule. It is well known that majoritarian aggregation is subject to irrationality, as the results in social choice theory and judgment aggregation show. The proposed logics for modelling collective attitudes are based on a substructural propositional logic that allows for circumventing inconsistent outcomes. Individual and collective propositional attitudes, such as beliefs, desires, obligations, are then modelled by means of minimal modalities to ensure a number of basic principles. In this way, a viable consistent modelling of collective attitudes is obtained.

This work contributes to the theory of judgment aggregation by discussing a number of significant non-classical logics. After adapting the standard framework of judgment aggregation to cope with non-classical logics, we discuss in particular results for the case of Intuitionistic Logic, the Lambek calculus, Linear Logic and Relevant Logics. The motivation for studying judgment aggregation in non-classical logics is that they offer a number of modelling choices to represent agents' reasoning in aggregation problems. By studying judgment aggregation in logics that are weaker than classical logic, we investigate whether some well-known impossibility results, that were tailored for classical logic, still apply to those weak systems.

The work of Arnon Avron and Ofer Arieli has shown a deep relationship between the theory of bilattices and the Belnap-Dunn logic E_fde. This correspondence has been interpreted as evidence that E_fde is "the" logic of bilattices, a consideration reinforced by the work of Yaroslav Shramko and Heinrich Wansing in which E_fde is shown to be similarly entrenched with respect to the theories of trilattices and, more generally, multilattices. In this paper, we export Melvin Fitting's "cut-down" connectives—propositional connectives that "cut down" available evidence—to the case of multilattices and show that two related first degree systems—Harry Deutsch's four-valued S_fde and Richard Angell's AC—emerge just as elegantly and are as intimately connected to the theories of bilattices and trilattices as the Belnap-Dunn logic.

How do philosophical accusations of talking nonsense relate to the layperson’s notions of meaning and meaningfulness?  If one were to explain carefully what philosophical nonsense was supposed to be, would one be greeted with incredulity?  How do people react to being accused of talking nonsense themselves?  Is anything analogous to an illusion of meaning recognised in everyday life?  In dream-reports perhaps?  Or in the utterances of schizophrenics or those under the influence of drugs?  Or in certain jokes and hoaxes?  Are contradictions felt to be meaningless?  I argue that, though it is clear that experimental philosophy could shed some light on these questions, their subject matter creates special difficulties.  First, it is more difficult than usual to formulate questions and produce ‘vignettes’ that do not subtly encourage certain responses at the expense of others.  Second, the fact that philosophical nonsense is a metaphilosophical concept ensures that its investigation is going to be more indirect than would be that of knowledge or intention, for example.

(I am sorry if Part Three seems rather fragmentary - a handful of questions left over from previous sections.  It really needs some kind of conclusion to draw the threads together, but for the moment inspiration eludes me.)

Under a proper translation, the languages of propositional (and quantified relevant logic) with an absurdity constant are characterized as the fragments of first order logic preserved under (world-object) relevant directed bisimulation. Furthermore,  the properties of pointed models axiomatizable by sets of propositional relevant formulas have a purely algebraic characterization. Finally, a form of the interpolation  property holds for the relevant fragment of first order logic.

World semantics for relevant logics include so-called non-normal or impossible worlds providing model-theoretic counterexamples to such irrelevant entailments as (A ∧ ¬A) → B, A → (B∨¬B), or A → (B → B). Some well-known views interpret non-normal worlds as information states. If so, they can plausibly model our ability of conceiving or representing logical impossibilities. The phenomenon is explored by combining a formal setting with philosophical discussion. I take Priest’s basic relevant logic N4 and extend it, on the syntactic side, with a representation operator, (R), and on the semantic side, with particularly anarchic non-normal worlds. This combination easily invalidates unwelcome “logical omniscience” inferences of standard epistemic logic, such as belief-consistency and closure under entailment. Some open questions are then raised on the best strategies to regiment (R) in order to express more vertebrate kinds of conceivability.

Raul Orayen's book _Lógica, significado y ontología_ is a deep study into a range of issues in the philosophy of logic, taking Quine as the main interlocutor. It goes into subjects such as Truth-bearerss, Logical Truth, Validity, Propositions, Quine's Extensionalism. Relevant Logic and disjunctive syllogism, and Castañeda's ontology of Guises.

KEYWORDS:
Raul Orayen, Logic, meaning, ontology, philosophy of logic, Quine, Truth-bearers, Logical Truth, Validity, Propositions, Extensionalism, Relevant Logic, disjunctive syllogism,  Castañeda, Guises.

In this paper we define and develop an algebraic structure associated with the concept of systems of relevant information (SRI), which is a variant of the semilattice semantics proposed by A. Urquhart in [6]. The propositional relevant logic RP is introduced as the syntactical counterpart of the SRI-structures. This logic has a primitive-recursive decision procedure. The idea behind this logic is that of "Relevant Deduction" , in which each premise is a block of information relevant to the conclusion. Finally, we prove that the class of SRI-structures is a sound and complete semantics for the logic RP .

An exact truthmaker for A is a state which, as well as guaranteeing A's truth, is wholly relevant to it. States with parts irrelevant to whether A is true do not count as exact truthmakers for A. Giving semantics in this way produces a very unusual consequence relation, on which conjunctions do not entail their conjuncts. This feature makes the resulting logic highly unusual. In this paper, we set out formal semantics for exact truthmaking and characterise the resulting notion of entailment, showing that it is compact and decidable. We then investigate the effect of various restrictions on the semantics. We also formulate a sequent-style proof system for exact entailment and give soundness and completeness results.

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.

The negative approach to a theory of rationality is primarily concerned with rejecting what is irrational,  which usually is equated with what is inconsistent. The positive approach tries to enhance patterns of reasoning practice. Different sorts of consistency and inconsistency are by no means equivalent. What is anyway required is positive reasoning. So, I espouse the positive view: to reason is to infer, and since inference is relative, so is reasoning. As an approximation to deductive reasoning, I sketch a system of logic close enough to the most widely publicized relevant system of entailment logic. Reasoning as thus implemented is of course somehow artificial. Some part of the remaining task can be afforded by a pragmatic rounding out of purely inferential logic.

KEYWORDS:
theory of rationality, irrationality, inconsistency,  reasoning, inference, relevant logic, entailment logic.

The axiom schema of Permutation may look like an innocent axiom but I want to argue in this paper why it should not be considered relevantly valid. A good reason to suspect of Permutation is the absence of the axiom in most relevance logic systems. To show how Permutation is in fact relevantly invalid, I will first talk about why Permutation may appear to be relevantly valid when we consider the usual relevantist properties. After, I will propose an alternative reading of those properties in which I treat them as specifications over proofs. Finally, I will suggest another relevantist property which will help me show why the axiom of Permutation is relevantly invalid.

ABSTRACT: This paper shows that, for the Hertz–Gentzen Systems of 1933 (without Thinning), extended by a classical rule T1 (from the Stoics) and using certain axioms (also from the Stoics), all derivations are analytic: every cut formula occurs as a subformula in the cut’s conclusion. Since the Stoic cut rules are instances of Gentzen’s Cut rule of 1933, from this we infer the decidability of the propositional logic of the Stoics. We infer the correctness for this logic of a “relevance criterion” and of two “balance criteria”, and hence (in contrast to one of Gentzen’s 1933 results) that a particular derivable sequent has no derivation that is “normal” in the sense that the first premiss of each cut is cut-free. We also infer that Cut is not admissible in the Stoic system, based on the standard Stoic axioms, the T1 rule and the instances of Cut with just two antecedent formulae in the first premiss. (OPEN ACCESS)

It is shown that the classes of Routley-Meyer models which are axiomatizable by a theory in a propositional relevant language with fusion and the Ackermann constant can be characterized by their closure under certain model-theoretic operations involving prime filter extensions, relevant directed bisimulations and disjoint unions.

In this paper I present Parconsistent Weak Kleene (PWK) and
I prove that its conjunction is definable in the Logic of Paradox LP introduced by Graham Priest, thus establishing connections between the two formalisms. I also discuss two different possible readings of PWK-conjunction and propose a relational semantics for it, along the lines of the Routley-Meyer semantics for Relevant Logic.

This paper provides an inferentialist motivation for a logic belonging in the connexive family, by borrowing elements from the bilateralist interpretation for Classical Logic without the Cut rule, proposed by David Ripley. The paper focuses on the relation between inferentialism and relevance, through the exploration of what we call relevant assertion and denial, showing that a connexive system emerges as a symptom of this interesting link. With the present attempt we hope to broaden the available interpretations for connexive logics, showing they can be rightfully motivated in terms of certain relevantist constraints imposed on assertion and denial.

Val Plumwood's 1993 paper, "The politics of reason: towards a feminist logic" (hence-forth POR) attempted to set the stage for what she hoped would begin serious feminist exploration into formal logic-not merely its historical abuses, but, more importantly, its potential uses. This work offers us: (1) a case for there being feminist logic; and (2) a sketch of what it should resemble. The former goal of Plumwood's paper encourages feminist theorists to reject anti-logic feminist views. The paper's latter aim is even more challenging. Plumwood's critique of classical negation (and classical logic) as a logic of domination asks us to recognize that particular logical systems are weapons of oppression. Against anti-logic feminist theorists, Plumwood argues that there are other logics besides classical logic, such as relevant logics, which are suited for feminist theorizing. Some logics may oppress while others may liberate. We provide details about the sources and context for her rejection of classical logic and motivation for promoting relevant logics as feminist. We begin with a discussion of various senses of "feminist logic" in order to position Plumwood's understanding of it with precision. Next, we examine Plumwood's motivation for promoting non-classical logic and connecting it with political ideology, contex-tualizing her ideas within the work of the Canberra Logic Group working in Australia. Unpacking what Plumwood means when she claims that classical logic is a logic of domination and her critique of classical negation requires examination of an overlooked, earlier publication, "Negation and contradiction" (NAC), co-authored with Richard Routley. This paper provides a much fuller account of classical negation and the liberation offered by relevant logics. We attempt to clarify their views through a perspective of "relevant