We present a new frame semantics for positive relevant and substructural propositional logics. This frame semantics is both a generalisation of Routley–Meyer ternary frames and a simplification of them. The key innovation of this semantics is the use of a single accessibility relation to relate collections of points to points. Different logics are modeled by varying the kinds of collections used: they can be sets, multisets, lists or trees. We show that collection frames on trees are sound and complete for (...) the basic positive distributive substructural logic $\mathsf {B}^+$, that collection frames on multisets are sound and complete for $\mathsf {RW}^+$ (the relevant logic $\mathsf {R}^+$, without contraction, or equivalently, positive multiplicative and additive linear logic with distribution for the additive connectives), and that collection frames on sets are sound for the positive relevant logic $\mathsf {R}^+$. The completeness of set frames for $\mathsf {R}^+$ is, currently, an open question. (shrink)

Logic is about reasoning, or so the story goes. This thesis looks at the concept of logic, what it is, and what claims of correctness of logics amount to. The concept of logic is not a settled matter, and has not been throughout the history of it as a notion. Tools from conceptual analysis aid in this historical venture. Once the unsettledness of logic is established we see the repercussions in current debates in the philosophy of logic. Much of the (...) battle over the ‘one true logic’ is conceptually talking past each other. The theory of logics-as-formalizations is presented as a conceptually open theory of logic which is Carnapian in flavour and grounding. Rudolf Carnap’s notions surrounding ‘external’ and ‘pseudo-questions’ about linguistic frameworks apply to formalizations, thus logics, as well. An account of what formalizations are, a more structured sub-set of modelling, is given to ground the claim that logics are formalizations. Finally, a novel account of correctness, the COFE framework, is developed which allows the notions of logical monism, pluralism and nihilism to be more precisely formulated than they currently are in the discourse. (shrink)

Sentences about logic are often used to show that certain embedding expressions are hyperintensional. Yet it is not clear how to regiment “logic talk” in the object language so that it can be compositionally embedded under such expressions. In this paper, I develop a formal system called hyperlogic that is designed to do just that. I provide a hyperintensional semantics for hyperlogic that doesn’t appeal to logically impossible worlds, as traditionally understood, but instead uses a shiftable parameter that determines the (...) interpretation of the logical connectives. I argue this semantics compares favorably to the more common impossible worlds semantics, which faces difficulties interpreting propositionally quantified logic talk. (shrink)

In his Lectures on the History of Philosophy, Hegel develops a subtle analysis of Megarian paradoxes: the Liar, the Veiled Man and the Sorites. In this paper, we focus on Hegel's interpretation of...

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)

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. (shrink)

This article is a preliminary presentation of conjunctive paraconsistency, the claim that there might be non-explosive true contradictions, but contradictory propositions cannot be considered separately true. In case of true ‘p and not p’, the conjuncts must be held untrue, Simplification fails. The conjunctive approach is dual to non-adjunctive conceptions of inconsistency, informed by the idea that there might be cases in which a proposition is true and its negation is true too, but the conjunction is untrue, Adjunction fails. While (...) non-adjunctivism is a well-known option, the other view is not so much studied nowadays, but it was not unknown in the tradition, and there are some positive suggestions, in recent literature, that the position is plausible and deserves to be developed. The article compares conjunctivism, non-adjunctivism and dialetheism, then focuses on some possible justifications, costs and benefits of the conjunctive view. (shrink)

We propose a new account of indicative conditionals, giving acceptability and logical closure conditions for them. We start from Adams’ Thesis: the claim that the acceptability of a simple indicative equals the corresponding conditional probability. The Thesis is widely endorsed, but arguably false and refuted by empirical research. To fix it, we submit, we need a relevance constraint: we accept a simple conditional 'If φ, then ψ' to the extent that (i) the conditional probability p(ψ|φ) is high, provided that (ii) (...) φ is relevant for ψ. How (i) should work is well-understood. It is (ii) that holds the key to improve our understanding of conditionals. Our account has (i) a probabilistic component, using Popper functions; (ii) a relevance component, given via an algebraic structure of topics or subject matters. We present a probabilistic logic for simple indicatives, and argue that its (in)validities are both theoretically desirable and in line with empirical results on how people reason with conditionals. (shrink)

An overview of hyperintensionality is provided. Hyperintensional languages have expressions with meanings that are more fine-grained than necessary equivalence. That is, the expressions may necessarily co-apply and yet be distinct in meaning. Adequately accounting for theories cast in hyperintensional languages is important in the philosophy of language; the philosophy of mind; metaphysics; and elsewhere. This entry presents a number of areas in which hyperintensionality is important; a range of approaches to theorising about hyperintensional matters; and a range of debates that (...) attention to hyperintensional constructions has generated. (shrink)

A. J. Ayer’s empiricist criterion of meaning was supposed to have sorted all statements into nonsense on the one hand, and tautologies or genuinely factual statements on the other. Unfortunately for Ayer, it follows from classical logic that his criterion is trivial—it classifies all statements as either tautologies or genuinely factual, but none as nonsense. However, in this paper, I argue that Ayer’s criterion of meaning can be defended from classical proofs of its triviality by the adoption of a relevant (...) logic—an idea which is motivated because, according to Ayer, the genuinely factual statements are those which observation is relevant to. (shrink)

Half a century later, a Dretskean stance on epistemic closure remains a minority view. Why? Mainly because critics have successfully poked holes in the epistemologies on which closure fails. However, none of the familiar pro-closure moves works against the counterexamples on display here. It is argued that these counterexamples pose the following dilemma: either accept that epistemic closure principles are false, and steal the thunder from those who attack classical logic on the basis of similarly problematic cases—specifically, relevance logicians and (...) like-minded philosophers—or stick with closure and surrender to relevantist claims of failure in truth-preservation aimed at classical rules of inference. Classicist closure advocates find the promise of a way out of the dilemma in the works of Roy Sorensen and John Hawthorne. The paper argues against their pro-closure move and renews Robert Audi’s call for a theory of closure-failure. (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)

Call an explanation in which a non-mathematical fact is explained—in part or in whole—by mathematical facts: an extra-mathematical explanation. Such explanations have attracted a great deal of interest recently in arguments over mathematical realism. In this article, a theory of extra-mathematical explanation is developed. The theory is modelled on a deductive-nomological theory of scientific explanation. A basic DN account of extra-mathematical explanation is proposed and then redeveloped in the light of two difficulties that the basic theory faces. The final view (...) appeals to relevance logic and uses resources in information theory to understand the explanatory relationship between mathematical and physical facts. 1Introduction2Anchoring3The Basic Deductive-Mathematical Account4The Genuineness Problem5Irrelevance6Relevance and Information7Objections and Replies 7.1Against relevance logic7.2Too epistemic7.3Informational containment8Conclusion. (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 this paper, a critical discussion is made of the role of entailments in the so-called New Paradigm of psychology of reasoning based on Bayesian models of rationality (Elqayam & Over, 2013). It is argued that assessments of probabilistic coherence cannot stand on their own, but that they need to be integrated with empirical studies of intuitive entailment judgments. This need is motivated not just by the requirements of probability theory itself, but also by a need to enhance the interdisciplinary (...) integration of the psychology of reasoning with formal semantics in linguistics. The constructive goal of the paper is to introduce a new experimental paradigm, called the Dialogical Entailment task, to supplement current trends in the psychology of reasoning towards investigating knowledge-rich, social reasoning under uncertainty (Oaksford and Chater, 2019). As a case study, this experimental paradigm is applied to reasoning with conditionals and negation operators (e.g. CEM, wide and narrow negation). As part of the investigation, participants’ entailment judgments are evaluated against their probability evaluations to assess participants’ cross-task consistency over two experimental sessions. (shrink)

There is a strong case to be made for thinking that an obscure logic, KR, is better than classical logic and better than any relevant logic. The argument for KR over relevant logics is that KR counts disjunctive syllogism valid, and this is the biggest complaint about relevant logics. The argument for KR over classical logic depends on the normativity of logic and the paradoxes of implication. The paradoxes of implication are taken by relevant logicians to justify relevant logic, but (...) considerations on the normativity of logic show that only some of the paradoxes of implication are genuine. KR avoids all the genuine paradoxes of implication, unlike classical logic. Overall, KR avoids the genuine paradoxes of implication and avoids the major objection to relevant logics. This combination of features provides strong reason to give KR a place in the conversation about the right logic. (shrink)

This paper presents a semantical analysis of the Weak Kleene Logics Kw3 and PWK from the tradition of Bochvar and Halldén. These are three-valued logics in which a formula takes the third value if at least one of its components does. The paper establishes two main results: a characterisation result for the relation of logical con- sequence in PWK – that is, we individuate necessary and sufficient conditions for a set.

Conditional logics were originally developed for the purpose of modeling intuitively correct modes of reasoning involving conditional—especially counterfactual—expressions in natural language. While the debate over the logic of conditionals is as old as propositional logic, it was the development of worlds semantics for modal logic in the past century that catalyzed the rapid maturation of the field. Moreover, like modal logic, conditional logic has subsequently found a wide array of uses, from the traditional (e.g. counterfactuals) to the exotic (e.g. conditional (...) obligation). Despite the close connections between conditional and modal logic, both the technical development and philosophical exploitation of the latter has outstripped that of the former, with the result that noticeable lacunae exist in the literature on conditional logic. My dissertation addresses a number of these underdeveloped frontiers, producing new technical insights and philosophical applications. -/- I contribute to the solution of a problem posed by Priest of finding sound and complete labeled tableaux for systems of conditional logic from Lewis' V-family. To develop these tableaux, I draw on previous work on labeled tableaux for modal and conditional logic; errors and shortcomings in recent work on this problem are identified and corrected. While modal logic has by now been thoroughly studied in non-classical contexts, e.g. intuitionistic and relevant logic, the literature on conditional logic is still overwhelmingly classical. Another contribution of my dissertation is a thorough analysis of intuitionistic conditional logic, in which I utilize both algebraic and worlds semantics, and investigate how several novel embedding results might shed light on the philosophical interpretation of both intuitionistic logic and conditional logic extensions thereof. -/- My dissertation examines deontic and connexive conditional logic as well as the underappreciated history of connexive notions in the analysis of conditional obligation. The possibility of interpreting deontic modal logics in such systems (via embedding results) serves as an important theoretical guide. A philosophically motivated proscription on impossible obligations is shown to correspond to, and justify, certain (weak) connexive theses. Finally, I contribute to the intensifying debate over counterpossibles, counterfactuals with impossible antecedents, and take—in contrast to Lewis and Williamson—a non-vacuous line. Thus, in my view, a counterpossible like "If there had been a counterexample to the law of the excluded middle, Brouwer would not have been vindicated" is false, not (vacuously) true, although it has an impossible antecedent. I exploit impossible (non-normal) worlds—originally developed to model non-normal modal logics—to provide non-vacuous semantics for counterpossibles. I buttress the case for non-vacuous semantics by making recourse to both novel technical results and theoretical considerations. (shrink)

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 and truth value gluts. Simple and natural semantic rules for negation and the conditional operator are formulated based on an incompatibility relation and a partial 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 intuitionistic 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. (shrink)

Neste artigo trato da questão sobre o que torna uma dedução confiável. Uma resposta satisfatória a tal questão nos ajudaria a entender como dedução pode expandir ou gerar conhecimento. Eu exploro duas respostas a tal questão. A primeira faz uso da noção de acarretamento lógico-formal, enquanto que a segunda faz uso da noção de acarretamento metafísico. A última é superior à primeira, pois nos permite explicar a confiabilidade de uma classe mais ampla de deduções.

This work contributes to the theory of judgement aggregation by discussing a number of significant non-classical logics. After adapting the standard framework of judgement 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 judgement aggregation in non-classical logics is that they offer a number of modelling choices to represent agents’ reasoning in aggregation problems. By studying judgement 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. (shrink)

I want to model a finite, fallible cognitive agent who imagines that p in the sense of mentally representing a scenario—a configuration of objects and properties—correctly described by p. I propose to capture imagination, so understood, via variably strict world quantifiers, in a modal framework including both possible and so-called impossible worlds. The latter secure lack of classical logical closure for the relevant mental states, while the variability of strictness captures how the agent imports information from actuality in the imagined (...) non-actual scenarios. Imagination turns out to be highly hyperintensional, but not logically anarchic. Section 1 sets the stage and impossible worlds are quickly introduced in Sect. 2. Section 3 proposes to model imagination via variably strict world quantifiers. Section 4 introduces the formal semantics. Section 5 argues that imagination has a minimal mereological structure validating some logical inferences. Section 6 deals with how imagination under-determines the represented contents. Section 7 proposes additional constraints on the semantics, validating further inferences. Section 8 describes some welcome invalidities. Section 9 examines the effects of importing false beliefs into the imagined scenarios. Finally, Sect. 10 hints at possible developments of the theory in the direction of two-dimensional semantics. (shrink)

Modern belief revision theory is based to a large extent on partial meet contraction that was introduced in the seminal article by Carlos Alchourrón, Peter Gärdenfors, and David Makinson that appeared in 1985. In the same year, Alchourrón and Makinson published a significantly different approach to the same problem, called safe contraction. Since then, safe contraction has received much less attention than partial meet contraction. The present paper summarizes the current state of knowledge on safe contraction, provides some new results (...) and offers a list of unsolved problems that are in need of investigation. (shrink)

This paper sets out a philosophical interpretation of the model theory of Mares and Goldblatt (The Journal of Symbolic Logic 71, 2006). This interpretation distinguishes between truth conditions and information conditions. Whereas the usual Tarskian truth condition holds for universally quantified statements, their information condition is quite different. The information condition utilizes general propositions . The present paper gives a philosophical explanation of general propositions and argues that these are needed to give an adequate theory of general information.

In Saving Truth from Paradox, Hartry Field presents and defends a theory of truth with a new conditional. In this paper, I present two criticisms of this theory, one concerning its assessments of validity and one concerning its treatment of truth-preservation claims. One way of adjusting the theory adequately responds to the truth-preservation criticism, at the cost of making the validity criticism worse. I show that in a restricted setting, Field has a way to respond to the validity criticism. I (...) close with some general considerations on the use of revision-theoretic methods in theories of truth. (shrink)

The pluralist sheds the more traditional ideas of truth and ontology. This is dangerous, because it threatens instability of the theory. To lend stability to his philosophy, the pluralist trades truth and ontology for rigour and other ‘fixtures’. Fixtures are the steady goal posts. They are the parts of a theory that stay fixed across a pair of theories, and allow us to make translations and comparisons. They can ultimately be moved, but we tend to keep them fixed temporarily. Apart (...) from considering rigour of proof as a fixture, I discuss fixed models, invariant notions and fixed information about objects across theories. There are other fixtures, but it is enough to start with these. (shrink)

An organizing theme of the dissertation is the issue of how to make philosophical theories useful for scientific purposes. An argument for the contention is presented that it doesn’t suffice merely to theoretically motivate one’s theories, and make them compatible with existing data, but that philosophers having this aim should ideally contribute to identifying unique and hard to vary predictions of their theories. This methodological recommendation is applied to the ranking-theoretic approach to conditionals, which emphasizes the epistemic relevance and the (...) expression of reason relations as part of the semantics of the natural language conditional. As a first step, this approach is theoretically motivated in a comparative discussion of other alternatives in psychology of reasoning, like the suppositional theory of conditionals, and novel approaches to the problems of compositionality and accounting for the objective purport of indicative conditionals are presented. In a second step, a formal model is formulated, which allows us to derive quantitative predictions from the ranking-theoretic approach, and it is investigated which novel avenues of empirical research that this model opens up for. Finally, a treatment is given of the problem of logical omniscience as it concerns the issue of whether ranking theory (and other similar approaches) makes too idealized assumptions about rationality to allow for interesting applications in psychology of reasoning. Building on the work of Robert Brandom, a novel solution to this problem is presented, which both opens up for new perspectives in psychology of reasoning and appears to be capable of satisfying a range of constraints on bridge principles between logic and norms of reasoning, which would otherwise stand in a tension. (shrink)

I present an approach to our conceiving absolute impossibilities—things which obtain at no possible world—in terms of ceteris paribus intentional operators: variably restricted quantifiers on possible and impossible worlds based on world similarity. The explicit content of a representation plays a role similar in some respects to the one of a ceteris paribus conditional antecedent. I discuss how such operators invalidate logical closure for conceivability, and how similarity works when impossible worlds are around. Unlike what happens with ceteris paribus counterfactual (...) conditionals, the closest worlds are relevantly closest belief-worlds: closest to how things are believed to be, rather than to how they are. Also, closeness takes into account apriority and the opacity of intentional contexts. (shrink)

Our beliefs utilize various sources: perception, memory, induction, etc. We trust these sources to provide reliable information about the world around us. My dissertation investigates how this trust could be justified. Chapter one introduces background material. I argue that justification rather than knowledge is of primary epistemological importance, discuss the internalism/externalism debate, and introduce an evidentialist thesis that provides a starting point/framework for epistemological theorizing. Chapter two introduces a puzzle concerning justification. Can a belief source provide justification absent prior justification (...) for believing it's reliable? Any answer appears to either make justifying the reliability of a source intellectually unsatisfying or all together impossible. Chapter three considers and rejects a plethora of proposed solutions to our puzzle. Investigating these solutions illustrates the need to further investigate evidence, evidence possession, and evidential support. Chapter four discusses the metaphysics of evidence. I argue that evidence always consists of a set of facts and that fact-proposition pairs stand in confirmation relations isomorphic to those holding between pairs of propositions. Chapter five argues that justification requires what I call actually connected possession of supporting evidence: a subject must be aware of supporting evidence and of the support relation itself. Chapter six argues that the relation constitutive of a set of facts being justificatory evidence is a sui generis and irreducible relation that is knowable a priori. Chapter seven begins by showing how Richard Fumerton's acquaintance theory meets the constraints on a theory of justification laid down in previous chapters. I modify the theory so as to: make room for fallible foundational justification, and allow inferential justification absent higher-order beliefs about evidential connections. Chapter eight applies the developed theory of justification to our initial puzzle. I show how my modified acquaintance theory is in a unique position to vindicate the idea that necessarily a source provides a person with justification only if she is aware of evidence for the reliability of that source. However, this awareness of evidence for a source's reliability falls short of a justified belief and thereby avoids impalement from our dilemma's skeptical horn. (shrink)

The system R, or more precisely the pure implicational fragment R›, is considered by the relevance logicians as the most important. The another central system of relevance logic has been the logic E of entailment that was supposed to capture strict relevant implication. The next system of relevance logic is RM or R-mingle. The question is whether adding mingle axiom to R› yields the pure implicational fragment RM› of the system? As concerns the weak systems there are at least two (...) approaches to the problem. First of all, it is possible to restrict a validity of some theorems. In another approach we can investigate even weaker logics which have no theorems and are characterized only by rules of deducibility. (shrink)

I provide an interpretation of Wittgenstein's much criticized remarks on Gödel's First Incompleteness Theorem in the light of paraconsistent arithmetics: in taking Gödel's proof as a paradoxical derivation, Wittgenstein was right, given his deliberate rejection of the standard distinction between theory and metatheory. The reasoning behind the proof of the truth of the Gödel sentence is then performed within the formal system itself, which turns out to be inconsistent. I show that the models of paraconsistent arithmetics (obtained via the Meyer-Mortensen (...) collapsing filter on the standard model) match with many intuitions underlying Wittgensteins philosophy of mathematics, such as its strict finitism and the insistence on the decidability of any meaningful mathematical question. (shrink)

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” in- ferences 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. (shrink)

This paper first shows that some versions of the logic R of Relevance do not satisfy the relevance principle introduced by Anderson and Belnap, the principle of which is generally accepted as the principle for relevance. After considering several possible (but defective) improvements of the relevance principle, this paper presents a new relevance principle for (three versions of) R, and explains why this principle is better than the original and others.

Counterpossibles are counterfactuals with necessarily false antecedents. The problem of counterpossibles is easiest to state within the "nearest possible world" framework for counterfactuals: on this approach, a counterfactual is true (roughly) when the consequent is true in the "nearest" possible world where the antecedent is true. Since counterpossibles have necessarily false antecedents, there is no possible world where the antecedent is true. On the approach favored by Lewis, Stalnaker, Williamson, and others, counterpossibles are all trivially true. I introduce several arguments (...) against the trivial approach. First, it is counter-intuitive to think that all counterpossibles are true. Second, if all counterpossibles were true, then we could not make sense of their use in logical, philosophical, or mathematical arguments. Making sense of the role of sentences like these requires that they not have vacuous truth conditions. The account of counterpossibles I ultimately favor is an extension of the "nearest possible world" semantics discussed above. The Lewis/Stalnaker account is supplemented with the addition of impossible worlds, and the nearness metric is extended to range over these impossible worlds as well as possible worlds. Thus, a counterfactual is true when its consequent is true in the nearest world where the antecedent is true; if the counterfactual's antecedent is impossible, then the nearest world in question will be an impossible world. Once the framework of impossible worlds and similarity is in place, we can put it to use in the analysis of other philosophical phenomena. I examine one proposal that makes use of a theory of counterpossibles to develop an analysis of the notion of metaphysical dependence. (shrink)

A logic is called 'paraconsistent' if it rejects the rule called 'ex contradictione quodlibet', according to which any conclusion follows from inconsistent premises. While logicians have proposed many technically developed paraconsistent logical systems and contemporary philosophers like Graham Priest have advanced the view that some contradictions can be true, and advocated a paraconsistent logic to deal with them, until recent times these systems have been little understood by philosophers. This book presents a comprehensive overview on paraconsistent logical systems to change (...) this situation. The book includes almost every major author currently working in the field. The papers are on the cutting edge of the literature some of which discuss current debates and others present important new ideas. The editors have avoided papers about technical details of paraconsistent logic, but instead concentrated upon works that discuss more 'big picture' ideas. Different treatments of paradoxes takes centre stage in many of the papers, but also there are several papers on how to interpret paraconistent logic and some on how it can be applied to philosophy of mathematics, the philosophy of language, and metaphysics. (shrink)

Lewisian Genuine Realism about possible worlds is often deemed unable to accommodate impossible worlds and reap the benefits that these bestow to rival theories. This thesis explores two alternative extensions of GR into the terrain of impossible worlds. It is divided in six chapters. Chapter I outlines Lewis’ theory, the motivations for impossible worlds, and the central problem that such worlds present for GR: How can GR even understand the notion of an impossible world, given Lewis’ reductive theoretical framework? Since (...) the desideratum is to incorporate impossible worlds into GR without compromising Lewis’ reductive analysis of modality, Chapter II defends that analysis against objections. The rest of the thesis is devoted to incorporating impossible worlds into GR. Chapter III explores GR-friendly impossible worlds in the form of set-theoretic constructions out of genuine possibilia. Then, Chapters IV-VI venture into concrete impossible worlds. Chapter IV addresses Lewis’ objection against such worlds, to the effect that contradictions true at impossible worlds amount to true contradictions tout court. I argue that even if so, the relevant contradictions are only ever about the non-actual, and that Lewis’ argument relies on a premise that cannot be nonquestion- beggingly upheld in the face of genuine impossible worlds in any case. Chapter V proposes that Lewis’ reductive analysis can be preserved, even in the face of genuine impossibilia, if we differentiate the impossible from the possible by means of accessibility relations, understood non-modally in terms of similarity. Finally, Chapter VI counters objections to the effect that there are certain impossibilities, formulated in Lewis’ theoretical language, which genuine impossibilia should, but cannot, represent. I conclude that Genuine Realism is still very much in the running when the discussion turns to impossible worlds. (shrink)

Tarski’s conceptual analysis of the notion of logical consequence is one of the pinnacles of the process of defining the metamathematical foundations of mathematics in the tradition of his predecessors Euclid, Frege, Russell and Hilbert, and his contemporaries Carnap, Gödel, Gentzen and Turing. However, he also notes that in defining the concept of consequence “efforts were made to adhere to the common usage of the language of every day life.” This paper addresses the issue of what relationship Tarski’s analysis, and (...) Béziau’s further generalization of it in universal logic, have to reasoning in the everyday lives of ordinary people from the cognitive processes of children through to those of specialists in the empirical and deductive sciences. It surveys a selection of relevant research in a range of disciplines providing theoretical and empirical studies of human reasoning, discusses the value of adopting a universal logic perspective, answers the questions posed in the call for this special issue, and suggests some specific research challenges. (shrink)

In "General Information in Relevant Logic" (Synthese 167, 2009), the semantics for relevant logic is interpreted in terms of objective information. Objective information is potential data that is available in an environment. This paper explores the notion of objective information further. The concept of availability in an environment is developed and used as a foundation for the semantics, in particular, as a basis for the understanding of the information that is expressed by relevant implication. It is also used to understand (...) the nature of misinformation. A form of relevant logic—called "LOI" for "logic of objective information"—is presented and the relationship between the justification of its proof theory and the semantics is discussed. This relationship is rather reciprocal. Intuitive features of the logic are used to interpret and justify aspects of the model theory and intuitive aspects of the model theory are used to interpret and justify features of the logic. Information conditions are presented for the connectives and the way that they fit into the theory of information is discussed. (shrink)

What is logical relevance? Anderson and Belnap say that the “modern classical tradition [,] stemming from Frege and Whitehead-Russell, gave no consideration whatsoever to the classical notion of relevance.” But just what is this classical notion? I argue that the relevance tradition is implicitly most deeply concerned with the containment of truth-grounds, less deeply with the containment of classes, and least of all with variable sharing in the Anderson–Belnap manner. Thus modern classical logicians such as Peirce, Frege, Russell, Wittgenstein, and (...) Quine are implicit relevantists on the deepest level. In showing this, I reunite two fields of logic which, strangely from the traditional point of view, have become basically separated from each other: relevance logic and diagram logic. I argue that there are two main concepts of relevance, intensional and extensional. The first is that of the relevantists, who overlook the presence of the second in modern classical logic. The second is the concept of truth-ground containment as following from in Wittgenstein’s Tractatus. I show that this second concept belongs to the diagram tradition of showing that the premisses contain the conclusion by the fact that the conclusion is diagrammed in the very act of diagramming the premisses. I argue that the extensional concept is primary, with at least five usable modern classical filters or constraints and indefinitely many secondary intensional filters or constraints. For the extensional concept is the genus of deductive relevance, and the filters define species. Also following the Tractatus, deductive relevance, or full truth-ground containment, is the limit of inductive relevance, or partial truth-ground containment. Purely extensional inductive or partial relevance has its filters or species too. Thus extensional relevance is more properly a universal concept of relevance or summum genus with modern classical deductive logic, relevantist deductive logic, and inductive logic as its three main domains. (shrink)

It is a venerable slogan due to David Hume, and inherited by the empiricist tradition, that the impossible cannot be believed, or even conceived. In Positivismus und Realismus, Moritz Schlick claimed that, while the merely practically impossible is still conceivable, the logically impossible, such as an explicit inconsistency, is simply unthinkable. -/- An opposite philosophical tradition, however, maintains that inconsistencies and logical impossibilities are thinkable, and sometimes believable, too. In the Science of Logic, Hegel already complained against “one of the (...) fundamental prejudices of logic as hitherto understood”, namely that “the contradictory cannot be imagined or thought” (Hegel 1931: 430). Our representational capabilities are not limited to the possible, for we appear to be able to imagine and describe also impossibilities — perhaps without being aware that they are impossible. -/- Such impossibilities and inconsistencies are what this entry is about... (shrink)

The epistemology of modality is gradually coming to play a central role in general discussions about modality. This paper is a contribution in this direction, in particular I draw a comparison between Lewis’s Modal realism and Timothy Williamson’s recent account of modality in terms of counterfactual thinking. In order to have criteria of evaluation, I also formulate four requirements which are supposed to be met by any theory of modality to be epistemologically adequate.

Up to now theories of semantic information have implicitly relied on logical monism, or the view that there is one true logic. The latter position has been explicitly challenged by logical pluralists. Adopting an unbiased attitude in the philosophy of information, we take a suggestion from Beall and Restall at heart and exploit logical pluralism to recognise another kind of pluralism. The latter is called informational pluralism, a thesis whose implications for a theory of semantic information we explore.

I discuss Smokrović’s work on the normativity of logic (Smokrović 2017, Smokrović 2018). I agree that the classical formal logic is not an adequate model for real-life reasoning. But I present some doubts about his notion of deductive logic and his proposal to model such reasoning in non-monotonic logic. No branch of formal logic by itself is likely to capture real-life inferential links (reasoned-inference). I use the logic of relevance as my case study and extend the pessimistic morals to modern (...) systems of non-classical logic. Finally, I propose a more lax conception of normativty: there is a connection between logical assessment in the broad sense (as sanctioned by the notion of cogency) and the evaluation and criticism of reasoning. (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)

What does it mean that some proposition follows from others? The standard way of spelling out the notion proceeds in modal terms: x follows from y iff necessarily, if y is true, so is x. But although this yields a useful and manageable account of consequence, it fails to capture certain aspects of our pre-theoretical understanding of consequence. In this paper, an alternative notion of logical consequence, based on the idea of grounding, is developed.