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)

This contributed volume includes both theoretical research on philosophical logic and its applications in artificial intelligence, mostly employing the concepts and techniques of modal logic. It collects selected papers presented at the Second Asia Workshop on Philosophical Logic, held in Guangzhou, China in 2014, as well as a number of invited papers by specialists in related fields. The contributions represent pioneering philosophical logic research in Asia.

I argue that there are Leibnizian-style cosmological arguments for the existence of God which start from very mild premises which affirm the mere possibility of a principle of sufficient reason. The utilization of such premises gives a great deal of plausibility to such types of argumentation. I spend the majority of the paper defending three major objections to such mild premises viz., a reductio argument from Peter van Inwagen and William Rowe, which proffers and defends the idea that a necessary (...) proposition cannot explain a contingent one. I, then, turn to an amelioration of the Rowe/van Inwagen argument which attempts to appeal to an entailment relation between explanans and explanandum that is fettered out in terms of relevance logic. Subsequent to dispelling with that worry, I tackle objections to the utilization of weak principles of sufficient reason that depend essentially upon agglomerative accounts of explanation. (shrink)

This paper sets out to evaluate the claim that Aristotle’s Assertoric Syllogistic is a relevance logic or shows significant similarities with it. I prepare the grounds for a meaningful comparison by extracting the notion of relevance employed in the most influential work on modern relevance logic, Anderson and Belnap’s Entailment. This notion is characterized by two conditions imposed on the concept of validity: first, that some meaning content is shared between the premises and the conclusion, and second, that the premises (...) of a proof are actually used to derive the conclusion. Turning to Aristotle’s Prior Analytics, I argue that there is evidence that Aristotle’s Assertoric Syllogistic satisfies both conditions. Moreover, Aristotle at one point explicitly addresses the potential harmfulness of syllogisms with unused premises. Here, I argue that Aristotle’s analysis allows for a rejection of such syllogisms on formal grounds established in the foregoing parts of the Prior Analytics. In a final section I consider the view that Aristotle distinguished between validity on the one hand and syllogistic validity on the other. Following this line of reasoning, Aristotle’s logic might not be a relevance logic, since relevance is part of syllogistic validity and not, as modern relevance logic demands, of general validity. I argue that the reasons to reject this view are more compelling than the reasons to accept it and that we can, cautiously, uphold the result that Aristotle’s logic is a relevance logic. (shrink)

In this article the standard philosophical method involving intuition-driven conceptual analysis is challenged in a new way. This orthodox approach to philosophy takes analysanda to be the specifications of the content of concepts in the form of sets of necessary and sufficient conditions. Here it is argued that there is no adequate account of what necessary and sufficient conditions are. So, the targets of applications of the standard philosophical method so understood are not sufficiently well understood for this method to (...) be dependable. (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.

Routley-Meyer type ternary relational semantics are defined for relevant logics including Routley and Meyer’s basic logic B plus the reductio rule and the disjunctive syllogism. Standard relevant logics such as E and R (plus γ ) and Ackermann’s logics of ‘strenge Implikation’ Π and Π ′ are among the logics considered.

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)

Combinator logics are a broad family of substructual logics that are formed by extending the basic relevant logic B with axioms that correspond closely to the reduction rules of proper combinators in combinatory logic. In the Routley-Meyer relational semantics for relevant logic each such combinator logic is characterized by the class of frames that meet a first-order condition that also directly corresponds to the same combinator's reduction rule. A second family of logics is also introduced that extends B with the (...) addition of propositional constants that correspond to combinators. These are characterized by relational frames that meet first-order conditions that reflect the structures of the combinators themselves. (shrink)

The results of this paper extend some of the intimate relations that are known to obtain between combinatory logic and certain substructural logics to establish a general characterization theorem that applies to a very broad family of such logics. In particular, I demonstrate that, for every combinator X, if LX is the logic that results by adding the set of types assigned to X (in an appropriate type assignment system, TAS) as axioms to the basic positive relevant logic B∘T, then (...) LX is sound and complete with respect to the class of frames in the Routley-Meyer relational semantics for relevant and substructural logics that meet a first-order condition that corresponds in a very direct way to the structure of the combinator X itself. (shrink)

Restall set forth a "consecution" calculus in his "An Introduction to Substructural Logics." This is a natural deduction type sequent calculus where the structural rules play an important role. This paper looks at different ways of extending Restall's calculus. It is shown that Restall's weak soundness and completeness result with regards to a Hilbert calculus can be extended to a strong one so as to encompass what Restall calls proofs from assumptions. It is also shown how to extend the calculus (...) so as to validate the metainferential rule of reasoning by cases, as well as certain theory-dependent rules. (shrink)

Dentro del conjunto de las lógicas no clásicas, las lógicas paraconsistentes han suscitado de manera particular el interés de diversos filósofos. Además de las definiciones tradicionales, en los últimos años, se han propuesto nuevas maneras de caracterizar a la paraconsistencia. Lo que tienen en común todas estas definiciones es que alguna forma de la regla o de la metarregla de explosión debe ser rechazada. En este artículo, presentaré dichas definiciones y evaluaré el rol que juegan la negación y la transitividad (...) en ellas. Finalmente, propondré una nueva caracterización de la paraconsistencia, la que llamaré paraconsistencia pura y mostraré que la regla de monotonía juega un rol crucial en todas las presentaciones de dicho concepto. (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)

Introduction to the Routley/Sylvan Issue.

The logic RM and its basic fragments (always with implication) are considered here as entire consequence relations, rather than as sets of theorems. A new observation made here is that the disjunction of RM is definable in terms of its other positive propositional connectives, unlike that of R. The basic fragments of RM therefore fall naturally into two classes, according to whether disjunction is or is not definable. In the equivalent quasivariety semantics of these fragments, which consist of subreducts of (...) Sugihara algebras, this corresponds to a distinction between strong and weak congruence properties. The distinction is explored here. A result of Avron is used to provide a local deduction-detachment theorem for the fragments without disjunction. Together with results of Sobociski, Parks and Meyer (which concern theorems only), this leads to axiomatizations of these entire fragments — not merely their theorems. These axiomatizations then form the basis of a proof that all of the basic fragments of RM with implication are finitely axiomatized consequence relations. (shrink)

The implicational fragment of the logic of relevant implication, $R_\to$ is known to be decidable. We show that the implicational fragment of the logic of ticket entailment, $T_\to$ is decidable. Our proof is based on the consecution calculus that we introduced specifically to solve this 50-year old open problem. We reduce the decidability problem of $T_\to$ to the decidability problem of $R_\to$. The decidability of $T_\to$ is equivalent to the decidability of the inhabitation problem of implicational types by combinators over (...) the base $\{\textsf{B},\textsf{B}',\textsf{I},\textsf{W}\}$. (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)

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.

Paradoxes have played an important role both in philosophy and in mathematics and paradox resolution is an important topic in both fields. Paradox resolution is deeply important because if such resolution cannot be achieved, we are threatened with the charge of debilitating irrationality. This is supposed to be the case for the following reason. Paradoxes consist of jointly contradictory sets of statements that are individually plausible or believable. These facts about paradoxes then give rise to a deeply troubling epistemic problem. (...) Specifically, if one believes all of the constitutive propositions that make up a paradox, then one is apparently committed to belief in every proposition. This is the result of the principle of classical logical known as ex contradictione (sequitur) quodlibetthat anything and everything follows from a contradiction, and the plausible idea that belief is closed under logical or material implication (i.e. the epistemic closure principle). But, it is manifestly and profoundly irrational to believe every proposition and so the presence of even one contradiction in one’s doxa appears to result in what seems to be total irrationality. This problem is the problem of paradox-induced explosion. In this paper it will be argued that in many cases this problem can plausibly be avoided in a purely epistemic manner, without having either to resort to non-classical logics for belief (e.g. paraconsistent logics) or to the denial of the standard closure principle for beliefs. The manner in which this result can be achieved depends on drawing an important distinction between the propositional attitude of belief and the weaker attitude of acceptance such that paradox constituting propositions are accepted but not believed. Paradox-induced explosion is then avoided by noting that while belief may well be closed under material implication or even under logical implication, these sorts of weaker commitments are not subject to closure principles of those sorts. So, this possibility provides us with a less radical way to deal with the existence of paradoxes and it preserves the idea that intelligent agents can actually entertain paradoxes. (shrink)

In 1942 Haskell B. Curry presented what is now called Curry's paradox which can be found in a logic independently of its stand on negation. In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes. In this article the non-classical resolution of Curry’s Paradox and Shaw-Kwei' sparadox without rejection any contraction postulate is proposed. In additional relevant paraconsistent logic C ̌_n^#,1≤n<ω, in fact,provide an effective way of circumventing triviality of da Costa’s paraconsistent Set Theories〖NF〗n^C.

Substructural pluralism about the meaning of logical connectives is best understood as the view that natural language connectives have all (and only) the properties conferred by classical logic, but that particular occurrences of these connectives cannot simultaneously exhibit all these properties. This is just a more sophisticated way of saying that while natural language connectives are ambiguous, they are not so in the way classical logic intends them to be. Since this view is usually framed as a means to resolve (...) paradoxes, little attention is paid to the logical properties of the ambiguous connectives themselves. The present paper sets out to fill this gap. First, I argue that substructural logicians should care about these connectives; next, I describe a consequence relation between a set of ambiguous premises and an ambiguous conclusion, and review the logical properties of ambiguous connectives; and finally, I highlight how ambiguous connectives can explain our intuitions about logical rivalry. (shrink)

This chapter presents a new semantics for inductive empirical knowledge. The epistemic agent is represented concretely as a learner who processes new inputs through time and who forms new beliefs from those inputs by means of a concrete, computable learning program. The agent’s belief state is represented hyper-intensionally as a set of time-indexed sentences. Knowledge is interpreted as avoidance of error in the limit and as having converged to true belief from the present time onward. Familiar topics are re-examined within (...) the semantics, such as inductive skepticism, the logic of discovery, Duhem’s problem, the articulation of theories by auxiliary hypotheses, the role of serendipity in scientific knowledge, Fitch’s paradox, deductive closure of knowability, whether one can know inductively that one knows inductively, whether one can know inductively that one does not know inductively, and whether expert instruction can spread common inductive knowledge—as opposed to mere, true belief—through a community of gullible pupils. (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)