The admissibility of Ackermann's rule? is one of the most important problems in relevant logic. While the?-admissibility of normal modal logics based on the relevant logic R has been previously discussed, the case for weaker relevant modal logics has not yet been considered. The method of normal models has often been used to prove the?-admissibility. This paper discusses which relevant modal logics admit? from the viewpoint of the method of normal models.

This paper challenges the soundness of the two-dimensional conceivability argument against the derivation of phenomenal truths from physical truths in light of a hyperintensional, ground-theoretic regimentation of the ontology of consciousness. The regimentation demonstrates how ontological dependencies between truths about consciousness and about physics cannot be witnessed by epistemic constraints, when the latter are recorded by the conceivability—i.e., the epistemic possibility—thereof. Generalizations and other aspects of the philosophical significance of the hyperintensional regimentation are further examined.

Philosophy and Literature is an internationally renowned refereed journal founded by Denis Dutton at the University of Canterbury, Christchurch. It is now published by the Johns Hopkins University Press. Since its inception in 1976, Philosophy and Literature has been concerned with the relation between literary and philosophical studies, publishing articles on the philosophical interpretation of literature as well as the literary treatment of philosophy. Philosophy and Literature has sometimes been regarded as iconoclastic, in the sense that it repudiates academic pretensions, (...) insidious jargon and institutional vogue. Dutton, who remains the editor, still writes a regular column. A distinctive feature of Philosophy and Literature was the annual Bad Writing Contest, held from 1995–98, which sought to identify (and publish) the ‘most stylistically lamentable passages’ of academic prose, often to great amusement. (shrink)

Here I show that the one-variable fragment of several first-order relevant logics corresponds to certain S5ish extensions of the underlying propositional relevant logic. In particular, given a fairly standard translation between modal and one-variable languages and a permuting propositional relevant logic L, a formula $$\mathcal {A}$$ A of the one-variable fragment is a theorem of LQ (QL) iff its translation is a theorem of L5 (L.5). The proof is model-theoretic. In one direction, semantics based on the Mares-Goldblatt [15] semantics for (...) quantified L are transformed into ternary (plus two binary) relational semantics for S5-like extensions of L (for a general presentation, see Seki [26, 27]). In the other direction, a valuation is given for the full first-order relevant logic based on L into a model for a suitable S5 extension of L. I also discuss this work’s relation to finding a complete axiomatization of the constant domain, non-general frame ternary relational semantics for which RQ is incomplete [11]. (shrink)

In this first volume of The Sylvan Jungle, the editors present a scholarly edition of the first chapter, "Exploring Meinong's Jungle," of Richard Routley's 1000-plus page book, Exploring Meinong's Jungle and Beyond. Going against the Quinean orthodoxy, Routley’s aim was to support Meinong’s idea that we can truthfully refer to non-existent and even impossible objects, like Superman, unicorns and the round-square cupola on Berkeley College. The tools of non-classical logic at Routley’s disposal enabled him to update Meinong’s project for a (...) new generation. This volume begins with an Introduction from Dominic Hyde, “The ‘Jungle Book’ in Context,” an essay that situates Exploring Meinong’s Jungle and Beyond historically. We provide the original Preface by Routley, followed by Chapter 1: “Exploring Meinong’s Jungle and Beyond.” In Chapter 2, Nicholas Griffin argues that Sylvan’s project was insufficiently radical with his essay, “Why the Original Theory of Items Didn’t Go Far Enough.” Sylvan revisits his position from this time in Chapter 3, with his article, “Re-Exploring Item-Theory.” Filippo Casati, who has worked in the Routley Archives then takes up the question of the future of Sylvan’s research program in his essay, “The Future Perfect of Exploring Meinong’s Jungle.” Iconic and iconoclastic Australian philosopher Richard Routley published Exploring Meinong’s Jungle and Beyond in 1980. This work has fallen out of print, yet without great fanfare it has influenced two generations of philosophers and logicians. (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.

This paper discusses the relation between the minimal positive relevant logic B and intersection and union type theories. There is a marvelous coincidence between these very differently motivated research areas. First, we show a perfect fit between the Intersection Type Discipline ITD and the tweaking BT of B, which saves implication and conjunction but drops disjunction . The filter models of the -calculus (and its intimate partner Combinatory Logic CL) of the first author and her coauthors then become theory models (...) of these calculi. (The logician's "theory" is the algebraist's "filter".) The coincidence extends to a dual interpretation of key particles—the subtype translates to provable , type intersection to conjunction , function space to implication, and whole domain to the (trivially added but trivial) truth T. This satisfying ointment contains a fly. For it is right, proper, and to be expected that type union should correspond to the logical disjunction of B. But the simulation of functional application by a fusion (or modus ponens product) operation on theories leaves the key Bubbling lemma of work on ITD unprovable for the -prime theories now appropriate for the modeling. The focus of the present paper lies in an appeal to Harrop theories which are (a) prime and (b) closed under fusion. A version of the Bubbling lemma is then proved for Harrop theories, which accordingly furnish a model of and CL. (shrink)

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

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.

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

In this paper I argue that there can be genuine (as opposed to merely verbal) disputes about whether a sentence form is logically true or an argument form is valid. I call such disputes ?cases of deviance?, of which I distinguish a weak and a strong form. Weak deviance holds if one disputant is right and the other wrong, but the available evidence is insufficient to determine which is which. Strong deviance holds if there is no fact of the matter. (...) In section 2 I argue that weak deviance need not be trivial and may even be interesting. Section 3 considers what it could mean to say that logic is determined by a theory, especially a theory of meaning, an idea that arises in section 2. In section 4 I discuss the dispute between classical and relevance logicians over entailment and argue that it is a case of strong deviance. Finally, in section 5 I show that the result of the previous section is not absolute but relative to the background logic used in reaching it. (shrink)

The knowability paradox is an instance of a remarkable reasoning pattern (actually, a pair of such patterns), in the course of which an occurrence of the possibility operator, the diamond, disappears. In the present paper, it is pointed out how the unwanted disappearance of the diamond may be escaped. The emphasis is not laid on a discussion of the contentious premise of the knowability paradox, namely that all truths are possibly known, but on how from this assumption the conclusion is (...) derived that all truths are, in fact, known. Nevertheless, the solution offered is in the spirit of the constructivist attitude usually maintained by defenders of the anti-realist premise. In order to avoid the paradoxical reasoning, a paraconsistent constructive relevant modal epistemic logic with strong negation is defined semantically. The system is axiomatized and shown to be complete. (shrink)

This paper defines a category of bounded distributive lattice-ordered grupoids with a left-residual operation that corresponds to a weak system in the family of relevant logics. Algebras corresponding to stronger systems are obtained by adding further postulates. A duality theoey piggy-backed on the Priestley duality theory for distributive lattices is developed for these algebras. The duality theory is then applied in providing characterizations of the dual spaces corresponding to stronger relevant logics.

The?-admissibility is one of the most important problems in the realm of relevant logics. To prove the 7-admissibility, either the method of normal models or the method using metavaluations may be employed. The?-admissibility of a wide class of relevant modal logics has been discussed in Part I based on a former method, but the?-admissibility based on metavaluations has not hitherto been fully considered. Sahlqvist axioms are well known as a means of expressing generalized forms of formulas with modal operators. This (...) paper shows that? is admissible for relevant modal logics with restricted Sahlqvist axioms in terms of the method using metavaluations. (shrink)

The admissibility of Ackermann’s rule γ is one of the most important problems in relevant logic. While the γ-admissibility of normal modal logics based on the relevant logic R has been previously discussed, the case for weaker relevant modal logics has not yet been considered. The method of normal models has often been used to prove the γ-admissibility. This paper discusses which relevant modal logics admit γ from the viewpoint of the method of normal models.

Halldén completeness closely resembles the relevance property. To prove Halldén completeness in terms of Kripke-style semantics, the van Benthem–Humberstone theorem is often used. In relevant modal logics, the Halldén completeness of Meyer–Fuhrmann logics has been obtained using the van Benthem–Humberstone theorem. However, there remain a number of Halldén-incomplete relevant modal logics. This paper discusses the Halldén completeness of a wider class of relevant modal logics, namely, those with some Sahlqvist axioms.

The admissibility of Ackermann's rule γ is one of the most important problems in relevant logics. The admissibility of γ was first proved by an algebraic method. However, the development of Routley-Meyer semantics and metavaluational techniques makes it possible to prove the admissibility of γ using the method of normal models or the method using metavaluations, and the use of such methods is preferred. This paper discusses an algebraic proof of the admissibility of γ in relevant modal logics based on (...) modern algebraic models. (shrink)

American-plan semantics with 4 values 1, 0, { {1, 0}} {{}}, interpretable as True, False, Both and Neither, are furnished for a range of logics, including relevant affixing systems. The evaluation rules for extensional connectives take a classical form: in particular, those for negation assume the form 1 (A, a) iff 0 (A, a) and 0 (A, a) iff 1 (A, a), so eliminating the star function *, on which much criticism of relevant logic semantics has focussed. The cost of (...) these classical features is a further relation (or operation), required in evaluating falsity assignments of implication formulae.Two styles of 4 valued relational semantics are developed; firstly a semantics using notions of double truth and double validity for basic relevant systemB and some extensions of it; and secondly, since the first semantics makes heavy weather of validating negation principles such as Contraposition, a reduced semantics using more complex implicational rules for relevant systemC and various of its extensions. To deal satisfactorily with elite systemsR,E andT, however, further complication is inevitable; and a relation of mateship (suggested by the Australian plan) is introduced to permit cross-over from 1 to 0 values and vice versa. (shrink)

Sylvan and Plumwood’s is the relevant De Morgan minimal logic in the Routley-Meyer semantics with a set of designated points. The aim of this paper is to define the logic and some of its extensions. The logic is the non-relevant De Morgan minimal logic in the Routley-Meyer semantics without a set of designated points.

As is known, a logic S is paraconsistent if the rule ECQ (E contradictione quodlibet) is not a rule of S. Not less well known is the fact that Lewis’ modal logics are not paraconsistent. Actually, Lewis vindicates the validity of ECQ in a famous proof currently known as the “Lewis’ proof” or “Lewis’ argument.” This proof essentially leans on the Disjunctive Syllogism as a rule of inference. The aim of this paper is to define a series of paraconsistent logics (...) included in S4 where the Disjunctive Syllogism is valid only as a rule of proof. (shrink)

Łukasiewicz three-valued logic Ł3 is often understood as the set of all 3-valued valid formulas according to Łukasiewicz’s 3-valued matrices. Following Wojcicki, in addition, we shall consider two alternative interpretations of Ł3: “well-determined” Ł3a and “truth-preserving” Ł3b defined by two different consequence relations on the 3-valued matrices. The aim of this paper is to provide (by using Dunn semantics) dual equivalent two-valued under-determined and over-determined interpretations for Ł3, Ł3a and Ł3b. The logic Ł3 is axiomatized as an extension of Routley (...) and Meyer’s basic positive logic following Brady’s strategy for axiomatizing many-valued logics by employing two-valued under-determined or over-determined interpretations. Finally, it is proved that “well determined” Łukasiewicz logics are paraconsistent. (shrink)

Łukasiewicz three-valued logic Ł3 is often understood as the set of all 3-valued valid formulas according to Łukasiewicz’s 3-valued matrices. Following Wojcicki, in addition, we shall consider two alternative interpretations of Ł3: “well-determined” Ł3a and “truth-preserving” Ł3b defined by two different consequence relations on the 3-valued matrices. The aim of this paper is to provide dual equivalent two-valued under-determined and over-determined interpretations for Ł3, Ł3a and Ł3b. The logic Ł3 is axiomatized as an extension of Routley and Meyer’s basic positive (...) logic following Brady’s strategy for axiomatizing many-valued logics by employing two-valued under-determined or over-determined interpretations. Finally, it is proved that “well determined” Łukasiewicz logics are paraconsistent. (shrink)

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.

Łukasiewicz 3-valued logic Ł3 is often understood as the set of all valid formulas according to Łukasiewicz 3-valued matrices MŁ3. Following Wojcicki, in addition, we shall consider two alternative interpretations of Ł3: ‘truth-preserving’ Ł3a and ‘well-determined’ Ł3b defined by two different consequence relations on the 3-valued matrices MŁ3. The aim of this article is to provide a Routley–Meyer ternary semantics for each one of these three versions of Łukasiewicz 3-valued logic: Ł3, Ł3a and Ł3b.

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)

Epistemic logics based on the possible worlds semantics suffer from the problem of logical omniscience, whereby agents are described as knowing all logical consequences of what they know, including all tautologies. This problem is doubly challenging: on the one hand, agents should be treated as logically non-omniscient, and on the other hand, as moderately logically competent. Many responses to logical omniscience fail to meet this double challenge because the concepts of knowledge and reasoning are not properly separated. In this paper, (...) I present a dynamic logic of knowledge that models an agent’s epistemic state as it evolves over the course of reasoning. I show that the logic does not sacrifice logical competence on the altar of logical non- omniscience. (shrink)

The system B+ is the minimal positive relevant logic. B+ is trivially extended to B+T on adding a greatest truth (Church constant) T. If we leave ∨ out of the formation apparatus, we get the fragment B∧T. It is known that the set of ALL B∧T theories provides a good model for the combinators CL at Level-I, which is the theory level. Restoring ∨ to get back B+T was not previously fruitful at Level-I, because the set of all B+T theories (...) is NOT a model of CL. It was to be expected from semantic completeness arguments for relevant logics that basic combinator laws would hold when restricted to PRIME B+T theories. Overcoming some previous difficulties, we show that this is the case, at Level I. But this does not form a model for CL. This paper also looks for corresponding results at Level-II, where we deal with sets of theories that we call propositions. We adapt work by Ghilezan to note that at Level-II also there is a model of CL in B∧T propositions. However, the corresponding result for B+T propositions extends smoothly to Level-II only in part. Specifically, only some of the basic combinator laws are proved here. We accordingly leave some work for the reader. (shrink)

In this article we review several paraconsistent logics from different authors to ‘close the gaps’ between them. Since paraconsistent logics is a broad area of research, it is possible that equivalent paraconsistent logics have different names. What we meant is that we provide connections between the logics studied comparing their different semantical approaches for a near future be able to obtain missing semantical characterization of different logics. We are introducing the term RC-type logics to denote a class of logics that (...) extends Cω and satisfies the RC rule. (shrink)

Modus ponens provides the central theme. There are laws, of the form A→C. A logic L collects such laws. Any datum A provides input to the laws of L. The central ternary relation R relates theories L,T and U, where U consists of all of the outputs C got by applying modus ponens to major premises from L and minor premises from T. Underlying this relation is a modus ponens product operation on theories L and T, whence RLTU iff LTU. (...) These ideas have been expressed, especially with Routley, as worlds semantics for relevant and other substructural logics.Worlds are best demythologized as theories, subject to truth-functional and other constraints. The chief constraint is that theories are taken as closed under logical entailment, which clearly begs the question if we are using the semantics to determine which theory L is Logic itself. Instead we draw the modal logicians’ conclusion—there are many substructural logics, each with its appropriate ternary relational postulates.Each logic L gives rise to a Calculus of L-theories, on which particular candidate logical axioms have the combinatorial properties expected from the well-known Curry–Howard isomorphism . We apply their bubbling lemma, updating the Fools Model of Combinatory Logic at the pure → level for the system BT. We make fusion an explicit connective, proving a combinator correspondence theorem. Having taken relevant → as a left residual for , we explore its right residual mate →r. Finally we concentrate on and prove a finite model property for the classical minimal relevant logic CB, a conservative extension of the minimal positive relevant logic B+. (shrink)

The system R## of "true" relevant arithmetic is got by adding the ω-rule "Infer VxAx from AO, A1, A2, ...." to the system R# of "relevant Peano arithmetic". The rule ⊃E (or "gamma") is admissible for R##. This contrasts with the counterexample to ⊃E for R# (Friedman & Meyer, "Whither Relevant Arithmetic"). There is a Way Up part of the proof, which selects an arbitrary non-theorem C of R## and which builds by generalizing Henkin and Belnap arguments a prime theory (...) T which still lacks C. (The key to the Way Up is a Witness Protection Program, using the ω-rule.) But T may be TOO BIG, whence there is a Way Down argument that produces a better theory TR, such that R## ⊆ TR ⊆ T. (The key to the Way Down is a Metavaluation, on which membership in T is combined with ordinary truth-functional conditions to determine TR.) The result is a theory that is Just Right, whence it never happens that A ⊃ C and A are theorems of R## but C is a non-theorem. (shrink)

The system R## of "true" relevant arithmetic is got by adding the ω-rule "Infer VxAx from AO, A1, A2, ...." to the system R# of "relevant Peano arithmetic". The rule ⊃E (or "gamma") is admissible for R##. This contrasts with the counterexample to ⊃E for R# (Friedman & Meyer, "Whither Relevant Arithmetic"). There is a Way Up part of the proof, which selects an arbitrary non-theorem C of R## and which builds by generalizing Henkin and Belnap arguments a prime theory (...) T which still lacks C. (The key to the Way Up is a Witness Protection Program, using the ω-rule.) But T may be TOO BIG, whence there is a Way Down argument that produces a better theory TR, such that R## ⊆ TR ⊆ T. (The key to the Way Down is a Metavaluation, on which membership in T is combined with ordinary truth-functional conditions to determine TR.) The result is a theory that is Just Right, whence it never happens that A ⊃ C and A are theorems of R## but C is a non-theorem. (shrink)

The aim of this paper is to investigate a Curry-Howard interpretation of the intersection and union type inference system for Combinatory Logic. Types are interpreted as formulas of a Hilbert-style logic L, which turns out to be an extension of the intuitionistic logic with respect to provable disjunctive formulas (because of new equivalence relations on formulas), while the implicational-conjunctive fragment of L is still a fragment of intuitionistic logic. Moreover, typable terms are translated in a typed version, so that --typed (...) combinatory logic terms are proved to completely codify the associated logical proofs. (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)

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

Motivated by weaknesses with traditional accounts of logical epistemology, considerable attention has been paid recently to the view, known as anti-exceptionalism about logic, that the subject matter and epistemology of logic may not be so different from that of the recognised sciences. One of the most prevalent claims made by advocates of AEL is that theory choice within logic is significantly similar to that within the sciences. This connection with scientific methodology highlights a considerable challenge for the anti-exceptionalist, as two (...) uncontentious claims about scientific theories are that they attempt to explain a target phenomenon and prove their worth through successful predictions. Thus, if this methodological AEL is to be viable, the anti-exceptionalist will need a reasonable account of what phenomena logics are attempting to explain, how they can explain, and in what sense they can be said to issue predictions. This paper makes sense of the anti-exceptionalist proposal with a new account of logical theory choice, logical predictivism, according to which logics are engaged in both a process of prediction and explanation. (shrink)

The logic CE (for "Classical E") results from adding Boolean negation to Anderson and Belnap's logic E. This paper shows that CE is not a conservative extension of E.

Sound and complete semantics for classical propositional logic can be obtained by interpreting sentences as sets. Replacing sets with commuting dense binary relations produces an interpretation that turns out to be sound but not complete for R. Adding transitivity yields sound and complete semantics for RM, because all normal Sugihara matrices are representable as algebras of binary relations.

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

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

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