Many systems of quantified modal logic cannot be characterised by Kripke's well-known possible worlds semantic analysis. This book shows how they can be characterised by a more general 'admissible semantics', using models in which there is a restriction on which sets of worlds count as propositions. This requires a new interpretation of quantifiers that takes into account the admissibility of propositions. The author sheds new light on the celebrated Barcan Formula, whose role becomes that of legitimising the Kripkean interpretation (...) of quantification. The theory is worked out for systems with quantifiers ranging over actual objects, and over all possibilia, and for logics with existence and identity predicates and definite descriptions. The final chapter develops a new admissible 'cover semantics' for propositional and quantified relevant logic, adapting ideas from the Kripke-Joyal semantics for intuitionistic logic in topos theory. This book is for mathematical or philosophical logicians, computer scientists and linguists. (shrink)

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

There is an intuition, notoriously difficult to formalise, that only some predicates express real properties. J. M. Dunn formalises this intuition with relevance logic, proposing a notion of relevant predication. For each first order formula Ax, Dunn specifies another formula that is intuitively interpreted as "Ax expresses a real property". Chapter I calls such an approach an object language approach, since the claim that Ax expresses a real property is rendered as a formula in the object language. On a (...) metalanguage approach, on the other hand, the claim that Ax expresses a real property would be metalinguistic, mentioning but not using the formula Ax. ;Our Introduction begins by investigating Dunn's motivation for relevant predication, and argues that it implicitly presupposes some interpretation of identity. Indeed, part of the dissertation's work is to use relevant predication as a key to a coherent account of relevant identity. ;Chapter I re-motivates relevant predication. We consider P. Geach's distinction between real and "Cambridge" change, and suggest that there is an underlying distinction between real and Cambridge predicates. We further argue that no classical object language approach can formalise this distinction. So we turn to relevance logic, remotivating relevant predication, with Geach's distinction in mind. ;In light of this new motivation, we argue that relevant prediction relies on a logically weak notion of identity, according to which "x = y" means "x and y share all relevant properties". Identity is taken up in a more technical setting in Chapters IV and V. ;Subsequent chapters investigate technical issues that flow from Chapter I. Chapter II investigates metalinguistic grammatical characterisations of relevant predication. Chapter III concerns the semantics of first order relevance logic. Chapter IV investigates the identity in R and its relationship to relevant predication. In particular, Chapter IV argues against introducing the axiom, $\to$ Ay), in relevance logic, since the resulting systems are unstable, in a sense motivated by the notion of relevant identity underlying Dunn's notion of relevant predication. Finally Chapter V investigates the possibility of a coherent theory of identity in R's modal relatives ${\bf R}\sp\square$ and E. (shrink)

In this paper, I set out a semantics for identity in relevant logic that is based on an analogy between the biconditional and identity. This analogy supports the semantics that Priest has set out for identity in basic relevant logic and it motivates a version of the Routley–Meyer semantics in which identities can be viewed as constraints on the ternary relation that is used to treat implication.

In this article, I consider the positive logic of Boolean groups (i.e. Abelian groups where every non-identity element has order 2), where these are taken as frames for an operational semantics à la Urquhart. I call this logic BG. It is shown that the logic over the smallest nontrivial Boolean group, taken as a frame, is identical to the positive fragment of a quasi-relevance logic that was developed by Robles and Méndez (an extension of this result where negation (...) is included is also discussed). It is proved that BG satisfies the variable sharing property and is Halldén complete, while failing to satisfy the disjunction property. Sound and complete subscripted tableaux are presented for BG. An axiomatization (in the positive language extended with fusion) is presented, which is sound with respect to BG and, with respect to a related ternary relational semantics, complete; the problem of identifying a complete axiomatization for BG itself is left open. Connections between BG and the quasi-relevance logic KR are also discussed. (shrink)

In this paper a system, RPF, of second-order relevance logic with S5 necessity is presented which contains a defined, notion of identity for propositions. A complete semantics is provided. It is shown that RPF allows for more than one necessary proposition. RPF contains primitive syntactic counterparts of the following semantic notions: (1) the reflexive, symmetrical, transitive binary alternativeness relation for S5 necessity, (2) the ternary Routley-Meyer alternativeness relation for implication, and (3) the Routley-Meyer notion of a prime intensional (...) theory, as well as defined syntactic counterparts, of the semantic notions of a possible world and the Routley-Meyer * operator. (shrink)

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

This essay discusses rules and semantic clauses relating to Substitution—Leibniz’s law in the conjunctive-implicational form s=t ∧ A(s) → A(t)—as these are put forward in Priest’s books "In Contradiction" and "An Introduction to Non-Classical Logic: From If to Is." The stated rules and clauses are shown to be too weak in some cases and too strong in others. New ones are presented and shown to be correct. Justification for the various rules are probed and it is argued that Substitution ought (...) to fail. (shrink)

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

I follow Giordani in distinguishing between a situational and informational conception of subject-matter and consider some of the different ways in which it might be explained within both an intensional and an hyperintensional account of propositional identity.

This book introduces the reader to relevant logic and provides the subject with a philosophical interpretation. The defining feature of relevant logic is that it forces the premises of an argument to be really used in deriving its conclusion. The logic is placed in the context of possible world semantics and situation semantics, which are then applied to provide an understanding of the various logical particles and natural language conditionals. The book ends by examining various applications of (...) relevant logic and presenting some interesting open problems. It will be of interest to a range of readers including advanced students of logic, philosophical and mathematical logicians, and computer scientists. (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)

This paper gives an outline of three different approaches to the four-valued semantics for relevant logics (and other non-classical logics in their vicinity). The first approach borrows from the 'Australian Plan' semantics, which uses a unary operator '⋆' for the evaluation of negation. This approach can model anything that the two-valued account can, but at the cost of relying on insights from the Australian Plan. The second approach is natural, well motivated, independent of the Australian Plan, and it (...) provides a semantics for the contraction-free relevant logic C (or RW). Unfortunately, its approach seems to model little else. The third approach seems to capture a wide range of formal systems, but at the time of writing, lacks a completeness proof. (shrink)

In this paper, the positive fragment of the logic math formula of contraction-less relevant implication is extended with the addition of a paraconsistent negation connective similar to the strong negation connective in Nelson's paraconsistent four-valued logic math formula. This extended relevant logic is called math formula, and it has the property of constructible falsity which is known to be a characteristic property of math formula. A Gentzen-type sequent calculus math formula for math formula is introduced, and the cut-elimination and decidability (...) theorems for math formula are proved. Two extended Routley-Meyer semantics are introduced for math formula, and the completeness theorems with respect to these semantics are proved. (shrink)

This paper continues the work of Priest and Sylvan in Simplified Semantics for Basic Relevant Logics, a paper on the simplified semantics of relevant logics, such as B⁺ and B. We show that the simplified semantics can also be used for a large number of extensions of the positive base logic B⁺, and then add the dualising '*' operator to model negation. This semantics is then used to give conservative extension results for Boolean negation.

This paper shows that the relevant logics E and Π′ are strongly sound and complete with regards to a version of the “simplified” Routley-Meyer semantics. Such a semantics for E has been thought impossible. Although it is impossible if an admissible rule of E – the rule of restricted assertion or equivalently Ackermann’s δ-rule – is solely added as a primitiverule, it is very much possible when E is axiomatized in the way Anderson and Belnap did. The simplified (...) semantics for E and Π′ requires unreduced frames. Contra what has been claimed, however, no additional frame component is required over and above what’s required to model other relevant logics such as T and R. It is also shown how to modify the tonicity requirements of theternary relation so as to allow for the standard truth condition for both fusion – the intensional conjunction◦ – as well as the converse conditional ←. (shrink)

We provide five semantic preservation properties which apply to the various rules -- primitive, derived and admissible -- of Hilbert-style axiomatizations of relevant logics. These preservation properties are with respect to the Routley-Meyer semantics, and consist of various truth- preservations and validity-preservations from the premises to the conclusions of these rules. We establish some deduction theorems, some persistence theorems and some soundness and completeness theorems, for these preservation properties. We then apply the above ideas, as best we can, to (...) the classical sentential and predicate calculi, to normal and non- normal modal logics, and to many- valued logics. (shrink)

Relevant Logics and their Rivals, Volume II extends the material of the first volume in two ways.

This paper has two aims. First, it sets out an interpretation of the relevant logic E of relevant entailment based on the theory of situated inference. Second, it uses this interpretation, together with Anderson and Belnap’s natural deduc- tion system for E, to generalise E to a range of other systems of strict relevant implication. Routley–Meyer ternary relation semantics for these systems are produced and completeness theorems are proven. -/- .

In this paper, we investigate neighbourhood semantics for modal extensions of relevant logics. In particular, we combine the neighbourhood interpretation of the relevant implication (and related connectives) with a neighbourhood interpretation of modal operators. We prove completeness for a range of systems and investigate the relations between neighbourhood models and relational models, setting out a range of augmentation conditions for the various relations and operations.

The Mares-Goldblatt semantics for quantified relevant logics have been developed for first-order extensions of R, and a range of other relevant logics and modal extensions thereof. All such work has taken place in the the ternary relation semantic framework, most famously developed by Sylvan and Meyer. In this paper, the Mares-Goldblatt technique for the interpretation of quantifiers is adapted to the more general neighbourhood semantic framework, developed by Sylvan, Meyer, and, more recently, Goble. This more algebraic semantics allows (...) one to characterise a still wider range of logics, and provides the grist for some new results. To showcase this, we show, using some non-augmented models, that some quantified relevant logics are not conservatively extended by connectives the addition of which do conservatively extend the associated propositional logics, namely fusion and the dual implication. We close by proposing some further uses to which the neighbourhood Mares-Goldblatt semantics may be put. (shrink)

This paper shows that the relevant logics E and Π′ are strongly sound and complete with regards to a version of the “simplified” Routley-Meyer semantics. Such a semantics for E has been thought impossible. Although it is impossible if an admissible rule of E  the rule of restricted assertion or equivalently Ackermann’s δ-rule  is solely added as a primitive rule, it is very much possible when E is axiomatized in the way Anderson and Belnap did. The (...) simplified semantics for E and Π′ requires unreduced frames. Contra what has been claimed, however, no additional frame component is required over and above what’s required to model other relevant logics such as T and R. It is also shown how to modify the tonicity requirements of the ternary relation so as to allow for the standard truth condition for both fusion – the intensional conjunction ◦ – as well as the converse conditional ←. (shrink)

This chapter contains sections titled: Non‐Sequiturs are Bad The Real Use of Premises Implication From Proof Theory to Semantics Adding Conjunction The Problem of Disjunction Routley and Meyer's Ternary Relation Rules for Disjunction The Semantics of Negation Rules for Negation Disjunctive Syllogism Logics Stronger than R Logics Weaker than R Relevant Logics and Natural Language Conditionals Theory of Properties Summary.

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.

It is shown how to model propositional constants within the simplified Routley-Meyer semantics. Various axioms and rules allowingthe definition of modal operators, implicative negations, enthymematical conditionals, and propositions expressing various infinite conjunctions anddisjunctions are set forth and shown to correspond to specific frame conditions. Two propositional constants which are both often designated as “the Ackermann constant” are shown to capture two such “infinite” propositions: The conjunction of every logical law and the conjunction of every truth – what Anderson (...) and Belnap called the “world” constant. (shrink)

Relevance logics are known to be sound and complete for relational semantics with a ternary accessibility relation. This paper investigates the problem of adequacy with respect to special kinds of dynamic semantics (i.e., proper relation algebras and relevant families of relations). We prove several soundness results here. We also prove the completeness of a certain positive fragment of R as well as of the first-degree fragment of relevance logics. These results show that some core ideas are (...) shared between relevance logics and relation algebras. Some details of certain incompleteness results, however, pinpoint where relevance logics and relation algebras diverge. To carry out these semantic investigations, we define a new tableaux formalization and new sequent calculi (with the single cut rule admissible) for various relevance logics. (shrink)

The quantified relevant logic RQ is given a new semantics in which a formula for all xA is true when there is some true proposition that implies all x-instantiations of A. Formulae are modelled as functions from variable-assignments to propositions, where a proposition is a set of worlds in a relevant model structure. A completeness proof is given for a basic quantificational system QR from which RQ is obtained by adding the axiom EC of 'extensional confinement': for all x(A (...) V B) -> (A V for all xB). with x not free in A. Validity of EC requires an additional model condition involving the boolean difference of propositions. A QR-model falsifying EC is constructed by forming the disjoint union of two natural arithmetical structures in which negation is interpreted by the minus operation. (shrink)

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

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

The semilattice relevant logics ∪R, ∪T, ∪RW, and ∪TW are defined by semilattice models in which conjunction and disjunction are interpreted in a natural way. For each of them, there is a cut-free labelled sequent calculus with plural succedents . We prove that these systems are equivalent, with respect to provable formulas, to the restricted systems with single succedents . Moreover, using this equivalence, we give a new Hilbert-style axiomatizations for ∪R and ∪T and prove equivalence between two semantics (...) for the contractionless logics ∪RW and ∪TW. (shrink)

This paper concerns the interface between relevant logics and recent developments in semantics for hyperintensional operators. In the latter area, the notion of topicality or aboutness has begun to play an explicit modeling role, with classes of models being studied which incorporate representations of topics, and of operations on topics. The idea is to use these topics in the evaluations of formulas including hyperintensional operators in order to explain why classically equivalent formulas may not be intersubstitutable in contexts using (...) those operators—though classically equivalent, if a pair of formulas can be interpreted to concern totally unrelated topics, then we should not be able to substitute one for the other in contexts which are sensitive to topics. Relevant logics, and the central relevance property of Variable Sharing, have been read as concerning something like topics in the past, and recently this angle on relevance has seen renewed interest. In this paper, I’ll develop and motivate a method for representing topics in the matrix models of logics, and apply this method to _R_ and its matrix models (given by De Morgan monoids) to argue that there is a clear semantic perspective on topicality according to which _R_ is topic-sensitive as a logic. (shrink)

It is shown how to model propositional constants within the simplified Routley-Meyer semantics. Various axioms and rules allowing the definition of modal operators, implicative negations, enthymematical conditionals, and propositions expressing various infinite conjunctions and disjunctions are set forth and shown to correspond to specific frame conditions. Two propositional constants which are both often designated as “the Ackermann constant” are shown to capture two such “infinite” propositions: The conjunction of every logical law and the conjunction of every truth –what (...) Anderson and Belnap called the “world” constant. (shrink)

This paper discusses three relevant logics that obey Component Homogeneity - a principle that Goddard and Routley introduce in their project of a logic of significance. The paper establishes two main results. First, it establishes a general characterization result for two families of logic that obey Component Homogeneity - that is, we provide a set of necessary and sufficient conditions for their consequence relations. From this, we derive characterization results for S*fde, dS*fde, crossS*fde. Second, the paper establishes complete sequent calculi (...) for S*fde, dS*fde, crossS*fde. Among the other accomplishments of the paper, we generalize the semantics from Bochvar, Hallden, Deutsch and Daniels, we provide a general recipe to define containment logics, we explore the single-premise/single-conclusion fragment of S*fde, dS*fde, crossS*fdeand the connections between crossS*fde and the logic Eq of equality by Epstein. Also, we present S*fde as a relevant logic of meaninglessness that follows the main philosophical tenets of Goddard and Routley, and we briefly examine three further systems that are closely related to our main logics. Finally, we discuss Routley's criticism to containment logic in light of our results, and overview some open issues. (shrink)

We examine the set of formula-to-formula valid inferences of Classical Logic, where the premise and the conclusion share at least a propositional variable in common. We review the fact, already proved in the literature, that such a system is identical to the first-degree entailment fragment of R. Epstein's Relatedness Logic, and that it is a non-transitive logic of the sort investigated by S. Frankowski and others. Furthermore, we provide a semantics and a calculus for this logic. The semantics (...) is defined in terms of a \-matrix built on top of a 5-valued extension of the 3-element weak Kleene algebra, whereas the calculus is defined in terms of a Gentzen-style sequent system where the left and right negation rules are subject to linguistic constraints. (shrink)

The logician's central concern is with the validity of argument. A logical theory ought, therefore, to provide a general criterion of validity. This book sets out to find such a criterion, and to describe the philosophical basis and the formal theory of a logic in which the premises of a valid argument are relevant to its conclusion. The notion of relevance required for this theory is obtained by an analysis of the grounds for asserting a formula in a (...) proof. (shrink)

This revised and considerably expanded 2nd edition brings together a wide range of topics, including modal, tense, conditional, intuitionist, many-valued, paraconsistent, relevant, and fuzzy logics. Part 1, on propositional logic, is the old Introduction, but contains much new material. Part 2 is entirely new, and covers quantification and identity for all the logics in Part 1. The material is unified by the underlying theme of world semantics. All of the topics are explained clearly using devices such as tableau proofs, (...) and their relation to current philosophical issues and debates are discussed. Students with a basic understanding of classical logic will find this book an invaluable introduction to an area that has become of central importance in both logic and philosophy. It will also interest people working in mathematics and computer science who wish to know about the area. (shrink)

Once upon a time, modal logic was castigated because it ‘had no semantics.’ Kripke, Hintikka, Kanger, and others changed all that. In a similar way, when Relevant Logic was introduced by Anderson and Belnap, it too was castigated for ‘having no semantics.’ The present overview marks a culmination of that effort. The semantic approach described here brings together a number of hitherto disparate efforts to set out formal systems for logics of relevant implication and entailment. It also makes (...) clear (despite some of our hopes and utterances) that the One True Logic does not exist. This is as true for relevant logics as Kripke et al., showed it to be for modal logics. In both cases, subtle (and not so subtle) variations on semantical postulates produce different logics in the same family. The question of which semantical postulates are correct makes no sense without further context, i.e., the questioner needs to answer the question: Correct for what? The question that does remain is: What motivates the relevant family of logics? And this is the question that is the main job for this chapter to investigate. (shrink)

Here, I combine the semantics of Mares and Goldblatt [20] and Seki [29, 30] to develop a semantics for quantified modal relevant logics extending ${\bf B}$. The combination requires demonstrating that the Mares–Goldblatt approach is apt for quantified extensions of ${\bf B}$ and other relevant logics, but no significant bridging principles are needed. The result is a single semantic approach for quantified modal relevant logics. Within this framework, I discuss the requirements a quantified modal relevant logic must satisfy (...) to be “sufficiently classical” in its modal fragment, where frame conditions are given that work for positive fragments of logics. The roles of the Barcan formula and its converse are also investigated. (shrink)

The purpose of teaching logic in philosophy is to enable us to evaluate arguments with respect to (formal) validity. Standard logics refer to a concept of validity which allows for the relation of implication to hold between premises and conclusion even in cases where there is no “relevant” connection between the premises and the conclusion. A prominent example for this is the rule “Ex-Falso-Quodlibet” (EFQ), which allows us to infer an arbitrary proposition from a contradiction. The tolerance of irrelevance endorsed (...) by standard logics unfortunately engenders that they cannot adequately fulfill their intended task of analyzing and evaluating philosophical, scientific and everyday-life arguments – instead, their application even gives rise to a multitude of artificial philosophical pseudoproblems (like the problem of the disposition predicates or the problem of counterfactuals). As alternatives to standard logics, there exist non-standard systems called “relevance logics” or “relevant logics” meant to avoid irrelevance. The problem with these systems, however, is that the mainstream relational semantics (“worlds semantics”) available for them is to be considered unintuitive and complex to a degree which is apt to render relevant logics unattractive to the majority of philosophers who are on the lookout not only for adequate, but also simple and efficient technical means for evaluating arguments. Therefore, the main aim of this treatise is to provide an alternative semantics (“rules semantics”) which is comparatively easy to grasp and simple in application. A second aim of the book is to extend the semantics as least as far as it takes to cover more or less all the logical notions philosophers need in their “everyday analyzing”. This includes first order predicate logic, higher order logic (for analyzing talk about “properties” etc.), identity, definite descriptions, abstraction principles and modal logic. This book can be read without having any more background than a good introductory course in classical logic provides. (shrink)

This anthology consists of forty-nine of Geach’s previously published papers on logic. He opens his Preface by writing: "I bring together here almost all my English articles that I have previously published and have not already collected or cannibalized in other books." It contains his first published paper "Designation and Truth" from Analysis 1947-48 as well as his elegant sketch of a decidable entailment system in the 1970 Philosophical Review. For the most part he has made only stylistic changes and (...) added some notes. So, he claims: "all articles appear without substantial change." However, he admits that he divided his "Quantification Theory and the Problem of Identifying Objects of Reference" from the now classic 1963 volume of Acta Philosophica Fennica. He divided it because he wanted to separate his strong advocacy of a substitution interpretation of the quantifiers from some views on quotation marks, about which he is less than certain. The papers are not in mere chronological order. They are organized by topic within chapters entitled: Historical Essays, Traditional Logic, Theory of Reference and Syntax, Intentionality, Quotation and Semantics, Set Theory, Identity Theory, Assertion, Imperatives and Practical Reasoning, and Logic in Metaphysics and Theology. The chapter titles are a fair guide to content. But note the following. Two lucid papers defending von Wright’s notion of entailment are in the Intentionality chapter. The essay "Ascriptivism" in the Assertion chapter is relevant to discussions of free will because he argues: "to ascribe an act to agent is a causal description of the act" and he urges a search for a non-Humean analysis of causality. The essays in the last chapter are: "Nominalism," "Some problems about time," and "God’s relation to the world.". (shrink)

The logic BN4 was defined by R.T. Brady in 1982. It can be considered as the 4-valued logic of the relevant conditional. E4 is a variant of BN4 that can be considered as the 4-valued logic of entailment. The aim of this paper is to define reduced general Routley-Meyer semantics for BN4 and E4. It is proved that BN4 and E4 are strongly sound and complete w.r.t. their respective semantics.

The challenge of giving a semantics for relevance logic in terms of worlds or situations intrigued several logicians. As a solution, Fine gave a two-sorted semantics. We overview the semantics as well as some further work of Fine in the area of relevance logic. Then we show that beyond supplying technical results such as soundness, completeness and the finite model property (fmp) for many logics, the operational–relational semantics provides footing for an informal interpretation and (...) it naturally leads to an interpretation in terms of relevance. (shrink)

The dissertation considers both metaphysical and logical issues related to the vagueness of natural language. The principle metaphysical claim is that the vagueness of language is, at least in some cases, a direct result of indeterminacy in the subject matter of the language, rather than any sort of flaw of the language. A limited defense of this claim is given, as well as criticism of alternative views. ;A number of logical issues are addressed. First, the relationship between the (...) notion of determinacy and the idea of an unsharp line is considered, and it is suggested that the relationship is not as simple as it may at first seem, and that the idea of the unsharp line may be irreducible. Next, it is urged that there is a methodological fork in the road for the systematic treatment of vague language, including formal semantics. On one path, we accept certain intuitively puzzling propositions, exemplified by "This is red or it is not the case that this is red, though it is indeterminate which." On the other path, we reject classical principles of reasoning in our own reasoning both in and about vague language. Some limited arguments are given for taking the former path, and a formal system relevant to this path is motivated and discussed. Finally, both the metaphysical and logical work of the prior parts of the dissertation are brought to bear on the subject of the indeterminacy of identity. (shrink)