Categorical-theoretic semantics for the relevancelogic is proposed which is based on the construction of the topos of functors from a relevant algebra (considered as a preorder category endowed with the special endofunctors) in the category of sets Set. The completeness of the relevant system R of entailment is proved in respect to the semantic considered.
In this paper we construct an extension, ℒ, of Anderson and Belnap's relevancelogic R that is classical in the sense that it contains p&p → q as a theorem, and we prove that ℒ is pretabular in the sense that while it does not have a finite characteristic matrix, every proper normal extension of it does. We end the paper by commenting on the possibility of finding other classical relevance logics that are also pretabular.
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: relevancelogic 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)
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 relevancelogic has been the logic E of entailment that was supposed to capture strict relevant implication. The next system of relevancelogic 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)
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 two deductive systems (i.e., two consequence relations) associated with relevancelogic are studied from an algebraic point of view. One is defined by the familiar, Hilbert-style, formalization of R; the other one is a weak version of it, called WR, which appears as the semantic entailment of the Meyer-Routley-Fine semantics, and which has already been suggested by Wójcicki for other reasons. This weaker consequence is first defined indirectly, using R, but we prove that the first (...) one turns out to be an axiomatic extension of WR. Moreover we provide WR with a natural Gentzen calculus (of a classical kind). It is proved that both deductive systems have the same associated class of algebras but different classes of models on these algebras. The notion of model used here is an abstract logic, that is, a closure operator on an abstract algebra; the abstract logics obtained in the case of WR are also the models, in a natural sense, of the given Gentzen calculus. (shrink)
Clark Glymour has argued that hypothetico-deductivism, which many take to be an important method of scientific confirmation, is hopeless because it cannot be reconstructed in classical logic. Such reconstructions, as Glymour points out, fail to uphold the condition of relevance between theory and evidence. I argue that the source of the irrelevant confirmations licensed by these reconstructions lies not with hypothetico-deductivism itself, but with the classical logic in which it is typically reconstructed. I present a new reconstruction (...) of hypothetico-deductivism in relevancelogic that does maintain the condition of relevance between theory and evidence. Hence, if hypothetico-deductivism is an important rationale in science, we have good reason to believe that the logic underlying scientific discourse is captured better by relevancelogic than by its classical counterpart. (shrink)
In this paper a system, RPF, of second-order relevancelogic 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)
Models are constructed for a variety of systems of quantified relevancelogic 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 RelevanceLogic (Journal of Philosophical Logic 17 (1988)).
Relevancelogic is ordinarily seen as a subsystem of classical logic under the translation that replaces arrows by horseshoes. If, however, we consider the arrow as an additional connective alongside the horseshoe and other classical connectives, another perspective emerges. Relevancelogic, specifically the system R, may be seen as a the output of a conservative extension of classical consequence into the language with arrow.
In previous work we gave a new proof-theoretical method for establishing upper-bounds on the space complexity of the provability problem of modal and other propositional non-classical logics. Here we extend and refine these results to give an O(n log n)-space decision procedure for the basic positive relevancelogic B+. We compute this upper-bound by first giving a sound and complete, cut-free, labelled sequent system for B+, and then establishing bounds on (...) the application of the rules of this system. (shrink)
This book introduces the reader to relevant logic and provides it with a philosophical interpretation. The defining feature of relevant logic is that it forces the premises of an argument to be really used ('relevant') 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 (especially implication and negation) 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)
The relevant modal logic G is a simple extension of the logic RT, the relevant counterpart of the familiar classically based system T. Using the Routley-Meyer semantics for relevant modal logics, this paper proves three main results regarding G: (i) G is semantically complete, but only with a non-standard interpretation of necessity. From this, however, other nice properties follow. (ii) With a standard interpretation of necessity, G is semantically incomplete; there is no class of frames that characterizes G. (...) (iii) The class of frames for G characterizes the classically based logic T. (shrink)
Four-valued semantics proved useful in many contexts from relevance logics to reasoning about computers. We extend this approach further. A sequent calculus is defined with logical connectives conjunction and disjunction that do not distribute over each other. We give a sound and complete semantics for this system and formulate the same logic as a tableaux system. Intensional conjunction (fusion) and its residuals (implications) can be added to the sequent calculus straightforwardly. We extend a simplified version of the earlier (...) semantics for this system and prove soundness and completeness. Then, with some modifications to this semantics, we arrive at a mathematically elegant yet powerful semantics that we call generalized Kripke semantics. (shrink)
Relevancelogic has become ontologically fertile. No longer is the idea of relevance restricted in its application to purely logical relations among propositions, for as Dunn has shown in his (1987), it is possible to extend the idea in such a way that we can distinguish also between relevant and irrelevant predications, as for example between “Reagan is tall” and “Reagan is such that Socrates is wise”. Dunn shows that we can exploit certain special properties of identity (...) within the context of standard relevancelogic in a way which allows us to discriminate further between relevant and irrelevant properties, as also between relevant and irrelevant relations. The idea yields a family of ontologically interesting results concerning the different ways in which attributes and objects may hang together. Because of certain notorious peculiarities of relevancelogic, however,1 Dunn’s idea breaks down where the attempt is made to have it bear fruit in application to relations among entities which are of homogeneous type. (shrink)
Charles S. Peirce’s pragmatist theory of logic teaches us to take the context of utterances as an indispensable logical notion without which there is no meaning. This is not a spat against compositionality per se , since it is possible to posit extra arguments to the meaning function that composes complex meaning. However, that method would be inappropriate for a realistic notion of the meaning of assertions. To accomplish a realistic notion of meaning (as opposed e.g. to algebraic meaning), (...) Sperber and Wilson’s Relevance Theory (RT) may be applied in the spirit of Peirce’s Pragmatic Maxim (PM): the weighing of information depends on (i) the practical consequences of accommodating the chosen piece of information introduced in communication, and (ii) what will ensue in actually using that piece in further cycles of discourse. Peirce’s unpublished papers suggest a relevance-like approach to meaning. Contextual features influenced his logic of Existential Graphs (EG). Arguments are presented pro and con the view in which EGs endorse non-compositionality of meaning. (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 presents an information-based logic that is applied to the analysis of entailment, implicature and presupposition in natural language. The logic is very fine-grained and is able to make distinctions that are outside the scope of classical logic. It is independently motivated by certain properties of natural human reasoning, namely partiality, paraconsistency, relevance, and defeasibility: once these are accounted for, the data on implicature and presupposition comes quite naturally.The logic is based on the family (...) of semantic spaces known as bilattices, originally proposed by Ginsberg (1988), and used extensively by Fitting (1989, 1992). Specifically, the logic is based on a subset of bilattices that I call evidential bilattices, constructed as the Cartesian product of certain algebras with themselves. The specific details of the epistemic agent approach of the logical system is derived from the work of Belnap (1975, 1977), augmented by the use of evidential links for inferencing. An important property of the system is that it has been implemented using an extension of Fitting's work on bilattice logic programming (1989, 1991) to build a model-based inference engine for the augmented Belnap logic. This theorem prover is very efficient for a reasonably wide range of inferences. (shrink)
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.
As many philosophers agree, the frame problem is concerned with how an agent may efficiently filter out irrelevant information in the process of problem-solving. Hence, how to solve this problem hinges on how to properly handle semantic relevance in cognitive modeling, which is an area of cognitive science that deals with simulating human's cognitive processes in a computerized model. By "semantic relevance", we mean certain inferential relations among acquired beliefs which may facilitate information retrieval and practical reasoning under (...) certain epistemic constraints, e. g., the insufficiency of knowledge, the limitation of time budget, etc. However, traditional approaches to relevance—as for example, relevancelogic, the Bayesian approach, as well as Description Logic—have failed to do justice to the foregoing constraints, and in this sense, they are not proper tools for solving the frame problem/relevance problem. As we will argue in this paper, Non-Axiomatic Reasoning System (NARS) can handle the frame problem in a more proper manner, because the resulting solution seriously takes epistemic constraints on cognition as a fundamental theoretical principle. (shrink)
In Change of View: Principles of Reasoning, Gilbert Harman argues that (i) all genuine reasoning is a matter of belief revision, and that, since (ii) logic is not "specially relevant" to belief revision, (iii) logic is not specially relevant to reasoning, either. Thus, Harman suggests, what is needed is a "theory of reasoning"-which, incidentally, will be psychologistic, telling us both how we do and how we should reason. I argue that Harman fails to establish the need for such (...) a theory, because (a) reasoning is not always a matter of belief revision, and (b) logic is, in fact, of the utmost relevance to both reasoning and belief revision. (shrink)
This article advances the view that propositional logic can and should be taught within general education logic courses in ways that emphasizes its practical usefulness, much beyond what commonly occurs in logic textbooks. Discussion and examples of this relevance include database searching, understanding structured documents, and integrating concepts of proof construction with argument analysis. The underlying rationale for this approach is shown to have import for questions concerning the design of logic courses, textbooks, and the (...) general education curriculum, particularly the sequencing of formal and informal logic courses. (shrink)
The implicational fragment of the relevancelogic "ticket entailment" is closely related to the so-called hereditary right maximal terms. I prove that the terms that need to be considered as inhabitants of the types which are theorems of $T_\rightarrow$ are in normal form and built in all but one casefrom B, B' and W only. As a tool in the proof ordered term rewriting systems are introduced. Based on the main theorem I define $FIT_\rightarrow$ - a Fitch-style calculus (...) (related to $FT_\rightarrow$ ) for the implicational fragment of ticket entailment. (shrink)
I examine a Canadian Supreme Court decision concerning the constitutionality of Canada's 1982 rape-shield legislation, and suggest how material from the decision might profitably be used in an informal-logic class in connection with the topics of relevance and conductive argument. I also consider theoretical matters related to the decision: first I develop two analyses of what I call an argument from 'unchasteness' and connect them to George Bowles's theory of propositional relevance; then I present Trudy Govier with (...) a problem in response to which she might revise her account of a conductive argument in a way I describe. (shrink)
Informal logic is a new sub-discipline of philosophy, roughly definable as the philosophy of argument. Contributors have challenged the traditional concept of an argument as a premiss-conclusion complex, in favour of speech-act, functional and dialogical conceptions; they have identified as additional components warrants, modal qualifiers, rebuttals, and a dialectical tier. They have objected that "soundness" is neither necessary nor sufficient for a good argument. Alternative proposals include acceptability, relevance and sufficiency of the premisses; conformity to a valid argument (...) schema; conformity to rules for discussion aimed at rational resolution of a dispute. Informal logic is a significant part of philosophy. (shrink)
We give a unified account of some results in the development of Polyadic Inductive Logic in the last decade with particular reference to the Principle of Spectrum Exchangeability, its consequences for Instantial Relevance, Language Invariance and Johnson's Sufficientness Principle, and the corresponding de Finetti style representation theorems.
This paper presents eight (previously unpublished) adaptive logics for belief revision, each of which define a belief revision operation in the sense of the AGM framework. All these revision operations are shown to satisfy the six basic AGM postulates for belief revision, and Parikh's axiom of Relevance. Using one of these logics as an example, we show how their proof theory gives a more dynamic flavor to belief revision than existing approaches. It is argued that this turns belief revision (...) (that obeys Relevance) into a more natural undertaking, where analytic steps are performed only as soon as they turn out to be necessary in order to uphold certain beliefs. (shrink)
We shed light on an old problem by showing that the logic LP cannot define a binary connective $\odot$ obeying detachment in the sense that every valuation satisfying $\varphi$ and $(\varphi\odot\psi)$ also satisfies $\psi$ , except trivially. We derive this as a corollary of a more general result concerning variable sharing.