This is an exploratory paper whose aim is to investigate the potentialities of bilattice theory for an adequate definition of the deduction apparatus for multi-valued logic. We argue that bilattice theory enables us to obtain a nice extension of the graded approach to fuzzy logic. To give an example, a completeness theorem for a logic based on Boolean algebras is proved.

Trying to overcome Dugundji’s result on uncharacterisability of modal logics by finite logical matrices, Kearns and Ivlev proposed, independently, a characterisation of some modal systems by means of four-valued multivalued truth-functions , as an alternative to Kripke semantics. This constitutes an antecedent of the non-deterministic matrices introduced by Avron and Lev . In this paper we propose a reconstruction of Kearns’s and Ivlev’s results in a uniform way, obtaining an extension to another modal systems. The first part of (...) the paper is devoted to four-valued Nmatrices, including Kearns’s and Ivlev’s. Besides proving with full details Kearns’s results for T, S4 and S5, we also obtain a characterisation of the system B by four-valued Nmatrices with level valuations. Concerning Ivlev’s results, two new modal systems are introduced and char.. (shrink)

We show that the existence of an infinite set can be reduced to the existence of finite sets “as big as we will”, provided that a multivalued extension of the relation of equipotence is admitted. In accordance, we modelize the notion of infinite set by a fuzzy subset representing the class of wide sets.

We introduce generalized quantifiers, as defined in Tarskian semantics by Mostowski and Lindström, in logics whose semantics is based on teams instead of assignments, e.g., IF-logic and Dependence logic. Both the monotone and the non-monotone case is considered. It is argued that to handle quantifier scope dependencies of generalized quantifiers in a satisfying way the dependence atom in Dependence logic is not well suited and that the multivalued dependence atom is a better choice. This atom is in (...) fact definably equivalent to the independence atom recently introduced by Väänänen and Grädel. (shrink)

It is part of the current wisdom that the Liar and similar semantic paradoxes can be taken care of by the use of certain non-classical multivalued logics. In this paper I want to suggest that bivalent logic can do just as well. This is accomplished by using a non-deterministic matrix to define the negation connective. I show that the systems obtained in this way support a transparent truth predicate. The paper also contains some remarks on the conceptual interest of (...) such systems. (shrink)

Demonstrating the different roles that logic plays in the disciplines of computer science, mathematics, and philosophy, this concise undergraduate textbook covers select topics from three different areas of logic: proof theory, computability theory, and nonclassical logic. The book balances accessibility, breadth, and rigor, and is designed so that its materials will fit into a single semester. Its distinctive presentation of traditional logic material will enhance readers' capabilities and mathematical maturity. The proof theory portion presents classical propositional logic and first-order logic (...) using a computer-oriented (resolution) formal system. Linear resolution and its connection to the programming language Prolog are also treated. The computability component offers a machine model and mathematical model for computation, proves the equivalence of the two approaches, and includes famous decision problems unsolvable by an algorithm. The section on nonclassical logic discusses the shortcomings of classical logic in its treatment of implication and an alternate approach that improves upon it: Anderson and Belnap's relevance logic. Applications are included in each section. The material on a four-valued semantics for relevance logic is presented in textbook form for the first time. Aimed at upper-level undergraduates of moderate analytical background, Three Views of Logic will be useful in a variety of classroom settings. Gives an exceptionally broad view of logic. Treats traditional logic in a modern format. Presents relevance logic with applications. Provides an ideal text for a variety of one-semester upper-level undergraduate courses. (shrink)

The serious formal investigation of n-valued systems of logic for n>2 dates back to Post's 1921 doctoral dissertation. The primary use for such structures, however, has been as model-theoretic devices in the investigation of systems of lower order. Ackermann's short book now comes as a welcome addition to the literature dealing with the formal properties and applications of n-valued systems in their own right. Ackermann begins with a general discussion of implicational calculi in which fundamental ideas of validity, well-formedness, and (...) other basic notions are set forth elegantly and clearly. While assuming the classical implicational calculus as an underlying substructure, Ackermann next motivates a consideration of possible modifications of the formal system in the direction of greater generality by means of the standard Lewis objections to the reading of "⊃" as "implies." This leads to the well known story of strict entailment, introduction of modal operators, and then a four-valued matrix as an underlying semantics for the expanded system. There then follows a good general discussion of the properties of many-valued logics, various nonstandard propositional calculi, and an admirable presentation of the Lukasiewicz-Tarski propositional calculus, accompanied by a number of important metatheorems. Ackermann also includes chapters on alternative multivalued [[sic]] systems and a glance at their applications. One would like to see, however, some discussion of such controversial matters as n-valued quantificational theory and the well-known objections to the entire program of nonstandard logistic theory. There is a good bibliography of important books and articles in the field.—H. P. K. (shrink)

Minimal Semantics asks what a theory of literal linguistic meaning is for - if you were to be given a working theory of meaning for a language right now, what would you be able to do with it? Emma Borg sets out to defend a formal approach to semantic theorising from a relatively new type of opponent - advocates of what she call 'dual pragmatics'. According to dual pragmatists, rich pragmatic processes play two distinct roles in linguistic comprehension: as (...) well as operating in a post-semantic capacity to determine the implicatures of an utterance, they also operate prior to the determination of truth-conditional content for a sentence. That is to say, they have an integral role to play within what is usually thought of as the semantic realm. Borg believes dual pragmatic accounts constitute the strongest contemporary challenge to standard formal approaches to semantics since they challenge the formal theorist to show not merely that there is some role for formal processes on route to determination of semantic content, but that such processes are sufficient for determining content. Minimal Semantics provides a detailed examination of this school of thought, introducing readers who are unfamiliar with the topic to key ideas like relevance theory and contextualism, and looking in detail at where these accounts diverge from the formal approach. Borg's defence of formal semantics has two main parts: first, she argues that the formal approach is most naturally compatible with an important and well-grounded psychological theory, namely the Fodorian modular picture of the mind. Then she argues that the main arguments adduced by dual pragmatists against formal semantics - concerning apparent contextual intrusions into semantic content - can in fact be countered by a formal theory. The defence holds, however, only if we are sensitive to the proper conditions of success for a semantic theory. Specifically, we should reject a range of onerous constraints on semantic theorizing (e.g., that it answer epistemic or metaphysical questions, or that it explain our communicative skills) and instead adopt a quite minimal picture of semantics. (shrink)

The urgent drive for vaccine development in the midst of the current COVID-19 pandemic has prompted public and private organisations to invest heavily in research and development of a COVID-19 vaccine. Organisations globally have affirmed the commitment of fair global access, but the means by which a successful vaccine can be mass produced and equitably distributed remains notably unanswered. Barriers for low-income countries include the inability to afford vaccines as well as inadequate resources to vaccinate, barriers that are exacerbated during (...) a pandemic. Fair distribution of a pandemic vaccine is unlikely without a solid ethical framework for allocation. This piece analyses four allocation paradigms: ability to develop or purchase; reciprocity; ability to implement; and distributive justice, and synthesises their ethical considerations to develop an allocation model to fit the COVID-19 pandemic. (shrink)

Organised into three sections covering the place of semantics in linguistics, the description of sentence and word meanings, and current theoretical approaches to semantics, this book is aimed at undergraduates as well as general readers.

The aim of this research work is to find out some results in fixed point theory for a pair of families of multivalued mappings fulfilling a new type of U -contractions in modular-like metric spaces. Some new results in graph theory for multigraph-dominated contractions in modular-like metric spaces are developed. An application has been presented to ensure the uniqueness and existence of a solution of families of nonlinear integral equations.

_Insensitive Semantics_ is an overview of and contribution to the debates about how to accommodate context sensitivity within a theory of human communication, investigating the effects of context on communicative interaction and, as a corollary, what a context of utterance is and what it is to be in one. Provides detailed and wide-ranging overviews of the central positions and arguments surrounding contextualism Addresses broad and varied aspects of the distinction between the semantic and non-semantic content of language Defends a distinctive (...) and explanatorily powerful combination of semantic minimalism and speech act pluralism Confronts core problems which not only run to the heart of philosophy of language and linguistics, but which arise in epistemology, metaphysics, and moral philosophy as well. (shrink)

This paper challenges the negative attitudes towards labelled proof systems, usually referred to as semantic pollution, by arguing that such critiques overlook the full potential of labelled calculi. The overarching objective is to develop a practice-based philosophical analysis of labelled calculi to provide insightful considerations regarding their proof-theoretic and philosophical value. To achieve this, successful applications of labelled calculi and related results will be showcased, and comparisons with other relevant works will be discussed. The paper ends by advocating for a (...) more practice-based approach towards the philosophical understanding of proof systems and their role in structural proof theory. (shrink)

ABSTRACT A modal logic interpretation of Dempster-Shafer theory is developed in the framework of multivalued models of modal logic, i.e. models in which in any possible world an arbitrary number (possibly zero) of atomic propositions can be true. Several approaches to conditioning in multivalued models of modal logic are presented.

We define the notion of “potential existence” by starting from the fact that in multi-valued logic the existential quantifier is interpreted by the least upper bound operator. Besides, we try to define in a general way how to pass from potential into actual existence.

Lexical Semantics is about the meaning of words. Although obviously a central concern of linguistics, the semantic behaviour of words has been unduly neglected in the current literature, which has tended to emphasize sentential semantics and its relation to formal systems of logic. In this textbook D. A. Cruse establishes in a principled and disciplined way the descriptive and generalizable facts about lexical relations that any formal theory of semantics will have to encompass. Among the topics covered (...) in depth are idiomaticity, lexical ambiguity, synonymy, hierarchical relations such as hyponymy and meronymy, and various types of oppositeness. Syntagmatic relations are also treated in some detail. The discussions are richly illustrated by examples drawn almost entirely from English. Although a familiarity with traditional grammar is assumed, readers with no technical linguistic background will find the exposition always accessible. All readers with an interest in semantics will find in this original text not only essential background but a stimulating new perspective on the field. (shrink)

Understanding Semantics offers an up-to-date, broad and thorough introduction to linguistic semantics, the field of linguistic meaning central to the understanding of language. The book takes a step-by-step approach, starting with the basic concepts and moving through central questions to an examination of the methods and results of the science of linguistic meaning.