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.

Whitehead, in two basic books, considers two different approaches to point-free geometry: the inclusion-based approach , whose primitive notions are regions and inclusion relation between regions, and the connection-based approach , where the connection relation is considered instead of the inclusion. We show that the latter cannot be reduced to the first one, although this can be done in the framework of multivalued logics.

Effective domain theory is applied to fuzzy logic. The aim is to give suitable notions of semi-decidable and decidable L-subset and to investigate about the effectiveness of the fuzzy deduction apparatus.

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.

In this transdisciplinary article which stems from philosophical considerations (that depart from phenomenology—after Merleau-Ponty, Heidegger and Rosen—and Hegelian dialectics), we develop a conception based on topological (the Moebius surface and the Klein bottle) and geometrical considerations (based on torsion and non-orientability of manifolds), and multivalued logics which we develop into a unified world conception that surmounts the Cartesian cut and Aristotelian logic. The role of torsion appears in a self-referential construction of space and time, which will be further (...) related to the commutator of the True and False operators of matrix logic, still with a quantum superposed state related to a Moebius surface, and as the physical field at the basis of Spencer-Brown’s primitive distinction in the protologic of the calculus of distinction. In this setting, paradox, self-reference, depth, time and space, higher-order non-dual logic, perception, spin and a time operator, the Klein bottle, hypernumbers due to Musès which include non-trivial square roots of ±1 and in particular non-trivial nilpotents, quantum field operators, the transformation of cognition to spin for two-state quantum systems, are found to be keenly interwoven in a world conception compatible with the philosophical approach taken for basis of this article. The Klein bottle is found not only to be the topological in-formation for self-reference and paradox whose logical counterpart in the calculus of indications are the paradoxical imaginary time waves, but also a classical-quantum transformer (Hadamard’s gate in quantum computation) which is indispensable to be able to obtain a complete multivalued logical system, and still to generate the matrix extension of classical connective Boolean logic. We further find that the multivalued logic that stems from considering the paradoxical equation in the calculus of distinctions, and in particular, the imaginary solutions to this equation, generates the matrix logic which supersedes the classical logic of connectives and which has for particular subtheories fuzzy and quantum logics. Thus, from a primitive distinction in the vacuum plane and the axioms of the calculus of distinction, we can derive by incorporating paradox, the world conception succinctly described above. (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.

Alex Oliver and Timothy Smiley provide a new account of plural logic. They argue that there is such a thing as genuinely plural denotation in logic, and expound a framework of ideas that includes the distinction between distributive and collective predicates, the theory of plural descriptions, multivalued functions, and lists.

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)

The logic of inexactness, presented in this paper, is a version of the Łukasiewicz logic with predicates valued in [0, ∞). We axiomatize multi-valued models of equality and ordering in this logic guaranteeing their imbeddibility in the real line. Our axioms of equality and ordering, when interpreted as axioms of proximity and dominance, can be applied to the foundations of measurement (especially in the social sciences). In two-valued logic they provide theories of ratio scale measurement. In multivalued logic they (...) enable us to treat formally errors arising in nominal and ordinal measurements. (shrink)

Alex Oliver and Timothy Smiley provide a new account of plural logic. They argue that there is such a thing as genuinely plural denotation in logic, and expound a framework of ideas that includes the distinction between distributive and collective predicates, the theory of plural descriptions, multivalued functions, and lists.

Alex Oliver and Timothy Smiley provide a new account of plural logic. They argue that there is such a thing as genuinely plural denotation in logic, and expound a framework of ideas that includes the distinction between distributive and collective predicates, the theory of plural descriptions, multivalued functions, and lists.

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)

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)

The article presents a brief analysis of how the existence of various logics became possi­ble. This is shown on the example of such well-known logical theories as syllogistics, temporal, multivalued, intuitionistic, paraconsistent and quantum logics. Each of them arose not on someone’s whim, but to solve specific problems. They are based on the most general ontological assumptions about the subject area under study. In formal logic onto­logical assumptions are refined in the concept of a model structure. Since (...) it is impossible to talk about logic in isolation from the language used, the most general epistemic as­sumptions about the nature of the relationship of linguistic expressions to those objects of extralinguistic reality that they represent are also accepted. One of the most important of these relationships is the concept of the truth of sentences, which was first formulated by Plato and Aristotle. Taking certain ontological and epistemic assumptions depending on the problem being solved, we obtain different logics. Process logic is primarily char­acterized by special ontological assumptions that are fundamentally different from the as­sumptions of other currently existing logics. The ontology of processes is an ontology of developing processes, not things. Historically, it was most vividly described in the writings of Heraclitus. In the overwhelming majority of modern approaches to the de­scription of processes, we see attempts to reduce them to sequences of states, which de­values the very concept of a process, just as a cinematic picture of the flow of time deval­ues the concept of time. Since logics are built on the basis of various ontological and epistemic assumptions, they are inherently theories of these accepted assumptions, and not universal reasoning tools that don’t depend on the characteristics of the study area and the categories of linguistic expressions. Universal logic is possible if one rises from the level of specific languages to a higher level of sign theory. (shrink)

We propose a distinction between two different concepts of time that play a role in physics: geometric time and creative time. The former is the time of deterministic physics and merely parametrizes a given evolution. The latter is instead characterized by real change, i.e. novel information that gets created when a non-necessary event becomes determined in a fundamentally indeterministic physics. This allows us to give a naturalistic characterization of the present as the moment that separates the potential future from the (...) determined past. We discuss how these two concepts find natural applications in classical and intuitionistic mathematics, respectively, and in classical and multivalued tensed logic, as well as how they relate to the well-known A- and B-theories in the philosophy of time. (shrink)

We extend the earlier results on the equivalence between the Boolean and the multivalued dependencies in relational databases and fragments of the Boolean propositional logic. It is shown that these equivalences are still valid for the databases that store complex data elements obtained from the recursive nesting of record, list, set and multiset constructors. The major proof argument utilises properties of Brouwerian algebras.The equivalences have several consequences. Firstly, they provide new insights into databases that are not in first normal (...) form. Secondly, they characterise the implication of data dependencies in nested databases in purely logical terms. The database designer can take advantage of these equivalences to reduce database design problems to well-studied problems in Boolean propositional logic. Furthermore, relational database design solutions can be reused to solve problems for nested databases. (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)

This book focuses on mathematical logic that was a prominent philosophical method in the Lvov-Warsaw School. Kazimierz Twardowski, who was the founder of the Lvov-Warsaw School, favoured psychology as a philosophical method. However, his pupil Jan Łukasiewicz opposed to him and claimed that the proper method is mathematical logic. Łukasiewicz’s work attracted many other members of the Lvov-Warsaw School and soon became one of its most important subjects of interest. The results of its members in the field of mathematical logic (...) have achieved world fame, but they have not yet been presented thoroughly and systematically in the Czech philosophical literature. The aim of this work is to partially fill this gap. The first chapter is a historical introduction to the issue. The subsequent three chapters deal directly with the logical systems created and developed by the logicians of the Lvov-Warsaw School: propositional calculus, multivalued logics, and Leśniewski’s system of logic and the foundations of mathematics. The last three chapters focus on the philosophical theories that have been influenced by mathematical logic: theories of truth, problems of deduction, induction, and methodology of science, as well as the original approach to the history of logic introduced by Łukasiewicz. (shrink)

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 give a new and elementary construction of primitive positive decomposition of higher arity relations into binary relations on finite domains. Such decompositions come up in applications to constraint satisfaction problems, clone theory and relational databases. The construction exploits functional completeness of 2-input functions in many-valued logic by interpreting relations as graphs of partially defined multivalued ‘functions’. The ‘functions’ are then composed from ordinary functions in the usual sense. The construction is computationally effective and relies on well-developed methods of (...) functional decomposition, but reduces relations only to ternary relations. An additional construction then decomposes ternary into binary relations, also effectively, by converting certain disjunctions into existential quantifications. The result gives a uniform proof of Peirce’s reduction thesis on finite domains, and shows that the graph of any Sheffer function composes all relations there. (shrink)

We determine the computational complexity of the Hahn-Banach Extension Theorem. To do so, we investigate some basic connections between reverse mathematics and computable analysis. In particular, we use Weak König's Lemma within the framework of computable analysis to classify incomputable functions of low complexity. By defining the multivalued function Sep and a natural notion of reducibility for multivalued functions, we obtain a computational counterpart of the subsystem of second-order arithmetic WKL0. We study analogies and differences between WKL0 and (...) the class of Sep-computable multivalued functions. Extending work of Brattka, we show that a natural multivalued function associated with the Hahn-Banach Extension Theorem is Sep-complete. (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.

Kulicki and Trypuz (2016) introduced three systems of multivalued deontic action logic to handle normative conflicts. The first system suggests a pessimistic view on normative conflicts, according to which any conflicting option represents something forbidden; the second system suggests an optimistic view, according to which any conflicting option represents something obligatory; finally, the third system suggests a neutral view, according to which any conflicting option represents something that is neither obligatory nor forbidden. The aim of the present paper is (...) to propose a fourth system in this family, which comes with a realistic view on normative conflicts: a normative conflict remains unsolved unless it is generated by two or more normative sources that can be compared. In accordance with this, we will provide a more refined formal framework for the family of systems at issue, which allows for explicit reference to sources of norms. Conflict resolution is thus a consequence of a codified hierarchy of normative sources. (shrink)

A problem is a multivalued function from a set of instances to a set of solutions. We consider only instances and solutions coded by sets of integers. A problem admits preservation of some computability-theoretic weakness property if every computable instance of the problem admits a solution relative to which the property holds. For example, cone avoidance is the ability, given a noncomputable set A and a computable instance of a problem ${\mathsf {P}}$, to find a solution relative to which (...) A is still noncomputable.In this article, we compare relativized versions of computability-theoretic notions of preservation which have been studied in reverse mathematics, and prove that the ones which were not already separated by natural statements in the literature actually coincide. In particular, we prove that it is equivalent to admit avoidance of one cone, of $\omega $ cones, of one hyperimmunity or of one non- $\Sigma ^{0}_1$ definition. We also prove that the hierarchies of preservation of hyperimmunity and non- $\Sigma ^{0}_1$ definitions coincide. On the other hand, none of these notions coincide in a nonrelativized setting. (shrink)

In this article, we develop tame topology over dp-minimal structures equipped with definable uniformities satisfying certain assumptions. Our assumptions are enough to ensure that definable sets are tame: there is a good notion of dimension on definable sets, definable functions are almost everywhere continuous, and definable sets are finite unions of graphs of definable continuous “multivalued functions.” This generalizes known statements about weakly o-minimal, C-minimal, and P-minimal theories.

We investigate the uniform computational content of the open and clopen Ramsey theorems in the Weihrauch lattice. While they are known to be equivalent to $\mathrm {ATR_0}$ from the point of view of reverse mathematics, there is not a canonical way to phrase them as multivalued functions. We identify eight different multivalued functions and study their degree from the point of view of Weihrauch, strong Weihrauch, and arithmetic Weihrauch reducibility. In particular one of our functions turns out to (...) be strictly stronger than any previously studied multivalued functions arising from statements around $\mathrm {ATR}_0$. (shrink)

A wide family of many-valued logics—for instance, those based on the weak Kleene algebra—includes a non-classical truth-value that is ‘contaminating’ in the sense that whenever the value is assigned to a formula φ⁠, any complex formula in which φ appears is assigned that value as well. In such systems, the contaminating value enjoys a wide range of interpretations, suggesting scenarios in which more than one of these interpretations are called for. This calls for an evaluation of systems with multiple (...) contaminating values. In this paper, we consider the countably infinite family of multiple-conclusion consequence relations in which classical logic is enriched with one or more contaminating values whose behavior is determined by a linear ordering between them. We consider some motivations and applications for such systems and provide general characterizations for all consequence relations in this family. Finally, we provide sequent calculi for a pair of four-valued logics including two linearly ordered contaminating values before defining two-sided sequent calculi corresponding to each of the infinite family of many-valued logics studied in this paper. (shrink)

Contemporary feminist theory is at an impasse: the project of reformulating concepts of self and social identity is thwarted by an association between identity and oppression and victimhood. In Sacrificial Logics, Allison Weir proposes a way out of this impasse through a concept of identity which depends on accepting difference. Weir argues that the equation of identity with repression and domination links "relational" feminists like Nancy Chodorow, who equate self-identity with the repression of connection to others, and poststructuralist feminists (...) like Judith Butler, who view any identity as a repression of nonidentity and difference. Through readings of Chodorow, Butler, Jessica Benjamin, Luce Irigaray, Jacqueline Rose and Julia Kristeva, Weir analyzes the relation of theories of self-identity to theories of women's identity, social identity, the identity of meaning in language and feminist solidarity. Drawing particularly on the work of Julia Kristeva, she argues for a reformulation of self-identity as a capacity to participate in a social world, and sketches a model of a self-identity which depends on a capacity to accept nonidentity, difference and connections to others. (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)

The class of equivalential logics comprises all implicative logics in the sense of Rasiowa [9], Suszko's logic SCI and many others. Roughly speaking, a logic is equivalential iff the greatest strict congruences in its matrices are determined by polynomials. The present paper is the first part of the survey in which systematic investigations into this class of logics are undertaken. Using results given in [3] and general theorems from the theory of quasi-varieties of models [5] we give (...) a characterization of all simple C-matrices for any equivalential logic C. In corollaries we give necessary and sufficient conditions for the class of all simple models for a given equivalential logic to be closed under free products. (shrink)

This paper is a contribution to the general study of consequence relations which contain (definable) connective of “disjunction”. Our work is centered around the “proof by cases property”, we present several of its equivalent definitions, and show some interesting applications, namely in constructing axiomatic systems for intersections of logics and recognizing weakly implicative fuzzy logics among the weakly implicative ones.

It is a classic result (McKinsey & Tarski, 1944; Rasiowa & Sikorski, 1963) that if we interpret modal diamond as topological closure, then the modal logic of any dense-in-itself metric space is the well-known modal system S4. In this paper, as a natural follow-up, we study the modal logic of an arbitrary metric space. Our main result establishes that modal logics arising from metric spaces form the following chain which is order-isomorphic (with respect to the ⊃ relation) to the (...) ordinalω+ 3:$S4.Gr{z_1} \supset S4.Gr{z_2} \supset S4.Gr{z_3} \supset \cdots \,S4.Grz \supset S4.1 \supset S4.$It follows that the modal logic of an arbitrary metric space is finitely axiomatizable, has the finite model property, and hence is decidable. (shrink)

This volume examines the concept of falsification as a central notion of semantic theories and its effects on logical laws. The point of departure is the general constructivist line of argument that Michael Dummett has offered over the last decades. From there, the author examines the ways in which falsifications can enter into a constructivist semantics, displays the full spectrum of options, and discusses the logical systems most suitable to each one of them. While the idea of introducing falsifications into (...) the semantic account is Dummett's own, the many ways in which falsificationism departs quite radically from verificationism are here spelled out in detail for the first time. The volume is divided into three large parts. The first part provides important background information about Dummett’s program, intuitionism and logics with gaps and gluts. The second part is devoted to the introduction of falsifications into the constructive account and shows that there is more than one way in which one can do this. The third part details the logical effects of these various moves. In the end, the book shows that the constructive path may branch in different directions: towards intuitionistic logic, dual intuitionistic logic and several variations of Nelson logics. The author argues that, on balance, the latter are the more promising routes to take. "Kapsner’s book is the first detailed investigation of how to incorporate the notion of falsification into formal logic. This is a fascinating logico-philosophical investigation, which will interest non-classical logicians of all stripes." Graham Priest, Graduate Center, City University of New York and University of Melbourne. (shrink)

Various four- and three-valued modal propositional logics are studied. The basic systems are modal extensions BK and BS4 of Belnap and Dunn's four-valued logic of firstdegree entailment. Three-valued extensions of BK and BS4 are considered as well. These logics are introduced semantically by means of relational models with two distinct evaluation relations, one for verification and the other for falsification. Axiom systems are defined and shown to be sound and complete with respect to the relational semantics and with (...) respect to twist structures over modal algebras. Sound and complete tableau calculi are presented as well. Moreover, a number of constructive non-modal logics with strong negation are faithfully embedded into BS4, into its three-valued extension B3S4, or into temporal BS4, BtS4. These logics include David Nelson's three-valued logic N3, the four-valued logic N4 bottom, the connexive logic C, and several extensions of bi-intuitionistic logic by strong negation. (shrink)

Multi-modal versions of propositional logics S5 or S4—commonly accepted as logics of knowledge—are capable of describing static states of knowledge but they do not reflect how the knowledge changes after communications among agents. In the present paper (part of broader research on logics of knowledge and communications) we define extensions of the logic S5 which can deal with public communications. The logics have natural semantics. We prove some completeness, decidability and interpretability results and formulate a general (...) method that solves certain kind of problems involving public communications—among them well known puzzles of Muddy Children and Mr. Sum & Mr. Product. As the paper gives a formal logical treatment of the operation of restriction of the universe of a Kripke model, it contributes also to investigations of semantics for modal logics. (shrink)

The proof theory of many-valued systems has not been investigated to an extent comparable to the work done on axiomatizatbility of many-valued logics. Proof theory requires appropriate formalisms, such as sequent calculus, natural deduction, and tableaux for classical (and intuitionistic) logic. One particular method for systematically obtaining calculi for all finite-valued logics was invented independently by several researchers, with slight variations in design and presentation. The main aim of this report is to develop the proof theory of finite-valued (...) first order logics in a general way, and to present some of the more important results in this area. In Systems covered are the resolution calculus, sequent calculus, tableaux, and natural deduction. This report is actually a template, from which all results can be specialized to particular logics. (shrink)

In 2016, Béziau introduces a restricted notion of paraconsistency, the so-called genuine paraconsistency. A logic is genuine paraconsistent if it rejects the laws $\varphi,\neg \varphi \vdash \psi$ and $\vdash \neg (\varphi \wedge \neg \varphi)$. In that paper, the author analyzes, among the three-valued logics, which of them satisfy this property. If we consider multiple-conclusion consequence relations, the dual properties of those above-mentioned are $\vdash \varphi, \neg \varphi$ and $\neg (\varphi \vee \neg \varphi) \vdash$. We call genuine paracomplete logics (...) those rejecting the mentioned properties. We present here an analysis of the three-valued genuine paracomplete logics. A very natural twist structures semantics for these logics is also found in a systematic way. This semantics produces automatically a simple and elegant Hilbert-style characterization for all these logics. Finally, we introduce the logic LGP which is genuine paracomplete is not genuine paraconsistent, not even paraconsistent and cannot be characterized by a single finite logical matrix. (shrink)

We characterise the intermediate logics which admit a cut-free hypersequent calculus of the form \, where \ is the hypersequent counterpart of the sequent calculus \ for propositional intuitionistic logic, and \ is a set of so-called structural hypersequent rules, i.e., rules not involving any logical connectives. The characterisation of this class of intermediate logics is presented both in terms of the algebraic and the relational semantics for intermediate logics. We discuss various—positive as well as negative—consequences of (...) this characterisation. (shrink)

This paper studies connections between two alternatives to the standard probability calculus for representing and reasoning about uncertainty: imprecise probability andcomparative probability. The goal is to identify complete logics for reasoning about uncertainty in a comparative probabilistic language whose semantics is given in terms of imprecise probability. Comparative probability operators are interpreted as quantifying over a set of probability measures. Modal and dynamic operators are added for reasoning about epistemic possibility and updating sets of probability measures.

We obtain modal completeness of the interpretability logics IL $\!\!\textsf {P}_{\textsf {0}}$ and ILR w.r.t. generalised Veltman semantics. Our proofs are based on the notion of full labels [2]. We also give shorter proofs of completeness w.r.t. the generalised semantics for many classical interpretability logics. We obtain decidability and finite model property w.r.t. the generalised semantics for IL $\textsf {P}_{\textsf {0}}$ and ILR. Finally, we develop a construction that might be useful for proofs of completeness of extensions of (...) ILW w.r.t. the generalised semantics in the future, and demonstrate its usage with $\textbf {IL}\textsf {W}^\ast = \textbf {IL}\textsf {WM}_{\textsf {0}}$. (shrink)

This article proposes the meeting of fuzzy logic with paraconsistency in a very precise and foundational way. Specifically, in this article we introduce expansions of the fuzzy logic MTL by means of primitive operators for consistency and inconsistency in the style of the so-called Logics of Formal Inconsistency (LFIs). The main novelty of the present approach is the definition of postulates for this type of operators over MTL-algebras, leading to the definition and axiomatization of a family of logics, (...) expansions of MTL, whose degree-preserving counterpart are paraconsistent and moreover LFIs. (shrink)

The paper presents a uniform proof-theoretic treatment of several kinds of free logic, including the logics of existence and definedness applied in constructive mathematics and computer science, and called here quasi-free logics. All free and quasi-free logics considered are formalised in the framework of sequent calculus, the latter for the first time. It is shown that in all cases remarkable simplifications of the starting systems are possible due to the special rule dealing with identity and existence predicate. (...) Cut elimination is proved in a constructive way for sequent calculi adequate for all logics under consideration. (shrink)

This is the continuation of the paper :417–446, 2010). We continue the abstract study of non-classical logics based on the kind of generalized implication connectives they possess and we focus on semilinear logics, i.e. those that are complete with respect to the class of models where the implication defines a linear order. We obtain general characterizations of semilinearity in terms of the intersection-prime extension property, the syntactical semilinearity metarule and the class of finitely subdirectly irreducible models. Moreover, we (...) consider extensions of the language with lattice connectives and generalized disjunctions, study their interplay with implication and obtain axiomatizations and further descriptions of semilinear logics in terms of disjunctions and the proof by cases property. (shrink)

2nd edition. Many-valued logics are those logics that have more than the two classical truth values, to wit, true and false; in fact, they can have from three to infinitely many truth values. This property, together with truth-functionality, provides a powerful formalism to reason in settings where classical logic—as well as other non-classical logics—is of no avail. Indeed, originally motivated by philosophical concerns, these logics soon proved relevant for a plethora of applications ranging from switching theory (...) to cognitive modeling, and they are today in more demand than ever, due to the realization that inconsistency and vagueness in knowledge bases and information processes are not only inevitable and acceptable, but also perhaps welcome. The main modern applications of (any) logic are to be found in the digital computer, and we thus require the practical knowledge how to computerize—which also means automate—decisions (i.e. reasoning) in many-valued logics. This, in turn, necessitates a mathematical foundation for these logics. This book provides both these mathematical foundation and practical knowledge in a rigorous, yet accessible, text, while at the same time situating these logics in the context of the satisfiability problem (SAT) and automated deduction. The main text is complemented with a large selection of exercises, a plus for the reader wishing to not only learn about, but also do something with, many-valued logics. (shrink)