Generalized Quantifiers and Modalities

Bisimulation is a central concept in the model theory of modal logic with extensive computational applications. It is a relation between two models in which related states have identical atomic properties and matching transition possibilities. Bisimulation captures the expressive power of propositional modal logic in the sense that bisimilar states are not distinguishable by any propositional modal formula. In recent years, fuzzy modal logic has received much attention because of the connection between uncertainty measures and fuzzy modalities. In this paper, we define the notion of fuzzy bisimulation and prove that, in a fuzzified version, it bears the same relationship to fuzzy modal logic that bisimulation bears to propositional modal logic.

As an extension of the traditional modal logic, the fuzzy first-order modal logic is discussed in this paper. A description of fuzzy first-order modal logic based on constant domain semantics is given, and a formal system of fuzzy reasoning based on the semantic information of models of first-order modal logic is established. It is also introduced in this paper the notion of the satisfiability of the reasoning system and some properties associated with the satisfiability are proved.

Modal logics and behavioral equivalences play an important role in the specification and verification of concurrent systems. In this paper, we first present a new notion of bisimulation for nondeterministic fuzzy transition systems, which is distribution based and coarser than state-based bisimulation appeared in the literature. Then, we define a distribution-based bisimilarity metric as the least fixed point of a suitable monotonic function on a complete lattice, which is a behavioral distance and is a more robust way of formalizing behavioral similarity between states than bisimulations. We also propose an on-the-fly algorithm for computing the bisimilarity metric. Moreover, we present a fuzzy modal logic and provide a sound and complete characterization of the bisimilarity metric. Interestingly, this characterization holds for a class of fuzzy modal logics. In addition, we show the nonexpansiveness of a typical parallel composition operator with respect to the bisimilarity metric, which makes compositional verification possible.

In the paper we introduce formal calculi which are a generalization of propositional modal logics. These calculi are called fuzzy modal logics. We introduce the concept of a fuzzy Kripke model and consider a semantics of these calculi in the class of fuzzy Kripke models. The main result of the paper is the completeness theorem of a minimal fuzzy modal logic in the class of fuzzy Kripke models.

Willard Van Orman Quine. On what there is. A reprint of XIX 134(1). From a logical point of view, by Willard Van Orman Quine, second, revised edition, Harvard University Press, Cambridge, Mass., 1961, and Harper Torchbooks, The Science Library, Harper & Row, New York and Evanston 1963, pp. 1–19. - Willard Van Orman Quine. Two dogmas of empiricism. A reprint of XIX 134(2). From a logical point of view, by Willard Van Orman Quine, second, revised edition, Harvard University Press, Cambridge, Mass., 1961, and Harper Torchbooks, The Science Library, Harper & Row, New York and Evanston 1963, pp. 20–46. - Willard Van Orman Quine. The problem of meaning in linguistics. A reprint of XIX 134(3). From a logical point of view, by Willard Van Orman Quine, second, revised edition, Harvard University Press, Cambridge, Mass., 1961, and Harper Torchbooks, The Science Library, Harper & Row, New York and Evanston 1963, pp. 47–64. - Willard Van Orman Quine. Identity, ostension, and hypostasis. A reprint of XIX 134(4). From a logical point of view, by Willard Van Orman Quine, second, revised edition, Harvard University Press, Cambridge, Mass., 1961, and Harper Torchbooks, The Science Library, Harper & Row, New York and Evanston 1963, pp. 65–79. - Willard Van Orman Quine. New foundations for mathematical logic. A reprint of XIX 134(5). From a logical point of view, by Willard Van Orman Quine, second, revised edition, Harvard University Press, Cambridge, Mass., 1961, and Harper Torchbooks, The Science Library, Harper & Row, New York and Evanston 1963, pp. 80–101. - Willard Van Orman Quine. Logic and the reification of universals. A revised reprint of XIX 135. From a logical point of view, by Willard Van Orman Quine, second, revised edition, Harvard University Press, Cambridge, Mass., 1961, and Harper Torchbooks, The Science Library, Harper & Row, New York and Evanston 1963, pp. 102–129. - Willard Van Orman Quine. Notes on the theory of reference. A reprint of XIX 136. From a logical point of view, by Willard Van Orman Quine, second, revised edition, Harvard University Press, Cambridge, Mass., 1961, and Harper Torchbooks, The Science Library, Harper & Row, New York and Evanston 1963, pp. 130–138. - Willard Van Orman Quine. Reference and modality. A revised reprint of XIX 137. From a logical point of view, by Willard Van Orman Quine, second, revised edition, Harvard University Press, Cambridge, Mass., 1961, and Harper Torchbooks, The Science Library, Harper & Row, New York and Evanston 1963, pp. 139–159. - Willard Van Orman Quine. Meaning and existential inference. A reprint of XIX 138. From a logical point of view, by Willard Van Orman Quine, second, revised edition, Harvard University Press, Cambridge, Mass., 1961, and Harper Torchbooks, The Science Library, Harper & Row, New York and Evanston 1963, pp. 160–167. - Volume 33 Issue 1

Boolean contact algebras constitute a suitable algebraic theory for qualitative spatial reasoning. They are Boolean algebras with an additional contact relation grasping the topological aspect of spatial entities. In this paper we present a logic that combines propositional logic, relevance logic, and modal logic to reason about Boolean contact algebras. This is done in two steps. First, we use the relevance logic operators to obtain a logic suitable for Boolean algebras. Then we add modal operators that are based on the contact relation. In both cases we present axioms that are equivalent to requirement that every frame for the logic is indeed a Boolean algebra resp. Boolean contact algebra. We also provide a natural deduction system for this logic by defining introduction and eliminations rules for each logical operator. The system is shown to be sound. Furthermore, we sketch an implementation of the natural deduction system in the functional programming language and interactive theorem prover Coq.

It is well known that an important task of the artificial intelligence is to make a computer simulate a human being in dealing with certainty and uncertainty in information. Logic gives a technique for laying the foundations of this task. Information processing dealing with certain information is based on the classical logic. Nonclassical logic including many-valued logic and fuzzy logic takes the advantage of the classical logic to handle information with various facets of uncertainty, such as fuzziness and randomness. Therefore, nonclassical logic has become a formal and useful tool for computer science to deal with fuzzy information and uncertain information. For investigation of several properties, such logics have been represented as algebras, that is, sets with one, two, or more algebraic operations satisfying some conditions inspired by these logics. This inspiration is illustrated by the similarities between the names. We have BCK-algebras and a BCK positive logic, BCI-algebras and a BCI positive logic, and BL-algebras and a basic logic. In many cases, the connection between such algebras and their corresponding logics is much stronger. In this case, one can give a translation procedure which translates all well-formed formulas and all theorems of a given logic L into terms and theorems of the corresponding algebras. In some cases, one can give also an inverse translation. In this case, we say that the given logic and class of such algebras are isomorphic. Nevertheless, the study of algebras motivated by logics is interesting and very useful also in the case when these structures are not isomorphic. An important class of algebras inspired by logic is BL-algebras introduced by Hajek in order to provide an algebraic proof of the completeness theorem of Basic Logic. A classical example of a BL-algebra is the interval [0,1] endowed with the structure induced by a continuous t-norm. Other important examples of BL-algebras are MV-algebras introduced earlier as a model of some infinitely valued Łukasiewicz logic. All these algebras are also strongly connected with residuated lattices. Hence, the filter theory plays an important role in the study of these algebras. From logic point of view, various filters correspond to various sets of provable formulae in the respective logic. Below we present several papers on filters in various algebras. In one of these papers, it is proved that there is a one-to-one correspondence between the set of all vague filters and all vague congruence of a lattice implication algebra. In another, the relations among fuzzy t-filters on residuated lattices are described. Another paper is devoted to the filter topology on lattice implication algebras. In further articles is characterized the relationship between the main types of fuzzy filters in BE-algebras and EQ-algebras. The role of ideals and soft sets in BL-algebras is described in another two papers. In papers on pseudo-weak-R0 algebras, the most simplified axioms system is presented and it is proved that pseudo-weak-R0 algebras are categorically isomorphic to pseudo-IMTL algebras. Pseudo-R0 algebras are categorically isomorphic to pseudo-NM algebras. Several presented papers are devoted to the possible applications of algebras inspired by logic and fuzzy sets. For example, a robust fuzzy logic controller is proposed for stabilization and disturbance rejection in nonlinear control systems of a particular type. The dynamic feedback controller is designed as a combination of a control law that compensates for nonlinear terms in a control system and a dynamic fuzzy logic controller that addresses unknown model uncertainties and an unmeasured disturbance. A mathematical derivation is used to prove that the controller is able to achieve asymptotic stability by processing state measurements. Properties of some hesitant triangular fuzzy aggregation operators based on Bonferroni means also are discussed. In one paper, the method of a construction of a ranking of the physical matches is proposed. This method is based on the fuzzy clustering analysis. Decisions are often made by many decision experts having preference of risks. In view of the grey situation group decision-making problems, a grey situation group decision-making method on the basis of prospect theory is shown. The method takes the positive and negative ideal situation distance as reference points, defines positive and negative prospect value function, and introduces decision experts' risk preference into grey situation decision-making to make the final decision more in line with decision experts' psychological behavior. One possibility of determination of the weight of each decision expert sets up comprehensive prospect value matrix for decision experts' evaluation, and finally determination of the optimal situation is presented. Jianming Zhan Bijan Davvaz Wieslaw A. Dudek Young Bae Jun Hee Sik Kim

We review, and in the process unify two techniques (due to Németi and Pigozzi), for proving results concerning amalgamation in several classes studied in algebraic logic. The logical counterpart of these results adress interpolation and definability properties in modal and algebraic logic. Presenting them in a functorial context as adjoint situations, we show that both techniques can indeed be seen as instances of the use of the Keisler-Shelah ultrapower Theorem in proving Robinson's Joint Consistency Theorem. Some new results are surveyed. The results of this paper are presented in such a way that further establishes the hitherto existing well developed duality theory between boolean algebras with operators and modal logic.

We explore the basic fuzzy logic BL as well as propositional fuzzy logics with modalities □ and ◊ and a total accessibility relation. Formulations and proofs are given to replacement theorems for BL. A basic calculus of modal fuzzy logic is introduced. For this calculus and its extensions, we prove replacement and deduction theorems.

This chapter deals with fuzzy logic in Zadeh’s ‘narrow sense’ [27], pointing to t-norm based truth functional logics, where the truth values model ‘degrees of truth’, identified with reals from the unit interval [0,1]. In particular, we are interested in Łukasiewicz logic Ł and some of its variants, but also in Godel logic G, and in Product logic P. The literature on formal deduction systems for these and related many-valued logics is vast and even more has been written about their algebraic background. We refer to the monographs [23], [22], and [10] for more information and references. Most authors take the usefulness of these logics in the context of approximate reasoning, i.e., reasoning with vague and imprecise notions for granted. However the corresponding proof systems are hardly ever explicitly related to models of correct reasoning with vague information. In other words: the challenge to derive particular fuzzy logics from first principles about approximate reasoning is not addressed explicitly. The reference to general models of reasoning and to theories of vagueness – a prolific discourse in contemporary analytic philosophy – is only implicit, if not simply missing, in most presentations of inference systems for fuzzy logics. Some notable exceptions, where an explicit semantic foundation for particular fuzzy logics is aimed at, are: Ruspini’s similarity semantics [35]; voting semantics [25]; ‘re-randomizing semantics’ [24]; measurement-theoretic justifications [7]; the Ulam-Renyi game based interpretation of D. Mundici [29]; and ‘acceptability semantics’ of J. Paris [31]. As we have argued elsewhere [12], these formal semantics for various t-norm based logics (in particular Łukasiewicz logic) should be placed in the wider discourse about adequate theories of vagueness, a prolific subfield of analytic philosophy.

Information flow control is a long-studied approach for establishing non-interference properties of programs. For instance, it can be used to prove that a secret does not interfere with some computation, thereby establishing that the former does not leak through the latter. Despite their potential as a holy grail for security reasoning and their maturity within the literature, information flow type systems have seen limited adoption. In practice, information flow specifications tend to be excessively complex and can easily spiral out of control even for simple programs. Additionally, while non-interference is well-behaved in an idealized setting where information leakage never occurs, most practical programs must violate non-interference in order to fulfill their purpose. Useful information flow type systems in prior work must therefore contend with a definition of non-interference extended with declassification , which often offers weaker modular reasoning properties. We introduce structural information flow , which both illuminates and addresses these issues from a logical viewpoint. In particular, we draw on established insights from the modal logic literature to argue that information flow reasoning arises from hybrid logic , rather than conventional modal logic as previously imagined. We show with a range of examples that structural information flow specifications are straightforward to write and easy to visually parse. Uniquely in the structural setting, we demonstrate that declassification emerges not as an aberration to non-interference, but as a natural and unavoidable consequence of sufficiently general machinery for information flow. This flavor of declassification features excellent local reasoning and enables our approach to account for real-world information flow needs without compromising its theoretical elegance. Finally, we establish non-interference via a logical relations approach, showing off its simplicity in the face of the expressive power captured.

This paper is concerned with a framework that combines sliding mode control and fuzzy logic systems. This controller is based on the variable structure system control theory and entails the advantages of sliding mode control and fuzzy logic control. The controlled system is a simulated two-links manipulator robot. It is found that the use of sliding mode controller gives good results yielding less coupling between freedom degrees. Moreover, the usage of fuzzy logic applied to the sliding mode controller leads a reduction of the size of the fuzzy inference system.

The article deals with Codd's relational model of data and its fuzzy logic extensions. Our main purpose is to examine, from the point of view of fuzzy logic in the narrow sense, some of the extensions proposed in the literature and the relationships between them. We argue that fuzzy logic in the narrow sense is important for the fuzzy logic extensions because it provides conceptual and methodological foundations, clarity and simplicity. We present several comparative observations as well as new technical results.

From logical point of view, it is worth discussing problem of legal gaps that is very important in the analysis of legal systems.The article belows contributes to the the analisys of so-called legal gaps from the perspective of deontic logic. In the beginning of the paper, after discussing certain properties of legal systems — completeness, consistency and contingency — on the basis of analytical methodology of legal positivism based on Dworkin-Hart’s debate about closure rules, in is describing different types of closure rules, which are employed for closing the legal systems. The paper compares the similarities and differences between different approaches to the studies of legal gaps. It is found that similarities between logical modalities and types of legalism makes possible to extend analysis of closure rules by introducing deontic modalities; the article explores the perspectives of such analysis. We introduce logical principles underlying two main types of legalism — Anglo-Saxon and continential ones. Drawning upon the basic principles of modal logic, we briefly sketch the prior history, development and current state of studies in the field of deontic logic. The paper suggests that it should be possible to develop a deontic assessement of the closure rules in legal systems; the first goal of the paper is achieved by spelling out differences between deontic modalities in open and closed legal systems.The final section of the paper is dedicated to analysis of deontic modalities employed for generating such closure rules.

Boolean algebra (BA) was introduced by George Boole in his first book The Mathematical Analysis of Logic (1847) and set forth more fully in his An Investigation of the Laws of Thought (1854). According to Huntington, the term “Boolean algebra” was first suggested by Sheffer in 1913. Fuzzy logic is a form of many-valued logic in which the truth values of variables may be any real number between 0 and 1, considered to be “fuzzy.” By contrast, in Boolean logic, the truth values of variables may only be the “crisp” values 0 or 1. Fuzzy logic has been employed to handle the concept of partial truth, where the truth value may range between completely true and completely false. Furthermore, when linguistic variables are used, these degrees may be managed by specific (membership) functions. The term fuzzy logic was introduced with the 1965 proposal of fuzzy set theory by Lotfi Zadeh. Fuzzy logic had however been studied since the 1920s, as infinite-valued logic—notably by Lukasiewicz and Tarski. In this chapter we look at both of these logics holistically; any extensive details are beyond the scope of this book, and we encourage our readers to refer to so many books and articles that can be found on the Internet or Amazon.