Self-Extensional Three-Valued Paraconsistent Logics

A logic \(\mathbf{L}\) is called self-extensional if it allows to replace occurrences of a formula by occurrences of an \(\mathbf{L}\)-equivalent one in the context of claims about logical consequence and logical validity. It is known that no three-valued paraconsistent logic which has an implication can be self-extensional. In this paper we show that in contrast, there is exactly one self-extensional three-valued paraconsistent logic in the language of \(\{\lnot ,\wedge ,\vee \}\) for which \(\vee \) is a disjunction, and \(\wedge \) is a conjunction. We also investigate the main properties of this logic, determine the expressive power of its language (in the three-valued context), and provide a cut-free Gentzen-type proof system for it.     

Empirical Negation

An extension of intuitionism to empirical discourse, a project most seriously taken up by Dummett and Tennant, requires an empirical negation whose strength lies somewhere between classical negation (‘It is unwarranted that. . . ’) and intuitionistic negation (‘It is refutable that. . . ’). I put forward one plausible candidate that compares favorably to some others that have been propounded in the literature. A tableau calculus is presented and shown to be strongly complete.     

Two Semantical Approaches to Paraconsistent Modalities

In this paper we extend the anodic systems introduced in Bueno-Soler (J Appl Non Class Logics 19(3):291–310, 2009) by adding certain paraconsistent axioms based on the so called logics of formal inconsistency, introduced in Carnielli et al. (Handbook of philosophical logic, Springer, Amsterdam, 2007), and define the classes of systems that we call cathodic. These classes consist of modal paraconsistent systems, an approach which permits us to treat with certain kinds of conflicting situations. Our interest in this paper is to show that such systems can be semantically characterized in two different ways: by Kripke-style semantics and by modal possible-translations semantics. Such results are inspired in some universal constructions in logic, in the sense that cathodic systems can be seen as a kind of fusion (a particular case of fibring) between modal logics and non-modal logics, as discussed in Carnielli et al. (Analysis and synthesis of logics, Springer, Amsterdam, 2007). The outcome is inherently within the spirit of universal logic, as our systems semantically intermingles modal logics, paraconsistent logics and many-valued logics, defining new blends of logics whose relevance we intend to show.     

Stability and General Logics

In this paper we make an attempt to study classes of models by using general logics. We do not believe that L_ww is always the best logic for analyzing a class of models. Let K be a class of models and L a logic. The main assumptions we make about K and C are that K has the L-amalgamation property and, later in the paper, that K does not omit L-types. We show that, if modified suitably, most of the results of stability theory hold in this context. The main difference is that existentially closed models of K play the role that arbitrary models play in traditional stability theory. We prove e. g. a structure theorem for the class of existentially closed models of K assuming that K is a trivial superstable class with ndop.     

Fuzzy Modal Logics

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.__________Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 1, pp. 201–230, 2003.     

Negation of binomial coefficients

We extend the concept of a binomial coefficient to all integer values of its parameters. Our approach is purely algebraic, but we show that it is equivalent to the evaluation of binomial coefficients by means of the Γ-function. In particular, we prove that the traditional rule of “negation” is wrong and should be substituted by a slightly more complex rule. We also show that the “cross product” rule remains valid for the extended definition.     

分划逻辑与着色卵石博奕

将Ｍａｌｉｔｚ量词Ｑ￣（ｍ，ｎ）的基数限制去掉，再作多分划拓广，便得到各种ｍｏｎａｄｉｃ型的分划量词。本文引入了分划逻辑的着色卵石博奕方法，证明了相应的Ｅｈｒｅｎｆｅｎｃｈｔ－Ｆｒａｓｓｅ定理，作为此方法的一个应用，证明了在表达能力上，ｍｏｎａｄｉｃ型的分划逻辑严格地弱于ｍｏｎａｄｉｃ二阶逻辑。     

Proofs and Countermodels in Non-Classical Logics

Proofs and countermodels are the two sides of completeness proofs, but, in general, failure to find one does not automatically give the other. The limitation is encountered also for decidable non-classical logics in traditional completeness proofs based on Henkin’s method of maximal consistent sets of formulas. A method is presented that makes it possible to establish completeness in a direct way: For any given sequent either a proof in the given logical system or a countermodel in the corresponding frame class is found. The method is a synthesis of a generation of calculi with internalized relational semantics, a Tait–Schütte–Takeuti style completeness proof, and procedures to finitize the countermodel construction. Finitizations for intuitionistic propositional logic are obtained through the search for a minimal derivation, through pruning of infinite branches in search trees by means of a suitable syntactic counterpart of semantic filtration, or through a proof-theoretic embedding into an appropriate provability logic. A number of examples illustrates the method, its subtleties, challenges, and present scope.     

Symmetric Generalized Galois Logics

Symmetric generalized Galois logics (i.e., symmetric gGls) are distributive gGls that include weak distributivity laws between some operations such as fusion and fission. Motivations for considering distribution between such operations include the provability of cut for binary consequence relations, abstract algebraic considerations and modeling linguistic phenomena in categorial grammars. We represent symmetric gGls by models on topological relational structures. On the other hand, topological relational structures are realized by structures of symmetric gGls. We generalize the weak distributivity laws between fusion and fission to interactions of certain monotone operations within distributive super gGls. We are able to prove appropriate generalizations of the previously obtained theorems—including a functorial duality result connecting classes of gGls and classes of structures for them.      

Admissibility in Positive Logics

The paper studies admissibility of multiple-conclusion rules in positive logics. Using modification of a method employed by M. Wajsberg in the proof of the separation theorem, it is shown that the problem of admissibility of multiple-conclusion rules in the positive logics is equivalent to the problem of admissibility in intermediate logics defined by positive additional axioms. Moreover, a multiple-conclusion rule \(\mathsf {r}\) follows from a set of multiple-conclusion rules \(\mathsf {R}\) over a positive logic \(\mathsf {P}\) if and only if \(\mathsf {r}\) follows from \(\mathsf {R}\) over \(\mathbf {Int}+ \mathsf {P}\).     

Barwise's Information Frames and Modal Logics

The article studies Barwise's information frames and settles the problem of Barwise dealing in finding axiomatizations for the modal logics generated by information frames. We find axiomatic systems for: (i) the modal logic of all complete information frames; (ii) the logic of all sound and complete information frames; (iii) the logic of all hereditary and complete information frames; (iv) the logic of all complete, sound, and hereditary information frames; (v) the logic of all consistent and complete information frames. The notion of weak modal logics is also proposed, and it is shown that the weak modal logics generated by all information frames and by all hereditary information frames are K and K4, respectively. Toward a general theory, we prove that any Kripke complete modal logic is a modal logic of a certain class of information frames, and that every modal logic generated by any given class of complete, rarefied, and fully classified information frames is Kripke complete.     

Minimally Generated Abstract Logics

In this paper we study an alternative approach to the concept of abstract logic and to connectives in abstract logics. The notion of abstract logic was introduced by Brown and Suszko (Diss Math 102:9–42, 1973)—nevertheless, similar concepts have been investigated by various authors. Considering abstract logics as intersection structures we extend several notions to their κ-versions (κ ≥ ω), introduce a hierarchy of κ-prime theories, which is important for our treatment of infinite connectives, and study different concepts of κ-compactness. We are particularly interested in non-topped intersection structures viewed as semi-lattices with a minimal meet-dense subset, i.e., with a minimal generator set. We study a chain condition which is sufficient for a minimal generator set, implies compactness of the logic, and in regular logics is equivalent to (κ-) compactness of the consequence relation together with the existence of a (κ-)inconsistent set, where κ is the cofinality of the cardinality of the logic. Some of these results are known in a similar form in the context of closure spaces, we give extensions to (non-topped) intersection structures and to big cardinals presenting new proofs based on set-theoretical tools. The existence of a minimal generator set is crucial for our way to define connectives. Although our method can be extended to further non-classical connectives we concentrate here on intuitionistic and infinite ones. Our approach leads us to the concept of the set of complete theories which is stable under all considered connectives and gives rise to the definition of the topological space of the logic. Topological representations of (non-classical) abstract logics by means of this space remain to be further investigated.     

Composition-Nominative Logics as Institutions

Composition-nominative logics (CNL) are program-oriented logics. They are based on algebras of partial predicates which do not have fixed arity. The aim of this work is to present CNL as institutions. Homomorphisms of first-order CNL are introduced, satisfaction condition is proved. Relations with institutions for classical first-order logic are considered. Directions for further investigation are outlined.     

Logics for stable and unstable mereological relations

In this paper we present logics about stable and unstable versions of several well-known relations from mereology: part-of, overlap and underlap. An intuitive semantics is given for the stable and unstable relations, describing them as dynamic counterparts of the base mereological relations. Stable relations are described as ones that always hold, while unstable relations hold sometimes. A set of first-order sentences is provided to serve as axioms for the stable and unstable relations, and representation theory is developed in similar fashion to Stone’s representation theory for distributive lattices.     

Intermediate Logics and the de Jongh property

We prove that all extensions of Heyting Arithmetic with a logic that has the finite frame property possess the de Jongh property.