Fuzzy inequational logic

We present a logic for reasoning about graded inequalities which generalizes the ordinary inequational logic used in universal algebra. The logic deals with atomic predicate formulas of the form of inequalities between terms and formalizes their semantic entailment and provability in graded setting which allows to draw partially true conclusions from partially true assumptions. We follow the Pavelka approach and define general degrees of semantic entailment and provability using complete residuated lattices as structures of truth degrees. We prove the logic is Pavelka-style complete. Furthermore, we present a logic for reasoning about graded if–then rules which is obtained as particular case of the general result.     

A relative interpolation theorem for infinitary universal Horn logic and its applications

In this paper we deal with infinitary universal Horn logic both with and without equality. First, we obtain a relative Lyndon-style interpolation theorem. Using this result, we prove a non-standard preservation theorem which contains, as a particular case, a Lyndon-style theorem on surjective homomorphisms in its Makkai-style formulation. Another consequence of the preservation theorem is a theorem on bimorphisms, which, in particular, provides a tool for immediate obtaining characterizations of infinitary universal Horn classes without equality from those with equality. From the theorem on surjective homomorphisms we also derive a non-standard Beth-style preservation theorem that yields a non-standard Beth-style definability theorem, according to which implicit definability of a relation symbol in an infinitary universal Horn theory implies its explicit definability by a conjunction of atomic formulas. We also apply our theorem on surjective homomorphisms, theorem on bimorphisms and definability theorem to algebraic logic for general propositional logic.     

Algebraic Characterizations for Universal Fragments of Logic

In this paper we address our efforts to extend the well-known connection in equational logic between equational theories and fully invariant congruences to other–possibly infinitary–logics. In the special case of algebras, this problem has been formerly treated by H. J. Hoehnke [10] and R. W. Quackenbush [14]. Here we show that the connection extends at least up to the universal fragment of logic. Namely, we establish that the concept of (infinitary) universal theory matches the abstract notion of fully invariant system. We also prove that, inside this wide group of theories, the ones which are strict universal Horn correspond to fully invariant closure systems, whereas those which are universal atomic can be characterized as principal fully invariant systems.     

Some Properties of Residuated Lattices

We investigate some (universal algebraic) properties of residuated lattices--algebras which play the role of structures of truth values of various systems of fuzzy logic.     

An Algebraic Approach to Knowledge Base Models Informational Equivalence

In this paper we study the notion of knowledge from the positions of universal algebra and algebraic logic. We consider first order knowledge which is based on first order logic. We define categories of knowledge and knowledge base models. These notions are defined for the fixed subject of knowledge. The key notion of informational equivalence of two knowledge base models is introduced. We use the idea of equivalence of categories in this definition. We prove that for finite models there is a clear way to determine whether the knowledge base models are informationally equivalent.     

Fuzzy Horn logic I

The paper presents generalizations of results on so-called Horn logic, well-known in universal algebra, to the setting of fuzzy logic. The theories we consider consist of formulas which are implications between identities (equations) with premises weighted by truth degrees. We adopt Pavelka style: theories are fuzzy sets of formulas and we consider degrees of provability of formulas from theories. Our basic structure of truth degrees is a complete residuated lattice. We derive a Pavelka-style completeness theorem (degree of provability equals degree of truth) from which we get some particular cases by imposing restrictions on the formulas under consideration. As a particular case, we obtain completeness of fuzzy equational logic.     

Surveyable sets

Call a set existentially (or universally) surveyable if existential (or universal) quantification over that set preserves decidability in the sense of intuitionistic logic. We study these notions of surveyability, their preservation properties, and their connections with each other and with the related notion of completeness of the two-element lattice.     

Kripke submodels and universal sentences

We define two notions for intuitionistic predicate logic: that of a submodel of a Kripke model, and that of a universal sentence. We then prove a corresponding preservation theorem. If a Kripke model is viewed as a functor from a small category to the category of all classical models with (homo)morphisms between them, then we define a submodel of a Kripke model to be a restriction of the original Kripke model to a subcategory of its domain, where every node in the subcategory is mapped to a classical submodel of the corresponding classical model in the range of the original Kripke model. We call a sentence universal if it is built inductively from atoms (including ? and ⊥) using ∧, ∨, ?, and →, with the restriction that antecedents of → must be atomic. We prove that an intuitionistic theory is axiomatized by universal sentences if and only if it is preserved under Kripke submodels. We also prove the following analogue of a classical model‐consistency theorem: The universal fragment of a theory Γ is contained in the universal fragment of a theory Δ if and only if every rooted Kripke model of Δ is strongly equivalent to a submodel of a rooted Kripke model of Γ. Our notions of Kripke submodel and universal sentence are natural in the sense that in the presence of the rule of excluded middle, they collapse to the classical notions of submodel and universal sentence. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Logic as a Science and Logic as a Theory: Remarks on Frege, Russell and the Logocentric Predicament

Since its publication in 1967, van Heijenoort??s paper, ??Logic as Calculus and Logic as Language?? has become a classic in the historiography of modern logic. According to van Heijenoort, the contrast between the two conceptions of logic provides the key to many philosophical issues underlying the entire classical period of modern logic, the period from Frege??s Begriffsschrift (1879) to the work of Herbrand, G?del and Tarski in the late 1920s and early 1930s. The present paper is a critical reflection on some aspects of van Heijenoort??s thesis. I concentrate on the case of Frege and Russell and the claim that their philosophies of logic are marked through and through by acceptance of the universalist conception of logic, which is an integral part of the view of logic as language. Using the so-called ??Logocentric Predicament?? (Henry M. Sheffer) as an illustration, I shall argue that the universalist conception does not have the consequences drawn from it by the van Heijenoort tradition. The crucial element here is that we draw a distinction between logic as a universal science and logic as a theory. According to both Frege and Russell, logic is first and foremost a universal science, which is concerned with the principles governing inferential transitions between propositions; but this in no way excludes the possibility of studying logic also as a theory, i.e., as an explicit formulation of (some) of these principles. Some aspects of this distinction will be discussed.     

On the Finite Model Property of Intuitionistic Modal Logics over MIPC

MIPC is a well-known intuitionistic modal logic of Prior (1957) and Bull (1966). It is shown that every normal intuitionistic modal logic L over MIPC has the finite model property whenever L is Kripke-complete and universal.     

Reverse contests

We study two forms of a reverse contest. In reverse contest A the designer imposes a punishment such that the agent with the highest effort who caused the greatest damage is punished. Conversely, in reverse contest B, the designer awards a prize to the agent with the lowest effort who caused the smallest damage. We analyze the behavior of the agents in both contest forms and demonstrate that asymmetry of the players' payoff functions does not necessarily yield different expected payoffs.     

The Logic with Truth and Falsehood Operators from a Point of View of Universal Logic

The logic with independent truth and falsehood operators TFL is proposed. In TFL(→) standard truth-conditions for the implication are adopted. Nevertheless the laws of classical logic are not valid. In this language more then 10⁷ different binary connectives can be defined. So this logic can be treated as universal logic relatively to the class of sentential logics.     

A paraconsistent extension of Sylvan’s logic

We deal with Sylvan’s logic CC_ω. It is proved that this logic is a conservative extension of positive intuitionistic logic. Moreover, a paraconsistent extension of Sylvan’s logic is constructed, which is also a conservative extension of positive intuitionistic logic and has the property of being decidable. The constructed logic, in which negation is defined via a total accessibility relation, is a natural intuitionistic analog of the modal system S5. For this logic, an axiomatization is given and the completeness theorem is proved. Supported by RFBR grant No. 06-01-00358 and by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-4787.2006.1. __________ Translated from Algebra i Logika, Vol. 46, No. 5, pp. 533–547, September–October, 2007.     

Computational complexity for bounded distributive lattices with negation

We study the computational complexity of the universal and quasi-equational theories of classes of bounded distributive lattices with a negation operation, i.e., a unary operation satisfying a subset of the properties of the Boolean negation. The upper bounds are obtained through the use of partial algebras. The lower bounds are either inherited from the equational theory of bounded distributive lattices or obtained through a reduction of a global satisfiability problem for a suitable system of propositional modal logic.     

A NOTE ON REGULAR AND EXTREYAL SUBOBJECTS

ABSTRACT

In the general setting of a complete, well-powered category A, we define and study two universal closure operations: regularization and extremalization, by means of regular and extremal subobjects of A. respectively. A general theorem of characterization of epimorphisms in A is given. When A is an epireflective subcategory of TOP, such operations are shown to coincide with A-closure [11] and epiclosure [2]. respectively. In the topological contest, regularization and extremalization are studied in detail and compared with r-closure, defined in [13].     

A two-person game of timing with random termination

We consider a marksmanship contest in which the first contestant to hit his target wins and the contest is to be terminated at a random timeT with cdfH(t). The model is evidently an extension of the classical discrete fire duel to the timing problem under an uncertain environment. It is shown that the uncertainty on the termination of the contest has influence on the equilibrium strategies and the equilibrium values.     

基于半马尔科夫过程的逻辑门可靠性分析

动态故障树分析方法是在静态故障树的基础上拓展而来的自上而下的图形化演绎技术,可以很好地对具有复杂失效行为和交互作用的系统进行建模,进而分析系统的可靠性。本文从动态故障树逻辑门的可靠性建模与分析入手,结合半马尔科夫过程原理,将动态逻辑门转化为半马尔科夫链。其次给出在半马尔科夫链中动态逻辑门输出事件的发生概率和系统可靠性的计算公式。提出各种逻辑门到半马尔科夫链的通用转化模型,通过更改通用模型中的相关参数,将逻辑门转化为半马尔科夫链。最后,基于半马尔科夫过程求解动态逻辑门输出事件的发生概率,以动态优先与门、顺序相关门和备件门为例,并给出系统可靠性的计算公式。     

Structuring the Universe of Universal Logic

How, why and what for we should combine logics is perfectly well explained in a number of works concerning this issue. But the interesting question seems to be the nature and the structure of the general universe of possible combinations of logical systems. Adopting the point of view of universal logic in the paper the categorical constructions are introduced which along with the coproducts underlying the fibring of logics describe the inner structure of the category of logical systems. It is shown that categorically the universe of universal logic turns out to be a topos and a paraconsistent complement topos. This work was supported by Russian Foundation for Humanities via the Project ”The structure of Universal Logics”, grant No 06-03-00195a.     

Product Ł ukasiewicz Logic

u logic plays a fundamental role among many-valued logics. However, the expressive power of this logic is restricted to piecewise linear functions. In this paper we enrich the language of u logic by adding a new connective which expresses multiplication. The resulting logic, P, is defined, developed, and put into the context of other well-known many-valued logics. We also deal with several extensions of this propositional logic. A predicate version of P logic is introduced and developed too.The work of the first author was supported by the Grant Agency of the Czech Republic under project GACR 201/02/1540, by the Grant Agency of the Czech Technical University in Prague under project CTU 0208613, and by Net CEEPUS SK-042.The work of the second author was supported by grant IAA1030004 of the Grant Agency of the Academy of Sciences of the Czech Republic.