A modal theorem-preserving translation of a class of three-valued logics of incomplete information

International audienceThere are several three-valued logical systems that form a scattered landscape, even if all reasonable connectives in three-valued logics can be derived from a few of them. Most papers on this subject neglect the issue of the relevance of such logics in relation with the intended meaning of the third truth-value. Here, we focus on the case where the third truth-value means unknown, as suggested by Kleene. Under such an understanding, we show that any truth-qualified formula in a large range of three-valued logics can be translated into KD as a modal formula of depth 1, with modalities in front of literals only, while preserving all tautologies and inference rules of the original three-valued logic. This simple information logic is a two-tiered classical propositional logic with simple semantics in terms of epistemic states understood as subsets of classical interpretations. We study in particular the translations of Kleene, Gödel, ᴌukasiewicz and Nelson logics. We show that Priest’s logic of paradox, closely connected to Kleene’s, can also be translated into our modal setting, simply by exchanging the modalities possible and necessary. Our work enables the precise expressive power of three-valued logics to be laid bare for the purpose of uncertainty management

Three-valued logics, uncertainty management and rough sets

This paper is a survey of the connections between three-valued logics and rough sets from the point of view of incomplete information management. Based on the fact that many three-valued logics can be put under a unique algebraic umbrella, we show how to translate three-valued conjunctions and implications into operations on ill-known sets such as rough sets. We then show that while such translations may provide mathematically elegant algebraic settings for rough sets, the interpretability of these connectives in terms of an original set approximated via an equivalence relation is very limited, thus casting doubts on the practical relevance of truth-functional logical renderings of rough sets

Deciding regular grammar logics with converse through first-order logic

We provide a simple translation of the satisfiability problem for regular grammar logics with converse into GF2, which is the intersection of the guarded fragment and the 2-variable fragment of first-order logic. This translation is theoretically interesting because it translates modal logics with certain frame conditions into first-order logic, without explicitly expressing the frame conditions. A consequence of the translation is that the general satisfiability problem for regular grammar logics with converse is in EXPTIME. This extends a previous result of the first author for grammar logics without converse. Using the same method, we show how some other modal logics can be naturally translated into GF2, including nominal tense logics and intuitionistic logic. In our view, the results in this paper show that the natural first-order fragment corresponding to regular grammar logics is simply GF2 without extra machinery such as fixed point-operators.Comment: 34 page

Universal Logic and the Geography of Thought - Reflections on logical pluralism in the light of culture

The aim of this dissertation is to provide an analysis for those involved and interested in the interdisciplinary study of logic, particularly Universal Logic. While continuing to remain aware of the importance of the central issues of logic, we hope that the factor of culture is also given serious consideration. Universal Logic provides a general theory of logic to study the most general and abstract properties of the various possible logics. As well as elucidating the basic knowledge and necessary definitions, we would especially like to address the problems of motivation concerning logical investigations in different cultures. First of all, I begin by considering Universal Logic as understood by Jean-Yves Béziau, and examine the basic ideas underlying the Universal Logic project. The basic approach, as originally employed by Universal Logicians, is introduced, after which the relationship between algebras and logics at an abstract level is discussed, i.e., Universal Algebra and Universal Logic. Secondly,I focus on a discussion of the translation paradox , which will enable readers to become more familiar with the new subject of logical translation, and subsequently comprehensively summarize its development in the literature. Besides helping readers to become more acquainted with the concept of logical translation, the discussion here will also attempt to formulate a new direction in support of logical pluralism as identified by Ruldof Carnap (1934), JC Beall and Greg Restall (2005), respectively. Thirdly, I provide a discussion of logical pluralism. Logical pluralism can be traced back to the principle of tolerance raised by Ruldof Carnap (1934), and readers will gain a comprehensive understanding of this concept from the discussion. Moreover,an attempt will be made to clarify the real and important issues in the contemporary debate between pluralism and monism within the field of logic in general. Fourthly, I study the phenomena of cultural-difference as related to the geography of thought. Two general systems in the geography of thought are distinguished, which we here call thought-analytic and thought-holistic. They are proposed to analyze and challenge the universality assumption regarding cognitive processes. People from different cultures and backgrounds have many differences in diverse areas, and these differences, if taken for granted, have proven particularly problematic in understanding logical thinking across cultures. Interestingly, the universality of cognitive processes has been challenged, especially by Richard Nisbett s research in cultural psychology. With respect to these concepts, C-UniLog can also be considered in relation to empirical evidence obtained by Richard Nisbett et al. In the final stage of this dissertation, I will propose an interpretation of the concept of logical translation, i.e., translations between formal logical mode (as cognitive processes in the case of westerners) and dialectical logical mode (as cognitive processes in the case of Asians). From this, I will formulate a new interpretation of the principle of tolerance, as well as of logical pluralism

Multi-Valued Verification of Strategic Ability

Some multi-agent scenarios call for the possibility of evaluating specifications in a richer domain of truth values. Examples include runtime monitoring of a temporal property over a growing prefix of an infinite path, inconsistency analysis in distributed databases, and verification methods that use incomplete anytime algorithms, such as bounded model checking. In this paper, we present multi-valued alternating-time temporal logic (mv-ATL*), an expressive logic to specify strategic abilities in multi-agent systems. It is well known that, for branching-time logics, a general method for model-independent translation from multi-valued to two-valued model checking exists. We show that the method cannot be directly extended to mv-ATL*. We also propose two ways of overcoming the problem. Firstly, we identify constraints on formulas for which the model-independent translation can be suitably adapted. Secondly, we present a model-dependent reduction that can be applied to all formulas of mv-ATL*. We show that, in all cases, the complexity of verification increases only linearly when new truth values are added to the evaluation domain. We also consider several examples that show possible applications of mv-ATL* and motivate its use for model checking multi-agent systems