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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.

Our article “A map of dependencies among three-valued logics” (Information Sciences, vol. 250, 161–177, 2013) contains some typing mistakes that may affect a proper understanding of the paper. They are corrected here.

Our article “A map of dependencies among three-valued logics” (Information Sciences, vol. 250, 161–177, 2013) contains some typing mistakes that may affect a proper understanding of the paper. They are corrected here.

There 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, Łukasiewicz 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 arise in several fields of computer science, both inspired by concrete problems (such as in the management of the null value in databases) and theoretical considerations. Several three-valued logics have been defined. They differ by their choice of basic connectives, hence also from a syntactic and proof-theoretic point of view. Different interpretations of the third truth value have also been suggested. They often carry an epistemic flavor. In this work, relationships between logical connectives on three-valued functions are explored. Existing theorems of functional completeness have laid bare some of these links, based on specific connectives. However we try to draw a map of such relationships between conjunctions, negations and implications that extend Boolean ones. It turns out that all reasonable connectives can be defined from a few of them and so all known three-valued logics appear as a fragment of only one logic. These results can be instrumental when choosing, for each application context, the appropriate fragment where the basic connectives make full sense, based on the appropriate meaning of the third truth-value

There are several three-valued logical systems. They give the impression of a scattered landscape. The majority of the works on this subject gives the truth tables, sometimes an Hilbert style axiomatization in a basic propositional language and a completeness theorem with respect to those truth tables. We show that all the reasonable connectives in three-valued logics can be built starting from few of them. Nevertheless, the issue of the usefulness of each system in relation with the third truth value is often neglected. Here, we review the interpretations of the third truth value. Then, we focus on the unknown case, suggested by Kleene. We show that any formula in three-valued logics can be encoded as a fragment of an epistemic logic (formulae of modal depth 1, with modalities in front of literals), preserving all tautologies and inference rules. We study in particular, the translation of Kleene, Godel, Lukasiewicz and Nelson logics. This work enables us to lay bare the limited expressive power of three-valued logics in uncertainty management.

In the literature, several 3-valued logics can be found. They differ from a syntactic and proof-theoretic point of view as well as on the interpretation given to the third value, which, nevertheless, often assumes an epistemic flavor. This paper is a preliminary step in the attempt to clarify the situation of three-valued logics from a semantic point of view. Logical operations on three-valued functions are studied and their relationships put forward. They are also linked to existing logics, pointing out their usage and interpretation of the third value. In the long range, the idea is to be able to relate as many three-valued calculi as possible to classes of applications where the third truth-value is naturally interpreted, and the basic connectives make full sense.

An algebraic model of a kind of modal extension of de Morgan logic is described under the name MDS5 algebra. The main properties of this algebra can be summarized as follows: (1) it is based on a de Morgan lattice, rather than a Boolean algebra; (2) a modal necessity operator that satisfies the axioms N, K, T, and 5 (and as a consequence also B and 4) of modal logic is introduced; it allows one to introduce a modal possibility by the usual combination of necessity operation and de Morgan negation; (3) the necessity operator satisfies a distributivity principle over joins. The latter property cannot be meaningfully added to the standard Boolean algebraic models of S5 modal logic, since in this Boolean context both modalities collapse in the identity mapping. The consistency of this algebraic model is proved, showing that usual fuzzy set theory on a universe U can be equipped with a MDS5 structure that satisfies all the above points (1)–(3), without the trivialization of the modalities to the identity mapping. Further, the relationship between this new algebra and Heyting-Wajsberg algebras is investigated. Finally, the question of the role of these deviant modalities, as opposed to the usual non-distributive ones, in the scope of knowledge representation and approximation spaces is discussed.

Environmental enrichment (EE) was shown recently to accelerate brain development in rodents. Increased levels of maternal care, and particularly tactile stimulation through licking and grooming, may represent a key component in the early phases of EE. We hypothesized that enriching the environment in terms of body massage may thus accelerate brain development in infants. We explored the effects of body massage in preterm infants and found that massage accelerates the maturation of electroencephalographic activity and of visual function, in particular visual acuity. In massaged infants, we found higher levels of blood IGF-1. Massage accelerated the maturation of visual function also in rat pups and increased the level of IGF-1 in the cortex. Antagonizing IGF-1 action by means of systemic injections of the IGF-1 antagonist JB1 blocked the effects of massage in rat pups. These results demonstrate that massage has an influence on brain development and in particular on visual development and suggest that its effects are mediated by specific endogenous factors such as IGF-1. Copyright © 2009 Society for Neuroscience.

This paper investigates the significance of truth-functional three valued logics of ill-known sets described by pairs of disjoint (or pairs of nested) subsets. In particular the case of three-valued logics of rough sets is studied in more details, showing that, while the mathematical underpinnings are sound, the relevance of these logics for reasoning about data-tables containing incomplete descriptions of objects is questionable. © 2010 IEEE.

The paper shows that a cube of opposition, a structure that generalizes the square of opposition invented in ancient logic, can be generated from the composition of a binary relation with a subset, by the effect of set complementation on the subset, on the relation, or on the result of the composition. Since the composition of relations is encountered in many areas, the structure of opposition exhibited by the cube of opposition has a universal flavor. In particular, it applies to information processing-oriented settings such as rough set theory, possibility theory, or formal concept analysis. We then discuss how this structure extends to a fuzzy relation and a fuzzy subset, and the graded cube of opposition thus obtained provides an organized view of the different existing compositions of fuzzy relations. The paper concludes by pointing out areas of research where the cube of opposition, or its graded version are of interest.

Most apicomplexan parasites possess a non-photosynthetic plastid (the apicoplast), which harbors enzymes for a number of metabolic pathways, including a prokaryotic type II fatty acid synthesis (FASII) pathway. In Toxoplasma gondii, the causative agent of toxoplasmosis, the FASII pathway is essential for parasite growth and infectivity. However, little is known about the fate of fatty acids synthesized by FASII. In this study, we have investigated the function of a plant-like glycerol 3-phosphate acyltransferase (TgATS1) that localizes to the T. gondii apicoplast. Knock-down of TgATS1 resulted in significantly reduced incorporation of FASII-synthesized fatty acids into phosphatidic acid and downstream phospholipids and a severe defect in intracellular parasite replication and survival. Lipidomic analysis demonstrated that lipid precursors are made in, and exported from, the apicoplast for de novo biosynthesis of bulk phospholipids. This study reveals that the apicoplast-located FASII and ATS1, which are primarily used to generate plastid galactolipids in plants and algae, instead generate bulk phospholipids for membrane biogenesis in T. gondii.

© 2015 Wiley Periodicals, Inc. This survey paper provides an overview of existing methods for building possibility distributions. We both consider the case of qualitative possibility theory, where the scale remains ordinal, and the case of quantitative possibility theory, where the scale is the real interval [0, 1]. Methods may be order-based or similarity-based for qualitative possibility distributions, whereas statistical methods apply in the quantitative case and then possibilities encode nested random epistemic sets or upper bounds of probabilities. But distance-based approaches, or expert estimates, may be also exploited in the quantitative case.

© Springer International Publishing Switzerland 2016. Sugeno integrals are aggregation functions defined on a qualitative scale where only minimum, maximum and order-reversing maps are allowed. Recently, variants of Sugeno integrals based on Gödel implication and its contraposition were defined and axiomatized in the setting of bounded chain with an involutive negation. This paper proposes a more general approach. We consider totally ordered scales, multivalued conjunction operations not necessarily commutative, and implication operations induced from them by means of an involutive negation. In such a context, different Sugeno-like integrals are defined and axiomatized.