An approximate Herbrand theorem is proved and used to establish Skolemization for first-order Łukasiewicz logic. Proof systems are then defined in the framework of hypersequents. In particular, extending a hypersequent calculus for propositional Łukasiewicz logic with usual Gentzen quantifier rules gives a calculus that is complete with respect to interpretations in safe MV-algebras, but lacks ...

It is well-known that satisfiability (and hence validity) in the minimal classical modal logic is a PSPACE-complete problem. In this paper we consider the satisfiability and validity problems (here they are not dual, although mutually reducible) for the minimal modal logic over a finite Lukasiewicz chain, and show that they also are PSPACE-complete. This result is also true when adding either t...

In the paper we explore the idea of describing Pawlak’s rough sets using three-valued logic, whereby the value t corresponds to the positive region of a set, the value f — to the negative region, and the undefined value u — to the border of the set. Due to the properties of the above regions in rough set theory, the semantics of the logic is described using a non-deterministic matrix (Nmatrix)....

are presented in the context of their applications to complex genetic network dynamics, highly complex systems, quantum automata [2]–[3] and quantum supercomputers. Our novel approach to the Categorical Ontology Theory of Levels impacts on Medical Bioinformatics and self-organizing, Highly-Complex Systems (HCS), such as living organisms and artificial intelligent systems (AIs). Quantum Automata...

Although it is well-known that every satisfiable formula in Lukasiewicz’ infinite-valued logic L∞ can be satisfied in some finitevalued logic, practical methods for finding an appropriate number of truth degrees do currently not exist. As a first step towards efficient reasoning in L∞, we propose a method to find a tight upper bound on this number which, in practice, often significantly improve...

A dialogue game based approach to the problem of providing a deeper semantic foundation for t-norm based fuzzy logics is explored. In particular, various versions, extensions and alternatives to Robin Giles’s dialogue and betting game for Lukasiewicz logic are re-visited and put in the context of other foundational research in logic. It emerges that dialogue games cover a wide range of topics r...

This paper is directed towards an infrastructure for handling both uncertainty and vagueness in the Rules, Logic, and Proof layers of the Semantic Web. More concretely, we present probabilistic fuzzy description logic programs, which combine fuzzy description logics, fuzzy logic programs (with stratified nonmonotonic negation), and probabilistic uncertainty in a uniform framework for the Semant...

In this paper we shall describe a correspondence between Rational à Lukasiewicz formulas and neural networks in which the activation function is the truncated identity and synaptic weights are rational numbers. On one hand to have a logical representation (in a given logic) of neural networks could widen the interpretability, amalgamability and reuse of these objects. On the other hand, neural ...

Paraconsistent logic and three-valued semantics: The term Paraconsistent was first used by the Peruvian philosopher Francisco Miró Quesada in the Third Latin America Conference on Mathematical Logic in 1976. A logic is called paraconsistent if there are formulas φ and ψ such that {φ,¬φ} 0 ψ. Besides other semantics, the three-valued semantics of (paraconsistent) logics have always received spec...

MV-algebras were introduced by Chang as an algebraic counterpart of the Lukasiewicz infinite-valued logic. D. Mundici proved that the category of MV-algebras is equivalent to the category of abelian l-groups with strong unit. A. Di Nola and A. Lettieri established a categorical equivalence between the category of perfect MV-algebras and the category of abelian l-groups. In this paper we investi...