Chris Fermüller has recently put forward a convincing interpretation of Lukasiewicz real valued logic based on Giles’s games (see, eg. Fermüller and Metcalfe, 2009; Fermüller, 2010). As the latter are based on probability, this solution can be seen as muddling up the clear and widely endorsed distinction between degrees of truth and degrees of belief. I argue that the distinction need not be gi...

We investigate what happens if PA LTr2, a co-inductive language, formalizes itself. We analyze the truth concept in fuzzy logics by formalizing truth degree theory in the framework of truth theories in fuzzy logics. Hájek-Paris-Shepherdson’s paradox [HPS00] involves that so called truth degrees do not represent the degrees of truthhood (defined by the truth predicate) correctly in Lukasiewicz i...

The notion of filter is established in fuzzy class theory. Graded properties of filters, prime filters and related concepts are investigated. In particular we study properties of filters relativized with respect to the basic three t-norms: Gödel, product, and Lukasiewicz. It is shown that our approach gives rise to a broad class of models including probability and necessity measures on fuzzy se...

Dominance is a relation on operations which are defined on a common poset. We treat the dominance relation on the set of ordinal sum t-norms which involve either exclusively the Lukasiewicz t-norm or exclusively the product t-norm as summand operations. We show that in both cases, the question of dominance can be reduced to a simple property of the idempotent elements of the dominating t-norm. ...

Canonical completeness results for Ł(C), the expansion of Łukasiewicz logic Ł with a countable set of truth-constants C, have been recently proved in [5] for the case when the algebra of truth constants C is a subalgebra of the rational interval [0, 1] ∩ Q. The case when C 6⊆ [0, 1] ∩ Q was left as an open problem. In this paper we solve positively this open problem by showing that Ł(C) is stro...

Fuzzy description logics (DLs) serve as a tool to handle vagueness in real-world knowledge. There is particular interest in logics implementing Łukasiewicz semantics, which has a number of favourable properties. Current decision procedures for Łukasiewicz fuzzy DLs work by reduction to exponentially large mixed integer programming problems. Here, we present a decision method that stays closer t...

It is known that determining the satisfiability of n-valued ?ukasiewicz rules is NP-complete for n?4, as well as that it can be solved in time linear in the length of the formula in the Boolean case (when n=2). However, the complexity for n=3 is an open problem. In this paper we formally prove that the satisfiability problem for 3-valued ?ukasiewicz rules is NP-complete. Moreover, we also prove...

It is known that determining the satisfiability of n-valued ?ukasiewicz rules is NP-complete for n?4, as well as that it can be solved in time linear in the length of the formula in the Boolean case (when n=2). However, the complexity for n=3 is an open problem. In this paper we formally prove that the satisfiability problem for 3-valued ?ukasiewicz rules is NP-complete. Moreover, we also prove...

Following M. Krachounov ([5]), max and min operations with fuzzy sets are considered instead of Lukasiewicz ones ([6], [7], [8], [9]). Her the domain F of a probability m : F → [0, 1] consists on IF-events A = (μA, νA) i.e. mappings from a measurable space (Ω,S) to the unit square ([1]). Local representation of sequences of M -observables by random variables is constructed and three kinds of co...

This paper presents a hybrid method to construct concise and comprehensible fuzzy rules from training data. The construction procedure consists of a genetic algorithm, which determines the rulebase, and a gradient based optimization for the tuning of the membership functions. An approximation of the well-known Lukasiewicz logic is used to describe both the fuzzy connectives and the memberships,...