We propose a parameterized framework based on a Heyting algebra and Lukasiewicz negation for modeling uncertainty for belief. We adopt a probability theory as mathematical formalism for manipulating uncertainty. An agent can express the uncertainty in her knowledge about a piece of information in the form of belief types: as a single probability, as an interval (lower and upper boundary for a p...

Towards sophisticated representation and reasoning techniques that allow for probabilistic uncertainty in the Rules, Logic, and Proof layers of the Semantic Web, we present probabilistic description logic programs (or pdl-programs), which are a combination of description logic programs (or dl-programs) under the answer set semantics and the well-founded semantics with Poole’s independent choice...

We explain Giles’s characterization of Lukasiewicz logic via a dialogue game combined with bets on results of experiments that may show dispersion. The game is generalized to other fuzzy logics and linked to recent results in proof theory. We argue that these results allow one to place t-norm based fuzzy logics in a common framework with supervaluation as a theory of vagueness.

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...

For two propositional fuzzy logics, we present analytic proof calculi, based on relational hypersequents. The logic considered first, called Mà L, is based on the finite ordinal sums of à Lukasiewicz t-norms. In addition to the usual connectives – the conjunction ̄, the implication →, and the constant 0 –, we use a further unary connective interpreted by the function associating with each truth...

We investigate the computational complexity of admissibility of inference rules in infinite-valued Lukasiewicz propositional logic ( L). It was shown in [13] that admissibility in L is checkable in PSPACE. We establish that this result is optimal, i.e., admissible rules of L are PSPACE-complete. In contrast, derivable rules of L are known to be coNP-complete.

In this paper we present a short history of logics: from particular cases of 2-symbol or numerical valued logic to the general case of n-symbol or numerical valued logic. We show generalizations of 2-valued Boolean logic to fuzzy logic, also from the Kleene’s and Lukasiewicz’ 3-symbol valued logics or Belnap’s 4-symbol valued logic to the most general n-symbol or numerical valued refined neutro...

Lukasiewicz’s infinite-valued logic is commonly defined as the set of formulas that take the value 1 under all evaluations in the Lukasiewicz algebra on the unit real interval. In the literature a deductive system axiomatized in a Hilbert style was associated to it, and was later shown to be semantically defined from Lukasiewicz algebra by using a “truthpreserving” scheme. This deductive system...

Fuzzy formal logics were introduced in order to handle graded truth values instead of only ‘true’ and ‘false’. A wide range of such logics were introduced successfully, like Monoidal T-norm based Logic, Basic Logic, Gödel Logic, Lukasiewicz Logic etc. However, in general, fuzzy set theory is not only concerned with vagueness, but also with uncertainty. A possible solution is to use intervals in...