Tommaso Flaminio

A logical and algebraic treatment of conditional probability

Abstract.This paper is devoted to a logical and algebraic treatment of conditional probability. The main ideas are the use of non-standard probabilities and of some kind of standard part function in order to deal with the case where the conditioning event has probability zero, and the use of a many-valued modal logic in order to deal probability of an event φ as the truth value of the sentence φ is probable, along the lines of Hájek’s book [H98] and of [EGH96]. To this purpose, we introduce a probabilistic many-valued logic, called FP(SŁΠ), which is sound and complete with respect a class of structures having a non-standard extension [0,1]⋆ of [0,1] as set of truth values. We also prove that the coherence of an assessment of conditional probabilities is equivalent to the coherence of a suitably defined theory over FP(SŁΠ) whose proper axioms reflect the assessment itself.

T-norm-based logics with an independent involutive negation

In this paper we investigate the addition of arbitrary independent involutive negations to t-norm-based logics. We deal with several extensions of MTL and establish general completeness results. Indeed, we will show that, given any t-norm-based logic satisfying some basic properties, its extension by means of an involutive negation preserves algebraic and (finite) strong standard completeness. We will deal with both propositional and predicate logics.

On the Logical Formalization of Possibilistic Counterparts of States over n-valued Łukasiewicz Events

The front panel of the trip device is formed by a flat screen display, for example liquid crystal, controlled by the trip device microprocessor. The display provides a permanent display of the information relating to the type of trip device, the tripping parameters and measured values, from data entered in the trip device memory. Display is in alphanumeric and/or graphic form.

Models for Many-Valued Probabilistic Reasoning

In this article, we compare models for many-valued probabilistic reasoning from the point of view of the sets of satisfiable formulas, positive satisfiable formulas, and tautologies. The results arising from this comparison will be used in the final part of the present article to provide results about the computational complexity for the problem of deciding if a formula belongs to one of the previously discussed sets.

Generalized rough approximations in Ł Π12

In this paper we will treat a generalization of inner and outer approximations of fuzzy sets, which we will call R-inner and R-outer approximations respectively (R being any finite set of rational numbers in [0,1]). In particular we will discuss the case of those fuzzy sets which are definable in the logic L@P12 by means of step functions from the hypercube [0,1]^k and taking value in an arbitrary (finite) subset of [0,1]@?Q. Then, we will show that if a fuzzy set is definable as truth table of a formula of L@P12, then both its R-inner and R-outer approximation are definable as truth table of formulas of L@P12. Finally, we will introduce a generalization of abstract approximation spaces and compare our approach with the notion of fuzzy rough set.

Logics for belief functions on MV-algebras

In this paper we present a generalization of belief functions over fuzzy events. In particular we focus on belief functions defined in the algebraic framework of finite MV-algebras of fuzzy sets. We introduce a fuzzy modal logic to formalize reasoning with belief functions on many-valued events. We prove, among other results, that several different notions of belief functions can be characterized in a quite uniform way, just by slightly modifying the complete axiomatization of one of the modal logics involved in the definition of our formalism.

The coherence of Łukasiewicz assessments is NP-complete

The problem of deciding whether a rational assessment of formulas of infinite-valued Lukasiewicz logic is coherent has been shown to be decidable by Mundici [1] and in PSPACE by Flaminio and Montagna [10]. We settle its computational complexity proving an NP-completeness result. We then obtain NP-completeness results for the satisfiability problem of certain many-valued probabilistic logics introduced by Flaminio and Montagna in [9].

Belief functions on MV-algebras of fuzzy events based on fuzzy evidence

Recently Kroupa has proposed a generalization of belief functions on MV-algebras, the latter being the chosen algebraic setting for fuzzy (or many-valued) events. However, Kroupas belief functions evaluate the degree of belief in the occurrence of fuzzy events by taking into account (weighted) evidence on classical subsets. In other words, the focal elements, used in determining the degree of belief, are classical sets. Within the MV-algebraic setting, the aim of the present work is to introduce a generalization of Kroupa belief functions that allows to deal with fuzzy events supported by evidence on fuzzy subsets.

A complete fuzzy logical system to deal with trust management systems

In this paper we approach trust management systems in a fuzzy logical setting. The idea is to provide a generalization of the classical framework, where trust is understood via the dichotomy “true--false”. In order to overcome the classical approach proposed by Weeks, following the ideas used by Hajek, Esteva, Godo and others to deal with probability, possibility, and necessity in a many-valued logical setting, we introduce the modal logic FTn(ŁΠ12) built up over the many-valued logic ŁΠ12. In particular, we enlarge the ŁΠ12 language by means of a binary modality says acting on pairs (pi,φ) of principals and assertions, where a principal is a propositional variable and an assertion is a propositional formula of a suited many-valued logic. The idea is to regard the evaluation of the modal formula says(pi,φ) as the degree of confidence the principalpiputs in the assertionφ. For FTn(ŁΠ12) we introduce a syntax, a semantic and we show completeness. Then we discuss the validity of generalized modus ponens rule in our setting. Finally we deal with a Pavelka-style extension of our logic, and we also extend FTn(ŁΠ12) to allow principals to be hierarchically organized.

A zero-layer based fuzzy probabilistic logic for conditional probability

This paper is devoted to a logical and algebraic treatment of conditional probability. Unlike the other approaches to this problem (cf [7],[10]) we base our work on the notion of zero-layer (cf [4]). Thus we define the fuzzy modal logic FPk(RŁΔ) with k modalities for non-conditional probably, built up over the many-valued logic RŁΔ (obtained by adding to the Rational Łukasiewicz logic the Baaz connective Δ). The main result of this paper tells us that it is possible to characterize the coherence of an assessment of conditional probability by the consistence of a suitable theory over FPk(RŁΔ).