Many-valued Events Research Articles

THREE CHARACTERIZATIONS OF STRICT COHERENCE ON INFINITE-VALUED EVENTS

AbstractThis article builds on a recent paper coauthored by the present author, H. Hosni and F. Montagna. It is meant to contribute to the logical foundations of probability theory on many-valued events and, specifically, to a deeper understanding of the notion of strict coherence. In particular, we will make use of geometrical, measure-theoretical and logical methods to provide three characterizations of strict coherence on formulas of infinite-valued Łukasiewicz logic.

STRICT COHERENCE ON MANY-VALUED EVENTS

AbstractWe investigate the property of strict coherence in the setting of many-valued logics. Our main results read as follows: (i) a map from an MV-algebra to [0,1] is strictly coherent if and only if it satisfies Carnap’s regularity condition, and (ii) a [0,1]-valued book on a finite set of many-valued events is strictly coherent if and only if it extends to a faithful state of an MV-algebra that contains them. Remarkably this latter result allows us to relax the rather demanding conditions for the Shimony-Kemeny characterisation of strict coherence put forward in the mid 1950s in this Journal.

Layers of zero probability and stable coherence over Łukasiewicz events

The notion of stable coherence has been recently introduced to characterize coherent assignments to conditional many-valued events by means of hyperreal-valued states. In a nutshell, an assignment, or book, $$\beta $$β on a finite set of conditional events is stably coherent if there exists a coherent variant $$\beta '$$βź of $$\beta $$β such that $$\beta '$$βź maps all antecedents of conditional events to a strictly positive hyperreal number, and such that $$\beta $$β and $$\beta '$$βź differ by an infinitesimal. In this paper, we provide a characterization of stable coherence in terms of layers of zero probability for books on źukasiewicz logic events.

Coherence in the aggregate: A betting method for belief functions on many-valued events

Betting methods, of which de Finetti's Dutch Book is by far the most well-known, are uncertainty modelling devices which accomplish a twofold aim. Whilst providing an (operational) interpretation of the relevant measure of uncertainty, they also provide a formal definition of coherence. The main purpose of this paper is to put forward a betting method for belief functions on MV-algebras of many-valued events which allows us to isolate the corresponding coherence criterion, which we term coherence in the aggregate. Our framework generalises the classical Dutch Book method.

Non-standard probability, coherence and conditional probability on many-valued events

The usual coherence criterion by de Finetti is extended both to many-valued events and to conditional probability. Special attention is paid to assessments in which the betting odds for conditioning events are zero. This case is treated by means of infinitesimal probabilities. We propose a rationality criterion, called stable coherence, which is stronger than coherence in the sense of no sure loss.

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.

Extension of belief functions to infinite-valued events

We generalise belief functions to many-valued events which are represented by elements of Lindenbaum algebra of infinite-valued ?ukasiewicz propositional logic. Our approach is based on mass assignments used in the Dempster---Shafer theory of evidence. A generalised belief function is totally monotone and it has Choquet integral representation with respect to a unique belief measure on Boolean events.

An algebraic treatment of imprecise probabilities

Abstract This is a survey paper about an algebraic approach to imprecise probabilities. In the first part of it, we outline the work by Walley on imprecise probabilities and the more algebraic approach of Fedel et al.. Then, in the second part we will present some work in progress about a general treatment of upper and lower probabilities over many-valued events and of upper and lower previsions of gambles, by means of Universal Algebra.

Partially Undetermined Many-Valued Events and Their Conditional Probability

A logic for classical conditional events was investigated by Dubois and Prade. In their approach, the truth value of a conditional event may be undetermined. In this paper we extend the treatment to many-valued events. Then we support the thesis that probability over partially undetermined events is a conditional probability, and we interpret it in terms of bets in the style of de Finetti. Finally, we show that the whole investigation can be carried out in a logical and algebraic setting, and we find a logical characterization of coherence for assessments of partially undetermined events.

A Notion of Coherence for Books on Conditional Events in Many-valued Logic

We introduce a new approach to conditional probability over many-valued events, which is based on bets. Then we show that this approach fits with Kroupa's approach, and we give two characterizations of coherence for books on conditional many-valued events, the first one based on states, and the second one based on logical coherence of a suitable theory on many-valued logic.

MV-algebras with internal states and probabilistic fuzzy logics

In this paper we enlarge the language of MV-algebras by a unary operation σ equationally described so as to preserve the basic properties of a state in its original meaning. The resulting class of algebras will be called MV-algebras with internal state (or SMV-algebras for short). After discussing some basic algebraic properties of SMV-algebras, we apply them to the study of the coherence problem for rational assessments on many-valued events. Then we propose an algebraic treatment of the Lebesgue integral and we show that internal states defined on a divisible MV Δ -algebra can be represented by means of this more general notion of integral.

Finitely many-valued regular events in a direct product of semigroups

Finitely many-valued regular events in a direct product of semigroups