Three Characterizations of Strict Coherence on Infinite-Valued Events

Abstract

This 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.

Boolean algebras of conditionals, probability and logic

“…, ϕ k } to [0, 1], we will equivalently regard it as a subset of either [0, 1] k or [0, 1] E . As shown in [16,Corollary 3.2], C E can be defined without considering all elements x ∈ [0, 1] n . Indeed, let ∆ be a regular complex 1 linearizing the McNaughton functions f 1 , .…”

“…By [16,Corollary 3.2], the definition of C E given above does not depend on the specific ∆ we choose to linearize the McNaughton functions f ϕi 's.…”

“…McNaughton functions are all P. However, [16,Proposition 5.3] provides an algorithm that for every rational polyhedron P determines a specific formula χ P with the above property. This argument clearly applies also to coherent sets as the latter are convex rational polyhedra.…”

“…) and the claim(2) are[16, Corollary 3.2] and[38, Corollary 5.4] respectively. It is hence left to show that every C E contains a Boolean point, that is a to say a vertex of the cube [0, 1] k .…”

Probability over Plonka sums of Boolean algebras: states, metrics and topology

“…We have shown (see Theorem 19) that states over involutive bisemilattices correspond to integrals on the dual space of the Booleanisation. It makes sense to ask whether this correspondence can be extended to faithful states, relying on the integral representation proved for faithful states over free MV-algebras in [16].…”