Subjective probability, trivalent logics and compound conditionals

TL;DR

This paper explores subjective probability through de Finetti's theory, examining coherence, conditional events in trivalent logic, and proposing a new approach to conjunction and disjunction of conditional events that preserves key logical and probabilistic properties.

Contribution

It introduces a novel approach to conjunction and disjunction of conditional events within trivalent logic that maintains essential logical and probabilistic properties.

Findings

Coherence criteria are equivalent and have a geometric interpretation.

Selected trivalent logics do not satisfy all logical and probabilistic properties.

The proposed approach preserves basic logical and probabilistic properties, including Fréchet-Hoeffding bounds.

Abstract

In this work we first illustrate the subjective theory of de Finetti. We recall the notion of coherence for both the betting scheme and the penalty criterion, by considering the unconditional and conditional cases. We show the equivalence of the two criteria by giving the geometrical interpretation of coherence. We also consider the notion of coherence based on proper scoring rules. We discuss conditional events in the trivalent logic of de Finetti and the numerical representation of truth-values. We check the validity of selected basic logical and probabilistic properties for some trivalent logics: Kleene-Lukasiewicz-Heyting-de Finetti; Lukasiewicz; Bochvar-Kleene; Sobocinski. We verify that none of these logics satisfies all the properties. Then, we consider our approach to conjunction and disjunction of conditional events in the setting of conditional random quantities. We verify that all the basic logical and probabilistic properties (included the Fr\'{e}chet-Hoeffding bounds) are preserved in our approach. We also recall the characterization of p-consistency and p-entailment by our notion of conjunction.