Extension of Boolean algebra by a Bayesian operator; application to the definition of a Deterministic Bayesian Logic

TL;DR

This paper introduces an algebraic extension of Boolean algebra incorporating a Bayesian conditional operator, enabling probability extension and leading to a new deterministic Bayesian logic.

Contribution

It proposes a novel algebraic extension of Boolean algebra with a Bayesian operator, addressing Lewis' triviality by constructing a conditional outside the event space.

Findings

Probability measures extend to the algebraic extension

A new bivalent logic based on the extension is developed

Properties of the new logic are systematically derived

Abstract

This work contributes to the domains of Boolean algebra and of Bayesian probability, by proposing an algebraic extension of Boolean algebras, which implements an operator for the Bayesian conditional inference and is closed under this operator. It is known since the work of Lewis (Lewis' triviality) that it is not possible to construct such conditional operator within the space of events. Nevertheless, this work proposes an answer which complements Lewis' triviality, by the construction of a conditional operator outside the space of events, thus resulting in an algebraic extension. In particular, it is proved that any probability defined on a Boolean algebra may be extended to its algebraic extension in compliance with the multiplicative definition of the conditional probability. In the last part of this paper, a new bivalent logic is introduced on the basis of this algebraic extension, and basic properties are derived.