On the Algebraic Structure of Conditional Events

Abstract

Abstract. This paper initiates an investigation of conditional measures as simple measures on conditional events. As a first step towards this end we investigate the construction of conditional algebras which allow us to distinguish between the logical properties of conditional events and those of the conditional measures which we can be attached to them. This distinction, we argue, helps us clarifying both concepts.

On Boolean Algebras of Conditionals and Their Logical Counterpart

“…This paper deepens the investigation on Boolean algebras of conditional events that we began in [4]. Here, we have presented a full description of the atomic structure of these algebras and, based on this, we have defined a corresponding logic, LBC, to reason with conditionals.…”

“…Boolean algebras of conditionals, introduced and investigated in [4], are built as follows. Let A = (A, ∧, ∨, ¬, , ⊥) be a Boolean algebra and let A = A \ {⊥}.…”

“…We refer the interested reader to [4] for further details, and limit ourselves to collect the properties required in this paper in the following proposition. Proposition 1.…”

“…This paper builds on [4], where a Boolean algebra structure for conditionals was proposed with the goal of clarifying the relationship between conditional probabilities and simple probabilities on conditional events. The approach of considering (measure-free) conditionals as Boolean objects departs from previous ones in the literature where conditionals are mainly considered as three-valued objects, proposing different definitions for the operations between consitionals, see e.g.…”

“…To this end, we make two main new contributions. First, we fully characterise the atomic structure of BAC algebras, a problem that was left open in [4]. This is done in Section 3.…”