Goodman–Nguyen–van Fraassen algebra

Construction of the algebra

Given set Ω, which is the set of possible outcomes, and set F of subsets of Ω—so that F is the set of possible events—consider an infinite Cartesian product of the form E₁ × E₂ × … × E_n × Ω × Ω × Ω × …, where E₁, E₂, … E_n are members of F. Such a product specifies the set of all infinite sequences whose first element is in E₁, whose second element is in E₂, …, and whose nth element is in E_n, and all of whose elements are in Ω. Note that one such product is the one where E₁ = E₂ = … = E_n = Ω, i.e., the set Ω × Ω × Ω × Ω × …. Designate this set as Ω ^ ; it is the set of all infinite sequences whose elements are in Ω.

A new Boolean algebra is now formed, whose elements are subsets of Ω ^ . To begin with, any event which was formerly represented by subset A of Ω is now represented by A ^ = A × Ω × Ω × Ω × ….

Additionally, however, for events A and B, let the conditional event A → B be represented as the following infinite union of disjoint sets:

[(A ∩ B) × Ω × Ω × Ω × …] ∪ [A′ × (A ∩ B) × Ω × Ω × Ω × …] ∪ [A′ × A ′ × (A ∩ B) × Ω × Ω × Ω × …] ∪ ….

The motivation for this representation of conditional events will be explained shortly. Note that the construction can be iterated; A and B can themselves be conditional events.

Intuitively, unconditional event A ought to be representable as conditional event Ω → A. And indeed: because Ω ∩ A = A and Ω′ = ∅, the infinite union representing Ω → A reduces to A × Ω × Ω × Ω × ….

Let F ^ now be a set of subsets of Ω ^ , which contains representations of all events in F and is otherwise just large enough to be closed under construction of conditional events and under the familiar Boolean operations. F ^ is a Boolean algebra of conditional events which contains a Boolean algebra corresponding to the algebra of ordinary events.

Definition of the extended probability function

Corresponding to the newly constructed logical objects, called conditional events, is a new definition of a probability function, P ^ , based on a standard probability function P:

P ^ (E₁ × E₂ × … E_n × Ω × Ω × Ω × …) = P(E₁)⋅P(E₂)⋅ … ⋅P(E_n)⋅P(Ω)⋅P(Ω)⋅P(Ω)⋅ … = P(E₁)⋅P(E₂)⋅ … ⋅P(E_n), since P(Ω) = 1.

It follows from the definition of P ^ that P ^ ( A ^ ) = P(A). Thus P ^ = P over the domain of P.

P(A → B) = P(B|A)

Now comes the insight which motivates all of the preceding work. For P, the original probability function, P(A′) = 1 – P(A), and therefore P(B|A) = P(A ∩ B) / P(A) can be rewritten as P(A ∩ B) / [1 – P(A′)]. The factor 1 / [1 – P(A′)], however, can in turn be represented by its Maclaurin series expansion, 1 + P(A′) + P(A′)² …. Therefore, P(B|A) = P(A ∩ B) + P(A′)P(A ∩ B) + P(A′)²P(A ∩ B) + ….

The right side of the equation is exactly the expression for the probability P ^ of A → B, just defined as a union of carefully chosen disjoint sets. Thus that union can be taken to represent the conditional event A→ B, such that P ^ (A → B) = P(B|A) for any choice of A, B, and P. But since P ^ = P over the domain of P, the hat notation is optional. So long as the context is understood (i.e., conditional event algebra), one can write P(A → B) = P(B|A), with P now being the extended probability function.