Conditional event algebra

A conditional event algebra (CEA) is an algebraic structure whose domain consists of logical objects described by statements of forms such as "If A, then B," "B, given A," and "B, in case A." Unlike the standard Boolean algebra of events, a CEA allows the defining of a probability function, P, which satisfies the equation P(If A then B) = P(A and B) / P(A) over a usefully broad range of conditions.

Standard probability theory

In standard probability theory, one begins with a set, Ω, of outcomes (or, in alternate terminology, a set of possible worlds) and a set, F, of some (not necessarily all) subsets of Ω, such that F is closed under the countably infinite versions of the operations of basic set theory: union (∪), intersection (∩), and complementation ( ′). A member of F is called an event (or, alternatively, a proposition), and F, the set of events, is the domain of the algebra. Ω is, necessarily, a member of F, namely the trivial event "Some outcome occurs."

A probability function P assigns to each member of F a real number, in such a way as to satisfy the following axioms:

For any event E, P(E) ≥ 0.

P(Ω) = 1

For any countable sequence E₁, E₂, ... of pairwise disjoint events, P(E₁ ∪ E₂ ∪ ...) = P(E₁) + P(E₂) + ....

It follows that P(E) is always less than or equal to 1. The probability function is the basis for statements like P(A ∩ B′) = 0.73, which means, "The probability that A but not B is 73%."

Conditional probabilities and probabilities of conditionals

The statement "The probability that if A, then B, is 24%" means (put intuitively) that event B occurs in 24% of the outcomes where event A occurs. The standard formal expression of this is P(B|A) = 0.24, where the conditional probability P(B|A) equals, by definition, P(A ∩ B) / P(A).

It is tempting to write, instead, P(A → B) = 0.24, where A → B is the conditional event "If A, then B." That is, given events A and B, one might posit an event, A → B, such that P(A → B) could be counted on to equal P(B|A). One benefit of being able to refer to conditional events would be the opportunity to nest conditional event descriptions within larger constructions. Then, for instance, one could write P(A ∪ (B → C)) = 0.51, meaning, "The probability that either A, or else if B, then C, is 51%."

Unfortunately, philosopher David Lewis showed that in orthodox probability theory, only certain trivial Boolean algebras with very few elements contain, for any given A and B, an event X such that P(X) = P(B|A) is true for any probability function P. Later extended by others, this result stands as a major obstacle to any talk about logical objects that can be the bearers of conditional probabilities.

The construction of conditional event algebras

The classification of an algebra makes no reference to the nature of the objects in the domain, being entirely a matter of the formal behavior of the operations on the domain. However, investigation of the properties of an algebra often proceeds by assuming the objects to have a particular character. Thus, the canonical Boolean algebra is, as described above, an algebra of subsets of a universe set. What Lewis in effect showed is what can and cannot be done with an algebra whose members behave like members of such a set of subsets.

Conditional event algebras circumvent the obstacle identified by Lewis by using a nonstandard domain of objects. Instead of being members of a set F of subsets of some universe set Ω, the canonical objects are normally higher-level constructions of members of F. The most natural construction, and historically the first, uses ordered pairs of members of F. Other constructions use sets of members of F or infinite sequences of members of F.

Specific types of CEA include the following (listed in order of discovery):

Shay algebras

Calabrese algebras

Goodman-Nguyen-van Fraassen algebras

Goodman-Nguyen-Walker algebras

CEAs differ in their formal properties, so that they cannot be considered a single, axiomatically characterized class of algebra. Goodman-Nguyen-van Frassen algebras, for example, are Boolean while Calabrese algebras are non-distributive. The latter, however, support the intuitively appealing identity A → (B → C) = (A ∩ B) → C, while the former do not.

References

Goodman, I. R., R. P. S. Mahler, and H. T. Nguyen. 1999. "What is conditional event algebra and why should you care?" SPIE Proceedings, Vol 3720.

Lewis, David K. 1976. "Probabilities of conditionals and conditional probabilities." Philosophical Review 85: 297-315.