Conditional event algebra

A conditional event algebra (CEA) is an algebraic structure whose domain consists of events, or states of affairs, described by statements of the form "If A, then B." Unlike the standard Boolean algebra of events, a CEA allows the defining of a probability function which satisfies the identity 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, $\Omega$ , of outcomes (or, in alternate terminology, a set of possible worlds) and a set, F, of some (not necessarily all) subsets of $\Omega$ , such that F is closed under the countably infinite versions of set-theoretic union ( $\cup$ ) and intersection ( $\cap$ ), and under complementation ( ’). A member of F is called an event (or, alternatively, a proposition). $\Omega$ is, necessarily, a member of F, namely the trivial event “Some outcome occurs.”

A probability function P assigns to each member of F a number on the interval [0, 1], in such a way as to satisfy the following axioms:

For any event E, P(E)

\geq

0.

P(

\Omega

) = 1

For any countable sequence E1, E2, ... of pairwise disjoint events, P(E1

\cup

E2

\cup

...) = P(E1) + P(E2) + ....

The probability function is the basis for statements like P(A $\cap$ 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 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 $\cap$ 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 ability to nest conditional event descriptions within larger constructions. Then, for instance, one could write P(A $\cup$ (B --> C)) = 0.51, meaning, “The probability that either A, or else if B, then C, is 51%.”

Unfortunately, the philosophical logician David Lewis showed that in orthodox probability theory, under normal circumstances there is, for a given A and B, no event X which reliably satisfies P(X) = P(B|A).(Ref needed) Later extended by others, this result stands as a major obstacle to any talk about conditional events.

Types of conditional event algebras

Conditional event algebras are designed to circumvent the obstacle just described. 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; for instance, "If A, then if B then C" is equivalent to "If A and B, then C" only in Shay and Calabrese algebras.