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. 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. 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: 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%.' 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.