The usual motivation for a CEA is to ground the definition of a probability function for events, P, that satisfies the equation P(if A then B) = P(A and B) / P(A).
In other words, and, customarily represented by the logical symbol ∧, is interpreted as set intersection: P(A ∧ B) = P(A ∩ B).
In standard practice, P(if A, then B) is not interpreted as P(A′ ∪ B), following the rule of material implication, but rather as the conditional probability of B given A, P(B | A) = P(A ∩ B) / P(A).
What would be needed, for consistency, is a treatment of if-then as a binary operation, →, such that for conditional events A → B and C → D, P(A → B) = P(B | A), P(C → D) = P(D | C), and P((A → B) ∧ (C → D)) are well-defined and reasonable.
Philosophers including Robert Stalnaker argued that ideally, a conditional event algebra, or CEA, would support a probability function that meets three conditions: However, David Lewis proved in 1976 a fact now known as Lewis's triviality result: these conditions can only be met with near-standard approaches in trivial examples.
With three or more possible outcomes, constructing a probability function requires choosing which of the above three conditions to violate.
[1] Tri-event CEAs take their inspiration from three-valued logic, where the identification of logical conjunction, disjunction, and negation with simple set operations no longer applies.
Ordinary events, which are never undecided, are incorporated into the algebra as tri-events conditional on Ω, the vacuous event represented by the entire sample space of outcomes; thus, A becomes Ω → A.
When negation is handled in the obvious way, with ¬A undecided just in case A is, this type of tri-event algebra corresponds to a three-valued logic proposed by Sobociński (1920) and favored by Belnap (1973), and also implied by Adams’s (1975) “quasi-conjunction” for conditionals.
Schay (1968) was the first to propose an algebraic treatment, which Calabrese (1987) developed more properly.
[2] The other type of tri-event CEA treats negation the same way as the first, but it treats conjunction and disjunction as min and max functions, respectively, with occurrence as the high value, failure as the low value, and undecidedness in between.
This type of tri-event algebra corresponds to a three-valued logic proposed by Łukasiewicz (1920) and also favored by de Finetti (1935).
Goodman, Nguyen and Walker (1991) eventually provided the algebraic formulation.
[3] With this convention, conditions 2 and 3 above are satisfied by the two leading tri-event CEA types.
Since the second factor is the Maclaurin series expansion of 1 / [1 – P(¬A)] = 1 / P(A), the infinite sum equals P(A ∧ B) / P(A) = P(B |A).
Thus the conditional probability P(B |A) is turned into simple probability P(B → A) by replacing Ω, the sample space of all ordinary outcomes, with Ω*, the sample space of all sequences of ordinary outcomes, and by identifying conditional event A → B with the set of sequences where the first (A ∧ B)-outcome comes before the first (A ∧ ¬B)-outcome.
This type of CEA was introduced by van Fraassen (1976), in response to Lewis’s result, and was later discovered independently by Goodman and Nguyen (1994).
Calabrese adopts the latter view, identifying (A → B) → (C → D) with ((¬A ∨ B) ∧ C) → D.[8] With a product-space CEA, nested conditionals call for nested sequence-constructions: evaluating P((A → B) → (C → D)) requires a sample space of metasequences of sequences of ordinary outcomes.
The initial impetus for CEAs is theoretical—namely, the challenge of responding to Lewis's triviality result—but practical applications have been proposed.
If, for instance, events A and C involve signals emitted by military radar stations and events B and D involve missile launches, an opposing military force with an automated missile defense system may want the system to be able to calculate P((A → B) ∧ (C → D)) and/or P((A → B) → (C → D)).
[11] Other applications range from image interpretation[12] to the detection of denial-of-service attacks on computer networks.
"A theory of conditional information for probabilistic inference in intelligent systems: II, Product space approach; III Mathematical appendix".
International Journal of Uncertainty, Fuzziness and Knowledge-based Systems 3(3): 247-339 Kelly, P. A., Derin, H., and Gong, W.-B.
"Some applications of conditional events and random sets for image estimation and system modeling".
"Abnormal network traffic detection based on conditional event algebra".