A Tacit Assumption Behind Lewis Triviality That Is Not Applicable to Product Space Conditional Event Algebra

Abstract

For a given (generally, assumed atomic) nontrivial Boolean algebra B and any a, b in B with b ≠ ∅, a conditional event (a|b) is said to exist if for all probability measures P over B, there exists some mapping P^(not necessarily a probability measure) and some algebra B^ (not necessarily Boolean) containing (a|b) such that if P(b) > 0, then P^((a|b)) = P(a|b), ordinary conditional probability of a given b. But, in turn, Lewis’ famous triviality result shows equivalently that if his forcing hypothesis holds—i.e., B and B^ at least overlap at B*(a, b, (a|b)), the Boolean subalgebra generated by a, b, and (a|b) (and for consistency P = P^)—then P must be trivial in the sense that a, b must be P-independent. The apparent informal implication of this, as gleaned from Lewis’ own comments and that of the numerous papers that are based on his result, is that this spells the “death knell” for Stalnaker’s thesis or even for any attempt at constructing a consistent nontrivial algebra of conditional events deriving from a Boolean algebra B. In this paper, we show two basic results contradicting this “conclusion”. First, fully consistent and nontrivial Boolean (in fact, sigma-) algebras of conditional events can always be constructed relative to any given Boolean algebra B, based upon a standard product probability space construction. This avoids Lewis’ forcing hypothesis—and thus Lewis triviality—yet satisfies natural isomorphic and homomorphic relations with B. Second, a key step in Lewis’ own proof, involving conditioning, when generalized to a setting where his forcing hypothesis no longer is imposed, leads directly to a simple criterion determining the structure of consistent nontrivial Boolean conditional event algebras. This criterion is also directly related to the well-known property of import/export in AI (Artificial Intelligence) systems and conditional logic.