Lewis's triviality result

In the mathematical theory of probability, David Lewis's triviality result is a theorem about the impossibility of systematically equating the conditional probability

with the probability of a so-called conditional event,

Beginning in the 1960s, several philosophical logicians—most notably Ernest Adams and Robert Stalnaker—floated the idea that one might also write

Part of the appeal of this move would be the possibility of embedding conditional expressions within more complex constructions.

, to express someone's high subjective degree of confidence ("75% sure") that either

Compound constructions containing conditional expressions might also be useful in the programming of automated decision-making systems.

[2] How might such a convention be combined with standard probability theory?

The most direct extension of the standard theory would be to treat

are probabilities allocated to the eight respective regions, such that

region's proportional share of the probability inside the

In general the equality will of course not be true, so that making it reliably true requires a new constraint on probability functions: in addition to satisfying Kolmogorov's probability axioms, they must also satisfy a new constraint, namely that

Lewis (1976) pointed out a seemingly fatal problem with the above proposal: assuming a nontrivial set of events, the new, restricted class of

-functions will not be closed under conditioning, the operation that turns probability function

This implies that if rationality requires having a well-behaved probability function, then a fully rational person (or computing system) would become irrational simply in virtue of learning that arbitrary event

Bas van Fraassen called this result "a veritable bombshell" (1976, p. 273).

, that are mutually exclusive but do not together exhaust all possibilities, so that

The proof derives a contradiction from the assumption that such a minimally non-trivial set of events exists.

A graphical version of the proof starts with Fig.

, which means that Conditioning on an event involves zeroing out the probabilities outside the event's region and increasing the probabilities inside the region by a common scale factor.

In a follow-up article, Lewis (1986) noted that the triviality proof can proceed by conditioning not on

but instead, by turns, on each of a finite set of mutually exclusive and jointly exhaustive events

He also gave a variant of the proof that involved not total conditioning, in which the probability of either

Separately, Hájek (1989) pointed out that even without conditioning, if the number of outcomes is large but finite, then in general

One way to put the point: standardly, any weighted average of two probability function is itself a probability function, so that between any two

holds for a minimally nontrivial set of events and for any

However, the proof relies on background assumptions that may be challenged.

It may be proposed, for instance, that the referent event of an expression like “

condition, has been to explore ways to model conditional events as something other than subsets of a universe set of outcomes.

Even before Lewis published his result, Schay (1968) had modeled conditional events as ordered pairs of sets of outcomes.

With that approach and others in the same spirit, conditional events and their associated combination and complementation operations do not constitute the usual algebra of sets of standard probability theory, but rather a more exotic type of structure, known as a conditional event algebra.