Using the methods of Russell and Restall’s paper on inference barriers, I will show that one can’t derive an “is” from a “probably”. That is, no consistent set of statements expressing only relations among the probabilities of statements expressible in a non-probabilistic object language can entail anything about the actual truth-values of such object language statements, unless the statements are either tautologies or contradictions.

Let S be a (consistent) set of probabilistic statements, and let O be the (non-tautological) object language statement that is said to follow from S. To show that O does not follow from S, start with a probability space (a set of “states”, together with an algebra of subsets of this set called “events”, a real-valued function on this algebra satisfying the probability axioms, and a specification of which state is “actual”) satisfying all of S. Call this space P. Now create a space P’ by adding a single state X in which O is false to the state space. A subset of this space will be an event iff either it doesn’t contain X and was an event in P, or it does contain X and removing X gives an event in P. Define the probability function on these events by assigning the same probability to any event in P’ that either it, or it without X, had in P. Let X be the “actual” state in the new space.

By construction, every probabilistic statement will have the same truth-value in the two spaces, because every proposition has exactly the same probability (effectively, all we did was add a single state with 0 probability). In particular, all of S is true in P’. Also, by construction, O is false in P’. Thus, S does not entail O, QED.

One way to block this argument would be to require that every non-empty event have non-zero probability, but this would block a lot of interesting probability spaces (in particular, any space with uncountably many mutually incompatible events). However, a very similar argument would go through if one allowed some small tolerance of epsilon in the probabilistic statements of S (assuming none of the statements are conditional probability statements, whose value can in fact deviate by much more than epsilon due to the addition of a single state of probability epsilon).

But in some sense, this shows the weakness of these “inference barrier” results, which Gillian Russell points out should really be called “implication barriers”. Under certain conditions, it’s certainly rational to infer that it will rain, given that there’s a 99% chance of rain. This result merely shows that no amount of probabilistic evidence will ever entail anything with certainty, even though it might entail it with probability 1. The distinction between probability 1 and certainty is something I’m thinking about right now for my dissertation.

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