Logical counterfactuals using optimal predictor schemes

We give a sufficient condition for a logical conditional expectation value defined using an optimal predictor scheme to be stable on counterfactual conditions.

Definition

Given an error space of rank , the stabilizer of , denoted is the set of functions s.t. for any we have .

Theorem

Consider an error space of rank 2, , a word ensemble and a -optimal predictor scheme for . Assume is s.t.

(i)

(ii)

Consider and , -optimal predictor schemes for . Then, .

Note

This result can be interpreted as stability on counterfactual conditions since the similarity is relative to rather than only relative to . That is, and are similar outside of as well.

Proof of Theorem

We will refer to the previously established results about -optimal predictor schemes by L.N where N is the number in the linked post. Thus Theorem 1 there becomes Theorem L.1 here and so on.

By Theorem L.A.7

On the other hand, by Lemma L.B.3

Combining the last two statements we conclude that

It follows that