Okasha on the Transmission of Epistemic Justification

In this blog, I consider a Bayesian analysis of Wright's principle of transmission failure, as defended by Okasha (2004).

Often, it appears that evidential support is transmitted across entailment. That is, the following principle seems right in many cases:

Transmission1: If E is evidence for P, and P entails Q, then E is evidence for Q.

Probabilistically put:

Transmission1: For all propositions, E, P, and Q, ⎕ ([Pr(P|E) > Pr(P) & P→Q] → Pr(Q|E) > Pr(Q))

According to Wright, the following seems to be an instance of transmission across entailment being satisfied:

(E) Jones consumed a large risotto of Boletus Satana.

(P) Jones consumed a lethal quantity of poison.

(Q) Jones is going to die.

E is evidence for P, P entails Q, and E is evidence for Q. Hence, E transmits support from P to Q. 

However, Wright argues that Transmission1 does not always hold and, hence, is false. Rather, in some cases, for E to support P at all, we must already have independent reasons for believing Q. This gives us:

Transmission2: E is evidence for P only if there are independent reasons for accepting Q, where P entails Q.

Okasha's Bayesian analysis of the above argument takes for granted the typical Bayesian understanding of evidential support:

Evidence: E is evidence for P iff Pr(P|E) > Pr(P).

Okasha then suggests that the following claims are involved in Transmission2: 

(i) Pr(P|E&Q) > Pr(P|Q)  "E supports P, only if we have independent reasons for accepting Q"

(ii) Pr(P|E) ≤ Pr(P)     "E does not support P otherwise"

(iii) Pr(Q|E) ≤ Pr(Q)     "E does not support Q"

This gives us a transmission-failure principle:

Transmission Failure: ([(i) Pr(P|E&Q) > Pr(P|Q) & (ii) Pr(P|E) ≤ Pr(P)] & P→Q) → (iii) Pr(Q|E) ≤ Pr(Q).

Okasha goes on to prove that, according to the probability calculus, Transmission Failure is true. So, if Transmission Failure adequately captures the content in Transmission2, then Transmission2 is correct, in certain cases. Even if P entails Q, there are cases where E supports P only when we already have independent reasons to accept Q.
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References:

Okasha, "Wright on the Transmission of Support -- a Bayesian Analysis" (2004).