(p.240) APPENDICES
(p.240) APPENDICES
1. A Refutation of Independence
Let us say that a belief is occurrent if it is the belief that you are consciously attending to, and let X's occurrent belief = the belief of X which is occurrent. There may, of course, be no such thing as X's occurrent belief, because either X has more than one occurrent belief, or X has none. Ditto for occurrent desire and X's occurrent desire.
Let P be the proposition: X's occurrent belief is false. Let Q be the proposition: X's occurrent desire is satisfied. If generalized independence is true, then any belief–desire pair is logically possible. So the following must also be possible: P = X's occurrent desire, and Q = X's occurrent belief. We show that this leads to a contradiction. First, let's go with a brief proof which assumes bivalence—that every proposition is either true or false.
Suppose P is true. Then X's belief is false. But Q = X's belief. So Q is false. So it is false that X's desire is satisfied. X's desire is not satisfied. But X's desire = P. So P is false. (Contradiction.)
Suppose P is false. Then it is false that X's belief is false. But Q = X's belief. So Q is true. If Q is true, then X's desire is satisfied. But X's desire = P. So P is true. (Contradiction.)
Bivalence, however, is not compatible with the analysis of presupposition I put forward in Chapter 1. There I argued for truth‐valuelessness. And truth–value gaps clearly can arise when we have definite descriptions which fail to pick out an object—definite descriptions like ‘the occurrent belief of X’. If there is no such thing as X's occurrent belief, then a de re claim like X's occurrent desire is false is truth‐valueless rather than true or false. And since truth and falsehood are total properties of propositions, the only way that X's occurrent desire is false can lack a truth value is by the non‐existence of X's occurrent belief. We can now give a proof that doesn't assume bivalence. Still, every proposition is either true, or false, or truth‐valueless.
Suppose P is true. Then X's belief is false. But Q = X's belief. So Q is false. So it is false that X's desire is satisfied. X's desire is not satisfied. But X's desire = P. So P is not true—it is either false or truth‐valueless. (Contradiction.)
(p.241) Suppose P is false. Then it is false that X's belief is false. But Q = X's belief. So Q is true or truth‐valueless. If Q is true, then X's desire is satisfied. But X's desire = P. So P is true. (Contradiction.) Suppose Q is truth‐valueless. Then there is no such thing as X's desire. (Contradiction.)
Suppose P is truth‐valueless. That can only be because there is no such thing as X's occurrent belief. (Contradiction.)^{1}
2. Seemings as Evidence
What we want to prove is that the proposition it seems that Q is prima facie evidence for Q—that is to say, it actually raises the probability of Q—if and only if there is a non‐zero probability that it seems that Q is a reliable indicator of the truth of Q.
Let S^{Q} be short for it seems that Q, and let Rel(P,Q) be short for P is a reliable indicator of the truth of Q. The reliability thesis says, plausibly, that the conjunction of the proposition S ^{Q} is a reliable indicator of the truth of Q and the proposition it seems that Q raises the (subjective) probability of Q.
Reliability: P(QS^{Q} & Rel(S^{Q},Q)) > P(Q).
The non‐reliability thesis says that the conjunction of the propositions that it seems that Q and that S ^{Q} is not a reliable indicator of the truth of Q leaves the probability of Q unchanged. Note that to say that S^{Q} is not a reliable indicator of the truth of Q is not to say that it is an indicator of Q's falsehood—that it is a somewhat reliable indicator of ˜Q. In other words, the negation of Rel(S^{Q},Q)—˜Rel(S^{Q},Q)—is not Rel(S^{Q},˜Q). Consequently, S^{Q}&˜Rel(S^{Q},Q) should not undermine your confidence in Q, and it certainly shouldn't enhance it, but simply leave it as it is. (Let's abbreviate “Rel(S^{Q},Q )” to “R^{Q}”.)
Non‐reliability: P(QS^{Q} &˜R^{Q}) = P(Q).
Clearly, whether or not it seems that Q is irrelevant to whether or not S^{Q} is a reliable indicator of the truth of Q.
Irrelevance: P(R^{Q}S^{Q}) = P(R^{Q}) (and P(˜R^{Q}S^{Q}) = P(˜R^{Q})).
Finally the following are standard properties of probability:
Additivity: P(AC) = P(A&BC) + P(A&˜BC).
Chain rule: P(A&BC) = P(AB&C) P(BC).
(p.242) To Prove: P(Q S^{Q}) > P(Q) iff P (R^{Q}) > 0.

1 P (Q S^{Q}) = P(Q & R^{Q}S^{Q}) + P(Q & ˜R^{Q}S^{Q}) (Additivity)

2 P(Q & R^{Q}S^{Q}) = P(Q S^{Q} & R^{Q})P(R^{Q}S^{Q}) (Chain rule)

3 P(Q & ˜R^{Q}S^{Q}) = P(Q S^{Q} & ˜R^{Q})P(˜R^{Q}S^{Q}) (Chain rule)

4 P(R^{Q}S^{Q}) = P(R^{Q}) (Irrelevance)

5 P(˜R^{Q}S^{Q}) = P(˜R^{Q}) (Irrelevance)

6 P(Q S^{Q}) = P(Q S^{Q} & R^{Q})P(R^{Q}) + P(Q S^{Q} & ˜R^{Q})P(˜R^{Q}) (1–5)

7 P(Q S^{Q} & R^{Q}) = P(Q) + δ (for some δ > 0) (Reliability)

8 P(Q S^{Q} & ˜R^{Q}) = P(Q) (Non‐reliability)

9 P(Q S^{Q}) = (P(Q) + δ)P(R^{Q}) + P(Q)P(˜R^{Q}) (6–8).

10 P(Q S^{Q}) = P(Q)[P(R^{Q}) + P(˜R^{Q})] + P(R^{Q})δ (9)

11 P(Q S^{Q}) = P(Q) + P(R^{Q})δ (10, Additivity)

12 P(Q S^{Q}) > P(Q) iff P (R^{Q}) > 0. (11)
Here is a possible objection to the conclusion. (The following argument is based on one in Cohen (2003).) Suppose it looks red confirms it is red. Then if X looks red, I have evidence that X is red and, if that is my total evidence, then I may be entitled to adopt the claim that X is red. So now I embrace the conjunction: X looks red and X is red. Now that conjunction raises the probability that my senses are reliable, and since my only evidence for that conjunction is the original claim that X looks red, my original observation alone—that X looks red—confirms the reliability of my senses. But that's ridiculous because it's just too easy. Knowledge of the reliability of the senses cannot come so cheaply!
My proof shows that there is a non sequitur lurking in this argument. Suppose that P(Rel (S^{Q}, Q))>0. Then it follows that P(Q S^{Q}) > P(Q). That is, that S^{Q} raises the probability of the truth of Q. But that is quite compatible with S^{Q} leaving the probability of Rel (S^{Q}, Q) exactly where it is. Indeed, that it leave it unchanged is a premiss of the argument—it is the premiss I have labelled Irrelevance.
(p.243) 3. The BAD Paradox
If we combine the DAB thesis (viz. D(A) = P(Å)), with the principle of updating by conditionalization (viz. where E is the new evidence, updated probability P^{+}(A) = P(AE)) we get a contradiction.

1 D^{+}(A) = P^{+}(Å). Hence D^{+}(A) = P(ÅA→ ˜Å) = P(Å & ˜A)/P(A→˜Å).
Since P(Å & ˜A) > 0, and Å&˜A entails (A→˜Å), P(A→ ˜Å) > 0, we have D^{+}(A) > 0.

2 Let P_{B}(A) be P(AB), and let D_{B} be the desiredness function based on P_{B}. The principle of desiredness tells us quite generally that D(A) = D_{A}(A), and so, D^{+}(A) = D^{+} _{A}(A). By DAB, D^{+} _{A}(A) = P^{+} _{A}(Å). Hence D^{+}(A) = P^{+} _{A}(Å) = P(ÅA & (A→˜Å)) = P(Å &˜Å & A)/P(˜Å & A). Since P(˜Å & A) > 0 and P(Å & ˜Å & A) = 0, we have D^{+}(A) = 0.
Notes:
(1) I am indebted to Paul Studtmann for forcing me to think about logical limitations on the generalized thesis of independence. Although this example is not one of those he suggested, it is certainly inspired by it.