Against Coherence: Truth, Probability, and Justification

By Erik J. Olsson | Go to book overview

Appendix B Proof of the Impossibility Theorem

We will consider a case of full agreement between independent reports that are individually credible, while respecting the ceteris paribus condition. We will show that there are no informative coherence measures that are truth conducive ceteris paribus in such a scenario which I will refer to as a basic Lewis scenario. The name is appropriate considering Lewis's reference to relatively unreliable witnesses telling the same story. A number of additional constraints will be imposed on the probabilities involved. The constraints are borrowed from a model proposed by Luc Bovens and his colleagues (2002). That model was in turn devised as an improvement of the model suggested in Olsson (2002b). The most salient feature of this sort of model is that the reliability profile of the witnesses is, in a sense, incompletely known. The witnesses may be completely reliable (R) or they may be completely unreliable (U), and initially we do not know which possibility holds. An interesting consequence of this sort of model is that, from a certain context-dependent level of prior improbability, the posterior probability will be inversely related to the prior: the lower the prior, the higher the posterior. This feature is exploited in the following.

Definition 1: A basic Lewis scenario is a pair 〈S,P〉 where S= {〈E1,H〉, 〈E2,H〉} and P a class of probability distributions defined on the algebra generated by propositions E1, E2, R1, R2, U1, U2, and H such that PP if and only if:

(i) P(Ri)+P(Ui)=1

(ii) 0<P(H)<1

(iii) P(E1/H, R1)=1=P(E2/H, R2)

(iv) P(E1H, R1)=0=P(E2H, R2)

(v) P(E1/H, U1)=P(H)=P(E2/H, Ui)

(vi) P(E1H, U1)=P(H)=P(E2H, U2)

(vii) P(Ri/H)=P(Ri)=P(RiH)

-211-

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