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Against CoherenceTruth, Probability, and Justification$

Erik J. Olsson

Print publication date: 2005

Print ISBN-13: 9780199279999

Published to British Academy Scholarship Online: July 2005

DOI: 10.1093/0199279993.001.0001

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(p.211) Appendix B Proof of the Impossibility Theorem

(p.211) Appendix B Proof of the Impossibility Theorem

Source:
Against Coherence
Publisher:
Oxford University Press

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= {〈E 1,H〉, 〈E 2,H〉} and P a class of probability distributions defined on the algebra generated by propositions E 1, E 2, R 1, R 2, U 1, U 2, and H such that PP if and only if:

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

(ii) 0<P(H)<1

(iii) P(E 1/H, R 1)=1=P(E 2/H, R 2)

(iv) P(E 1H, R 1)=0=P(E 2H, R 2)

(v) P(E 1/H, U 1)=P(H)=P(E 2/H, Ui)

(vi) P(E 1H, U 1)=P(H)=P(E 2H, U 2)

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

(p.212)

(viii) P(Ui/H)=P(Ui)=P(UiH)

(ix) P(E 1/H)=P(E 1/H, E 2)

(x) P(E 1H)=P(E 1H,E 2)

(xi) P(R 1)=P(R 2)>0

It can be shown that basic Lewis scenarios satisfy the conditions of individual credibility and independence.

Lemma 1: (Theorem 3 in Bovens et al. 2002) Let 〈S,P〉 be a basic Lewis scenario. Letting h=P(H), h ¯ = P ( ¬ H ) , and r = P(R i),

P ( H / E 1 , E 2 ) = h * = ( h + r h ¯ ) 2 ( h + r 2 h ¯ )

Lemma 2: (Bovens et al. 2002: 547) Let 〈S, P〉 be a basic Lewis scenario. For all r, h * as a function of h has a unique global minimum for h ∈ ]0,1[ which is reached at

h min = r 1 + r
By calculating the first derivative one can see that h * increases (decreases) strictly monotonically for h> (<) h min.

Observation 1: 0<h *<1

Observation 2: h * → 1 as h → 0

Observation 3: h min → 0 as r → 0

Observation 4: h min → 1/2 as r → 1

Definition 2: Let C be a coherence measure. C is informative in a basic Lewis scenario 〈S,P〉 if and only if there are P, P′ ∈ P such that CP(S)≠CP′(S).

Definition 3: A coherence measure C is truth conducive ceteris paribus in a basic Lewis scenario 〈S,P〉 if and only if: if CP(S)>CP′(S), then P(S)>P′(S) for all P,P′ ∈ P such that P(Ri)=P′(Ri).

The stipulation that P(Ri)=P′(Ri) is part of the ceteris paribus condition. The other part, concerning independence, is guaranteed already by the fact that we are dealing with Lewis scenarios that, so to speak, have independence built into them.

I will make frequent use in the following of the fact that a probability distribution in P is uniquely characterized by the probability it assigns to H and Ri. Furthermore, for every pair 〈r,h〉 there is a probability distribution Pr,h in P such that P(Ri)=r and P(H)=h.

Observation 5: P r, hmin(r)(H/E 1,E 2) → 0 as r → 0

(p.213) Impossibility theorem: There are no informative coherence measures that are truth conducive ceteris paribus in a basic Lewis scenario.

Proof: We will seek to establish that if C is truth conducive ceteris paribus in a basic Lewis scenario, then C is not informative in such a scenario. We recall that the degree of coherence of an evidential system S={〈E 1,H〉, 〈E 2,H〉} is the coherence of the pair 〈H, H〉. Moreover, if C is a coherence measure then C(〈H, H〉) is defined in terms of the probability of H and its Boolean combinations, as explained in section 6.1 above. In other words, CP(〈H, H〉)=C(h) where h=P(H). From what we just said it is clear that in order to show that C is not informative, in the sense of CP(S)=CP′(S) for all P, P′ ∈ P, it suffices to prove that C(h) is constant for all h ∈ ]0,1[. We will try to accomplish this in two steps, by first showing that C(h) is constant in I=]0, ½[ and then extending this result to the whole interval ]0,1[.

Suppose, then, that C is not constant in I. Hence, there are h 1, h 2I such that C(h 1)≠C(h 2). We may assume h 1<h 2.

Case 1: C(h 1)>C(h 2). By Observation 3, h min goes to 0 as r goes to 0. Since h 1>0, it follows that there is a probability of reliability r such that h min<h 1. Consider distributions P r , h 1 and P r , h 2 in P. By Lemma 2, h min is a unique global minimum and h * is monotonically decreasing for h>h min. Hence, P r , h 1 ( h 1 / E 1 , E 2 ) < P r , h 2 ( h 2 / E 1 , E 2 ) . Hence, C is not truth conducive (see Figure B1).

Case 2: C(h 1)<C(h 2). By Observation 4, h min goes to 1/2 r goes to 1. It follows that there is a probability of reliability r such that h 2< h min < 1/2. Consider distributions P r , h 1 and P r , h 2 in P. By Lemma 2, h min is a unique global minimum and h * is monotonically increasing for h<h min. Hence, P r , h 1 ( h 1 / E 1 , E 2 ) > P r , h 2 ( h 2 , E 1 , E 2 ) . It follows that C is not truth conducive (see Figure B2).

Appendix B Proof of the Impossibility Theorem

Figure B1. C(h1)>C(h2). By choosing r such that hmin<h1 we can construct a counter example to the truth conduciveness of C in the interval I =]0, ½[.

(p.214) What has been shown so far is that, if C is truth conducive, C is constant in I.

We will proceed to show that, if C is truth conducive, then C is constant in I′=[½, 1[ as well. Suppose C is truth conducive but not constant in I′. Since C is truth conducive, C(h)=c for all hI. Since C is assumed not constant in I′, there is an hI′ such that C(h)≠c.

Case 1: C(h)>c.

By Observation 2, Pr,h(H/E 1,E 2) goes to 1 as h goes to 0. Since Pr,h(H/E 1,E 2)<1, there is a h′ ∈ I such that Pr,h′(H/E 1,E 2)>Pr,h(H/E 1,E 2), whereas C(h′)=c<C(h). This contradicts the assumption of C's truth conduciveness (see Figure B3).

Appendix B Proof of the Impossibility Theorem

Figure B2. C(h1)<C(h2). By choosing r such that hminε]h2, ½[ we can construct a counter-example to the truth conduciveness of C for h ε]0, ½[.

Appendix B Proof of the Impossibility Theorem

Figure B3. C(h)>c. There is then a point h′ such that C(h′)=c<c(h) but Pr,h′(H/E1, E2) > Pr,h′(H/E1, E2).

(p.215)
Appendix B Proof of the Impossibility Theorem

Figure B4. C(h) < c. By choosing r so that Pr,hmin(H/E1, E2) < Pr,h(H/E1, E2) we get a counter example to the truth conduciveness of C.

Case 2: C(h)<c. By Observation 5, P r , h min ( H / E 1 , E 2 ) goes to 0 as r goes to 0. By Observation 3, h min goes to 0 as r goes to 0. It follows by these two observations and the fact that P(h)>0 that there is an r such that P r , h min ( H / E 1 , E 2 ) < P r , h ( H / E 1 , E 2 ) with h minI. Since h minI, C(h min)=c>C(h). We have shown that there is an h′ such that C(h)<C(h′) and yet Pr,h(H/E 1,E 2)>Pr,h′(H/E 1,E 2). Again, we have a clash with the assumption that C is truth conducive (see Figure B4).

We have reached a contradiction and may conclude that, if C is truth conducive, then C is constant not only in I but also in I′ so that C is in fact constant in the whole interval ]0,1[. As we said in the beginning, this is sufficient to establish that, if C is truth conducive ceteris paribus for a basic Lewis scenario, then C is not informative in such a scenario.