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

# (p.211) 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 (2002*b*). 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.

Definition1:A basic Lewis scenariois a pair 〈S,P〉 whereS= {〈E_{1},H〉, 〈E_{2},H〉} andPa class of probability distributions defined on the algebra generated by propositionsE_{1},E_{2},R_{1},R_{2},U_{1},U_{2}, andHsuch thatP∈Pif and only if:(i)

P(R)+_{i}P(U)=1_{i}(ii) 0<

P(H)<1(iii)

P(E_{1}/H,R_{1})=1=P(E_{2}/H,R_{2})(iv)

P(E_{1}/¬H,R_{1})=0=P(E_{2}/¬H,R_{2})(v)

P(E_{1}/H,U_{1})=P(H)=P(E_{2}/H,U)_{i}(vi)

P(E_{1}/¬H,U_{1})=P(H)=P(E_{2}/¬H,U_{2})(vii)

(p.212)P(R/_{i}H)=P(R)=_{i}P(R/¬_{i}H)(viii)

P(U/_{i}H)=P(U)=_{i}P(U/¬_{i}H)(ix)

P(E_{1}/H)=P(E_{1}/H,E_{2})(x)

P(E_{1}/¬H)=P(E_{1}/¬H,E_{2})(xi)

P(R_{1})=P(R_{2})>0It 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*), $\overline{h}=P\left(\neg H\right)$, and *r* = *P*(*R* _{i}),

**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*

^{*}increases (decreases) strictly monotonically for

*h*> (<)

*h*

_{min}.

The stipulation that

Observation 1:0<h^{*}<1

Observation 2:h^{*}→ 1 ash→ 0

Observation 3:h_{min}→ 0 asr→ 0

Observation 4:h_{min}→ 1/2 asr→ 1

Definition 2: LetCbe a coherence measure.Cisinformativein a basic Lewis scenario 〈S,P〉 if and only if there areP,P′ ∈Psuch that C(_{P}S)≠C(_{P′}S).

Definition 3: A coherence measure C istruth conducive ceteris paribusin a basic Lewis scenario 〈S,P〉 if and only if: ifC(_{P}S)>C(_{P′}S), thenP(S)>P′(S) for allP,P′ ∈Psuch thatP(R)=_{i}P′(R)._{i}

*P*(

*R*)=

_{i}*P′*(

*R*) is part of the

_{i}*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 *R _{i}*. Furthermore, for every pair 〈

*r,h*〉 there is a probability distribution

*P*in

_{r,h}**P**such that

*P*(

*R*)=

_{i}*r*and

*P*(

*H*)=

*h*.

Observation 5:P_{r, hmin(r)}(H/E_{1},E_{2}) → 0 asr→ 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, *C _{P}*(〈

*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

*C*(

_{P}**S**)=

*C*(

_{P′}**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* _{2} ∈ *I* 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, ${P}_{r,{h}_{1}}\left({h}_{1}/{E}_{1},{E}_{2}\right)<{P}_{r,{h}_{2}}\left({h}_{2}/{E}_{1},{E}_{2}\right)$. 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, ${P}_{r,{h}_{1}}\left({h}_{1}/{E}_{1},{E}_{2}\right)>{P}_{r,{h}_{2}}\left({h}_{2},{E}_{1},{E}_{2}\right)$. It follows that *C* is not truth conducive (see Figure B2).

(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 *h* ∈ *I*. Since *C* is assumed not constant in *I*′, there is an *h* ∈ *I*′ such that *C*(*h*)≠*c*.

*Case 1*: *C*(*h*)>*c*.

By Observation 2, *P _{r,h}*(

*H*/

*E*

_{1},

*E*

_{2}) goes to 1 as

*h*goes to 0. Since

*P*(

_{r,h}*H*/

*E*

_{1},

*E*

_{2})<1, there is a

*h*′ ∈

*I*such that

*P*(

_{r,h′}*H*/

*E*

_{1},

*E*

_{2})>

*P*(

_{r,h}*H*/

*E*

_{1},

*E*

_{2}), whereas

*C*(

*h′*)=

*c*<

*C*(

*h*). This contradicts the assumption of

*C*'s truth conduciveness (see Figure B3).

*Case 2*: *C*(*h*)<*c*. By Observation 5, ${P}_{\text{r},{h}_{min}}\left(H/{E}_{1},{E}_{2}\right)$ 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 ${P}_{\text{r},{h}_{min}}\left(H/{E}_{1},{E}_{2}\right)<{P}_{r,h}\left(H/{E}_{1},{E}_{2}\right)$ with *h* _{min} ∈ *I*. Since *h* _{min} ∈ *I*, *C*(*h* _{min})=*c*>*C*(*h*). We have shown that there is an *h*′ such that *C*(*h*)<*C*(*h*′) and yet *P _{r,h}*(

*H*/

*E*

_{1},

*E*

_{2})>

*P*(

_{r,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.