# (p.205) Appendix A Counter-example to the Doxastic Extension Principle

# (p.205) Appendix A Counter-example to the Doxastic Extension Principle

Let **S** and **S**′ be two testimonial systems. **S**′ is a non-trivial extension of **S** if and only if (i) **S** ⊂ **S**′ and (ii) the content of **S**′ is logically stronger than the content of **S** (i.e. there are logical consequences of the contents of **S**′ that are not logical consequences of the contents of **S**). The Testimonial Extension Principle says this:

As a special case, we have the Doxastic Extension Principle:(

TEP) IfS′ is a non-trivial extension ofS, thenS′ is less probable thanS.

As I have argued, Klein and Warfield need ((

DEP) IfSandS′ are doxastic systems andS′ is a non-trivial extension ofS, thenS′ is less probable thanS.

*DEP*) to be true for their counter-example to the truth conduciveness of coherence to work. I will now give a counter-example to this principle. The example shows that a non-trivial extended doxastic system can in fact be more probable than the original system.

Let us consider the following simple variation on the Dunnit example. Suppose that there has been a robbery. A conscientious detective would like to know whether Dunnit committed the robbery (*R*) and consults independent witnesses in order to gather evidence. Although the witnesses need not be fully reliable, they all have a track record of being sufficiently reliable, and our detective routinely adopts the belief that some item of evidence holds just in case there is a witness report to this effect.

As was noted in the very beginning of this book, we often adopt beliefs in a routine-like manner. As Isaac Levi (1991: 71) describes the process, ‘[i]n routine expansion, the inquirer expands according to a program for adding new information to his state of full belief or corpus in response to external stimulation’. A characteristic feature of such routine expansion is that it does not rely on inference. While routine expansion does begin with certain (p.206) assumptions or premisses—namely, assumptions about the reliability of the programme—the expansion adopted is not inferred from these premisses (ibid.: 74). On Levi's view, routine expansion includes consulting witnesses (ibid.: 75). As we have seen, BonJour emphasizes the importance of such automatically acquired (‘cognitively spontaneous’) beliefs for a coherence theory.

We will assume that each item of evidence is reported by one single witness. After querying a bystander, the detective adopts the belief that Dunnit was driving his car away from the crime scene at high speed (*C*). After querying one of Dunnit's neighbours, he adopts the belief that Dunnit is in the possession of a gun of the same type as the one used in the robbery (*G*). The original doxastic system **S** contains the pairs 〈*BelC,C*〉 and 〈*BelG,G*〉. Subsequently, a new witness steps forward: after querying the bank clerk in Dunnit's bank, the detective adopts the belief that Dunnit deposited a large sum of money in his bank the day after the robbery (*M*). Dunnit's non-trivially extended doxastic system **S**′ now contains the additional pair 〈*BelM,M*〉.

One might object that we could have discussed an extension from one to two beliefs, rather than from two to three beliefs. From a mathematical point of view, this would indeed have been sufficient for the purpose of rejecting (*DEP*). However, from an epistemological point of view, the one-proposition case is, as we noted already in Chapter 2, problematic. The reason is that we want to compare two belief systems with respect to their coherence, and one belief hardly qualifies as a belief *system*. More fundamentally, coherence is, as we also saw, a concept that simply does not apply to singletons. This is what Rescher's Principle says. Hence, we cannot compare a singleton with a set of two or more propositions with respect to coherence. The relation ‘more coherent than’ is undefined if one of the relata is a singleton. In order to avoid such conceptual problems, we consider an extension of a belief system from two to three propositions.

It is easy to construct a case in which (*DEP*) is false on the grounds that the extended doxastic system **S**′ is *equally* probable as the original doxastic system **S**. Suppose that the witnesses are all *fully* reliable. Then, obviously,

*DEP*) as it stands.

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But (*DEP*) is actually stronger than it needs to be. What Klein and Warfield need is only a weaker principle to the effect that a non-trivially extended doxastic system is not more probable than the original. Let us refer to this as the Weak Doxastic Extension Principle (*WDEP* Kern). That is the principle we need to rebut. The challenge, then, is to construct a case in which the extended doxastic system **S**′ is *more* probable than the original doxastic system **S**.

First, let us assume that there is a large number *n* of suspects and that each suspect stands an equal chance of having committed the robbery, so that

*P*as a variable for propositions and

*p, q, s, t*, and

*u*as numerical variables):

*Independent Evidence*. The respective items of evidence are probabilistically independent of any other items of evidence, conditional on the
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hypothesis. What does this mean? Suppose that we actually know whether Dunnit committed the robbery. Then there is a certain chance that he was speeding on the motorway away from the crime scene. Now suppose that we learn in addition that he is in possession of a gun of the same type as the one that was used in the robbery. Then learning this new item of evidence will not affect the chance that Dunnit was speeding on the motorway. Note that this assumption is not always fulfilled: the items of evidence may be of a nature that does not warrant this assumption. For instance, the fact that Dunnit came into the repair shop the day before the robbery for a tune-up so that his car would perform optimally at high speed would also constitute an item of evidence that he committed the robbery, but it would not constitute an independent item of evidence.

*Independent Witnesses*. The detective's routinely acquired belief about some item of evidence is probabilistically independent of any other item of evidence or of any other of his routinely acquired beliefs, conditional on the evidence. What does this mean? Suppose that we actually know whether Dunnit was speeding on the highway. Then there is a certain chance that a reliable witness would step forward with a report to this effect and that the detective would adopt this report as a routinely acquired belief. Now suppose that we learn in addition that Dunnit was in possession of a gun of the same type as the one that was used in the robbery or that there was a witness report to this effect. Then this will not affect the chance that a reliable witness would step forward with a report to the effect that Dunnit was speeding. This assumption stipulates that each witness is focused on the items of evidence that he reports on and does not attend to other items of evidence or to reports about other items of evidence. What would it take for this condition not to be fulfilled? Suppose that the witnesses have been doing their own detective work: they checked out other items of evidence or talked to witnesses who reported on these items of evidence. Then their judgements of whether it was really Dunnit they saw speeding in the car could well be affected by whether there was other evidence to the effect that Dunnit committed the crime or by whether there were any reports of such evidence.

*Obliquely Testable Hypothesis*. The detective's routinely acquired beliefs about the evidence are probabilistically independent of the hypothesis, conditional on the evidence. What this means is that none of the witnesses has any direct access to whether Dunnit committed the robbery or not. This question remains hidden in the black box: the witnesses' only access to it is through the items of evidence. Suppose that we actually know that the items of evidence obtain (or that some or none obtain). Then there is a
(p.209)
certain chance that reliable witnesses will step forward and that the detective would come to acquire beliefs to the effect that the evidence obtains. Now suppose that we learn in addition that Dunnit actually committed the crime. Then this will not affect the chance that the detective would come to acquire beliefs to the effect that the evidence obtains. What would it take for this condition not to be fulfilled? Suppose that the witnesses got a quick glimpse of the robbery scene. Then their judgements of whether it was really Dunnit who was speeding on the motorway, of whether Dunnit really has the same gun as the one used in the crime scene, and so on, may be coloured by what they were able to gather from the robbery scene.

These independence assumptions can be expressed formally in the style of Dawid (1979) and Spohn (1980). For the technical details, I refer to Bovens and Olsson (2002).

We can now show that the Weak Doxastic Extension Principle is false for this particular example for some plausible values of *p, q, s, t*, and *u*. The probability of the original doxastic system **S** is:

*Prob*

_{S}):

**can take on either of the two possible truth-values of**

*R**R*, and so on. We apply the chain rule:

*Prob*

_{S′}) we can derive

*p, q, s, t*, and

*u*, we can compute

*Prob*

_{s}.4 and

*Prob*

_{s′}.4. Setting all parameters at .90 yields

*Prob*

_{s′}=

*P*(

*C, G, M*/

*BelC, BelG, BelM*)=.8910>.7562=

*P*(C, G/

*BelC*,

*BelG*)=

*Prob*

_{s}, which is precisely the desired result.

We may conclude that if, under certain independence assumptions, we gain ‘coherent’ information from highly but not fully reliable witnesses, then our extended belief system may well be more probable than our original belief system, in which case we have a counter-example not only to the Doxastic Extension Principle but also to the Weak Doxastic Extension Principle.