It so rarely happens that witnesses of the same transaction perfectly and entirely agree in all points connected with it, that an entire and complete coincidence in every particular, so far from strengthening their credit, not unfrequently engenders a suspicion of practice and concert.
Thomas Starkie, A Practical Treatise of the Law of Evidence
Abstract
While it is natural to assume that contradiction between alleged witness testimonies to some event disconfirms the event, this generalization is subject to important qualifications. I consider a series of increasingly complex probabilistic cases that help us to understand the effect of contradictions more precisely. Due to the possibility of honest error on a difficult detail even on the part of highly reliable witnesses, agreement on such a detail can confirm H much more than contradiction disconfirms H. It is also possible to model scenarios where we strongly suspect ahead of time that one source has copied another. In these cases, contradiction on a detail due to witness error can even confirm H by disconfirming collusion or copying. Finally, still more complex scenarios show that indirect confirmation, as opposed to exact agreement, provides the “best of both worlds,” simultaneously disconfirming suspected copying while permitting the statements of both sources to be true.
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Notes
This corresponds to asking, in the evaluation of historical arguments from silence, how probable it is that a speaker or author would mention an event if it really took place. If that probability is low in absolute terms, that will make the argument from silence weak (T. McGrew 2014), ceteris paribus. The argument from silence is only an analogue to an argument from disagreement on detail, but it is a useful comparison.
A reviewer for this journal has pointed out the generalization that, when P(E|H) and P(~ E|~ H) are equidistant from 0.5, E favors H exactly as much as ~ E favors ~ H. This is true. It is, in effect, the same probabilistic fact as the one just noted: Namely, by the Bayes factor analysis used in this paper, when P(E|H) = P(E|~ H), E favors H just as much as ~ E favors ~ H. Which is to say, neither E nor ~ E favors either hypothesis over the other. The connection between the two points is as follows: For any hypothesis H and any univocal evidence E, P(E|H) and P(~ E|H) must sum to 1. Therefore, P(E|H) and P(~ E|H) are always equidistant from 0.5, in opposite directions. E.g. if P(E|H) is 0.7, P(~ E|H) is 0.3, and so forth. Moreover, if P(E|H) = P(E|~ H), then P(~ E|H) = P(~ E|~ H). Therefore, P(E|H) and P(~ E|~ H) are equidistant from 0.5, in opposite directions, just in case P(E|H) = P(E|~ H).
This example illustrates an important part of the reason for my use of the likelihood ratio as the measure of confirmation in this paper. My goal is to focus on the force of the evidence in itself, independent of the prior probabilities of H and ~ H. The ratio P(H|E)/P(H), a strong competitor in the popularity contest among measures of confirmation (see Fitelson, 1998), is sensitive to the prior probabilities, which obscures the point in question. Suppose that in the case just discussed the prior probability of H were 0.99 in the distribution in question. Then, when E is the agreement of the alleged witnesses on the detail and the likelihoods are as stated—P(E|H) = 0.4, P(E|~ H) = .01—P(H|E) ≈ 0.9997 and P(H|E)/P(H) ≈ 1.0098. When E is the witnesses’ contradicting each other on the detail and we have P(H) = 0.99 and the stated likelihoods—P(E|H) = 0.6, P(E|~ H) = 0.99—P(~ H|E) ≈ 0.01639, and P(~ H|E)/P(~ H) ≈ 1.6393. In other words, if we use this measure with these priors, contradiction confirms ~ H more than agreement confirms H. But this is a function of the priors, as can be seen if we keep the likelihoods the same and simply reverse the priors, so that the prior of H is 0.01 instead. Then, for E = agreement, P(H|E)/P(H) ≈ 28.776, while for E = contradiction, P(~ H|E)/P(~ H) ≈ 1.003. In other words, in that case, by that measure, agreement confirms H far more than contradiction confirms ~ H, but this is because H rather than ~ H has the low prior probability. In this paper I will sometimes compare the impact of differing evidence on the same hypothesis, for which the prior probability can be held equal. At other times I will compare the impact of different evidence upon hypotheses with different prior probabilities. So it would be impossible to illustrate all of the interesting effects I wish to discuss while using a measure that is sensitive to prior probability; by choice I am leaving prior probabilities out of consideration as much as possible.
This does not have to mean that partial copying has the same probability given H that it has given ~ H, especially not in the distribution representing a situation where both witnesses have attested to H. In fact, I will assume throughout that, once we have conditionalized on the fact that both documents or witnesses describe the main event in H, the absolute probability of both dependence hypotheses given H is lower than it is given ~ H. This is because, if H is true, we could have independent attestation to a real event, whereas that is impossible if H is false.
I am not assuming that the evidence thus far about the main event has equal probability given Echo Chamber and Partial Copying, either given H or given ~ H. This is important, since they would have to have the same proportions to one another given H and ~ H respectively in the current distribution if they both had equal likelihood vis a vis the evidence thus far and if they both had the same prior probabilities before any of the current evidence was taken into account. Our illustrative numbers in this distribution are P(EC|~ H) = 0.8 and P(PC|~ H) = 0.2, but those same respective conditional probabilities given H are 0.5 and 0.1. 8/2 is not the same proportion as 5/1. But, since EC and PC did not constitute a partition of either H or ~ H in the initial distribution, before any testimony was taken, then it is possible for them to stand in different proportions to one another in the current distribution given H and ~ H.
A contradiction in this complex scenario confirms H about as much as a contradiction confirms ~ H in the simpler scenario in Sect. 3 where we had a forthcomingness requirement and some degree of fallibility given H (namely, a probability that the second source agrees with the first on the specific number, given H, of 0.4).
The reader may wonder about the origin of these highly specific numbers. When one is trying to allow for four different possibilities which must all sum to one in a given portion of the distribution, and when one is trying to model particular epistemic intuitions, some numbers will appear over-precise so as to result in a coherent distribution. As explained in more detail below, for Partial Copying the goal is to model what we might picture happening if H is true but the second document has no independent access to the relevant events and is not going to copy the detail from the first document. (In this sense Partial Copying is similar to what was called “sloppy collusion,” above.) The author might well be cautious on such matters of detail, leading to silence on this one. He might try to guess, leading to contradiction. I discuss below why I have made silence more probable than contradiction. He would be very unlikely to produce a delicate indirect confirmation of the detail. And he would be even more unlikely to give the exact same detail without copying it. Trying to model all of this for four possibilities is what produces such artificially specific numbers.
Because there are more than two exclusive and exhaustive possible outcomes, it is possible for more than one of them to favor H. When we have given such a detailed analysis of multiple possible outcomes, we cannot make epistemically enlightening generalizations about the effect of the mere negation of one outcome—e.g., “not silence.” The negation of silence must favor ~ H if silence favors H. “Not silence” does very slightly favor ~ H, overall, due to the impact of exact agreement in further confirming copying. But “not silence” includes a possible outcome that significantly favors H on a Bayes factor analysis (indirect confirmation) and one that slightly favors it (contradiction).
Scenarios involving all four options (exact agreement, indirect confirmation, silence, and contradiction) are so complex that a distribution is even possible in which both exact agreement and contradiction disconfirm H, counterintuitive as that may seem. This is a result of the fact that there are several possible outcomes given H that take portions of the probability space. Suppose that Partial Copying rather than Echo Chamber is dominant, though both are possible. Suppose that P(Partial Copying|~ H) = 0.8, P(Echo Chamber|~ H) = 0.2, P(Partial Copying|H) = 0.2, P(Echo Chamber|H) = 0.05, and P(Independent Access|H) = 0.75, and all other conditional probabilities remain as stated above in Sect. 5. Then both contradiction and exact agreement confirm ~ H somewhat, though the confirmation from exact agreement is negligible. Indirect confirmation remains the strongest evidence for H. Proof omitted.
Proof omitted.
With these conditional numbers in place, something similar is true of silence. Silence absolutely rules out Echo Chamber but is more probable given Partial Copying than given Independent Access. Hence, once we have one case in hand where the second witness does not address the detail at all, a further instance of the same kind will necessarily disconfirm H at least somewhat.
My thanks to Timothy McGrew for useful conversations in which my ideas on these issues were tested and clarified.
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McGrew, L. Confirmation, Coincidence, and Contradiction. Synthese 199, 6981–7002 (2021). https://doi.org/10.1007/s11229-021-03102-x
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DOI: https://doi.org/10.1007/s11229-021-03102-x