Abstract
In the argument by reductio ad absurdum, we prove the conclusion by showing that its negation is inconsistent with the premises. I extend this argument pattern to probabilistic support, viz., if the negation of the hypothesis is incoherent with the body of evidence (in the sense of mutual reduction of the probabilities), then the body of evidence confirms (raises the probability of) the hypothesis. The following comparative form of the principle also holds: If the body of evidence is more coherent with the hypothesis than it is with the negation of the hypothesis, then the body of evidence confirms the hypothesis. The principle reveals that the charge of circularity that is still common in epistemology is misguided—for example, it is perfectly legitimate to confirm the reliability of memory by memorial evidence and the reliability of sense perception by sense perceptual evidence.
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Notes
- 1.
- 2.
See Shogenji (2006) for an extended discussion of epistemic circularity involving testimonial evidence.
- 3.
See Alston (1989, pp. 326–329, 1993, pp. 15–17) for the distinction between logical circularity and epistemic circularity.
- 4.
I ignore the other type of degenerate cases in which the conclusion is logically true, by assuming that the conclusion is contingent. The reason for the different treatment is that probabilistic support, to which I will extend the argument pattern of reductio ad absurdum, only concerns a hypothesis that is not certain. If the hypothesis is logically true, then the question does not arise whether a body of evidence confirms (raise the probability of) the hypothesis.
- 5.
The principle is not biconditional so as to allow for the possibility that an inconsistent set of premises may deductively support the conclusion, for example, when the conclusion is one of the premises.
- 6.
In case some people worry that we cannot form the conjunction p1 ∧…∧ pn if the set of premises is infinite, they may recall that by the compactness theorem if an infinite set of premises logically entails a conclusion, there is a finite subset of it that also logically entails the conclusion, so that we can form the conjunction of the premises. The issue is of little relevance anyway to our actual reasoning since our epistemic resource is finite.
- 7.
Section 7 addresses the role of the background assumptions in suspected cases of epistemic circularity.
- 8.
- 9.
Those who like the neutral value 0 may prefer the log-ratio measure ConLR(h, e) = log[P(h|e)/P(h)] to the ratio measure. They are ordinally equivalent, but the neutral value for the log-ratio measure is 0.
- 10.
- 11.
Fitelson’s (2003) measure circumvents the problem by making an exception for the case of logically true members, but as Meijs (2006) points out, Fitelson’s measure behaves strangely when the members are almost logically true. The reason is that almost logically true members should have an almost neutral degree of coherence and an almost maximal degree of coherence at the same time.
- 12.
I am assuming here (and in what follows) that the conditional probabilities P(x2|x1) and P(x1|x2) are defined, that is, P(x1) ≠ 0 and P(x2) ≠ 0.
- 13.
CohR(x1, x2) is a restricted (to n = 2) version of the measure of coherence proposed in (Shogenji (1999). Those who dislike the neutral value 1 of the ratio measure may prefer its logarithmic variant CohLR(x1, x2) = log [P(x1 ∧ x2)/P(x1) × P(x2)]. The logarithmic variant is ordinally equivalent to the ratio measure, but its neutral value is 0.
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Shogenji, T. (2013). Reductio, Coherence, and the Myth of Epistemic Circularity. In: Zenker, F. (eds) Bayesian Argumentation. Synthese Library, vol 362. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5357-0_9
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