Probability, Abduction, and Peirce's Contrary Results Condition

In this blog, I assess different interpretations of Peirce's early formuation of abductive inference along Bayesian lines. In his 1878 paper, "Illustrations of the logic of Science," C.S. Peirce maintains that there are at least three types of inferences: deductive, inductive, and hypothesis (what he later called abduction). Peirce's later formulation of the abductive schema goes as follows:

1. The surprising fact E is observed.

2. If H were true, E would be a matter of course.

∴ H.

In his 1878 paper, Peirce makes two important qualifications regarding this sort of reasoning. First, abduction is a "weak kind of argument". Regarding the degree of support conferred to the conclusion by the premises, Peirce thinks "we cannot say that we believe the latter to be true; we only surmise that it may be so" (p. 473). In what follows, I will read Peirce as saying that Pr(E) << .5 and Pr(E|H) >> .5. However, it is more difficult to say exactly how confident Peirce thinks we should be about the conclusion, H. Three readings are possible: either H is now slightly greater than .5, or H is within a small margin around .5, or H is less than .5 but not negligible. In what follows, I will simply express the weight of the abductive support for H as 'Pr(H|E) > pᵢ', allowing that pᵢ can be interpreted in any of the following ways:

     (i) p₁ > .5 (slightly)

     (ii) p₂ ≈ .5

     (iii) p₃ < .5 but not negligible. 

I consider the merits of Peirce's abduction, as interpreted by (i). Also, regardless of how we interpret the claim that abduction only allows us to "surmise" the truth of the conclusion, it seems fitting to assume that at the very least Peirce thinks abductive inference confirms the claim in question, where E confirms H iff Pr(H|E) > Pr(H) (I exclude K for simplicity). I work with this assumption in the foregoing discussion.

The second of Peirce's qualifications concerns non-deductive inference broadly construed, of which abduction is a species. Peirce writes: "When we adopt a certain hypothesis, it is not alone because it will explain the observed facts, but also because the contrary hypothesis would probably lead to results contrary to those observed" (1878, p. 474). Adopting H on the basis of E requires the following:

(3) If ~H were the case, E would be very unlikely.

This can be formulated into the following condition for non-deductive inference:

Contrary Results Condition (CRC): for all non-deductive inference, H is made epistemically acceptable on the basis of E iff a) H explains (or makes likely) E and b) ~H makes E very unlikely.

CRC takes (3) as essential for non-deductive epistemic support. I now assess the merits of viewing (3) as either a necessary condition of abductive inference or, in conjunction with (1)-(2), as a sufficient condition for abductive inference. Consider what Peirce's abduction amounts to if the latter is true. Formally:

     (a) if Pr(E) << .5, Pr(E|H) >> .5, and Pr(E|~H) << .5, then Pr(H|E) > pᵢ. 

On interpretation (i), (a) is false. While it is true on any interpretation of pᵢ that E confirms H (this is because Pr(E|H)/Pr(E) > 1. See my blog on the relevance quotient), it does not follow that this confirmation raises H above .5. What is true is that Pr(H|E) > .5 only if the priors ratio —Pr(H)/Pr(~H) — is at an appropriate value, r, such that Pr(E|H)/Pr(E|~H) = q, and r x q > .5. But this is just a more complicated way of stating what is trivial from the Bayesian point of view: both likelihoods and priors matter with respect to posteriors.

Now, consider (3) as a necessary condition for inferring that Pr(H|E) > pᵢ. Formally:

     (b) If (1)-(2) justify believing that Pr(H|E) > pᵢ, then Pr(E|~H) << .5.

Again, on (i) this is incorrect. Assuming Pr(H)/Pr(~H) = 1, the probability calculus tells us that Pr(H|E) can be greater than .5, even if Pr(E|~H) > .5. What matters is that Pr(E|H)/Pr(E|~H) > 1, and this inequality can hold even if Pr(E|~H) > .5. Hence, (b) is false. 

On the strongest reading of pᵢ, interpreting Peirce as arguing that (3) is both necessary and (jointly) sufficient for abductive inference raises Bayesian eyebrows. Although I won't go into the details, (ii) and (iii) face similar problems. This is not an indictment of abductive reasoning, however. At most, it is an indictment of Peirce's claim that (3) is germane to abductive inference (or non-deductive inference in general). The CRC is mistaken.