Contrastive Bayesianism

One way to argue against Elliot Sober’s Contrastive Empiricism is to claim that the approach violates an important principle of confirmation theory. Branden Fitelson (2010) has done this in a recent paper by providing a counterexample to Sober’s Law of Likelihood (LL) for ‘favoring’ relations. According to (LL) evidence E favors hypothesis H1 over hypothesis H2 if and only if H1 confers greater probability on E than H2 does. You compare the likelihoods of alternative hypotheses. Fitelson argues against (LL) by proposing a counterexample. The counterexample includes the following facts:

  • E: The card is a spade.
  • H1: The card is the ace of spades.
  • H2: The card is black.

This makes the probabilities run as follows: Pr(E|H1) = 1 and Pr(E|H2) = 1/2. So, on (LL), E favors H1 over H2. However, this seems wrong. The truth of E guarantees (or necessitates) the truth of H2, but the truth of E doesn’t guarantee the truth of H1. This makes E conclusive evidence for H2 but not H1. So, it seems E favors H2 over H1 despite the likelihoods pointing the other direction. For Fitelson this suggests the following confirmation principle, called Conclusive Evidence (CE), a principle which (LL) violates:

  • (CE) If E constitutes conclusive evidence for H2, but E constitutes less than conclusive evidence for H1, then E favors H2 over H1.

In email correspondence with Fitelson I argued against (CE) by claiming that it causes probability and entailment to come apart. In this post I want to propose a different idea.

Carnap (1962) proposes two independent confirmation measures. The first is confirmation as firmness. Confirmation(f) tracks both truth and entailment. The second is confirmation as increase in firmness. Confirmation(i) tracks entailment in addition to an intuitive relevance. I think this suggests a Relevance (R) principle:

  • (R) If E constitutes conclusive evidence for H2, but E constitutes less than conclusive evidence for H1, then E is more relevant to H2 than H1.

If Fitelson’s use of ‘conclusive evidence’ is to work with regard to confirms(i) it should work with regard to relevance. In the card-drawing counterexample is the fact that the card is a spade (E) more relevant to the fact that the card is black (H2), or is (E) more relevant to the fact that the card is the ace of spades (H1)?

For one thing, relevance cannot be assimilated into entailment. This is because confirms(i) tracks entailment AS WELL AS relevance, according to Fitelson. If E is conclusive evidence for H, then Pr(H|E) = 1. In the counterexample, E is conclusive evidence for H2, as E establishes H2 with certainty, yet Pr(E|H2) = 1/2 and Pr(E|H1) = 1. This means H1 entails E, but H2 doesn’t. However, H2 seems more relevant to E than H1 even though H1 entails E. That the card is black is more relevant to the claim that the card is a spade than that the card is the ace of spades is relevant to the claim that the card is a spade. The latter claim is redundant whereas the former claim delimits, and adds to, the semantic content on the table. “The card is black, and by the way, the card is a spade not a club.” On this account, relevance provides non-redundant semantic information in helping the inquirer delimit the space of logical possibilities.

By contrast, it is true that if the card is a spade, then the card is black. But this is running things in the opposite direction from what I stated above. This makes it the case that the card is a spade (E) is more relevant to the claim that the card is the ace of spades (H1) than (E) is relevant to the claim that the card is black (H2). In this direction the color of the card in terms of meaning is already bound-up in the card being a spade. A spade in a standard deck is by definition a black card. My basic idea is that (intuitive) relevance is a measure of the degree to which something doesn’t participate in semantic redundancy.

In conclusion, I claimed (R) is false. Intuitive relevance, when it comes to conclusive evidence, works in the opposite direction of probabilistic strength and entailment. This means (CE), which implies (R) in order to qualify under a confirmation(i) measure, is also false. If confirmation(i) only tracks entailments, and relevance is dropped from the measure, then confirmation(i) can be assimilated into confirmation(f). This is because confirmation(i) involves necessary preservation of truth. If confirmation(i) isn’t different than confirmation(f) by virtue of tracking a type of relevance, then confirmation(i) is merely tracking truth-preservation. That is, confirmation(i) is merely tracking truth and, as such, is not independent from confirmation(f).

One Comment

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  1. Hi Christopher,

    I agree with you that Branden’s putative counterexample to LL cuts no ice. But, in contrast to you, I am perfectly happy with CE. For what it is worth, the root of my unhappiness with the cards case is simply that the probability of the intersection of the hypotheses is non-zero. For this reason, I find it counterintuitive to claim that the evidence favours one of the hypotheses over the other, or, equivalently, that it provides evidence that one hypothesis is true rather than the other. Once LL is appropriately restricted, it demonstrably satisfies CE. (I have made this observation in print, in a forthcoming Synthese piece: ‘Contrastive Confirmation: Some Competing Accounts’. There I adduce various further arguments in favour of LL.).

    All the best,

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