The Paradox of Confirmation 1

Hempel first introduced the paradox of confirmation in 1937. Since then, a very extensive literature on the paradox has evolved (Vranas 2004). Much of this literature can be seen as responding to Hempel’s subsequent discussions and analyses of the paradox (Hempel 1945). Recently, it was noted that Hempel’s intuitive (and plausible) resolution of the paradox was inconsistent with his official theory of confirmation (Fitelson and Hawthorne 2006). In this article, we will try to explain how this inconsistency affects the historical dialectic about the paradox and how it illuminates the nature of confirmation. In the end, we will argue that Hempel’s intuitions about the paradox of confirmation were (basically) correct, and that it is his theory that should be rejected, in favor of a (broadly) Bayesian account of confirmation. 1. The Original Formulation of the Paradox Informally and pre-theoretically, confirmation is a relation of “support” between statements or propositions. So, when we say that p confirms q , what we mean (roughly and intuitively) is that the truth of p provides ( some degree of ) support 2 for the truth of q . These are called qualitative confirmation claims. And, when we say that p confirms q more strongly than p confirms r , we mean (roughly and intuitively) that the truth of p provides better support for the truth of q than it does for the truth of r . These are called comparative confirmation claims. Confirmation theory aims to provide formal explications of both the qualitative and comparative informal “support” concepts involved in such claims. The paradox of confirmation is a paradox involving the qualitative relation of confirmation, but some of its contemporary resolutions appeal also to the comparative concept. We begin with the original formulation of the paradox. Traditionally, the Paradox of Confirmation (as introduced in Hempel 1937) is based on the following two assumptions about the qualitative confirmation relation: • Nicod Condition (NC) : For any object a and any properties F and G , the proposition that a has both F and G confirms the proposition that 96 The Paradox of Confirmation © Blackwell Publishing 2006 Philosophy Compass 1/1 (2006): 95–113, 10.1111/j.1747-9991.2006.00011.x every F has G . A more formal version of (NC) is the following claim expressed in monadic predicate-logical symbolism: For any individual term ‘ a ’ and any pair of predicates ‘ F ’ and ‘ G ’ ( Fa · Ga ) confirms ( ∀ x )( Fx ⊃ Gx ). In slogan form, (NC) might be expressed as “universal claims are confirmed by their positive instances.” It is called the Nicod Condition because it was first endorsed by Nicod (1970). We will say much more about (NC) below. • Equivalence Condition (EC) : For any propositions H 1, H 2 , and E , if E confirms H 1 and H 1 is (classically) logically equivalent to H 2 , then E confirms H 2 . The intuition behind (EC) is that if H 1 and H 2 are (classically) logically equivalent, then they make exactly the same predictions (indeed, they say the same thing ), and so anything that counts as evidence for H 1 should also count as evidence for H 2 . 3 From (NC) and (EC), we can deduce the following, “paradoxical conclusion”: • Paradoxical Conclusion (PC) : The proposition that a is both nonblack and a non-raven, ( ∼ Ba · ∼ Ra ), confirms the proposition that every raven is black, ( ∀ x )(Rx ⊃ Bx). The canonical derivation of (PC) from (EC) and (NC) proceeds as follows: 1. By (NC), (∼Ba · ∼Ra) confirms (∀x)(∼Bx ⊃ ∼Rx). 2. In classical logic, (∀x)(∼Bx ⊃ ∼Rx) is equivalent to (∀x)(Rx ⊃ Bx). 3. By (1), (2), and (EC), (∼Ba · ∼Ra) confirms (∀x)(Rx ⊃ Bx). QED. The earliest analyses of this infamous paradox were offered by Hempel, Goodman, and Quine. Next, we will discuss how each of these philosophers attempted to resolve the paradox. 2. Early Analyses of the Paradox due to Hempel, Goodman, and Quine 2.1 The Analyses of Hempel and Goodman Hempel (1945) and Goodman (1954) didn’t view (PC) as paradoxical. Indeed, Hempel and Goodman viewed the argument above from (1) and (2) to (PC) as sound. So, as far as Hempel and Goodman are concerned, there is something misguided about whatever intuitions may have lead some philosophers to see “paradox” here. As Hempel explains (Goodman’s discussion is very similar on this score), one might be misled into thinking that (PC) is false by conflating (PC) with a different claim – a claim that is clearly false. Hempel warns us that [our emphasis]