Goodman's "New Riddle"

First, a brief historical trace of the developments in confirmation theory leading up to Goodman’s infamous “grue” paradox is presented. Then, Goodman’s argument is analyzed from both Hempelian and Bayesian perspectives. A guiding analogy is drawn between certain arguments against classical deductive logic, and Goodman’s “grue” argument against classical inductive logic. The upshot of this analogy is that the “New Riddle” is not as vexing as many commentators have claimed (especially, from a Bayesian inductive-logical point of view). Specifically, the analogy reveals an intimate connection between Goodman’s problem, and the “problem of old evidence”. Several other novel aspects of Goodman’s argument are also discussed (mainly, from a Bayesian perspective). 1. Prehistory: Nicod & Hempel In order to fully understand Goodman’s “New Riddle,” it is useful to place it in its proper historical context. As we will see shortly, one of Goodman’s central aims (with his “New Riddle”) was to uncover an interesting problem that plagues Hempel’s [17] theory of confirmation. And, Hempel’s theory was inspired by some of Nicod’s [25] earlier, inchoate remarks about instantial confirmation, such as: Consider the formula or the law: A entails B. How can a particular proposition, [i.e.] a fact, affect its probability? If this fact consists of the presence of B in a case of A, it is favourable to the law . . . on the contrary, if it consists of the absence of B in a case of A, it is unfavourable to this law. By “is (un)favorable to,” Nicod meant “raises (lowers) the inductive probability of”. In other words, Nicod is describing a probabilistic relevance conception of confirmation, according to which it is postulated (roughly) that positive instances are probability-raisers of universal generalizations. Here, Nicod has in mind Keynesian [21] inductive probability (more on that later). While Nicod is not very clear on the logical details of his probability-raising account of instantial confirmation (stay tuned for both Hempelian and Bayesian precisifications/reconstructions of Nicod’s remarks), three aspects of Nicod’s conception of confirmation are apparent: • Instantial confirmation is a relation between singular and general propositions/statements (or, if you will, between “facts” and “laws”). Date: 01/02/08. Penultimate draft (final version to appear in the Journal of Philosophical Logic). I would like to thank audiences at the Mathematical Methods in Philosophy Workshop in Banff, the Reasoning about Probabilities & Probabilistic Reasoning conference in Amsterdam, the Why Formal Epistemology? conference in Norman, the philosophy departments of New York University and the University of Southern California, and the Group in Logic and the Methodology of Science in Berkeley for useful discussions concerning the central arguments of this paper. On an individual level, I am indebted to Johan van Benthem, Darren Bradley, Fabrizio Cariani, David Chalmers, Kenny Easwaran, Hartry Field, Alan Hájek, Jim Hawthorne, Chris Hitchcock, Paul Horwich, Franz Huber, Jim Joyce, Matt Kotzen, Jon Kvanvig, Steve Leeds, Hannes Leitgeb, John MacFarlane, Jim Pryor, Susanna Rinard, Jake Ross, Sherri Roush, Gerhard Schurz, Elliott Sober, Michael Strevens, Barry Stroud, Scott Sturgeon, Brian Talbot, Mike Titelbaum, Tim Williamson, Gideon Yaffe, and an anonymous referee for the Journal of Philosophical Logic for comments and suggestions about various versions of this paper. 2 BRANDEN FITELSON • Confirmation consists in positive probabilistic relevance, and disconfirmation consists in negative probabilistic relevance (where the salient probabilities are inductive in the Keynesian [21] sense). • Universal generalizations are confirmed by their positive instances and disconfirmed by their negative instances. Hempel [17] offers a precise, logical reconstruction of Nicod’s naïve instantial account. There are several peculiar features of Hempel’s reconstruction of Nicod. I will focus presently on two such features. First, Hempel’s reconstruction is completely non-probabilistic (we’ll return to that later). Second, Hempel’s reconstruction takes the relata of Nicod’s confirmation relation to be objects and universal statements, as opposed to singular statements and universal statements. In modern (first-order) parlance, Hempel’s reconstruction of Nicod can be expressed as: (NC0) For all objects x (with names x), and for all predicate expressions φ and ψ: x confirms [(∀y)(φy ⊃ ψy)\ iff [φx &ψx\ is true, and x disconfirms [(∀y)(φy ⊃ ψy)\ iff [φx &∼ψx\ is true. As Hempel explains, (NC0) has absurd consequences. For one thing, (NC0) leads to a theory of confirmation that violates the hypothetical equivalence condition: (EQCH ) If x confirms H, then x confirms anything logically equivalent to H. To see this, note that — according to (NC0) — both of the following obtain: • a confirms “(∀y)(Fy ⊃ Gy),” provided a is such that Fa &Ga. • Nothing can confirm “(∀y)[(Fy & ∼Gy) ⊃ (Fy & ∼Fy)],” since no object a can be such that Fa &∼Fa. But, “(∀y)(Fy ⊃ Gy)” and “(∀y)[(Fy & ∼Gy) ⊃ (Fy & ∼Fy)]” are logically equivalent. Thus, (NC0) implies that a confirms the hypothesis that all Fs are Gs only if this hypothesis is expressed in a particular way. I agree with Hempel that this violation of (EQCH ) is a compelling reason to reject (NC0) as an account of instantial confirmation. But, I think Hempel’s reconstruction of Nicod is uncharitable on this score (more on that later). In any event, my main focus here will be on the theory Hempel ultimately ends-up with. For present purposes, there is no need to examine Hempel’s [17] ultimate theory of confirmation in its full logical glory. For our subsequent discussion of Goodman’s “New Riddle,” we’ll mainly need just the following two properties of Hempel’s confirmation relation, which is a relation between statements and statements, as opposed to objects and statements. 1Traditionally, such conditions are called “Nicod’s Criteria”, which explains the abbreviations. Historically, however, it would be more accurate to call them “Keynes’s Criteria” (KC), since Keynes [21] was really the first to explicitly endorse the three conditions (above) that Nicod later championed. 2Not everyone in the literature accepts (EQCH ), especially if (EQCH ) is understood in terms of classical logical equivalence. See [28] for discussion. I will simply assume (EQCH ) throughout my discussion. I will not defend this assumption (or its evidential counterpart, which will be discussed below), which I take to be common ground in the present historical dialectic. (NC) is clearly a more charitable reconstruction of Nicod’s (3) than (NC0) is. So, perhaps it’s most charitable to read Hempel as taking (NC) as his reconstruction of Nicod’s remarks on instantial confirmation. In any case, Hempel’s reconstruction still ignores the probabilistic aspect of Nicod’s account. This will be important later, when we discuss probabilistic approaches to Goodman’s “New Riddle”. At that time, we will see an even more charitable (Bayesian) reconstruction of Nicod’s remarks. GOODMAN’S “NEW RIDDLE” 3 (NC) For all constants x and for all (independent) predicate expressions φ, ψ: [φx &ψx\ confirms [(∀y)(φy ⊃ ψy)\ and [φx &∼ψx\ disconfirms [(∀y)(φy ⊃ ψy)\. (M) For all constants x, for all (consistent) predicate expressions φ, ψ, and for all statements H: If [φx\ confirms H, then [φx &ψx\ confirms H. As we will see, it is far from obvious whether (NC) and (M) are (intuitively) correct confirmation-theoretic principles (especially, from a probabilistic point of view!). Severally, (NC) and (M) have some rather undesirable consequences, and jointly they face the full brunt of Goodman’s “New Riddle,” to which I now turn. 2. History: Hempel & Goodman Generally, Goodman [14] speaks rather highly of Hempel’s theory of confirmation. However, Goodman [14, ch. 3] thinks his “New Riddle of Induction” shows that Hempel’s theory is in need of rather serious revision/augmentation. Here is an extended quote from Goodman, which describes his “New Riddle” in detail: Now let me introduce another predicate less familiar than “green”. It is the predicate “grue” and it applies to all things examined before t just in case they are green but to other things just in case they are blue. Then at time t we have, for each evidence statement asserting that a given emerald is green, a parallel evidence statement asserting that that emerald is grue. And the statements that emerald a is grue, that emerald b is grue, and so on, will each confirm the general hypothesis that all emeralds are grue. Thus according to our definition, the prediction that all emeralds subsequently examined will be green and the prediction that all will be grue are alike confirmed by evidence statements describing the same observations. But if an emerald subsequently examined is grue, it is blue and hence not green. Thus although we are well aware which of the two incompatible predictions is genuinely confirmed, they are equally well confirmed according to our present definition. Moreover, it is clear that if we simply choose an appropriate predicate, then on the basis of these same observations we shall have equal confirmation, by our definition, for any prediction whatever about other emeralds—or indeed about anything else. . . . We are left . . . with the intolerable result that anything confirms anything. Before reconstructing the various (three, to be exact) arguments against Hempel’s theory of confirmation that are implicit in this passage (and the rest of the chapter), I will first introduce some notation, to help us keep track of the logic of the arguments. Let Ox Ö x is examined before t, Gx Ö x is green, Ex Ö x is an emerald. Now, Goodman defines “grue” as follows: Gx Ö Ox ≡ Gx. With these definitions in hand, we can state the two salient universal statements (hypotheses): (H1) All emeralds are green. More formally, H1 is: (∀x)(Ex ⊃ Gx). 4Here, “≡” is the material biconditional. I am thus glossing the detail that unexamined-beforet grue emeralds are blue on Goodman’s official definition. This is okay, since all that’s needed for Goodman’s “New Riddle” (at least, for its most important aspects) is the assumption that unexaminedbefore-t grue emeralds are non-green. Quine [26] would have objected to this reconstruction on the grounds that “non-green” is not a natural kind term (and therefore not “projectible”). Goodman did not agree with Quine on this score. As such, there is no real loss of generality here, from Goodman’s point of view. In any event, I could relax this simplification, but that would just (unnecessarily) complicate most of my subsequent arguments. In the final section (on Goodman’s “triviality argument”), I will return to this “blue vs non-green” issue (since in that context, the difference becomes important). 4 BRANDEN FITELSON (H2) All emeralds are grue. More formally, H2 is: (∀x)(Ex ⊃ Gx). Even more precisely, H2 is: (∀x)[Ex ⊃ (Ox ≡ Gx)]. And, we can state three salient singular (evidential) statements, about an object a: (E1) Ea &Ga. [a is an emerald and a is green] (E2) Ea & (Oa ≡ Ga). [a is an emerald and a is grue] (E) Ea &Oa &Ga. [a is an emerald and a is both green and grue] Now, we are in a position to look more carefully at the various claims Goodman makes in the above passage. At the beginning of the passage, Goodman considers an object a which is examined before t, an emerald, and green. He correctly points out that all three of E1, E2, and E are true of such an object a. And, he is also correct when he points out that — according to Hempel’s theory of confirmation — E1 confirms H1 and E2 confirms H2. This just follows from (NC). Then, he seems to suggest that it therefore follows that “the observation of a at t” confirms both H1 and H2 “equally.” It is important to note that this claim of Goodman’s can only be correct if “the observation of a at t” is a statement (since Hempel’s relation is a relation between statements and statements, not between “observations” and statements) which bears the Hempelian confirmation relation to both universal statements H1 and H2. Which instantial, evidential statement can play this role? Neither E1 nor E2 can, since Hempel’s theory entails neither that E1 confirms H2 nor that E2 confirms H1. However, the stronger claim E can play this role, since: (†) E confirms H1 and E confirms H2. That Hempel’s theory entails (†) is a consequence of (NC), (M), and the following evidential equivalence condition, which is also a consequence of Hempel’s theory: (EQCE) If E confirms H, then anything logically equivalent to E also confirms H. Figure 1, below, contains a visually perspicuous proof of (†) from (NC), (M), and (EQCE). The single arrows in the figure are confirmation relations (annotated on the right with justifications), and the double arrows are entailment relations. (∀x)(Ex ⊃ Gx) (∀x)[Ex ⊃ (Ox ≡ Gx)] Ea &Ga Ea & (Oa ≡ Ga) (Ea & (Oa ≡ Ga)) &Oa (Ea &Ga) &Oa Ea &Oa &Ga = E ↑ ↑ (NC) (NC)