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Evidential Relevance and the Grue Paradox

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Evidential Relevance and the Grue Paradox 科 学 哲 学31-1(1998) Evidential Relevance and the Grue Paradox Robert T. Pennock Philosophy Department The University of Texas at Austin Abstract Goodman's Grue Paradox may be intransigent as a version of the problem of induction, but may be resolved within the more lim ited context of confirmation theory in which the task is to explicate the basic notion of evidential relevance. Although the green and grue hypotheses are equivalently confirmed if we follow Goodman's use of the Hempelian instance confirmation relation, there are asym metries than can be exploited if we adopt an "ontic" confirmation theory that uses a causal notion of evidential relevance. I sort out a variety of interpretive confusions about the intended content of the definition of grue and show how the causal approach resolves each in a way that is not paradoxical. 1. Introduction If we are to make progress in confirmation theory, we must distinguish local problems having to do with the nature of evidence from the global problem of induction. Nelson Goodman posed the Grue Paradox as "The New Riddle of In- duction," but his presentation set up the problem within a Hempelian model of confirmation. In what follows I do not address the aspect of the paradox that participates in the old problem of induction but focus upon the aspect that deals with the explication of the confirmation relation, that is, the relation having to do with what counts as evidence of what. In •˜2-•˜5 1 highlight features of the para- 101 dox that are problematic when viewed with this particular question of evidential relevance in mind and in •˜6 I show how we may avoid the problems by moving from a syntactic/semantic to an ontic/causal relevance relation.' 2. Will the Real Grue Paradox Please Stand Up? The canonical statement of Goodman's paradox occurs in Fact, Fiction and Forecast: [The predicate 'grue' ] 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 emer- ald is green a parallel evidence statement asserting 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. (Goodman 1983 p. 74) Much confusion arises in the literature because this definition of 'grue' is ambiguous and because philosophers sometimes discuss the paradox without speci- fying a precise form. I follow Putnam's interpretation of the definition "some- thing [is] grue if it is either observed before a certain date and is. green, or is not observed before that date and is blue." (Putnam 1983, p. vii) The same form is used by Scheffler (1958, p. 23), Quine (1970, p. 41), Sober (1994), Stalker (1994, p. 2), and many others. Not only does this continue to be the most prevalent form, there are also good reasons to think that it is correct. First, we may take a cue from Goodman's original formulation of the para- dox in "A Query on Confirmation" that used a similar predicate 'S' to mean " is drawn by VE day and is red, or is drawn later and is non-red." (Goodman 1946 p. 383) This also lets us see that the intended contrast is not "examined" versus "not examined" but rather "before time T" and "at or after time T." Indeed, recent presentations often omit the former element as inessential.2 Formally, the prob- lem relies on definitions of the following sort: (1) Ex •¬ x is an emerald (2) Tx •¬ x is before time T (3) Ux •¬ x is at or after time T (4) Gx •¬ x is green (at t) 102 Evidential Relevance and the Grue Paradox (5) Bx•¬x is blue (at t) (6) x is grue (at t) [(Tx & Gx) •É (Ux & Bx)] (7) x is bleen (at t) •¬ [(Tx & Bx) •É (Ux & Gx)] A second reason for considering the standard definition (6) is that other defi- nitions do not generate the paradox in the way Goodman posed it. Frank Jackson (1975) and others have shown this for the following alternative definitions: (8) x is grue iff x is green before T and blue thereafter which was used by Kyburg (1970), Hacking (1965), and Barker (1957), and for (9)xis grue at tiff (xis green att& t<T) or(x isblueatt& t>T) which was used by Skyrms (1966), Salmon (1963), and Barker and Achinstein (1960). It would be useful to have a systematic survey that sorts out confusions of other formulations in the literature, but I'll proceed using definition (6). When philosophers discuss the Grue Paradox with other ends in mind, they omit the framework of Hempelian confirmation theory in which Goodman pre- sented the paradox. I include it because it is central to how the paradox relates to my question about evidential relevance. On Hempel's view, a scientific law is symbolized by a universally generalized hypothesis-statement and is confirmed by a true evidence-statement that is an instance of it. Hempel (1965) defined a special instance relation to solve some technical problems that we need not get into. For simplicity we may take an "instance" to be defined syntactically such that both the antecedent and the consequent of the conditional stating the universal generalization are satisfied. For example, "Ea & Ga" is an instance of the hypoth- esis "(x)(Ex •¬ Gx)." Thus, to provide the intended interpretation, the evidence- statement "a is an emerald and a is green" confirms the hypothesis "All emeralds are green." It is by virtue of the syntactically defined instance relation that the former statement counts on Hempel's confirmation theory as evidentially relevant to the given hypothesis. However, if this instance relation is sufficient for confirmation, then we are in trouble, said Goodman, for the observation that a is a green emerald, if made before time T, also confirms the hypothesis "All emeralds are grue," and it is ab- surd to think that this observation also confirms generalizations containing predi- cates like "grue," which imply the "incompatible" prediction that "if an emerald subsequently examined is grue, it is blue and hence not green." (Goodman 1983 p. 74) He draws the moral that some generalizations are "projectible" (i.e., induc- 103 tively confirmable) and some are not, and that we must find a way to tell them apart, proposing his notion of "entrenchment" of a predicate to do this. Contrar- ily, I will argue that instead of trying to disqualify predicates like grue we should reexamine the confirmation relation that licensed the same datum as equally rel- evant to both the green and grue hypotheses. 3. Confusions about Goodman's Symmetry Argument A typical reaction when one first hears Goodman's paradox is to conclude that it arises because of the peculiar positional nature of the predicate "grue." Carnap (1947) first made this point, arguing that we can disqualify grue-like predi- cates because they are not a "purely qualitative" (i.e., because they cannot be de- termined without consulting a time-piece). However, Goodman countered, if we take "grue" and "bleen" as the "'basic terms," then it would be "green" and "blue" that would be defined by reference to time T. (Goodman 1983) This might be done in the following manner: (10): Rx •¬ x is grue (11): Lx •¬ x is bleen (12): x is green •¬ [(Tx & Rx) •É (Ux & Lx)] (13): x is blue •¬ [(Tx & Lx) •É (Ux & Rx)] Thus qualitativeness is relative, and our preference for green and blue over grue and bleen is merely linguistic chauvinism. Carnap aimed to show that green was privileged over grue because the latter contained an illegitimate time-reference, but Goodman parried by pointing out a symmetry in the definitions of the predicates, since the presence of the time refer- ence depended upon which were taken as primitive. Goodman shows how to re- write the definition of green so that it contains a time-reference "if we start with ' grue' and 'bleen.' " (Goodman 1983 p. 79) However, he does not justify the implicit claim that in this way "grue" and "bleen" do not still require a time-refer- ence, which is needed for a complete symmetry with the original case in which they did but "green" and "blue" did not. Goodman has us "start with" grue and bleen as primitives because otherwise their definitions (10) and (11) would depend upon definitions (6) and (7), which are explicitly positional, so the sym- metry would obviously fail. But can they work as "timeless" primitives? 104 Evidential Relevance and the Grue Paradox Someone defending Goodman's parry might ask that we imagine ourselves having been born on Grue-World and having learned its predicates as children, probably from Kermit the Frog on Sesame Street singing "It's not easy being grue." In philosophy class we would have discussed the problem of the missing shade of bleen. Someone on Grue-World, the story goes, would think of grue and bleen in exactly the same way that we think of green and blue; a Grueling would no more look at his time-piece to tell colors than we do. However, this conclusion is incor- rect. Imagine an inhabitant of Grue-World who has grown up in just the way de- scribed above•\his name is Rip Van Winkle. One fine October day in 1986, say, he was hunting in the Catskills, admiring the leaves as they changed from grue to rellow, and the clear bleen sky.
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