A SOLUTION TO GRUE: THE FIRST STEP
To begin with, it is untrue that every property that mentions a specific time cannot be projected. Suppose, for example, that there is a certain necktie produced by Harvard University emblazoned with the letters MCMLVI, and that all the owners of such ties whom we have interviewed were graduated from Harvard in 1956.
We might then legitimately infer that all owners of this type of tie were graduated from Harvard in 1956, despite the fact that “were graduated from Harvard in 1956” expresses a temporal property.We need to be more selective with the temporal properties we say cannot be projected. Grue is a very special type of temporal property. It is a disjunctive one having this form:
(1) x has property P at time t if and only if x has property Q1 at t and t is prior to a specific time T or x has property Q2 at t and t is T or later.
This is not yet a sufficient characterization. Two provisos must be added. First, the properties Q1 and Q2 must be incompatible, for example, green and blue (something cannot have both at once). Second (this will become important later), the properties Q1 and Q2 (e.g., green and blue) must not be thought of as disjunctive properties satisfying (1).[28]
Grue, and indeed any property of type (1), is a property of an even more general type that is not necessarily temporal. Consider disjunctive properties with this form:
(2) x has P if and only if x has Q1 and condition C obtains or x has Q2 and condition C does not obtain.
Again the properties Q1 and Q2 must be incompatible. And they must not be disjunctive properties satisfying (2). In the grue case, Q1 is green and Q2 is blue. Condition C is that the time at which x has whatever color it has is before 2500. Goodman's paradox can be generated with respect to any property of type (2), whether temporal or not, if all the Ps examined have been Q1 and condition C obtains.
For example, again let Q1 be green, let Q2 be blue, but let condition C be that x's temperature is less than some fixed value M. Suppose that all emeralds so far examined have been green and have been at temperatures below M degrees. Then if we project P with respect to emeralds, we generate a conclusion that entails that at temperatures reaching or exceeding M emeralds are blue. We generate the paradox even when the property P in question is nontemporal.[29]How can Goodman's puzzle be resolved? Suppose that information e says that all P1s so far examined are P2, and hypothesis h states that all P1s are P2. Under what conditions can we project P2 relative to P1? Let us look at the grue case first.
We can project the property grue relative to the property of being an emerald, but only if evidence e reports on times both before and after T (2500), that is, only if e reports that the emeralds examined before T are grue (and hence green) and that emeralds examined at T or later are grue (and hence blue).
More generally, if P2 (e.g., grue) is a disjunctive property of types (1) or (2), with Q1 and Q2 (e.g., green and blue) as disjuncts, and if P1 (e.g., being an emerald) is not a disjunctive property of types (1) or (2), and if e reports that all the P1s examined are P2 in virtue of being Q1 and none in virtue of being Q2, then from e we cannot conclude that all P1s are P2.
I will speak of a selection procedure as a rule for determining how to test, or obtain evidence for or against, a hypothesis. Consider the following two types of selection procedures for a hypothesis h of the form “All P1s are P2”:
SP1: Select P1s to observe at times that are both before and after T (Or more generally, for properties of form (2), select P1s to observe at times that satisfy condition C and also ones that fail to satisfy C.)
SP2: Select P1s to observe at times that are only before T.
(Select only P1s to observe at times that satisfy C.)Where P1 and P2 are properties of the sort described in the previous paragraph, from e (the fact that all observed P1s are P2) we can infer h (all P1s are P2) only if the selection procedure SP1 is employed, not SP2.
The basic idea derives from an injunction to “vary the instances.” A disjunctive property P of the type depicted in (1) and (2) applies to two different sorts of cases: ones in which an item that is P (e.g., grue) has property Q1 (green) before time T (condition C is satisfied) and ones in which an item that is P has an incompatible property Q2 (blue) at or after T (condition C is not satisfied). Since property P, when projected, is supposed to apply to items of both types, where these types are incompatible, items of both types need to be obtained as instances of the generalization. That is, SP1 is to be followed, not SP2.
For example, projecting the property grue, in the case of emeralds, requires that some emeralds be examined before 2500 to determine whether they are then green, and that some emeralds be examined after 2500 to determine whether they are then blue. Only if both of these determinations are made, and the emeralds examined before 2500 are green and those examined later are blue, can the resulting information e warrant a generalization to the hypothesis that all emeralds are grue.
Contrast this case with one in which the property green is projected with respect to emeralds. This property is not being construed as one that applies to two different sorts of cases: ones in which a green item has some nondisjunctive property Q1 before time T and ones in which a green item has some incompatible nondisjunctive property Q2 after T. So projecting the property green, in the case of emeralds, does not require that some emeralds be examined before T to determine whether they have such a property Q1 and that some emeralds be examined after T to determine whether they have Q2.
Accordingly, it is not the case that only if both determinations are made and the emeralds examined before T have Q1 while those examined after T have Q2 can the resulting information (that all the examined emeralds are green) warrant the conclusion that all emeralds are green. In this case selection procedure SP2 (as well as SP1) can be used. One can select emeralds to observe at times that are only before some T, or at times that are before T and after.The important point is not that grue, unlike green, is a temporal property. That is not enough to prevent grue from being projected. The important point is that grue is a certain type of disjunctive property, while green is not. To project this type of disjunctive property P with respect to some type of item, one needs to vary the instances observed by examining items that satisfy the condition C and items that do not.
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