https://plato.stanford.edu/entries/goodman/#OldNewRidIndTheSol

5.3 The New Riddle of Induction

Before presenting Goodman’s solution, we first have to discuss Goodman’s own challenge, the so-called “New Riddle of Induction”.

Consider the following two (supposedly true) statements:

• (B1)This piece of copper conducts electricity.

• (B2)This man in the room is a third son.

B1 is a confirmation instance of the following regularity statement:

• (L1)All pieces of copper conduct electricity.

But does B2 confirm anything like L2?

• (L2)All men in this room are third sons.

Obviously, it does not. But what makes the difference? Both regularity statements (L1 and L2) are built according to the exact same syntactical procedure from the evidence statements. Therefore, it does not seem to be for a syntactical reason that B1 confirms L1 but B2 fails to confirm L2. Rather, the reason is that statements like L1 are lawlike, whereas statements like L2 at best express accidentally true generalizations. Lawlike statements, in contrast to accidentally true general statements, are confirmed by their instances and support counterfactuals. L1 supports the counterfactual claim that if this thing I have in my hand were a piece of copper, it would conduct electricity. In contrast, supposing that it is indeed true, L2 would not support that if an arbitrary man were here in the room, he would be a third son. To tell which statements are lawlike and which statements are not is therefore of great importance in the philosophy of science. A satisfactory account of induction (or corroboration -- see the discussion of Popper in the entry on the problem of induction) as well as explanation and prediction requires this distinction. Goodman, however, shows that this is extremely hard to get.

Here comes the riddle. Suppose that your research is in gemology. Your special interest lies in the color properties of certain gemstones, in particular, emeralds. All emeralds you have examined before a certain time tt were green (your notebook is full of evidence statements of the form “Emerald xx found at place yy date z(z≤t)z(z≤t) is green”). It seems that, at tt, this supports the hypothesis that all emeralds are green (L3).

Now Goodman introduces the predicate “grue”. This predicate applies to all things examined before some future time tt just in case they are green but to other things (observed at or after tt) just in case they are blue:

• (DEF1)xx is grue =df=df xx is examined before tt and green ∨ xx is not so examined and blue 

Until tt it is obviously the case that for each statement in your notebook, there is a parallel statement asserting that the emerald xx found at place yy date z(z≤t)z(z≤t) is grue. Each of these statements is analytically equivalent with the corresponding one in your notebook. All these grue-evidence statements taken together confirm the hypothesis that all emeralds are grue (L4), and they confirm this hypothesis to the exact same degree as the green-evidence statements confirmed the hypothesis that all emeralds are green. But if that is the case, then the following two predictions are also confirmed to the same degree:

• (P1)The next emerald first examined after tt will be green.

• (P2)The next emerald first examined after tt will be grue.

However, to be a grue emerald examined after tt is not to be a green emerald. An emerald first examined after tt is grue iff it is blue. We have two mutually incompatible predictions, both confirmed to the same degree by the past evidence. We could obviously define infinitely many grue-like predicates that would all lead to new, similarly incompatible predictions.

The immediate lesson is that we cannot use all kinds of weird predicates to formulate hypotheses or to classify our evidence. Some predicates (which are the ones like “green”) can be used for this; other predicates (the ones like “grue”) must be excluded, if induction is supposed to make any sense. This already is an interesting result. For valid inductive inferences the choice of predicates matters. 

It is not just that we lack justification for accepting a general hypothesis as true only on the basis of positive instances and lack of counterinstances (which was the old problem), or to define what rule we are using when accepting a general hypothesis as true on these grounds (which was the problem after Hume). The problem is to explain why some general statements (such as L3) are confirmed by their instances, whereas others (such as L4) are not. Again, this is a matter of the lawlikeness of L3 in contrast to L4, but how are we supposed to tell the lawlike regularities from the illegitimate generalizations?

An immediate reply is that the illegitimate generalization L4 involves a temporal restriction, just as L2 was restricted spatially (see e.g., Carnap 1947). The idea would be that predicates that cannot be used for induction are analytically “positional”, i.e., their definitions refer to individual constants (for places or times). A projectible predicate, i.e., a predicate that can be used for induction, has no definition which would refer to such individual constants but is purely qualitative (e.g., because it is a basic predicate). The trouble is that this reply makes it relative to a language whether or not a predicate is projectible. If we begin with a language containing the basic predicates “green” and “blue” (as in English), “grue” and “bleen” are positional. “Bleen” is defined as follows:

• (DEF2)xx is bleen =df=df xx is examined before tt and blue ∨ xx is not so examined and green

But if we start with a language that has “bleen” and “grue” as basic predicates, “green” and “blue” are positional:

• (DEF3)xx is green =df=df xx is examined before tt and grue ∨ xx is not so examined and bleen

• (DEF4)xx is blue =df=df xx is examined before tt and bleen ∨ xx is not so examined and grue

Both languages are symmetrical in all their semantic and syntactical properties. So the positionality of predicates is not invariant with respect to linguistically equivalent transformations. But if this is the case, there is no semantic or syntactic criterion on whose basis we could draw the line between projectible predicates and predicates that we cannot use for induction.

5.4 Goodman’s Solution

Goodman’s solution to the new riddle of induction resembles Hume’s solution in an important way. Instead of providing a theory that would ultimately justify our choice of predicates for induction, he develops a theory that provides an account of how we in fact choose predicates for induction and projection. Goodman observes that predicates like “green” are favored over predicates like “grue”, because the former are much better entrenched, i.e., in the past we projected many more hypotheses featuring “green” or predicates co-extensional with “green” than hypothesis featuring the predicate “grue”. If two hypotheses are the same with respect to their empirical track-record, then the hypothesis that uses the better entrenched predicates overrides the alternatives. On the basis of these considerations, Goodman defines projectibility(and cognates) for hypotheses (FFF, 108):

A hypothesis is projectible iff it is supported, unviolated, and unexhausted, and all hypotheses conflicting with it are overridden.

A hypothesis is unprojectible iff it is unsupported, exhausted, violated, or overridden.

A hypothesis is nonprojectible iff it and a conflicting hypothesis are supported, unviolated, unexhausted, and not overridden.

The last definition takes care of situations when we are confronted with two hypotheses that are in conflict and neither has a better entrenched predicate. Entrenchment can even be further refined to account for cases in which a predicate inherits the entrenchment from another of which it is derivative. (The critical literature on Goodman’s New Riddle is too extensive to do it justice here; see Stalker 1994 and Elgin 1997c for selections of important essays on the topic. Stalker 1994 also contains an annotated bibliography, comprising over 300 entries. The discussion of course continued after the 1990s and the literature is still growing.)

Goodman’s solution makes projectibility essentially a matter of what language we use and have used to describe and predict the behaviour of our world. However, this language-, or better version-relativism is just another aspect of Goodman’s irrealism.