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SOLUTION OF GOODMAN’S RIDDLE

Is there still a judge in Berlin?

ACKOWLEDGMENT

I wish to thank prof. J.D.Norton. He thinks that I did not understand Goodman’s riddle, I think that he did not understand my solution. The future will say who is right. In the meanwhile I acknowledge that his comment offered me a good opportunity to improve the exposition of my own ideas on the subject.

ABSTRACT

I spoke of a solution, but perhaps better I could speak of a confutation, since my radical claim is that Goodman’s riddle arises from a logical mistake; a very subtile one, indeed, yet a logical mistake concerning the ranges of quantification. Here below I propose what firmly seems to me a satisfying reset of the subject.

CURRENT ACADEMIC APPROACH

A premise. The occurrence of asterisks and inverted commas as diacritical symbols is not a typographical negligence, but the consequence of intentional reasons (here left out).

As for the current academic approach I draw my concise track from Stanford Enciclopedia of Philosophy, entry "Nelson Goodman, §5.3".

DEFINITION:

(i)      something is grue iff (it is examined before t° and is green) or (it is examined after t° and is blue)

(the disjunction meant by “or” is inclusive and superfluous brackets facilitate the argument below).

Owing to the above definition, before t° for each evidence statement asserting that a given emerald is green, there is a parallel statement asserting that the very emerald is grue. Thus the two inductive hypotheses *all emeralds are green* and *all emeralds are grue* result equally confirmed. But as soon as we survey an emerald after t°, we knock against a contradiction, for the two hypotheses lead to incompatible results.

INCONSISTENCY OF THE CURRENT ACADEMIC APPROACH

CONSISTENCY CONDITION. No evidence can support at the same time incompatible inductive generalizations.

THEOREM. It is inconsistent to claim that *being green if examined before t°* ought to entail *being blue if examined after t°*.

PROOF . By reductio ad absurdum. Exactly as (i) defines *grue*,

      something is grink iff (it is examined before t° and green) or (it is examined after t° and pink)

defines *grink*, and so on for every chromatic nuance we like (*grellow*, *grack* , *grurple*…). Therefore, if we assume that the observation before t° of some green emerald is an inductive evidence for the generalization that all emeralds are grue, we must also assume that the same observation is an inductive evidence for the generalization that all emeralds are grink (then pink after t°) and grellow (then yellow after t°) and so on. But as all these generalizations are reciprocally incompatible, we must conclude that our initial assumption leads to inconsistency.

SOLUTION OF THE RIDDLE

Once ascertained that the current derivation

*green before t°*, ergo *grue*, ergo *blue after t°*

is logically untenable, the arising question is: where hides the mistake? And the right answer follows simply from a less hasty and superficial analysis of the matter. Let me enter into details

INITIAL CONTEXT. No definition of *grue* is considered. The set of emeralds is accounted only accoring to a chromatic criterion. Since

the emerald a is green

the emerald b is green

the emerald c is green

and so on, the inductive generalization we reach is

(ii)                                                                                                                         all emeralds are green

where of course the range of “all” is the whole set of emeralds.

the emerald a is green and examined before t°

the emerald b is green and examined before t°

the emerald c is green and examined before t°

and so on. According with the inductive rule applied to the initial context above

(iii)                                                                                                    all emeralds are green and examined before t°

is the generalization supported by our evidences. Thus we attain the quite crucial point: what is the range of “all”. The answer is constrained for maintaining to “all” the same range of the initial context above (the whole set of emeralds), entails that no emerald can be examined after t°: a manifest absurdity. The way out to avoid this misinterpretation is to read (iii) as

(iv)                                                                                                       all emeralds examined before t° are green

that is to restrict the range of quantification. And effectively (iv) says what we learn from the mentioned evidences, which do not even speak, so to write, of the after t°. In this sense we can interpret (iii) as: within the temporal borders established by t°, all emeralds are green. (an elementar Venn diagram, where the circle representing the whole set of emeralds is partitioned in four sectors helps to understand, for it visualize that all points representing our evidences belong to one only sector).

Let me insist. Since our analysis must account for *examined before/after t°*, and since all our evidences of green emeralds refer to the before t°, they cannot tell us that ALL emeralds are green, but only that ALL emeralds EXAMINED BEFORE t° are green.

GOODMAN’S CONTEXT. The definition of *grue* is now considered. It depends strictly on the chronologic dimension; as a matter of fact the first couple of brackets in (i) tells us that the predicates inclusively disjunct are not *green* and *blue*. yet *green AND EXAMINED BEFORE t°*.and *blue AND EXAMINED AFTER t°*. In compliance with (i)

(v)                                                                                                        all emeralds examined before t° are grue

is unexceptionably entailed by (iv), but of course, exactly as (iv) cannot tell us anything about the greenness of the emeralds examined after t°, (v) does not tell us anything about their grueness. A definition is not an evidence; it is a means of giving a certain expressive form to information otherwise acquired. And the information acquired is still the one adduced by (iv). First of all, in order to expect that all emeralds examined after t° be blue we ought to know that all emeralds examined after t° are grue, which is not for we have no evidence about the after t°.

MARGINAL CONSIDERATIONS

Given that we have no integrative evidences to overcome this intrinsic lack, if we like we can proceed through hypothetical assumptions; free to choose the assumptions we prefer, provided they do not contradict each other. The most reasonable one is to assume that

(vi)                                                                                                         all emeralds examined after t° are green

for it results scarcely plausible (though not illogical) that at t° emeralds change their color. Anyhow as soon as we decide to assume (vi), we must ban the assumption

(vii)                                                                                                        all emeralds examined after t° are grue

for (vii) is an hypothesis contradicting (vi). And viceversa if we (rather oddly, indeed, but legitimately) decide to assume that all emeralds examined after t° are grue, we must ban (vi). The riddle would arise only if our evidences entailed (ii) instead of (iv). The subtile mistake hides just in the lack of distinction between *all emeralds* and *all emeralds EXAMINED BEFORE t°*, that is it hides in the confusion between the two ranges of quantification. Once the mistake is well understood, the riddle vanishes. The assumption of (vi) AND (vii) at the same time would not yield any riddle, since it would not involve any puzzling question; it would only represent a manifest and coarse contradiction. The crowded debates about more or less entrenched predicates or about lawlikeness or about unnatural properties or about the insidiousness of languages and so on collapse authomatically as they all fail to focus the basic mistake.

Finally I remark that my analysis overcomes the inconsistency affecting the definitions of *grink* et cetera, for as soon as *all emeralds* is restricted to *all emeralds examined before t°*, the requirement to be grue identifies itself with the requirement to be grink and grellow and so on.

Other reflections on inductive generalizations might be proposed, yet I stop here because I hope to have been clear enough to convince every open minded logician. Anyway

italo@italogandolfi.com

is my address.

I corroborate the solution above through an example reproposing the logical scheme of the riddle in a quite common context.

Let us consider a set of cats, whose temper is valued at two different times of the day. We call “meek” a cat of tender attitude and “capricious” a cat whose attitude is tender at noon but aggressive at nigth. And just now at noon we are observing that a and b and c and so on have a tender attitude: therefore the generalizations

all cats are tender

all cats are capricious

are equally supported by the same evidence. But then what about the presumable temper at midnight? The germ of the impasse, once more, is the range of “all”. A definition is not an evidence, I repeat. As soon as a definition introduces a predicative opposition, it reduces the range of the respective generalizations. That is: no generalization can rule situations concerning predicative oppositions which are not supported by any evidence.

Above I spoke of predicative oppositions without specifying the temporal dimension because other predicative oppositions are likewise good. For instance we could ground the opposition between *meek* and *capricious* upon a topographic criterion (observed at home vs observed outside). Anyhow the reduction of the range the generalization refers to is sufficient to overcome any impasse.

An easy appeal to Venn’s diagrams visualizes the whole matter.