Logical and Spiritual REFLECTIONS

Book 1. Hume’s Problems with Induction

Chapter 9. Goodman’s paradox of prediction

Nelson Goodman[1] proposed in 1955 a “riddle of induction” (as he called it[2]) or “paradox of prediction” (as others have characterized it), which seemed to demonstrate a formal difficulty in generalization. This may be stated as follows:

“Goodman … introduce[d] the color grue, which applies to all things examined before a certain time t just in case they are green, but also to other things just in case they are blue and not examined before time t. If we examine emeralds before time t and find that emerald a is green, emerald b is green, and so forth, each will confirm the hypothesis that all emeralds are green. However, emeralds a, b, c, … etc. also confirm the hypothesis that all emeralds are grue. In this case emeralds a,b,c, … examined after time t should be grue, and therefore blue!” (Emphasis mine)[3]

The significance of this artifice, according to its proponents, is that although green and “grue” have the same linguistic form, and so should be subject to the same logical processes (in this case, the inductive process of generalization), they are internally quite different types of concepts, since the first implies a similarity between its past and future instances, while the second suggests a change of color over time, so that the result is paradoxically quite different if we generalize with reference to the one or the other.

However, as I shall now formally demonstrate, this is merely a sleight of hand, for though the act of generalization is equally valid for green and for grue, it does not follow that we can infer any emeralds to be blue from the induced general proposition that all emeralds are grue. That is to say, the conclusion “and therefore blue” in the above presentation is an erroneous deduction.

To expose this simple error, the given scenario must be reformulated more carefully (the symbols X, A, B, C are mine):

· Say we examine all available emeralds (X), till a certain time (t), and finding them all to be green (A), we ordinarily conclude by generalization that All emeralds are green (All X are A), although we know [from past experience with induction in general] that the next emerald we find, after time t, might well turn out to be blue (B) [or indeed, to be some other color[4]]!

· Let us now following Goodman introduce a new concept “grue” (C) to be defined as the class grouping all things that were examined before a certain time t and found to be green (A) and all things not examined before time t which happen to be blue (B) [or indeed, to be some other color].[5]

· Applying this definition, all X (emeralds) examined before t were found A (green) are also C (grue); i.e. by syllogism we can infer Some X are C. As for remaining eventual cases of X, those not examined till after time t [if ever], each will either be found be to be A (green) or to be B (blue) [or indeed, to be some other color]; in that sense, the latter X too are C. Hence, All X are C would seem a reasonable conclusion.

· But it certainly does not logically follow from the preceding that any emeralds will indeed be found to be any color other than green, i.e. that any X are B [i.e. blue, or whatever non-green color]! For, properly understood, the category C is not formulated as a disjunction of A or B that is bound to actualize both some cases of X-A and some cases of X-B.

· If you look closely, you will see that C includes on the one hand things already known to be A (green emeralds already observed) and on the other hand a palette of things of still unknown qualification, i.e. either A or B (blue) [or even some other color]. The latter is a disjunction of conceivable outcomes, not one of inevitable outcomes. To infer X-B as an actual outcome would therefore be a non sequitur.

· The fact that we do not know whether any future X will be found A does not allow us to infer from this disjunction of possibilities that some future X will necessarily be B. We do not yet know whether any future X will be found B, either. We may well find that All X are A (All emeralds are green) remains forever applicable after time t as before time t (as predicted in the initial ordinary generalization).

· The premises ‘All X are C’ and ‘All C are A or B’ indeed yield the syllogistic conclusion ‘All X are A or B’. But the disjunction ‘A or B’ here cannot be interpreted differently in the major premise and in the conclusion. The disjunction in the premise not being extensional, the disjunction in the conclusion cannot be treated as extensional[6]. To do so would be to commit the fallacy of four terms.

It is thus clear from our exposition that the introduction of the concept “grue” has changed nothing whatsoever in the inductive possibilities offered by the given data. The correct inductive conclusion remains unaffected by Goodman’s fun and games. All Goodman has succeeded in doing is artfully conceal his fallacious deductive reasoning (misinterpretation of the kind of disjunction involved); it is all just sophistry.

In the thick smoke of Goodman’s rhetoric, it is made to appear as if blue emeralds are as easy to predict as green ones. But that is not at all the logical conclusion according to inductive logic. Why? Because in the case of the hypothesis that future emeralds observed will be found green, we have some concrete data to support it, namely that all present and past emeralds observed have been found green.

Whereas, in the support of Goodman’s hypothesis that blue emeralds will appear, we have no experiential evidence whatever so far. All we can say is that it is not inconceivable that blue emeralds might one day be found, but that does not imply that any ever will. ‘Not inconceivable’ does not justify actual prediction. It just means ‘imaginable in the present context of knowledge’.

That is, all we have is a general epistemological principle to remain open-minded to all eventual outcomes, based on past experience relating to all sorts of objects, that novelty does appear occasionally. But such scientific open-mindedness is not equivalent to a positive prediction of specific changes. It is just a call, in the name of realism, to avoidance of prejudice and rigidity.

A question we ought to ask is whether Goodman’s “grue” construct is a well-formed concept?

An ordinary concept of “grue” (or green-blue) would simply be formulated as “green and/or blue”. We may well find it valuable to introduce such a concept, perhaps to stress that green and blue are close in the range of colors, or that some things are partly green and partly blue, or sometimes green and sometimes blue, or that some hues in between are hard to classify as clearly green or clearly blue. The dividing line between these colors is after all pretty arbitrary.

Given that some emeralds are green, we could then deduce that some emeralds are grue. It would be equally valid to induce thence that all emeralds are green or that all emeralds are grue. This would imply no inherent self-contradiction, because to say that all emeralds are grue does not imply (or exclude) that any emeralds are blue. All emeralds are grue is formally compatible with the eventuality that all emeralds are green. So there is no “paradox of prediction” in fact.

Goodman’s “grue” construct is no different from this ordinary concept with respect to such logical implication. Its difference is not in the involvement of disjunction (green or blue), since such disjunction is quite commonplace; for example, the concept “colored” means (roughly) “red, orange, yellow, green, blue, indigo or violet”. The significant difference in Goodman’s construct is its involvement of temporal-epistemic conditions. This serves the rhetoric purpose of clouding the issues.

Defining the concept “grue” as the class of all things examined before time t and found to be green and all things not examined before t that happen to be blue – involves a self-contradiction of sorts. If I have not yet examined the things after time t, how can I positively say of any of them that they are blue? I could only make such a statement ex post facto, after having examined some of the things after time t and found them blue.

Alternatively, it would have to be said by a ‘third party’ looking on, who has examined some of the things before time t and found them blue, and who is observing my situation before I have done the same. But as regards all current observers taken together, they cannot logically adopt such a hypothesis, about things that happen to be blue although they have not yet been observed to be so. We can only consistently talk about things that might yet be found blue. For this reason, Goodman’s grue concept is not well-formed.

Grue is primarily defined as the union of green things and blue things; but it does not follow from such definition that if some things (such as emeralds) are green, then other such things (i.e. other emeralds) must be blue. To say that a kind of thing (emeralds) is grue is not to intend that its instances must cover the whole range of possibilities included under grue. The concept of grue remains legitimate provided we find the predicates it collects together (green, blue) scattered in various kinds of thing (emeralds, the sea, etc).

Thus, every ordinary predicate involves some uncertainty as to its application to specific subjects. Moreover, this applicability may vary with time: according to our context of knowledge, and according to changes occurring in the objects observed. Therefore, there is no need to involve such epistemic and temporal factors in the definition of any of the concepts we propose. Such factors are inherent to conceptualization.

The reason Goodman introduced such complications in his definition of “grue” was because he wanted to refute (or give the impression he was refuting) the process of generalization we commonly use to develop our knowledge on the basis of limited observation.

According to inductive logic, observing that some X are A, and so far seeking and not finding any X that are not A, we may generalize and say All X are A. This remains effectively true for us so long as we have no evidence of any X that is not A. Generalization involves prediction, i.e. saying something about cases of X we have not yet observed and maybe never will.

Goodman wished to demonstrate that we are equally justified in predicting a negative outcome (i.e. not A, e.g. B) as a positive outcome (i.e. A)[7]. He did not realize the logical justification of our generalizations[8]. We are not arbitrarily predicting that the cases of X we observe in the future will be A rather than not A. We are just sticking to the same polarity (A), because it is the only polarity we have any empirical evidence for so far. Comparatively, to predict the opposite polarity (not A, in this context) would be purely arbitrary – a wild assertion. Specifically for X, the first move has some empirical support, whereas the second has none at all.

Goodman simply did not realize this difference in justification between the two courses, though it is obvious to anyone who takes the time to reflect. He thus failed to apply the inductive principle that a confirmed hypothesis is always to be preferred to an unconfirmed one. Moreover, as we saw, in his eagerness to invalidate inductive reasoning, he committed one of the most elementary errors of deductive reasoning!

Underlying Goodman’s riddle is another important question for inductive logic: how far up any scale of classification can generalizations legitimately be taken? Having for a given subject generalized a certain predicate, why not generalize further up the scale to a larger predicate?[9]

Consider a subject X and any two predicates S and G, related as species and genus, i.e. such that all S are G but not all G are S (i.e. some nonS are also G). Here, note well, S and G are both ordinary concepts, like green and colored.

· If all cases of X that we have observed so far are found to be S, and we have looked out for and not encountered any X that are not S, we may inductively infer that All X are S. This generalization remains valid so long as no cases of X that are not S are found; but if any X-nonS eventually do appear, we are required by inductive logic to revise our previous judgment, and particularize it to Some X are S and some X are not S. For induction proceeds conditionally[10].

· The same reasoning applies to G[11]. Alternatively, granting that All S are G, we can from All X are S deductively infer that All X are G, by syllogism (1st Figure, AAA). Thus, we might postulate, if we are justified to generalize, for a given subject X, as far as the specific predicate S, we are also justified to do so higher still on the scale of classification, as far as the more general predicate G. This is logically okay if properly understood and applied.

· However, it would be a gross error of judgment[12] to infer from such valid generalization that there might be some X that are G but not S (even if we know there are things other than X that are G but not S). At this stage, the actual content All X are G is identical (in extension and implicitly in intension) to the All X are S from which it was derived[13]. How the two statements differ is only with regard to eventual corrective particularization

· Suppose tomorrow we discover an X that though still G is not S (for example, an emerald of some color other than green). In such event, we would have to particularize the first (more specific) statement to ‘Some X are S and some X are not S’; but the second (more generic) statement ‘All X are G’ would remain unchanged.[14]

· But as a result of such particularization All X are G has a vaguer meaning, since G no longer for us refers only to the S species of G but equally to some other (nonS) species of it. Thus, though the inductive rule would be to generalize as far up the scale as we indeed can go, we must keep in mind that the further up the scale we go, the more we dilute the eventual significance of our generalization.[15]

Thus, although in principle generalization up the scale is unfettered, in practice we proceed relatively slowly so as to maintain the noetic utility of our ideas and statements. To give a formal example: the proposition All X are S might be used as minor premise in a syllogism where S is the middle term, whereas the proposition All X are G – even if still identical in extension and intension to the preceding – would be useless in that same context (i.e. with S as middle term).

Moreover, to regard All X are G as a more profitable generalization than All X are S, in the sense of providing us with information about more things for the same price in terms of given data, signals a confusion[16] between generalization for a given subject from a narrower predicate to a wider predicate, and generalization of a given predicate from a narrower subject to a wider subject.

The latter case is the truly profitable form of generalization. Suppose All X are P, and Y is an overclass of X (i.e. All X are Y, though not all Y are X), then this would consist in inducing that All Y are P — of course, unless or until some Y that is not P is discovered. The rules of such generalization are dealt with fully in my work Future Logic under the heading of Factorial Induction (Part VI).

[1] USA, 1906-98.

[2] Or more pretentiously, “the new problem of induction”.

[3] Here I’m quoting: http://en.wikipedia.org/wiki/Nelson_Goodman. Elsewhere, we are informed that “applies to all things examined before t just in case they are green but to other things just in case they are blue” is Goodman’s own wording in his original presentation in Fact, Fiction, and Forecast (http://en.wikipedia.org/wiki/Grue_%28color%29).

[4] This is my own interpolation, to make Goodman’s thesis more accurate. For there is no reason to suppose a priori that only blue emeralds might eventually be found. We are only guessing the possibility of blue emeralds, not basing it on any specific observations – therefore any other color is equally probable (or improbable). Nevertheless, my refutation of Goodman works just as well without this added comment.

[5] Note that the latter things are stated to be merely “not examined until time t [yet, if ever]”; this is not to be confused (as some commentators have done) with “examined after time t”, for no matter how many things we do eventually examine, we will obviously never achieve (or know we have achieved) a complete enumeration of all such things in the universe. Note also that the concept grue is here defined as a general predicate for any eventual subject (“things”), rather than specifically for emeralds.

[6] That is, a base of the given disjunction is Some C might be B, whereas the corresponding base of the allegedly inferred disjunction is Some X are B. But to imagine something happening is not proof it has to in fact happen sometimes. The conclusion does not follow from the premise.

[7] To do so, he needed to construct a concept that would include both A and notA, so that generalization could be formally shown to be able to go either way. However, since a concept including contradictories in non-informative, he included contraries, viz. ‘A or B’ (where B is not A). This slightly conceals the issue, but does not in fact change it.

[8] See my Future Logic, chapter 50.

[9] I have touched upon this topic (indirectly, with regard to ethical logic) in my Judaic Logic, chapter 13.3.

[10] For adduction or generalization is justified by two essential principles: (1) confirmation of a hypothesis by a positive instance, and (2) the non-rejection of the same hypothesis by any negative instance, and both principles must be equally obeyed for it to proceed logically. There are of course many other conditions involved – see my essay “Principles of Adduction” in Phenomenology (chapter VII.1).

[11] That is, given Some X are G (or deducing this from Some X are S), we can generalize to All X are G, provided there is no known negative instance (X-nonG) to belie it.

[12] This is as we saw one of the errors Goodman committed in formulating his “riddle”. This error is of a deductive rather than inductive nature.

[13] This is obvious if we consider that we may equally well obtain All X are G: (a) by generalization from Some X are G, which we deduce from Some X are S, or (b) by deduction from All X are S, which we generalize from Some X are S. In truth, it could be argued that these two are slightly different, since (a) requires that we make sure that there are no instances of X that are not G, whereas (b) requires that we make sure that there are no instances of X that are not S. This difference is however brought out in the ensuing stage of eventual particularization.

[14] Note that if we discover an X that is not G, it is necessarily also not S, given All S are G. In that event, both general propositions would of course have to be particularized.

[15] In this context, we could compare Goodman’s “grue” concept to Feynman’s concept of “oomph”. The latter, defined (tongue-in-cheek) as “a kind of tendency for movement” might seem useful to “explain” various phenomena, but it is so vague that it cannot predict anything and is therefore worthless (p. 19).

[16] Which Goodman was guilty of in formulating his “riddle”, incidentally.

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