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Goodman's paradox

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Goodman's paradox

For the linguistic term "grue", used for translation from natural languages, see Distinction of blue and green in various languages.

Grue and bleen are predicates coined by Nelson Goodman in Fact, Fiction, and Forecast to illustrate "the new riddle of induction". These predicates are unusual because their application to things are time dependent. For Goodman they illustrate the problem of projectable predicates and ultimately, which empirical generalizations are law-like and which are not.[1] Goodman's construction and use of grue and bleen illustrates how philosophers use simple examples in conceptual analysis.

Grue and bleen defined

Goodman defined grue relative to an arbitrary but fixed time t as follows: An object is grue just in case it is observed before t and is green, or else is not so observed and is blue. An object is bleen just in case it is observed before t and is blue, or else is not so observed and is green.[2]

To understand the problem Goodman posed, it is helpful to imagine some arbitrary future time t, say January 1, 2024. For all green things we observe up to time t, such as emeralds and well-watered grass, both the predicates green and grue apply. Likewise for all blue things we observe up to time t, such as bluebirds or blue flowers, both the predicates blue and bleen apply. On January 2, 2024, however, emeralds and well-watered grass are now bleen and bluebirds or blue flowers are now grue. Clearly, the predicates grue and bleen are not the kinds of predicates we use in everyday life or in science, but the problem is that they apply in just the same way as the predicates green and blue up until some future time t. From our current perspective (i.e., before time t), how can we say which predicates are more projectable into the future: green and blue or grue and bleen?

The New Riddle of Induction

In this section, Goodman's new riddle of induction is outlined in order to set the context for his introduction of the predicates grue and bleen and thereby illustrate their philosophical importance.[1][3]

The Old Problem of Induction and Its Dissolution

Goodman poses Hume's problem of Induction as a problem of the validity of the predictions we make. Since predictions are about what has yet to be observed and because there is no necessary connection between what has been observed and what will be observed, what is the justification for the predictions we make? We cannot use deductive logic to infer predictions about future observations based on past observations because there are no valid rules of deductive logic for such inferences. Hume's answer was that our observations of one kind of event following another kind of event result in our minds forming habits of regularity (i.e., associating one kind of event with another kind). The predictions we make are then based on these regularities or habits of mind we have formed.

Goodman takes Hume's answer to be a serious one. He rejects other philosophers' objection that Hume is merely explaining the origin of our predictions and not their justification. His view is that Hume is on to something deeper. To illustrate this, Goodman turns to the problem of justifying a system of rules of deduction. For Goodman, the validity of a deductive system is justified by their conformity to good deductive practice. The justification of rules of a deductive system depends on our judgements about whether to reject or accept specific deductive inferences. Thus, for Goodman, the problem of induction dissolves into the same problem as justifying a deductive system and while, according to Goodman, Hume was on the right track with habits of mind, the problem is more complex than Hume realized.

In the context of justifying rules of induction, this becomes the problem of confirmation of generalizations for Goodman. However, the confirmation is not a problem of justification but instead it is a problem of precisely defining how evidence confirms generalizations. It is with this turn that grue and bleen have their philosophical role in Goodman's view of induction.

Projectable Predicates

The new riddle of induction, for Goodman, rests on our ability to distinguish lawlike from non-lawlike generalizations. Lawlike generalizations are capable of confirmation while non-lawlike generalization are not. Lawlike generalizations are required for making predictions. Using examples from Goodman, the generalization that all copper conducts electricity is capable of confirmation by a particular piece of copper whereas the generalization that all men in a given room are third sons is not lawlike but accidental. The generalization that all copper conducts electricity is a basis for predicting that this piece of copper will conduct electricity. The generalization that all men in a given room are third sons, however, is not a basis for predicting that a given man in that room is a third son.

What then makes some generalization lawlike and other accidental? This, for Goodman, becomes a problem of determining which predicates are projectable (i.e., can be used in lawlike generalizations that serve as predictions) and which are not. Goodman argues that this is where the fundamental problem lies. This problem, known as Goodman's paradox, is as follows. Consider the evidence that all emeralds examined thus far have been green. This leads us to conclude (by induction) that all future emeralds will be green. However, whether this prediction is lawlike or not depends on the predicates used in this prediction. Goodman observed that (assuming t has yet to pass) it is equally true that every emerald that has been observed is grue. Thus, by the same evidence we can conclude that all future emeralds will be grue. The new problem of induction becomes one of distinguishing projectable predicates such as "green" and "blue" from non-projectable predicates such as "grue" and bleen.

Hume, Goodman argues, missed this problem. We do not, by habit, form generalizations from all associations of events we have observed but only some of them. Lawlike predictions (or projections) ultimately are distinguishable by the predicates we use. Goodman's solution is to argue that Lawlike predictions are based on projectable predicates such as "green" and "blue" and not on non-projectable predicates such as "grue" and bleen and what makes predicates projectable is their entrenchment, which depend on their past use in successful projections. Thus, "grue" and bleen function in Goodman's arguments to both illustrate the new riddle of induction and to illustrate the distinction between projectable and non-projectable predicates via their relative entrenchment.


The most obvious response is to point to the artificially disjunctive definition of grue. The notion of predicate entrenchment is not required. Goodman, however, noted that this move will not work. If we take grue and bleen as primitive predicates, we can define green as "grue if first observed before t and bleen otherwise", and likewise for blue. To deny the acceptability of this disjunctive definition of green would be to beg the question.

Another proposed resolution of the paradox (which Goodman addresses and rejects) that does not require predicate entrenchment is that "x is grue" is not solely a predicate of x, but of x and a time t—we can know that an object is green without knowing the time t, but we cannot know that it is grue. If this is the case, we should not expect "x is grue" to remain true when the time changes. However, one might ask why "x is green" is not considered a predicate of a particular time t—the more common definition of green does not require any mention of a time t, but the definition grue does. As we have just seen, this response also begs the question because definition blue can be defined in terms of grue and bleen, which explicitly refer to time.[4]

Swinburne gets past the objection that green be redefined in terms of grue and bleen by making a distinction based on how we test for the applicability of a predicate in a particular case. He distinguishes between qualitative and locational predicates. Qualitative predicates, like green, can be assessed without knowing the spatial or temporal relation of x to a particular time, place or event. Locational predicates, like grue, cannot be assessed without knowing the spatial or temporal relation of x to a particular time, place or event, in this case whether x is being observed before or after time t. Although green can be given a definition in terms of the locational predicates grue and bleen, this is irrelevant to the fact that green meets the criterion for being a qualitative predicate whereas grue is merely locational. He concludes that if some x's under examination—like emeralds—satisfy both a qualitative and a locational predicate, but projecting these two predicates yields conflicting predictions, namely, whether emeralds examined after time t shall appear blue or green, we should project the qualitative predicate, in this case green.[5]

Similar Predicates Used in Philosophical Analysis


In his book Wittgenstein on Rules and Private Language, Saul Kripke proposed a related argument that leads to skepticism about meaning rather than skepticism about induction, as part of his personal interpretation (nicknamed "Kripkenstein" by some[6]) of the private language argument. He proposed a new form of addition, which he called quus, which is identical with "+" in all cases except those in which either of the numbers added are equal to or greater than 57; in which case the answer would be 5, i.e.:

\text{x quus y}= \begin{cases} \text{x + y} & \text{for }x,y <57 \\[12pt] 5 & \text{for } x,y \ge 57 \end{cases}

He then asks how, given certain obvious circumstances, anyone could know that previously when I thought I had meant "+", I had not actually meant quus. Kripke then argues for an interpretation of Wittgenstein as holding that the meanings of words are not individually contained mental entities.

See also


Further reading

  • Goodman, Nelson (1955). Fact, Fiction, and Forecast. Cambridge, Massachusetts: Harvard UP, 1955. 2nd edition, Indianapolis: Bobbs-Merrill, 1965. 3rd. edition Indianapolis: Bobbs-Merrill, 1973. 4th edition, Cambridge, Massachusetts: Harvard UP, 1983.
  • Franceschi, Paul, Une solution pour le paradoxe de Goodman, Dialogue, vol.40, 2001, pp. 99–123, English translation.
  • Elgin, Catherine, ed. (1997). The Philosophy of Nelson Goodman: Selected Essays. Vol. 2, Nelson Goodman's New Riddle of Induction. New York: Garland. ISBN 0-8153-2610-6.
  • Goodman's original definition of grue
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