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Goodman’s Paradox (New Riddle of Induction)

Goodman’s paradox refers to a problematic statement concerning the scientific method commonly referred to as induction. Goodman referred to the problem posed by his statement as the “New Riddle of Induction”. The principle of scientific induction is that by making several observations of a certain class of thing, and finding that each such observation shows that each observed thing of that class has a certain well-specified property, then one might hypothesize that every thing of that class has that certain property. The classic example is the observation by someone that all ravens that he has observed are black, and so he makes the inductive hypothesis that all ravens are black. But the inductive method can never prove with absolute certainty that every thing of the class must have the specific property that is hypothesized.

 

In a book entitled “Fact, fiction, and forecast ”, (Footnote: Nelson Goodman, Fact, fiction, and forecast, Harvard University Press, 4th edition, 1983, ISBN: 9780674290709) Nelson Goodman introduced his “new riddle of induction”, and used it to supposedly demonstrate the difficulty of distinguishing predicates (Footnote: A predicate is a definition of a property that a thing can have.) that are acceptable for the purposes of scientific induction, and those that are not. He creates a definition, and claims that, although he says that he does consider the definition unacceptable as the basis for an inductive hypothesis, he also claims that his definition does not conflict with any rule for the use of definitions for inductive hypotheses. His definition was called ‘grue’, and he described ‘grue’ as:

 

“applies to all things examined before t just in case they are green but to other things just in case they are blue.”

 

Applying this definition as the basis of an inductive hypothesis to a certain class of things means that one is making the hypothesis that all things of that class are ‘grue’, regardless of whether or when they are examined. The supposed paradox is that, before the time t, there is no difference between an inductive hypothesis that all emeralds are green and an inductive hypothesis that all emeralds are ‘grue’ - both are equally supported by individual examinations of emeralds - but the supposed riddle is that after the time t, by the inductive hypothesis that all emeralds are ‘grue’, all emeralds should now be blue, which is clearly wrong - hence the “paradox”.

 

But there is no paradox at all here. The conundrum arises directly from Goodman’s definition which is unfortunately rather vaguely formulated. But regardless of whatever interpretation is put on Goodman’s original wording, it can be shown that there is still no paradox involved, as will be demonstrated in this article.

 

Interpretations of Goodman’s description

Our first consideration regarding Goodman’s original definition of ‘grue’ is that a thing clearly cannot be green and blue at the same time(Footnote: Goodman is clearly referring to relatively large systems, not to systems at the quantum level.) so it must be the case, regardless of interpretation, that Goodman is implying the exclusive or in his definition. This will be applied in the following as “or else”.

 

Possible Interpretation 1

A commonly accepted re-wording of Goodman’s description is that a thing is ‘grue’:

“if it is examined before time t and is green or else if it is not examined before time t and is blue.”

 

One of the principles of a scientific inductive hypothesis is that, given any one individual of the class of things that are the subject of an inductive hypothesis, one may examine it to see if it satisfies the hypothesis or not. Another principle is that there is no difference between a thing of that class that happens to be examined, and one that is not examined.

 

But the above definition turns that principle on its head by claiming that if a thing of a class of things that are hypothesized to be ‘grue’ is examined before time t, it is green, and if not examined before time t, it is blue.

 

Despite Goodman’s claim that there is nothing that can be pinpointed as being inherently unscientific in his definition of ‘grue’, this definition certainly is unscientific and cannot be a valid basis for a scientific inductive hypothesis..

 

Furthermore, one could also consider that the above definition is not falsifiable until after the time t, and it is clear that it is impossible to obtain any evidence whatsoever either to support or refute the hypothesis that unexamined ‘grue’ things are blue. Since the definition is not falsifiable it cannot be used as a valid basis for a scientific inductive hypothesis before the time t. And after the time t, the definition would then simply be that a ‘grue’ thing:

“is green (in which case it was examined before time t) or else it is blue.”

 

So that after the time t, the definition would be a perfectly valid basis for an inductive hypothesis (if a rather useless one) - and this would involve no contradiction or paradox - but before the time t the definition could not be a valid basis for an inductive hypothesis.

 

Furthermore, we might note that one of Goodman’s claims is that, before time t, there is no essential difference between using the hypothesis that all emeralds are green as a scientific inductive hypothesis and using the hypothesis that all emeralds are ‘grue’ as a scientific inductive hypothesis. This means (if the above mentioned definition was the intended definition) that Goodman must have been assuming that if any emerald is examined before time t, it must instantly miraculously change color from being blue to being green. The only other possibility is that it is somehow predetermined which emeralds would be examined before time t, and such emeralds are and were always green, while all other emeralds are and always were blue. Either way, by this definition, Goodman’s claim that there is some difficulty in finding a valid reason for rejecting the definition as a basis for a scientific inductive hypothesis is preposterous.

 

Possible Interpretation 2

One could also interpret Goodman’s original wording as intending that a thing is ‘grue’:

“if it is examined before time t and is green or else if it is blue.”

 

If we hypothesize, for example, that all emeralds are ’grue, then if an emerald has not been examined before time t, it must be blue, but if it is examined before time t, it must be either blue or green.

 

But the above definition only states that ‘grue’ things are either green or blue, and since it makes no mention of any possibility that a ‘grue’ thing can change color, the only logical conclusion is that the hypothesis is that some ‘grue’ things are examined before time t and found to be green, and others are examined before time t and found to be blue, and all ‘grue’ things that remain unexamined before time t are blue.

 

As in the previous case, the definition claims that a thing that is ‘grue’ and examined before time t can be green or blue, but if not examined before time t, it can only be blue. This violates the principle that there is no difference between a thing of a class that happens to be examined, and one that is not examined.

 

And the condition that all ‘grue’ things that remain unexamined before time t are blue, generates exactly the same problem as in the previous case. It is impossible to obtain any evidence whatsoever either to support or refute the hypothesis that ‘grue’ things that are not examined before time t are blue, and this renders the definition unacceptable for use as a basis for a scientific inductive hypothesis.

 

Possible Interpretation 3

Perhaps we might suppose that Goodman actually intended that a thing is ‘grue’:

“if it is examinable before time t and is green or else if it is blue.”

 

The way to deal with this definition is the same as for Interpretation 1 above. In the definition above, if before time t a thing is not examinable, but is nevertheless blue, then it is ‘grue’.

 

But, as previously noted, one of the principles of a scientific inductive hypothesis is that, given any one individual of the class of things that are the subject of an inductive hypothesis, one examines it to see if it satisfies the hypothesis or not. And another is that the hypothesis assumes that given a specific class of things, there is no difference between a thing of that class that is amenable to examination, and one that is not. But the above definition would be claiming that if a thing of a class of things that are hypothesized to be ‘grue’ is examinable before time t, it is green, and if not possible to examine it, then it is blue.

 

As for Interpretation 1, it is clear that such a definition is not a valid basis for a scientific inductive hypothesis.

 

Possible Interpretation 4

So perhaps Goodman actually intended that a thing is ‘grue’:

“if it is examinable before time t and is green or else if it is examinable and is blue.”

 

By this definition a thing that is ‘grue’ and which is green before time t may still be green after time t. And a thing that is ‘grue’ and blue before time t is also blue after time t. So we might apply the definition to a class of things and we could have a thing, and if we examine it before time t, it might be either blue or green. And if we don’t examine it before time t, it also might be blue or green. After time t we decide to examine it, and again it might be blue or green.

 

However, if we might suggest that we say that, for example, emeralds are ‘grue’, in the same way as we might say that emeralds are, that is clearly incorrect, since any emerald that is impossible to examine before time t is excluded from this definition of ‘grue’, even though it satisfies the definition of green.

 

As in the previous case, the definition violates the principle that, given a specific class of things, there is no difference between a thing of that class that is amenable to examination, and one that is not.

 

The time constraint

Several commentators have remarked on the fact that Goodman’s definition of ‘grue’ relies on a constraint defined in terms of a specific point in time t. Note that there is an important difference between:

  • the claim that all things of a certain class have a certain property over a specific time-span, after which they do not have that property
    and
  • the claim that all things of a certain class have a certain property up to one specific point in time t and thereafter do not have that property.

 

The former in general is amenable to a scientific inductive hypothesis, and can be tested on individual things indefinably into the future. But the latter is at odds with the principle of scientific induction, which is that any inductive hypothesis is a hypothesis that applies indefinitely into the future.

 

Goodman’s response to this criticism that his definition of ‘grue’ involves a reference to a specific point in time includes the use of another definition - a definition of ‘bleen’:

 

“applies to all things examined before t just in case they are blue but to other things just in case they are green.”

 

Given the criticism that while the definitions of “green” and “blue” are purely qualitative but that the definitions of ‘grue’ and ‘bleen’ are not, since they involve a reference to a specific point in time, Goodman’s reply is:

But the argument that the former but not he latter are purely qualitative seems to me quite unsound. True enough, if we start with ‘blue’ and ‘green’, then ‘grue’ and ‘bleen’ will be explained in terms of ‘blue’ and ‘green’ and a temporal term. But equally truly, if we start with ‘grue’ and ‘bleen’, then ‘blue’ and ‘green’ will be explained in terms of ‘grue’ and ‘bleen’ and a temporal term; ‘green’, for example, applies to emeralds examined before time t just in case they are grue, and to other emeralds just in case they are bleen. Thus qualitativeness is an entirely relative matter and does not by itself establish any dichotomy of predicates. This relativity seems to be completely overlooked by those who contend that the qualitative character of a predicate is a criterion for its good behavior.

 

It is surprising that it seems that no-one (Footnote: For example, see articles on Goodman’s New Riddle of Induction by:
a) William Warren Bartley, ‘Goodman’s paradox: A simple-minded solution’, Philosophical Studies 19.6 (1968), pp 85-88.
b) Rudolf Carnap, ‘On the Application of Inductive Logic’. Philosophy and Phenomenological Research, Vol. 8, No. 1 (Sept 1947), pp. 133-148.
c) Karl Popper, Realism and the aim of science: From the postscript to the logic of scientific discovery, Routledge, (2013).
d) W V Quine, ‘Natural Kinds’, in the book Ontological Relativity and other Essays No.1, Columbia University Press (1969).
e) R G Swinburne, ‘Grue’, Analysis, Vol. 28, No. 4 (Mar 1968), pp. 123-128.)
has pointed out the obvious flaw in this reply. Goodman claims that we can start with ‘grue’ and ‘bleen’, and says:

‘green’, for example, applies to emeralds examined before time t just in case they are ‘grue’, and to other emeralds just in case they are ‘bleen’.

but if ‘green’ is to be defined in terms of ‘grue’ and ‘bleen’, rather than being defined conventionally in terms of the wavelength of light coming from an object, then the word ‘green’ cannot be used in the definition of ‘grue’ and ‘bleen’, and the same applies to the word ‘blue’. But the only information we have of the terms ‘grue’ and ‘bleen’ are in terms of ‘green’ and ‘blue’ and time. Logically, if we are not relying on the definitions of ‘green’ and ‘blue’ then the only information that we have for a thing being ‘grue’ is:

“if it is examined before time t and is ‘googoo’ or else if it is ‘booboo’.”

and for a thing being ‘bleen’ is:

“if it is examined before time t and is ‘booboo’ or else if it is ‘googoo’.”

where ‘googoo’ and ‘booboo’ are terms that must be undefined, since Goodman asserts that we are starting our definitions with ‘grue’ and ‘bleen’, and so we may not use the predefined definitions of green and blue to define ‘grue’ and ‘bleen’. Clearly, using the words ‘green’ and ‘blue’ would include the implicit definitions of blue and green. So, where Goodman says:

‘green’, for example, applies to emeralds examined before time t just in case they are ‘grue’, and to other emeralds just in case they are ‘bleen’.

becomes:

‘googoo’ applies to emeralds examined before time t just in case they are ‘grue’, and to other emeralds just in case they are ‘bleen’.

which, inserting for the terms ‘grue’ and ‘bleen’, gives:

‘googoo’ applies to emeralds examined before time t and are ‘googoo’, or else they are ‘booboo’, and to other emeralds just in case they are examined before time t and are ‘booboo’, or else they are ‘googoo’.

 

This is a meaningless circular self-referential expression that doesn’t define anything, never mind ‘green’, besides the obvious fact that the only information we have on what ‘grue’ and ‘bleen’ might be still includes a reference to a specific time.

 

Summary

It can be seen that whatever interpretation is applied to Goodman’s rather vague description, it belies Goodman’s claim that there is no logical method of differentiating his definition from a definition which is acceptable as the basis for a inductive hypothesis. There’s no paradox, and there’s no riddle.

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Footnotes:

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The Lighter Side

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There is now a paper that deals with the matter of language and the diagonal proof, see On Considerations of Language in the Diagonal Proof.

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Links

For convenience, there are now two pages on this site with links to various material relating to Gödel and the Incompleteness Theorem

 

– a page with general links:

Gödel Links

 

– and a page relating specifically to the Gödel mind-machine debate:

Gödel, Minds, and Machines

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