Thenew riddle of induction was presented byNelson Goodman inFact, Fiction, and Forecast as a successor toHume's original problem. It presents the logicalpredicatesgrue andbleen which are unusual due to their time-dependence. Many have tried to solve the new riddle on those terms, butHilary Putnam and others have argued such time-dependency depends on the language adopted, and in some languages it is equally true for natural-sounding predicates such as "green". For Goodman they illustrate the problem of projectible predicates and ultimately, which empirical generalizations arelaw-like and which are not.[1][2] Goodman's construction and use ofgrue andbleen illustrates how philosophers use simple examples inconceptual analysis.
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Goodman defined "grue" relative to an arbitrary but fixed timet:[a] an object is grueif and only if it is observed beforet and is green, or else is not so observed and is blue. An object is "bleen" if and only if it is observed beforet and is blue, or else is not so observed and is green.[3]
For some arbitrary future timet, say January 1, 2035, for all green things observed prior tot, such asemeralds and well-watered grass, both the predicatesgreen andgrue apply. Likewise for all blue things observed prior tot, such asbluebirds orblue flowers, both the predicatesblue andbleen apply. On January 2, 2035, however, emeralds and well-watered grass arebleen, and bluebirds or blue flowers aregrue. The predicatesgrue andbleen are not the kinds of predicates used in everyday life or in science, but they apply in just the same way as the predicatesgreen andblue up until some future timet. From the perspective of observers before timet it is indeterminate which predicates are future projectible (green andblue orgrue andbleen).
In this section, Goodman's new riddle of induction is outlined in order to set the context for his introduction of the predicatesgrue andbleen and thereby illustrate theirphilosophical importance.[2][4]
Goodman posesHume's problem of induction as a problem of the validity of thepredictions 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, there is no objective justification for these predictions. Deductive logic cannot be used 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 observations of one kind of event following another kind of event result in habits of regularity (i.e., associating one kind of event with another kind). Predictions are then based on these regularities or habits of mind.
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 has identified something deeper. To illustrate this, Goodman turns to the problem of justifying asystem of rules of deduction. For Goodman, the validity of a deductive system is justified by its 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 thatgrue andbleen have their philosophical role in Goodman's view of induction.
The new riddle of induction, for Goodman, rests on our ability to distinguishlawlike fromnon-lawlike generalizations.Lawlike generalizations are capable of confirmation whilenon-lawlike generalizations 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 notlawlike 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.
The question, therefore, is what makes some generalizationslawlike and others accidental. This, for Goodman, becomes a problem of determining which predicates are projectible (i.e., can be used inlawlike generalizations that serve as predictions) and which are not. Goodman argues that this is where the fundamental problem lies. This problem is known asGoodman's paradox: from the apparently strong evidence that allemeralds examined thus far have been green, one may inductively conclude that all future emeralds will be green. However, whether this prediction islawlike or not depends on the predicates used in this prediction. Goodman observed that (assumingt has yet to pass) it is equally true that every emerald that has been observed isgrue. Thus, by the same evidence we can conclude that all future emeralds will begrue. The new problem of induction becomes one of distinguishing projectible predicates such asgreen andblue from non-projectible predicates such asgrue andbleen.
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. All past observed emeralds were green, and we formed a habit of thinking the next emerald will be green, but they were equally grue, and we do not form habits concerning grueness.Lawlike predictions (or projections) ultimately are distinguishable by the predicates we use. Goodman's solution is to argue thatlawlike predictions are based on projectible predicates such asgreen andblue and not on non-projectible predicates such asgrue andbleen and what makes predicates projectible is theirentrenchment, which depends on their successful past projections. Thus,grue andbleen function in Goodman's arguments to both illustrate the new riddle of induction and to illustrate the distinction between projectible and non-projectible predicates via their relative entrenchment.
One response is to appeal to the artificiallydisjunctive definition of grue. The notion of predicateentrenchment is not required. Goodman said that this does not succeed. If we takegrue andbleen as primitive predicates, we can define green as "grue if first observed beforet andbleen otherwise", and likewise for blue. To deny the acceptability of this disjunctive definition of green would be tobeg the question.
Another proposed resolution that does not require predicateentrenchment is that "x is grue" is not solely a predicate ofx, but ofx and a timet—we can know that an object is green without knowing the timet, 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" isnot considered a predicate of a particular timet—the more common definition ofgreen does not require any mention of a timet, but the definitiongrue does. Goodman also addresses and rejects this proposed solution asquestion begging becauseblue can be defined in terms ofgrue andbleen, which explicitly refer to time.[5]
Richard Swinburne gets past the objection that green may be redefined in terms ofgrue andbleen 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 ofx to a particular time, place or event. Locational predicates, likegrue,cannot be assessed without knowing the spatial or temporal relation ofx to a particular time, place or event, in this case whetherx is being observed before or after timet. Although green can be given a definition in terms of the locational predicatesgrue andbleen, this is irrelevant to the fact that green meets the criterion for being a qualitative predicate whereasgrue is merely locational. He concludes that if somex'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 timet shall appear grue or green, we should project the qualitative predicate, in this case green.[6]
Rudolf Carnap responded[7] to Goodman's 1946 article. Carnap's approach to inductive logic is based on the notion ofdegree of confirmationc(h,e) of a given hypothesish by a given evidencee.[b] Bothh ande are logical formulas expressed in a simple languageL which allows for
Theuniverse of discourse consists of denumerably many individuals, each of which is designated by its own constant symbol; such individuals are meant to be regarded as positions ("like space-time points in our actual world") rather than extended physical bodies.[9] A state description is a (usually infinite) conjunction containing every possible ground atomic sentence, either negated or unnegated; such a conjunction describes a possible state of the whole universe.[10] Carnap requires the following semantic properties:
Carnap distinguishes three kinds of properties:
To illuminate this taxonomy, letx be a variable anda a constant symbol; then an example of 1. could be "x is blue orx is non-warm", an example of 2. "x =a", and an example of 3. "x is red and notx =a".
Based on his theory of inductive logic sketched above, Carnap formalizes Goodman's notion of projectibility of a propertyW as follows: the higher the relative frequency ofW in an observed sample, the higher is the probability that a non-observed individual has the propertyW. Carnap suggests "as a tentative answer" to Goodman, that all purely qualitative properties are projectible, all purely positional properties are non-projectible, and mixed properties require further investigation.[16]
Willard Van Orman Quine discusses an approach to consider only "natural kinds" as projectible predicates.[17]He first relates Goodman's grue paradox toHempel'sraven paradox by defining two predicatesF andG to be (simultaneously) projectible if all their shared instances count toward confirmation of the claim "eachF is aG".[18] Then Hempel's paradox just shows that the complements of projectible predicates (such as "is a raven", and "is black") need not be projectible,[g] while Goodman's paradox shows that "is green" is projectible, but "is grue" is not.
Next, Quine reduces projectibility to the subjective notion ofsimilarity. Two green emeralds are usually considered more similar than two grue ones if only one of them is green. Observing a green emerald makes us expect a similar observation (i.e., a green emerald) next time. Green emeralds are anatural kind, but grue emeralds are not. Quine investigates "the dubious scientific standing of a general notion of similarity, or of kind".[19] Both are basic to thought and language, like the logical notions of e.g.identity,negation,disjunction. However, it remains unclear how to relate the logical notions tosimilarity orkind;[h] Quine therefore tries to relate at least the latter two notions to each other.
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Relation between similarity and kind
Assuming finitely manykinds only, the notion ofsimilarity can be defined by that ofkind: an objectA is more similar toB than toC ifA andB belong jointly to more kinds[i] thanA andC do.[21][j]
Vice versa, it remains again unclear how to definekind bysimilarity. Defining e.g. the kind of red things as the set of all things that are more similar to a fixed "paradigmatical" red object than this is to another fixed "foil" non-red object (cf. left picture) isn't satisfactory, since the degree of overall similarity, including e.g. shape, weight, will afford little evidence of degree of redness.[21] (In the picture, the yellow paprika might be considered more similar to the red one than the orange.)
An alternative approach inspired byCarnap defines a natural kind to be aset whose members are more similar to each other than each non-member is to at least one member.[22][k]However, Goodman[23] argued, that this definition would make the set of all red round things, red wooden things, and round wooden things (cf. right picture) meet the proposed definition of a natural kind,[l] while "surely it is not what anyone means by a kind".[m][24]
While neither of the notions of similarity and kind can be defined by the other, they at least vary together: ifA is reassessed to be more similar toC than toB rather than the other way around, the assignment ofA,B,C to kinds will be permuted correspondingly; and conversely.[24]
Basic importance of similarity and kind
In language, every general term owes its generality to some resemblance of the thingsreferred to.Learning to use a word depends on a double resemblance, viz. between the present and past circumstances in which the word was used, and between the present and past phonetic utterances of the word.[25]
Every reasonable expectation depends on resemblance of circumstances, together with our tendency to expect similar causes to have similar effects.[19] This includes any scientific experiment, since it can be reproduced only under similar, but not under completely identical, circumstances. AlreadyHeraclitus' famous saying "No man ever steps in the same river twice" highlighted the distinction between similar and identical circumstances.
| Birds' similarity relations | |
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| Tinbergen and Lorentz demonstrated a coarse similarity relation of inexperienced turkey chicks.[26][27][28]Upper row: real hawk (left) and goose (right) in flight.Lower row: cardboard dummies releasing similar reactions as their originals. | |
Genesis of similarity and kind
In abehavioral sense, humans and other animals have an innate standard of similarity. It is part of our animal birthright, and characteristically animal in its lack of intellectual status, e.g. its alienness to mathematics and logic,[29] cf. bird example.
Induction itself is essentiallyanimal expectation or habit formation.Ostensive learning[30] is a case of induction, and a curiously comfortable one, since each man's spacing of qualities and kind is enough like his neighbor's.[31] In contrast, the "brute irrationality of our sense of similarity" offers little reason to expect it being somehow in tune with the unanimated nature, which we never made.[n] Why inductively obtained theories about it should be trusted is the perennial philosophicalproblem of induction. Quine, followingWatanabe,[32] suggestsDarwin's theory as an explanation: if people's innate spacing of qualities is a gene-linked trait, then the spacing that has made for the most successful inductions will have tended to predominate throughnatural selection.[33] However, this cannot account for the human ability to dynamically refine one's spacing of qualities in the course of getting acquainted with a new area.[o]
In his bookWittgenstein 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[34]) of theprivate language argument. He proposed a new form of addition, which he calledquus, 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.:
He then asks how, given certain obvious circumstances, anyone could know that previously when I thought I had meant "+", I had not actually meantquus. Kripke then argues for an interpretation ofWittgenstein as holding that the meanings of words are not individually contained mental entities.
