New riddle of induction: Difference between revisions

The '''new riddle of induction''' was presented by [[Nelson Goodman]] in ''[[Fact, Fiction, and Forecast]]'' as a successor to [[problem of induction|Hume's original problem]]. It presents the logical [[Predicate (mathematical logic)|predicates]] '''grue''' and '''bleen''' which are unusual due to their time-dependence. Many have tried to solve the new riddle on those terms, but [[Hilary 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 are [[Scientific law|law-like]] and which are not.<ref name="Goodman.1946">{{cite journal |author=Nelson Goodman |title=A Query on Confirmation |journal=The Journal of Philosophy |date=Jul 1946 |volume=43 |number=14 |pages=383–385 |url=http://wordsmatter.caltech.edu/~franz/Confirmation%20and%20Induction/PDFs/Nelson%20Goodman%20-%20A%20Query%20on%20Confirmation.pdf |doi=10.2307/2020332 |jstor=2020332}}</ref><ref name="Goodman.1983">{{cite book sfn|author=Nelson Goodman |title=Fact, fiction, and forecast |url=https://books.google.com/books?id=i97_LdPXwrAC |access-date=8 March 2012 |year=1983 |publisher=Harvard University Press |isbn=978-0-674-29071-6 |pagep=74}}</ref> Goodman's construction and use of ''grue'' and ''bleen'' illustrates how philosophers use simple examples in [[analytic philosophy|conceptual analysis]].

Goodman defined "grue" relative to an arbitrary but fixed time ''t'':<ref group="note">{{efn|Historically, Goodman used ''"[[V-E day]]"'' and ''"a certain time t"'' in ''A Query on Confirmation'' (p.&nbsp;383) and ''Fact, fiction, and forecast'' (3rd ed. 1973, p.&nbsp;73), respectively.</ref>}} an object is grue [[if and only if]] it is observed before ''t'' and is green, or else is not so observed and is blue. An object is "bleen" if and only if it is observed before ''t'' and is blue, or else is not so observed and is green.<ref>{{cite SEP|title=Nelson Goodman|url-id=goodman|date=Mar 25, 2019}}</ref>

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 [[Philosophy|philosophical importance]].<ref name="{{sfn|Goodman.|1983"/>|p=74}}<ref name="Godfrey-Smith">{{cite book|author=Peter Godfrey-Smith|title=Theory and Reality|url=http://www.press.uchicago.edu|access-date=23 October 2012|year=2003|publisher=University of Chicago Press|isbn=978-0-226-30063-4|page=53}}</ref>

Another proposed resolution 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. Goodman also addresses and rejects this proposed solution as [[Begging the question|question begging]] because ''blue'' can be defined in terms of ''grue'' and ''bleen'', which explicitly refer to time.<ref>{{sfn|Goodman, |1983|p. =79.</ref>}}

[[Rudolf Carnap]] responded<ref>{{cite journalsfn| author=Rudolf Carnap| title=On the Application of Inductive Logic| journal=Philosophy and Phenomenological Research| year=1947| volumep=8| issue=1| pages=133–148| url=http://www.hss.caltech.edu/~franz/Confirmation%20and%20Induction/PDFs/Rudolf%20Carnap%20-%20On%20the%20Application%20of%20Inductive%20Logic.pdf| doi=10.2307/2102920| jstor=2102920| access-date=2014-01-27| archive-url=https://web.archive.org/web/20060920034402/http://www.hss.caltech.edu/~franz/Confirmation%20and%20Induction/PDFs/Rudolf%20Carnap%20-%20On%20the%20Application%20of%20Inductive%20Logic.pdf| archive-date=2006-09-20| url-status=dead139}} Here: p. 139.</ref> to Goodman's 1946 article. Carnap's approach to inductive logic is based on the notion of ''degree of confirmation'' ''c''(''h'',''e'') of a given hypothesis ''h'' by a given evidence ''e''.<ref group=note>p. 138; later on p. 143f, {{efn|he uses another variant, ''c''^*(''h'',''e''), for which he gives a formula to compute actual values;{{sfn|Carnap|1947|p=138, 143f}} different from Laplace's [[Rule of Succession]]. See Carnap's book ''Studies in inductive logic and probability'', Vol.1. University of California Press, 1971, for more details, in particular sect.IV.16 for ''c'', and app.A.1 for ''c''^*.</ref>}} Both ''h'' and ''e'' are logical formulas expressed in a simple language ''L'' which allows for

The [[universe 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.<ref>{{sfn|Carnap (|1947), |p. =134.</ref>}} 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.<ref>This might be seen as corresponding to [[Wittgenstein]]'s [[Tractatus Logico-Philosophicus#Proposition 1|Tractatus]], Nr.1.11.</ref> Carnap requires the following semantic properties:

* Atomic sentences must be logically independent of each other.<ref>cf. Tractatus Nr.1.21</ref> In particular, different constant symbols must designate different and entirely separate individuals.<ref group=note>{{efn|For example, if ''a'' and ''b'' had a part in common, then "''a'' is warm and ''b'' is not warm" would be an impossible combination.</ref>}} Moreover, different predicates must be logically independent.<ref group=note>{{efn|For example, "is a raven" and "is a bird" cannot both be admitted predicates, since the former would exclude the negation of the latter. As another example, "is warm" and "is warmer than" cannot both be predicates, since "''a'' is warm and ''b'' is warmer than ''a'' and ''b'' is not warm" is an impossible combination.</ref><ref group=note>}}{{efn|Carnap argues ({{sfn|Carnap|1947|p. =135)}} that logical independence is required for deductive logic as well, in order for the set of [[analytic-synthetic distinction#Frege and Carnap revise the Kantian definition|analytical sentences]] to be decidable.</ref>}}

* The qualities and relations designated by the predicates must be simple, i.e. they must not be analyzable into simpler components.<ref>{{sfn|Carnap (|1947), |p. =136.</ref> }} Apparently, Carnap had in mind an [[irreflexive]], [[partial order|partial]], and [[well-founded]]<ref group{{sfn|Carnap|1947|loc=note>p. 137: "... carry the analysis [of complex predicates into simpler components] to the end", p. 137.</ref>}} [[order theory|order]]<ref group=note>{{efn|Carnap doesn't consider predicates that are mutually definable by each other, leading to a [[preorder]].</ref>}} ''is simpler than''.

* The set of primitive predicates in ''L'' must be complete, i.e. every respect in which two positions in the universe may be found to differ by direct observation, must be expressible in ''L''.<ref>{{sfn|Carnap (|1947), |p. =138.</ref>}}

Based on his theory of inductive logic sketched above, Carnap formalizes Goodman's notion of projectibility of a property ''W'' as follows: the higher the relative frequency of ''W'' in an observed sample, the higher is the probability that a non-observed individual has the property ''W''. 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.<ref>{{sfn|Carnap (|1947), |p. =146.</ref>}}

He first relates Goodman's grue paradox to [[Carl Gustav Hempel|Hempel]]'s [[raven paradox]] by defining two predicates ''F'' and ''G'' to be (simultaneously) projectible if all their shared instances count toward confirmation of the claim "each ''F'' is a ''G''".<ref>{{sfn|Quine (|1970), |p. =41</ref>}} Then Hempel's paradox just shows that the complements of projectible predicates (such as "is a raven", and "is black") need not be projectible,<ref group=note>{{efn|Observing a black raven is considered to confirm the claim "all ravens are black", while the [[contraposition|logically equivalent]] claim "all non-black things are non-ravens" is not considered to be confirmed by observing e.g. a green leaf.</ref>}} while Goodman's paradox shows that "is green" is projectible, but "is grue" is not.

Next, Quine reduces projectibility to the subjective notion of ''similarity''. 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 a ''natural kind'', but grue emeralds are not. Quine investigates "the dubious scientific standing of a general notion of similarity, or of kind".<ref name="{{sfn|Quine (|1970), |p. =42">Quine (1970), p. 42.</ref>}} Both are basic to thought and language, like the logical notions of e.g. [[identity (philosophy)|identity]], [[negation]], [[disjunction]]. However, it remains unclear how to relate the logical notions to ''similarity'' or ''kind'';<ref group=note>{{efn|Defining two things to be similar if they have all, or most, or many, properties in common doesn't make sense if properties, like [[mathematical set]]s, take things in every possible combination. {{sfn|Quine (|1970), |p. =43.}} Assuming a finite universe of ''n'' things, any two of them belong to exactly 2^''n''-2 sets, and share exactly that number of [[extensional]] properties. [[Satosi Watanabe|Watanabe]] called this the "[[Ugly duckling theorem]]".</ref>}} Quine therefore tries to relate at least the latter two notions to each other.

Assuming finitely many ''kinds'' only, the notion of ''similarity'' can be defined by that of ''kind'': an object ''A'' is more similar to ''B'' than to ''C'' if ''A'' and ''B'' belong jointly to more kinds<ref group=note>{{efn|Rather than arbitrary sets</ref>}} than ''A'' and ''C'' do.<ref name="{{sfn|Quine (|1970), |p. =44">Quine (1970), p. 44</ref><ref group=note>}}{{efn|Quines uses this ternary relation in order to admit different levels of similarity, such that e.g. red things can be more similar to each other than just colored things.</ref>}}

Vice versa, it remains again unclear how to define ''kind'' by ''similarity''. 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.<ref name="{{sfn|Quine (|1970), |p. =44"/>}} (In the picture, the yellow paprika might be considered more similar to the red one than the orange.)

An alternative approach inspired by [[Carnap]] defines a natural kind to be a [[set (mathematics)|set]] whose members are more similar to each other than each non-member is to at least one member.<ref>{{sfn|Quine (|1970), |p. 44–45.</ref><ref group=note>44-45}}{{efn|Formally: A set ''K'' is a kind if ∀''Y'' ∉ ''K''. ∃ ''X''₁ ∈ ''K''. ∀ ''X''₂ ∈ ''K''. (''X''₁ is more similar to ''X''₂ than to ''Y'').</ref>}}

However, Goodman<ref>{{cite booksfn| author=Nelson Goodman| title=The Structure of Appearance1951| yearp=1951163f}} Here: p. 163f.</ref> 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,<ref group=note>{{efn|Each member of the set resembles each other member in being red, or in being round, or in being wooden, or even in several of these properties.</ref>}} while "surely it is not what anyone means by a kind".<ref group=note>{{efn|The set contains e.g. yellow [[croquet]] balls and red rubber balls, but not yellow rubber balls.</ref><ref name="}}{{sfn|Quine (|1970), |p. =45">Quine (1970), p. 45.</ref>}}

While neither of the notions of similarity and kind can be defined by the other, they at least vary together: if ''A'' is reassessed to be more similar to ''C'' than to ''B'' rather than the other way around, the assignment of ''A'', ''B'', ''C'' to kinds will be permuted correspondingly; and conversely.<ref name="{{sfn|Quine (|1970), |p. =45"/>}}

In language, every general term owes its generality to some resemblance of the things [[Reference#Semantics|referred]] to. [[Language acquisition|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.<ref>{{sfn|Quine (|1970), |p. =42, 45–48.</ref>45-48}}

Every reasonable expectation depends on resemblance of circumstances, together with our tendency to expect similar causes to have similar effects.<ref name="{{sfn|Quine (|1970), |p. =42"/>}} This includes any scientific experiment, since it can be reproduced only under similar, but not under completely identical, circumstances. Already [[Heraclitus]]' famous saying "No man ever steps in the same river twice" highlighted the distinction between similar and identical circumstances.

In a [[Behaviorism|behavioral]] 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,<ref>{{sfn|Quine (|1970), |p. =46.</ref>}} cf. bird example.

Induction itself is essentially [[Classical conditioning|animal expectation]] or habit formation. [[Ostensive definition|Ostensive learning]]<ref>Investigated in more detail in {{cite book|author=Willard Van Orman Quine |title=The Roots of Reference |url=https://archive.org/details/rootsofreference00wvqu |url-access=registration | location=La Salle, Illinois | publisher=Open Court Publishing Co. |year=1974|isbn=9780812691016 }} Sect. 11.</ref> 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.<ref>{{sfn|Quine (|1970), |p. =47.</ref>}} 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.<ref group="note">{{efn|Quine seems to allude to Vico's [[Giambattista Vico#The verum factum principle|verum factum principle]] here.</ref>}} Why inductively obtained theories about it should be trusted is the perennial philosophical [[problem of induction]]. Quine, following [[Satosi Watanabe|Watanabe]],<ref>{{cite book| editor1=Stanislas I. Dockx |editor2=Paul Bernays |editor2-link=Paul Bernays |title=Information and Prediction in Science |chapter-url=https://www.ncbi.nlm.nih.gov/nlmcatalog/41130 |location=New York |year=1965 |publisher=Academic Press | oclc=522269 | lccn=64-24655 | author=Satosi Watanabe |author-link=Satosi Watanabe |chapter=Une Explication Mathématique du Classement d'Objets | pages=39–76}} Here: p. 41.</ref> suggests [[Charles Darwin|Darwin]]'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 through [[natural selection]].<ref>{{sfn|Quine (|1970), |p. =48.</ref>}} 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.<ref group="note">{{efn|Demonstrated by psychological experiments e.g. about classification of previously unseen artificial objects, like "[[Greeble (psychology)|Greebles]]".</ref>}}