Editing New riddle of induction (section)

==Responses==
One response is to appeal to the artificially [[Logical disjunction|disjunctive]] definition of grue.  The notion of predicate ''entrenchment'' is not required. Goodman said that this does not succeed.  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 [[Begging the question|beg the question]].

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.{{sfn|Goodman|1983|p=79}}

===Swinburne===
[[Richard Swinburne]] gets past the objection that green may 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 grue or green, we should project the qualitative predicate, in this case green.<ref>R. G. Swinburne, 'Grue', Analysis, Vol. 28, No. 4 (March 1968), pp. 123-128.</ref>

===Carnap===
[[Rudolf Carnap]] responded{{sfn|Carnap|1947|p=139}} 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''.{{efn|he uses another variant, ''c''<sup>*</sup>(''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''<sup>*</sup>.}} Both ''h'' and ''e'' are logical formulas expressed in a simple language ''L'' which allows for
* multiple quantification ("for every ''x'' there is a ''y'' such that ..."),
* unary and binary predicate symbols (properties and relations), and
* an equality relation "=".
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.{{sfn|Carnap|1947|p=134}} 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.{{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.}} Moreover, different predicates must be logically independent.{{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.}}{{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.}}
* The qualities and relations designated by the predicates must be simple, i.e. they must not be analyzable into simpler components.{{sfn|Carnap|1947|p=136}} Apparently, Carnap had in mind an [[irreflexive]], [[partial order|partial]], and [[well-founded]]{{sfn|Carnap|1947|loc=p. 137: "... carry the analysis [of complex predicates into simpler components] to the end"}} [[order theory|order]]{{efn|Carnap doesn't consider predicates that are mutually definable by each other, leading to a [[preorder]].}} ''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''.{{sfn|Carnap|1947|p=138}}
Carnap distinguishes three kinds of properties:
# Purely qualitative properties; that is, properties expressible without using individual constants, but not without primitive predicates,
# Purely positional properties; that is, properties expressible without primitive predicates, and
# Mixed properties; that is, all remaining expressible properties.
To illuminate this taxonomy, let ''x'' be a variable and ''a'' a constant symbol; then an example of 1. could be "''x'' is blue or ''x'' is non-warm", an example of 2. "''x'' = ''a''", and an example of 3. "''x'' is red and not ''x'' = ''a''".

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.{{sfn|Carnap|1947|p=146}}

===Quine===
[[Willard Van Orman Quine]] discusses an approach to consider only "[[natural kind]]s" as projectible predicates.{{sfn|Quine|1970}}
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''".{{sfn|Quine|1970|p=41}} Then Hempel's paradox just shows that the complements of projectible predicates (such as "is a raven", and "is black") need not be projectible,{{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.}} 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".{{sfn|Quine|1970|p=42}} 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'';{{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<sup>''n''-2</sup> sets, and share exactly that number of [[extensional]] properties. [[Satosi Watanabe|Watanabe]] called this the "[[Ugly duckling theorem]]".}} Quine therefore tries to relate at least the latter two notions to each other.

{| style="float:right"
| [[File:GoodmansCounterexampleNaturalKind.gif|thumb|150px|Goodman's counter-example against a definition of "natural kind" based on Carnap]]
|}
{| style="float:right"
| [[File:Quine's qualitative sphere_svg.svg|thumb|150px|Failed attempt to define a kind as the set of all objects ''x'' that are more similar to a "paradigm" object ''p'' than ''p'' is to a "foil" object, in analogy to the definition of a [[circle]] area in geometry]]
|}

'''Relation between similarity and kind'''

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{{efn|Rather than arbitrary sets}} than ''A'' and ''C'' do.{{sfn|Quine|1970|p=44}}{{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.}}

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.{{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.{{sfn|Quine|1970|p=44-45}}{{efn|Formally: A set ''K'' is a kind if ∀''Y'' ∉ ''K''. ∃ ''X''<sub>1</sub> ∈ ''K''. ∀ ''X''<sub>2</sub> ∈ ''K''.  (''X''<sub>1</sub> is more similar to ''X''<sub>2</sub> than to ''Y'').}}
However, Goodman{{sfn|Goodman|1951|p=163f}} 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,{{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.}} while "surely it is not what anyone means by a kind".{{efn|The set contains e.g. yellow [[croquet]] balls and red rubber balls, but not yellow rubber balls.}}{{sfn|Quine|1970|p=45}}

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.{{sfn|Quine|1970|p=45}}

'''Basic importance of similarity and kind'''

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.{{sfn|Quine|1970|p=42, 45-48}}

Every reasonable expectation depends on resemblance of circumstances, together with our tendency to expect similar causes to have similar effects.{{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.

{| align="right" class="collapsible collapsed" style="flush:right"
|-
! colspan=2 | Birds' similarity relations
|- valign="bottom"
| [[File:Cooper's Hawk 2 edited.jpg|thumb|150px]] || [[File:Goose-flying.jpg|thumb|150px]]
|- valign="top"
| [[File:Falcon scheme of chickens and ducks.gif|thumb|150px]] || [[File:Goose scheme of chickens and ducks.gif|thumb|150px]]
|- 
| colspan=2 width=300px | Tinbergen and Lorentz demonstrated a coarse similarity relation of inexperienced turkey chicks.{{sfn|Hoffman|1998|loc=Chapter 1}}{{sfn|Tinbergen|1951|loc=Chapter IV}}{{sfn|Tinbergen|1948|loc=p. 34, Fig. 21C}} '''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 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,{{sfn|Quine|1970|p=46}} cf. bird example.

==== Habit formation ====
Induction itself is essentially [[Classical conditioning|animal expectation]] or habit formation. [[Ostensive definition|Ostensive learning]]{{sfn|Quine|1974|loc=Sect. 11}} 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.{{sfn|Quine|1970|p=47}} 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.{{efn|Quine seems to allude to Vico's [[Giambattista Vico#The verum factum principle|verum factum principle]] here.}} Why inductively obtained theories about it should be trusted is the perennial philosophical [[problem of induction]].  Quine, following [[Satosi Watanabe|Watanabe]],{{sfn|Watanabe|1965|p=41}} 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]].{{sfn|Quine|1970|p=48}} 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.{{efn|Demonstrated by psychological experiments e.g. about classification of previously unseen artificial objects, like "[[Greeble (psychology)|Greebles]]".}}