982:"Suppose that one is to list the attributes that plums and lawnmowers have in common in order to judge their similarity. It is easy to see that the list could be infinite: Both weigh less than 10,000 kg (and less than 10,001 kg), both did not exist 10,000,000 years ago (and 10,000,001 years ago), both cannot hear well, both can be dropped, both take up space, and so on. Likewise, the list of differences could be infinite… any two entities can be arbitrarily similar or dissimilar by changing the criterion of what counts as a relevant attribute."
970:"Presumably, people's perceptual and conceptual processes have evolved that information that matters to human needs and goals can be roughly approximated by a similarity heuristic... If you are in the jungle and you see a tiger but you decide not to stereotype (perhaps because you believe that similarity is a false friend), then you will probably be eaten. In other words, in the biological world stereotyping based on veridical judgments of overall similarity statistically results in greater survival and reproductive success."
72:
962:
had sufficient weight. Of course, if these feature weights were fixed, then these similarity relations would be constrained". Yet the property "striped" as a weight 'fix' or constraint is arbitrary itself, meaning: "unless one can specify such criteria, then the claim that categorization is based on
957:
A possible way around the ugly duckling theorem would be to introduce a constraint on how similarity is measured by limiting the properties involved in classification, for instance, between A and B. However Medin et al. (1993) point out that this does not actually resolve the arbitrariness or bias
565:
However, the choice of boolean features to consider could have been somewhat arbitrary. Perhaps there were features derivable from the original features that were important for identifying the ugly duckling. The set of booleans in the vector can be extended with new features computed as
958:
problem since in what respects A is similar to B: "varies with the stimulus context and task, so that there is no unique answer, to the question of how similar is one object to another". For example, "a barberpole and a zebra would be more similar than a horse and a zebra if the feature
214:
objects. One can use that to measure the similarity between two objects, and one would see how many sets they have in common. However, one cannot. Any two objects have exactly the same number of classes in common if we can form any possible class, namely
990:, which states that all algorithms for learning of boolean functions from input/output examples have the same overall generalization performance as random guessing. The latter result is generalized by Woodward to functions on countably infinite domains.
624:
Proof. Let x and y be two vectors. If they are the same, then their completed vectors must also be the same because any
Boolean function of x will agree with the same Boolean function of y. If x and y are different, then there exists a coordinate
290:
As all possible choices of zeros and ones are there, any two bit-positions will agree exactly half the time. One may pick two elements and reorder the bits so they are the first two, and imagine the numbers sorted lexicographically. The first
974:
Unless some properties are considered more salient, or 'weighted' more important than others, everything will appear equally similar, hence
Watanabe (1986) wrote: "any objects, in so far as they are distinguishable, are equally similar".
1179:
came to the same conclusion: "But importance is a highly volatile matter, varying with every shift of context and interest, and quite incapable of supporting the fixed distinctions that philosophers so often seek to rest upon
466:
or on half of all the cases, no matter which two elements one picks. So if we have no preconceived bias about which categories are better, everything is then equally similar (or equally dissimilar). The number of
471:
simultaneously satisfied by two non-identical elements is constant over all such pairs. Thus, some kind of inductive bias is needed to make judgements to prefer certain categories over others.
175:
about what sorts of categories are "natural" or "normal" and what are not. So one has to consider all the possible classes that could be, all the possible ways of making a set out of the
536:
621:
features. The ugly duckling theorem states that there is no ugly duckling because any two completed vectors will either be equal or differ in exactly half of the features.
949:
will agree on exactly one of the two functions. If they agree on one, they must disagree on the other and vice versa. (This proof is believed to be due to
Watanabe.)
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In a weaker setting that assumes infinitely many properties, Murphy and Medin (1985) give an example of two putative classified things, plums and lawnmowers:
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135:", respectively. Since F happens to imply W, each predicate that can be formed from F and W coincides with another one, hence there are only 8
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booleans each. The ugly duckling is the vector which is least like the others. Given the booleans, this can be computed using
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integer), with a zero for each element not in the class and a one for each element in the class. As one finds, there are
40:
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36:
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Stamos (2003) remarked that some judgments of overall similarity are non-arbitrary in the sense they are useful:
248:(half the total number of classes there are). To see this is so, one may imagine each class is represented by an
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things in the universe, and one wants to put them into classes or categories. One has no preconceived ideas or
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44:
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Boolean variables, with each one exactly once. Viewing these
Boolean functions as polynomials in
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1308:
32:
1529:
1514:
1429:
1389:
1298:
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1016:), but there cannot be separate objects or entities that have all their properties in common.
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Douglas L. Medin and R.L. Goldstone and Dedre
Gentner (1993). "Respects for similarity".
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1336:
Proceedings of the 1994 International
Conference on Machine Learning (San Mateo/CA)
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100:
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Nelson
Goodman (1972). "Seven Strictures on Similarity". In Nelson Goodman (ed.).
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1256:
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distinct possible predicates, each shown on an own line. The white ducklings
16:
Argument that classification is not really possible without some sort of bias
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207:
1221:
1204:
986:
According to
Woodward, the ugly duckling theorem is related to Schaffer's
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original features. The only canonical way to do this is to extend it with
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1287:
International
Conference on Intelligent Computing and Intelligent Systems
113:
88:
52:
24:
27:
showing that classification is not really possible without some sort of
1045:
Knowing and
Guessing: A Quantitative Study of Inference and Information
31:. More particularly, it assumes finitely many properties combinable by
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594:
possible Boolean functions. The resulting completed vectors have
1012:– Classification (discernibility) is possible (with or without a
1064:
725:. Now the completed features contain every Boolean function on
172:
56:
1358:
361:
will have it set to one. Within each of those blocks, the top
1280:"Computable and Incomputable Functions and Search Algorithms"
1209:
Annals of the Japan Association for Philosophy of Science
765:
variables over GF(2), segregate the functions into pairs
87:, and properties F ("first"), W ("white"). "0", "1", "
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numbers will have bit #1 set to zero, and the second
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35:, and finitely many objects; it asserts that any two
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without that linear term. Now, for every such pair
431:
will have it as one, so they agree on two blocks of
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1397:
1329:"A conservation law for generalization performance"
1236:Gregory L. Murphy and Douglas L. Medin (Jul 1985).
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59:as two swans are to each other. It was derived by
963:attribute matching is almost entirely vacuous".
147:agree on 4 of them (line 2, 3, 4, 8), but so do
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1034:
988:Conservation Law for Generalization Performance
980:
968:
1238:"The Role of Theories in Conceptual Coherence"
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8:
1165:. New York: Bobbs-Merrill. pp. 437–446.
396:will have bit #2 set to zero and the other
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43:) properties. The theorem is named after
1000:No free lunch in search and optimization
531:{\displaystyle x_{1},x_{2},\dots ,x_{n}}
70:
1030:
7:
1338:. Morgan Kaufmann. pp. 259–265.
1334:. In Willian, H.; Cohen, W. (eds.).
837:-th coordinate as a linear term and
155:, too (line 3, 5, 7, 8), and so do
75:Watanabe's example, using objects
39:objects share the same number of (
14:
896:
884:
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772:
1:
1295:10.1109/ICICISYS.2009.5358045
1278:John R. Woodward (Nov 2009).
1535:Metaphors referring to birds
1205:"Epistemological Relativity"
1148:10.1037/0033-295x.100.2.254
206:such ways, the size of the
51:", because it shows that a
1561:
1349:Woodward (2009), p. 875 lf
1289:. IEEE. pp. 871–875.
1257:10.1037/0033-295x.92.3.289
1193:. Lexington Books. p. 344.
1010:Identity of indiscernibles
1446:The Ugly Duckling and Me!
1124:, F, and W, respectively.
1327:Cullen Schaffer (1994).
1203:Satosi Watanabe (1986).
1041:Satosi Watanabe (1969).
55:is just as similar to a
1386:Hans Christian Andersen
1020:New riddle of induction
538:be a set of vectors of
459:{\displaystyle 2^{n}/4}
424:{\displaystyle 2^{n}/4}
389:{\displaystyle 2^{n}/4}
354:{\displaystyle 2^{n}/2}
319:{\displaystyle 2^{n}/2}
241:{\displaystyle 2^{n-1}}
45:Hans Christian Andersen
1493:Ugly Duckling fountain
1222:10.4288/jafpos1956.7.1
1189:Stamos, D. N. (2003).
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1488:Ugly duckling theorem
1163:Problems and Projects
1005:No free lunch theorem
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902:{\displaystyle (f,g)}
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790:{\displaystyle (f,g)}
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280:{\displaystyle 2^{n}}
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199:{\displaystyle 2^{n}}
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21:ugly duckling theorem
1422:Downhearted Duckling
1245:Psychological Review
1136:Psychological Review
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67:Mathematical formula
1191:The Species Problem
1049:. New York: Wiley.
179:objects. There are
33:logical connectives
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167:Suppose there are
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163:(line 1, 3, 6, 8).
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1481:The Ugly Duckling
1430:The Ugly Duckling
1414:The Ugly Duckling
1406:The Ugly Duckling
1390:The Ugly Duckling
1304:978-1-4244-4754-1
942:{\displaystyle y}
922:{\displaystyle x}
870:{\displaystyle f}
850:{\displaystyle g}
830:{\displaystyle i}
810:{\displaystyle f}
758:{\displaystyle k}
738:{\displaystyle k}
718:{\displaystyle y}
698:{\displaystyle i}
685:differs from the
678:{\displaystyle x}
658:{\displaystyle i}
638:{\displaystyle i}
583:{\displaystyle k}
568:boolean functions
551:{\displaystyle k}
475:Boolean functions
49:The Ugly Duckling
1552:
1540:1960s neologisms
1525:Machine learning
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47:'s 1843 story "
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1468:(1993 musical)
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1438:The Daydreamer
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1177:Nelson Goodman
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1473:Ugly Duckling
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130:
124:
118:
112:
108:
104:
84:
80:
76:
20:
18:
1484:(audiobook)
1476:(2009 play)
1215:(1): 1–14.
1077:Watanabe's
41:extensional
1509:Categories
1495:(fountain)
953:Discussion
645:where the
469:predicates
103:" denote "
1520:Arguments
513:…
231:−
208:power set
63:in 1969.
37:different
1530:Ontology
1515:Theorems
1392:" (1843)
1313:14473304
1065:68-56165
994:See also
129:", and "
99:", and "
53:duckling
25:argument
1265:4023146
960:striped
570:of the
1449:(2006)
1441:(1966)
1425:(1953)
1417:(1939)
1409:(1931)
1311:
1301:
1263:
1105:, and
1063:
1053:
797:where
254:string
173:biases
23:is an
1465:Honk!
1457:Other
1398:Films
1332:(PDF)
1309:S2CID
1283:(PDF)
1241:(PDF)
1026:Notes
252:-bit
105:false
1545:Bias
1388:'s "
1299:ISBN
1261:PMID
1180:it".
1061:LCCN
1051:ISBN
1014:bias
929:and
479:Let
256:(or
159:and
151:and
143:and
123:", "
117:", "
111:", "
109:true
107:", "
95:", "
91:", "
57:swan
29:bias
19:The
1291:doi
1253:doi
1217:doi
1144:doi
1140:100
857:is
592:all
210:of
120:and
114:not
1511::
1307:.
1297:.
1285:.
1259:.
1249:92
1247:.
1243:.
1211:.
1207:.
1138:.
1120:,
1116:,
1098:,
1091:,
1084:,
1059:.
1033:^
909:,
562:.
126:or
83:,
79:,
1378:e
1371:t
1364:v
1315:.
1293::
1267:.
1255::
1225:.
1219::
1213:7
1150:.
1146::
1122:A
1118:B
1114:C
1110:2
1107:y
1103:1
1100:y
1096:3
1093:x
1089:2
1086:x
1082:1
1079:x
1067:.
937:y
917:x
897:)
894:g
891:,
888:f
885:(
865:f
845:g
825:i
805:f
785:)
782:g
779:,
776:f
773:(
753:k
733:k
713:y
693:i
673:x
653:i
633:i
607:k
603:2
578:k
546:k
524:n
520:x
516:,
510:,
505:2
501:x
497:,
492:1
488:x
454:4
450:/
444:n
440:2
419:4
415:/
409:n
405:2
384:4
380:/
374:n
370:2
349:2
345:/
339:n
335:2
314:2
310:/
304:n
300:2
273:n
269:2
250:n
234:1
228:n
224:2
212:n
192:n
188:2
177:n
169:n
161:C
157:B
153:C
149:A
145:B
141:A
101:⊕
97:∨
93:∧
89:¬
85:C
81:B
77:A
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