Abstract
Intuitively speaking, a statement is truthlike or verisimilar if it is like the truth’ or ‘similar to the truth’. In other words, a truthlike statement has to ‘resemble the truth’ or to be ‘close to the truth’, but it need not be true or even probable (i.e., likely to be true).1 The notion of truthlikeness or verisimilitude can thus be regarded as a special case of the more general concept of similarity or resemblance. For this reason, we shall give in this chapter a survey of the work on similarity and dissimilarity that has been done in the fields of philosophy, logic, mathematics, statistics, information theory, computer science, biology, anthropology, psychology, and linguistics.
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Notes
According to John Locke, “probability is likeness to be true” (An Essay Concerning Human Understanding, 1689, Book 4, Ch. XV). In many modern languages, the concept of probability is derived from the Latin verisimilitudo by combining the words ‘true’ (Latin verum) and ‘similitude’ (cf. Hartigan, 1971), but this should not lead us to confuse these concepts with each other. For the connection between truthlikeness and probability, see Chapter 5.2 below. For an analysis of the first component in the notion of truthlikeness — i.e., the truth — see Chapter 4.3 below.
The standard abbreviation ‘iff’ for ‘if and only if’ is used in this book.
log denotes here the natural logarithm.
Boorman and Olivier (1973) note — without giving any details — that the definition (28) can be generalized to cases where A 1 and A 2 are associated with different label sets V 1 and V 2 and where all the modes in A 1 and A 2 are labeled.
For the basic concepts and results of topology, see Bourbaki (1966) and Simmons (1963).
Y is compact if every class of closed subsets of Y with empty intersection has a finite subclass with empty intersection.
F. Bacon’s (1620) method of eliminative induction can be viewed as a systematic way of exploring analogies, as Keynes (1921) observes.
See the summary in Klug (1966). For the treatment of analogy in legal reasoning, see also Niiniluoto (1983b).
See, for example, Heywood (1976), Ruse (1973), Sneath and Sokal (1973). A nice illustration of taxonomical methods is given in the exhibition Dinosaurs and their living relatives, opened at the Natural History Museum, London, in 1979. It examines the question: Which living animals are most closely related to dinosaurs? On the basis of evidence about ‘homologies’ (i.e., “characteristics that are similar because they are inherited from a common ancestor”), it comes to the conclusion: “Birds are the closest living relatives of the dinosaurs. But some scientists think that birds and crocodiles are more closely related to each other than either of them is to dinosaurs”. For example, fossil birds, fossil crocodiles, dinosaurs, pretosaurs, and thecodontians share uniquely among them the homology that they have a hole in the skull in the front of the eye socket. (See Niiniluoto, 1980b.)
Dinosaurs and Their Living Relatives, p. 17. See also Sneath and Sokal (1973), Ch.6.
Multistate qualitative characters are treated in the same way — without assuming that the states of O i can be ordered (see Sneath and Sokal, 1973, p. 115). In J. C. Gower’s (1971) general similarity coefficient Math the value of 0 ≤ S ijk ≤ 1 expresses the similarity between the states x ij and x ik and W ijk is the weight for character O i . If S ijk = |x ij – x ik | for quantitative states, Gower’s coefficient is a complement of a weighted mean character difference (36). In Chapter 3.2 below we generalize this idea to cases where Q i may be multistate qualitative characters. For a treatment of similarity relation with vague properties, see Bugajski (1983).
For pattern recognition, see Fu (1977, 1982), Chen (1973).
For cluster analysis, see Chen (1973), Sneath and Sokal (1973), van Ryzin (1977), Rao (1977), Zadeh (1977).
See Rosenfeld (1969, 1976, 1979).
See also Carnap (1971b, 1980), Hilpinen (1973), Goodman (1972), and Eberle (1975). For a non-psychological, physicalistic interpretation of qualities, see Armstrong (1978a, b).
See Tversky (1977), Beals, Krantz, and Tversky (1968), Krantz and Tversky (1975), Sattath and Tversky (1977), Schwarz and Tversky (1980).
Condition (47) need not be interpreted to imply that all metrics d can be used in defining the concept of similarity.
D. Lewis (1973), who uses as a primitive concept the notion of overall similarity between possible worlds, also argues that similarity is a directional relation.
For criticism of nominalism, see Armstrong (1978a, b), Loux (1978), and Bunge (1977).
The atoms correspond to Carnap’s Q-predicates (see Chapter 2.2).
For a criticism of negative and disjunctive properties, see Armstrong (1978b).
While I agree with Armstrong’s (1978a, b) criticism of nominalism and platonism, I am inclined to accept (against his view) a ‘Stoutian’ realism, where each particular has its own individual qualities or ‘tropes’ — and universals exist only as abstractions in the man-made World 3. However, I shall not develop these ideas here further, since nothing in this book depends on a solution to the classical problem of universals.
For these terms, see Stalnaker (1968).
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© 1987 D. Reidel Publishing Company, Dordrecht, Holland
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Niiniluoto, I. (1987). Distance and Similarity. In: Truthlikeness. Synthese Library, vol 185. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-3739-0_1
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