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Similarity relations and measures of distance

More formally, provided that set S is non-empty (S Ф 0), a similarity relation can be defined as a relation from set S onto itself:

such that

for every object o, o. e S, we have oRO.

The above definition does not yet provide a complete picture of the internal structure of the similarity relation. This can be obtained as follows.11 Let ot and o. be vectors belonging to circumscription set In general, similarity between any two objects presupposes that a minimum number a* of features is common between the objects under consideration.12 This condition may be expressed by assuming that for every pair o, Oj e Rs, we have a number of common features a such that:

The above definition entails that Rs is a binary relation in set S. However, the way in which the internal structure of Rs is generated makes clear that the domain and range of Rs depends on existing ontological (structural) and epistemic conditions. No similarity function and statistics, can be introduced if there are no common elements between sets П and E. Indeed, the domain in which similarity functions can be defined is subject to change depending on changes in either the ontology or the epistemic rules of the universe under consideration. For example, the switch from a more circumscribed to a less circumscribed ontology (that is, to an ontology less wedded to the principle of individuation) may entail a lower threshold a*, and thus assignment to binary set Rs of objects that might not have been sufficiently close under the previous similarity threshold. Alternatively, the switch from a less to a more circumscribed ontology (that is, an ontology paying more attention to individuation principles) may entail a higher threshold a *, and thus withdrawal from binary set Rs of objects included under the previous threshold. The internal structure of the similarity relation may also change as a result of epistemic variation. For instance, the introduction of a larger set of categories (that is, the expansion of the E set in Figure 5.3) would increase the intersection area with circumscription set П, and thus allow the expansion of binary set Rs. Alternatively, contraction in the number of categories available to describe phenomena (that is, contraction of set E ) would decrease the intersection area with set П, and thus induce contraction of binary set Rs . It is worth noting that both latter processes take place on the assumption of a given ontology (a given П set).

The above argument rests on the following assumptions: (i) any given similarity relation is associated with a measure of distance between objects or situations; (ii) distance may change due to a change in the way we identify those objects or situations, and (iii) adequate categories must be available in order to make full use of the opportunities inherent to any given ontology.13 In general, a reduced circumscription may be associated with a relatively larger Rs set as long as the intersection П П E is sufficiently expanded. This means that adequate epistemic enrichment is a necessary condition of higher-cardinality Rs sets. Alternatively, enhanced circumscriptions are associated with relatively smaller Rs sets unless the epistemic set E is also sufficiently expanded to compensate for the introduction of a richer ontology. In short, epistemic enrichment may induce higher cardinality of the binary Rs set both with an expansion and a contraction of the underlying ontology. But the conditions for that enrichment are very different in the two cases: (i) expansion of the П set makes ontology more sensitive to individual objects and must be compensated by epistemic expansion (categorization) enabling recognition of increasingly 'fine' similarity structures; (ii) contraction of the П set makes ontology less sensitive to individual objects and must be associated with epistemic expansion (categorization) enabling recognition of increasingly 'abstract' similarity structures. To sum up, a change of epistemic conditions may induce a change of similarity relations, but that change would be different depending on the structure and dynamics of the П set (see above).

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