Uncertainty under similarity structures
Orders of similarity and similarity judgements
John Maynard Keynes argues in A Treatise on Probability (1973 , p. 39; hereafter TP) that '[w]e say that one argument is more probable than another (i.e. nearer to certainty) in the same kind of way as we can describe one object as more like than another to a standard object of comparison'.14 In the similarity case, according to Keynes, '[w]hen we say of three objects A, B, and C that B is more like A than C, we mean, not that there is any respect in which B is in itself quantitatively greater than C, but that, if the three objects are placed in an order of similarity, B is nearer to A than C is' (TP, p. 39). In the uncertainty case 'certainty, impossibility, and a probability, which has an intermediate value, for example, constitute an ordered series in which the probability lies between certainty and impossibility' (TP, p. 37; emphasis in original). The above passages show that, in Keynes's view, the analogy between similarity and probability derives from the serial nature of both types of judgement and from the need to adopt in both cases a specific benchmark on which to base judgement (the standard object of comparison in the similarity case, certainty in the probability case). In both cases we place the relevant elements (be they objects or events) along an ordered series identified by a particular precedence relation. Objects are more or less similar depending on the position they hold along the serial order between identity and unlikeness; events are more or less probable depending on the position they hold along the serial order between certainty and impossibility.15 Keynes's argument suggests a formal analogy between judgements of similarity (in the case of objects) and judgements of likelihood (in the case of events) that is worth further exploration. In practice, according to Keynes, a judgement of likelihood is a special judgement of similarity in which certainty is the standard object of comparison. This point of view has far-reaching implications in view of the argument developed in the previous sections. For a similarity relation as defined above (definition 1) is not immediately conducive to an order of similarity. Given a binary relation Rs in set S, we can only say that for every pair of objects (or situations) o, o? e S, we have ot Rs o. However, we cannot say 'how similar' ot and o. are in each case, nor can we unambiguously identify what the 'standard object of comparison' is as we move from one pair (o, oj) to another. In practice, the definition of similarity as a binary relation is not sufficient to ground similarity in the domain of serial comparisons (these are comparisons leading to the arrangement of objects along an ordered series).
The above discussion of the internal structure of similarity (see section 5.3) and Keynes's concept of 'standard object of comparison' provide a clue to what may be done in order to shift from pairwise similarity to similarity of the serial type. This can be seen as follows. First, one should identify the specific similarity domain relevant to the comparison in view. This may not be an easy task. For any two objects (or situations) ot, oj may belong to a given similarity relation Rs on the condition that they have a number of common features о such that: о > о*. This means one has to identify a certain number of dimensions relevant to the comparison and a certain threshold (о*) below which no similarity can be identified. Indeed, similarity for any given dimension presupposes the possibility of assessing commonality of features across different objects. A practical way to go about that may be to introduce equivalence intervals within which the value taken by any given feature is considered to be the same. For example, if we compare objects (or situations) ot and oj along the dimensions of height, length and colour, it may be useful to consider the following equivalence intervals for height, length and colour respectively: h = [hl, hu], w = [wl, wu], c = [cl, cu]. Any value for h, w and c falling within the corresponding equivalence interval will be considered the same for the purpose of similarity comparison. We are now in the position to outline a stepwise procedure that allows assignment of objects to similarity relations. First, one has to identify a list of similarity dimensions (the dimensions for which similarity is assessed). Second, one should determine the equivalence intervals for any given, dimension of similarity. Third, it would be necessary to identify for which dimensions the objects under consideration are actually similar. Finally, one would be in the position to assess whether objects ot and Oj actually meet condition a > a*.
The above argument entails consideration of the internal structure of similarity. In particular, it entails the view that any given judgement of overall similarity derives from a composition of partial similarity judgements associated with particular dimensions of the objects or situations to be considered. To sum up, objects or situations ot and Oj may be compared along manifold similarity dimensions (in our example, height, length and colour), and any such dimension gives rise to a particular order of similarity (in our example, the orders of height, length and colour similarity, respectively). It is important to emphasize that, in principle, partial similarity on one dimension (say, weight) is independent of partial similarity on other dimensions (say, length and colour).16 In other terms, objects or situations ot and Oj may be close to one another when similarity is assessed in terms of height but not when similarity is assessed in terms of length or colour.17 As a result, ot and oj may be associated with different degrees of similarity depending on which similarity order is considered. On the other hand, the same degree of similarity may be associated with more than one order of similarity (for example, objects ot and oj may be 'equally similar' if their height, length and colour fall at exactly the same distance from the two polar situations of perfect likeness and unlikeness). Figure 5.4 shows the relationship between different orders of similarity for any two objects.
Let point 0 be associated with (complete) unlikeness and point 1 with perfect likeness. Any given order of similarity allows the placing of objects
Figure 5.4 Orders of similarity and similarity overlaps
(or situations) on the [0, 1] scale, and the same object (situation) is likely to be located at different points on that scale depending on which particular order of similarity is considered. For example, object ot would be associated with point A on similarity order h, B on similarity order w, and C on similarity order c relatively to object o, which we may take as our standard object of comparison. Only under exceptional circumstances would ot be located at the same point on different orders of similarity (this would require the special case of qualities such as h, w and c gradually fading into one another, like a colour of different shades). The opposite case would be the one in which a given position on the similarity scale [0, 1] were occupied by a certain object by virtue of a perfect alignment of the corresponding qualities. Here, different values for qualities h, w, and c would be associated with the same position on the similarity scale. The most common situation would be the one in which different values for qualities h, w, and c would be associated with different positions on the similarity scale. In the latter case, each similarity order provides a different similarity assessment for the same object and (at least in principle) it is impossible to associate the object under consideration with a single position on the [0, 1] scale.
As we let features h, w, and c to vary, we can explore the associated changes in the similarity position of object or situation ot relatively to the same object of comparison o. In this case, similarity of features is matter of degree within the [0, 1] continuum. This means that, for any given feature к (that is, for any given order of similarity), we may take a conventional similarity value, say ak*, and consider ak* as the specific similarity threshold for the feature under consideration. In other words, any particular order of similarity is associated with a condition ak > ak*. Here, overall similarity would depend on whether the number of features for which the latter condition holds is greater than or equal to a* (see condition 2, section 5.3.2 above). It is interesting to note that, by the above argument, partial similarity would vary within range [0, 1], while overall similarity could be defined as the number of relevant similarity features across different similarity domains. This allows the assessment of overall similarity independently of a comparison of features across different orders of similarity.
The above discussion highlights the fact that the internal structure of similarity may have an important influence upon the nature of similarity and ultimately upon whether a similarity relation can be identified. For the distinction between different orders of similarity suggests that, in general, any given object or situation ot may be 'similar' to another object (the 'standard object of comparison') only if we consider some of its features (that is, some of the associated orders of similarity) but not if we consider some other features (that is, some other orders of similarity). In short, consideration of the internal structure of similarity makes partial similarities more likely but, at the same time, makes overall similarity highly conventional and often extremely difficult to achieve.