by Gerald Gaus
Note that in an important sense {a, b, c} is equivalent to {c, b, a} insofar as, at least initially, we do not care whether our similarity ordering runs from right to left or from left to right. As long as we stick with a direction once one is imposed there is no worry about which way the dimension runs. (In an analogous way {c, a, b} is equivalent in this sense to {b, a, c}.) Suppose, then, that we choose the directionality imposed by {a, b, c}. We can thus set aside the {c, b, a} ordering and turn our attention to the second equivalent pair, {c, a, b} and {b, a, c}, that is consistent with our initial intuitive judgment. Here we can choose {c, a, b}, dropping the opposite “direction” of {b, a, c}, and, having done this, we will have imposed a direction on our dimension, leaving us with the judgment that (a∼b)>(a∼c) is consistent with {a, b, c} OR {c, a, b}.
Now suppose a second intuitive similarity judgment is made by a perspective: (c∼b)>(c∼a). This is not consistent with {c, a, b}. (Nor would it be consistent with {b, a, c}, the “other direction” of the {c, a, b} ordering.) So we are left with {a, b, c}. Notice that we have taken multiple similarity judgments and derived an ordered triplet that arrays the options along a dimension. We thus have generated “in-betweenness” (b is in between a and c) from “more similar than” relations. We can then build out from the ordered triplet to yield an entire dimension in terms of similarity (or, in terms of our model, an ordered set of all social worlds in {X}). Note that this is more enriched metric information than a simple ordinal ranking. If we have {a, b, c} we can find out whether b is more similar to a or c, giving us more than in-betweenness, as we also get “in between but closer to.”
That we tend to array domains along a single dimension is well documented in empirical research. Subjects from around the world gravitate to single-dimensional classification systems when they are forced to think formally about classifications. When asked to sort elements in a domain {X}, participants repeatedly select some single common dimension and sort according to it, avoiding the problem of aggregating different dimensions of similarity.1 Indeed, studies indicate that people have difficulty learning categories that depend on even two dimensions of similarity (e.g., the length and the orientation of a line).2 However, in their informal classifications people seem to employ family resemblances, which allow them to sort on multiple dimensions. Moreover, Philip Tetlock’s studies indicate that given optimal conditions virtually everyone can make decisions commensurating different dimensions of evaluation, which lends support to our intuitive procedure.3
(II) KEYNES’S WORRY AND MULTIDIMENSIONAL WEIGHTING SYSTEMS
John Maynard Keynes argues that it is implausible to expect a complete ordering by aggregating individual pairwise similarity judgments, along the lines of the procedure in (I). As do I, Keynes presents a pairwise comparisons of pairs analysis of similarity:
When we say of three objects A, B, and C that B is more like A than C is, 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. There are also, as in the case of probability, different orders of similarity. For instance, a book bound in blue morocco [f1, h1] is more like a book bound in red morocco [f2, h1] than if it were bound in blue calf [f1, h2] and a book bound in red calf [f2, h2] is more like the book in red morocco [f2, h1] than if it were in blue calf [f1, h2].4
It is easy to misread Keynes here because of the rather confusing “than if it were bound” locution, as the reference to “it” is not pellucid. However, we should read Keynes as saying that blue morocco [f1, h1] is more similar to red morocco [f2, h1] than “it” (the blue morocco book [f1, h1]) would be if it were a blue calf [f1, h2] book; thus the claim is that a book bound in red morocco is more similar to one bound in blue morocco than a red morocco book is to a blue calf book. So [(f2, h1)∼(f1, h1)] > [(f2, h1)∼(f1, h2)]. And that seems quite right. Keynes also holds that it makes perfect sense to say that “a book bound in red calf [f2, h2] is more like the book in red morocco [f2, h1] than if it [the red calf book] were in blue calf [f1, h2],” so [(f2, h1)∼(f2, h2)] > [(f2, h1)∼(f1, h2)]. However, Keynes goes on to argue that these comparisons might not be completable. Continuing the above example, he remarks that “there may be no comparison between the degree of similarity which exists between books bound in red morocco [f2, h1], and blue morocco [f1, h1], and that which exists between books bound in red morocco [f2, h1] and red calf [f2, h2].”5 To say whether (f2, h1) is more similar to (f1, h1), or to (f2, h2), would require judging whether morocco (h1) is more similar to calf (h2) than blue (f1) is to red (f2). Keynes’s claim here is that red-blue and morocco-calf constitute “different orders of similarity.”6 To make a similarity judgment over these pairs would require comparing the similarity of two colors to the similarity of two leathers, but these involve similarities of different types.
In this example, the comparison of which he is skeptical does not seem overly perplexing—if we include the idea of a perspective, which inherently categorizes and stresses some features of the world over others. Suppose I am a leather fancier when it comes to bindings; what really is salient to me is leather. If so, I will have no difficulty at all saying that a book in red morocco is more similar to a book in blue morocco than it is to a book in red calf. On my perspective, leather is the salient similarity, and the organization of my books in my library will show this.
However, a more fundamental worry remains. Suppose that the leathers become more and more dissimilar so that one book is bound in rat (h3). So now I have to ask is the red morocco book more similar to the blue rat book or to the red calf book? At some point as we compare more and more dissimilar leathers it could seem that I will eventually think similar color determines overall similarity, so that I might well conclude that the red morocco book is more similar to the red calf book than it is to the blue rat book. Thus, in arriving at overall similarity I am using the leather order of similarity until some point where I jump to the color order of similarity, because I am saying that at this point that the difference in leather is so great that the similarity in color is determinative of the overall similarity judgment. It is this judgment about which Keynes is skeptical.7
I think, in turn, we should be skeptical of this skepticism. For one thing, similarity of “leather” is itself a multidimensional criterion: suppleness, sheen, durability are all qualities of leather; to make similarity judgments of the “order of similarity” of leather implies that one has made similarity judgments along several dimensions and managed to commensurate them, so that one can say “the difference in suppleness is greater than the difference in sheen.” So to really stay true to the incommensurability of all different orders of similarity claim, we would have to resort to truly unidimensional orders, perhaps only atomic properties, with no comparisons except along one and only one dimension. That we do something is good evidence that it can be done.
However if we remain worried by Keynes’s argument, and so if we still need to assuage it to ensure that those comparisons are completable, we can go beyond the simple pairwise comparison procedure in (I). Following the lead of Martin Weitzman,8 we can suppose that the features of members of a domain (social worlds, buildings, ecosystems) can be measured on multiple dimensions. For example, classifications of similarity in architecture might focus on classification systems involving “the period of the building, its style, distinguishing features, location, and so forth.”9 For each of these classifications, a specific building can be classified into a number of subcategories (recognized periods, styles, sizes, etc.). One procedure would be to determine similarity in terms of shared characteristics; those that share many subcategory classifications are very similar, those that have few common subclassifications less similar. Of course a perspective can get more fine grained with richer similarity/diversity measures.10 More sophisticated perspectives develop richer conceptions of the various dimensions by which objects are sorted—say not only counting shared featu
res (as in our architectural example) but generating cardinal measures of the extent to which two objects, a and b, share a given feature f. Whereas in our original architectural case we may say that buildings a and b are similar because they share, say, five out of six classifications (same style, same period, etc.), we could develop a richer scalar/cardinal system where buildings rate, say, 0–100 on whether they are classical, whether they have Gothic influences, and so on, such that 100 represents the maximal possible difference within a subcategory while 1 represents the minimum.11
Figure A-1. A similarity weighting system
If a given perspective develops such a rich conception of the relevant dimensions, there is no special problem in the perspective aggregating these subscores into an overall score, given the way it views the relative importance of these dimensions. We must remember that the understanding of similarity and its underlying attributes is specific to a particular perspective on justice. There is no claim that the underlying taxonomy or classification system for generating a complete ordering of the domain {X} is shared by all plausible perspectives or is uncontroversial. To better see how a perspective might commensurate different dimensions consider a very simple aggregation system. Suppose that a, b, and c are three social worlds, and suppose that perspective Σ identifies three types of features, f, g, and h, that are relevant to justice and that Σ has diversity measures for each feature along the lines we have been discussing. Thus we might have judgments as in figure A-1.
So, assuming that we simply sum up the weights of differences, we would have the similarity of (a, b) = .6(5) + .3(15) + .1(4) = 3.85; (b, c) = .6(7) + .3(5) + .1(2) = 4.85; (a, c) + .6(12) + .3(10) + .1(6) = 10.8, so [(a∼b)>(a∼c)], [(b∼a)>(b∼c)]. This, of course, is just a toy example; there is no need for such a simple additive weighting system, nor need we suppose such precise measures. (Note that since the overall result is similarity judgments in the form of pairwise comparison of overlapping pairs, the overly precise scoring system is not reproduced in the final judgments.) In any event, the aim is simply to show how such comparisons might be accomplished.
Michael Morreau has presented an Arrow-inspired impossibility proof, showing that it is impossible to arrive at these sort of aggregative, overall similarity judgments by aggregating various dimensions of similarity and meet versions of the standard Arrovian conditions.12 It is essential to realize that such Arrow-inspired impossibility proofs depend on restricting the various “dimensional” information to pairwise ordinal information. Perhaps the most fundamental insight of axiomatic social choice theory is that the possibility of producing a complete, transitive, overall ordering of a domain of options (such as the overall similarity of social worlds) fundamentally depends on the richness of dimensional information that can be drawn on in generating the ordering of that domain.13 In standard cases involving social choice/welfare functionals, the “dimensional” information is each individual’s cardinal rankings of the options (each individual’s cardinal ranking is “a dimension”) from which a social, overall ordering can be generated. In our case of similarity judgments, the dimensional information is the different scalar measures of each dimension of similarity (e.g., features of social worlds), and then a perspective employs this to generate an overall similarity ordering of the domain {X}. As Sen has demonstrated, we can certainly generate complete, transitive orderings of a set of options by aggregating multiple dimensions of evaluation when the dimensional information is rich (say, cardinal) rather than mere ordering information, and we can at least approximately commensurate this richer dimensional information.14 Arrow-like results, showing deep problems with aggregating information over a number of dimensions, are powerful, but much of their power derives from the severe limitations on the sorts of information they allow: information is restricted to purely pairwise, ordinal judgments on all dimensions, thus only noncomparative ordinal information can be used to generate overall similarity judgments. While in some contexts this severe information restriction supposed by Arrow-type theorems is reasonable, in the case of a single perspective on justice seeking to understand the structure of domain {X} of possible social worlds, it is extraordinarily constraining. A perspective seeks to organize and evaluate a set of possible social worlds: we have no reason to posit severe restrictions on the sorts of information such that, when sorting worlds on some feature (say, income distribution or freedom), it can use only pairwise ranking judgments rather than richer, scaling information.
In this regard it is worth noting that the basic similarity orderings using the procedure in (I) above, yield superordinal information: if we have a triplet {a, b, c} we can find out not only whether b is “in-between” a and c (basic ordinal information) but also whether b is closer to (“more similar to”) a than to c; this superordinal information would be ignored in translating judgments of “more similar to” into simple pure ordinal judgments for an Arrow-type theorem. An Arrow-inspired proof must reduce this information to strictly ordinal, which measures only “in-betweeness. This highlights the way in which Arrow-like theorems are restrictive in the information they allow.
(III) N-DIMENSIONAL SPACE
Instead of an ordering approach, we could have developed a more complex multidimensional model, in which a perspective Σ would locate worlds a, b, and c in an N-dimensional similarity space, and then employ an N-dimensional metric, which would measure the N-dimensional distance between a and b and a and c to yield an overall conclusion whether a is closer to b or to c. The more formally minded may find this a more satisfying analysis, though much more would depend on the distance metric (DM) than in the simple ordering model, where the similarity ordering (SO) is much more important, as I think it should be. In a more complex model the topology of the N-dimensional space would be a critical element of a perspective: distance would be critical, and that could be measured differently. For example, arguably the two most familiar metrics are the Euclidean metric and the taxi-cab metric (sometimes referred to as the Manhattan norm or box metric).15 The Euclidean metric defines distance according to the familiar formula for the shortest linear distance between two points:
Figure A-2. Comparison of Euclidean and taxi-cab metrics
(which in two dimensions simplifies to the formula for the length of the hypotenuse of a right triangle). The taxi-cab metric, on the other hand, defines distance in two-dimensional space according to the shortest distance between two points from the perspective of taxi navigating a rectangular street grid (hence the name), which in higher dimensional space can be represented as:
As figure A-2 indicates the Euclidean and taxi-cab metrics will sometimes disagree about which of two points is closer to a third.
N-dimensional similarity space thus raises a number of interesting complexities and possibilities for increased diversity. However, I believe that the simple basic ordering model is sufficient to make my general points; a more complex topology of similarity would not make a great deal of difference to the core philosophical points, and would be much more complex.
1 The classic study is Medin, Wattenmaker, and Hampson, “Family Resemblance, Conceptual Cohesiveness, and Category Construction.” See also Ahn and Medin, “A Two-Stage Model of Category Construction”; Canini et al., “Revealing Human Inductive Biases for Category Learning by Simulating Cultural Transmission.” This does not seem to be simply a feature of Western reasoners; for a cross-cultural study, see Norenzayan et al., “Cultural Preferences for Formal versus Intuitive Reasoning.”
2 Ashby and Maddox, “Human Category Learning”; Ashby, Queller, and Berretty, “On the Dominance of Unidimensional Rules in Unsupervised Categorization.”
3 Tetlock, “Coping with Trade-Offs: Psychological Constraints and Political Implications.” See Fred D’Agostino’s discussion in Incommensurability and Commensuration, pp. 68ff.
4 Keynes, A Treatise on Probability, p. 36, where f1 = blue; f2 = red; h1 = morocco; h2 = calf.
5 Ibid.
6 Ibid. Emphasis in original.
7 As is Morreau, “It Simply Does Not Add Up,” esp. pp. 480–81.
8 Weitzman, “On Diversity.” See also Page, Diversity and Complexity, pp. 58ff.
9 Weitzman, “On Diversity,” pp. 365ff.
10 For details on how this can be done, see ibid.
11 Weitzman shows how such richer information can be developed from the basic idea of pairwise similarity judgments. Ibid.
12 Morreau, “It Simply Does Not Add Up,” pp. 483ff.
13 It must be stressed that though this second procedure employs cardinal/scalar information concerning the various dimensions of similarity, with regard to the SO element of the model, we are still seeking only to derive an ordering of the similarity of social worlds. This is formally similar to Sen’s “Social Welfare Functionals,” which generate an overall ordering from richer, more-than-ordering, dimensional information. “On Weights and Measures.”
14 On this critical point, see ibid.
15 I am drawing here on Gaus and Hankins, “Searching for the Ideal.”
Appendix B
On Predictive Diversity
THE KEY TO THE DIVERSITY PREDICTION THEOREM IS TO DETERMINE (i) the average error of individual predictive models and (ii) the collective error of the models taken together, and see how (i) compares with (ii).1 First, a measure of (i), average individual error, is required. Taking a cue from Francis Galton’s famous weight-judging competition (and somewhat simplifying the numbers), suppose we are trying to guess (A) the weight of an ox which actually weighs 1,000 pounds2 (let us simply use thousands of pounds, so call this “1K”), and then (B) the weight of a bull, which actually weighs 2,000 pounds (2K). Suppose an individual employs his or her predictive model for both competitions; assume it predicts the same weight for the ox and for the bull, 1,500 pounds (1.5K). We cannot compute its average error by simply taking each error and averaging. That would give us an error of .5 + (–.5) = 0. But obviously the model is manifestly error prone. We avoid this by first squaring the errors, [(.52) + (–.52)] = .25 + .25 = .5, giving us the average error of this model. Figure B-1 gives the individual errors for three predictive models, α, β, and γ, applied to our two weight-judging contests.