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Mathematics of Quality, Preference Logic and the Rationality of Desire
Associate Professor, PhD., Institute for Investigation of the Societies and Knowledge, Bulgarian Academy of Sciences
§1.An ancient philosophical dogma still reigns in (at least some) philosophical faculties. Its roots are more readily observable in the teachings of the scholastics in the Middle Ages and have gained mature, but at the same obscure expression in the writings of famous representatives of continental philosophy. Still (at least for the time being) we may dub it “the threefold dogma of Aristotelianism”. Indeed, if you are eager to vindicate a set of confused claims, comprising a particular dogma, then the best way to do this is to show that it is an old dogma. The principles of this venerable preconception, nested deeply in the collective unconscious of philosophy, are the following:
(1) Mathematics deals with Quantity, not Quality, therefore its compass is limited to the province of the effectively measurable. All things pertaining to Quality proper are not subject to mathematical treatment, they are essentially un-mathematical. The systematic correlation of qualitative attributes and mathematical entities is unfeasible.
(2) Since qualitative existence is intrinsically non-quantifiable, subjective preferences relate to the qualitative determinations of objects of experience, and quantitative determinateness is a precondition for formalization in general, logic does not deal (and should not deal) with preferences. In short, there is no logic of preference.
(3) Finally, if subjective preferences are not susceptible to formal-logical treatment, then our desires (i.e. the systematic expression of our preferences for qualitatively determined objects) are inevitably irrational, they are simply products of the lower, purely vegetative part of the soul.
Several observations readily suggest themselves in this connection:
(1) The first claim cannot be attributed directly to Aristotle, since in his Metaphysics he clearly stated that the “unmovable objects of mathematics”, i.e. numbers, also have qualities (for example, divisibility), therefore it would be misleading to claim that mathematics does not deal with qualities (cf. Metaphysica , 14, 1020b). As James Franklin has already noted, the “mathematics as a science of quantity” paradigm was developed much later, through a long chain of mediators whose most easily detectable branches end somewhere around 17th century (Franklin 2011, 5).
(2) As is well-known, the third book of Aristotle’s Topics deals primarily with the question “which is the more desirable” (Topica III, 1, 116a), and the second book of his Prior Analytics even attempted to lay bare some formal principles relating to the logic of preference (Analytica Priora II, 22, 68a). Nevertheless, the second dogma was clearly in operation, since the development of formally precise and technically elaborated preference logic began in the second half of 20th century.
(3) Concerning the third claim we also need to make a qualification. In „On the Soul” Aristotle commented on the apparently sound division of the soul into two parts – rational and irrational, but hastened to add that “wish is found in the calculative part and desire and passion in the irrational” (De Anima III, 9, 432b). Moreover, in his famous treatise on ethics, he commented in the same vein that even the so-called irrational part “seems to have a share in a rational principle” as shows on inspection the decorous figure of the “continent man” (Ethica Nicomachea I, 13, 1102b).
Furthermore, the advent of mathematical disciplines such as measurement theory and game theory has shaken, at least to some extent, the plausibility of (2) and (3), but (1) is still virtually unaffected by the contemporary development of mathematics. The design of this short note is to cast a blow on this primary misconception which would inevitably shake the ground of the other two.
§2.The locus classicus on the discrimination between quality and quantity is Aristotle’s Categories. There the great ancient master of logic noted that the characteristic property of quantities is that it is meaningful to predicate equality for them, while the characteristic property of qualities is that they are said to be similar, or that “likeness and unlikeness can be predicated with reference to them only” (see respectively Categoriae. vi, 6a and viii, 11a). This difference is decisive, since from our present vantage point of view it is easy to see that while equality is an equivalence relation (it is reflexive, symmetric, and transitive), similarity is not an equivalence relation (at least since it is obviously not transitive; the meaning of this fact shall become apparent later in this essay). We don’t need to reflect much in order to see, that it is perfectly possible that A is similar to B (in some respect), B is similar to C (in the same respect), C is similar to D … and Y is similar to Z, but A is not at all similar to Z (after all, this is the upshot of Wittgenstein’s theory of family resemblance). Therefore, since at the times of Aristotle there was no conceivable mathematical theory of intransitive indifference, he was completely justified to claim that qualities are not susceptible to mathematical treatment. As a matter of fact, for the development of such a theory we had to wait for more than two millennia – it was introduced in the second half of the 20th century by Duncan Luce, who invented the concept of semiorder and commenced the development of a mathematically elaborated theory of intransitive indifference. As we shall see in a moment, his algebraic theory can claim to be a true “mathematics of quality”, manifestly defeating the age-old preconception, limiting the scope of mathematical treatment to discrete and continuous quantity, i.e. to number and figure.
§3. In its present form, the axiomatic definition of semiorder is pretty simple and comprises a list of three axioms: a binary relation P on a set S is said to be a semiorder on S iff (1) P is irreflexive (there is no element of S, a, such that aPa); (2) for any quadruple of elements of S, a, b, c and d, if aPb and cPd, then either aPd, or cPb; (3) for any quadruple of elements of S, a, b, c and d, if aPb and bPc, then either aPd, or cPd (Luce and Suppes 1965, 280, def. 12). Plainly, any semiorder gives rise to a weak order (a transitive and connected relation W) by letting aWb iff for all c, bPc implies aPc and cPa implies cPb (Roberts 1985, 256). Moreover, “among the structures which generalize the classical notion of ranking (linear order), [semiorder is] the one closest to a linear order” (Pirlot and Vincke 1997, 4). From the computational side, it is important to note that semiorders are tractable mathematical objects: they are representable by intervals on the real line, their order dimension is always less than or equal to three, the number of non equivalent semiorders on a set of n elements is given by the n-th Catalan number, hence recognizing whether a binary relation is a semiorder can be done in linear time (Pirlot and Vincke 1997, 170). What makes semiorders important in the present context is the fact that they induce a natural indifference relation I on the set S (where aIb iff not aPb and not bPa), which is not necessarily transitive: aIb and bIc does not imply aIc. (Of course, intransitive indifference can be obtained in may non-equivalent ways and in different mathematical settings, but order theory and semiorders in particular offer an extremely convenient and manifestly natural way to do this.) This sanctions the claim that the notion of semiorder captures the mathematical meaning of similarity – any quality Q induces a semiorder on the set of objects with the property that aPb iff a is more Q-ish than b, where “Q” stands for some paradigmatic representative (prototype) of the property P. Then the axioms listed above acquire a completely natural interpretation: taking “white” for Q, we can render them as saying that (1) nothing is whiter than itself; (2) any two pairs of colored objects comparable with respect to their whiteness can be juxtaposed in a way that respects their order relation; (3) any triple of colored objects can be juxtaposed with a fourth in a way that respects their order relation (the intuitive meaning of (2) and (3) are more readily observed in their Hasse diagrams which say that semiorders exclude the configurations 2+2 and 3+1). That is why, semiorders can be considered as providing a mathematical theory of qualitative structures, just as other well-known relations and in particular simple orders (or transitive antisymmetric orders, i.e. transitive order relations with the further property that if aRb and bRa, then a=b), presupposing the notion of equality (transitive indifference) are viewed as expressing the mathematical properties of numerically interpretable quantitative structures.
§4. All that was said above suggests that if we understand the difference between quantity and quality in the traditional Aristotelian way, then presently it is completely unjustifiable to claim that there is no mathematical study of qualitative structures since we do have a nontrivial theory of intransitive indifference. Now we come to the second dogma of Aristotelianism which concerns the concept of preference. It is not at all coincidental that this particular field of philosophical logic was also conceived in the late 50-ies and early 60-ies, i.e. just after the introduction of the concept of semiorder. Preference is generally viewed as “subjective betterness” (Hansson 1989, 2), i.e. as a quality pertaining to states of affairs, which can be viewed as favorable or adverse with respect to subject's desires, needs and ends. That the historical order is not at all coincidental is proved by the fact that two of the originators of the field – Halldén and von Wright, clearly took part in the same project that already gave birth to semiorders. Indeed, Halldén explicitly alluded to the paper of Luce and Suppes which presents the classical definition of semiorder (Halldén 1966, 320), while von Wright not only used the typical language of order theory (discriminating between the asymmetric part of a relation P and its symmetric complement I) but built his whole approach to preference logic on a principle (the so-called principle of value-comparability) which is closely connected with axiom (3) above (von Wright 1972, 151-153). What is even more important, the semiorder axioms provide an obvious starting point in the formal study of preference, since if we interpret “aPb” as “a is preferable to b” (the reason why the letter “P” was used in the first place), then the axioms above become sound principles from which a rudimentary theory of preference can be easily developed. Therefore, the second dogma is just as easily refutable as the third.
§5. Does this mean that our desires are rational? What should “rational” mean here? According to my preferred rendering of this term, S is rational (w.r.t. his preferences) iff S’s preferences are susceptible to change in a coherent and regular manner in parallel with the influx of S’s experience. In other words, S is rational iff her preferences “evolve in the manner dictated by experience” (Gupta 2006, 155). Do our desires evolve in the prescribed way over time? Do we really want what proves to be the best option for us in the long run? This is obviously not a philosophical question which can be answered on the basis of a priori considerations. On the other hand, what can be rigorously proved is this: if (a) our desires are minimally rational in the sense that they conform to the semiorder axioms and (b) our experience is stable enough, then nothing prevents our preferences from evolving in the way just described. This is an easy consequence of a remarkable theorem which shows that semiorders are well-graded relations, i.e. “[a]ny semiorder on a finite set can be reached from any other semiorder on the same set by elementary steps consisting either in the addition or in the removal of a single ordered pair, in such a way that only semiorders are generated at every step, and also that the number of steps equals the distance between the two semiorders” (Doignon and Falmagne 1997, 35). In other words, any configuration of minimally rational preferences can be reached from any other by a stepwise process of revision which shows that even if ab initio our desires are unreasonable (i.e. not conductive for our overall aims and principal values), nothing precludes their constant rationalization, i.e. their stable approximation to a semiorder of preferences which is uniquely determined by the feedback loop of experience. Rationality is not some transcendent faculty of the soul, it constantly reassures its existence in a manifest way. As far as rationality is concerned, being rational and looking rational are not different things.
Aristotle 1925. Ethica Nicomachea. In: Ross, W. D. (ed.) The Collected Works of Aristotle, vol. IX. Oxford: Clarendon Press.
Aristotle. 1928a. Analytica Priora. In: Ross, W. D. (ed.) The Collected Works of Aristotle, vol. I. Oxford: Clarendon Press.
Aristotle. 1928b. Categories. In: Ross, W. D. (ed.) The Collected Works of Aristotle, vol. I. Oxford: Clarendon Press.
Aristotle. 1928c. Metaphysica. In: Ross, W. D. (ed.) The Collected Works of Aristotle, vol. VIII. Oxford: Clarendon Press.
Aristotle. 1928d. Topica and De Sophisticis Ellenchus. In: Ross, W. D. (ed.) The Collected Works of Aristotle, vol. I. Oxford: Clarendon Press.
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Franklin, J. 2011. Aristotelianism in the Philosophy of Mathematics. Studia Neoaristotelica 8(1), 3-14.
Gupta, A. 2006. Empiricism and Experience. New York: Oxford University Press.
Halldén, S. 1966. Preference logic and theory choice. Synthese 16, 307-320.
Hansson, S. O. 1989. A New Semantical Approach to the Logic of Preference. Erkenntnis 31(1), 1-42.
Luce, R. D. 1956. Semiorders and a Theory of Utility Discrimination. Econometrica 24(2), 178-191.
Luce, R. D. and P. Suppes. 1965. Preference, Utility, and Subjective Probability. In: Luce, R. D. et al. (eds.) Handbook of Mathematical Psychology, vol. III, pp. 252-410. New York: Wiley.
Pirlot, M. and Ph. Vincke. 1997. Semiorders: Properties, Representations, Applications. Dordrecht: Kluwer Academic Publishers.
Roberts, F. S. 1985. Measurement Theory. Cambridge: Cambridge University Press.
Tversky, A. 1977. Features of similarity. Psychological Review 84, 327-352.
Von Wright, G. H. 1972. The Logic of Preference Reconsidered. Theory and Decision 3, 140-169.
 For the original definition of the term, cf. the seminal paper “Semiorders and a Theory of Utility Discrimination” (Luce 1956, 181).
 As a matter of fact, the mathematics of quality is still in the embryonic phase of its development since the formal properties of the concept of similarity are not unpacked fully into axioms (cf. Tversky 1977).