Wednesday, July 18, 2007

Parity of Value: a formal model

Over at Ethics Etc, S. Matthew Liao presents Ruth Chang's argument for a fourth relation of comparative value - 'on a par' - to supplement the standard 'better than', 'worse than', and 'equal to' relations:
1. Mozart is neither better nor worse than Michelangelo, with respect to Creative Genius (CG).
2. A Mozart who has some small improvement that bears on CG (Mozart+) is better than Mozart, with respect to CG.
3. Mozart+ is not better than Michelangelo, with respect to CG.
4. Therefore, Mozart and Michelangelo are not related by any of the standard trichotomy of relations, with respect to CG. (This is the Small Improvement Argument)

5. Mozart is better than Talentlessi, a very bad music composer, with respect to CG.
6. Michelangelo is also better than Talentlessi, with respect to CG.
7. Therefore, Mozart and Michelangelo are comparable, with respect to CG. (This is the Chaining Argument)

8. Given 4 and 7, there must be a fourth comparative relation; Chang calls it the parity relation. (This is the Parity Conclusion)

The premises all seem intuitively plausible, yet it may be initially puzzling how they could all be true -- at least if we conceive of value as a point on a scale.

The idea seems to be that CG is some kind of holistic value, constructed from a composite of various partly-commensurate dimensions (e.g. music and art). That could explain why Mozart+ beats Mozart (the slight increase is on the same dimension, so Mozart+ strictly dominates Mozart, being better in some ways and worse in none) yet Mozart+ does not beat Michelangelo, being better in some ways but worse in others, with the tradeoff being, in some sense, "too close to call". But note also that the two dimensions are at least comparable at the extremes: Michelangelo's artistic genius outweighs Talentlessi's sorry musical skills. In sum: Mozart and Michelangelo both excel along different dimensions of Creative Genius, which places them 'on a par' in such a way as that a minor improvement to either would not affect their relative standing.

It's an intuitive enough picture, but is it theoretically consistent? Liao expressed doubts. But I think we can construct a formal model which exhibits all the theoretical properties Chang needs here, i.e. showing the premises (1, 2, 3, 5, 6) to be mutually consistent. Here is my model:

(A) Let 'x' and 'y' denote two dimensions of Creative Genius, and let Proto-CG be composite value combining x and y but with some vagueness as to their relative weightings.

Assign base values:
* Mozart = 100x + 0y
* Mozart+ = 101x + 0y
* Michelangelo = 0x + 100y
* Talentlessi = 1x + 0y

Hence, the following facts hold concerning ordering relations with respect to the proto-CG scale:
1-p. It is not determinate whether Mozart is either better or worse than Michelangelo.
2-p. Mozart+ is determinately greater than Mozart.
3-p. It is not determinate that Mozart+ is better than Michelangelo.
5-p. Mozart is determinately greater than Talentlessi.
6-p. Michelangelo is also determinately greater than Talentlessi.

Liao raised an important objection to my model at this point:
If it is vague as to whether Mozart is better or worse than Michaelangelo or equally good, then, it is not true that Mozart is neither better nor worse than Michaelangelo or equally good.

Granting this point, it is important for me to emphasize that the #-p facts hold merely with respect to proto-CG, and do not yet speak to the ultimate CG relations which we are interested in. What we need is some schema to translate these vague Proto-CG relations into the determinate CG relations stated in the original premises. That is the role of the second part of my model.

(B) We may now construct CG orderings from Proto-CG orderings as follows:

For the standard trichotomy of positive ordering relations ('better than', 'worse than', and 'equal to'), let us say that the relation holds with respect to CG iff it is determinate that the relation holds with respect to proto-CG. (I'll call this the "axiom of determination" unless anyone can think of a spiffier name.)

This axiom establishes entailment relations from each #-p to the corresponding original premise #. For example, from the fact (1-p) that it is not determinate whether Mozart is either better or worse than Michelangelo with respect to proto-CG, we can infer from the axiom of determination that (1) Mozart is neither better nor worse than Michelangelo, with respect to CG.

Closing Remarks: My formalization raises some intriguing questions of philosophical methodology. E.g. what philosophical interest can such a formal model have? What does this style of argument really show? It's not as though the process I've described is meant to literally reflect the fundamental metaphysics of values. It's merely a model. (In particular, it seems implausible that my internal 'proto-CG' variable corresponds to any significant value in reality! I employ it as a purely technical 'fix', to get my model to yield the right outputs.)

But I think it has philosophical worth in the following respect: it establishes that Chang's premises about value are mutually consistent. This model shows one possible way that they could all be true. Perhaps reality provides another. But at least we can dispel our initial skepticism about whether they were consistent at all.

1 comment:

  1. Richard,

    You need not grant Liao his point. Many believe that some sort of supervaluationism plus indeterminacy is correct, and therefore there simply is no fact about what relation Mozart and Mich bear in terms of CG. Metaphysics is silent on the matter. It isn't one of the three; it is none of the above.

    Also, I find (1) very questionable. It is far more plausible if it read: Mozart is neither clearly more CG nor clearly less CG than Mich. To assume that he is neither more nor less is a very strong assumption--for some equivalent to assuming that he is exactly equal in CG.


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