The problem with moral realism is widely supposed to be its ontological extravagance. So suppose we begin with maximal ontological parsimony: say, with a metaphysics like the one expressed in Democritus’ ‘nothing but atoms and the void’.1 Or even, since Democritus apparently allowed there to be infinitely many atoms, suppose we begin with something more minimal still—the hyper-Democritean thesis that nothing exists except the void, and in it, the following sixteen-dot matrix:
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What ontology could be more minimal (except, presumably, one with fewer dots)? But even this hyper-Democritean ontological minimalism sustains an indefinite multiplicity and complexity of patterns. If these sixteen dots exist thus arranged, then it is immediately not the case that only these sixteen dots exist. There also exists every pattern that these sixteen dots constitute.
And goes on
There is no limit to the patternsthat we can find in the diagram—beyond those set by our ingenuity, imagination, or mathematical/geometrical aptitude.... [a lot of essay later]... It is evident from my discussion of matrices like the sixteen-dot one of section 1 that I admit the reality of all sorts of patterns. This, together with the claim that properties including moral properties are patterns, gets me the result that moral properties are no less real than all the other patterns that we might perceive in reality. Unfortunately it also gets me the result that indefinitely many other sorts of other patterns are there in reality too.
My basic idea is that the set of all possible patterns in a situation is in fact equal to the power set of the set of all the elements (factors to be taken into consideration) in that situation. Let's return to the 16 dot matrix used in the quote above to make the point clear; each dot can be named a - p, with the top row being a, b, c, d and the first column being a, e, i, m and the total set {a...p}=X. We can now represent any organisational schema using some subset of P(X). For instance, if one saw the matrix as 4 groups of 4 dots each, that could be represented with the set Y = {{a,b, e, f}, {c, d, g, h}, {i, j, m, n}, {k, l, o, p}}, or if one saw a cross brought to the foreground with the corners merely back grounding it one could use the set Z={b, c, e, f, g, h, i, j, k, l, n, o}. Note that Y ⊂P(X) and Z ⊂ P(X). This should generalise to any other pattern one cares to think of in the matrix.
The primary advantage of such a set theoretic analysis is that it allows us to dispute the grim conclusion that Prof. Chappell comes to; namely we have a fairly definite proof that there are not an infinite amount of possible manners of regarding a situation whatever the skeptic may say. For P(X) will be of cardinality 2^16, and even if that is a very large number it is certainly a lot less than infinite! (It occurs to me that, in fact, it might be P[P(X)] which I am interested in, hence giving me a much greater cardinality of possibilities - anybody got any thoughts on this?) The practical importance of this is seen when one considers what it would have been like if the relativist had an infinite amount of patterns to cast upon a situation; it would mean that no matter how many times the absolutist demonstated a particular way of seeing the world a bad way to do so, there would be an infinite amount remaining! If, on the other hand, the possibilities beside the morally upright (or, set of morally upright possibilities for regarding a situation - just because the path of virtue is straight and narrow, we needn't assume it is single file!) are finite then we are at least fighting an - in principle - winnable war against the relativist. And to know this will be good for the troops morale!
