Exotic Computational Principles

Exotic computational principles are those not derivable from the theory of proportions constructed in Chapter 2. The most important of these principles are (PFREQ) and (AGREE). The purpose of this chapter is to show that these principles can be derived from a strengthened theory of proportions—what we might call ‘the exotic theory of proportions’. The principles of the theory of proportions constructed in Chapter 2 seem completely unproblematic, and accordingly the derivations of principles of nomic probability can reasonably be regarded as proofs of those principles. That ceases to be the case when we turn to the exotic theory of proportions and the corresponding exotic principles of nomic probability. Although quite intuitive, the exotic axioms for proportions are also very strong and correspondingly riskier. Furthermore, although the exotic axioms are intuitive, intuitions become suspect at this level. The problem is that there are a large number of intuitive candidates for exotic axioms, and although each is intuitive by itself, they are jointly inconsistent. This will be illustrated below. It means that we cannot have unqualified trust in our intuitions. In light of this, (PFREQ) and (AGREE) seem more certain than the exotic principles of proportions from which they can be derived. As such, those derivations cannot reasonably be regarded as justifications for the probability principles. Instead they are best viewed as explanations for why the probability principles are true given the characterization of nomic probabilities in terms of proportions. The derivations play an explanatory role rather than a justificatory role. The exotic principles of proportions concern proportions in relational sets. Recall that we can compare sizes of sets with the relation ┌X ⇆Y┐, which was defined as ┌מ(X/X∪Y) = מ(Y/X∪Y) ┐. Our first exotic principle relates the size of a binary relation (a “two-dimensional set”) to the sizes of its one-dimensional segments. If x is in the domain D(R) of R, let Rx be the Rprojection of x, i.e., {y| Rxy}. Suppose D(R) = D(S), and for each x in their domain, Rx ⇆ Sx. Then their “linear dimensions” are everywhere the same.