Sets, relations, functions

When we start doing serious logic, we very quickly find ourselves wanting to talk about sets. ‘Domains of quantification’ are sets, the ‘extensions of predicates’ are subsets of the domain of quantification. And so forth. But note that this does not immediately entangle us with the heavy duty notion of set that it is the business of hard-core mathematical theories like Zermelo-Fraenkel Set Theory to regiment and explore. Let me explain. A mathematical theory like ZF concerns the hierarchical universe you get if you take some things – or maybe just the empty set – and form all the possible sets of those. And then form all the possible sets whose members are drawn from the thing(s) you started off with and the sets you’ve just built; and then form all the possible sets whose members are drawn from what you’ve constructed so far; and then go up another level and form every possible collection out of what you have available now. Keep on going for ever . . . ; and when you have done all that, put together all the members from every set you have constructed, and now start set-building again, transfinitely. Keep on going again . . . . And that’s just the beginning! Evidently, we are going to end up with a vast hierarchical universe of sets, a universe rich enough to model more or less any kind of mathematical structure we might dream up – which is why we can in effect do more or less any mathematics inside ZF – or more accurately, within ZFC, which is ZF plus the Axiom of Choice – the canonical theory of this wildly proliferating universe. And equally evidently, when we interpret some version of QL, and talk about its ‘domain of quantification’ and the ‘extensions of predicates’ and so on we are – in the general case – not concerned with wild universes at all. For example, the domain of quantification might be, very tamely, the set of people. The extension of the one-place predicate ‘W’ might be just the set of wise people (a subset of the domain). The extension of the two-place predicate ‘L’ might be just the set of pairs of people such that the first loves the second (a subset of the totality of arbitrary pairs from the domain) And so it goes. So all we need to worry about for interpreting a first-order language is normally some humdrum set of things (for the domain, e.g. the set of people, the set of natural numbers), subsets of domains (for extensions of monadic predicates), sets of pairs from the domain, and so on. There is no hierarchical building up from the initial domain to ever bigger sets; rather there is a cutting down of the domain to its subsets, or at any rate to sets of pairs culled from the domain, and so on. OK, things get more complicated when we turn from thinking about this or that interpretation, and start generalizing over all possible interpretations – as when we say a quantificational argument is valid if there is no interpretation which makes the premisses