Existence is not only a profession of ontological commitment. It is also, as we saw in Chapter Seven,1 a counter in argumentative discourse. To exist is to be talkable about. It licenses discussion by warding off the conversation stopper “But there is no such thing”.2 Its force, therefore, often depends on context, and the objection anticipated. The extreme case-e.g. the greatest prime number-is where the referring expression is a contradiction in terms. Against such an objection, bare consistency is enough. In first-order logic the Completeness Theorem establishes that any consistent set of
well-formed formulae has a model, and consistency proofs for non-Euclidean geometry, non-Desarguian geometry and non-standard models of Peano’s axioms are all we need for us to be able to talk about them with a clear conscience. But it is only in few cases that bare consistency suffices.3 Non-standard models of Peano’s axioms do not rate high ontologically, lacking even w-consistency. Fictions, too, can be talked about, and shadows, but though discourse about them is allowed to be intelligible, it is not taken seriouslyit is not for real.