I am reading Horwich's little book on truth. Horwich's minimalism is generated by one axiom:
- Only propositions are true,
- <p> is true if and only if p.
Two objections come to mind (the first may be answered later on in the book—I haven't finished the book). First, what if some other worlds contain propositions that our world does not contain? We should be able to say that in such a world it, say, still the case that every proposition which is true is true, and so on. But minimalism as it stands does not seem to let us say that. One could make the axioms vary between worlds, but then it would be unclear that we are talking about the same thing, truth, in all of them.
The second is it's going to be pretty tricky to restrict the range of propositions p it's permissible to substitute in (2) in order to avoid liar paradoxes. The big problem will be with contingent liar paradoxes, like the proposition p that Plantinga's favorite proposition is not true, which proposition is unproblematic, unless it contingently happens to be Plantinga's favorite. It seems that if one is going to handle liar paradoxes by restricting the range of instance of instances of (2), one will have a different axiom schema in different worlds.
Of course, contingent liars will be a problem for everybody, and it may be that the minimalist may be able to adopt a different solution. I rather like approaches on which paradoxical "sentences" (contrary to the mainstream, I don't think they actually are sentences) don't express propositions. Let's see if that works for the minimalist. I don't know what actually is Plantinga's favorite proposition. Maybe it's the proposition that if Christianity is true, then Christian belief is justified. Now, let p be the proposition that Plantinga's favorite proposition is not true. Then, p is not actually Plantinga's favorite proposition. Moreover, p cannot be Plantinga's favorite proposition. For if p were Plantinga's favorite proposition, it would be true if and only if not true. But that is a very strange result. After all, couldn't p be inscribed somewhere, and couldn't Plantinga somehow form an odd liking for "the proposition expressed by the words inscribed there", even without reading these words?
One approach one could take here is this: Deny that p exists in those worlds in which it is Plantinga's favorite. But that is not available to the minimalist, because the minimalist appears to be committed to propositions being necessary beings. So not every solution is available to the minimalist. But I am not sure this is the best solution anyway. It might be better simply to deny the possibility of p becoming Plantinga's favorite proposition. This would have the consequence that in any world such that on page 17 of some book, on line three, it is written "Plantinga's favorite proposition is false", then that "sentence" fails to express a proposition in any world in which Plantinga's favorite proposition is "the proposition written on page 17, on line three, of that book" (note: a favorite proposition can be favorite under a description; it need not be grasped to be favorite). And this solution is available to the minimalist.
I rather like the following view, by the way: Minimalism is basically true as a theory of the truth of propositions; but an inflationary view of the meaning, and hence truth, of sentences is also correct.