Is the empty set false?
In "Two Concepts of Modality", Alvin Plantinga argues that propositions aren't sets of worlds, because "you can't believe a set, and a set can't be either true or false" [208]. I think this argument is better than it might appear in the rather Ungerian context of Plantinga's paper, where he uses several arguments of the same kind to support completely crazy views, like that Lewis is an antirealist about possible worlds.
The traditional job description for propositions says that they are a) the ultimate bearers of truth-values, b) the content/object of propositional attitudes, and c) the meanings of declarative sentences. Plantinga is right that sets aren't the most intuitive candidates for this job: Is the empty set an 'ultimate bearer' of the truth-value false? Is it the content of Frege's belief in Axiom 5? Is it what you have to know in order to understand Axiom 5? Well, intuitively not, but I don't think intuition is to judge questions like these. More importantly, there are reasons against the identification of sets with propositions.
For example, as so often in set theory, there are too many candidates. Instead of sets of worlds we can equally well take propositions to be, say, functions from worlds to truth values. I'm inclined to believe that this doesn't make any difference. But why not? Why doesn't it make a difference whether the empty set has a truth value? Does it or doesn't it?
For another example, the content of propositional attitudes seems to be causally efficacious, but can a set or function be so? I think not. Sure, many philosophers think that events are sets or tuples of some sort or other, and certainly events have causes and effects. But does anyone really mean this literally? Take Lewis. He once identified events with sets of regions of spacetime. He also thought, at least at one time, that sets are genuine entities that exist outside any possible world. Did he really believe that entities that don't exist at our world have causal effects in our world? No. Strictly speaking, it's not the events that are causally efficacious, but the occurrence of the events. An event occurrs at a region of a world iff that region is a member of the event.
Fine, so what is the occurrence? Is it the membership relation between region and event? Or the fact that the region is a member? Or the region-qua-member? And whatever it is, why not identify this thing with the event, so that events themselves have effects, just as the folk says?
(In fact, I think Lewis has changed his views on events recently and came to identify them with regions-qua-members, or, actually, occupants-of-regions-qua-members. I have no direct evidence for this, but the discussion in "Void and Object" makes much more sense on this reading. Any clarification from insiders are highly welcome.)
A reasonable answer might be that we don't really need occurrences, that is, we don't need them as entities. We would need them if causation was an ordinary relation. But maybe Lewis is right and it isn't. Maybe 'cause' is not a relational expression but a binary sentential operator, so that "the rain caused the fire" really means something like "the rain occurred, and that caused: a fire occurred", which in turn could be given a counterfactual or regularity analysis. So strictly speaking, there are no events, and there is no relation of causation. But this isn't as outrageous as it sounds, since all our ordinary talk about events and causation remains true.
Uhm, right, I wanted to say something about propositions. The situation is similar here, I think. As it turns out, we don't have much use of propositions as entities. We need an analysis of "proposition p is true", "Alvin believes proposition p", and "S1 and S2 both express proposition p", but again a contextual analysis will suffice: Let s be the set of worlds in which p. Then "proposition p is true" comes out as "the actual world is an Element of s"; "Alvin believes proposition p" as "s is a subset of Alvin's doxastic possibilities"; etc. We don't need to identify s with p. If we don't, we avoid Plantinga's puzzles. For example, we needn't claim that the empty set is false, or that Frege believed the empty set until Russell showed him the error of his ways. Plantinga is right: Strictly speaking, there are no propositions. But this isn't as outrageous as it sounds, since all our ordinary talk about propositions remains true. The only legitimate worry is that our semantics of sentences about propositions might be slightly counter-intuitive.