Truthmakers for Negative Existentials

Some assumptions:

- Atomic singular sentences of the form a is F encode atomic Russellian propositions.
- Atomic sentences that contain non-referring names encode atomic gappy propositions.
- Atomic sentences that contain a name for a property that does not exist encode atomic gappy propositions.
- Atomic sentences that include 'exists' or cognates encode the first-order property of existence.
- Non-singular existence statements encode a second-order property of existence.
- All atomic gappy propositions are false. All of their negations are true.

That a proposition is true or false is not a fundamental fact. Truthmaker theorists want to capture this nonfundamentality by holding that for every true proposition there is some state of affairs that makes it true. (I'm ignoring the trope option here. I'm also joining truthmakerists in ignoring the issue of falsemakers. After all, that something is false is no more fundamental than that something is true. But falsemakers never seem to come up. Maybe the assumption is that we could do falsemakers in terms of truthmakers if we could just nail truthmakers. I'm not sure this assumption is true, but I'll bracket it here.) True negative existentials are a major bugaboo for truthmaker theorists. Two main constraints on truthmakers is that they must necessitate the relevant truth and the truths must be about them. Another is that they are supposed to not be "suspicious". (This is related to the power of truthmakers to "catch cheaters".)

It is widely held that the best shot for being a truthmaker for a singular or non-singular true negative existential is something like the entire world plus the fact that there's nothing more. Opponents of truthmaker complain that this sort of truthmaker does bad on all counts. I wonder if there is not a better one to be had. (I don't know the truthmaker literature very well though.)

Here's the idea: the gappy proposition < __, smokes > is false and it represents nothing at all as smoking. This is to say that it represents the gappy state of affairs [ __, smoking ] as obtaining. Paradoxes aside, suppose a liberal account of states of affairs, including gappy ones. To be true is to represent a state of affairs that is among the states of affairs that obtain. To be false is to fail to do that.

So < NEG, < __, exists > > is true because [ __, exists ] is not among the states of affairs that obtain.

< NEG, < hobbits, exist > > is true because nothing has the property of being a hobbit. That is to say, [ __, is a hobbit ] is not among the states of affairs that obtain, and neither is any state of affairs [ o, is a hobbit ] for any o that exists.

Following Kripke, suppose 'is a unicorn' does not express a property. Then 'Unicorns don't exist' encodes something like < NEG, < __, exist > > where the existence property is second-order. This is true for reasons parallel to the first-order case.

This view seems to beat others in terms of necessitation and aboutness, though perhaps gappy states of affairs are suspicious. I don't know.

Perhaps the fact that the gappy states of affairs are not among those that obtain is not itself a fundamental fact. Maybe it holds in virtue of the totality of things plus the fact that there is nothing more. Maybe that eases suspicion while maintaining aboutness. But it does not merely collapse into the usual account. If I say that Joshua's life is longer than JonBenet Ramsey's, I am talking about their lives. But that is consistent with their lives being nonfundamental. So I think this is okay.

Perhaps one could object to the intrinsic weirdness of gappy states of affairs. But if one accepts gappy propositions, one should not have this problem with gappy states of affairs. Russellian propositions are so very states-of-affairs-like that I can't think of a principled reason for accepting one and not the other. The link is even tighter if we accept the identity theory of truth (I don't): the true propositions just are the states of affairs that obtain. Then gappy propositionists get gappy states of affairs "for free".

What's not to like?

On the proposition that there are a plurality of worlds

David Lewis thinks that there are many concrete possible worlds (at least one for each de dicto possibility) and propositions are sets of those worlds. I disagree. I will present two dilemmas against this view.

It seems clear that the following is true:

1. Possibly, there are talking donkeys.

On Lewis's view, this means that there is a world and there are talking donkeys in that world. But, this implies that:

2. There are (quantifiers wide open) talking donkeys.

This is a proposition. So, according to Lewis, it must be a set of possible worlds. But which set is it? As I see it, there are only two options. If propositions are sets of worlds, then either (2) is the set of worlds that contain (within their worldly boundaries) talking donkeys, or (2) is the set of all worlds.

Consider the first option: that (2) is the set of worlds that contain (within their worldly boundaries) talking donkeys. This option can't be right. After all, (2) is true. Moreover, for any proposition p, if p is true and propositions are sets of worlds, then the actual world is a member of p. The actual world, though, is not a member of the set of worlds that contain talking donkeys. So, if propositions are sets of worlds, then (2) is not the set of worlds that contain talking donkeys (within their worldly boundaries).

The only other plausible option is that (2) is the set of all worlds. This seems plausible since (given that the quantifiers in (2) are wide open) it is true at any world. But, then (2) is the same as the proposition that arithmetic is incomplete. But, that is absurd. So, (2) is not the set of all worlds.

Even if you deny the absurdity of identifying (2) with the proposition that arithmetic is incomplete, there are still problems for the second option. Consider the following proposition:

3. There are (quantifiers wide open) cats.

Since the quantifiers in (3) are wide open, just like those in (2), then it seems we have no reason to treat them differently. No reason to say, for example, that (2) is a set of worlds that contain cats whereas (3) is the set of all worlds. So, we should treat them the same. If (2) is the set of all worlds, (as the second option indicates), then it must be that (3) is the set of all worlds too. So, if (2) is the set of all worlds, then (2) is identical to (3). But, clearly those are different propositions. I believe (3) yet I think that (2) is false.

To recap, if propositions are sets of worlds, then (3) is either the set of all worlds that contain talking donkeys or the set of absolutely all worlds. But, it is not the set of all worlds that contain talking donkeys and it is not the set of absolutely all worlds. So, propositions are not sets of worlds.

That's the end of the first dilemma against the Lewisian view. Here, then, is the second dilemma. Consider Lewis's modal realist thesis itself:

PW: There are (quantifiers wide open) many concrete possible worlds.

If (PW) is true and propositions are sets of worlds, then either (PW) is the set of worlds that contain (within their worldly boundaries) many concrete possible worlds or it is the set of all worlds. If (PW) is a set of worlds that contain (within their worldly boundaries) many concrete possible worlds, then it is the empty set (since no world contains many concrete possible worlds). But, if it is the empty set, then it is false (since on the sets-of-worlds view of propositions, a proposition is true iff the actual world is a member of the set that is that proposition). So, if (PW) is true, (PW) is not the set of possible worlds that contain many concrete possible worlds.

Perhaps (PW) is the set of all worlds. But, if that is the case, then (PW) is identical to the proposition that 2+2=4. But, it if it is identical to the proposition that 2+2=4, then I believe (PW) (since I believe that 2+2=4). But, I don't believe (PW). So, (PW) is not a set of all worlds.

It follows that either (PW) is not true or propositions are not sets of worlds. If propositions are sets of worlds, then (PW) is false. But, since the view that propositions are sets of worlds is true only if (PW) is true (the view makes sense only if (PW) is true), it follows that propositions are not sets of worlds.