I've uploaded another revision of the emperors paper. The best thing about this version is that it's four pages shorter than the previous ones.
Unrelatedly, I've removed the Herbrand restriction from my tree prover. The restriction says that a Gamma node should not be expanded more often than there are closed terms on the branch. But currently, the prover doesn't keep track of the number of closed terms, it only keeps track of the number of function symbols (including 0-place function symbols, i.e. individual constants). So if a lot of s(0), s(s(0)), etc. are on a branch (as in this proof, where I noticed the bug), the prover wrongly applied the Herbrand restriction, thinking all of them are only two closed terms.
All my testcases work just as well without the Herbrand restriction. If you find a case where the performance got worse, please let me know. It shouldn't be difficult to fix it (rather than removing it completely).
Suppose there are at least proper-class many possibilia. Does it follow that some fusions of possibilia are not members of any set? For the last two years or so I thought it does. My reasoning was that if some of the possibilia correspond one-one with all the sets, then some atoms of possibilia also correspond one-one with all the sets (for there cannot be proper-class many fusions of set-many atoms); but since there are always more fusions of atoms than atoms, it follows that there must be more fusions of atoms of possibilia than sets, and hence that some (in fact, most) of these fusions lack a singleton. This does not take into account atomless possibilia, but I always thought the reasoning would easily carry over, by something like the fact that even with gunk
Well, I know what Counterpart Theory is not: it is not a theory according to which ordinary things do not really exist at other possible worlds.
There are two readings of "ordinary things do not exist at other worlds". The first is a neutral reading on which things exist at another world in the way they sleep at another world or win elections at another world: whatever possible worlds are, they somehow represent things as existing and sleeping and winning. In this sense, something exists at a world iff the world represents it as existing. Anyone who accepts possible worlds talk at all accepts that ordinary things exist at other worlds in this sense.
My fellow Germans have donated very generously to the tsunami relief effort. That's good. But it's remarkable that we have donated so much to this cause, and far less to other good causes. 100 Euros given to the tsunami victims could also have been spent, say, to help the refugees in Darfur, or to support the reconstruction of war-torn Uganda or Sierra Leone, to provide medical care for people in Ethiopia or Bangladesh, to prevent deforestation, overfishing and soil erosion, to fight climate change, and so on. Donations are urgently needed all the time.
David Chalmers has an interesting post on the differences between his and Frank Jackson's versions of two-dimensionalism. It turns out that my reading of a certain passage in "Why we need A-intensions" was right: Jackson believes that truth at a world considered as actual is somehow reducible via de-rigidification to truth at a world considered as counterfactual.
The difference between linguistic ersatzism, where possible worlds are replaced by sets of sentences, and modal fictionalism, where the pluriverse of all worlds is replaced by a large set of sentences describing all worlds at once, appears to be small. Nevertheless, I (still) think the analytic power of fictionalism is greatly diminished compared to that of linguistic ersatzism.
One of the great advantages of possibilia is that they provide a unified framework to reduce lots of kinds of things: properties can be identified with sets of possibilia, propositions with sets of worlds, meanings with functions from worlds to extensions, events with functions from worlds to regions, and so on. But suppose possibilia don't really exist, but exist only according to some fiction. Then properties can't be sets of possibilia. By the usual rule of interpreting statements about fictional entities, it will at most be true that according to the fiction, properties are sets of possibilia. But that doesn't help us if we're looking for a unified ontology. We'd like to know what properties really are, not what they are according to some fiction. If as fictionalists we think that properties really are sets of possibilia, then we have to conclude that properties don't really exist, just as the (other-worldly) possibilia don't really exist.
According to the Principle of Recombination,
for any things at any worlds there is a world containing a duplicate
of each of these things and nothing else (that is, nothing that is not
a part of the fusion of the duplicates).
Applied to the mereological fusion of David Hume and David Lewis, this says that there is a world containing nothing but a duplicate of the fusion of Hume and Lewis. This duplicate presumably has a part that is a duplicate of Hume and another that is a duplicate of Lewis. How are these parts spatiotemporally related?
Brian points to Gabriel Uzquiano's Cardinality Puzzle about Mereology and Set Theory (PDF), which he (Gabriel) introduced a while ago in the now-deceased Philosophy from the 617 weblog. I still don't know enough set theory and mereology to competently discuss the matter, but anyway, it seems to me that perhaps the puzzle can be strengthened, as follows.
The whole four-dimensional universe, including past, present and future times, does not change; it will not be different tomorrow; it remains the same at all times.
If the whole four-dimensional universe remains the same at all times, then presumably no part of it will ever fail to exist or has ever failed to exist.
So for example, the apple I'm just about to eat will never fail to exist. It will exist forevermore. As will I, and you, and this weblog.
There is but one totality of worlds; it is not a world; it could not have been different. (Lewis, Plurality, p.80)
If the totality of worlds could not have been different, then presumably no possible world could have failed to exist.
Then in particular, the actual world, @, could not have failed to exist.
So there is an actually existing thing, namely @, that could not have failed to exist.
Even worse, arguably @ has some of its parts essentially. So there are some actually existing things besides @ that could not have failed to exist.
One might even say that all worlds have all their parts essentially, simply because worlds do not exist at other worlds. Then it follows that no actually existing thing could have failed to exist.