At the Central APA, I attended a presentation by Greg Fowler and Chris Tillman in which they showed the inconsistency of the following claims relating to the mereological sum of absolutely everything, call it ‘U’, and the proposition that U exists:
1) U is part of the proposition that U exists.
2) The proposition that U exists is part of U.
3) U is not identical to the proposition that U exists.
4) Parthood is anti-symmetric (i.e. if x is part of y, and y is part of x, x=y).
Their discussion was framed (roughly) as an argument against (4), on the basis of (1)-(3), but I think it is more useful to think of it as an inconsistent tetrad.
If we grant that there is such a thing as the mereological sum of absolutely everything, and we grant the existence of propositions, then (2) would be hard to deny. If everything is a part of U, and there is a proposition that U exists, it is part of U. So, the likely culprits are (1), (3) and (4). But, to me at least, (4) seems to be on better footing than the assumption that there is a mereological sum of absolutely everything, so I’m unlikely to give that up to resolve the tension.
As to (3), I find the following to be a reasonably compelling argument against giving it up: U is not truth evaluable, but the proposition that U exists is truth evaluable, so they are not identical. That said, I think fleshing out a denial of (3) would be among the more interesting responses to the puzzle.
At any rate, I am left with a rejection of (2). Now, Chris and Greg argued that giving this up would cause trouble for explaining the structure of structured propositions, but they only considered denying (2) by denying that structured propositions have any (proper) parts whatsoever. This way of denying (2) is pretty strong, since (2) only asserts that one particular thing is a part of the proposition that U exists. In other words, Chris and Greg argued (compellingly) that parthood is needed in the analysis of constituency, but used that as a basis for concluding that the constituents of a proposition are parts of that proposition.
Here is my flippant argument that constituency neither is nor requires parthood (i.e. that being a constituent of something does not entail being a part of it):
DD1) As a resident of Illinois, I am one of Dick Durbin’s constituents.
DD2) I am not one of Dick Durbin’s parts.
DDC) So, constituency neither is nor requires parthood.
And here is my almost-as-flippant explanation of why this notion of constituency is relevant to our discussion of propositions:
It is in virtue of being one of Dick Durbin’s constituents that I am represented by him in the senate. So, the Dick Durbin argument shows that one can explain why something represents its constituents without the constituents being parts of the thing doing the representing.
So, I’m inclined to think that propositional constituents are represented by the parts of propositions, but need not themselves be parts of propositions.