Wednesday, February 24, 2010

Constituency neither is nor requires Parthood

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.


Anonymous said...

I suppose someone might respond that the point of saying that objects are parts of propositions is that it's then clear how the propositions represent. I.e. the proposition expressed by 'Aristotle is bald' contains Aristotle, which is why it's about him/represents him. That probably motivates some direct reference theorists.

Now you are building representation into your account of what propositions are (structured objects made up of things that represent objects and properties, or whatever). That looks like it costs you the representation explanation, if you liked it to begin with.

Lewis Powell said...

I'm not entirely sure why lagadonian representation gets a free pass. There are many things I am part of that do not represent me. My hand is part of my body, but it is not clear that my body represents my hand.

I take it that direct reference theories are concerned with whether or not the object in question is a genuine constituent of the proposition, which, on my view, they still can be.

One an still hope to explain representation. All I've done is ensure that propositions succeed in representing. One way to think about it is this: a suitably general statement of the approach I advocated in the post is compatible with the lagadonian representational scheme you suggest, since the one you suggest still has propositions being structured objects made up of things that represent objects and properties.

In other words, both views fall under this kind: "Propositions are structured objects representing their constituents in virtue of having parts that stand in relation R to those constituents." One view takes relation R to be identity, while I am skeptical about the plausibility of using identity to do the work. I haven't offered a specific alternate proposal for the relation yet.

Andrew Bailey said...

This is, roughly, Armstrong's view about states of affairs. They have proper constituents but not proper parts. The chief difference between 'is a proper constituent of' and 'is a proper part of' is that the latter but not the former obeys familiar mereological axioms.

Lewis Powell said...

I want to be careful to distinguish my view, which permits that propositions are still mereologically complex, from the one you attribute to Armstrong about states of affairs.

My view is simply that being a constituent does not require being a part. I may still be okay with the view that having a constituent entails having a part.

Anonymous said...

The question is, why does having the part it does mean that the proposition has the constituent it does?

I was thinking that spelling out R as anything other than identity requires you to say something like (x)(R(Aristotle,x)) iff x represents Aristotle. Therefore you have to explain representation.

Your point is that saying that the lagadonian view doesn't have to answer this question, or that 'because they're identical' is just an obviously good answer, gives it a free pass. That might be right.

By the way, I agree that not just any object with Aristotle as a part represents Aristotle.

Lewis Powell said...

If it is possible for X to be a part of Y, but not represented by Y (as you concede) then one cannot simply analyze the representational type of constituency as parthood.

Perhaps we could analyze representational constituency as being part of a proposition?

It turns out this will not work either. Assume that X is a part of the proposition that X exists, and that X is a composite object.

Since X's nose is a part of X, and X is a part of the proposition, X's nose is a part of the proposition, but not a constituent (in the sense where the nose is represented by the proposition). So, on the assumption that constituents are parts, neither parthood alone, nor part-of-a-proposition-hood can explain constituency.

Now, this doesn't show that no lagadonian account is plausible, but it does demonstrate that everyone, including the lagadonian, has some substantive explanatory work to do in accounting for constituency.

This observation, combined with the Tillman/Fowler argument, leads me to conclude that lagadonian constituency is not worth the cost of jettisoning the antisymmetry of parthood.

Anonymous said...

I see your point about there being a requirement for everybody to explain representation. Everybody has to explain what holds a bunch of objects together and makes them a proposition (especially when there are distinct propositions with the same parts, think of 'Jack kissed Jill'/'Jill kissed Jack'). That should also explain why Aristotle's nose isn't a constituent of the proposition expressed by 'Aristotle is bald'.

I was thinking that once that problem is solved, there then isn't a further problem for the lagadonian. But you do have a point that I was just assuming that.

Anyway, it might be agree to disagree time in the absence of a good theory of propositions.

Cory said...

I'm not going to respond to the discussion (sorry!) because I don't have the time to process all of it right now, but I would like to respond to the post.

I would agree that (2) is the "culprit", but for rather different reasons; in any set theoretic (or otherwise "mathematical") universe, the language being used to describe the universe is inherently outside the universe. As soon as you put the descriptive language within the universe, it can discuss itself directly, and lead immediately to paradox. So, if you would like to avoid a paradox here, don't let the propositions into the universe.