Friday, December 5, 2008

Deustch and Russel's paradox

Consider the following philosophical rule of etiquette:
(E) If a paradox is bad for everybody, don't use it to refute your opponents unless you personally have a solution.

I believe Deustch has violated (E). That's bad not only because it's rude, but because it's begging for a tu qoque response. I'll give that here.

Deustch thinks that any structured proposition theory is inconsistent because of the following argument:
Consider a set of propositions w, such that a proposition p is in w just in case the following obtains:
a) for some set of propositions m, p is the proposition that everything in m is true
b) p is not in m
He then asks that you consider the following proposition:
(P) everything in w is true

The argument goes as follows
1) Propositions have constituents
2) (1) -> (3)
3) Those constituents must exist if the proposition exists
4) (P) exists
5) w exists (3&4)

I'll divide the argument up for clarity. Here's to establish that if P is in w, then p is not in w, therefore p cannot be in w:
6) P is in w (assume for reductio)
7) The set of propositions such that P is the proposition that everything in it is true is w
8) P is not in w (6&7, by the conditions under which a proposition is in w)

To establish that P is in w:
9) P is not in w
10) there's a set of propositions such that (a) and (b) hold of (P) (9, w satisfies a and b for P)
11) P is in w (10, conditions under which a proposition is in w)

main argument:
12) P is in w and P is not in w (8, 11)
13) ~(1) (closed reductio)

Deustch contrasts this with the main objection against unstructure proposition theory, which is Soames' objection we saw earlier this term. Soames' objection applies directly to unstructured proposition theorists and nobody else. Deustch can only frame this as the main objection to structure proposition theory if it only applies to them. If it applies to unstructured proposition theory as well, his game is over.
This argument, of course, is a version of russel's paradox. This paradox springs up pretty much any place in which there's some principle of unrestricted composition (things can always combine to make bigger/more complex things). We see this in set theory, mereology, when formulating what properties are, possible worlds, everywhere. As for this particular formulation, the unstructured proposition theorists would deny (1) (of course). However we don't need a premise as strong as (1) to get the paradox going. Consider the same argument except replace premises (1), (2) and (3) with the following:
(1') Propositions are about things
(2') (1') -> (3')
(3') The things propositions are about exist
The argument goes through exactly as before. So the unstructured proposition theorist can't simply deny one. In fact, Deustch takes a different route when resisting the paradox. Here's what Deustch says about this:
"The essential assumption is that if a and b are distinct objects, then the propositions having them as constituents differ.[15] This is not true of possible worlds semantics, since e.g. the propositions expressed by "Jones wears a hat or he doesn't" and "Smith wears a hat or he doesn't" will express the same proposition whether or not Smith and Jones are identical.[16]"
This can be seen as a rather convoluted rejection of (7). Just because (P) is 'about' w over here, doesn't mean it's 'about' w over there. However, if one is going to be wishy-washy with aboutness facts, then plausibly propositions aren't (intrinsically) about anything (perhaps they're about something relative to a mode of presentation or something). This leads to a denial of (1'). This is already biting the bullet big time, but I can make things worse. Consider just talking about sentences, not propositions. Let (P) just be the sentence, not the proposition expressed. Let w be a set of sentences, m be a set of sentences etc. Replaces (1'), (2'), (3') and (4) with:
(1'') Proper names have referents
(2'') (1') -> (3')
(3'') If S is a well-formed sentence of non-fiction, the referents of all the proper names in S exist.
(4) (P) is well-formed sentence of non-fiction

With the proper modifications in the rest of the argument (just chanings 'proposition' to 'sentence' in each case) the argument goes through. Again, the denial of modified (7) would pretty quickly lead to the denial of (1''). Alternately the unstructured proposition theorist could deny (4), or (2''). However, the denial of these is a hefty cost.

A proponent of structured proposition theory may deny (2) (meinongianism, gappy proposition theory), or they could deny (4). For dealing with the modified arguments, they have similar options as the unstructured proposition theorist. These would also be costs, but the costs would be comparable to the costs of unstructured proposition theory.

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