Thursday, July 5, 2007

Propositions and Sets

Today we mentioned some reasons for thinking that propositions are not reducible to set-like entities. The objection explicitly considered by Divers is one from Plantinga:

1. Some propositions have truth values and no sets have truth values.
2. If (1), then some propositions are not sets. (by Leibniz's Law)
3. If some propositions are not sets, then not all propositions are sets.
4. If not all propositions are sets, then propositions are not reducible to sets.
5. So if (1), then propositions are not reducible to sets. (2-4)
6. So propositions are not reducible to sets. (1,5)

Plantinga contends that (1) is obvious. Divers thinks it is not obvious. Let's concede the point to Divers unless someone can come up with another way to support (1).

Another objection, from Jeff King's SEP entry on structured propositions, is as follows:

7. If some sets are propositions, then some sets have truth values (modal properties, etc) and others do not.
8. If some sets have truth values and others do not, then there is an explanation of why this is the case.
9. So if some sets are propositions, then there is an explanation of why some sets have truth values and others do not. (7,8)
10. There is no explanation of why some sets have truth values and others do not.
11. So it's not the case that some sets are propositions. (7,10)

I think someone like Lewis can resist (10) with some plausibility. Here's a view that Lewis and some of his opponents, like Salmon and Soames, both seem to hold:

Propositions are pieces of information semantically encoded by well-formed declarative sentences. They are truth-apt objects of cognitive attitudes (like belief, etc).

This characterization serves to specify the role of propositions. Something is a proposition iff it's the best candidate for that role. If that turns out to be shoes or fish or whatever, then propositions may be identified with shoes, fish, whatever. Now Lewis holds that certain sets occupy this role. (For what it's worth, Salmon and Soames give their theories of propositions in set-theoretic terms but are not explicit about whether the set-theoretic entities are supposed to be propositions or if they merely represent them.) If he's right about that, then it seems he has a not implausible explanation of why (e.g.) some sets are true and others have no truth value. There's more that can be said about this objection, but I'll leave it at that for now.

King's second objection is a version of the Benacerraf problem. (Link requires JSTOR access.) It requires a bit of set-up. Consider sentence *:

* Brendan loves Adam

Suppose we held that propositions were ordered n-tuples. Consider the following ordered triples:

(i) bLa
(ii) aLb
(iii) Lab
(iv) Lba
(v) abL
(vi) baL

Furthermore, there are many ways to construct ordered n-tuples. For each way, there is a non-equivalent set that corresponds to each of (i)-(vi). Let's suppose there are only seven ways. Then there are 42 sets: each of (i)-(vi) constructed in each of the seven ways. Here's the objection:

12. If propositions are sets, then there is a unique most eligible candidate among the sets for being the proposition expressed by * in English.
13. There are (at least) 42 sets that are equally eligible candidates for being the proposition expressed by * in English.
14. If there are (at least) 42 sets that are equally eligible candidates for being the proposition expressed by * in English, then there is no unique most eligible candidate among the sets for being the proposition expressed by * in English.
15. So there is no unique most eligible candidate among the sets for being the proposition expressed by * in English. (13,14)
16. So it's not the case that propositions are sets. (12,15)

Carl offered a reason for denying (1): given a multiplicity of equally eligible candidates, a proponent of the "propositions are sets" view could hold that it's indeterminate which of the 42 sets is the proposition that Brendan loves Adam. One could add that picking any of the 42 to represent the information that Brendan loves Adam is harmless as long as one makes the appropriately uniform choices for representing other propositions. One could also hold that it's appropriate to talk about the proposition that Brendan loves Adam iff according to any legitimate way of eliminating the indeterminacy, there is only one candidate for the proposition.

Some thoughts:

A. The sharpenings would have to be done with care. Suppose one pursued the same tack for numbers. There may be admissible sharpenings for propositions according to which S is a proposition and admissible sharpenings for numbers according to which S is a number; 0, for instance. Then 0 would have a truth value and it would be possible to believe 0. That's no good. But it seems like it could be prevented by adding the relevant constraints on admissible sharpenings.

B. I worry on the indeterminacy proposal that it would be true that, were we to have decided on a different sharpening, then the proposition expressed by 'Brendan loves Adam' would have been the proposition expressed by 'Adam loves Brendan' (while all the facts about the English sentences 'Brendan loves Adam' and 'Adam loves Brendan' remain fixed). The counterfactual strikes me as false. There are probably ways around this too: there are similar views about vagueness according to which there are several admissible sharpenings for 'red' and 'orange' and some things that are red under one sharpening are orange on another, but under no sharpening are some things both red and orange. But note a lack of parallel: all of the set-theoretic candidates for being the proposition that Brendan loves Adam are the set-theoretic candidates for the proposition that Adam loves Brendan. In spite of the disanalogy, I confess that the objection does not strike me as especially serious.

C. It would be self-refuting for me to believe that there are no beliefs. On one usage of 'belief', the word refers to the objects of belief. On this understanding, 'I believe there are no beliefs' expresses a proposition that entails that I bear a relation to the proposition that there are no propositions. Contrast this with my (pretend) belief that there are no sets. This does not seem similarly self-refuting. But it would be on the "propositions are sets" view. This is the basis for a Leibniz's Law objection, but I think it's better than Plantinga's because it does not rest on the contention that it's just obvious that sets don't have truth-values. Divers will cry "hyperintensionality" here, but I don't buy it. The two beliefs really strike me as different in the way described.

D. There are cardinality problems for the view that propositions are sets. There are several ways to state these. Here's one. The proposition that absolutely everything is self-identical is (logically) true. But there is no ordered pair with absolutely everything as one member and the property of being self-identical as the other. That is because there is no set that has as a proper subset absolutely everything. That is because if sets are things, there are too many things for all of them to be a subset (even an improper subset) of a set. (Given any set, the set of all of its subsets has a strictly greater cardinality. So any candidate for being a set that has absolutely everything as a subset is such that there's a "bigger" set: the set of all of its subsets.) Furthermore, if there were a proposition that absolutely everything is self-identical, and it was a set, then it would be a proper subset of itself (since it, too, is one of absolutely everything). This violates standard axioms of set theory ("well-foundedness"). Upshot: if the "propositions are sets" view is true, then there is no proposition that absolutely everything is self-identical. So if propositions are sets, then some logical truth is not true.

I take the last sort of problem to be the most serious. But note it will not do to rest with the claim that propositions are not sets. A positive theory is needed. And part of the burden of the proponent of the positive theory is to show that propositions don't run into cardinality problems anyway. More work is called for.

At any rate, my main purpose in posting this was to recap some of the discussion and elicit further thoughts on the thesis that all propositions are sets.


Dan said...

C strikes me as a hooded man style paradox. That is, it's possible for me to believe there are no sets while still believing that there are beliefs. In a similar way I can believe Old Dirty Bastard rocks, while Russel Jones does not rock. It seems to me that once you choose your favorite solution to the ODB/RJ problem (Fregian senses, ways of taking propositions etc.), you have a free solution to the sets/propositions problem.

I have a proposal for D, which I'm sure Chelsey will consider cheating. On her behalf, I'm not sure that that's not the case.
It's true that there is no set of absolutely everything. However, give a name to absolutely everything, call it A. A is not a set and can't be a set. However, prima facie, I see no problem with having A as an element within a set. So a proposition "absolutely everything is green" would look like this:
{A,{all green things}}.
This looks like cheating, but consider how the paradox arises. There's something incoherent about a set that has absolutely everything as a member. A doesn't have any members at all! If you must, consider A the plurality of all things. Alternately, you can consider it the individual called "reality". To reduce ad-hocery, you can stipulate that the subject of any proposition is not a set, but a mereological sum, or a plurality. (pluralities would be better, otherwise "all football players have one head" would come out false). Or you can alternate between having a plurality as the subject, or a mereological sum as the subject depending on the grammer of the sentence.
The problems arise again when trying to characterize the predicate "is self identical". Since everything is in fact self identical, and predicates are (under Lewis's view) the set of all things which satisfy the predicate, being self identical would be the set of all things. To get around this one would have to do some fancy footwork. If I was interested in saving a bad theory, I'd try and pass off predicates that are universally satisfied (the existence predicate, being self identical etc.) as a different sort of thing than ordinary predicates. An oddity of this move would be that any proposition that has absolutely everything as the subject would be necessarily false.

Dan said...

Note too, this formulation of propositions as set-like entities eases the pressure of Bennaciraf since it does not need to invode ordered pairs. In a sense, ordering is implicit since the subject is never a set, while the predicate is always a set.

Ben said...

What's the cardinality problem with the proposition expressed by 'Absolutely everything is self-identical'? It doesn't contain absolutely everything as a constituent. Rather, it contains the semantic content of 'absolutely everything'. Maybe that's some quantifier or something. There might be a problem doing the semantics for this proposition -- e.g. if we wanted to assign a set to the quantifier –- but that has nothing to do with the structure or the constituents of the proposition.

Am I missing something?

Chris Tillman said...

Okay, you're right, bad example. Better example: Consider each singular proposition that has some thing or other and the property of being self-identical as constituents. A plausible thought is that a universal proposition that everything is F is true only if every propositional conjunction of its propositional instances is true. But that won't be the case if propositions are sets since some conjunctions formed by the instances would be too big to form sets themselves. To put the problem another way, propositional conjunction introduction fails if propositions are sets.