Propositions and Sets
Are propositions sets? I don't think so, but I'm worried about how good the arguments are against the view.
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)
Here's 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. (The view is underdescribed, but hopefully the idea is clear.)
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. A proponent of "propositions are sets" 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.
Any other thoughts on general reasons for/against the view that propositions are sets?
There are several reasons in the literature for thinking that propositions are not reducible to set-like entities. Here 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. Those who endorse the view that propositions are sets will of course think 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)
Here's 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. (The view is underdescribed, but hopefully the idea is clear.)
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. A proponent of "propositions are sets" 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.
Any other thoughts on general reasons for/against the view that propositions are sets?
posted by Chris Tillman at 12:30 PM
1 Comments:
Chris,
I agree that the cardinality arguments are the best arguments against the view that propositions are sets. But, I wonder whether these problems could be avoided. Suppose we think, as Lewis does, that sets are fusions of individuals. On Lewis' view, a set is a fusion of simples which correspond to the various individuals in the world. One key to Lewis' view is that individuals correspond to particular mereological simples known as singletons. We all know that the combination of this with classical extensional mereology runs into trouble. So, one thing that a follower of Lewis might have to do is reject CEM. But, there are still problems lurking.
What I wonder is whether we could combine a mereological system that has the structure making properties that Armstrong needs with a Lewis-like view of mereology. We could, for example, divide the objects of the world into kinds. and have members of one kind correspond to members of another kind (regardless of whether they are mereologically simple). THe members of a particular kind will then be the singletons of the members of another kind. Since these singeltons cold be mereologically complex, we might be able to avoid various cardinality problems associated with views that have a one-one correspondence between simples and objects. But, the view might have other benefits as well. If it is done just right, maybe it will avoid all the cardinality worries that have been mentioned in the literature.
Unfortunately, I don't have anything more to say except for this sketch of a project. The sketch suggests a project similar to the one that Ben was workign one when we saw him talk in Syracuse. But, I wonder if the right use of kinds might help to avoid the problems he ran into. I really, don't know. But I think it might be worth thinking about.
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