Simples and "Occupation"

Two of the main competing views of simples are occupation accounts. These two competing views are the Pointy View (PV) and MaxCon. Each of these views was originally formulated by Ned Markosian in his paper "Simples". Kris McDanial has argued, however, that occupation accounts are false on the grounds that there can be fusions of co-located individuals. This, in combination with the fact that non-occupation accounts of simples are false, suggests a brutal view of simples. I am inclined to think that McDaniel's arguments are sound. My inclination is as strong as my inclination toward accepting co-location. But, if it should turn out that co-location is impossible, then I think that these occupation accounts of simples are the strongest views thus far discussed. However, I think that there is a serious worry for PV and MaxCon that has not yet been addressed. In this post, I wish to spell out this worry and suggest alternative occupation accounts of simples.

I will take the pointy view as my main example. But, notice that everything I say about the pointy view can be said (with very slight modification) about MaxCon. Thus, the problem that I pose is a problem for both views. According to the pointy view,

Necessarily: for any x: x is a simple (at t) iff x is a pointy object (at t).

x is a pointy object (at t) =df the region occupied by x (at t) contains exactly one point.

This seems clear enough. However, there are some terms that we have been using that has not been fully defined. These terms are are related to the verb 'to occupy'. Markosian borrows from van Inwagen the following definition (I have added subscripts for help in later disambiguations):

O occupies1 R (at t) =df R is the set containing all and only those points that lie within O (at t).

Here are some comments that I would like to make at this point. First, this definition puts the burden of understanding this view of simples squarely on the primitive 'lie within'. I think I have an intuitive grasp of this primitive. However, there is some room for confusion. For example, one might believe that for one thing to lie within another it must be a part of the other. But, this is not the correct sense of 'lie within' for van Inwagen's definition. Another way that one object might lie within another is when the first is completely surrounded by the second. For example, a ring might lie within a box. But, again, this is not the correct sense of 'lie within'. For the record, even though I think I understand how van Inwagen is trying to use 'lie within' I am inclined to think that it is a deviant use of the phrase. If it is not a deviant use of the phrase, then it is certainly a restricted use and the restriction should be made explcit by ruling out the kinds of examples I suggested above.

The more important worry that I wish to address is the following. This definition seems to suggest that the relation picked out by 'occupies1' is functional. That is, every material object occupies1 exactly one region. This suggests that the definition of ' O occupies1 R' is equivalent to Parson's definition of 'O is exactly located at R' (correct me if I am mistaken here). This fact makes it clear why Markosian uses the definite article in his formulation of PV. But, there is a problem with this assumption.

Here is a case. Suppose that there is a single point sized object that persits for an hour. After an hour of its life has passed by, it appears as if another point sized particle appears along side the first. Then it seems like these two particles persist together until the first particle dissappears. One seemingly metaphysically possible event that fits this discription is a case of time travel. It could be that a single particle persists for a while until it materializes alongside itself. This materialization is the result of the particle time traveling from the future. It seems appropriate to say that his particle is a mereological simple. After all, it doesn't seem like an object can gain parts simply by time traveling in the way that this particle did. But, under PV, this particle is not a simple (during the time when it sits alongside itself). This is because the particles occupies a region that contains two points during those times. This suggests that the pointy view is false. (this example should also show that MaxCon is false since a region containing exactly two points cannot be continuous).

It is not enough to modify the pointy view by getting rid of the definite article. This is because 'occupies1' picks out a relation that is functional. So, eleminating the definite article will not solve the problem. We must also reinterpret the word 'occupies'. McDaniel (and independantly our very own Rock*) has suggested that there is a primitive occupation relation. This occupation relation is not functional. Let's use 'occupies2' to pick out this new relation and use 'might' in some kind of epistemic sense. When an object occupies2 a region it might also occupy2 another region. Moreover, when it occupies2 two regions, it need not occupy2 the union of those two regions. Finally, when we say that one thing occupies2 a region it might be that that thing or parts of that thing occupy2 subregions of that region and it might not. This is not entirely clear, but I think that we get the idea.

Now, we can simply reinterpret PV with 'occupies2'. But, this will not solve the problem. The definite article has to go. Here is one variant of the pointy view which turns out to be inadequate.

PV*: Necessarily: for any x: x is a simple (at t) iff a region occupied2 by x (at t) contains exactly one point.

Here is why this account is inadequate. Suppose we accept DAUP and the Liberal View of Recepticles. Suppose also that there is an object that occupies a pointy region and also occupies an extended region. Then, by the combination of DAUP and the Liberal View of Recepticles, this object has parts. It will have parts that occupy each of the subregions of the extended region it occupies. However, according to PV* it is a simple since one of the regions it occupies contains exactly one point.

Here, I think, is the best reformulation of the pointy view:

PV**: Necessarily: for any x: x is a simple (at t) iff x occupies2 (at t) a region and every region occupied2 by x (at t) contains exactly one point.

There is a similar reformulation of MaxCon that avoids the time travel counterexample given above. However, I worry about cases of time travel where a MaxCon simple comes into contact with itself. But, I have not thought through that example yet.

Propositions and Sets

Are propositions sets? I don't think so, but I'm worried about how good the arguments are against the view.

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?