Friday, April 15, 2011

One-Category Abundant Platonism

Standard Abundant Platonism (SAP) holds that to every predicate there corresponds a property, and items satisfy the predicate if and only if they exemplify the property.  Moreover, it holds that exemplifiers are not explanatorily prior to what they exemplify.  Normally, we think of SAP as a two-category theory: individuals and properties.

But here is a suspicion I have.  Little if any explanatory work is being done by the distinction between individuals and properties.  The serious explanatory work is all being done by the relation of exemplification.  Here are two examples.

1. Standard Platonists say that x and y are exactly alike in some respect if and only if there is some property P such that x exemplifies P and y exemplifies P.  But drop the word "property" from the previous sentence, and we have an account of exact alikeness that is even better: x and y are exactly alike in some respect if and only if there is a z such that x exemplifies z and y exemplifies z.  This is extensionally just as good, but simpler. (One can do more complex stuff about determinates and determinables to get resemblance in some specific respect, but again that doesn't need the concept of property, just the relation of being a determinable of.)

2. Standard Platonists say that to each predicate F there corresponds a property Fness, and that x is F if and only if, and if so because, x exemplifies Fness (we should probably have an exception to the "because" clause when Fness is exemplification).  But change "there corresponds a property Fness" to "there corresponds an entity Fness", and this works just as well as an account of predication.

Besides, the concepts of "individual" and "property" are foggy.  (We might try to say: "x is an individual if and only if x cannot be exemplified."  But that doesn't work for abundant Platonism, as abundant Platonism will have properties like being a square circle.)

So, if you're going to be a Platonist, why be a two-category abundant Platonist?  Why not be a one-category abundant Platonist instead?


  1. Jeff Sanford RussellApril 17, 2011 at 11:05 AM

    This is parallel to issues that come up in axiomatizing set theory (use both "set" and "member", or just "member"?). One minor wrinkle that comes up in that case is how to characterize the empty set—you probably don't want to just say the empty set is the thing with no members (though mathematicians get away with it when they're doing pure set theory) since that doesn't distinguish it from the non-sets.

    So you might similarly worry about unexemplified (or necessarily unexemplified) properties, if there are such things (as the abundant Platonist you describe will presumably believe). How can you characterize these? I suppose you might try to do it using "correspondence", but you might also want to say that there are inexpressible unexemplified properties. This is going to be difficult to say. I don't think this is a very serious concern, but it might be something to worry about a little.

  2. Little if any explanatory work is being done by the distinction between individuals and properties. The serious explanatory work is all being done by the relation of exemplification.

    I think the SAPist would reply that the distinction does some explanatory work. It explains why some entities don't and can't exemplify others. Individuals cannot be exemplified by either individuals or properties. Granted, this does not provide us with a sufficient condition for being an individual, but the idea that an individual can be exemplified by something else (unlike the idea that unexemplifiable properties can) seem to be a category mistake. What do you think?

  3. This is a minor point, but I'm unclear about why this move takes your Platonist from positing two categories to positing one rather than from positing four categories to positing two. I would have thought that the Platonist has four categories corresponding to the positions into which she is willing to quantify. The original Platonist claims that there are (1) properties, (2) particulars, (3) exemplifiers, and (4) exemplified things. After adopting your proposal she posits only (1*) exemplifiers and (2*) exemplified things. One might reply that only monadic predicates (and not relational predicates such as 'exemplifies') require us to posit categories in our ontology. But that can't be right, because your modified Platonist doesn't use (as yet) any monadic predicates (such as 'is a particular' and 'is a universal') to express her theory. So, relational predicates generate don't require positing categories, then it looks like your modified Platonist has a zero-category ontology.

  4. Jeff:

    The set theory analogy is helpful, thanks. I suppose the worry is that you lose extensionality if you just characterize the empty set as an object with no members.

    I suppose, though, you could stipulate "sets" like this: x is a set iff x=Quine or x has a member. Then there is a unique empty "set", namely Quine. The predicate "is a set" is then pretty unnatural, but I am not worried about that, since the applications of set theory don't require naturalness, I think. The empty set does seem intuitively different from other sets. :-)

    Anyway, back to Platonism. So, one way to put the worry is that on the modified view, I have to say that it's just a brute fact that Socrates can't be exemplified. I don't know if this is a very powerful worry, because maybe the SAPist's explanation simply shifts the bump under the rug. For the modified view, the puzzle is: Why does Socrates exemplify impossibility of exemplification? For SAP, the puzzle is: Why does Socrates exemplify particularity? If the answer to the latter is "because of Socrates' nature", the modified view can say that, too.

    But maybe the way to put the objection is this. The SAPist can group a large class of unexemplifiables together, and say that they all have the same kind of reason for their unexemplifiability--namely, that they have particularity. This does provide a kind of unity.

    Still, isn't the SAPist likely to have lots of brute unexemplifiables anyway, like being a self-cause? I suppose it's better to have fewer brutes, though.

    Note, though, that a lot of cases of unexemplifiability might actually be explicable in the modified theory. Maybe only non-spatiotemporal entities can be exemplified. Maybe only necessary entities can be exemplified. That might not cover all the cases of brute non-exemplifiables, but will reduce the field.


    You'll recall from my little contest that I don't think it's at all absurd that something might exemplify a person. I don't see the category mistake. :-)


    Mike Rea raised the zero-category point against Laurie Paul at INPC. I don't think anything hangs on whether one describes this as a one-category or zero-category view.

  5. You'll recall from my little contest that I don't think it's at all absurd that something might exemplify a person. I don't see the category mistake. :-)

    I didn't say that that's what you would say; I said that that's what they (the SAPists) would say (and, for what it's worth, they seem to be in very good company in thinking so). Personally, I don't see what it would mean for a person to exemplify a person more than I know what it would mean for a number to be purple. And just as I don't know how to explain what's wrong with the latter except that numbers are just not the kinds of things that can be purple, I don't know what's wrong with the former except to say that persons are just not the kinds of things that can be exemplified.

    Of course, if you don't believe that there is any such distinction then the distinction cannot do any work, but how does that differ from saying 'I don't see what the distinction between the mental and the physical is doing for dualists'?

  6. Gabriele:

    1. "I don't see what it would mean for a person to exemplify a person more than I know what it would mean for a number to be purple."

    Platonists think exemplification is a primitive relation (or quasi-relation or whatever). What it would be for a rose to be purple is for the rose to stand in exemplification to purpleness. This is a primitive relation. And then we can say that what it would be for a number to be purple is for the number to stand in exemplification to purpleness.

    I suspect if we think that the latter is puzzling in a way in which you don't think the former is, that's because you aren't thinking through-and-through in Platonist terms of exemplification as a primitive relation. We are tempted to say things like: for a rose to be purple is for its surface to reflect light thus and so; but how could a number do that? But that Platonist says: for a rose to be purple is for it to exemplify purpleness, and it exemplifies purpleness if and only if it exemplifies various reflectivity properties.

    Likewise, what it would be for a rose to exemplify Socrates is the same--it would be for the rose to stand in the same primitive relation of exemplification to Socrates. There is nothing further to standing in that relation (according to the Platonist), so there is nothing further to be said or understood about what it would be like for the rose to exemplify Socrates just as there is nothing further to be said or understood about what it would be like for the rose to exemplify purpleness. It's just that the rose can't exemplify Socrates (but that's not special to Socrates--the rose can't exemplify immateriality, either).

    2. The dualism example gives me pause. The idea, I think, would be that the dualist looks at the physical things around us, and thinks that because they are physical, they can't account for mental activity, and so introduces a new kind of entities. (But not a new category, I assume. But I don't know what exactly categories are and how they differ from more ordinary kinds.)

    Now, we might give a parallel just-so story about Platonism. We're puzzled about how predication and similarity are to be explained on the hypothesis that all there are are the things around us, the particulars. So we introduce a new category of things, by relation to which predication and similarity are to be explained.

    But I don't know how essential to the story it is that it be a new category. Consider this elaboration of the just-so story. We realize that predication and similarity can't be explained by relation to changing or material or contingent things. So we posit enough unchanging, immaterial and necessary things to explain predication and similarity. On this picture, there is no new category of property introduced--we are just introducing things that lack changingness, materiality and contingency. This version of the story isn't that far from the historical Plato, so it's not so crazy.

    (The preceding story would be parallel to some version of the dualist story like this. Maybe a dualist argues that thoughts cannot be spatiotemporally located, and hence there must be objects that lack materiality.)

    Another just-so story is that our language quantifies over entities that are exemplified, e.g., in saying that two things are the same color, and it is difficult to nominalize this quantification away. So we suppose that there really are entities for these quantified claims to be about. But on this story, there doesn't seem to be much need for a category difference.

    Still, the dualism analogy and the first just-so story I gave do give me pause. Thanks!

    1. I'm interested in this idea of connecting the primitive relation-hood of exemplification to whether or not it's intelligible for a rose to exemplify Socrates. I'm a fan of the idea that exemplification is primitive. I don't know the literature but it's hard to fathom how it could be otherwise. Could there really be an analysis/metaphysical reduction of exemplification? Wouldn't it involve some things being some ways? Which would involve exemplification?

      But I digress. I don't think the fact that exemplification is primitive really has anything to do with the intelligibility of a rose exemplifying Socrates. The notion seems to be somehow that if exemplification is primitive then anything goes - as far as intelligibility is concerned - with respect to what exemplifies what. A primitive relation - just to use the handy expression - can limit what stands in it no less than non-primitive relations or non-primitive whatevers.

      One might believe that causation is primitive but still think only events are causal; one might think belief is primitive but still think only propositions are believed.

  7. It's true I have a hard time thinking like a SAP (in fact, I just can't understand how to think like one) but, as far as I can see, the problem is exactly that, if exemplification is a primitive relation, OAP is less explanatory powerful than SAP in that it cannot explain why some entities are never exemplified by others and why some entities (the properties) are so different from others (the particulars). In other words, I can't see how the gain in simplicity is worth the loss in explanatory power.

  8. I don't get it -- you can eliminate the word "property" -- but for this chair to be red and that table to be red will still be different from this chair being that table. There will still be something, "being red", which is common to many things that still are not common to each other.

    We live in a world in which we come up against things which are the same as each other without being each other -- and this makes us distinguish two ways of being or two kinds of being -- and leads to the question how there are beings which many beings are, though the beings themselves are not those many beings.

    That would be so far as simple predicates. As for relations, we at least don't want to say they are no more than the things related -- for Sally and Fido might both be, without it being that Sally pets Fido. There is something else that brings them together in the right way, such that it is true to say, "Sally pets Fido."

  9. I feel I wasn't entirely clear in the last comment, so let me try to be more precise.

    You seem to be suggesting we could do without the distinction of what is (beings) into two kinds, with tables and chairs on the one hand, and being a table or being a chair on the other. Instead we would just have things, some of which are related to each other in this way, that one partakes of the other.

    But if X and Y are tables, we can say X is a table and we can say Y is a table, and yet, though X and Y are the same, they are not the same.

    So we resolve the paradox by proposing that "being a table" is a different kind of being from X and Y, because whereas it is not true that many things are either X or Y, but rather only one thing in each case (X,Y), it is true that many things are a table.

    And yet if we suppose that being a table is still *something* besides those beings which are tables, we are led to the view that there are two kinds of beings, and not simply beings which are related in various ways.