Accidents and truthmakers

It is difficult to hold (a) Aquinas’ idea that in transubstantiation the accidents of bread and wine continue existing after the bread and wine have perished together with (b) the idea that accidents are truthmakers for predications.

For if the accident of the whiteness of the bread is a truthmaker for the proposition that the bread is white, then it is (absurdly) true to say that the bread is white even after transubstantiation, since when the truthmaker exists, the proposition it makes true is true.

So, if one wants to hold on to the logical possibility that accidents could outlast their substance, one has to modify the thesis that accidents are truthmakers for predications. Instead, perhaps, one could say that the truthmaker for the proposition that x is F is x’s Fness together with x. This solves the problem of the bread being white after transubstantiation, since after transubstantiation there is no bread, and so if the truthmaker is the accident of whiteness together with the bread, then after transubstantiation the bread part of the truthmaker doesn’t exist. So all is well.

But here is a further puzzle. Intuitively, if God can detach the bread’s accidents from the bread when the bread ceases to exist, why can’t God detach the bread’s accidents from the bread while the bread continues to exist? But if God could detach the bread’s accidents from the bread while the bread continued to exist, then God could detach, say, the whiteness W of a bread from a bread B, and then the bread could be dyed black. Were that possible, it couldn’t be true that W and B are a truthmaker for the proposition that the bread is white, since W and B could continue to exist without the bread being white any more.

So, holding that the substance and its accident is a truthmaker for the predication, while accepting the logical possibility of Aquinas-style transubstantiation, requires one to hold that God can only detach the bread’s accidents from the bread while annihilating the bread. That seems counterintuitive.

Another move is this. Posit an “attachment” trope. Thus, when x is F, there are three particular things: x, x’s accident of Fness, and an attachment trope between x and x’s accident of Fness. Further, posit that in transubstantiation the ordinary accidents continue to exist, but the attachment tropes perish. And now we can say that the truthmaker of “The bread is white” is B, W and the attachment trope between B and W. (There is no infinite regress, since we can suppose that the attachment trope cannot exist detached.) But God can make W exist without the attachment trope, and either with or without B.

But it is an unpleasant thing that the attachment trope is a metaphysical ingredient posited solely to save transubstantiation. Moreover, the attachment trope would be a counterexample to the Thomistic principle that God can supply whatever creatures do. For it is essential to the story that the attachment trope cannot possibly exist in the absence of bread.

Probably, the Thomist’s best move is to deny that accidents (whether with or without the underlying substance) provide truthmakers for predications. If we did that, then a nice bonus is that we can have accidents moving between substances, which would provide a nice metaphysical account of why it is that flamingos turn pink after eating pink stuff.

"On the same grounds"

Each of Alice and Seabiscuit is a human or a horse. But Alice is a human or a horse “on other grounds” than Seabiscuit is a human or a horse. In Alice’s case, it’s because she is a human and in Seabiscuit’s it’s because he’s a horse.

The concept of satisfying a predicate “on other grounds” is a difficult one to make precise, but I think it is potentially a useful one. For instance, one way to formulate a doctrine of analogical predication is to say that whenever the same positive predicate applies to God and a creature, the predicate applies on other grounds in the two cases.

The “on other/same grounds” operator can be used in two different ways. To see the difference, consider:

1. Alice is Alice or a human.

2. Bob is Alice or a human.

In one sense, these hold on the same grounds: (1) is grounded in Alice being human and (2) is grounded in Bob being human. In another sense, they hold on different grounds: for the grounds of (1) also include Alice’s being Alice while the grounds of (2) do not include Bob’s being Alice (or even Bob’s being Bob).

Stipulatively, I’ll go for the weaker sense of “on the same grounds” and the stronger sense of “on different grounds”: as long as there is at least one way of grounding “in the same way”, I will count two claims as grounded the same way. This lets me say that Christ knows that 2 + 2 = 4 on the same grounds as the Father does, namely by the divine nature, even though there is another way in which Christ knows it, which the Father does not share, namely by humanity.

Even with this clarification, it is still kind of difficult to come up with a precise account of “on other/same grounds”. For it’s not the case that the grounds are literally the same. We want to say that the claims that Bob is human and that Carl is human hold on the same grounds. But the grounding is literally different. The grounds of the former is Bob’s possession of a human nature while the grounds of the latter is Carl’s possession of a human nature. Moreover, if trope theory is correct, then the two human natures are numerically different. What we want to say is something like this: the grounds are qualitatively the same. But how exactly to account for the “qualitatively sameness” is something I don’t know.

There is a lot of room for interesting research here.

Yet another bundle theory of objects

I will offer a bundle theory with one primitive symmetric relationship. Moreover, the primitive relationship is essential to pairs. I don’t like bundle theories, but this one seems to offer a nice and elegant solution to the bundling problem.

Here goes. The fundamental entities are tropes. The primitive symmetric relationship is partnership. As stated above, this is essential to pairs: if x and y are partners in one world, they are partners in all worlds in which both exist. If x and y are tropes that exist and are partners, then we say they are coinstantiated.

Say that two possible tropes, existing in worlds w₁ and w₂ respectively, are immediate partners provided that there is a possible world where they both exist and are partners. Then derivative partnerhood is defined to be the transitive closure of immediate partnerhood.

The bundles in any fixed world are in one-to-one correspondence with the maximal non-empty pluralities of pairwise-partnered tropes, and each bundle is said to have each of the tropes that makes up the corresponding plurality. We have an account of transworld identity: a bundle in w₁ is transworld identical with a bundle in w₂ just in case some trope in the first bundle is a derivative partner of some trope in the second bundle. (This is a four-dimensionalist version. If we want a three dimensionalist one, then replace worlds throughout with world-time pairs instead.) So we have predication (or as good as a trope theorist is going to have) and identity. That seems enough for a reductive story about objects.

We can even have ersatz objects if we have the ability to form large transworld sets of possible tropes: just let an ersatz object be a maximal set of pairwise derivately partnered tropes. An ersatz object then is said to ersatz-exist at a world w iff some trope that is a member of the ersatz object exists at w. We can then count objects by counting the ersatz objects.

This story is compatible with all our standard modal intuitions without any counterpart theoretic cheats.

Of course, the partnership relationship is mysterious. But it is essential to pairs, so at least it doesn’t introduce any contingent brute facts. And every story in the neighborhood has something mysterious about it.

There are two very serious problems, however:

1. On this story we don’t really exist. All that really exist are the tropes.

2. This story is incompatible with transsubstantiation—as we would expect of a story on which there is no substance.

So what’s the point of this post? Well, I think it is nice to develop a really good version of an opposing theory, so as to be able to focus one’s critique on what really matters.

An argument against heavy-weight Platonism

Heavy-weight Platonism explains (or grounds) something's being green by its instantiating greenness. Light-weight Platonism refrains form making such an explanatory claim, restricting itself to saying that something is green if and only if it instantiates greenness. Let's think about a suggestive argument against heavy-weight Platonism.

It would be ad hoc to hold the explanatory thesis for properties but not for relations. The unrestricted heavy-weight Platonist will thus hold that for all n>0:

1. For any any n-ary predicate F, if x₁,...,x_n are F, this is because x₁,...,x_n instantiate Fness.

(One might want to build in an ad hoc exception for the predicate "instantiates" to avoid regress.) But just as it was unlikely that the initial n=1 case would hold without the relation cases, i.e., the n>1 cases, so too:

2. If (1) holds for each n>0, then it also holds for n=0.

What is the n=0 case? Well, a 0-ary predicate is just a sentence, a 0-ary property is a proposition, the "-ness" operator when applied to a sentence yields the proposition expressed by the sentence, and instantiation in the 0-ary case is just truth. Thus:

3. If (1) holds for each n>0, then for any sentence s, if s, then this is because because of the truth of the proposition that s.

(The quantification is substitutional.) For any sentence s, let <s> be the proposition that s. The following is very plausible:

4. For any sentence s, if s, then <s> is true because s.

But (4) conflicts with (3) (assuming some sentence is true). In fact, to generate a problem for (3), we don't even need (4) for all s just for some, and surely the proposition <The sky is blue> is true because the sky is blue, rather than the other way around: the facts about the physical world explain the relevant truth facts about propositions. Thus:

5. It is false that (1) holds for each n>0.

The above argument is compatible, however, with a restricted heavy-weight Platonism on which sometimes instantiation facts explain the possession of attributes. Perhaps, for instance, if "is green" is a fundamental predicate, then Sam is green because Sam instantiates greenness, but this is not so for non-fundamental predicates. And maybe there are no fundamental sentences (a fundamental sentence would perhaps need to be grammatically unstructured in a language that cuts nature at the joints, and maybe a language that cuts nature at the joints will require all sentences to include predication or quantification or both, and hence not to be unstructured). If so, that would give a non-arbitrary distinction between the n>0 cases and the n=0 case. There is some independent reason, after all, to think that (1) fails for complex predicates. For instance, it doesn't seem right to say that Sam is green-and-round because he instantiates greenandroundness. Rather, Sam is green-and-round because Sam is green and Sam is round.

Towards a Thomistic theory of fundamental distributional properties

Recently, various metaphysicians (e.g., Parsons, and Arntzenius and Hawthorne) have tried to give an account of spatially nonuniform properties that would work for extended simples or gunky objects (i.e., ones that have no smallest parts). I think there is an interesting account that has in an important way a Thomistic root, and that's no surprise because Aquinas did not believe that substances had substantial parts, so he faced the problems that people thinking about extended simples face. I will develop a partial account for shape, location and color. The version I will give in moderate detail is Pythagorean, because mathematical objects are involved in physical reality itself. The Pythagorean account is easier to wrap one's mind around. I think it may be possible to use Category Theory to de-Pythagorize the account, but I will only sketch the beginning of that line of thought.

A basic insight Aquinas has is that material objects have a special accident called "dimensive quantity", which accident in turn provides a basis for further accidents, such as color. Moreover, objects normally are located in a place by having their dimensive quantity be located there.

On to the Pythagorean account. Suppose that each extended object O has fundamentally associated with it a manifold G of some appropriate smoothness type (we may in the end want to generalize this, perhaps to a metric space, perhaps a topological space, but let's stick to manifolds for now). This manifold I will call the object's (internal) geometry. The fundamental relation between the object and the manifold that associates the manifold to the object is being geometrized by: the object is geometrized by the manifold. This manifold is a purely mathematical object existing in the Platonic heaven. Nonetheless, which manifold an object is geometrized by significantly affects its nomic interaction with other objects. The shape properties of an object are grounded in the fact that the object is geometrized by such-and-such a manifold.

Next, we need location. There is a fundamental relation between an object O and a function L from the object's geometry G to another mathematical manifold called "spacetime", which relation I will call being located by. The function L describes how the object's geometry is located within spacetime. We can now say that two objects O₁ and O₂ overlap if and only if there are L₁ and L₂ such that O_i is located by L_i, for i=1,2, and the ranges of the functions L₁ and L₂ overlap.

Now, let's add some color into the picture. There is an abstract object which is a colorspace. Maybe it's some kind of a three-dimensional manifold. There is a fundamental relation between an object O and a function c from the object's geometry to the colorspace, which we may call being colored by. This function describes the distribution of color over the object's geometry.

This is the Pythagorean version of the view. Now we should de-Pythagorize it. Suppose a fundamental determinable of objects: being geometrized. A maximally specific determinate of being geometrized will be called a geometrization. And then—this is getting sketchier—one makes the geometrizations, and maybe other Platonic things like geometrizations, into a category isomorphic to an appropriate category of manifolds. I don't know what, if any, classical ontological category arrows correspond to. Maybe some kinds of token relations. If we're substantivalists about spacetime, we can suppose a special object, S, the spacetime. And then there is a fundamental relation of being located by between an object O and a morphism L of the category of geometrizations whose domain is O's geometrization and whose codomain is S's geometrization. It's harder to bring colors into the picture. This is far as I got. And even if I finish the de-Pythagorization, I will still want to de-Platonize it.

More generally, the de-Pythagorization proceeds by replacing mathematical objects associated with an object with maximally specific determinates of a determinable that, nonetheless, stand in the same structural relations as the mathematical objects did. Category Theory is a promising way to capture that structural sameness, but it might not be the only way.

Two problems of temporary intrinsics

I've been thinking about the problem of temporary intrinsics and don't see much of a problem.  There are two kinds of ways of formulating the problem, and I think they are basically different problems, and neither is particularly compelling.


Formulation 1:

1. Socrates at t0 is bent.
2. Socrates at t1 is straight.
3. Socrates at t0 = Socrates at t1
4. So, Socrates at t0 is bent and is straight.  (Which is absurd.)

I think this is a linguistic paradox rather than a metaphysical problem, and hence deserving of being linguistically defused.  "Socrates at t0 is bent" is awkward English.  The normal word order is "Socrates is bent at t0."  But if we rephrase (1) this way (and (2) analogously), the argument becomes invalid.  Nothing untoward follows from Socrates being bent at t0 and Socrates being straight at t1.  All that follows is that he is bent at t0 and straight at t1.

For the argument to be valid, we need to parse (1) as: "Socrates-at-t0 is bent", and (2)-(4) analogously.  But what is this Socrates-at-t1?  Suppose we say that it's just Socrates under another name.  Then we should deny (1), since Socrates isn't bent--he's dead (unless by "is bent" we mean "is bent at some time or other", in which case (4) tells us that "Socrates-at-t0 is bent at some time or other and is straight at some time or other", which is unproblematic).  Of course, if "Socrates-at-t0" is just another name for Socrates, we can say "Socrates-at-t0 is bent at t0".  But no untoward consequences follow from the claims that Socrates-at-t0 is bent at t0 and that Socrates-at-t1 is straight at t1.  We end up saying that Socrates-at-t0 is straight at t1, which sounds weird, but that weirdness only comes from this weird "Socrates-at-t0" name we've used.  It's like the weirdness of saying: "Ivan the Terrible was actually a pretty nice kid" (which for all I know is true).

I think what is going on here is this.  We sometimes speak in the historical present with a contextually implicit time.  We say things like: "September 1, 1939.  Germany invades Poland.  The Polish defenses crumble."  The two sentences following the contextual introduction of September 1, 1939 are to be understood as saying that Germany invades Poland and the Polish defenses crumble on that date.  We do the same thing spatially.  For instance, we can be describing the course of the (imaginary) Borogove River which comes from Oklahoma to Texas.  We've just described it in Oklahoma.  We now say: "Texas.  The Borogove is very silty."  We mean that it is silty in Texas.  In the case of "Socrates-at-t0", the "-at-t0" determines the context of evaluation for the historical present "is bent."  So all we are saying is that Socrates-at-t0 is bent at t0.  And no paradox ensues.

Now, there is another reading.  We sometimes adopt a metaphor of a individual being split into multiple individuals, either by means of time or role.  Thus we say things like:

5. Late Plato disagrees with Middle Plato on whether all the serious problems of philosophy are solved by positing the Forms.
6. Smith the Rhetorician loves this argument, but Smith the Philosopher hates it.

When we adopt this fiction, we do not allow intersubstitution--that would be inappropriate mixing of metaphor with reality, like when someone says that the lights came on for her after she read so-and-so's paper and we ask if they were incandescent or fluorescent.  In other words, on this metaphorical reading of (1) and (2), we will reject (3).

Granted, the perdurantist can take "Socrates-at-t0" and "Socrates-at-t1" to literally refer to two entities, and then reject (3).  But that kind of metaphysics is not at all required by the argument.

So, in the first formulation, the argument can be defused purely on linguistic grounds.  This point applies also to my favorite formulation of the problem:

7. The young Socrates is ignorant.
8. The old Socrates is wise.
9. The young Socrates is the old Socrates.
10. So the young Socrates is wise and the old Socrates is ignorant.

Formulation 2:

11. Presentism is true or the application of a temporarily applicable predicate to x is never correctly explained in terms of x's instantiation of a non-relational monadic property whose choice is dependent only the predicate (and not on the time of application).
12. The predication of shape (say) predicates is correctly explained in terms of the object's instantiation of corresponding shape properties.

Notice that while the first formulation could grip a non-philosopher, (11) is simply a constraint on philosophical theories of predicate application.  There seems to be very little cost in denying (12) and its parallels, since (12) and its parallels simply do not state any sort of ordinary intuition--they are a substantive claim about how to explain predication.

Pluralist theories of predication

According to anti-pluralist theories of predication, there is only a small handful of fundamental predicates and they are all of a highly general and abstract nature.  Sometimes there is only one.  For instance, strong Platonism has as fundamental only the multigrade predicate Instantiates.  All other predications should be analyzed in terms of it.  Resemblance nominalism has the fundamental predicate ResemblesInRespect and then needs some story about respects (which story may involve one or two more fundamental predicate).  Bundle theory will have the fundamental predicate CobundledWith, plus perhaps the predicates of set theory (∈ and IsASet) or of some other highly general theory for constructing objects out of bundles.

According to pluralist theories of predication, there are many fundamental predicates and many of them are of a very concrete nature.  For instance, the pluralist is likely to have predicates like Horse, Daphnia and NegativelyCharged.  She may also have highly abstract predicates like ∈ as well.

Ostrich nominalists are pluralists.  But one can also be a weak Platonist and a pluralist.  I am inclined to think that the solution to the problem of the unity of form and matter given in Metaphysics H.6 commits Aristotle to pluralism.

The big insight of the pluralist is that the puzzle of predication is no less of a puzzle when that puzzle concerns a small handful of fundamental predicates.  There may be theoretical simplicity grounds to prefer particular anti-pluralist theories of predication over particular pluralist theories, but I suspect these will result in a stalemate.  And then the pluralist will win, as her fundamental predicates fit better with our intuitions, I think.

Basic entities and predication

Suppose that trope theory is correct. Then what it is for x to have a given property P is to have a trope, say P_x, associated with it. But suppose now that x is a reducible entity—one facts about which reduce to the existence and functioning of other entities (e.g., x might be a table—table-facts reduce to facts about particles and societies). In that case, it is surely not the case that what it is for x to have P is for x to have associated with it P_x. For if x has P_x associated with it, then x is no longer reducible. For consider the fact that x has P. For this fact is the same as the fact that x is associated with P_x. But that x is associated with P_x does not reduce to facts about how, say, the components of x are arranged. For the latter facts are constituted by association with certain tropes of the components; but the fact we are interested in involves P_x. The only way x's having P could reduce would be if facts about the existence of P_x somehow reduced to facts about other things. But then P_x wouldn't really be a trope. The point of tropes is that they are ontologically basic—facts about them don't reduce.

Therefore, if trope theory is correct, then it does not apply to cases where we predicate something of a reducible entity. This, I think, gives one good reason to say that the reducible entity does not really exist in the same sense of "exist" that the other entities do. After all, if predication means something different in its case from what what it means in the other cases, it seems plausible its entitihood is not univocal with theirs.

I ssupect that the same argument might work with other theories of predication as well. If so, then reducible entities don't really exist in the full sense of the word.

A regress for bundle theory

According to bundle theory, each individual is a bundle of properties. But what is an individual? Presumably, individuals are existing entities that we can individuate, quantify over and predicate things of. Take that as a sufficient condition for being an individual. But then a property is also an individual. (If one balks at this, I expect that it is simply because one has stipulated this fact away, say by defining individuals as existing entities that we can individuate, quantify over and predicate things of which are not properties. If so, then the class of "individuals" is gerrymandered. But we needn't worry about words. Call something that we can individuate, quantify over and predicate things of an "individual*", and construe what I say below as about individuals*.)

But now the regress is obvious. Socrates, let us say, is a bundle of humanity, maleness, snubnosedness, smartness, hellenicity, etc. Fine. But humanity is also an individual. So, humanity is itself a bundle of properties. What properties? I don't know. Maybe properties like propertyhood, unchangingness, animal-kind-hood, rational-being-kind-hood, etc. Already it gets weird—we have no idea what to say. And then the problem returns for the properties that humanity is a bundle of. What, say, is propertyhood a bundle of? What is animal-kind-hood a bundle of? Obviously, a regress ensues. Is it vicious? I suspect so. We have bundles of bundles of bundles of .... If the bundling is done set-wise, then the sets will violate regularity.

And in any case if the individuation of bundles is by their members, that never bottoms out. On an abundant theory of properties, our non-property individuals all look like {{{...},{...},{...},...},{...},{...}},{...},...}, with exactly the same structure of braces. (That's for the set-theoretic construction. Otherwise, replace the braces by whatever bundling method we have.) On a sparse theory of properties, if it turns out that non-property individuals have finitely many properties, and that properties all have finitely many properties, maybe then we can differentiate these things by the structure of the rooted property-tree (individual x has 7 properties at the first level, the first of which branches has 19 properties, etc.) But that's as crazy as Pythagoreanism.

So maybe the bundle theorist will limit her theory of predication to individuals that are not themselves properties. But if she does that, she still needs a story about how we manage to predicate things of properties. For we do. Humanity is a universal. It is unchanging. It is non-spatio-temporal. It is different from hellenicity. And so on. And whatever non-bundle theory of predication that we give for the properties of properties, the opponent of bundle theory will say: Why not just simplify and give that for the individuals that are not properties?

Or one might take first-order properties to be bundles of individuals. But that's terrible. First, they'd have to be bundles of possible individuals. Second, we now have circularity in place of regress, which is worse.

This arguments seems to force the bundle theorist to say that eventually we get to properties that have no properties (what about their abstractness? their propertyhood? maybe we say that these are not genuine properties--maybe abstractness is just the denial of concreteness). Todd Buras then points out to me that these properties with no properties are just like the bare particulars avoiding which was one of the main motives for bundle theory!

Note: The argument works equally well if we have bundles of tropes instead of bundles of universals.

I know that these issues have been worked over, and regress-finding is a fun game for the philosophical family, so this regress is quite likely known. (If you have a reference, please let me know.)

Kind relative predication

It is a broadly Aristotelian doctrine that many predicates apply to individuals in a kind-relative way. Call such predicates k-predicates. (We can stipulatively say that God is the sole member of a kind membership in which is identical with himself, or something like that.) For a k-predicate F, what exactly it is for an x to be F depends on what kind of an entity x is. If it is the same thing for x to be such that Fx as for y to be such that Fy, then kinds of x and y are either the same or have something in common (e.g., a higher genus).

Examples of k-predicates are easy to find. To determine whether a given individual "has legs", we first have to see what counts as legs for an individual of that kind. Thus, Peter has legs in virtue of a particular pair of limbs. Which limbs count as legs? That is defined in part by his human nature, or maybe more generally his nature as a member of Tetrapoda. What it would be for an amoeba to have legs is a different, and more poorly defined, question. What it is for a table is fairly well defined in terms of the nature of the table (we could imagine a table with four upward projecting horns at the corners; the nature of a table being to stand on solid ground on its legs, if it has any, would prevent these from counting as legs).

Whether an entity is n inches tall is even more clearly kind-relative. It depends on which axis counts as the "vertical" axis—remember that the entity might be lying on its side for much of the day.

Another kind of predicate is an r-predicate. An r-predicate is a predicate that can only be had by members of one particular (perhaps higher level) kind. Thus, "is a mammal" is an r-predicate, since it can only be had by mammals. And "is Socrates" is also an r-predicate, since it can only be had by a human.

We can perhaps form complex predicates that are neither k- nor r-predicates. Thus, "is not Socrates" is not an r-predicate (all horses and chairs, and most humans satisfy it) and may not be a k-predicate either. Though on the other hand, it may a k-predicate: maybe for non-humans, it holds in virtue of kind difference, while for humans, it holds in virtue of numerical difference within a kind.

There are also some non-contentful predicates, like "is a substance" or "is self-identical" that are neither k- nor r-predicates.

Thesis: All simple, contentful predicates are either k- or r-predicates.

I don't know if the thesis is true. There seem to be counterexamples. Having a particular shape does not seem to be a k- or r-predicate. Likewise, having a certain mass does not seem to be such, either. I suspect such apparent counterexamples can be overcome—but that may be matter for another post.