Presentism and the intrinsicness of past tensed properties

Many presentists think that objects have past-tensed properties. Thus an object that is now straight but was bent has the property of having been bent. (Some such presentists use these properties to ground facts about the past.)

But assuming for simplicity that being bent is an intrinsic property, we can argue that having been bent is an intrinsic property as well. Here’s why. If being bent doesn’t describe an object in relation to the existence, non-existence or features of any other object (assuming being bent is intrinsic), neither does having been bent. Nor is having been bent “temporally impure”—it does not describe the object in terms of anything happening at other times, since nothing can happen at other times on presentism. It does not describe the object in relation its past or future temporal slices or past or future events involving the object, since on presentism there are no past or future objects, and there are no past or future events.

But if having been bent is an intrinsic property of an object, it seems that, by a plausible patchwork principle or by intuitions about the omnipotence of God, an object could come into existence just for one instant and yet have been bent at that instant. Which is absurd.

Property inheritance

There seems to be such a thing as property inheritance, where x inherits a property F from y which has F in a non-derivative way. Here are some examples of this phenomenon on various theories:

1. I inherit mass from my molecules.

2. A person inherits some of their thoughts from the animal that constitutes the person.

3. A four-dimensional whole inherits its temporary properties from its temporal parts.

These are all cases of upward inheritance: a thing inheriting a property from parts or constituent. There can, however, be downward inheritance.

4. When a whole has the property of belonging to you, so do its parts, and often the parts inherit the property of being owned from the whole, though not always (you can buy a famous chess set piece by piece).

There may also be cases of sideways inheritance.

5. A layperson possesses the concept of a quark by inheritance from an expert to whom they defer with respect to the concept.

There seems to be some kind of a logical connection between property inheritance and property grounding, but the two concepts are not the same, since x’s possession of a property can be grounded in y’s possession of a different property—say, a president’s being elected is grounded in voters’ electing—while inheritance is always of the same property.

It is tempting to say:

6. An object x inherits a property F from an object y if and only if x’s having F is grounded in y’s having the same property F.

That’s not quite right. For if p grounds q, then p entails q. But this bundle of molecules’ having mass may not not entail my having mass, since it might be a contingent feature of the bundle that they are my molecules, so there is a possible world where the bundle exists and has mass, but I don’t (if only because I don’t exist). It seems that what we need in (6) is something weaker than grounding. But partial grounding seems too weak to plug into an account of property inheritance. Consider my property of knowing something. One of my pieces of knowledge is that you know something. So my knowing something is partially grounded in your knowing something, but I do not think that this counts as property inheritance. (Suppose one bites the bullet and says that my knowing something is inherited from you. Then, oddly, I have the property of knowing something both by inheritance and not by inheritance—inherited and non-inherited property possession are now compatible. I don’t know if that’s right, but at least it’s odd.)

I think we can at least say:

7. An object x inherits a property F from an object y only if x’s having F is grounded in y’s having the same property F.

But I don’t know how to turn this into a necessary and sufficient condition.

The mind-world similarity thesis

Eventually, the modern tradition becomes very suspicious the idea that there can be a similarity between the contents of the mind and characteristics of things in the external world. First, we have Locke denying the possibility of the similarity thesis for secondary qualities like red and sweet, and then we have others, like Berkeley and Reid, denying the possibility of the similarity thesis for primary qualities, like triangularity. In the case of primary qualities, it just seems absurd to think that the mind should hold something like a triangle.

This denial of the possibility of the similarity thesis seems to me to be a massive failure of the philosophical imagination, and a neglect of a sympathy to the history of philosophy. The allegation of the absurdity of thinking that triangularity should be present in the mind and in the world seems to come from thinking that the only way triangularity can be present in an entity is by the entity’s having triangularity. But why should having be the only possible relation by which triangularity could be present in a thing?

Here are some ways in which a property could be in a thing without the thing having the property.

• Let S be the set of the polygonality properties. Thus, the members of S are triangularity, quadrilaterality, etc. Triangularity is then in S qua member of S, but S is not a triangle—it does not have triangularity.

• On divine simplicity, God is identical with his divinity. But God can be present in Francis without Francis having God’s divinity—i.e., without Francis being divine.

• Suppose that I have a wood triangle in a steel box. The triangle’s triangularity is in the triangle, and the triangle is in the box, so the triangularity is in the box.

• Say that my fingernail is pointy. The properties of a thing are parts of a thing. So, the fingernail has its pointiness as a part. But the fingernail is a part of me, and parthood is transitive. So the pointiness of the fingernail is a part of me. But I am not pointy, even though I have a pointiness in me.

There is nothing absurd, then, about there being triangularity in the mind without the mind being itself triangular.

Moreover, having triangularity in the mind is not even a necessary condition for there to be a relevant similarity between the mind and a wooden triangle outside of me. It could be that the triangularity in the triangle is not a simple entity, but is composed of two components, T and P, where the P component is common (either as type or as token) between all properties, and the T component distinguishes triangularity from other properties. Thus, squareness might consist of S and P, and redness might consist of R and P. Well, then, we can suppose that when I think of or perceive a triangle as a triangle, then T comes to be in my mind without P doing so. Perhaps T comes to be “elementally” present in my mind, or perhaps it comes to be compounded with something else. (Here is a Thomistic version: triangularity has an essence T and a natural esse P; when present in the mind, the essence is there, but instead comes to have a different thing from the natural esse, say an intentional esse.) In either case, we have something importantly in common between the mind and the triangle qua triangular, namely T, without having triangularity in the mind, but only a component of triangularity.

There is no paucity of options. Indeed, we have an embarrassment of riches—many, many ways of making the similarity thesis true.

Theism and abundant theories of properties

On abundant theories of properties (whether Platonic universals or tropes), for every predicate, or at least every predicate satisfied by something, there is a corresponding property expressed by the predicate.

Here is a plausible sounding argument:

1. The predicate “is morally evil” is satisfied by someone.

2. So, on an abundant theory of properties, there exists a property of being morally evil.

3. The property of being morally bad, if it exists, is thoroughly evil.

4. So, on an abundant theory of properties, there exists something that is thoroughly evil.

5. If theism is true, nothing that exists is thoroughly evil (since every entity is the perfect God or created by the perfect God).

6. So if theism is true, an abundant theory of properties is false.

If I accepted an abundant theory of properties, I would question (3). For instance, maybe properties are concepts in the mind of God. A concept of something morally evil is not itself an evil concept.

Still, it does seem to me that this argument provides a theist with a little bit of a reason to be suspicious of abundant theories of properties.

Properties, relations and functions

Many philosophical discussions presuppose a picture of reality on which, fundamentally, there are objects which have properties and stand in relations. But if we look to how science describes the world, it might be more natural to bring (partial) functions in at the ground level.

Objects have attributes like mass, momentum, charge, DNA sequence, size and shape. These attributes associate values, like 3.4kg, 15~kg m/s north-east, 5C, TTCGAAAAG, 5m and sphericity, to the objects. The usual philosophical way of modeling such attributes is through the mechanism of determinables and determinates. Thus, an object may have the determinable property of having mass and its determinate having mass 3.4kg. We then have a metaphysical law that prohibits objects from having multiple same-level determinates of the same determinable.

A special challenge arises from the numerical or vector structure of many of the values of the attributes. I suppose what we would say is that the set of lowest-level determinates of a determinable “naturally” has the mathematical structure of a subset of a complete ordered field (i.e., of something isomorphic to the set of real numbers) or of a vector space over such a field, so that momenta can be added, masses can be multiplied, etc. There is a lot of duplication here, however: there is one addition operator on the space of lowest-level momentum determinates and another addition operator on the space of lowest-level position determinates in the Newtonian picture. Moreover, for science to work, we need to be able to combine the values of various attributes: we need to be able to divide products of masses by squares of distances to make sense of Newton’s laws of gravitation. But it doesn’t seem to make sense to divide mass properties, or their products, by distance properties, or their squares. The operations themselves would have to be modeled as higher level relations, so that momentum addition would be modeled as a ternary relation between momenta, and there would be parallel algebraic laws for momentum addition and position addition. All this can be done, one operation at a time, but it’s not very elegant.

Wouldn’t it be more elegant if instead we thought of the attributes as partial functions? Thus, mass would be a partial function from objects to the positive real numbers (using a natural unit system) and both Newtonian position and momentum will be partial functions from objects to Euclidean three-dimensional space. One doesn’t need separate operations for the addition of positions and of momenta any more. Moreover, one doesn’t need to model addition as a ternary relation but as a function of two arguments.

There is a second reason to admit functions as first-class citizens into our metaphysics, and this reason comes from intuition. Properties make intuitive sense. But I think there is something intuitively metaphysically puzzling about relations that are not merely to be analyzed into a property of a plurality (such as being arranged in a ball, or having a total mass of 5kg), but where the order of the relata matters. I think we can make sense of binary non-symmetric relations in terms of the analogy of agents and patients: x does something to y (e.g. causes it). But ternary relations that don’t reduce to a property of a plurality, but where order matters, seem puzzling. There are two main technical ways to solve this. One is to reduce such relations to properties of tuples, where tuples are special abstract objects formed from concrete objects. The other is Josh Rasmussen’s introduction of structured mereological wholes. Both are clever, but they do complicate the ontology.

But unary partial functions—i.e., unary attributes—are all we need to reduce both properties and relations of arbitrary finate arity. And unary attributes like mass and velocity make perfect intuitive sense.

First, properties can simply be reduced to partial functions to some set with only one object (say, the number “1” or the truth-value “true” or the empty partial function): the property is had by an object provided that the object is in the domain of the partial function.

Second, n-ary relations can be reduced to n-ary partial functions in exactly the same way: x₁, ..., x_n stand in the relation if and only if the n-tuple (x₁, ..., x_n) lies in the domain of the partial function.

Third, n-ary partial functions for finite n > 1 can be reduced to unary partial functions by currying. For instance, a binary partial function f can be modeled as a unary function g that assigns to each object x (or, better, each object x such that f(x, y) is defined for some y) a unary function g(x) such that (g(x))(y)=f(x, y) precisely whenever the latter is defined. Generalizing this lets one reduce n-ary partial functions to (n − 1)-ary ones, and so on down to unary ones.

There is, however, an important possible hitch. It could turn out that a property/relation ontology is more easily amenable to nominalist reduction than a function ontology. If so, then for those of us like me who are suspicious of Platonism, this could be a decisive consideration in favor of the more traditional approach.

Moreover, some people might be suspicious of the idea that purely mathematical objects, like numbers, are so intimately involved in the real world. After all, such involvement does bring up the Benacerraf problem. But maybe we should say: It solves it! What are the genuine real numbers? It's the values that charge and mass can take. And the genuine natural numbers are then the naturals amongst the genuine reals.

Distinguishing between properties

Some philosophers worry about “principles of individuation” that make two things of one kind be different from another. Suppose we share that worry. Then we should be worried about Platonism. For it is very hard to say what make two fundamental Platonic entities of the same sort different, say being positively charged from being negatively charged, or saltiness from sweetness.

However, the light-weight Platonist, who denies that predication is to be grounded in possession of universals, has a nice story to tell about the above kinds of cases. For here is a qualitative difference between saltiness and sweetness:

• saltiness is necessarily had by all and only salty things, but

• sweetness is not necessarily had by all and only salty things.

But for the heavy-weight Platonist to tell this story would involve circularity, for what it is for a thing to be salty will be to exemplify saltiness.

Of course, this story only works for properties that aren’t necessarily coextensive. But it’s some progress.

What are properties?

A difficult metaphysical question is what makes something be a property rather than a particular.

In general, heavy-weight Platonism answers the question of what makes x be F, when being F is fundamental, as follows: x instantiates the property of Fness.

It is hard to see what could be more fundamental on Platonism than being a property. So, a heavy-weight Platonist has an elegant answer as to what makes something be a property: it instantiates the second-order property of propertyhood.

Essences

Some properties that a thing has partially or wholly explain other properties the thing has or doesn’t have. For instance, my having a body partially explains my being in Waco and wholly explains my having a body or horns. Some properties that a thing has do not explain, even partially, what other properties the thing has or doesn’t have. Call such properties “explanatorily fundamental”.

So, here’s a theory. The primary essential properties of a thing are the explanatorily fundamental properties of the thing. The primary essential properties are both essential in the medieval explanatory sense and the contemporary modal sense (properties a thing cannot exist without).

What about the case of Christ, who is essentially divine and essentially human, and yet prior (in the order of explanation) to the incarnation was not human? Here’s what we could say: Divinity is the one and only primary essential property of Christ. But humanity is a secondary essential property. A secondary essential property of a thing is the sort of property that (a) is not a primary essential property of that thing, but (b) normally is the primary essential property of its possessor. In the case of Christ, his divinity is explanatorily prior to his humanity, but normally a thing’s humanity does not have any property of that thing explanatorily prior to it.

Could God be divinity?

Here's a plausible thesis:

1. If it is of x's essence to be F, then Fness is prior to x.

This thesis yields a fairly standard argument against the version of divine simplicity which identifies God with the property of divinity. For if God is divinity, then divinity is prior to divinity by (1), which is absurd.

But (1) is false. For, surely:

2. It is of a property's essence to be a property.

But propertyhood is a property, so it is of propertyhood's essence to be a property, and so propertyhood is prior to propertyhood if (1) is true, which is absurd. So, given (2), we need to reject (1), and this argument against the God=divinity version of divine simplicity fails.

What else might properties do?

Suppose that we think of properties as the things that fulfill some functional roles: they are had in common by things that are alike, they correspond to fundamental predicates, etc. Then there is no reason to think that these functional roles are the only things properties do. It is prima facie compatible with fulfilling such functional roles that a property do many other things: it might occupy space, sparkle, eat or think.

Can we produce arguments that the things that fulfill the functional roles that properties are defined by cannot occupy space, sparkle, eat or think? It is difficult to do so. What is it about properties that rules out such activity?

Here's one candidate: necessity. The functional roles properties satisfy require properties to exist necessarily. But all things that occupy space are contingent. And all things that sparkle or eat also occupy space. So no property occupies space, sparkles or eats. (Yes, this has nothing to say about thinking.) Yeah, but first of all it's controversial that all properties are necessary. Many trope theorists think that typical tropes are both contingent and properties. Moreover, it may be that my thisness is a property and yet as contingent as I am. Second, it is unclear that everything that occupies space has to be contingent. One might argue as follows: surely, for any possible entity x, it could be that all space is vacant of x. But it does not follow that everything that occupies space has to be contingent. For we still have the epistemic possibility of a necessary being contingently occupying a region space. Christians, for instance, believe that the Second Person of the Trinity contingently occupied some space in the Holy Land in the first century--admittedly, did not occupy it qua God, but qua human, yet nonetheless did occupy it--and yet the standard view is that God is a necessary being. (Also, God is said to be omnipresent; but we can say that omnipresence isn't "occupation" of space, or that all-space isn't a region of space.)

So the modal argument isn't satisfactory. We still haven't ruled out a property's occupying space, sparkling or eating, much less thinking. In general, I think it's going to be really hard to find an argument to rule that out.

Here's another candidate: abstractness. Properties are abstract, and abstracta can't occupy space, sparkle, eat or think. But the difficulty is giving an account of abstracta that lets us be confident both that properties are abstract and that abstract things can't engage in such activities. That's hard. We could, for instance, define abstract things as those that do not stand in spatiotemporal relations. That would rule out occupying space, sparkling or eating--but the question whether all properties are abstracta would now be as difficult as the question whether a property can occupy space. Likewise, we could define abstract things as those that do not stand in causal relations, which would rule out sparkling, eating and thinking, but of course anybody who is open to the possibility that properties can do these activities will be open to properties standing in causal relations. Or we could define abstractness by ostension: abstract things are things like properties, propositions, numbers, etc. Now it's clear that properties are abstracta, but we are no further ahead on the occupying space, sparkling, eating or thinking front--unless perhaps we can make some kind of an inductive argument that the other kinds of abstracta can't do these things, so neither can properties. But whether propositions or numbers can do these things is, I think, just as problematic a question as whether properties can.

All in all, here's what I think: If we think of the Xs (properties, propositions, numbers, etc.) as things that fulfill some functional roles, it's going to be super-hard to rule out the possibility that some or all Xs do things other than fulfilling these functional roles.

For more related discussion, see this old contest.

Thoughts on theistic Platonism

Platonists hold that properties exist independently of their instances. Heavy-weight Platonists add the further thesis that the characterization of objects is grounded in or explained by the instantiation of a property, at least in fundamental cases. Thus, a blade of grass is green because the blade of grass instantiates greenness (at least assuming greenness is one of the fundamental properties).

Heavy-weight Platonism has a significant attraction. After all, according to Platonism (and assuming greenness is a property),

1. Necessarily (i) an object is green if and only if (ii) it instantiates greenness.

The necessary connection between (i) and (ii) shouldn’t just be a coincidence. Heavy-weight Platonism explains this connection by making (ii) explain or ground (i). Light-weight Platonism, which makes no claims about an explanatory connection between (i) and (ii), makes it seem like the connection is a coincidence.

Still, I think it’s worth thinking about some other ways one could explain the coincidence (1). There are three obvious formal options:

2. (ii) explains (i)
3. (i) explains (ii)
4. Something else explains both (i) and (ii).

Option (2) is heavy-weight Platonism. But what about (2) and (3)? It’s worth noting that there are available theories of both sorts.

Here’s a base theory that can lead to any one of (2)–(4). Properties are conceptions in the mind of God. Furthermore, instantiation is divine classification: x’s instantiating a property P just is God classifying x under conception P. It is natural, given this base theory, to affirm (3): x’s instantiating greenness just is God’s classifying x under greenness, and God classifies x under greenness because x is green. Thus, x instantiates greenness because x is green.

But, interestingly, this base theory can give other explanatory directions. For instance, Thomists think that God’s knowledge is the cause of creation. This suggests a view like this: God’s classifying x under greenness (which on the base theory just is x’s instantiating greenness) causes x to be green. On this view, x is green because x instantiates greenness. If the “because” here involves grounding, and not just causation, this is heavy-weight Platonism, with a Thomistic underpinning. Either way, we get (2).

And here is a third option. God wills x to be green. God’s willing x to be green explains both x’s being green and God’s classifying x as green. The latter comes from God’s willing as an instance of what Anscombe calls intentional knowledge. This yields (4).

So, interestingly, a theistic conceptual Platonism can yield any one of the three options (2)–(4). I think the version that yields (3)—interestingly, not the Thomistic one—is the one that best fits with divine simplicity.

Divine aseity and light-weight Platonism

Here's a standard theistic argument against Platonism: If Platonism is true, then God is dependent on properties like divinity, goodness, omniscience and omnipotence. But God is not dependent on anything. So, Platonism is false.

I think it's worth noting that this argument only works given heavy-weight Platonism. The light- and heavy-weight Platonists agree that, at least if F is fundamental, x is F if and only if x instantiates Fness. But the heavy-weight Platonist adds the claim that if x is F, it is F because it instantiates Fness. The light-weight Platonist--van Inwagen is the most prominent example--makes no such explanatory claim.

Without the explanatory claim, the dependence argument for a conflict between Platonism and theism fails. For while it may be true on light-weight Platonism (assuming "is divine" is fundamental--something that Jon Jacobs at least will deny--or an abundant Platonism) that God is divine if and only if God instantiates divinity, we cannot conclude that God's being divine depends on God's instantiating divinity or on any other property. Indeed, the light-weight Platonist could (but does not have to) even make the opposite claim, that God instantiates divinity (or goodness, omniscience and omnipotence) because he is divine (and good, omniscient and omnipotent).

Of course, the aseity argument isn't the only reason to deny Platonism. God is the creator of everything other than himself, and that causes problems for properties, too.

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.

Fundamental logical relations

One might think the fundamental logical relations are between propositions. But I now think they are between relations (and propositions are a special case: 0-ary relations). Why? Well, in quantified logic (think: universal introduction and existential elimination) we need to talk about logical relationships between either (a) sentences with arbitrary names, (b) sentences with irrelevant names or (c) open formulas. Now ordinary sentences represent propositions, and the logical relations between sentences plausibly are grounded in the logical relations between the corresponding propositions. But sentences with arbitrary names don't represent anything, and so we have this unsatisfying grounding discontinuity: some logical relations between sentence-like entities in proofs are grounded in relations between proposition-like entities and others aren't. For somewhat similar reasons, if we want our logic to mirror the logical structure of reality, (b) isn't an option.

So we have philosophical reason to use a logic where there are logical relationships between open formulas. But an open formula represents a relation of arity equal to the number of free variables. It seems, thus, that some logical connections between propositions hold in virtue of logical connections between relations. Thus, the proposition that all humans are mortal follows from the propositions that all humans are animals and that all animals are mortal because the property (a property is a unary relation) of being mortal-if-human follows from the properties of being animal-if-human and being mortal-if-animal.

OK, time to stop procratinating grading the modal logic assignments!

From properties to sets

If we have abundant properties in our ontology, do we need to posit a second kind of entities, the sets?

Properties are kind of like sets. If P is a property, write x∈P if and only if x has P. A whole bunch of the Zermelo-Fraenkel axioms then are quite plausible. But not all. The most glaring failure is extensionality. The property of being human and the property of being a member of a globally dominant primate species have the same instances, but are not the same property.

We can get extensionality by a little trick and an axiom. Assume the following Axiom of Choice for Properties:

1. If R is any symmetric and transitive relation, then there is a property P such that (a) if x has P, then x stands in R to itself, and (b) for all x if x stands in R to itself, there exists a unique y such that x stands in R to y and y has P.

Like the ordinary Axiom of Choice, this is a kind of principle of plenitude. Apply (1) to the relation C of coextensionality that holds between two properties if and only if they have the same instances. This generates a property S₁ that is had only by properties and is such that for any property P there exists exactly one property Q such that P and Q are coextensive and Q has S₁. In other words, S₁ selects a unique property coextensive with a given property.

To a first approximation, then, we can think of those entities that have S₁ as sets. Then every set is a property, but not every property is a set. We certainly have extensionality, with the usual restriction to allow for urelements (i.e., extensionality only applies to sets). All the other axioms of Zermelo-Fraenkel with urelements minus Separation, Foundation and Choice are pretty plausibly true (they follow from plausible analogues for properties on an abundant view of properties). We get Choice for sets for free from (1).

Unfortunately, we cannot have Separation, however. For the property S₁ is coextensive to some set U by our assumptions. And the members of U will just be the instances of S₁, i.e., all the sets. And so we have a universal set, and of course a universal set plus Separation implies Comprehension, and hence the Russell Paradox.

So matters aren't so easy. The Axiom of Foundation is also not so clear. Might there not be a self-instancing property?

Thus the above simple approach gives us too many sets. But there is a solution to this problem, and this is simply to postulate the following second axiom about properties:

2. There is a property S₂ of properties such that (a) concreteness has S₂, and (b) all the axioms of Zermelo-Fraenkel Set Theory with Urelements minus Extensionality are satisfied when we stipulate that (i) a set is anything that has S₂ and (ii) A∈B if and only if A is an instance of B.

This axiom is fairly plausible, I think.

Now suppose that S₁ is as before, and let S₂ be any property satisfying (2). Then let S be the conjunction of S₁ and S₂. It is easy to see that if we take our sets to be those properties that have S, we will have all of Zermelo-Fraenkel with Choice and Urelements (ZFCU). Or at least so it seems to me—I haven't written out formal proofs, and maybe I need some further plausible assumptions about what abundant properties are like.

Of course, we cannot expect S₁ and S₂ to be unique. So there will be multiple candidates for sets. That's fine with me.

The big question is whether (1) and (2) are true. But if the theoretical utility of positing sets is a reason to think sets exist, then theoretical utility plus parsimony plus the reasons to believe in properties are a reason to think (1) and (2) are true.

Leibniz and the Gaifman-Hales Theorem

The basic idea behind Leibniz's characteristique is that all concepts are generated out of simple concepts, and there are no non-trivial logical relations between the simple concepts.

Today I was thinking about how to model this mathematically. Concepts presumably form a Boolean algebra. But infinities are very important to Leibniz. For instance, to each individual there corresponds a complete individual concept, which is an infinite concept specifying everything the individual does. So an ordinary Boolean algebra with binary conjunction and disjunction won't be good enough for Leibniz. We need concepts to form a complete Boolean algebra, one where an arbitrary set of elements has a conjunction and a disjunction.

So we want the space of concepts to be a complete Boolean algebra. We also want it to be generated by—built up out of—the set of simple concepts. Finally, we don't want there to be any nontrivial logical relations between the simple concepts. We want the theory to be entirely formal. This is one of Leibniz's basic intuitions. It seems to me that the way to formalize this condition is to say that the complete Boolean algebra is freely generated by the simple concepts.

Pity, though. The Gaifman-Hales Theorem implies that if there are infinitely many simple concepts, there is no complete Boolean algebra generated by them (this assumes a quite weak version of the Axiom of Choice, namely that every infinite set contains a countably infinite subset).

It looks, thus, like the Leibniz project is provably a failure.

Perhaps not, though. Apparently if one relaxes the requirement that the complete Boolean algebra be a set and allows it to be a proper class, but keeps the idea that the simple concepts form a set, one can get a complete Boolean algebra freely generated by the simple concepts.

Still, it's interesting that from an infinite set of simple concepts, one generates a proper class of concepts.

Particularizers instead of haecceities

A haecceity of x is a property that, necessarily, x and only x has. For instance, it might be the property of being identical with x. If a particularly strong converse to the essentiality of origins holds, a good choice for a haecceity would be a complete history of the coming-into-existence of x. Haecceities are a useful tool. For instance, they let one replace de re modality with de dicto. For another, they help explain what God deliberates about when he deliberates which individuals to create.

There is a different tool that can do some of the same work: a particularizer. We can think of an x-particularizer as equivalent to the second order property of being instantiated by x. Thus, if A is an x-particularizer, then necessarily a property Q has A if and only if x has Q. I will occasionally read "Q has A" as "A particularizes Q". The main trick to using particularizers is to note that, necessarily, x exists if and only if x instantiates some property. Thus, if A is an x-particularizer, then, necessarily, x exists if and only if some property has A.

Suppose that any two distinct things differ in some property and that particularizers exist necessarily.

Then we can use particularizers for de re modals. Suppose A is an x-particularizer. Then, Q is an essential property of x if and only if necessarily: if any property has A, then Q has A. If we have an abundant account of properties, we can then account for more complex modals. And we can likewise account for God's creative deliberation about individuals: God deliberates about which particularizers should be instantiated.

A particularly neat thing about particularizers is that with some generalization they allow us to reduce quantification over particulars to quantification over properties. We need the primitive predicate P where P(A) if and only if A is a particularizer. If A is a property, I will use A(y) to abbreviate: y has A. Use E(A) to abbreviate ∃B(A(B)). If A is a particularizer of x, then E(A) holds if and only if x exists. Use A~B to abbreviate ∀C(A(C) iff B(C)). If A and B are particularizers, then A~B means that they are co-particularizers—i.e., there is an x such that they are both x-particularizers. Suppose now we want to say that there are exactly two dogs. Let D be the property of being a dog. We say:

• ∃A∃B(P(A)&P(B)&A(D)&B(D)&~(A~B)&∀C((P(C)&C(D))→(A~C or B~C)).

I.e., there are particularizers that (a) particularize doghood, (b) are not co-particularizers, and (c) any particularizer that particularizes doghood is a co-particularizer of one of them.

If we want to deal with relations, and not just unary properties, then we need to generalize the notion of particularizers. One way to do this would to be suppose a primitive "multiplication" operation that forms an n-ary particularizer A₁A₂...A_n out of a sequence A₁,A₂,...,A_n, where an n-ary relation B has A₁A₂...A_n if and only if x₁,x₂,...,x_n stand in B, where A_i is an x_i-particularizer.

Instead of names of particulars, we will then work with names of their particularizers. Note that if in a Fregean way we think of quantifiers as corresponding to second-order properties, then particularizers will correspond to quantifiers (and remember the Montague way of thinking of names as quantifiers—this all fits neatly together).

Abundant Platonists who think that for every predicate there is a corresponding property should not balk at the existence of particularizers. We can define a particularizer either in terms of an entity x, as the property of being instantiated by x, or in terms of a haecceity H, as the property of having an instantiator in common with H. Likewise, we can define a haecceity in terms of a particularizer. If A is a particularizer, then the property of having all the properties that are particularized by A will make a fine haecceity. Or we can take particularizers to be primitive, whether we have abundant or sparse Platonism.

The above shows that we could do without first-order quantification and without talking of particulars. Now I think that nobody should simplify their ontology by getting rid of objects. Yet the above shows that we can do so. How to resist this simplifying reduction? I think the best way is to say that it does not sit well with the fundamentality of claims such as "I exist" and "I am conscious." For on the above reduction, these claims end up being reducible to E(A) and A(consciousness), where A is a me-particularizer. But only someone with an ontology on which "I exist" or "I am conscious" can resist the reduction in this way.

Two kinds of Platonism

There are two kinds of Platonism. Both hold that there are properties. But they differ as to the grounding relation that holds between predication and property possession (I will also assume that what goes for properties goes for relations, but sometimes formulate things just in terms of properties for simplicity). Both agree that if there is a property of Fness (there might not be if F is gerrymandered or negative, on sparse Platonisms), then x is F if and only if x instantiates Fness. Deep Platonism further affirms:

1. If there is a property of Fness, then the fact that x is F is grounded in the fact that x instantiates Fness.

Shallow Platonism denies (1). It is likely to instead affirm:

2. If there is a property of Fness, then the fact that x is F partly grounds or explains the fact that x instantiates Fness.

Deep Platonism faces two problems. The first is the Regress Problem. For if Deep Platonism is true, then "instantiates" seems non-gerrymandered and positive, and so it should correspond to a Platonic entity, the relation of instantiation. Then, the fact that x instantiates Fness will be grounded in the fact that x and Fness instantiate instantiation. But this leads to a vicious regress where each instantiation relation is grounded in the next.

The second is the Creation Problem. Everything that exists and is distinct from God is created by God. If the properties are all distinct from each other, then at most one is identical with God, and hence all but at most one property are created by God. But explanatorily prior to creating anything, will have multiple properties such as that he is able to do something and that he knows something. But how can he have those properties when there is at most one property at this point in the explanatory story?

Both problems have Deep Platonist solutions. For instance, one might say that "instantiates" is the unique non-gerrymandered and positive predicate that has no Platonic correspondent, or one might say that (1) has an exception in the case of instantiation. And one might say that God has at most one property, say divinity, and he is identical to that property. (But this, too, seems to lead to exceptions for (1), or perhaps an implausible view of what predicates correspond to properties. For there sure seem to be many other non-gerrymandered and positive predicates, like "is wise" and "is powerful", that apply to God.)

But Shallow Platonism has a particularly neat solution to both problems. There either is no regress, or if there is a regress, it is an unproblematic forward regress: because x is F, x and Fness instantiate instantiation, and because of that x, Fness and instantiation instantiate instantiation, and so on. Forward regresses are not at all problematic. And while it may be explanatorily prior to the creation of properties (or of all but one property) that God is wise, it is not explanatorily prior to the creation of properties that God instantiates wisdom.

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.

Grounding inhomogeneity and analogy

I ought to respect innocent human life. So I ought not feed cyanide to the innocent. I ought to respect the legitimate intellectual autonomy of others. So I ought not force my students to believe all my metaphysical views.

So, some ought claims are grounded, in part or whole, in other ought claims, and sometimes in further non-normative claims (such as that cyanide kills). This is familiar in many other cases. Thus, it's a standard libertarian view about freedom that some exercises of freedom are only derivately free: they are free insofar as they flow from a character that was formed by other free actions.

It would generate a vicious regress to suppose that all free actions are derivatively free. (In this case, the impossibility of the regress is obvious from the fact that we've only performed finitely many actions in our history.) Likewise, it would be a vicious regress to suppose all ought claims are grounded in further ought claims.

So there are some thing that are derivatively obligatory and some that are non-derivatively obligatory. (The two categories might overlap. For if I promise to fulfill a non-derivative obligation, then that obligation is both non-derivatively obligatory and obligatory by derivation from the duty to keep promises.) Likewise for freedom and many other properties. The non-derivative cases may be brute and ungrounded, or they may be grounded in a different kind of fact (e.g., maybe non-derivative freedom is grounded in alternate possibilities or non-derivative ought is grounded in divine commands—I am not advocating either option as it stands).

Here is a maxim I find plausible: Properties that exhibit this kind of grounding inhomogeneity—sometimes being grounded in one kind of fact and sometimes either ungrounded or grounded in a different kind of fact—are in fact non-fundamental.

This may lead one to say that properties that exhibit this kind of inhomogeneity are really disjunctive. That (or the related suggestion that they are existentially quantified) may be true, but I think it isn't the whole truth. Maybe freedom just is the disjunction of non-derivative and derivative freedom. But it's not a mere disjunction, in the way that being red or cubical is. It's a disjunction between related properties. In the cases of freedom and obligation, the relationship here seems to me to be precisely that of Aristotelian analogy: there is a focal sense of freedom and obligation—the non-derivative case—and there is a non-focal sense as well.

Conjecture: When we are dealing with a somewhat natural property that exhibits grounding inhomogeneity, we are precisely dealing with a disjunction (or quantificational combination) between analogous more fundamental properties.

Aquinas' thinking on divine names fits well here. St Thomas thinks that when we predicate wisdom of God and Socrates, we do so analogically, because God's wisdom is God and Socrates' wisdom is accidental to him. But this difference is precisely a grounding inhomogeneity in the property wisdom, with God being the focal case.