The filioque

Suppose we model the Trinity as three vertices with arrows between them representing relations of procession, such that:

1. No two vertices have two arrows between them and arrows only go between distinct vertices.

2. The arrows define the vertices in the sense that there is no way to relabel the vertices while preserving which vertex has an arrow to which vertext (i.e., there is no non-trivial directed graph automorphism).

There are seven different structures possible subject to (a).

Condition (b) is a way of capturing part of the traditional Western idea that the Trinity is relational, so that the distinctions between the persons arise from the processional structure. Four of the graphs violate this condition: in (1) we can rearrange the labels in any way without changing the relational structure; in (4) and (5) we can swap A and C; in (7), we can rotate the whole graph by 120 degrees.

Aquinas captures the relationality idea by the principle that person x is distinguished from person y if and only if there is a (real) processional relation between x and y. I think this principle is too strong in one way and too weak in another way. It is too strong, because (2) would allow the persons to be distinguished, even though there is no processional relation between A and C (or B and C): A would be the unique person from whom someone proceeds; B would be the unique person that proceeds; C would be the unique person unrelated by procession. It is too weak, because in (7) we have a processional relation between each person, but it fails to distinguish the three persons. Of course, (2) and (7) have other flaws.

The following options satisfy condition (b): (2), (3) and (6).

Graph (2) is theologically unacceptable, as it has a person not related to any other. And so while the relationality in (2) would be sufficient to define the persons, because C would be unrelated to A and B, the graph does not represent a fully relational Trinity. Besides which, (2) has two persons, A and C, who do not proceed from another, and the East and West agree that only the Father does not proceed.

Graph (3) is interesting. If we adopted this model of the Trinity, we would have to take A to be the Father, since everyone agrees that the Father does not proceed for anyone. Moreover, since everyone agrees that the Son proceeds from the Father, we would hae to

In graph (3), we would have to take A to be the Father, since the Father does not proceed from any other person, as the East and West agree. Likewise, the East and West agree that the Son comes from the Father. So B would have to be the Son. That would make the Holy Spirit be C, and proceed only from the Son. Neither the East nor the West will like this. And it would violate the principle that the Son eternally has nothing beyond what the Father has, excepting what follows from the Son’s being generated by the Father, since now the Son has the Spirit proceeding from him, but the Father does not. I suppose one might try to reconcile (3) with the Western view by saying that the Spirit proceeds mediately from the Father and immediately from the Son. I don’t like this.

That leaves graph (6), the Western view of the Trinity, with A being the Father, C being the Son and B being the Holy Spirit.

This isn’t really a great argument against the Eastern view of the Trinity, because the East will deny (b), on the grounds that one can distinguish the persons not merely by the directions of the procession, but by the types of processions, with generation and spiration being different types. One can think of this Eastern move as allowing the arrows to have different colors, and then instead of (b) having the condition that there is no way to relabel the vertices while preserving which vertex has an arrow of what color to what vertex.

The point here is rather to allow us to state with some precision what principles force the western view. Namely, we have:

Theorem: Suppose:

1. There are exactly three persons in the Trinity

2. The (binary) procession relation is asymmetric: if y proceeds from x, x does not proceed from y

3. At most most one person does not proceed from another

4. There is one person from whom both of the other persons proceed

5. There is no way to relabel the persons while preserving the procession relations.

Then there is a way of respectively naming the persons “Father”, “Son” and “Holy Spirit” such that it is correct to say: “The Father does not proceed, the Son proceeds from the Father, and the Holy Spirit proceeds from the Father and the Son.”

Symmetric relations and logic

Suppose Alice and Bob are friends, and that friendship is a fundamental relation. Consider the facts expressed by these two sentences:

1. Alice and Bob are friends.

2. Bob and Alice are friends.

It is implausible that these are different facts. For if they were different facts, they would both be fundamental facts (otherwise, which of them would be the fundamental one?), and we would be multiplying fundamental facts beyond necessity—only one of the two is needed in the totality of fundamental facts.

Furthermore, I think that the propositions expressed by (1) and (2) are the same. Here’s one reason to think this. Imagine a three-dimensional written language where plural symmetric predicates like “are friends” are written (say, laser-inscribed inside a piece of glass) with “and are friends” on one horizontal layer, with “Alice” on a layer below the “and” and “Bob” on a layer above it. If (1) and (2) express different propositions, we would have to ask which of them is a better translation of the three-dimensional language. But surely there is no fact about that.

If this is right, then First Order Logic (FOL) fails to accurately represent propositions about fundamental relations, by having two atomic sentences, F(a,b) and F(b,a), where there is only one fundamental fact. Moreover, FOL will end up having non-trivial proofs whose conclusion expresses the same proposition as the premise, since we will presumably have an axiom like ∀x∀y(F(x,y)→F(y,x)) that lets us prove F(b,a) from F(a,b). This is not the only example of this phenomenon. Take the proof that ∀xF(x) follows from ∀yF(y), even though surely the two express the same proposition, namely that everything is F.

In particular, the logic of sentences appears to differ from the logic of propositions, since the proposition that Bob and Alice are friends follows by reiteration from the proposition that Alice and Bob are friends if they are the same proposition, but sentence (2) does not follow from sentence (1) by reiteration (nor is this true for the FOL versions).

If we think there is a One True Logic, it presumably will be a logic of propositions rather than sentences, then. But what it will be like is a difficult question, to answer which we will have to a worked out theory of when we have the same proposition and when we have different ones.

Consciousness and plurality

One classic critique of Descartes’ “cogito ergo sum” is that perhaps there can be thought without a subject. Perhaps the right thing to say about thought is feature-placing language like “It’s thinking”, understood as parallel to “It’s raining” or “It’s sunny”, where there really is no entity that is raining or sunny, but English grammar requires a subject so we through in an “it”.

There is a more moderate option, though, that I think deserves a bit more consideration. Perhaps thought has an irreducibly plural subject, and in a language that expresses the underlying metaphysics better, we should say “The neurons are (collectively) thinking” or maybe even “The particles are (collectively) thinking.” On this view, thought is a relation that holds between a plurality of objects, without these objects making up a whole that thinks. This, for instance, is a very natural view for physicalist who is a compositional nihilist (i.e., thinks that only simples exists).

It seems to me that it is hard to reject this view if one’s only data is the fact of consciousness, as it is for Descartes. What kills the three-dimensionalist version of this view, in my opinion, is that it cannot do justice to the identity of the thinker over time, since there would be different pluralities of neurons or particles engaged in the thinking over time. And a four-dimensionalist version cannot do justice to the identity of the thinker in counterfactual scenarios. However, this data isn’t quite as self-evident as what Descartes wants.

In any case, I think this is a view that non-naturalists like me need to take pretty seriously.

Substances and their existences

I used to think that:

1. x is a substance just in case  <x exists> is not grounded in any fact about any other entity other than x.

But it is plausible that a finite creature’s existing is its being created. And that Alice is created seems to be grounded in God creating Alice, which seems to be a fact about God.

There is a nice response to this worry. On standard medieval views, creation is a one-way relation. When God creates Alice, there is a relation of being-created-by-God in Alice but no relation of creating-Alice in God. We can say, then, that in an important sense the fact that God creates Alice is not a fact about God, but about Alice, where we say that a fact is about x provided the fact is in part a fact of x’s existing or there being some property or relation in x.

It’s interesting that the very plausible account (1) of substance combined with a theistically plausible view that the esse of a finite thing is its being-created yields the rather abstruse one-way relation thesis.

This line of thought does not, however, fit well with the claim that I made in The Principle of Sufficient Reason that for a caused entity, its esse is its being caused. For Alice is also caused by her parents. And while divine causation may be a one-way relation, it seems unlikely that creaturely causation is.

There are three ways out of this worry. (i) We could say that creaturely causation is also a one-way relation. (ii) We could say that I was slightly wrong, and for a caused entity, its esse is its being primarily caused, i.e., caused by God. (iii) We could modify (1) to:

2. x is a substance just in case  < x exists> has a grounding in a fact that is neither about any entity other than x nor grounded in a fact about any entity other than x.

For we can then say that while Alice’s being-caused is grounded in her parents’ activity, which is a fact about her parents, it is also grounded in God’s causing Alice, which is not a fact about God in the sense of being grounded in a relation or property of God’s.

I like both (ii) and (iii). What is especially attractive about (ii) is that if the esse of Alice is her being caused, then the esse of Alice is highly disjunctive, being multiply grounded—in God’s causing Alice, in her parents causing her, in her parents’ gametes causing her, maybe even in her grandparents’ causing her, etc. But it doesn’t seem right to say that Alice’s esse is highly disjunctive. So a focus on primary causation seems attractive. And I think—but without a careful examination—that the arguments in Principle of Sufficient Reason work with that modification still.

Eliminating contingent relations

Here’s a Pythagorean way to eliminate contingent relations from a theory. Let’s say that we want to eliminate the relation of love from a theory of persons. We suppose instead that each person has two fundamental contingent numerical determinables: personal number and love number, both of which are positive integers, with each individual’s personal number being prime and an essential property of theirs. Then we say that x loves y iff x’s love number is divisible by y’s personal number. For instance suppose we have three people: Alice, Bob and Carl, and Alice loves herself and Carl, Bob loves himself and Alice, and Carl loves no one. This can be made true by the following setup:

	Alice	Bob	Carl
Personal number	2	3	5
Love number	10	6	1

While of course my illustration of this in terms of love is unserious, and only really works assuming love-finitism (each person can only love finitely many people), the point generalizes: we can replace non-mathematical relations with mathematizable determinables and mathematical relations. For instance, spatial relations can be analyzed by supposing that objects have a location determinable whose possible values are regions of a mathematical manifold.

This requires two kinds of non-contingent relations: mathematical relations between the values and the determinable–determinate relation. One may worry that these are just as puzzling as the contingent ones. I don't know. I've always found contingent relations really puzzling.

Spacetime and Aristotelianism

For a long time I’ve been inclining towards relationalism about space (or more generally spacetime), but lately my intuitions have been shifting. And here is an argument that seems to move me pretty far from it.

Given general relativity, the most plausible relationalism is about spacetime, not about space.

Given Aristotelianism, relations must be grounded in substances.

Here is one possibility for this grounding:

1. All spatiotemporal relations are symmetrically grounded: if x and y are spatiotemporally related, then there is an x-to-y token relation inherent in x and a y-to-x token relation inherent in y.

But this has the implausible consequence that there is routine backwards causation, because if I walk a step to the right, then that causes different tokens of Napoleon-to-me spatiotemporal relations to be found in Napoleon than would have been found in him had I walked a step to the left.

So, we need to suppose:

2. Properly timelike spatiotemporal relations are grounded only in the later substance.

But what about spacelike spatiotemporal relations? Presumably, they are symmetrically or asymmetrically grounded.

If they are symmetrically grounded, then we have routine faster-than-light causation, because if I walk a step to the right, then that causes different tokens of x-to-me spatiotemporal relations to be found in distant objects throughout the universe.

Moreover, on the symmetric grounding, we get the odd consequence that it is only the goodness of God that guarantees that you are the same distance from me as I am from you.

If they are asymmetrically grounded, then we have arbitrariness as to which side they are grounded on, and it is a regulative ideal to avoid arbitrariness. And we still have routine faster-than-light causation. For presumably it often happens that I make a voluntary movement and someone on the other side of the earth makes a voluntary movement spacelike related to my movement (because there are so many people!), and now wherever the spatiotemporal relations is grounded, it will have to be affected by the other’s movement.

I suppose routine faster-than-light causation isn’t too terrible if it can’t be used to send signals, but it still does seem implausible. It seems to me to be another regulative ideal to avoid nonlocality in our theories.

What are the alternatives to relationalism? Substantivalism is one. We can think of spacetime as a substance with an accident corresponding to every point. And then we have relationships to these accidents. There is a lot of technical detail to work out here as to how the causal relationships between objects and spacetime points and the geometry of spacetime work out, and whether it fits with an Aristotelian view. I am mildly optimistic.

Another approach I like is a view on which spacetime position is a nonrelational position determinable accident. Determinable accidents have determinates which one can represent as values. These values may be numerical (e.g., mass or charge), but they may be more complex than that. It’s easiest in a flat spacetime: spacetime position is then a determinable whose determinates can be represented as quadruples of real numbers. In a non-flat spacetime, it’s more complicated. One option for the values of determinate positions is that they are “pointed spacetime manifold portions”, i.e., intersections of a Lorentzian manifold with a backwards lightcone (with the intended interpretation that the position of the object is at the tip of the lightcone). (What we don’t want is for the positions to be points in a single fixed manifold, because then we have backwards causation problems, since as I walk around, the shifting of my mass affects which spacetime manifold Napoleon lived in.)

Self-locating beliefs in the Trinity

Here is a difficulty for the doctrine of the Trinity that I don’t remember coming across before:

1. The Father and the Son have the numerically same divine mind.

2. If x and y have the numerically same divine mind, then x and y have the same divine beliefs.

3. The Father has an “I am the Father” divine belief.

4. So, the Son has an “I am the Father” divine belief. (1–3)

5. An “I am the Father” divine belief in the Son would be false.

6. There are no false divine beliefs.

7. So, the Son has no “I am the Father” divine belief. (5–6)

8. Contradiction!

Here, premise (1) follows from the heuristic that what there are two of in Christ, there is one of in the Trinity: there are two minds in Christ, so one mind in the Trinity. Non-heuristically, if there are two minds in Christ—the human and the divine mind—the mind must be a function of the nature, and as there is one divine nature in the Trinity, there is one mind in the Trinity.

There is a quick way out of the paradox: Restrict premise (2) to propositional beliefs rather than de se or self-locating beliefs. The belief that would be expressed in English by “I am the Father” is a de se or self-locating belief. There are corresponding propositional beliefs, such as the belief that the Father is the Father and the Son is the Son, but these are unproblematically had in common by the Father, the Son and the Holy Spirit.

However, while this quick way gets one out of the argument, it nonetheless leaves raises the difficult question of how it is the Father knows de se that he is the Father and the Son knows de se that he is not the Father, while yet there is one mind.

The solution had better be in terms of the relations between the divine persons, for there is no difference between the persons of the Trinity except the relational. I am reminded here of Thomas’s discussion of creation and the Trinity:

And therefore to create belongs to God according to His being, that is, His essence, which is common to the three Persons. Hence to create is not proper to any one Person, but is common to the whole Trinity.

Nevertheless the divine Persons, according to the nature of their procession, have a causality respecting the creation of things. For as was said above, when treating of the knowledge and will of God, God is the cause of things by His intellect and will, just as the craftsman is cause of the things made by his craft. Now the craftsman works through the word conceived in his mind, and through the love of his will regarding some object. Hence also God the Father made the creature through His Word, which is His Son; and through His Love, which is the Holy Ghost. And so the processions of the Persons are the type of the productions of creatures inasmuch as they include the essential attributes, knowledge and will.

Thus, each divine person is fully the Creator, but is fully the Creator in a way that takes into account the relationship that defines the person in the Trinity: the Father creates in a Fatherly way, the Son as the Logos through which creation is done, and the Spirit as the Love in which creation is inspired. What makes it be the case that the Father creates in a Fatherly way is just that the Father creates and he stands in the relations that constitute him as Father; what makes it be the case that the Son creates in a Filial way is just that the Son creates and he stands in the relations that constitute him as Son; and similarly for the Holy Spirit.

We might thus imagine the following story. There is a state F of the divine mind such that the Father’s Fatherly instantiation of F constitutes F into a belief that he (de se) is the Father. The Son instantiates the numerically same state F in a Filial way. But while a Fatherly instantiation of F is correctly described in English as constituting an “I am the Father” belief, a Filial instantiation of F is not aptly so described. Perhaps, a Filial instantiation of F is aptly described as a believing of “I am the Son of the one who is the Father.” Thus, the de se beliefs of the persons of the Trinity are constituted by mental states common to the Trinity and the relations constituting the persons.

Causation and Thomism

Assume a Thomistic metaphysics, including the primary/secondary causation model from Aquinas. Thus, whenever a created cause has an effect, it has the effect it does only because God, through primary causation, cooperates with the created cause. If God did not cooperate with the created cause, the creature’s secondary causal power would be impotent to produce the effect. On the other hand, God can directly cause something by primary causation without the secondary causes doing anything.

Suppose that I strike a match, but God doesn’t cooperate in the frictional causing of fire. Then the match is struck, and does everything a struck match does, except that no fire results. Now imagine these two scenarios:

1. I strike a match which, in the ordinary way and with God’s cooperation, causes fire.

2. I strike a match, but God does not cooperate with me; however, God miraculously causes a fire just like the one in (1).

These two scenarios are different. So they must differ in something. They cannot differ in God, since that would violate divine simplicity, a core commitment of Thomistic metaphysics. So they must differ on the side of creatures. If so, they differ in the match striking or in the fire (or, more precisely, in the match and in that which is on fire, given a substance-accident ontology), God’s cooperation or lack thereof making the match-striking or the fire different.

Suppose God’s cooperation makes the match-striking different. Then in scenario (1), created reality includes the event of divinely-cooperated-match-striking. This event surely doesn’t need any further divine cooperation, or else we’d have a regress. But no item in created reality is sufficient on its own to produce an effect witout God’s cooperation being added to it on the primary/secondary causation model.

So, it is the fire that must be made different by God’s cooperation. In scenario (1), the fire is caused by the match and God while in (2), it is caused by God alone, and that makes the fire different between the scenarios. I would like to say that the esse of the fire is different in the two cases: in the ordinary case its esse is at least in part being caused by the match with God’s cooperation while in the other case the match doesn’t enter into its esse. But the details don’t matter for this post: what matters is that the fire is different between scenarios (1) and (2).

But this has an unfortunate consequence: If the fire must be different in some metaphysical way in cases (1) and (2), it follows that God cannot directly and independently of creation cause the same effect as the match caused. And this violates the Thomistic principle that whatever a finite cause suffices for can be produced by God directly. God cannot directly produce a fire-caused-by-the-match; that would be a contradiction.

So we have a conflict between a number of Thomistic principles:

3. Divine simplicity

4. Divine omnipotence

5. The primary/secondary causation model

6. God can directly cause anything he can cause in cooperation with a creature.

It seems to me that 3-5 are more central to Thomism than 6. So, I am inclined to reject 6, perhaps replacing it with the weaker principle that for any item x that God can cause in cooperation with a creature, God can cause an item x^* which is qualitatively just like x.

Perhaps I was too quick when I said that (1) and (2) must differ in the match or the fire. Perhaps they differ in something “in between” the match and the fire, a token causal relation. I think this is a problematic solution for two reasons. First, it is central to Aristotelianism that all that exists are substances and their accidents. The token causal relation, if it’s not “in” any creature, would violate this. Second, it seems that the match strike plus this “in between” thingy are now sufficient to produce the fire, or else we can run the above argument with the match strike and the “in between” thingy in place of the match strike. But no mere creature is sufficient to produce an effect without God’s cooperation.

Eliminativist relational dualism

Here is a combination of views that, as far as I know, is missing from the literature:

• eliminativism about persons

• multigrade relational dualism.

Let me explain the view. There are no people on earth. There are just particles arranged humanwise. No particle by itself has mental properties and there is no whole having mental properties. But there is irreducibly collective activity mental activity. The humanwise-arranged particles responsible for this post stand in non-physical mental relations, such as collectively being aware of the smoothness of the spacebar and collectively intending to communicate a novel philosophy of mind view. These relations are multigrade in the sense that there is no specific number of particles that are needed to stand in such a relation.

There are about 10²⁸ (give or take an order of magnitude, depending on whether we count just the brain or the whole body) particles jointly responsible for this post, but some particles are always flying off and terminating their participation in the relations and others are joining, which is why the mental relation is multigrade.

The availability of the view shows that arguments, like those of van Inwagen and Merricks, for the existence of complex wholes based on our mental function need more work. As does Descartes’ “I think therefore I am”: perhaps all we can say at the outset is is “There is thinking so there is one or more things that individually or jointly engage in thought.”

The view has the very attractive feature that it is compatible with nihilism about parthood: there need be no part-to-whole relation. This allows for a very nice and simple ideology.

The view is a dualist view, and hence puzzles about phenomenal properties, the unity of consciousness and intentionality can be solved much like in property dualism. (E.g., consciousness is unified by joint possession of a consciousness property.)

But the typical property dualist has two things to explain: why a single entity arises from a bunch of particles arranged a certain way and why that entity gains mental properties. The eliminative relational dualist only needs to explain the latter.

At the same time, some of the difficulties with the more normal kinds of eliminativism about persons (think of Unger and the Churchlands) are neatly solved. The idea of thought without any subject of thought is absurd. But we can draw on reflections in social epistemology to get a model for thought with an irreducibly plural subject of thought.

As I think is often the case with metaphysical views, the main difficulties arise in ethics. There is no particular difficulty about being responsible. The particles that engaged in a joint action are jointly responsible. The problem, however, is with holding responsible. Particles are always leaving and entering the mental relations. Many of those particles that were responsible for the robbery last year are now scattered across the city, and while (at least for last year’s robbery) a bunch of them are still clumped together, it’s impossible to punish them without punishing a plurality of particles that includes a number of particles that had nothing to do with the robbery. And this is nothing compared to the difficulty of punishing cannibalism, since we will end up punishing many victimized particles.

Multigrade relations

One strategy for avoiding ontological commitment to sets is to deal with pluralities and multigrade relations. Multigrade relations are relations that can be had by a variable number of things. Instead of, say, saying of the books on my shelf that there is a set of them whose total number of pages is exactly 800, one says that there are xs such that each of them is a book on my shelf and the xs stand in the multigrade relation of jointly having 800 pages. Let’s say these books are x₁, x₂ and x₃. Then we express their jointly having 800 pages as:

• Has800Pages(x₁, x₂, x₃).

We do not need a set of them to express this. And the Has800Pages(x₁, ...) predicate flexibly can take as many arguments as one wishes, corresponding to the multigradeness of the property it expresses.

But now consider a different statement: there are two pluralities of books on my shelf, having no books in common, where each plurality has the same total number of pages as the other. Can we make sense of this using multigrade relations instead of sets?

I don’t see how. Let’s say that the plurality x₁, x₂, x₃ and the plurality y₁, y₂ of my books each have the same total number of pages. So we introduce a predicate with variable arity and say:

• HasSameTotalNumberOfPages(x₁, x₂, x₃, y₁, y₂).

But that doesn’t work! For how can we tell if it is says that x₁, x₂, x₃ have the same number of pages as y₁, y₂ rather than saying that x₁, x₂ have the same number of pages as x₃, y₁, y₃?

We could multiply predicates with fixed arity and say:

• TheFirst3HaveTheSameTotalNumberOfPagesAsTheLast2(x₁, x₂, x₃, y₁, y₂).

But that won’t work with quantification, since we don’t know ahead of time how many xs and how many ys we are dealing with.

Maybe we should do this:

• Count(3,x₁, x₂, x₃) and Count(2,y₁, y₂) and SameNumberOfPages(3,x₁, x₂, x₃,2,y₁, y₂)

where the SameNumberOfPages variable arity predicate takes a number, then a plurality of that number of objects, then another number, and then another plurality of that number of objects.

But these kinds of solutions won’t work for infinite pluralities. For instance, suppose we want to say that the xs cause the ys, where there are ℵ₂ xs and ℵ₃ ys. Then I guess we say something like:

• Cause(ℵ₂, x₁, x₂, ..., ℵ₃, y₁, y₂, ...).

There are serious technical problems here, however. I will leave it to the reader to explore these.

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.

Intrinsic attribution

1. If heavyweight Platonism is true, all attribution of attributes to a subject is grounded in facts relating the subject to abstracta.

2. Intrinsic attribution is never grounded in facts relating a subject to something distinct from itself.

3. There are cases of intrinsic attribution with a non-abstract subject.

4. If heavyweight Platonism is true, each case of intrinsic attribution to a non-abstract subject is grounded in facts relating that object to something other than itself. (By 1 and 2)

5. So, if heavyweight Platonism is true, there are no cases of intrinsic attribution to a non-abstract subject. (2 and 4)

6. So, heavyweight Platonism is not true. (By 2 and 5)

Here, however, is a problem with 3. All cases of attribution to a creature are grounded in the creature’s participation in God. Hence, no creature is a subject of intrinsic attribution. And God’s attributes are grounded in a relation between God and the Godhead. But by divine simplicity, God is the Godhead. Since the Godhead is abstract, God is abstract (as well as being concrete) and hence God does not provide an example of intrinsic attribution with a non-abstract subject.

I still feel that there is something to the above argument. Maybe the sense in which a creature’s attributes are grounded in the creature’s participation in God is different from the sense of grounding in 2.

Minimizing the number of fundamental relations between concreta

An interesting metaphysics project that could use more work is to minimize the number of fundamental non-intentional relations between substances. How few can we do with? I think it would be really great if one could reduce the number of such relations to a handful of relations, or at least determinable families of relations. There is only one candidate really clear to me: causation. I think it would be an interesting research project to adopt the working hypothesis that causation is the one and only such relation and see how things go.

A lot of people will want to add parthood to this list, but I don't think a substance can be a part of another substance. (Parts are grounded in wholes, and substances are not grounded in other things.) Spatial relations like being seven meters apart (on relationalism about locations) and being located at (on substantivalism about locations) are a family of plausible candidates.

Relational gender essentialism

It might turn out to be like this: There is no significant difference between matter and antimatter, except insofar as they are related to one another. A proton is attracted to antiproton, while each is repelled by its own kind. Our universe, as a contingent matter of fact, has more matter than antimatter. But, perhaps, if one swapped the matter and antimatter, the resulting universe wouldn't be different in any significant way. If we this is true, we might say that there is a relational matter-antimatter essentialism. It is of great importance to matter and to antimatter that they are matter and antimatter, respectively, but it is important only because of the relation between the two, not because of intrinsic differences.

I don't know if it's like that with matter and antimatter, but I do know that it's like that with the Father, the Son and the Holy Spirit. The only important non-contingent differences are those constituted by the relationships between them. (There are also contingent extrinsic differences.)

Could it be like that with men and women? The special relation between men and women--say, that man is for woman and woman for man, or that one of each is needed for procreation--is essential and important to men and women. But there are no important non-contingent intrinsic differences on this theory.

There might, however, be important contingent theological differences due to some symmetry-breaking contingent event or events. Maybe, when the Logos became one human being, the Logos had to become either a man or a woman. If the relation between men and women is important, the decision whether to become a man or to become a woman, might have been a kind of symmetry-breaking, with other differences in salvation history following on it. In itself, that decision could have been unimportant. If the Logos had become a woman, we would have a salvation history that was very much alike, except now Sarah would have been asked to sacrifice a daughter, we would have had an all-female priesthood, and so on.

Or perhaps the symmetry-breaking came from the contingent structure of our sinfulness. Perhaps the contingent fact that men tended to oppress women more than the other way around made it appropriate for the Logos to become a man, so as to provide the more sorely needed example of a man becoming the servant of all and sacrificing himself for all, and in turn followed the other differences.

I don't know if relational gender essentialism is the right picture. But it's a picture worth thinking about.

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!

A Platonic theory of determinables

In an earlier post, I explored, without endorsing, a Platonic theory of spacetime, on which spacetime is an abstract Platonic entity, and objects are located by virtue of standing in a relation to abstract points of that entity.

This could extend to other determinables.  Consider, for instance, mass, and simplify by supposing presentism or lack of time variation.  An object o could have mass x, where x is some real number (we need a natural unit system for that), precisely in virtue of o's being M-related to the real number x, a Platonic entity, where M is a natural "mass relation".  This works even for much more complex determinables like wavefunctions.  Thus, an object o could have a wavefunction in virtue of being W-related to some abstract function from R³ to C, again assuming presentism or lack of time variation.  To get time variation into the picture, we could suppose that the mass relation relates objects to functions from a time sequence (an internal time sequence?) to reals.
This would help with regard to the epistemology of abstracta even if (contrary to fact, I am inclined to say) abstracta are causally inert.  For even if the number x is causally inert, the event of o being M-related to x is not causally inert (it causes gravitational influences, for instance).

One intuitive difficulty for this theory is that it is now looking logically possible for an object to have two masses or two wavefunctions at any given time.  I do not think this consequence absurd myself.  If the second person of the Trinity became incarnate as two different humans at the same time, which Aquinas thinks is possible (a possibility that we may care about if it turns out that there are fallen non-human rational beings), he might have two different masses at a given time.  Alternately, one can just say that there are brutely necessary restrictions here.

Notice an interesting consequence of this theory.  If a naturalist were to adopt this theory, it might make it easier to get her to accept a non-reductionist theory of mind on which for us to believe a proposition just is to stand in an irreducible belief relation to a proposition.  After all, it is no more philosophically puzzling how one can stand in an irreducible mass relation to a number or function than it is how one can stand in an irreducible belief relation to a proposition.  And it is no more philosophically puzzling how one's standing in a belief relation to a proposition could causally affect one's behavior than how one's standing in a mass relation could.

What bothers me about this theory, as well as the earlier theory of spacetime, is that abstracta are divine ideas.  But it seems wrong to say that mass and location facts are constituted by a relation to God.  That sounds too panentheistic.  But here's one interesting philosophical/theological question.  Aquinas insists that things are the way they are by participation in God.  Thus, Socrates is wise by (natural) participation in God (and Paul is wise by supernatural participation in God).  Does this mean that (a) Socrates' accidental form of wisdom is identical with a participating in God or does it mean that (b) Socrates' accidental form of wisdom is something distinct from but dependent on Socrates' participating in God?  If the former, then the Platonic theory I offered will be no more problematic than Aquinas' view (but of course I'll want to say something like what Aquinas says about one-sided relations, so that the mass relation is a relation to God but there is no corresponding relation of God to the object--maybe the suggestion in this post helps), and in fact Aquinas' view might just be a variant of the Platonic theory.  If the latter, then the Platonic theory is more panentheistic than Aquinas', and insofar as Aquinas seems to me to be as close as one can orthodoxly come to panentheism, I would then reject the Platonic theory.

There is also going to be some trickiness coordinating the location determinable with the other determinables.  We want to be able to say things like "x is beige on its left side".  Working this out may require me to abandon the heuristic that there is nothing special about location--that it's just another relation.