A difference among relations

I want to get at a certain difference between relations like love and hate, on the one hand, and relations like causation and parthood, on the other.

One way to see the difference is that there are significant formal constraints on the distribution of causation and parthood, but no formal contraints on the distribution of love and hate. For instance, there cannot be cycles of causation and parthood must be transitive. But there are no such constraints on love and hatred: any combination of loves and hatreds is possible for ordinary human beings.

Another way to see what seems to be the same difference is this. Bracketing considerations of God’s goodness, God can just directly freely insert and remove instances of love and hatred within a bunch of people without changing anything else. But it seems that God can’t just directly freely insert and remove instances of causation and parthood within a bunch of entities. It seems he can’t, for instance, reverse the direction of causation between my father and myself, or will me to be a part of my heart.

I find it a very mysterious feature of causation and parthood that it has such formal constraints.

The lack of free recombinability for a relation is sometimes explained by the relation being non-fundamental. For instance, “darker than” must be transitive, but the reason is that its obtaining depends on more fundamental facts, such as how dark the two relata are, and this dependence induces the logical features of the relation. It could turn out that causation and parthood do depend on more fundamental things, though this does not seem very plausible. But at the same time, it seems pretty plausible that whatever more fundamental things causation and parthood depend on, these will include relations, and relations of a sort that have the non-recombinability features of causation and parthood.

Anyway, it seems to me that there is some fundamental difference between fundamental relations that have no formal constraints and fundamental relations that do have formal constraints, and I don’t know what accounts for it. I feel that this difference is a mere indicator of some deeper difference in kind. Causation and parthood “look” very different from love and hatred.

Causal well-foundedness

I have defended causal finitism at length, mainly (but not only) as a way of getting out of paradoxes.

I want to think about a weaker thesis than causal finitism today. Causal finitism says that there is no item that has infinitely many things causally prior to it.

A weaker thesis is that causal priority (which I will assume to be transitive) is causal well-foundedness, namely that any set S of items contains a minimal item—an item that has nothing in S causally prior to it. Assuming the Axiom of Choice (and its applicability to sets of “items”), this is equivalent to saying there are no backwards infinite regresses ... < A₋₃ < A₋₂ < A₋₁ < A₀, where < indicates causal priority.

The arrangement of the Grim Reapers appears not to be well-founded: each one’s actions respond to the actions of the one before. So causal well-foundedness seems to rule out that paradox just as well as causal finitism.

At the same time, just as causal finitism implies the existence of a first cause (and hence yields a kalaam argument), so does causal well-foundedness: just take the set of all items that are causally prior to something, and a minimal item will be a first cause.

So far, no differences.

But alas once we start looking at other paradoxes, differences arise. Thomson’s lamp is unaffected by causal well-foundedness: the story is well-founded. Similarly, it seems one can shuffle an infinite deck of cards using a well-founded supertask. Likewise, there are paradoxes arising from from infinitely many things affecting a single result without any chain, like the paradox where infinitely many people roll independent and fair dice without seeing the results, and an angel announces that all but finitely many got sixes.

So you get what you pay for. Causal well-foundedness is a weaker assumption than causal finitism, but it also rules out fewer paradoxes.

Beginnings and causation

Some people think that it’s way more obvious that a contingent thing that has a beginning has to have a cause than that a contingent that has no beginning has to have a case.

Here’s one reason to be suspicious of this intuition. Whether something contingent needs to have a cause should not depend on its orientation. But whether something has a beginning is a function of its orientation in spacetime. Take an object that has extends backward in time to infinity but forwards only finitely, and hence has no beginning. Consider this as a four-dimensional object. You can then rotate it so it extends forwards in time to infinity but only a finite distance backward. You’ve merely changed its orientation, but now the object has a beginning.

This is very much a four-dimensionalist argument, of course.

Theoretical parsimony

As is usual, distinguish ontological parsimony, namely positing fewer beings, from ideological parsimony, namely positing fewer fundamental undefined concepts.

The principle that we should assign higher probabilities to more ontologically parsimonious theories is wacky. It violates the Principal Principle. If you are flipping a coin and planning to have children if and only if you get heads, then you should assign a slightly higher probability to tails. (Granted, this argument fails if eternalism is false. But you can tweak it by supposing that a hundred years ago Alice flipped a fair coin with that intention, and now you should again assign a slightly higher probability to tails.)

What about ideological parsimony? It initially sounds a lot better to assign higher probabilities to more ideologically parsimonious theories. Indeed, I think there has to be something right about such a principle of simplicity. But I don’t at present fully know how to formulate it in a way that escapes objection.

For suppose that we are non-reductionists about biological species, so we think that species are fundamental undefined concepts. You are working with a bacterium in the lab, and on a whim plan to flip a coin. On heads, you will irradiate the bacterium, and on tails you will do no such thing. Irradiated, the bacterium is more likely to produce a new species when it reproduces. So on ideological parsimony it seems we should assign a slightly higher probability to tails, as then it is more likely that the true theory of the world posits fewer species, and hence fewer fundamental undefined concepts.

Objection 1: Reductionism is the right theory about biological species.

Response: Maybe it is. But it’s not the case that it certainly is. And as long as there is a non-zero probability that non-reductionism about species is right, we should assign a slightly higher probability to tails—for conditionally on reductionism, the probabilities are equal, while conditionally on non-reductionism, we should assign a higher probability on tails.

Objection 2: There are so many species that the difference from positing one more is negligible.

Response: Maybe it is. But now parsimony cannot do the work it’s designed to do, and hence it is largely unmotivated. Here’s an example of the kind of work it’s designed for. You observe the world’s koalas and see no significant differences in behavior or appearance. You could suppose there are two species of koalas and you just haven’t been able to tell them apart. But that’s not parsimonious. So you suppose one species. But given how large the number of species on earth is, positing one more makes little difference—assuming the objection is correct.

Objection 3: A full story of the world will include the causal powers in the world. If the radiation has the causal power C_S to produce a new species S of bacteria, then a full description of the world will contain a description of C_S, and hence of S, even if S has no members. Thus it makes no difference to the ideological complexity of the world whether the radiation is applied or not.

Response: This seems the most powerful objection. But it might undercut the applications of parsimony. Plausibly, the world has a first cause behind everything else. Now either the first cause’s causal powers are such that their complete description includes all possible kinds or not (this may depend on questions involving the first cause’s simplicity). If it has such causal powers, then ideological parsimony does rather less work than we thought, since the total description of reality already includes the concepts of the two species of koalas as per my previous response. Suppose now the first cause’s causal powers are not such that their complete description includes all possible kinds. Imagine that you make the following plan. You will flip a fair coin. On heads you will pray that the first cause produce a new kind of thing. Now, the first cause might not fulfill your prayer—indeed, perhaps you are not even confident the first cause is a person. But it is at least slightly more likely that the first cause will make a new kind of thing if you pray than if you don’t. And that’s enough to change the epistemic probabilities for your coin flip, in an implausible way.

Irreducibly joint activity

I’ve long wondered what it means to say that a bunch of things are “irreducibly jointly producing an effect.” One place this comes up in contemporary metaphysics is strongly non-reductivist materialist theories of mind on which the particles of our bodies working together produce thought.

This irreducible production—if there is such a thing—is supposed to be different from the reducible way of how, say, four rocks can depress an analog spring-based scale. Yet I’ve always found it hard to grasp the difference. After all, no single one kilogram rock has the power to depress the scale to the four kilogram point. You need all four. Why is that not an irreducibly joint production of an effect?

A paradigm example of irreducible production would be something like this, however. Suppose the laws of nature ensure that when you arrange four spherical objects of equal size in a tetrahedron, they begin to whistle (I am adapting an example by Trenton Merricks). You arrange three or five or any other number of marbles, nothing like that happens. What’s the difference?

In the case of the rocks and the scale, it is tempting to say that the outcome is just a matter of “summing forces”. But as we learn from Nancy Cartwright and by reflecting on Hume’s insights, summing forces is not as innocent as it sounds. The world could have been so that when multiple forces impinge, they average, or the greatest force carries the day, or the weakest does. Thus the laws of nature seem to have to include some clause like

1. When a number of objects act on an object with forces, the latter accelerates in proportion to the vector sum of the forces.

But now consider.

2. When a number of spherical objects are arranged, whistling results with an intensity equal to one if the objects are four in number, equal in size, and tetrahedrally arranged, and intensity zero otherwise.

The main difference seems to be that “vector sum” is a simpler and more mathematically natural description than “one if the objects are four in number, equal in size, and tetrahedrally arranged, and zero otherwise”. And that’s just a difference of degree, while the difference between reductively and non-reductively joint action is not a difference of degree.

Here is a thought. Perhaps the worry arises out of a mistaken way to think about the case of the scale. The force on the scale’s platform (subsequently transmitted to the spring) is not the sum of the pressures imposed by the four rocks. In the Newtonian setting, each particle of the platform is affected by gravitational attraction to every other particle with mass in the universe, as well as affected by interaction via other forces (e.g., electromagnetic) with countless particles. There are no small closed systems. Perhaps, then, we should rethink the ordinary case: it isn’t a case of reducibility either. On the contrary, it is a case of an irreducible interaction with, basically, everything else—or, in a relativistic context, everything else in its past light cone.

The difference between the whistling tetrahedron of balls and the scale’s depression then isn’t that the whistling is irreducibly jointly effected by multiple things and the scale’s depression isn’t. It is, rather, that the scale’s depression is jointly effected by vastly more things, most of which have only a slight effect, while the magical-seeming whistling is produced only by the four things.

On this picture, what would reducible joint influence be like? It would have to be a case of independent production, where maybe object x produces property P in object z while object y independently produces property Q in z, so that x and z reducibly jointly produce P ∧ Q in z. But it’s not clear that this ever actually happens in our world—all we have are better and worse approximations to such causally independent production.

Some non-reductivist materialists will want to say that our particles, through irreducibly joint activity, produce mental states. Can we accommodate this in the above story? Yes, but only if we suppose that the mental states are in no way directly affected by anything outside our body.

Here is a worry about what I said. Suppose that the four rocks are sufficient to push the scale to a position where the indicator needle is right around the middle of the 4 kg dashmark, while the sum total of the other forces in the universe is insufficient to shift the needle to the outside of the 4 kg dashmark. Then while the exact position of the needle isn’t effected by the four rocks alone, the fact that the needle points somewhere within the 4 kg dashmark on the scale seems to be accounted for by the four rocks alone. Thus, with respect to the coarse grained event of the needle being within the 4 kg dashmark, the four rocks are sufficient to produce the effect. And if that’s right, then we’ve lost the distinction between the four rocks depressing the scale and four marbles causing whistling.

Conjecturally, I answer as follows. The coarse-grained event of a needle’s being in some range (given by the thickness of the 4 kg dashmark) has an identity dependence on fine-grained event of the needle’s being exactly where it is. Thus, had the fine-grained event been different, the coarse-grained event would have been a numerically different event. Hence the coarse-grained event does actually depend on vast numbers of things.

Causal countabilism

Causal finitism says that every item has a finite causal history.

A weaker view is causal countabilism, that all the causal histories of items are at most countably infinite.

Is there any evidence for causal countabilism? Well, of course, any argument for causal finitism is an argument for causal countabilism. But is there any other evidence?

I can think of one piece. It would be good if we could come up with a metaphysical account of the countable, maybe to resolve the Lowenheim-Skolem theorem, or to help provide a metaphysical characterization of the natural numbers. A causal countabilism that isn’t a causal finitism could do that: a set is countable if and only if it is the same size as a possible causal history of an event. If a theory can help accomplish an important task, maybe that’s some evidence for the theory. (That said, causal finitism can also account for countability, as I discuss in my infinity book.)

Are there any known applications for causal countabilism? Before today, I would have said “no”. But I now know two. One is that the “technical assumption” in my last version of the Meyer cosmological argument is entailed by causal countabilism. A second will be in my next post today.

Arguing for a first cause without assuming transitivity of causation

This is another technical research note.

Here is a cosmological argument that does not assume causation is transitive.

Suppose the set of causes forms a directed acyclic graph where the edges are cases of causation. Say that x weakly causes y provided that there is a chain of causation from x to y.

Suppose that any internal chain C of causes that is reverse well-ordered under weak causation is founded, i.e., there is a weak cause c of everything in the chain except c itself. Assume there are no causal cycles. Suppose the following technical condition:

• There is no sequence of causes considered with respect to weak causation that is reverse-order isomorphic to ω₁.

Finally assume there is at least one cause. It follows (assuming the Axiom of Choice) that there is an uncaused cause.

The technical assumption seems pretty plausible. Nobody to my knowledge has proposed a model of the universe where the causal history contains a reverse copy of ω₁. It would be like thinking the past has not only an infinite but an uncountable number of days. (An uncountable number of past moments is tame—though of course it violates causal finitism—but an uncountable number of past days would be wild.)

Sketch of proof: Let G be the directed acyclic graph whose vertices are causes and where (a,b) is an edge iff b causes a. Let R be the partial order generated by the edge directions. Our technical assumption says that ω₁ does not order embed into G. Let C be any well-ordered R-chain of vertices in G. If we can show in general that C has an R-upper bound, by a standard refinement of Zorn’s Lemma it will follow that G has an R-maximal element, and that’s an caused cause.

Since ω₁ does not embed in G, C is countable. Passing to a cofinal (with respect to R) subset of C that is isomorphic to ω, we can assume C has order type ω. Now insert finite paths between all pairs of successive vertices of C (this uses AC, and the acyclicity implies that the paths all run in the same direction as the order on C). The resulting set C′ of vertices is an internal chain with respect to graph direction, and so it has an R-upper bound by the foundedness assumption on chains.

Two kinds of causal chains

Some of my posts are basically research notes to self, stored publicly. This is one of those. It is unlikely to be of much interest to others.

Over the last couple of days, I’ve been trying to make the assumptions in the Meyer cosmological argument as weak as I could. In particular, I’ve been trying to drop the assumption that causation is transitive. A crucial assumption in the Meyer argument is that every chain C of causes is founded—i.e., there is a cause x that is causally prior to every item in C other than perhaps x itself (in case x is in C).

In the transitive case, the notion of a chain is straightforward: a chain of causes is a set C of causes such that for any distinct x and y in C, either x causes y or y causes x. But without transitivity, we have a choice between two notions of a chain:

1. Ambient chain: For any distinct x and y in C, there is a finite sequence c₁, ..., c_n such that c_i causes c_i + 1 and either c₁ = x and c_n = y or c₁ = y and c_n = x.

2. Internal chain: For any distinct x and y in C, there is a finite sequence c₁, ..., c_n of items in C such that c_i causes c_i + 1 and either c₁ = x and c_n = y or c₁ = y and c_n = x.

Any ambient chain is an internal chain, but the converse is not true. Let say that we have a sequence of causes c₁, ..., c₁₀₀ where c_i causes c_i + 1 but due to a bad failure of transitivity there is no other causation—the only time c_i causes c_j is if j = i + 1. Then the even- (or odd-) numbered causes are an ambient chain but not an internal chain.

So I now had a decision point: Do I assume that every ambient chain is founded or merely that every internal chain is founded? I was able to make the argument go in the ambient case in the blog post I linked above, but then I tried to prove it in the internal case.

To that end, I needed to apply Zorn’s Lemma in the case of a relation that need not be transitive. Assuming the foundedness of ambient chains, this was easy: I just applied Zorn’s Lemma to the transitive closure of the causal relation. But in the internal case, what I needed was a genuine generalization of Zorn’s Lemma. This (or maybe an extension that doesn’t assume acyclicity) was what I needed:

Conjecture. Let G be a nonempty directed acyclic graph. Let R be the partial order generated by the connection relation. Suppose that the vertices of every unilaterally connected subgraph linearly ordered under R have an R-upper bound in G. Then G has an R-maximal element.

Here, a unilaterally connected directed graph is one where there is a unidirectional path between any pair of vertices.

Given this conjecture, I would be able to take the edges of the graph to be the reverses of causal relations, assume the founded internal chain condition, and show there is an uncaused cause.

Alas, today I finished proving that the conjecture is false. For the record, here is the counterexample. Let G be a directed graph which has two types of vertices: members of ω₁ and pairs (a,b) with a < b where a, b ∈ ω₁. Let the edges of G run from a to (a,b) and (a,b) to b, and let there be no other edges. Then it turns out that any unilaterally connected subgraph linearly ordered under R is countable (indeed has order type ω), and hence has an upper bound in G. The proof isn’t hard if you can visualize the graph.

(Two days ago, I wasted many hours trying to nudge (free) AI to settle the conjecture. The AI was very confident that the conjecture was true, and gave me a sequence of fallacious proofs!)

Is the counterexample to this conjecture anything one should worry about if one is trying to prove the existence of a first cause? I don’t think so. Causally, it corresponds to a case where we have an infinite regress with order type being the reverse of ω₁ (already weird), massive failures of transitivity (more weirdness), and maybe (assuming one interprets “cause” as full cause) massive systematic overdetermination or at least overcausation (yet more weirdness). Nothing like this is a plausible model of reality. It would be nice to formulate more precisely just how weird a causal system would need to be to provide a counterexample to the conjecture.

The Meyer cosmological argument revisited

I’m playing with an alternative way to run the Meyer cosmological argument (with slightly weaker assumptions).

Suppose there is at least one cause, and there is a set of all causes.

Suppose causation is transitive. (If it’s not, replace “x causes y” with “there is a finite chain of causes running from x to y” throughout the argument.)

Define a causal circularity as a set of at least two causes such that for any distinct x and y in the set, x causes y and y causes x.

Define a causal chain as a set of causes such that for any distinct x and y in the chain, x causes y or y causes x but not both.

A causal chain S is reverse-well-ordered provided that any non-empty subset of it contains a unique effect, i.e., for any non-empty subset U of S, there is a z in U such that everything else in U is a cause of z.

The reverse well-order condition ensures, among other things, that each cause in the chain that is not the first cause in the chain has an immediate predecessor, and hence rules out “continuous chains”. This makes the chains more intuitively “chain-like”, and hence should make us feel more secure about applying our intuitions to them.

Say that a set S of causes is unfounded provided that there is no cause x that causes all the items in S perhaps other than itself.

Say that it is externally unfounded provided that there is no such cause outside the set S.

Then, given the Axiom of Choice, at least one of the following three is true:

1. There is an unfounded reverse-well-ordered causal chain.

2. There is an externally unfounded causal circularity.

3. There is an uncaused cause.

Hence we get an argument for an uncaused cause assuming all reverse-well-ordered causal chains are founded and any causal circularities are externally founded.

Proof: Suppose (1) and (2) are false. Write x ≤ y if x causes y or x = y. Write x ∼ y if x ≤ y and y ≤ x. For x a cause, let [x] be the equivalence class of x under ∼ in the set of all causes. Let C be the set of all equivalence classes of causes, and extend ≤ to C in the obvious way to provide a partial order on C.

Let U be a reverse-well-ordered ≤-chain in C. By the Axiom of Choice, let V be a set containing exactly one member from each ∼-equivalence class that is a member of U. This is a reverse-well-ordered causal chain. It must be founded by the falsity of (1). Thus, there is an x in V such that x ≤ y for all y in V. Thus [x] is a ≤-lower bound for U. By a slight generalization of Zorn’s Lemma requiring only well-ordered chains (this proof yields this generalization), there is a [u] in C that is ≤-minimal. If [u] is a singleton, then u is an uncaused cause. Suppose [u] has more than one member. Then [u] is a causal circularity, and hence has an external cause v by the falsity of (2). Then [v] ≤ [u] and [v] ≠ [u], contrary to [u] being ≤-minimal. Hence [u] cannot have more than one member, and we are done.

Proper parts and dependence

Consider this initially plausible thesis:

1. If x has a proper part, and all of x’s proper parts depend on y, then x depends on y.

Add:

2. An effect depends on its cause.

3. Nothing depends on itself.

We conclude:

4. Nothing that has proper parts is a cause of all of them.

This has a nice theological application.

5. God is the cause of everything that is not God.

6. Therefore, if God has proper parts, he is the cause of all of them.

7. So, God has no proper parts.

However, I am dubious of premise 1. I think (1) depends on a story about parthood on which a whole is made of its parts. But if we don’t have that story, we could imagine a simple thing that then goes on to produce a proper part for itself. And so we lose the nice argument for divine simplicity. Which is too bad, but there are others.

How do we get to an unmoved mover?

I just realized that there is a difficulty in Aquinas’ First Way that I think hasn’t been noted, which builds on the difficulty noted yesterday.

Put the First Way in the following simpleminded way, which I think captures the central ideas:

1. Causes of change are either passers-on of change or originators of change.

2. It can’t be that the causes of change are all passers-on of change.

3. So there must be an originator of change.

4. And this is an unmoved mover (or, more precisely, unchanged changer).

There is a fair amount of detail one can fill in to argue for (2), but that’s not what I want to focus on. I want to focus on the move from (3) to (4).

What licenses us in thinking that an originator of change is itself unchanging?

The idea seems to be that if a cause of change is itself changing, then it is merely a passer-on of change. But this need not be. In my previous post, I imagined an unchanging and timeless demiurge endowed by God with the power to originate change, and noted that in that scenario the demiurge is an unmoved mover but isn’t God.

But now let’s build up and modify my demiurge story. First, let’s be concrete about what motion the demiurge causes. The demiurge has been gifted by God with the power to directly will Bob to rotate around a fixed axis, and the demiurge changelessly exercises this power. (Maybe Bob is one of the Aristotelian heavenly spheres). Second, let’s specify that the demiurge, albeit unchanging, is in time and has an unchanging body in addition to a mind and will. If we can’t have a changeless thing in space and time, don’t worry about it. The third step will fix that. The third step is this. The demiurge itself is caused by God to slowly orbit the sun in a way that the demiurge does not notice.

Thus, we have a demiurge with a mental power to make Bob rotate, and the demiurge exercises this mental power changelessly. At the same time, and completelessly coincidentally to the demiurge’s exercise of the mental power to make Bob rotate, the demiurge orbits the sun.

The demiurge is thus a moved mover. But it is also an originator rather than passer-on of change. In Aristotelian terminology, the demiurge is a moved mover per accidens: its own movement around the sun is coincidental to its origination of Bob’s axial rotation.

We cannot, thus, assume from the existence of an originator of change that there is an unmoved mover.

Does Aquinas have the resources to fill in the gap? Of course, if the (accidentally) moved originator of change coming out of step (3) of the argument is itself changing, that change has to have a cause, and we can then run the argument again. If we can rule out an ungrounded infinite sequence of accidentally moved originators of change—ones that like our demiurge happen to be moving in one respect but produce change by a coincidental exercise of power—then we can get to a genuinely unmoved mover.

But Aquinas’ main tool for avoiding regresses in cosmological arguments is the idea that there cannot be an infinite regress in a per se causal sequence. And while there are complications in the notion of a per se causal sequence, I think it is pretty clear that the sequence I am imagining is not a per se causal sequence. The demiurge’s moving of Bob is coincidental to the demiurge’s own motion. Suppose that a demiurge makes Bob spin by an unchanging exercise of a mental power, and a tetartourge by an unchanging exercise of a mental power makes the demiurge coincidentally slowly orbit the sun while the tetartourge coincidentally orbits the moon. Then the tetartourge is not a cause of Bob’s motion. But in a per se causal sequence we have transitivity: the earlier items are always causes of the later ones. So this is not a per se causal sequence.

Of course, we might ask what explains the demiurge’s (and tetartourge’s) possession and exercise of the mental power to make something else move. But now we are deviating from the First Way: we are asking for explanations of something other than change. I think one can fill this in, by an argument about causation and regresses rather than change and motion. But if we do that, then the stuff about change and motion that is at the heart of the First Way can simply drop out.

I titled my previous post “An easily patched hole in the First Way”. But I think I was too glib there, too. In that context, too, I think one needs to go beyond the resources of the First Way to patch the hole.

Maybe we can create spacetime

Suppose a substantivalist view of spacetime on which points of spacetime really exist.

Suppose I had taken a different path to my office today. Then the curvature of spacetime would have been slightly different according to General Relativity.

Question: Would spacetime have had the same points, but with different metric relations, or would spacetime have had different points with different metric relations?

If we go for the same points option, then we have to say that the distance between two points is not an essential property of the two points. Moreover it then turns out that spacetime has degrees of freedom that are completely unaccounted for in General Relativity, degrees of freedom that specify ``where’’ (with respect to the metric) our world’s points of spacetime would be in counterfactual situations. This makes for a much more complicated theory.

If we go for the different points option, then we have the cool capability of creating points of spacetime by waving our arms. While this is a little counterintuitive, it seems to me to be the best answer. Perhaps the best story here is that points of spacetime are individuated by the limiting metric properties of the patches of spacetime near them and by their causal history.

Classical mereology and causal regresses

Assume classical mereology with arbitrary fusions.

Further assume two plausible theses:

1. If each of the ys is caused by at least one of the xs and there is no overlap between any of the xs and ys, then the fusion of the ys is caused by a part of the fusion of the xs.

2. It is impossible to have non-overlapping objects A and B such that A is caused by a part of B and B is caused by a part of A.

It follows that:

3. It is impossible to have an infinite causal regress of non-overlapping items.

For suppose that A₀ is caused by A₋₁ which is caused by A₋₂ and so on. Let E be a fusion of the even-numbered items and O a fusion of the odd-numbered ones. Then by (1), a part of E causes O and a part of O causes E, contrary to (2).

This is rather like explanatory circularity arguments I have used in the past against regresses, but it uses causation and mereology instead.

Per se and per accidens multiplication of causes

Can there be an infinite sequence of efficient causes? Famously, Aquinas says both “No” and “Yes”, and makes a distinction between a per se ordering (“No”) and an accidental ordering (“Yes”). But it is difficult to reconstruct how the distinction goes, and whether there is good reason to maintain given modern physics.

Here is the central passage from Summa Theologiae I.46.2 reply 7, in Freddoso’s translation:

It is impossible to proceed to infinity per se among efficient causes, i.e., it is impossible for causes that are required per se for a given effect to be multiplied to infinity—as, for instance, if a rock were being moved with a stick, and the stick were being moved by a hand, and so on ad infinitum.

By contrast, it is not impossible to proceed to infinity per accidens among agent causes, i.e., it is not impossible if all the causes that are multiplied to infinity belong to a single order (ordinem) of causes and if their multiplication is incidental (per accidens)—as, for instance, if a craftsman were to use many hammers incidentally, because one after another kept breaking. In such a case, it is incidental to any given hammer that it acts after the action of a given one of the other hammers. In the same way, it is incidental to this man, insofar as he generates, that he himself was generated by another. For he generates insofar as he is a man and not insofar as he is the son of some other man, since all the men who generate belong to the same order (gradum) of efficient causality, viz., the order of a particular generating cause. In this sense, it is not impossible for man to be generated by man ad infinitum.

However, it would indeed be impossible for the generation of this man to depend upon that man, and upon an elemental body [a corpore elementari], and upon the sun, and so on ad infinitum.

What’s going on here? Re-reading the text (and double-checking against the Latin) I notice that per se and per accidens are introduced not as modifying the causal relations, but the infinite multiplication of causes. No indication is given initially that the causation functions differently in the two cases. Further, it is striking that both of the examples of per accidens multiplication of causes involve causes of the same type: hammers and humans (Freddoso’s “man” translates homo throughout the text).

To a first approximation, it seems then that what is forbidden is a regress of infinitely many types of causes, whereas a regress of infinitely many tokens is permitted. But that is too simple. After all, if an infinite causal sequence of humans generating humans were possible, it would surely also be possible for each of these humans to be qualitatively different from the others—say, in exact shade of eye color—and hence for there to be infinitely many types among them. In other words, not just any type will do.

Let’s focus in on two other ingredients in the text, the observation that the humans all “belong to the same order of efficient causality”, and the sun–elementary body–human example. Both of these rang a bell to me, because I had recently been writing on the Principle of Proportionate Causality. At Summa Theologiae I.4.2, St Thomas makes a different distinction that distinguishes between the human–human and the sun–body–human cases:

whatever perfection exists in an effect must be found in the effective cause: either in the same formality, if it is a univocal agent—as when man reproduces man; or in a more eminent degree [eminentiori modo], if it is an equivocal agent—thus in the sun is the likeness of whatever is generated by the sun’s power.

Here is a suggestion. In distinguishing per se and per accidens infinite multiplication of causes, Aquinas is indeed distinguishing counting types and tokens. But the types he is counting are what one might call “causal types” or “perfections”. The idea is that we have the same causal type when we have univocal agency, “as when man reproduces man”, and different causal type when we have equivocal agency, as when the sun generates something, since on Aquinas’ astronomical theory the sun is sui generis and hence when the sun generates, the sun is quite different from what it generates. In other words, I am tentatively suggesting that we identify the gradus of efficient causality of I.46.2 with the modus of perfection of I.4.2.

The picture of efficient causation that arises from I.4.2 is that in a finite or infinite causal regress we have two types of moves between effect and cause: a lateral move to a cause with the same perfection as the effect and an ascending vertical move to a cause that has the perfection more eminently.

The lateral moves only accidentally multiply the explanations, because the lateral moves do not really explain the perfection. If I got my humanity from another human, there is a sense in which this is not really an explanation of where my humanity comes from. The human I got my humanity from was just passing that humanity on. I need to move upwards, attributing my humanity to a higher cause. On this reading, Aquinas is claiming that there can only be finitely many upwards moves in a causal regress. Why? Maybe because infinite passing-on of more to less eminent perfections is just as unexplanatory as finite passing on of the same perfection. We need an ultimate origin of the perfections, a highest cause.

I like this approach, but it fits better with the sun–elemenatary body–human example than the hand–stick–rock example. It seems, after all, that in the hand–stick–rock example we have the same relevant perfection in all three items—locomotion, which is passed from hand to stick and then from stick to rock. This would thus seem like a per accidens multiplication rather than a per se one. If so, then it is tempting to say that Aquinas’ hand–stick–rock example is inapt. But perhaps we can say this. Hand-motion is probably meant to be a voluntary human activity. Plausibly, this is different in causal type from stick-motion: going from stick to hand is indeed an explanatory ascent. But it’s harder to see the progression from rock to stick as an explanatory ascent. After all, a rock can move a stick just as much as a stick can move a rock. But perhaps we can still think we have an ascent from rock-moving to stick-moved-by-hand, since a stick-moved-by-hand maybe has more of the perfection of the voluntary hand motion to it? That sounds iffy, but it’s the best I can do.

I wish Aquinas discussed a case of stick–stick–stick, where each stick moves the next? Would he make this be a per se multiplication of causes like the hand–stick–rock case? If so, that’s a count against my reading. Or would he say that it’s an accidental multiplication? If so, then my tentative reading might be right.

It’s also possible that Aquinas’ examples of hand–stick–rock and sun–elementary body–human are in fact more unlike than he noticed, and that it is the latter that is a better example of per se multiplication of causes.

Towards a solution to the "God as author of evil" problem for the Thomistic model of meticulous providence

On the Thomistic primary/secondary causation model of meticulous divine providence, when we act wrongly, God fully determines the positive aspects of the action with primary causation, and we in parallel cause the action with secondary causation.

Like many people, I worry that this makes God the author of sin in an objectionable way.

Alice and Bob are studying together for a calculus exam that will be graded on a curve. In order that she may do terribly on the exam, and thus that he might do better, and hence be more likely to get into his dream PhD program in ethics, Bob lies to Alice, who has missed three weeks of class, that the derivative of the logarithm is the exponential.

What does God cause in Bob’s action on the Thomistic model? It seems that all of the following are positive aspects:

1. The physical movements in Bob’s mouth, throat, and lungs.

2. The sounds in the air.

So far we don’t have a serious theological problem. For (1) and (2) are not intrinsically bad, since Bob could virtuously utter the same sounds while playacting on stage. But let’s add some more aspects:

3. Bob’s intention that the speech constitute an assertion of the proposition that the derivative of the logarithm is the exponential.

4. Bob’s intention that the asserted proposition be a falsehood that Alice comes to believe and that leads to her doing terribly on the exam.

Perhaps one can argue that falsity a negative thing—a lack of conformity with reality. However, intending falsity seems to be a positive thing, a positive (but wicked) act of the will. Thus it seems that (3) and (4) are positive things. But once we put together all of (1)–(4), or even just (3) and (4), then it’s hard to deny that what we have is something wicked, and so if God is intending all of (1)–(4), it’s hard to avoid the idea that this makes God responsible for the sin in a highly problematic way.

There may be a way out, however. In both written and spoken language, meaning is normally not constituted just by the positive aspects of reality but also by negative ones. In spoken language, we can think of the positive aspects as the peaks of the soundwaves (considered as pressure waves in the air). But if you remove the troughs from the soundwaves, you lose the communication. In print, on the other hand, the meaning depends not just on the ink that’s there, but on the ink that’s not there. A page wholly covered with ink means nothing. We only have meaningful letters because the inked regions are surrounded by non-inked regions.

It could well turn out that the language of the mind in discursively thinking beings like us is like that as well, so that a thought or intention is constituted not only by ontologically positive but also by ontologically negative aspects. Now you could be responsible for the ink within the print inscription

5. The derivative of the logarithm is the exponential

without being responsible for the inscription. For instance, you and a friend might have had a plan to draw a black rectangle and you divided up the labor as follows: you inked the region of rectangle covered by the letters of “The derivative of the logarithm is the exponential” and then your friend would ink the rest of the containing rectangle—i.e., everything outside the letters. But your friend didn’t do the job. Similarly, then, if intentions are constituted by both positive and negative features, God could intend the positive features of an intention without being responsible for the intention as such.

This does place constraints on the language of the mind, i.e., on the actual mental accidents that constitutes our thoughts, and specifically our intentions. Note, though, that we don’t need that all intentions have a negative constituent. Only intentions to produce negative things, like falsehood, need to have a negative constituent for us to avert the problem of God willing intentional sin. We could imagine a written language where positive phrases are written in two colors of ink, one for the letters and the other for the surrounding rectangle, and their negations are written by omitting the ink for the letters. In such a language, statements involving positive phrases are purely positive, while those involving negative phrases are partly negative.

I am not very happy with this solution. I still worry that being responsible for the ink in (5) makes one responsible for (5) when one chooses not to have the rest of the rectangle filled in.

Three fixity principles

In debates about free will and foreknowledge as well as about compatibility and incompatibilism, fixity-of-history theses come up. Here is such a thesis:

1. If a decision is causally or logically necessitated by the history behind the decision, then one could not have decided otherwise.

But now we have a crucial question as to what is meant by “the history behind the decision”. There are at least two takes on this. On the temporal version, the history behind the decision is the sum total of what happened temporally prior to the decision. On the causal version, the history is the sum total of what happend causally prior to the decision.

This is not just a nitpicking question. Linda Zagzebski for instance nicely shows that if we go for the causal-history version of (1), then the main argument for the incompatibility of free will and foreknowledge does not get off the ground assuming God’s forebelief is not causally prior to the action. On the other hand, if we go the temporal-history version, then we have a prima facie argument for such incompatibility (though I think it’s blockable).

I am pretty confident that we should go for the causal-history version, and this has to do with the fact that the temporal-history version is not strong enough to capture our fixity intuitions. Suppose that we live in a world with simultaneous causation—say, a Newtonian world with rigid objects such that if you push object A and A pushes B, then B begins to move at exactly the same time as you start pushing (rather than with a delay caused by then need for a compression wave traveling through nonrigid materials at less than the speed of light). Then we could imagine cases where someone’s decision is causally necessitated by something outside the agent that is simultaneous with the decision. Such causal necessitation would just much make it true that one could not have decided otherwise as would causal necessitation by something in the past.

Furthermore, if backwards causation is possible, then a neurosurgeon in the future who used a backwards-causing machine to determine your decision would clearly prevent you from deciding otherwise, even though the neurosurgeon’s action was not in the temporal history. We may not believe backwards causation is possible, but it is clear that if it were possible, then deterministic backwards causation would be just as threatening for free decisions as deterministic forwards causation. This shows that causal determination is indeed a threat.

Of course, my above argument only shows that if we need to choose between the causal and temporal history versions of (1), we should definitely go for the causal one. But perhaps we don’t need to choose. We could accept both versions. But if we think we accept both versions, I think what we really should accept is an even stronger principle, where “history” is causal-cum-temporal (cct). On that stronger principle, event A counts as in the cct history of event E provided that it is either temporally or causally prior to E. The resulting fixity principle is pretty strong principle, but also a bit gerrymandered. And I think accepting this principle not that plausible, because the much simpler causal version captures our intuitions about all the ordinary cases (not involving God, or backwards or simultaneous causation), since in all ordinary cases causal and temporal history coincide, and we should not go for a more complex principle without pretty good reason.

Causal histories and freedom

Linda Zagzebski proposes the plausible principle that one is able to ϕ only if ϕing is compatible with one’s causal history relevant to ϕing.

Suppose Alice is considering whether to rob a bank. While she is doing so, God loudly announces to her nearby friend Bob that Alice will not rob the bank. God’s announcement is in Swahili, which Bob knows and Alice used to know in childhood but completely forgot. But the sound of the language her loving parents spoke to her as a child leads to Alice putting more emphasis on virtue in her deliberation, and she freely decides not to rob the bank.

Since God cannot lie, and since God’s announcement is a part of Alice’s causal history in her deliberation, Alice’s robbing the bank is incompatible with causal history and by Zagzebski’s principle, Alice cannot rob the bank. Yet it is unclear how God’s announcement removes her freedom to rob. After all, had God announced in Swahili that Alice will have breakfast, that would have influenced her deliberation in the same way, and yet obviously she would still have been free to rob the bank. But since Alice doesn’t know Swahili, the content seems causally irrelevant.

I think there are two ways out of this. First, we might cut events very finely. There is (a) God’s saying something or other in Swahili and there is (b) God’s saying in Swahili that Alice will not rob the bank. To determine the causal history, we pare away from the events all that’s causally irrelevant, and so we include (a) but not (b) in the causal history.

Alternately, we might say this. Whether or not Alice knows Swahili, her decision is affected by the detailed facts about the sounds in God’s announcement. Indeed, by essentiality of origins, her deliberation is a numerically different process because of the difference of sounds. And now we can say that God cannot make the announcement, because doing so would result in a circularity in the explanatory order: God would be making the announcement because Alice is not going to rob and Alice is not going to rob because God is making the announcement because she is not going to rob. So it is not so much that the announcement takes away God’s freedom, but that God cannot produce explanatory circularities.

It’s worth noting that Molinism does not seem to help. Sure, the subjunctive conditional of free will

1. Were God to announce in Swahili that Alice won’t rob, Alice wouldn’t rob

is true. But it is necessarily true independently of Molinism!

Permanence and meaning

Consider this strong meaning-permanence thesis:

1. There being a permanent end to all humanly relevant events would render all of our present activities meaningless.

And this weak one:

2. There being a permanent end to all humanly relevant events would render some of our present activities meaningless.

Here is a quick and easy argument that both are false. Let’s imagine that we believe in a narrative N where there are humanly relevant events that are go on forever and that render some of our present activities meaningful. After all, if there is no such narrative, then it is odd to say that a permanent end to humanly relevant events renders some or all of our present activities meaningless, since these activities would necessarily be meaningless even if there were no such end.

Now, let’s imagine that we came to think that the events and experiences in N exponentially speed up with respect to objective time, in such a way that the first “year”, by human reckoning (revolutions of the earth about the sun, say), described by N takes an objective year, but the second “year” takes half a year, the third “year” takes a quarter of a year, and so on. Thus, we come to think that all the events and experineces in N take place objectively in two years. This is then followed by a clean wipe of reality, and a new creation that has no meaningful connection to any humanly relevant events. Call this story N^*. I think it makes little human difference whether reality is described by N or by N^*. In terms of subjective time, the humanly relevant events of N^* take infinitely long. The only difference is that after the humanly relevant events there are other events that are not humanly relevant. Enriching reality with these events surely does not take away meaning.

So, none of our present activities lose meaning on N^*. But on N^* there is a permanent end of humanly relevant events. Thus, (1) and (2) are both false.

Perhaps this was too quick, though. What if your life project is to fill as much of time with humanity as you can? Then on N, if there are humans always, your project is successful, But on N^*, your project is not successful, because there is infinite humanless time after the end of humanity in two objective years, and so humans occupy only an infinitesimal fraction of time.

But I think it’s mistaken to think that it should be our project to fill up time or space with humans or human events. In other words, the filling-up project is meaningless regardless of success. Take the spatial analogue. Suppose somehow we didn’t know about other galaxies (maybe there are dust clouds shielding them from our view) and we have filled up our galaxy with humans. Would we lose any real meaning in our activities if we found out that reality is richer than we thought, and contains other galaxies beyond our reach? I don’t think so.

The above argument is compatible with a modified version of (1):

3. There being a permanent end to all humanly relevant events after a finite number of events would render all of our present activities meaningless.

For we might think that the reason ordinary stories about a permanent end have a tendency to make us think our activities are meaningless does not have to do with time, but with the idea that the narrative structure for humans requires infinity.

Divine speech acts

Suppose random quantum processes result in deep marks on a stone that spell out:

• Thou shalt not eat goat. – God

What would need to be true for it to be the case that God said (or wrote) that, thereby forbidding us to eat goat?

I assume that God always cooperates with creaturely causation, so divine causation is involved in the above production. However, such divine cooperation with the production of something that looks like an inscription or sounds like an utterance does not suffice to make it be the case that God said the thing. Imagine that a cult leader makes the above inscription. God is still cooperating with the cult leader’s causality, but we don’t want to attribute the inscription to God’s authorship.

One obvious answer is by analogy to our language. A part of what makes a performance a speech act of a particular sort is a certain kind of intention, e.g., that the performance be taken to be that sort of speech act. So maybe it just depends on God’s intentions. If God merely intends cooperation with quantum processes, there is no inscription, just random marks on stone that happen to look like an inscription. But if God intends the marks to be taken to be an inscription, they are an inscription.

This solution, however, is unhelpful given divine simplicity. The intention is a contingent feature of God, and on divine simplicity the contingency of contingent divine features is always grounded in some contingent arrangement of creatures. There cannot be two worlds that are exactly alike in their created aspects but where God has different intentions in the two worlds. So given divine simplicity, there has to be a characterization of what makes the marks a divine command in terms of what creation is like. (My view of divine intentions is, roughly, that God intends F in doing A iff intending F would be a good reason for God to do A. This presupposes divine omnirationality.)

Here is one possibility.

1. Something that looks or sounds like a speech act is a divine speech act if and only if it was directly produced by God without secondary causes.

But this seems mistaken. Imagine that in the sight of a tribe, God created a stone and a stylus ex nihilo, and then miraculously moved the stylus in such a way as to inscribe the prohibition on eating goat. Then, surely, the members of the tribe upon seeing the stylus moving through the air and gouging clear text in the stone would be right to attribute the message to God. But the inscription was not directly produced by God: it was produced by means of a stylus.

Perhaps:

2. Something that looks or sounds like a speech act is a divine speech act if and only if it was a deterministic result of something done by God without secondary causes.

This still seems a bit too restrictive. Imagine that while God used the stylus to inscribe the stone in our previous story, he nonetheless allowed for ordinary quantum randomness in the interaction between the hard stylus and the softer stone, which randomness ensured that there was a tiny probability that no inscription would result—that, say, stylus atoms would quantum tunnel through the stone atoms.

One might replace “deterministic” with “extremely probable”. But just how probable would it have to be?

Here is a different suggestion that seems to me more promising.

3. Something that looks or sounds like a speech act is a divine speech act to humans if and only if a normal human who knew all the metaphysical and physical facts about the production of this act, as well as the human social context of the production, would reasonably take it to be a divine speech act.

This suggestion allows for the possibility that a normal human would be mistaken about whether something is a divine speech act—but the mistake would then be traced back to a mistake about the relevant metaphysical, physical and social facts.

The applicability of (3) is still difficult. Take the initial example where the apparent divine prohibition on eating goat appears from quantum randomness. Would a reasonable and normal human who knew it to have appeared from quantum randomness with ordinary divine cooperation of the sort found in all creaturely causation think it to be a divine speech act? I don’t know. I don’t know that I am a reasonable and normal human, and I don’t actually know what to think about this.

Causation and the grounding problem for presentism

The past-grounding problem for presentism is of explaining what grounds facts about the past. The tensed-property solution is that presently existing objects have past-tensed properties like “Existing a hundred million years after a dinosaur” which ground the facts about the past.

Here is a problem. The presently existing objects exist at least partly because of how the world was a hundred million years ago. If how the world was a hundred million years ago is grounded in the properties of presently existing things, then we have a circularity in the order of explanation: the present objects’ existence is partly-explained by how the world was, and how the world was is grounding-explained by the objects’ possession of the properties, while the objects’ possession of the properties is partly ontologically explained by the objects’ existence.

Objection 1: This won’t bother one if one thinks one can have explanatory circularity as long as the explanations are of different sorts. But I think explanations of different sorts are still explanations, and circularity is still bad.

Objection 2: It seems that B’s being caused by A is explanatorily prior to B’s existing, so sometimes an instance of property possession is prior to existence. But I think this is mistaken. What’s prior to B is A’s exercise of causality, not B’s being caused by A.

Objection 3: If we solve the past-grounding problem by making use of past-tensed properties of God, then the problem disappears. For God doesn’t exist now because of how the world was a hundred million years ago. God exists now because God is a necessary being. I think this is a good response if one doesn’t believe in divine simplicity, but I am convinced of divine simplicity, which prohibits God from having contingent properties.