Causal finitism/countabilism and 4D unrestricted composition

Assume four dimensional unrestricted composition (or mereological universalism). Plausibly, there is a world where on Monday there are infinitely many things, and on Tuesday Alice and Bob cause a sofa to go into the living room.

Also plausibly:

1. If x and y cause z at t, then anything composed of x and y at t causes z at t.

Why? Well, anything composed of x and y at t has the causal powers of x and y at t, and these causal powers are activated, and their activation results in z, so it seems that anything so composed causes z.

But on Monday there were infinitely many things. Thus, by four dimensional unrestricted composition, there are infinitely many things composed on Monday of some of the infinitely many things that exist on Monday and composed on Tuesday of Alice and Bob. By (1), each of these things is the cause of the sofa’s entry into the living room.

Thus, infinitely many things are the cause of the sofa’s entry into the living room. This violates causal finitism. Hence we have an argument from causal finitism against 4D unrestricted composition.

In fact, we have an argument from causal countabilism against 4D unrestricted composition, since there are uncountably many things composed on Monday of the infinitely many things that exist then and composed on Tuesday of Alice and Bob.

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.

An argument against 4D unrestricted composition

1. How many objects exist at t is not even partly grounded in facts about how things are at times later than t in worlds without backwards causation.

2. If standard four-dimensional unrestricted composition is true, then how many objects exists at t is partly grounded in fact about how things are at times other than t.

3. Standard four-dimensional unrestricted composition is false.

I think (1) is very plausible. What about (2)? Bracket God. Consider a world w₂ without backwards causation whose timeline consists of two moments, t₁ and t₂, and where there is a simple a existining only at t₁ aand a different simple b existing only at t₂. On standard four-dimensional unrestricted composition, there are three objects in existence in w₂: a, b and a + b. Of these, two objects, namely a and a + b, exist at t₁. Now consider a world w₁ just like w₂ but without b, and hence without anything at t₂. There is only one object, namely a, at t₁ in w₁.

What grounds the difference between the number of objects at t₁ in the worlds is whether b exists at t₂, contrary to (1).

The best escape from this argument is to suppose what I call five-dimensional unrestricted composition, where all modal profiles are filled. On that view, in both worlds there are infinitely many objects—probably beyond cardinality—at t₁. But that has other plausibility problems.

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.

Guerilla projection

Suppose I make a drawing on a one inch square of transparency, and then project it to cover one side of an 40 foot cubical building.

How big is my work of art? Is it a 2D one inch square or a 2D 40 foot square, or a 3D square frustum with one side being a one inch square and the other a 40 foot square, all filled with light?

I think normally we would think of the artwork here as a 2D 40 foot square. But the artwork is made of light. The artwork would then be a thin slice of the light reflected from the wall, right by the wall. But the artwork is something we see. And if it’s the light that we see here, then why is it the thin slice right around the wall that we see, rather than, say, a thin slice a foot in front of the eyes? In both cases, the light in the slice will travel to the eyes. It’s hard to say why the slice by the wall would be privileged.

This suggests that the piece of art is actually the 3d square frustum. But then we don’t actually see all of the piece at any given time—we don’t see the light as filling the intervening space—which is counterintuitive.

Perhaps what we see is the one inch square, through the magnificatory mediation of the wall? But that means that the large size of the piece is an illusion.

If artifacts exist, then sometimes the size of an artifact seems hard to specify.

Yet another argument against artifacts

Suppose I find on the ground a thin layer of sand. Painstakingly, I remove piece of sand after piece of sand, until what is left is a very fragile picture of a cat made out of grains of sand none of which were moved by me. If artifacts exist, this picture is an artifact and exists. But note that in the story I have not causally affected any of the grains of sand that remain in the picture. And, plausibly:

1. If an object is composed of some parts, it is not possible to cause the object to exist without causally affecting any of the parts.

I conclude that the picture here doesn’t exist. And neither do artifacts in general.

Choice and Zorn for plurals

Here’s one version of the axiom of choice for pluralities in terms of superplurals:

• AC(plural,superplurals): For a superplurality xxx of disjoint pluralities, there is a plurality zz that contains exactly one item from each plurality yy in xxx.

And here’s Zorn’s Lemma for pluralities and and partial order formulas:

• Zorn(plural,formulas): Given a partial order formula ϕ(y,z) and plurality xx, if every subplurality of xx that is a ϕ-chain has a ϕ-upper bound in xx, then xx has a ϕ-maximal element.

(A partial order formula ϕ(x,y) is a formula such that the expression ϕ(x,y) satisfies the axioms of x ≤ y for a partial order ≤. A ϕ-chain and ϕ-upper bound are defined in the natural way.)

Interestingly, AC(plural,superplurals) does not entail Zorn(plural,formulas). For consider a model of plurals and superplurals where the domain is the natural numbers, the plurals are interpreted as non-empty sets of naturals, the superplurals as non-empty sets of non-empty sets of naturals, and we have all the arithmetical predicates, all within a model of ZF where the Axiom of Dependent Choice fails for some relation and there is a predicate expressing that relation. Then AC(plural,superplurals) holds in this model, because the naturals are well-ordered. However, Zorn(plural,formulas) with the naturals being the domain yields Dependent Choice for that relation (since we can encode finite sequences of naturals as naturals).

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.

Molinism and prophecy

Suppose Bob offers a cookie to Alice who is free to accept or reject it. God would like to manifest his omniscience to Alice by announcing to her whether she will accept or reject the cookie. To that end, God uses his middle knowledge to consider the following conditionals:

1. Were God to announce that Alice will accept, Alice would accept.

2. Were God to announce that Alice will reject, Alice would reject.

God then follows the following algorithm:

1. If (i) is true and regardless of (ii), God announces that Alice will accept.

2. If (i) is false but (ii) is true, God announces that Alice will reject.

3. If both (i) and (ii) are false, God refrains from making an announcement.

But there is a problem. Both (i) and (ii) seem to be necessary truths! For God cannot lie and cannot be mistaken. Thus, following the algorithm, God will always announce that Alice will accept, and indeed she will. Hence God can with certainty get Alice to freely accept the cookie—which seems wrong! So there is something wrong with Molinism.

However, I wonder whether the Molinist has to say that (i) and (ii) both have to be true. Granted, their necessity follows from the following plausible rule for subjunctive conditionals:

4. Necessarily: If p is possible, and p entails q, then were p to hold, q would hold,

assuming that the antecedents of (i) and (ii) are possible and given S5. (And one can avoid S5 by assuming that necessarily the antecedents of (i) and (ii) are possible, and that the entailment between God announcing something and its being true is not only necessary but necessarily necessary.)

But perhaps (4) is not as obviously true as it seems. Consider a case where free will is not involved. There is a perfectly reliable sound-activated light. God is debating whether to announce “There will be no miracle and yet the light will not turn on.” It seems that it would be reasonable for God to reason that were he to make that announcement, the light would turn on (since it’s perfectly reliable, it would take a miracle for it not to), and hence the announcement would be false. Yet God’s making the announcement is possible (there is no problem about God making the announcement in a world where the lightbulb is broken).

So perhaps the Molinist can say that there can be subjunctive conditionals of free that are true even though the antecedent entails the negation of the consequent. If so, then (i) and (ii) could still be contingent.

I am still dubious of a logic of subjunctives where the subjunctive can violate an entailment. But perhaps it’s not as bad as I once thought it (compare this old paper of mine).

Molinism and correlations

Assume Molinism.

Suppose T is a set of character traits that makes one be equipoised between freely taking and refusing a $5000 bribe for a city contract. Let C₁, C₂, ... be distinct and highly specific possible circumstances of Curley being offered such a bribe, differing in morally unimportant ways that are not very relevant to Curley’s character. Perhaps he is offered the bribe in an Italian restaurant, or in a Thai restaurant, or while walking in the park. Perhaps the bribe is being offered by a man or by a woman. Maybe it’s in the morning or the evening. Etc.

Consider the counterfactual of free will:

• Q_i: Were Curley to have T and be offered a $5000 bribe for a city contract in C_i, he would accept the bribe.

Intuitively, P(Q_i) ≈ 1/2 for any i. Let’s accept that intuition. But now here is an interesting question: What kinds of statistical correlation is there between Q₁, Q₂, ...?

The intuition behind van Inwagen’s re-run thought experiment says that if Curley were offered a sequence of exactly similar bribes, with memory erased in between, then Curley’s acceptance/rejection decisions would behave like an independent sequence of random variables. That intuition suggests that Q₁, Q₂, ... are also statistically independent.

On the other hand, one might have the conflicting intuition that Q_i and Q_j are more correlated when the circumstances C_i and C_j are more similar.

On the third hand, one might think that although each of Q_i has probability around 1/2, the probabilities of conjunctions of the Q_i are undefined, and hence it makes no sense to talk about their correlations.

These seem to me to be the three most plausible views on the correlation question. Call these the Independence, Similarity, and Undefined views.

On Molinism plus Independence, God has a vast amount of providential power. He is nearly certain to be able to get Curley (or any other free agent) to freely do anything he wants, simply by choosing minor and unimportant features of circumstances. Given independence, it is extremely unlikely that all the Q_i have the same truth value. Thus, God can just choose the circumstances to get the truth value he wants—to get Curley to accept or to get Curley to reject the bribe. In particular, on Independence, free will does not do much to help the theist.

On the Similarity view, there are some more serious constraints of God’s providential power. However, I think there is another problem with the Similarity view. On Similarity, there is presumably some complicated function from the degree of similarity between circumstances to the degree of correlation (say, measured by covariance) between the truth values of the corresponding counterfactuals. Where does this function come from? Normally correlations between events are explained by laws of nature. But counterfactuals of free will are prior to laws of nature. I suppose the correlation function would have to be something necessary. But it does seem mysterious.

I think that if I were a Molinist, I would find the Undefined view somewhat appealing. The version of it that seems most plausible would be that P(Q_iQ_j) is not defined as a number, but can be represented as the interval [max(0,P(Q_i)+P(Q_j)−1),min(P(Q_i),P(Q_j))]. But I still find it a bit odd that P(Q_i) and P(Q_j) are defined but P(Q_iQ_j) is not. Nonetheless, this seems possible.

Defining parthood in terms of life

A couple of days ago I blogged that van Inwagen is right that we should not expect an answer to the General Composition Question.

Now I am not so sure. Suppose van Inwagen is right about the Special Composition Question having the answer he thinks it has: Necessarily, the xs compose another thing if and only they have a life together.

The following is then plausibly true:

1. Necessarily, y is a part of z if and only if y = z or there is a life L and xs such that (a) the xs live life L together, (b) z itself lives life L, and (c) y is among the xs.

But if so, then we have a non-mereological definition of parthood, and if substitute that into the definition of composition, we get a non-mereological account of when the xs compose y.

Maybe the problem with (1) is that it commits one to an ontology that quantifies over lives (which are presumably some kind of event), rather than just having some plural predicate like “are jointly lifewise active”.

But one can perhaps escape even that. Perhaps we can suppose a plural predicate S(xx,yy) which expresses the idea that the xx and yy same-co-live. One might want to paraphrase that as “the xx live the same life as the yy do”, but that would commit one to quantifying over lives. And then we can say:

2. Necessarily, y is a part of z if and only if y = z or there are xx that have y among them such that S(xx,z).

Yet another argument against Molinism

Assume Molinism. Let maximal transworld depravity be the claim that for any circumstances C in which an agent x would be freely choosing between options A₁, A₂, ..., the agent would choose an option than which they could not choose a worse. If Molinism is true, maximal transworld depravity seems possible.

Suppose God finds there to be maximal transworld depravity, and consequently chooses not to create anything. Nonetheless, wouldn’t it be a very bad thing that maximal transworld depravity holds? After all, it prevents God from having all sorts of great possibilities. But how could there be anything bad in a world where only God exists?

So, we have an argument against Molinism. If Molinism is true, it is possible that God alone exists and yet reality has something very bad about it.

Theism and the Special Composition Question

Given theism, I suspect there is no answer to van Inwagen’s Special Composition Question, which asks for non-mereological necessary and sufficient conditions for a plurality of things to compose a whole.

The basic idea is this. Proper parthood is a real relation. God can simply insert or delete new instances of a real relation into the world as long as he does not violate the logic of this relation. Then imagine that God simply deletes all instances of proper parthood from this world. The logic of proper parthood is not violated by this. But we cannot non-mereologically describe the difference between our world and the resulting mereologically nihilistic world, and yet our world is one where (I assume) there is non-trivial composition, and the modified world is one where there isn’t any. Hence, we cannot give non-mereological necessary and sufficient conditions for a plurality to compose.

One might object that the whole depends on the parts (a similar argument can be given if the parts depend on the whole, with the same response), and so God cannot delete the parthood relation between me and my parts, as that will make my parts disappear from reality. But as Aquinas argues, what creatures can do, God can do it without help from creatures. So if my parts can keep me in existence, God can keep me in existence without any reliance on these parts.

Here is a theistic argument directly aimed at van Inwagen’s organicism. Any creature that exists must be sustained by God. God could keep on sustaining my parts without sustaining me, so I drop out of existence. It would still be true that my former parts are engaging in the kind of activity that define life for van Inwagen, but they would no longer compose a whole.

Composition and laws of nature

I’ve for a while wondered if mereology can’t be nomically strongly emergent, namely whether there couldn’t be laws of nature of the form:

1. If the xs satisfy F, then the xs cause there to be a y composed of the xs.

This is a species of Markosian’s brute composition theory: There is no set of finite informative conditions that are necessary and sufficient for composition.

One might think that above nomic story has an informative necessary and sufficient condition for composition:

2. The xs compose something if and only if there is only one x or there is a law of the form (1) that the xs satisfy the antecedent of.

Condition (2) wouldn’t count as an answer to van Inwagen’s Special Composition Question because it uses mereological vocabulary on its right hand side (since (1) uses the word “composed”). However, (2) would still be pretty informative. Nonetheless, (2) shouldn’t be acceptable to a theist, because the kind of thing that can happen by a law of nature could also be directly produced by God apart from any law: God can make the xs compose y.