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.