Persons (and a remark on fine-tuning)

Here’s a speculative theory of persons:

• A person is an individual of a sort that pursues the good as such.

What’s interesting on this theory of persons is that mental life is not directly a part of the definition. For pursuit as such does not require a mind: plants pursue particular goods like nutrition and reproduction (but while they pursue particular goods, they don’t pursue the good as such).

I conjecture that pursuit of the good as such requires a mind and will. If this is right, then that brings my teleological definition closer to intuitive accounts of persons. Interestingly, though, I also conjecture that pursuit of the good does not require a conscious mind—just a mind with intention, intentionality and intensionality (funny how different the meanings of these three similar-sounding words are). Since personhood implies dignity, it follows that dignity does not require consciousness. (An important observation for thinking about people in persistent vegetative states.)

But this is all super conjectural.

What if it turns out that one can pursue the good as such without mind? Should I then modify the definition of a person to replace “pursues” with “intentionally pursues”?

I think it depends on whether the kind of being that pursues the good as such without mind still has the kind of dignity that persons have. If so, then I am happy to leave my definition unchanged. But if dignity requires mind, and mind doesn’t follow from pursuit of the good, then I will insert “intentionally” into my account.

As I said, this is all speculative. But one thing I am fairly confident of: being the sort of individual that pursues the good as such is a necessary condition for being a person.

An interesting consequence is this. One could have a kind of being with advanced agency, problem-solving ability, and abstract thought, without its being a person. Indeed, I think such a being might even have the abstract concept of the good, but if it wasn’t of a sort to pursue the instances of that concept as such, it wouldn’t be a person.

If I am right at least about the necessary condition, I think we make some progress on the fine-tuning argument for theism. One of the objections to the fine-tuning argument for theism is this: “Why should we think that a personal designer would want there to be life?” Well, if a person is the sort of being that pursues the good in general, the good as such, and if life is eminently valuable, it is not astronomically unlikely that a personal designer would want there to be life. And given how ridiculously narrow some of the life-compatible ranges in the fine-tuning argument are, “not astronomically unlikely” is all we need.

Existing for all times is still puzzling

Let’s assume causal finitism is false, and that the past can be infinite.

If an object pops into existence ex nihilo at a specific time t, that feels objectionably arbitrary, since we want to know why did it pop into existence then rather than earlier or later (or if it couldn’t have done that, for essentiality of origins reasons, why didn’t something just like it do that).

But if the object has existed for infinite time, that does not feel arbitrary, and some people have the intuition that the object does not call out for an explanation even if it is metaphysically contingent.

This failure to be puzzled is I think based on a mistake. For even if the object existed for an infinite time, we can ask why it didn’t exist for a longer infinite time. That question makes sense. After all, if the object existed on days ...,  − 3,  − 2,  − 1, 0, where 0 is today, then we can ask why before all those days it didn’t exist on an earlier day, one which we might denote  − ω, where ω is the first infinite ordinal.

One might answer: “Well, it didn’t exist on day  − ω, because time does not include day  − ω.” But that doesn’t solve the puzzle. After all, imagine that on day  − 18 an object popped into existence, and when we are puzzled by this, we are told: “Well, day  − 18 is the first day of time. The object couldn’t have existed on day  − 19 because there was no such day.” But now we want to know: Why did time itself begin on day  − 18 rather than on day  −19 or day −10⁹⁴? And similarly, we should want to know why time extends only to the dates ...,  − 3,  − 2,  − 1, 0 and not to day  − ω, or day  − ω − 1, or day  − 2^ω₉₄.

In fact, there is a sense in which the first of the following questions might be less puzzling than the second:

• Why did time begin on day  − 18 rather than day  − 10⁹⁴?

• Why did time extend only to days ...,  − 3,  − 2,  − 1, 0 rather than beginning with day  − ω, inclusive?

For if time is relational, then there might be no real metaphysical difference between time beginning on day  − 18 and time beginning on day  − 10⁹⁴, but even so there is a metaphysical difference between time extending only to days ...,  − 3,  − 2,  − 1, 0 and its starting on  − ω. If time extends only to days ...,  − 3,  − 2,  − 1, 0, then it has no first day, but if it starts on day  − ω, then it has a first day, namely  − ω.

In fact, there are infinities of different possible temporal orders, all of them linear (i.e., reflexive, transitive and total). Why is time one of these rather than another? If times are representable as real numbers (as classical physics assumes), why? Why not merely as rational numbers? Or, more expansively, why not as hyperreals? Or why not as an order that looks like the real numbers with zero removed?

Whatever is the time sequence occupied by an object, even if the object fills the time sequence, it calls out for an explanation why the time sequence that it fills is this one rather than another one.

Restricting the PSR to things with a finite past

A correspondent asked me why not to simply believe in a restricted Principle of Sufficient Reason (PSR) that says that everything contingent that has a beginning in time has an explanation. (Graham Oppy has once proposed such an idea.)

One of the reasons for accepting the PSR is that without it we fall into skepticism. If violations of the PSR are possible, they have no meaningful probabilities, and in particular cannot be said to be unlikely. E.g., without the PSR, we have to say it is not unlikely that we just popped into existence five minutes ago for no reason at all, and so we cannot say that skeptical hypotheses about the past are improbable.

There are two types of skepticism. On weak skepticism about X, we conclude we can’t know about X. On strong skepticism about X, we conclude that we can’t have a reasonable belief about X. Weak skepticism is not a big deal. Strong skepticism is. This is strong skepticism about the past.

The specific case of the five minute past doesn’t survive if we have the PSR restricted to things with a beginning. But another argument does. Suppose the full PSR is false but the restricted PSR is true. Now consider the scenario where reality consists of a large number of persons who for an infinite amount of time have been caught in an endless five minute loop of experiences just like mine. Violations of the PSR have no meaningful probabilities, so we cannot say that this hypothesis is unlikely. So, we get skepticism. Of the strong kind.

Randomness and identity

Suppose an amoeba biologically symmetrically splits into two cells, A and B. Here are the main options for what happens metaphysically:

1. The parent perishes and the child cells are new organisms.

2. The parent becomes a scattered two-celled organism.

3. The parent becomes A, and B is a new organism.

4. The parent becomes B, and A is a new organism.

Typically, philosophers dismiss options (3) and (4) as arbitrary in cases where the splitting is symmetric. Option (2) is implausible, so that leaves option (1).

But now consider an option that the above argument for (1) does not consider:

5. Indeterministically, the parent has chance 1/2 of becoming A, with B being new, and chance 1/2 of becoming B, with A being new.

(Technically, this is not another option, but a further specification of (3) or (4).) Rule (5) is perfectly symmetric between A and B, so there is no arbitrary preference in the rule—though of course the random result may be said to be arbitrary.

I expect that the main reason (5) is usually not considered is that it is assumed that the answer as to which of (1)–(4) happens supervenes on qualitative biological features of the amoeba and offspring system. But if we have a “further fact” view about identity, or if we are dualists about amoebae—say, because we think that an amoeba has a non-physical soul or form that—then I think (5) is not at all crazy. There could well be indeterministic laws of how the “further fact” behaves or where the soul goes.

Of course, (5) is not the only indeterministic symmetric option that avoids scattered individuals. One could have a view where there is a 1/3 chance for each of options (1), (3) and (4). Or where there is a 2/3 chance of (1) and a 1/6 chance of each of (3) and (4).

Some people will think that amoebae do not have “further facts” or souls, but are entirely physically reducible. I am far from confident of that. But in any case, whether the indeterministic solution works for amoebae, it could well work for humans, whether in actual cases of early twinning or in science-fictional brain splitting thought experiments.

Here is another metaphysical difficulty where an indeterministic answer might work. Consider beginning and end of life issues: when do constituents come together as a new organism and when do organisms depart from life? It feels arbitrary to suppose, say, that when the damage exceeds some threshold, the organism must die there and then. But what if we suppose that there is a continuous distribution—say, something like an exponential one—with parameters continuously depending on the degree of damage, governing the random point at which the organism departs from this life if the damage is not changed? Of course, this kind of a story only makes sense on a non-naturalistic picture where there is some further unobservable fact about life.

I wonder what other metaphysical questions might have random answers.

The antisymmetry of temporary parthood

Sider suggests this four-dimensionalist characterization of “temporary parthood” in terms of temporal parts:

1. Necessarily, x is part of y at t iff x and y each exist at t, and x’s temporal part at t is part of y’s temporal part at t.

Somehow I never noticed something that is probably pretty obvious to people who work on this: temporary parthood defined thus is not antisymmetric (where a relation R is antisymmetric provided that xRy and yRx implies x = y). For consider myself, x, and my temporal part, x_t, at t. Then x_t ≠ x, but x_t is a part of x and x is a part of x_t, assuming plausibly that the temporal part of x_t at t is x_t (otherwise, we get a weird infinite regress where each temporal part at t has a distinct temporal part at t).

Is this a problem? I think so. We do after all seem to have an ordinary notion of parthood at a time, and that notion is antisymmetric: two distinct objects can’t be mutually parts of each other at the same time. If you are a part of a club at t, the club isn’t a part of you at t, even if you are the only member at that time.

We can avoid the above temporal part counterexample to asymmetry by modifying (1) to be:

2. Necessarily, x is part of y at t iff x and y each exist at t, and x’s temporal part at t is part of y’s temporal part at t, and y is not a proper part of x.

But that’s not good enough. For consider non-instantaneous temporal parts, such as myself from time a to time b (both inclusive): x_[a,b]. Then x_[0,3] and x_[1,4] have the same temporal part at t, and so by (2) they are still mutually parts of each other at t, since neither is a proper part of the other simpliciter.

But perhaps we can argue that temporary parthood is not antisymmetric. Consider this story:

• t₁: you have a lump of unbaked clay

• t₂: the clay has been formed into a statue

• t₃: the statue has just been gilded

• t₄: the gilding has just been removed

• t₅: a piece of the clay has been pinched off in order to improve the appearance of the statue’s nose.

Consider the cloud of molecules of the original lump of clay. This cloud persists through the whole story. Initially, it does not form a statue, and at the end it forms more than the statue (since the pinched off bit is a part of the cloud but not the statue then). It seems clear that:

1. The cloud is a part of the statue at t₃, with the gold being the other part at t₃

But the cloud did not become a temporary part of the statue when it was gilded. Thus:

2. The cloud is already a part of the statue at t₂.

Conversely:

3. The statue is a part of the cloud at t₅.

But the statue does not become a temporary part of the cloud when a bit is pinched off. Thus:

4. The statue is a part of the cloud at t₄.

But there is surely no difference in the temporary relationships between the statue and the cloud at t₂ and t₄, so:

22. The statue is a part of the cloud at t₂.

But since the statue is not the cloud, (ii) and (v) imply that parthood at t₂ is not antisymmetric.

If we have a strong intuition that temporary parthood should be antisymmetric, the above example suggests that we should have a restricted view of composition, one on which at least one of the cloud and the statue does not exist.

An argument for reincarnation

Here’s a fun argument:

1. We are responsible for something.

2. To be responsible for anything that is not a choice, we have to have been responsible for choosing it.

3. To be responsible for anything that is choice, we have to be responsible for the features of our character by which we made the choice.

4. So, if we are responsible for something, we have made an infinite sequence of choices.

5. So, we have made an infinite sequence of choices.

6. We have not made an infinite sequence of choices in any one life.

7. So, we have had an infinite sequence of lives.

Famously, Galen Strawson makes the 2–4 part of this argument, and concludes that since we have not made an infinite sequence of choices, we are not responsible for anything. But I think it is more reasonable to believe in reincarnation (or some other way of having had infinitely many choices—say, a supertask) than to disbelieve in responsibility.

That said, I don’t think we should take the above as providing significant evidence for reincarnation. For even if we had made an infinite number of choices, I don’t think that would get us out of the point Strawson is making. There would still be a sense in which we wouldn’t be responsible “for the regress”. This observation has applications beyond free will: an infinite regress of causes doesn’t explain the universe.

My own preferred way out of Strawson’s argument does not involve infinite regresses, but simply denying premise (3). Premise (3) is plausible when our character determines the choice, but less so when it does not. Take a paradigm example of someone freely and indeterministically making a choice with conflicting desires and reasons for different options. Then imagine that they were brainwashed into having these very desires and reasons. That doesn’t affect the freedom with which they choose on the basis of those desires and reasons.

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.