Alexander Pruss's Blog

Choice and Zorn for plurals

noreply@blogger.com (Alexander R Pruss) — Wed, 16 Sep 2026 17:14:13 +0000

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

noreply@blogger.com (Alexander R Pruss) — Wed, 16 Sep 2026 15:24:05 +0000

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 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:

There is an unfounded reverse-well-ordered causal chain.
There is an externally unfounded causal circularity.
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

noreply@blogger.com (Alexander R Pruss) — Tue, 15 Sep 2026 15:59:04 +0000

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:

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

God then follows the following algorithm:

If (i) is true and regardless of (ii), God announces that Alice will accept.
If (i) is false but (ii) is true, God announces that Alice will reject.
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:

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

noreply@blogger.com (Alexander R Pruss) — Tue, 15 Sep 2026 15:22:07 +0000

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

noreply@blogger.com (Alexander R Pruss) — Mon, 14 Sep 2026 22:02:31 +0000

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:

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:

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

noreply@blogger.com (Alexander R Pruss) — Mon, 14 Sep 2026 19:27:35 +0000

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

noreply@blogger.com (Alexander R Pruss) — Mon, 14 Sep 2026 17:10:28 +0000

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

noreply@blogger.com (Alexander R Pruss) — Mon, 14 Sep 2026 16:42:52 +0000

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:

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:

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.

Promises and permissibility

noreply@blogger.com (Alexander R Pruss) — Fri, 11 Sep 2026 19:16:46 +0000

Here is an initially plausible idea:

A promise can take an action that absent the promise is merely permissible and turn it into an action that is obligatory, but it cannot take an action that absent the promise is impermissible and turn it into an action that is obligatory.

I wanted to use this idea in an argument. But I have since come to a fairly definitive refutation. Suppose last year I promised you that today I would drive to Waco airport and sit for one minute on a bench and come back home. (Maybe we were playing some game, and this was the agreed-upon penalty for losing.) Plausibly, I am obligated to do so. But absent the promise, it would be impermissible for me to do it, because it would be pointless, and it is wrong to drive pointlessly, since each time one drives one (a) endangers human life and (b) contributes to environmental degradation.

Of course, a promise can’t take an action that is intrinsically wrong and make it obligatory.

Withholding and withdrawing treatment, once again

noreply@blogger.com (Alexander R Pruss) — Fri, 11 Sep 2026 18:59:49 +0000

Consider one more pair of cases to challenge the standard view that there is no significant moral difference between withholding and withdrawing treatment.

A patient’s knee became inflexible with age. The only treatment available is an artificial replacement. The patient is competent, informed, free, but eccentric. Consider two cases:

The patient doesn’t agree to an artificial replacement, because the patient thinks that if their leg is flexible, they will be expected to do half of the chores in their two-person household, whereas right now they are getting away with doing only a quarter on account of their bad knee.
The patient has already received an artificial replacement, but is dismayed by the fact that they are now expected to do half of the chores, so they demand that a physician inject epoxy into the artificial knee to restore the inflexibility they enjoyed prior to the replacement.

In case 1, it is clear that it is a violation of the patient’s bodily integrity and autonomy to force the knee replacement on them, even if one thinks that the patient’s reason for refusal is terrible.

But in case 2, it would be reasonable for a physician to refuse to inject the epoxy, as this would violate the do-no-harm principle which forbids performing medical procedures, even ones requested by the patient, that on balance harm the patient.

Refraining from implanting an artificial knee is an instance of withholding medical treatment. Refraining from injecting epoxy into the artificial knee is an instance of withdrawing medical treatment by disabling a medical device. It’s a different way of disabling than just by throwing a switch, but it is the least damaging way to disable the artificial knee (removing the artificial knee would be tantamount to amputating the leg above the knee).

The judgment that in case 1, acceding to the patient’s refusal is clearly right, while in case 2, refusing the patient’s request for discontinuation of the artificial knee treatment for joint inflexibility at the very least can be right. Hence withholding and withdrawing are not morally equivalent.

Withholding, withdrawing, and the magic of medicine

noreply@blogger.com (Alexander R Pruss) — Fri, 11 Sep 2026 16:49:06 +0000

Consider two cases where a patient ended up with an implanted device D that benefits the patient by ameliorating an otherwise fatal medical condition.

Alice got in a fight with Carl the hospital janitor. The janitor grabbed the first weapon at hand—an injector for device D—and stabbed Alice with it, thereby ensuring that Alice now has D implanted in her.
Alice’s twin sister Brenda got D duly implanted by Carl’s sister Dr. Diana, a physician at the hospital.

Now suppose that Alice and Brenda both demand a physician to remove D.

It seems that Alice’s demand is completely on par with the demand to remove a bullet or a kidney stone. But normally a physician needs to make a judgment that a procedure requested by a patient is likely to on-balance benefit the patient, and ought to refuse otherwise. If a physician is asked to remove a bullet or a kidney stone, but professionally judges that the patient will die without the bullet or kidney stone, they must refuse the request. It thus appears that the physician ought to refuse to remove D from Alice, unless there are significant burdens from D to Alice (maybe D is not a burden at all, but Alice just wants to die).

But in the case of Brenda, removing D is not just “a procedure”, but it constitutes the withdrawing of a medical treatment. On a standard view, just as one must withhold a medical treatment when a patient competently demands withholding it, even if the treatment would save the patient’s life, a demand for withdrawing is exactly on par, and hence must be honored, even if D does not impose any significant burden on Brenda.

Suppose the standard view is right. Then the above shows you can have two patients who are in exactly the same physical state, and are making exactly the same demand, but the demands are treated differently by standard medical ethics. The reason they are being treated differently is that in Alice’s case, D’s presence does not constitute a medical treatment because D was injected by the janitor in the course of a fight, while in Brenda’s case, D;’s presence does constitute a medical treatment because D was duly injected by a physician.

I find this troubling. It seems to be taking medicine as something magical, as if the status of D inside Alice’s and Bob’s became different when D came to be present “medically”.

Maybe one way to avoid the “magical” thinking is to think that consensual medical implantations come with an explicit or implicit agreement between the patient and the provider (the medical professional or maybe the medical institution) to withdraw the item should one so demand it? Then the magic is not due to medicine but due to promising. (I don’t think this completely resolves the questions, because not all promises need to be kept.)

A problem for Van Inwagen's life condition for composition

noreply@blogger.com (Alexander R Pruss) — Thu, 10 Sep 2026 15:47:08 +0000

Van Inwagen thinks that a plurality of xs composes something if and only if either there is only one thing among the xs or the xs have a life together.

He also agrees that the fundamental particles making up a cell in a multicellural organism have a life together, and so there are cells.

Here is a problem. The following seem true:

It is possible to have two overlapping cells in a multicellular organism neither of which is a part of the other.
The cells of a multicellular organism together with any intercellular fundamental particles have a life together.

Why think (1)? Because, first, conjoint twins show that multicellular organisms can have “proper overlap”: i.e., they have a part in common but neither is a part of the other, and it seems hard to deny that cells, too, could in principle be like that. Second, it seems plausible that mid-way through a process of mitosis we have two cells that properly overlap.

It seems hard to deny (2), assuming there are cells.

But now consider a multicellular organism which happens to have two overlapping cells. Its cells, together with any intercellular fundamental particles, have a life together. Hence, by van Inwagen’s famous account of composition, the cells and the intercellular fundamental particles compose something. But van Inwagen’s definition of composition includes a no-overlap condition: if the xs compose y, no two xs overlap. And yet in our case we have life and overlap.

What should van Inwagen do? One move would be to weaken the Special Composition Question. Instead of asking when the xs compose something, he could ask when non-overlapping xs compose something. But that makes his view rather less informative than we might think, if overlap between cells is fairly common. Presumably in a body of my size, at any given time, mitosis is occuring in a number of cells (I haven’t done the math, but it seems likely).

Withholding and withdrawing treatment

noreply@blogger.com (Alexander R Pruss) — Thu, 10 Sep 2026 15:14:24 +0000

I have heard that there is a consensus on the equivalence of the actions of withholding medical treatment and withdrawing it, even when the withdrawal requires a positive action from the medical professional (e.g., disconnecting a machine).

Here is a reason to doubt this consensus.

Consider first this pair of cases. In both, the following facts hold. A patient will die within hours without a ventilator, and competently requests that they be on a ventilator. However, an enemy of the patient has very credibly threatened the doctor with death if the patient is on a ventilator tomorrow. The only way the enemy can get at the patient is through the doctor, and there is no way to protect the doctor from the enemy. The difference between the two cases, however, is that in one case, the patient is already on a ventilator and in the other the patient is not yet on it.

If the patient is already on the ventilator, the doctor should refuse to disconnect the patient because of the enemy’s threat. Disconnecting the patient would be a wrongful cooperation in murder, and one should die rather than do that. Disconnecting the patient would be tantamount to murder. On the other hand, if the patient is not yet on the ventilator, then it would be supererogatory for the doctor to ventilate the patient at the expense of the doctor’s life. The doctor does not have the duty to save the patient’s life at the nearly certain expense of their own life. And even if one thinks it is a duty, failure in this duty is far below the wickedness of murder.

But disconnecting is withdrawal and failure to connect is withholding. Thus the two are not equivalent actions, because they can differ in moral evaluation even when done for the same reason (avoidance of the threat).

One might think the two become equivalent when the patient refuses to consent to the treatment (whether its continuation or initiation). It seems odd to think that the patient’s refusal somehow turns both actions equivalent.

But anyway, consider a reversal of the threat case. A “friend” of the refusing patient informs the doctor that they will cut off the doctor’s thumbs unless the patient is on a ventilator tomorrow, and suppose that the ventilation in question is invasive. It seems pretty plausible that in the case where the patient is not yet on the ventilator, the doctor should refuse to invasively ventilate them. For to invasively ventilate the patient would be an assault on the patient’s bodily integrity, and one should suffer a greater loss to one’s own bodily integrity rather than impose a lesser loss on another.

But in the case where the patient is already on the ventilator, it seems that the doctor has no obligation to suffer a greater loss themselves in order to respect the patient’s wish to be disconnected from the ventilator.

Thus, even in the case where the patient refuses consent, withholding and withdrawing are not equivalent.

Length and the memory theory of personal identity

noreply@blogger.com (Alexander R Pruss) — Thu, 10 Sep 2026 03:21:47 +0000

That there is an entity that is 25 inches long in 1973 and 72 inches long in 1993 is not even partly grounded in facts about anybody’s memory.
If the memory theory of personal identity is true, the existence of such an entity is at least partly grounded in facts about my memory.
So, the memory theory of personal identity is false.

Premise (1) is a raw appeal to intuition.

Premise (2) follows from the following:

Normally an existentially quantified fact is grounded in each of its instances.
I was 25 inches long in 1973 and 72 inches long in 1993.
I was a person.
The case at hand is a relevantly normal case.

I suppose the memory theorist just needs to deny (1).

Van Inwagen is right: We should not expect an answer to the General Composition Question

noreply@blogger.com (Alexander R Pruss) — Wed, 09 Sep 2026 22:40:50 +0000

Van Inwagen’s General Constitution Question (GCQ) is to provide necessary and sufficient conditions for the xs to compose y in non-mereological terms. Van Inwagen is skeptical of whether the GCQ has any non-trivial answer. Here is a reason to think he’s right to be skeptical.

It might well be true that atomism is necessarily true, namely that every object has a simple part. But given atomism, the following is true:

Necessarily, z is a part of y if and only if there are xs that include z and that compose y.

(Proof: The right to left part is trivial: if the xs compose y, they are parts of y. Conversely, suppose z is a part of y. Let the xs consist of z and all the atomic parts of y that aren’t parts of z. It is easy to see that the xs compose y.)

Thus, if atomism is true, an answer to GCQ will provide a non-mereological account of all of mereology. That seems to be too much to expect.

Even if atomism is true, an answer to GCQ will provide a non-mereological account of the parthood of all wholly non-gunky objects (where an object is partly gunky provided that it has some part that has no simple parts). That is also more than we should expect.

Why does God create those he knows will reject him?

noreply@blogger.com (Alexander R Pruss) — Tue, 08 Sep 2026 21:57:38 +0000

Question: Suppose Alice goes to hell, having conclusively rejected God’s grace. Given that God knew that Alice will reject his grace, why did this knowledge not lead him to refrain from creating Alice?

Answer: It is impossible that the following two claims are true:

God refrains from creating Alice.
God knows that Alice will reject his grace.

For 1 entails:

Alice won’t exist

while 2 entails:

Alice will reject God’s grace

which in turn entails:

Alice will exist.

(Of course, if Molinism is true, one can ask a very similar question to the one I started with. But Molinism is false.)

Animalism without biologism

noreply@blogger.com (Alexander R Pruss) — Tue, 08 Sep 2026 19:16:41 +0000

Animalism holds that we are animals. This is a pretty common-sense view. We are clearly mammals, and mammals are animals.

However, there is a common objection: cerebrum transplants. As the objection goes, if your cerebrum is transplanted from your skull to a different skull (or to a supportive vat), you follow the cerebrum, while the animal stays with the lower brain and the rest of the body. So you are not the animal.

Let me say something that will sound silly: This objection is guilty of biologism about animals. The biologism here consists in the claim that we should take how biologists think about the continuity of animal life to be normative for how we should think about the identity of animals. You might think it’s obvious that we should engage in biologism about animals. But biologists are not metaphysicians, and the question of the metaphysical identity over time of animals does not really enter into the empirical investigations biologists engage in.

On what I think is the correct metaphysics of animals, the identity of an organism over time is grounded in the persistence of the animal’s soul or form, a metaphysical feature of the animal which is largely beyond the interest of contemporary biologists. Moreover, the animal’s form is responsible for its teleological structure, which defines what the animal’s flourishing is, and what is and is not central to that flourishing. It is plausible—though not necessary—to think that the form tends to remain with those components of the animal that are more central to its flourishing as the kind of thing it is.

In the case of homo sapiens, central to its flourishing is the exercise of the excellences of intellect and will. And the cerebrum is more directly involved in these exercises than any other part of the body, so it is plausible to think that the human animal goes with the cerebrum in a transplant.

Biologistically speaking, it is tempting to think of the intellect and will as subserving the animal functions of homeostatic maintenance, growth and reproduction. But that is a mistake. In a human, the functioning of the intellect and will is more central to flourishing than the animal functions.

This may seem discontinuous from other animals. Some will be happy with the discontinuity and some will think it’s a reductio ad absurdum of the view. But I think there is less discontinuity than seems to be the case. Think of dogs. It seems that their social life, with one another and with humans, constitutes more of their flourishing than their performance of the functions that define biological life. The social life presumably evolved because it promoted nutrition, growth and reproduction, but the social life appears more valuable. Given the likely ties between this social life and the cerebrum, it would not surprise me if dogs went with their cerebra in a cerebrum transplant.

The questions here are ones of value, and we should avoid biologism about value, even in non-human animals.

Classical theists should take externalism in philosophy of mind more seriously

noreply@blogger.com (Alexander R Pruss) — Mon, 07 Sep 2026 18:22:24 +0000

After Kripke, pretty much everyone nowadays accepts moderate mental externalism: the content of our thoughts depends on the extra-mental world. On earth where the predominant colorless liquid with such-and-such properties, the word “water” refers to water (i.e., H₂O), while on twin-earth where it is XYZ that has these properties, the word “water” refers to XYZ, and the thoughts that are expressed with sentences using “water” have a content involving water and XYZ, respectively, even if they are grounded in the same internal states of the brain and/or soul.

But there are stronger forms of mental externalism. First, we could have much more radical mental content externalism. Perhaps in one world the very same internal state means that there is water in the lake and in another world it means that the lake is dry but full of elephants. Second, we could have externalism about the subjective experience—the qualia—of mental activity. Thus, perhaps, depending on how extra-mental reality is, in one world one has the experience of seeing a red cube and in other that of smelling a daisy, despite having the same internal state in both worlds.

These stronger forms of mental externalism are rather counterintuitive and strange.

The point of this post, however, is that those of us who accept classical theism should take them much more seriously than we have tended to do. For on classical theism, God is simple and yet knows contingent extra-mental reality. This means that in different worlds God has different knowledge, while yet by simplicity remaining exactly internally the same. Hence we have an extremely strong mental content externism in the case of God: the very same state could mean that the universe consists of a lake with water and that it consists of a mountain covered by elephants. Moreover, if we add—as surely we should—that God’s knowledge is a conscious “vision” of reality, we also have to admit that externalism about subjective exerience is true of God.

Now, we cannot simply go from these externalisms being true of God to their being true of us. But I think we should take the externalisms in our case to be a very serious possibility.

A first attempt at exploring this kind of externalism in the case of subjective experience is here.

Subjective time and self-locating probability

noreply@blogger.com (Alexander R Pruss) — Mon, 31 Aug 2026 18:07:46 +0000

Suppose I am sluggish mentally in the morning and function fast mentally in the afternoons. I find myself in a dark room with no information about what time it is. Consider the two hypotheses MORN, that it is between 7am and 8am and AFT, that it is between 1pm and 2pm. Which is more likely, or are they equally likely?

It seems obvious that if I function mentally the same way in the morning and afternoon, MORN and AFT should be equally likely.

But now suppose I live in a world where every morning all processes in the solar system slow down by a factor of two, returning to the normal speed in the afternoon. Then, surely, for all practical purposes, the period from 7am to 8am is half an hour long! And so it seems right to say P(MORN) = (1/2)P(AFT). But the case of my own mental sluggishness seems to be just a more localized version of the solar system slowing down. Thus, in my original story we should say that likewise P(MORN) < P(AFT).

Here is one way to imagine mental functioning slowing down: my thinking is divided into discrete moments (like a computer’s internal clock, where basic operations take a clock cycle), and in the morning there are fewer of these discrete moments of thinking. If so, then it seems reasonable to say that the probability that the current time is between x and y is proportional to the number of discrete moments of thought between x and y, and hence P(MORN) < P(AFT), there being fewer moments of thought in the morning.

But I have a hard time getting good intuitions about the continuous case.

Preference and choice disposition

noreply@blogger.com (Alexander R Pruss) — Mon, 31 Aug 2026 17:31:33 +0000

Here is an initially plausible thesis about deliberation that I've long felt is right:

Normally, when choosing between options, you are more likely to choose an option X than an option Y if and only if you prefer X to Y.

Now consider a normal case where you must choose between four options:

Cookies and Earl Gray
Cookies and chamomile
Crackers and Earl Gray
Crackers and chamomile.

Suppose that you mildly prefer cookies to crackers and Earl Gray to chamomile, so that you have a 60% chance of choosing cookies over crackers and 60% chance of choosing Earl Gray over chamomile. Further, suppose you have no preferences about the combination as such. Then the chances of your four options are:

P(A) = 36%
P(B) = 24%
P(C) = 24%
P(D) = 16%.

Clearly, you can choose A. But notice an odd thing. You can organize your deliberation to start as follows: First choose between A and the disjunction B or C or D. (Maybe then if you chose B or C or D, you decide between B and C or D. And finally if you chose C or D, you decide between C and D.) However, here is another plausible principle of rationality:

Normally, if you prefer A to every disjunct of a disjunction, you prefer A to the disjunction.

Organizing your deliberation as above is not abnormal. So, (2) should still apply. You prefer A to B, and you prefer A to C, and you prefer A to D. So you should prefer A to the disjunction B or C or D, and hence you prefer A over the disjunction B or C or D. Thus, by (1) you are more likely to choose A than the disjunction B or C or D. In particular, the chance of your choosing A isn’t 36%, but must be more than 50%.

Hence, (1) predicts that the probabilities of our choices greatly depend on how one coarse-grains the options in one’s deliberation. This shouldn’t be. I thus reject (1). Which makes me sad, as it was a very neat thesis.

Explanatory infinitism

noreply@blogger.com (Alexander R Pruss) — Fri, 28 Aug 2026 19:41:01 +0000

I wish one could formulate and defend a version of explanatory finitism, roughly saying that nothing has infinitely many explanations or an infinite chain of explanations, even if these explanations are not causal (causal finitism of course rules out the causal case). But it seems hard to do so. There do seem to be counterexamples to explanatory finitism. Here some I’m thinking about right now:

Why are there infinitely many primes? Because there are infinitely many primes bigger than two. Why are there infinitely many primes bigger than two? Because there are infinitely many primes bigger than three. And so on.
Alice has just gone to heaven. Why is she going to experience a day of bliss? Because she will have a day of bliss tomorrow. And also because she will have a day of bliss the day after tomorrow. And also because she will have a day of bliss the day after that. And so on.
Alice has just gone to heaven. Why is she going to experience bliss on every future day? Partly because she will have bliss tomorrow. And partly because she will have bliss the day after tomorrow. And so on.

Determining camera pose from three landmarks and accelerometer: initial notes

noreply@blogger.com (Alexander R Pruss) — Fri, 28 Aug 2026 16:56:42 +0000

Suppose we have three landmarks at known locations in three dimensional space, not all on one line, and a camera that is not located at any landmark but sees all three landmarks, with known optical parameters such as focal length. The problem of determining the camera pose—the camera position and angles—from the positions of the landmarks in the camera image is known as the P3P problem. It is known that in general there will be at most four solutions for camera pose. And indeed sometimes there will be four solutions.

What if we have some additional information, namely we know how the camera is oriented with respect to gravity (e.g., because the camera is held horizontally or it’s equipped with an accelerometer)? Call the problem of reconstructing the camera image from n landmarks and gravity data PnPA. I recently showed that with just two landmarks, i.e., P2PA, there will be either one, two or infinitely many solutions, and geometrically characterized exactly which case occurs when.

Question: What can we say about the number of solutions to P3PA?

In this post I will make some slight progress on this question.

First we characterize when there are more than two solutions. Note that once we know the camera position, we can calculate the direction it’s pointing from the camera image (Lemma 4 in my paper). So we only need to look at the number of solutions for camera position.

Fact 1: There are at most two solutions for P3PA, except in the case where the three landmarks and camera all lie on one horizontal circle, in which case there are infinitely many solutions.

Proof: For P2PA, we have more than two solutions in precisely the following cases: (a) the two landmarks are on a single vertical line; (b) the two landmarks are in the same horizontal plane and so is the camera; and (c) the two landmarks and the camera are all on one line. To have more that two solutions for P3PA, each pair of landmarks must satisfy at least one of (a)–(c). Suppose this is so.

Suppose first that two landmarks, say m₁ and m₂, satisfy (a). Next suppose that no two landmarks lie on the same horizontal plane, so (b) is satisfied for no pair of landmarks. Then the third landmark m₃ does not lie on the same line as both m₁ and m₂, and hence neither the pair m₁ and m₃ nor th epair m₂ and m₃ satisfies (a), and at least one of these pairs fails to satisfy (c). Hence we have a pair that fails to satisfy any of (a)–(c), and we have at most two solutions by my P2PA result.

Now, continuing to suppose m₁ and m₂ satisfy (a), suppose that some pair of landmarks lies on the same horizontal plane. It can’t be m₁ and m₂ (as then they will be at the same point, and hence all three landmarks will be on one line). Without loss of generality, suppose m₁ and m₃ lie on the same horizontal plane H. The pair m₁ and m₃ cannot satisfy (a) (or else m₃ is at the same point as m₁). If it satisfies either (b) or (c), the camera is on the plane P, and hence in any case we have (b).

Furthermore, if no two landmarks satisfy (a), then since it can’t be that every pair of landmarks satisfies (c) as that would put all the landmarks on one line, at least one pair of landmarks must satisfy (b).

We thus have reduced to the case where a pair of landmarks satisfies (b): they are on the same horizontal plane H as the camera C. Let’s say that these landmarks are m₁ and m₂. Let m′₃ be the projection of m₃ to this plane. From the camera’s optical parameters and the camera image, we can calculate the angles m₁Cm₂, m₁Cm′₃ and m₂Cm′₃. It is known that the locus of points in a plane that subtend the same angle to two fixed points is an arc through these points. Thus, if we are to have more than two solutions, the arcs respectively through m₁Cm₂, m₁Cm′₃ and m₂Cm′₃ must intersect in at least three points. This would require m₁, m₂, m′₃ and C to all lie on the same circle T.

Now, suppose m₃ lies off the plane H. Then given the camera image and the gravity vector, we can measure the angle between m₃, the camera and the plane H, and given the position of m₃ we can compute the distance from m′₃ to the camera. This constrains the camera to lie on the circle T as well as on a second circle T′ around m′₃. Since m′₃ lies on T, these two circles intersect in at most two points. Thus, we have at most two solutions.

On the other hand, when m₃ is in the same plane, so m₁, m₂, m₃ and the camera lie on the same horizontal circle, we will have infinitely many solutions. For if A and B are two fixed points on a circle, and C is a third point on the same side as A and B, the angle ACB will be constant regardless of the choice of C. The landmarks m₁, m₂ and m₃ split T into three arcs, and the camera could be anywhere in the arc it’s in as far as the image goes.

Fact 2: There are cases where there are exactly two solutions.

Proof by picture: Suppose m₁ and m₂ lie on the same plane and m₃ lies off the plane. Suppose the camera is horizontally oriented, and pointed at m₁. Wherever the camera is on the blue arc, it sees m₁ and m₂ the same way (because the angle indicated by the dotted lines does not change, as discussed above). The image of m₃ in the camera is always directly above or below the image of m₁, and as long as the distance from camera to m₁ is the same, the image of m₃ does not move. Thus, the image of m₃ does not change as the camera moves on the red circle. Hence, at the two points where the red and blue circles intersect, the camera sees the same thing, and hence we have two solutions.

Against Arianism

noreply@blogger.com (Alexander R Pruss) — Thu, 27 Aug 2026 15:50:54 +0000

The Christian’s deepest desire is for direct union with God.
Our union with the Father must always be mediated by the Son.
So, if only the Father is God, then the Christian’s deepest desire cannot be fulfilled.
The Christian’s deepest desire can be fulfilled.
So, it is false that only the Father is God.

An Aristotelian argument for Causal Finitism

noreply@blogger.com (Alexander R Pruss) — Wed, 26 Aug 2026 16:55:17 +0000

The Aristotelian account of individuation for beings in the same species proceeds by matter. But to fill out this account, we may need to talk about the times at which the matter is different.

One reason for talking about this is the possibility on some versions of Aristotelianism that a chunk of matter can survive substantial change, and hence can compose different substances—albeit at different times. If one wolf kills and eats another, there is in principle no reason why eventually the cannibal wolf’s matter might not entirely be the same as the matter of the victim, then.

So, perhaps, we may want to say that what distinguishes members x and y of the same material species is that they have different matter at the same time. But that’s not quite right. For it could be that x and y don’t ever exist at the same time! We might thus want to say that what grounds the distinction between x and y is:

at some time t, either x exists and y doesn’t or x doesn’t exist and y does or x and y both exist and have different matter.

This has at least two problems. First, if I travel backwards in time to a time when I already existed, then I will have different matter from my past self at the same time (since I’ve changed some cells in the meanwhile).

But the second is probably the more serious. According to the story in (1), past, present and future differences in matter all ground the distinction between x and y. But it is implausible that future differences in matter be relevant to my present distinction from you. So we should minimally restrict (1) to talking of past and present times t. But if we do that, then what grounds our difference changes over time, and does so pointlessly, because as soon as our distinction has once been established, that’s surely all we need. In fact, the last point is particularly important given the soul’s survival past death: what makes your soul past death different from mine is presumably the past difference in matter. This suggests that if we want to get at what truly grounds our distinction, we have to go back in time as far as we can, to the very beginnings of our existence, and say:

either x and y differ in the time at which they began to exist, or x and y began at the same time with different matter.

There is another reason to prefer (2) to (1). We not only want an account of the distinction of substances within a world, but between possible worlds. But there are many possible worlds where I exist but where my current matter is different from the matter I now have in the actual world (namely, because my diet was different). Thus, (1) has no hope of being extended to an account of transworld distinction. But (2) has some hope of such an extension. At the very least, it gives a sufficient condition for distinctness in the transworld case, while (1) does not even do that.

But now here is an interesting thing. If (2) is the true story about the distinction between material substances of the same species, it follows that:

It is impossible for there to be two material substances of the same species that have no beginning in time.

But if Causal Finitism (the doctrine that one effect cannot have infinitely many causes) is false, it seems like it should be possible to have an organism that has lived for an infinite amount of time, and likewise possible to have two such organisms in one species. So (2) gives us a new reason to accept Causal Finitism.

However, personally, I don’t like the Aristotelian account of distinctions between substances.

A theological argument against Causal Finitism

noreply@blogger.com (Alexander R Pruss) — Wed, 26 Aug 2026 16:09:01 +0000

Some people are attracted to this idea:

God gives infinitely many chances for everyone to be saved.

At the same time, by and large (with the main exception being Origen) the Christian tradition accepts:

Whether you are saved or not is determined at the time of death.

It’s interesting to note that if you deny Causal Finitism, you can have both (1) and (2). You can suppose, for instance, that God miraculously sets up a supertask for you in the millisecond prior to death, miraculously accelerating the functioning of your soul in such a way that in the first half millisecond you have one choice whether to be saved or not, in the next quarter millisecond you have another choice, in the next eighth you have another, and so on. And then you’re saved if any of these infinitely many choices was made for God, and damned only if all them were against God.

While I buy Causal Finitism, and hence probably have to deny the conjunction of (1) and (2) since on (1) and (2) your state post-death appears to be determined by infinitely many causes (the choices), I think this is a somewhat attractive reason to deny Causal Finitism.