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.

Promises and permissibility

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

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:

1. 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.

2. 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

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

1. 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.

2. 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

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:

1. It is possible to have two overlapping cells in a multicellular organism neither of which is a part of the other.

2. 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

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.