Wednesday, May 23, 2018

Substantial change

The following seems to me to be a central tenet of classical Aristotelianism:

  1. The identity of a parcel of matter is grounded in form.

But it seems to me that matter is introduced by Aristotelianism primarily to solve the problem of distinguishing (a) one substance changing into another (or into several others) from (b) one substance perishing and a new substance coming to be. The solution seems to be that in case (a) the matter persists, but not so in case (b).

But if the identity of a parcel of matter comes from form, then it is very puzzling indeed how a parcel of matter can remain selfsame while a change of form occurs. In other words, there is a tension between (1) and the motive for the introduction of matter into the ontology.

I am inclined to hold on to (1) in some sense, but reject the idea that matter solves the problem of substantial change.

Here is my currently best deflationary solution to the problem of substantial change. Certain kinds of causal interactions are described as “transfers of qualities”. For instance, when billiard ball A strikes billiard ball B in such a way that A stops moving and B begins moving, the momentum of A is “transferred” to B. However, we certainly do not want a metaphysics of momentum transfer on which there exists an entity m that previously was in A and later the numerically same m is present in B. That would be taking the talk of “transfer” too literally. Similarly, we talk of heat transfer.

I do not have an account of quality transfer, but a rough necessary condition for it is that there is a causal interaction where A causes B to gain a property that it itself is losing. There is an obvious difference between the momentum transfer story and the case where A is miraculously stopped by God from moving while B is simultaneously and miraculously set by God in motion.

Now, a special case of quality transfer is when a causal interaction not only transfers a quality but also creates one or more new substances. For instance, suppose a gecko chased by a predator drops its tail, whose writhing confuses the predator. In doing so, the gecko transfers some of its mass, extension, color, motion and other qualities to a new substance (or aggregate of substances), a substance that comes to exist as a result of the same causal interaction.

The technical neo-Aristotelian term for the gecko’s loss of its tail is excretion. Excretion comes in two sorts. The kind of excretion in the case of the gecko’s autotomy is productive excretion, where qualities, notably including mass and extension (understood broadly to include temporal extension for aspatial temporal things), are transfered to one or more substances that are produced in the same causal interaction. Another kind of excretion is accretive excretion, where qualities are transferred to one or more substances that exist independently of this causal interaction. For instance, if an animal were to swallow a living plant, perhaps the plant in the animal’s digestive system could be accretively excreting: its qualities, notably including mass and extension, would come to be gradually lost to the plant while gained by the animal. (This is a bit more complicated in real life, I expect: I doubt the digested bits immediately become parts of the animal.)

Substantial corruption of a material substance, then, is total excretion, a causal interaction where all of a substance’s extension and mass is excreted to one or more substances. This comes in two basic varieties: substantial change where the the beneficiary substances result from the same causal interaction and accretive substantial corruption where the beneficiary substances exist independently of this causal interaction (and typically are preexistent). And one can have a combination case where some of the beneficiaries result from the interaction and some are not dependent on it.

But there is nothing metaphysically deep about substantial corruption.

Wednesday, May 16, 2018

Possibly giving a finite description of a nonmeasurable set

It is often assumed that one couldn’t finitely specify a nonmeasurable set. In this post I will argue for two theses:

  1. It is possible that someone finitely specifies a nonmeasurable set.

  2. It is possible that someone finitely specifies a nonmeasurable set and reasonably believes—and maybe even knows—that she is doing so.

Here’s the argument for (1).

Imagine we live an uncountable multiverse where the universes differ with respect to some parameter V such that every possible value of V corresponds to exactly one universe in the multiverse. (Perhaps there is some branching process which generates a universe for every possible value of V.)

Suppose that there is a non-trivial interval L of possible values of V such that all and only the universes with V in L have intelligent life. Suppose that within each universe with V in L there runs a random evolutionary process, and that the evolutionary processes in different universes are causally isolated of each other.

Finally, suppose that for each universe with V in L, the chance that the first instance of intelligent life will be warm-blooded is 1/2.

Now, I claim that for every subset W of L, the following statement is possible:

  1. The set W is in fact the set of all the values of V corresponding to universes in which the first instance of intelligent life is warm-blooded.

The reason is that if some subset W of L were not a possible option for the set of all V-values corresponding to the first instance of intelligent life being warm-blooded, then that would require some sort of an interaction or dependency between the evolutionary processes in the different universes that rules out W. But the evolutionary procesess in the different universes are causally isolated.

Now, let W be any nonmeasurable subset of L (I am assuming that there are nonmeasurable sets, say because of the Axiom of Choice). Then since (3) is possible, it follows that it is possible that the finite description “The set of values of V corresponding to universes in which the first instance of intelligent life is warm blooded” describes W, and hence describes a nonmeasurable set. It is also plainly compossible with everything above that somebody in this multiverse in fact makes use of this finite description, and hence (1) is true.

The argument for (2) is more contentious. Enrich the above assumptions with the added possibility that the people in one of the universes have figured out that they live in a multiverse such as above: one parametrized by values of V, with an interval L of intelligent-life-permitting values of V, with random and isolated evolutionary processes, and with the chance of intelligent life being warm-blooded being 1/2 conditionally on V being in L. For instance, the above claims might follow from particularly elegant and well-confirmed laws of nature.

Given that they have figured this out, they can then let “Q” be an abbreviation for “The set of all values of V corresponding to universes wehre the first instance of intelligent life is warm-blooded.” And they can ask themselves: Is Q likely to be measurable or not?

The set Q is a randomly chosen subset of L. On the standard (product measure) understanding of how to probabilistically make sense of this “random choice” of subset, the event of Q being nonmeasurable is itself nonmeasurable (see the Sawin answer here). However, intuitively we would expect Q to be nonmeasurable. Terence Tao shares this intuition (see the paragraph starting “Intuitively”). His reason for the intuition is that if Q were measurable, then by something like the Law of Large Numbers, we would expect the intersection of Q with a subinterval I of L to have a measure equal to half of the measure of I, which would be in tension with the Lebesgue Density Theorem. This reasoning may not be precisifiable mathematically, but it is intuitively compelling. One might also just have a reasonable and direct intuition that the nonmeasurability is the default among subsets, and so a “random subset” is going to be nonmeasurable.

So, the denizens of our multiverse can use these intuitions to reasonably conclude that Q is nonmeasurable. Hence, (2) is true. Can they leverage these intuitions into knowledge? That’s less clear to me, but I can’t rule it out.

Monday, May 14, 2018

Simultaneous and diachronic causation

The main problem with the idea that all causation is simultaneous is to make sense of the obvious fact of diachronic causation, as when setting an alarm in the evening causes it to go off in the morning. Here is a theory that has both simultaneity and diachronicity that bears further examination:

  • All causation between substances is simultaneous

  • There is diachronic causation within a substance.

We now have a model of how setting the alarm works, on the simplifying assumption that the alarm clock is a substance. In the evening, by simultaneous causation, I cause the clock to have a certain state. A sequence of diachronic causal interactions within the clock—accidents of the clock causing other accidents of the clock, say—then causes the alarm to go off. The alarm’s going off then, by means of simultaneous causation between substances, causes particles in the air to move, etc. In other words, the diachronicity of the causation is all internal to the substances.

An even more interesting theory would hold that:

  • All causation between substances is simultaneous

  • All causation within a substance is diachronic.

If we were willing to swallow this, then we would have a very elegant account of the internal time of a substance as constituted by the causal relations within the substance (presumably, the causal relations between the accidents of the substance).

Why are there infinitely many abstracta rather than none?

It just hit me how puzzling Platonism is. There are infinitely many abstract objects. These objects are really real, and their existence seems not to be explained by the existence of concreta, as on Aristotelianism. Why is there this infinitude of objects?

Of course, we can say that this is just a necessary fact. And maybe it’s just brute and unexplained why necessarily there is this infinitude of objects. But isn’t it puzzling?

Augustinian Platonism, on which the abstract objects are ideas in the mind of God, offers an explanation of the puzzle: the infinitely many objects exist because God thinks them. That still raises the question of why God thinks them. But maybe there is some hope that there is a story as to why God’s perfection requires him to think these infinitely many ideas, even if the story is beyond our ken.

I suppose a non-theistic Platonist could similarly hope for an explanation. My intuition is that the Augustinian’s hope is more reasonable.

Friday, May 11, 2018

Fun with desire satisfaction

Suppose that desire satisfaction as such contributes to happiness. Then it makes sense to pay a neuroscientist to induce in me as many desires as possible for obvious mathematical truths: the desire that 1+1=2, that 1+2=3, that 1+3=4, etc.

Or if desire has to be for a state of affairs in one’s control, one can pay the neuroscientist to induce in me as many desires as possible for states of affairs like: my not wearing a T-shirt that has a green numeral 1, my not wearing a T-shirt that has a green numeral 2, etc. Then by not wearing a T-shirt with any green numerals, I fulfill lots of desires.

Thursday, May 10, 2018

Provability and numerical experiments

A tempting view of mathematics is that mathematicians are discovering not facts about what is true, but about what is provable from what.

But proof is not the only way mathematicians have of getting at truth. Numerical experiment is another. For instance, while we don’t have a proof of Goldbach’s Conjecture (each even number bigger than two is the sum of two primes), it has been checked to hold for numbers up to 4 ⋅ 1018. This seems to give significant inductive evidence that Goldbach’s Conjecture is true. But it does not seem to give significant evidence that Goldbach’s Conjecture can be proved.

Here’s why. Admittedly, when we learned that that the conjecture holds for some particular number n, say 13, we also learned that the conjecture can be proved for that specific number n (e.g., 13 = 11 + 2 and 11 and 2 are prime, etc.). Inductively, then, this gives us significant evidence that for each particular number n, Goldbach’s conjecture for n is provable (to simplify notation, stipulate Goldbach’s Conjecture to hold trivially for odd n or n < 4). But one cannot move from ∀n Provable(G(n)) to Provable(∀n G(n)) (to abuse notation a little).

The issue is that the inductive evidence we have gathered strongly supports the claim that Goldbach’s Conjecture is true, but gives much less evidence for the further claim that Goldbach’s Conjecture is provable.

The argument above is a parallel to the standard argument in the philosophy of science that the success of the practice of induction is best explained by scientific realism.

Monday, May 7, 2018

Heaven and materialism: The return of the swollen head problem

Plausibly, there is a maximum information density for human brains. This means that if internal mental states supervene on the information content of brains and there is infinite eternal life, then either:

  1. Our head grows without bound to accommodate a larger and larger brain, or

  2. Our brain remains bounded in size and either (a) eventually we settle down to a single unchanging internal mental state (including experiential state) which we maintain for eternity, or (b) we eternally move between a finite number of different internal mental states (including experiential states).

For if a brain remains bounded in size, there are only finitely many information states it can have, because of the maximum information density. Neither of options 2a and 2b is satisfactory, because mental (intellectual, emotional and volitive) growth is important to human flourishing, and a single unchanging internal mental state or eternal repetition does not fit with human flourishing.

Note, too, that on both options 2a and 2b, a human being in heaven will eventually be ignorant of how long she’s been there. On option 2b, she will eventually also be ignorant of whether it is the first time, the second time, or the billionth that she is experiencing a particular internal mental state. (I am distinguishing “internal mental states” from broad mental states that may have externalist semantics.) This, too, does not fit with the image of eternal flourishing.

This is, of course, a serious problem for the Christian materialist. I assume they won’t want to embrace the growing head option 1. Probably the best bet will be to say that in the afterlife, our physics and biology changes in such a way as to remove the information density limits from the brain. It is not clear, however, that we would still count as human beings after such a radical change in how our brains function.

The above is also a problem for any materialist or supervenientist who becomes convinced—as I think we all should be—that our full flourishing requires eternal life. For the flourishing of an entity cannot involve something that is contrary to the nature of a being of that sort. But if 2a and 2b are not compatible with our flourishing, and if 1 is contrary to our nature, then our flourishing would seem to involve something contrary to our human nature.

This is a variant of the argument here, but focused on mental states rather than on memory.

Friday, May 4, 2018

Medical and spacecraft ventilators

Some thinking that to turn off a patient’s ventilator would not be to kill but “to let die”. But it seems obvious that to turn off a spacecraft’s ventilation system would be to kill the astronauts through suffocation.

Of course, there are differences between the two cases. One difference is that the medical ventilator is more intimately connected to the patient. This difference, however, would seem to make turning off the ventilator be more of a killing.

A perhaps more promising difference is that when the patient’s ventilator is turned off, the patient dies from a disease that renders unassisted breathing impossible, while the astronauts die from the turning off of the air system. Maybe there is something to this, but I am doubtful. For we can also say that just as the patient would die from a disease, the astronauts would die from the airlessness of space. It is true that one of these is a disease and the other is an environmental condition, but why should that make a difference with respect to what is a killing?

Moreover, if an engineer turns off the ventilation system on the spacecraft before an astronaut reveals that the technician’s doctoral dissertation was plagiarized, that’s murder. And similarly if a doctor turns off a ventilator before the patient reveals that the doctor cheated in medical school, that’s clearly murder, too.

Similarly, if the death penalty is ever permissible, it could in some cases be administered by disconnecting a ventilator—and it would clearly still be an execution, and hence a killing.

But what if the doctor turns off the ventilator for some reason other than to cause the patient’s death, say to prevent an electrical overload to the hospital’s system which would kill many other patients? Changing the intentions with which an act of killing is done can change whether the act is an intentional killing, whether the act is wrong and whether the act is a murder, but I do not think it changes whether the act is a killing. Thus, the doctor who turns off the ventilator for a reason other than to cause death is still killing, but not intentionally.

Nor does it make a difference with respect to killing whether the disconnection is thought of as causing or hastening death. The doctor who turns off the ventilator to prevent the doctor’s medical school cheating from coming to light could think of the activity as hastening death—making the patient die before revealing the secret. But it’s still murder, and hence it’s still killing. Similarly, the plagiarist engineer would be a murderer even if the air system on the spacecraft were failing and the astronauts would die anyway within a week.

Of course, the judgment that turning off the ventilator is killing does not imply that it is murder or even impermissible. But if we grant that it is always murder to intentionally kill the innocent, the turning off a ventilator in order to cause or hasten death is murder.

Wednesday, May 2, 2018

Time as the measure of change?

Aristotle says that time is the measure of change.

Suppose a pool of liquid changes from fragrant to putrid. We can quantify or measure such features as:

  • the spatial extent of the change

  • the value (in multiple senses) of the change

  • the probability of the change

  • the temporal extent of the change.

Obviously, when we talk of time as the measure of change, we have in mind the last of these four. But to define time as the temporal measure of change is blatantly circular. So the Aristotelian needs to non-circularly specify the sort of measurement of change that time provides. (I am not saying this can’t be done. But it is a challenge.)

Tuesday, May 1, 2018

Time and clocks

Einstein said that time is what clocks measure.

Consider an object x that travels over some path P in spacetime. How long did the travels of x take? Well, if in fact x had a clock traveling with it, we can say that the travels of x took the amount of time indicated on the clock.

But what if x had no clock with it? Surely, time still passed for x.

A natural answer:

  • the travels of x took an amount of time t if and only if a clock would have measured t had it been co-traveling with x.

That can’t be quite right. After all, perhaps x would have traveled for a different amount of time if x had a clock with it. Imagine, for instance, that x went for a one-hour morning jog, but x forgot her clock. Having forgot her clock, she ended up jogging 64 minutes. But had she had a clock with her, she would have jogged exactly 60 minutes.

That seems, though, a really uncharitable interpretation of the counterfactual. Obviously, we need to fix the spacetime path P that x takes. Thus:

  • the travels of x over path P took an amount of time t if and only if a clock would have measured t had it been co-traveling with x over the same path P.

But this is a very strange counterfactual if we think about it. Clocks have mass. Like any other massive object, they distort spacetime. The spacetime manifold would thus have been slightly different if x had a clock co-moving with it. In fact, it is quite unclear whether one can make any sense of “the same path P” in the counterfactual manifold.

We can try to control for the mass of the clock. Perhaps in the counterfactual scenario, we need to require that x lose some weight—that x plus the clock have the same mass in the counterfactual scenario as x alone had in the actual scenario. Or, more simply, perhaps we can drop x altogether from the counterfactual scenario, and suppose that P is being traveled by a clock of the same mass as x.

But we won’t be able to control for the mass of the clock if x is lighter than any clock could be. Perhaps no clock can be as light as a single electron, say.

I doubt one can fix these counterfactuals.

Perhaps, though, I was too quick to say that if x had no clock with it, time still passed for x. Ordinary material substances do have clocks in them. These clocks may not move perfectly uniformly, but they still provide a measure of length of time. Alice’s jog took 396,400 heartbeats. Bob’s education took up 3/4 of his childhood. Maybe the relevant clocks, then, are internal changes in substances. And where the substances lack such internal changes, time does not pass for them.

Monday, April 30, 2018

Avoiding double counting of culpabilities

Here’s an interesting double-counting problem for wrongdoing. Alice stands to inherit a lot of money from a rich uncle in Australia. Bob thinks he stands to inherit a lot of money from a rich uncle in New Zealand. Both of them know that it’s wrong to kill rich uncles for their inheritance, but each of them nonetheless hires a hitman with the instruction to kill the rich uncle. Both hitmen run off with the money and do nothing. But Bob in fact has no uncles—he was misinformed.

Here are some plausible observations:

  1. Alice culpably committed two wrongs: she violated her conscience and she wronged her uncle by hiring a hitman to kill him.

  2. Bob culpably committed only one of these wrongs: he violated his conscience.

  3. Bob is just as morally culpable as Alice.

Here is one way to reconcile these observations. We should distinguish between something like moral failings of the will, on the one hand, and wrongdoings, on the other. It is the moral failings of the will that result in culpability. This culpability then will qualify one or more wrongdoings. But the amount of culpability is not accounted by looking at the culpable wrongdoings, but at the moral failings of the will. A being that executes unalloyed perfect justice will look only at these failings of the will. Alice and Bob each morally failed in the same way and to the same degree (as far as the stories go), and so they are equally culpable. But, nonetheless, Alice has two culpable wrongdoings—culpable through the same moral failing of the will, which should not be double counted for purposes of just punishment.

Friday, April 27, 2018

Love and deontology

Sometimes it wrongs a person to intentionally do them what is known to be in their own best interest. If by torturing you for 60 minutes I can prevent you from being tortured in the same way for 70 minutes by someone else, it may be in your best interest that I torture you. But it is still wrong for me to torture you. Cases of this sort can be multiplied, though of course only deontologists will find any of them plausible.

(One can also analyze these cases as ones where the action is wrong because it is a violation of the agent’s own human dignity. I think the actions are violations of the agent’s own dignity, but they are violations of the agent’s dignity because they wrong the other party.)

These are cases where your action wrongs someone but causes them on balance benefit. This means that to be wronged does not entail being on balance harmed.

Here is how I think we should think of these cases. The true ethics is an ethics of love: I should love everyone. But benevolence is only one of the three fundamental aspects of love, with the other two being union and appreciation. To wrong someone is to violate one or more of the three aspects of love. If I intentionally do something that is known to be in your best interest, I do not violate the benevolence aspect of love. But I may violate one of the other two aspects. In the cases I am thinking of, like torture, the act is an affront to your human dignity, and by affronting your human dignity I am directly acting against the appropriate kind of unitive relationship between human beings—hence, I violate the unitive aspect of love.

It may seem, however, that these are cases where I have a real moral dilemma. For if I refuse to do the act, then it seems I am violating the benevolence of love. But this is mistaken. To fail to be benevolent is not to oppose benevolence. Some cases are obvious. If I fail to be benevolent to you because someone just as close to me has a greater need, I may have done something not in your best interest, but I have not violated the benevolence of love. Now, if I intentionally did to you what was not in your best interest because it was not in your best interest, then I have violated love.

Thursday, April 26, 2018

Alethic Platonism

I’ve been thinking about an interesting metaphysical thesis about arithmetic, which we might call alethic Platonism about arithmetic: there is a privileged, complete and objectively correct assignment of truth values to arithmetical sentences, not relative to a particular model or axiomatization.

Prima facie, one can be an alethic Platonist about arithmetic without being an ontological Platonist: one can be an alethic Platonist without thinking that numbers really exist. One might, for instance, be a conceptualist, or think that facts about natural numbers are hypothetical facts about sequences of dashes.

Conversely, one can be an ontological Platonist without being an alethic Platonist about arithmetic: one can, for instance, think there really are infinitely many pluralities of abstracta each of which is equally well qualified to count as “the natural numbers”, with different such candidates for “the natural numbers” disagreeing on some of the truths of arithmetic.

Alethic Platonism is, thus, orthogonal to ontological Platonism. Similar orthogonal pairs of Platonist claims can be made about sets as about naturals.

One might also call alethic Platonism “alethic absolutism”.

I suspect causal finitism commits one to alethic Platonism.

Something close to alethic Platonism about arithmetic is required if one thinks that there is a privileged, complete and objectively correct assignment of truth values to claims about what sentence can be proved from what sentence. Specifically, it seems to me that such an absolutism about proof-existence commits one to alethic Platonism about the Σ10 sentences of arithmetic.

Wednesday, April 25, 2018

We aren't just rational animals

I think some Aristotelian philosophers are inclined to think that our nature is to be rational animals, so that all rational animals would be of the same metaphysical species. Here is a problem with this. Our nature—or form or essence—specifies the norms for our structure. Our norms specify that we should be bipedal: there is something wrong with us if we are incapable of bipedality. But an intelligent squid would be a rational animal, and its norms would surely not specify that it is supposed to be bipedal. So, it seems, that the hypothetical intelligent squid would have a different nature from ours.

But that was too quick. For it could be that our nature grounds conditionals like:

  1. If you’re human, you should have two arms and two legs

  2. If you’re a squid, you should have eight arms and two tentacles.

We have some reason to think there are such conditional normative facts even if we take our metaphysical species narrowly to be something like human or even homo sapiens, since presumably our nature grounds normative conditionals about bodily structure with antecedents specifying whether we are male or female.

But there is a hitch here: if humans and intelligent squid have the same form, what makes it be the case that for me the antecedent of 1 is true while for Alice (say) the antecedent of 2 is true? I think our best story may be that it is facts about DNA, so in fact the antecedents of 1 and 2 are abbreviations for complex facts about DNA.

That might work for DNA-based animals, which are all the animals we have on earth, but it probably won’t work for all possible animals. For surely there nomically could be animals that are not based on DNA, and it is implausible that we carry in our nature the grounds for an array of conditionals for all the nomically (at least) possible genetic encoding schemes.

I suppose we could take our nature to be rational members of the Animalia, with the assumption that the kingdom Animalia necessarily includes only DNA-based organisms (but not all of them, of course). But Animalia seems a somewhat arbitrary choice of classification to tack on to rationality. It doesn’t have the exobiological generality of animal, the earthly generality of DNA-based organism, or the specificity of human.

It seems to me that

  • rational DNA-based organism, or

  • rational member of genus Homo

are better options for where to draw the lines of our metaphysical species, assuming “rationality” is the right category (as opposed to, say, St. John Paul II’s suggestion that we are fundamentally self-givers), than either rational animal or rational member of Animalia.

Tuesday, April 24, 2018

Balancing between theism and atheism

The problem of evil consists of three main parts:

  • The problem of suffering.

  • The problem of evil choices.

  • The problem of hiddenness (which is an evil at most conditionally on God’s existing).

The theist has trouble explaining why there is so much suffering. The atheist, however, has trouble explaining why there is any suffering, given that suffering presupposes consciousness, and the atheist has trouble explaining why there is any consciousness.

Of course, there are atheist-friendly naturalistic accounts of consciousness. But they all face serious difficulties. This parallels the fact that theists have theodical accounts of why God permits so much suffering, accounts that also face serious difficulties.

So, on the above, considerations of suffering are a net tie between theism and atheism.

The theist does not actually have all that much trouble explaining why there are evil choices. Libertarian free will does the job. Of course, there are some problems with libertarian accounts of free will. These problems are not, I think, nearly as serious as the problems that theists have with explaining why there is so much suffering or atheists have with explaining why there is consciousness. Moreover, there is a parallel problem for the atheist. Evil choices can only exist given free will. Prima facie the most plausible accounts of free will are libertarian agent-causal ones. But those are problematic for the atheist, who will find it difficult to explaining where libertarian agents come from. The atheist probably has to embrace a compatibilist theory, which has at least as many problems as libertarian agent-causalism.

So, considerations of evil choices look at best as a net tie for the atheist.

Finally, there is the problem of hiddenness for the theist. But while the theist has trouble explaining how we don’t all know something so important as the existence of God, the atheist has epistemological trouble of her own: she has trouble explaining how she knows that there is no God. After all, knowledge of the highly abstract facts that enter into arguments regarding the existence of God is not the sort of knowledge that seems to be accessible to evolved natural beings.

So, considerations of knowledge of the existence or non-existence of God look as a net tie.

The problem of evil, however, exhausts the powerful arguments for atheism. But the above considerations far from exhaust the powerful arguments for theism.

The above reasoning no doubt has difficulties. But I want to propose it as a strategy for settling disputes in cases where it's hard to assign probabilities. For even if it's hard to assign probabilities, we can have good intuitions that two considerations are a wash, that they provide equal evidence. And if we can line up arguments in such a way, being more careful with issues of statistical dependence than I was above, then we can come to a view as to which way some bunch of evidence points.