Merging Lewisian worlds

According to Lewis, any pair (or, more generally, plurality) of concrete (he doesn't even restrict it this way) of objects has a mereological sum. Now, suppose that x and y are concrete objects in worlds w₁ and w₂ respectively. Let z be the mereological sum of x and y. According to Lewis, worlds are maximal spatiotemporally connected sums of objects. Now, here are some plausible principles:

1. Spatiotemporal connection is transitive and symmetric.
2. If a is spatiotemporally connected to a part of b, then a is spatiotemporally connected to b.

Consider any concrete objects a and b in w₁ and w₂, respectively. Then a is connected with x, since all objects in a world are connected. And y is connected with b. Moreover, by 2, a is connected with z since x is a part of z. And by 2, b is connected with z. Thus, by 1, a is connected with b. Thus, all objects in w₁ and w₂ are mutually connected, and so by Lewis's account of worlds, there is only one world. Which is absurd.

Explaining the simplicity of theories

The following is a basic presupposition of science:

1. If two scientific theories equally well fit our observations, and one of them is by far simpler than the other, then the simpler theory is more likely to be true.

Granted, we don't have a good account of "far simpler" or even of "equally well fit", but nonetheless something like (1) is surely true. And that is an amazing fact about the world. What explains that fact?

Note that we cannot really explain (1) simply by citing the fundamental physical laws of nature. For (1) is true in reality as discovered across the disciplines, not just in fundamental physics. It is surely true of biological, geological, astronomical and sociological theories.

Fact (1) suggests that the laws and other structure of our world are generated in a way that tends towards simplicity given the same empirical outcomes. Why? Well, I see three stories.

Theism: Simplicity is good, either intrinsically for aesthetic reasons or instrumentally because it helps agents like us get the good of empirical knowledge, and so a perfect being will prefer simpler structures when they can produce the same empirical outcomes.

Axiarchism: Simplicity is good, as above, and there is a fundamental law of metaphysics that all must be for the best.

Logocentrism: The world is generated by something like a random process that randomly generates a complete coherent descriptive sentence, in a non-gerrymandered language, with longer sentences having lower probability.

I find Logocentrism incredible: Why should the length of a linguistic expression matter except where there is a mind?

Faculty opening at Baylor Philosophy

BAYLOR UNIVERSITY, Waco, TX announces a tenure-track Assistant Professor position in the Department of Philosophy beginning in the fall of 2014. AOS and AOC:  Open. Salary is competitive. Teaching load and scholarly expectations are consistent with those of a research university. Review of applications will begin immediately and will continue until the position is filled. To ensure full consideration, the completed application should be received by November 1, 2013.

Baylor, the world's largest Baptist University, holds a Carnegie classification as a "high-research" institution.  Baylor's mission is to educate men and women for worldwide leadership and service by integrating academic excellence and Christian commitment within a caring community.  Because Baylor aspires to become a top tier research university while reaffirming and deepening its distinctive Christian mission, Baylor is actively recruiting new faculty with a strong commitment to scholarly activity and an equally strong commitment to teaching.

The letter of application should respond to Baylor's most recent mission statement Pro Futuris (available on the web at http://www.baylor.edu/vision) and include an account of the applicant's own religious views. In addition to a letter of application, the candidate should submit a CV, a professional writing sample, three letters of recommendation, and official transcripts.  Send applications to Dr. C. Stephen Evans, Chair, Search Committee, Department of Philosophy, Baylor University, One Bear Place #97273, Waco, Texas, 76798-7273.Baylor is affiliated with the Baptist General Convention of Texas and as an AA/EEO employer; Baylor encourages minorities, women, veterans, and persons with disabilities to apply.

Progress report: Positive results for invariant Popper functions

This is a very technical note. I've spent a fair amount of time this week thinking about invariant Popper functions. Say that a group G is neatly supramenable if there is a Popper function P on G with every non-empty subset normal and satisfying the strong invariance condition P(gA|B)=P(A|B) whenever gA∪A⊆B. Neatly supramenable groups are supramenable: every non-empty subset A has an invariant finitely additive measure m such that m(A)=1. Anyway, I think I can prove—I now have two proofs drafted, so that makes me more confident—that every exponentially bounded group is neatly supramenable. Thus, every elementary supramenable group is neatly supramenable.

One philosophically interesting upshot of all this is that n-dimensional Euclidean space supports a Popper function with all non-empty subsets normal that is invariant under all translations as well as under single-coordinate reflections ((x₁,...,x_i,...,x_n) going to (x₁,...,−x_i,...,x_n)). But when one adds rotations into the mix, this is false for n≥2. So there is something philosophically problematic about rotations for the notion of uniform probability.

Don't quote the result yet as the proofs use mathematics that I am not very familiar with (ultrafilters, non-standard analysis, etc.).

Oh, and all of this uses the Axiom of Choice.

Soul-body interaction

I have a post on soul-body interaction on Biola's Center for Christian Thought blog that may be of interest to my readers here.

Trust and lies

You promise to meet me for dinner at 7. We say that the promise normally makes it appropriate for me to trust you'll show up at 7. But that's not quite right. What is more appropriate to trust is that you'll meet me for dinner at 7 or have good moral reason not to be there. This point applies even if I know that you won't have such good moral reason. For that you won't isn't s matter of trust of you, but of prediction.
By the same token, if it can ever be permissible to lie, and you assert something, I never ought to trust you that you are being truthful. Instead at most I ought to trust that you either are being truthful or have good moral reason to lie.
So if it is ever appropriate to take it on trust alone that you are being truthful, lying is always wrong.

An argument for incommensurable goods

One of the upshots of a number of my posts on the limitations of probability theory is that there are events that are probabilistically incomparable—neither can be said to be more likely than the other. (For instance, this post.) But an objective chance at a good is good, and better the greater the chance and worse the lower the chance. Chances p and q at the same good G will, then, be incommensurably good when the chances p and q are incomparable. Hence, if there are incomparable objective chances, there can be incommensurable goods. But it's plausible that there can be incomparable chances (see the post I linked to above, for instance). So there can be incommensurable goods.

Doctrine and morals

Some Christians think the Christian scriptures and tradition more reliable on doctrine than on morals. But Christianity came to change our lives rather than to satisfy our curiosity. It is more a way of life rather than an intellectual creed. And given this, Christians would expect the Christian scriptures and tradition to be, if anything, more reliable on morals than on doctrine. And since the center of Christian life is love, Christians should expect the greatest reliability to be on those parts of morality most intimately connected with love.

Regularity on the circle

Suppose that a point is uniformly chosen on the circumference of the circle T. Write A≤B for "the point is at least as likely to be in B as in A" and say A<B when A≤B but not B≤A. Here are some very plausible axioms:

1. If A≤B and B≤C, then A≤C. (Transitivity)
2. A≤A. (Reflexivity)
3. Either A≤B or B≤A (or both). (Totality)
4. If A is a proper subset of B, then A<B. (Regularity)

Moreover, we have a very plausible invariance condition:

5. If r is any reflection in a line going through the center of the circle T and A≤B, then rA≤rB,

i.e., the probability comparison holds between A and B if and only if it holds between their reflections.

Proposition. There is no relation ≤ satisfying (1)-(5) for all countable subsets A, B and C of T.

I do not as yet know if the Proposition is true if we replace reflections by rotations in (5).

Totality and/or Regularity should go. Other cases suggest to me that both should go.

Proof of Proposition: Suppose ≤ satisfies (1)-(5). Say that A~B if and only if A≤B and B≤A. It is easy to see that ~ is transitive since ≤ is transitive and if A~B then rA~rB. Now observe that rA~A. For either rA≤A or A≤rA by totality. If rA≤A then A=r²A≤rA (the square of a reflection is the identity). If A≤rA then rA≤r²A=A. In both cases, thus, A~rA.

Therefore, if A≤B, then rA~A≤B and so rA≤B. Now, any rotation can be written as the composition of a pair of reflections (a rotation by angle θ equals the composition of reflections in lines subtending angle θ/2). Thus, for every every rotation r, if we have A≤B, then we have rA≤B and rA≤rB. It follows easily that A<B if and only if rA<B.

Now, let r be a rotation by an angle which is an irrational number of degrees and let x₀ be any point on the circle. Let A be the set {x₀,rx₀,r²x₀,r³x₀,...}. Observe that rA={rx₀,r²x₀,r³x₀,r⁴x₀,...} is a proper subset of A (x₀ is not equal to rⁿx₀ for any positive integer n as r was a rotation by an irrational number of degrees). Thus, rA<A by Regularity. Thus, A<A, which is a contradiction.

One Body book on sale

Notre Dame University Press has my One Body book (and various other philosophy books) on sale at 30% off with code NDE813 until September 15, 2013.

A variant on Plantinga's evolutionary self-defeat argument

If naturalism is true, we would not expect our metaphysics to be reliable beyond the natural realm, since dealing with the natural realm is all our reasoning ability evolved for. But naturalism is a metaphysical thesis that goes beyond the natural realm—namely, it claims that there is nothing beyond it. Thus, if naturalism is true, we would not expect to be reliable in getting to claims like naturalism.

Endangerment and harm

Suppose I deliberately endanger you, but the danger doesn't befall you. Then there is a sense in which I do you no harm, but there is also a sense in which imposing the danger on your was a harm to you. You have a claim against me for my endangerment of you.

But one can also endanger people who never exist. For instance, if I give you a drug that has a high probability of physically harming your future children if you have any (let's say I assign a certain moderate probability to your having children), but you never actually have any children. There I might be harming you in some way, but I don't harmed them, since they never exist to be harmed. One can tweak the case so there are no parents to be harmed. Maybe I expect intelligent life to evolve on some planet with moderate probability, and I set up a device to harm some intelligent beings on that planet once they evolve, but no life evolves there.

There are thus two probabilities in endangerment. There is the probability that there is going to be potential victims at all and the conditional probability that a potential victim will be harmed given that there is going to be a potential victim at all. And the probability of harm is the product of these two probabilities.

It is a very interesting question whether there is a significant moral difference between a case where

• I deliberately cause a probability 1/4 of harm to a person I know for sure to exist

versus a case where

• I deliberately cause a probability 1/2 of (same as above) harm to a person I assign probability 1/2 to the existence of (i.e., I deliberately cause it to be the case that if that person exists, she has chance 1/2 of suffering that harm)

when we suppose that in the first case the danger did not in fact befall the person while in the second case the person did not in fact exist.

The consequentialist intuitions that we all have to some degree pull one to saying that there is no difference. On the other hand, in the first case there is a person that I have failed to love and respect her in the way that she deserves, while there is no such failure of love and respect in the second case. In fact, if one has a picture of morality as essentially involving interpersonal relations, it is difficult to see how any wrongdoing has happened in the second case if in fact the person never comes to exist.

A theist might be able to maintain both something like the consequentialist intuition and the idea that moral failures are primarily failures of interpersonal relations. There is a deep and mysterious message in Scripture expressed by the Psalmist saying to God: "Against you, you alone, have I sinned" (Ps 51.4). The Psalm heading connects this with David's sin against Uriah, which makes this message particularly puzzling, since it seems clear that David sinned against both Uriah and God. But suppose we take really seriously the idea that all positive attributes are acts of participation in God. Uriah's dignity, then, is an act of participation in God's dignity, and its value entirely derivative from God's infinite dignity. In some sense, then, David's wrongdoing against Uriah really just is a wrongdoing against God. Now suppose that David had been wrong, and there never had been a Uriah. (Maybe Bathsheba was an unmarried woman who created a myth of an Uriah in order to protect herself from unwanted advances.) The wrongdoing against God's dignity would have been just the same. The wrongdoing against Uriah wouldn't have been there, but that wrongdoing's "culpatory force" was entirely derivative from the culpatory force of the wrongdoing against God, since Uriah's dignity was an act of participation in God's dignity. If we have something like this picture, then we really can say that all moral failures are primarily failures of interpersonal relations and yet hold the two cases, the one where there is an endangered victim and the one where there turns out not to be one, to be morally on par. For all respect and love is ultimately and implicitly for God, though perhaps God qua participated in or participable in by a creature.

The value of punishment

Boethius gives a striking thesis:

The wicked are happier in undergoing punishment than if no penalty of justice chasten them. And I am not now meaning what might occur to anyone—that bad character is amended by retribution, and is brought into the right path by the terror of punishment, or that it serves as an example to warn others to avoid transgression [...] .
Surely, then, the wicked, when they are punished, have a good thing added to them, the punishment which by the law of justice is good [...]. (Consolation, IV)

The brief argument seems to be:

1. What justice calls for is good.
2. Justice calls for punishment.
3. So, punishment is good.

Unfortunately, the conclusion that we want is not (3) but:

4. Punishment is good for the person undergoing it.

Can we fill in some plausible steps between (3) and (4)?

Perhaps Boethius takes it as clear that justice does not call for punishment simply because "bad character is amended by retribution ... or as an example to warn others" or in any other easy reductive account that "might occur to anyone" (e.g., Nietzsche's "account" on which punishment gives compensatory pleasure to the victims, or accounts on which criminals are simply taken out of circulation by being jailed, for the protection of society). Such benefits are there, but they aren't the benefit that justice is primarily aimed at.

Plausibly, the benefits of justice are to persons. Well, the relevant persons seem to be:

• the criminal
• the victims
• other members of society
• the punisher
• God.

Which of these are such that justice's primary aim is at a benefit for them?

Let's start by ruling out options from the bottom of the list. Everyone benefits, in an extrinsic but important way, when those they love benefit. God loves all. So any benefit to anyone is an extrinsic benefit to God. God is love and is immutable and simple. It plausibly follows that all contingent benefits to God are such extrinsic benefits. Thus, a contingent benefit to God will require a benefit to someone else. So choosing the option of God doesn't get us out of the puzzle. Moreover, when we aim at a benefit for God in this way, we should also be aiming non-instrumentally at a benefit for the creature.

While the punisher may receive some pleasure or the good of excellence in a job well done, surely that's not what justice aims for. It seems clear that just punishment does not require other members of society. Imagine someone has killed every other member of her society. She deserves punishment—say, from another society, or from a self-imposed life of penance.

The victims? There is an intuition that punishment is a way of honoring victims, perhaps posthumously. One problem with this is that criminal punishment appears just even when the criminal was forgiven by the victims. But when the criminal was forgiven by the victims, the criminal should not be punished for the sake of the victims, since by forgiveness the victims relinquish claims to punishment on their behalf. But I worry that this argument is not sufficient. After all, it could be that all of society counts as a victim in the case of a crime, since society's laws have been unjustly violated, so forgiveness by the more particular victims is not yet forgiveness by all of society. (This also shows how "victimless" crimes aren't victimless.)

Here is perhaps a more telling counterexample to the victim theory. Suppose Patricia notices a planet with rudimentary unicellural life but where she has very good reason to think intelligent life will evolve. She leaves behind a device which will kill off the intelligent life on the planet as soon as there are a million intelligent beings. A week later, Darth Vader tests the Death Star on the planet. Patricia has committed attempted murder, but there are no victims: there never was and never will be any intelligent life on this planet. Yet Patricia is clearly deserving of punishment as having committed a species of attempted genocide. Moreover, even if she broke no society's law, there is some sense in which justice calls for her punishment—this is the sort of thing that there ought to be a law against, and this "ought" is an "ought" of justice.

Another worry about locating the benefit in the victims is that may seem problematic to impose such great intrinsic burdens on the living for the sake of what appear to be merely extrinsic benefits to the dead (apart from somewhat dubious theories of the afterlife on which the dead rejoice in seeing their malefactors punished).

That leaves the criminal as the recipient of the benefits of punishment.

I think the most serious challenge to the argument is one on which justice leads to goods that are not the goods to any persons. Maybe justice benefits "the moral order of the universe". I am sceptical, though, of goods that are not derivative from goods to fundamental entities, and on my ontology the universe and its moral order are not fundamental entities.

Normal Popper functions and comparative probability

Let Ω be a non-empty set and F be a field of subsets (i.e., set of subsets closed under finite unions and complements). A Popper function on F is a real-valued function P defined for pairs of members of F such that:

1. 0≤P(X|Y)≤P(Y|Y)=1
2. if P(Ω−Y|Y)<1, then P(−|Y) is a finitely additive probability on F
3. P(X∩Y|Z)=P(X|Z)P(Y|X∩Z)
4. if P(X|Y)=P(Y|X)=1, then P(Z|X)=P(Z|Y).

(This is van Fraassen's axiomatization.) A member B of F is normal provided that P(Ω−B|B)<1. (This condition is equivalent to saying that P(∅|B)<1.) A Popper function is normal provided that every non-empty member of F is normal.

On the other hand, a comparative probability function on F (Paul Bartha has talked about things like this) is a function from pairs of members of F to [0,∞] such that:

5. C(X,X)=1
6. C(X,Y)C(Y,Z)=C(X,Z) provided the left-hand side is defined (0 times ∞ and ∞ times zero are the undefined cases)
7. C(−,Y) is a finitely-additive measure if Y is non-empty.

Alan Hajek, I think, has suggested that one can define a comparative probability function in terms of a Popper function. We can do it as follows. If P is a normal Popper function, then let C_P(X,Y)=P(X|X∪Y)/P(Y|X∪Y). And given a comparative probability function, we can define a normal Popper function by P_C(X|Y)=C_P(X∩Y|Y).

I haven't written out the details, but it looks like we then have:

Proposition. If P is a normal Popper function, then C_P is a comparative probability function, and if C is a comparative probability function, then P_C is a normal Popper function. Moreover, if P is a normal Popper function and C=C_P, then P_C=P, while if C is a comparative probability function and P=P_C, then C_P=C.

Thus, there is a nice one-to-one correspondence between comparative probabilities and normal Popper functions.

Personally, I find comparative probabilities to be easier to prove theorems about than Popper functions, because I find (6) much easier to remember than (3).

Problems for isometrically invariant Popper functions in the plane

Popper functions encode finitely-additive conditional probabilities. The hope of Popper functions is to allow conditionalizations on sets that classically have null probability, such as finite sets in a continuous context. A Popper function allows for non-trivial conditionalization on a set A only if A is normal, i.e., P(∅|A)=0. A Popper function P is weakly invariant under isometries (the group generated by rotations, translations and reflections, in the case of the plane) provided that P(gA|gB)=P(A|B) for any isometry g and A and B such that A,B,gA,gB are subsets of our probability space Ω. (Strong invariance would also say that P(gA|B)=P(A|B) and P(A|gB)=P(A|B) under appropriate circumstances.)

In an earlier post I basically proved (without the Axiom of Choice) that any weakly isometrically invariant Popper function on a solid three-dimensional ball that makes all countable sets measurable also makes all finite sets abnormal. This triviality result of course generalizes to higher dimensions.

I am now able to prove that if P is a weakly isometrically invariant Popper function on a subset Ω of the plane that contains a solid disc, with P defined at least for all countable sets, then there is a countable abnormal set. Again, since subsets of an abormal set are abnormal, it follows that some singleton is abnormal, and hence all singletons are abnormal by invariance, and hence all finite sets are abnormal as finite unions of abnormal sets are abnormal.

A proof is very roughly sketched here. It uses Just's construction of a bounded paradoxical subset of the plane (cf. this paper which gives another construction that could be used).

It also follows that there is no weakly isometrically invariant comparative probability function defined for all nonempty pairs of subsets of Ω (for definition of comparative probability, see my answer here).

On the other hand, Parikh and Parnes have shown that there is a Popper function defined for all pairs of subsets of the unit interval on the line making all nonempty subsets normal. Moreover I think their proof can be used to show that there is a translation-invariant Popper function defined for all pairs of subsets of n-dimensional Euclidean space making all non-empty subsets normal. We can get invariance under reflection in any finite set of orthogonal hyperplanes for free just by averaging combinations.

So the problem really is with rotations. Something important changes when one goes from dimension one to higher dimensions: rotations become available.

An addendum on consciousness, evolution and God

An addendum to my last post. If the theory in that post is correct, then all representative states depend on or are conscious mental states. Now, plausibly, some primitive organisms in our evolutionary history had representative states but not conscious mental states. If so, then it follows that their representative states were derivative from conscious mental states of something else. But presumably these organisms did not depend on more complex, conscious organisms for their mentality--when these organisms first evolved, there were no conscious organisms. So the dependence of their representative states must have been on something other than the mental states of an organism. One plausible theory is that their representative states depended for their representativeness on God's conscious mind (e.g., God consciously intended that such-and-such a mental state should represent such-and-such an environmental state).

So the theory of the preceding post is not friendly to naturalism, it looks like.

It's just a theory, though. I don't have much of an argument for it, except one based on its simplicity.

Consciousness

I used to think that the problem of consciousness can be plausible reduced to the problem of representation: that to have a quale of red might just be to represent something as being a certain way. But this can't be right. For we have both conscious and unconscious representations. My mind represents Beijing as the capital of China even when I am not consciously thinking about Beijing, capitals or China. And the point seems to generalize: the representative content that a quale of red is supposed to correspond to can surely be had unconsciously, e.g., as when I have a non-occurrent belief that paradigmatic tomatoes are red.

But perhaps there is something in the vicinity that might work. Let me try this. There are two kinds of representative mental states: fundamental and derivative ones. Non-fundamental representative states get their representative content from other representative states (ultimately reaching back to the fundamental ones) in the way that a book gets its representative content from the representative states of human language users.

Then to have a quale is nothing but to have a fundamental representation. Thus, when I see the tomato as red, I am having a fundamental representation of redness (or maybe of the reflectivity/emmissiveness with respect to a certain range of wavelengths of light). But when I have the non-occurrent belief that a tomato is red, my non-occurrent belief is a derivative representation, which derives its content from conscious and hence fundamental representative states. Perhaps it derives its content from my having perceived tomatoes and red things. Or perhaps it derives its content from a propensity to produce certain kinds of conscious images, including red ones, in my imagination. (The imagination seems to me to have played a rather bigger role in philosophy of mind in past centuries. Perhaps that role should return.) Or perhaps, if I have never seen or imagined anything red, it derives its content from someone else's fundamental representative states—say, from someone else's having seen red things.

On this picture, we have two axes of differentiating qualia. There is a representative content axis, which differentiates the quale of red from that of green or of squeaky. And there is a mode of representation axis, which differentiates non-occurrent beliefs involving redness from occurrent perceptions of red things. On the above story, mode of representation difference reduces solely to the difference between fundamental and derivative representations.

But maybe a further differentiation is needed: to imagine (or remember) a red tomato is phenomenologically different from seeing a red tomato, yet both are conscious. What accounts for this difference? If there is a difference of mode without a difference of content here, my above account won't account for this, since it only licenses a binary distinction, not a ternary one between non-conscious representation, imaginary representation and perceptual representation. But maybe the difference between the imaginary and perceptual is simply one of detail of content? Given that we so easily drift between imagination and dream, and that dream is phenomenologically like the perceptual but with less detail, this is not implausible.

So this suggests the following way to make the distinctions:

1. Thought of red without consciousness of the quale of red = A representative state whose red-content component is only derivatively representative.
2. Perception of red = A fundamental representative state with detailed red content.
3. Imagination/memory of red = A fundamental representative state with non-detailed red content (and maybe some truth-canceling "this isn't real/present" content? Spinoza has something like that).

Deep question: Are fundamental representative states metaphysically fundamental or only representatively fundamental?

The logician's daughter

Yesterday, my eldest daughter burned the dessert she was baking. You see, the recipe at one point (after a fair amount of baking at a lower temperature) said to bake "for fifteen minutes or until golden brown." She knew the second disjunct was already true at the beginning of that period, but decided to opt for following the first disjunct, as apparently allowed by the recipe. The dessert was black when she was done (though some of the inside was edible).

Grounding grounding

Suppose Bill is a bachelor and Marcus is married. I claim that <Bill is a bachelor> stands in the same relation to <Bill is a never-married marriageable man> as <Marcus is a bachelor> stands to <Marcus is a never-married marriageable man>. But the propositions about Bill are true while those about Marcus are false. Since grounding is a relation that holds only between truths, the relevant relation that the two pairs of propositions have in common is not the grounding relation. It is something else. Call it ontological explanation, following Dan Johnson's dissertation. (Since, I think, explanation is factive, so that only truths can be explained, and they can only be explained by truths, ontological explanation isn't explanation strictly speaking.)

Abbreviate "<x is a bachelor>" as b_x and "<x is a never-married marriageable man>" as n_x. Then, necessarily, for every human being (at least) x, n_x ontologically explains b_x. Let B be Bill and M be Marcus. Then, n_B ontologically explains and grounds b_B, while n_M ontologically explains but does not ground b_M.

Moreover, we are in a position to offer a grounding for the proposition <n_B grounds b_B>. This grounding is given by the contingent truth n_B and the necessary truth <n_B ontologically explains b_B>. So at least in this case, the grounding truth is itself grounded in a truth about Bill together with a necessary truth of ontological explanation.

So at least sometimes we can find a grounding for grounding truths partly in terms of ontological explanation truths. This gives some evidence that ontological explanation facts are more primitive than grounding facts.

Is this pattern in general true? Is it the case that if p grounds q, then p together with <p ontologically explains q> grounds <p grounds q>? Not if ontological explanation is like Johnson thinks it is. For Johnson thinks that that if a ontologically explains b, then a is metaphysically necessary and sufficient for b. But p can ground q without being necessary for q: that I am sitting grounds that I am sitting or standing.

Perhaps we can modify Johnson's account by holding on to the sufficiency while dropping the necessity. Then we will have something like ontological explanation where a ontologically explains b only if a is metaphysically sufficient for b. In that case, the general pattern might hold. What grounds that <<I am sitting> grounds <I am sitting or standing>>? It is <I am sitting> and <<I am sitting> ontologically explains <I am sitting or standing>>. Of course, the falsehood <I am standing> also ontologically explains that I am sitting or standing.

If this is right, then we can get below the hood on grounding: the more primitive notion is ontological explanation (modified from Johnson's account as above). If Johnson is right to require necessity, we still can get below the hood on grounding in some cases.

Here is one reason all this might matter. Consider propositional desires other than beliefs. Let's say Marcus wishes he were a bachelor. It is important, both to Marcus and to the analysis of the situation, that <Marcus is a bachelor> is ontologically explained by <Marcus is a never-married marriageable man>. There is something about being never-married, or being marriageable, or being a man, or a combination of these that implicitly appeals to Marcus. (Likewise, ontological explanation seems potentially relevant to Double Effect.)

One could try to handle the stuff about ontological explanation by using counterfactual grounding. The relation between n_x and b_x is that n_x would ground (or would necessarily ground) b_x were n_x true. But it is implausible that such a counterfactual fact is prior to the grounding fact if x is Bill.

Posthumous benefits

Here's an interesting principle.

1. If a human being x exists in worlds w₁ and w₂ and x's lifetime occupies the same times in the two worlds, and at every time t in this lifetime, x is no better off in w₂ than in w₁, then x is no better off in w₂ than in w₁.

This principle implies that strictly posthumous benefits—i.e., benefits that do not make one better off at any time before death—are only benefits to one if there is life after death. Hence, if there are strictly posthumous benefits, there is life after death.

Are there good candidates for strictly posthumous benefits? Well, of course, such things as having a joyous afterlife might be examples, but those examples wouldn't be helpful for arguing that there is an afterlife.

What about such things as someone's posthumous keeping of a promise or fulfillment of a request, or maybe a writer's gain in reputation after death? I am not sure the benefit is strictly posthumous in these three cases. If you keep your promise to me, you bring it about that a promise that won't be kept wasn't made to me. And so, arguably, by keeping your promise you make me have been better off at the time the promise was made. Likewise, if my request was fulfilled or my writing gained in reputation, I did not request or write in vain, so I was better off at the time of the request or writing.

Posthumous forgiveness by you of my wrongdoing against you might be a better example of a strictly posthumous benefit. It seems that the benefit of being forgiven accrues not at the time of wrongdoing but at the time of forgiveness. If so, then that posthumous forgiveness would be a strictly posthumous benefit gives an argument for an afterlife.

But perhaps the benefit of forgiveness somewhat accrues at the time of wrongdoing. Maybe one is worse off there and then if one does a wrong that will never be forgiven. That sounds right to me. However, I don't think this accounts for the entirety of the benefit of being forgiven. Being forgiven removes guilt. However, the argument is now weakened. For the denier of an afterlife can claim that posthumous forgiveness only gives one the benefit of not having committed a wrong that won't be forgiven. It is definitely a benefit, but not as great one as forgiveness while one is alive.

I still think that consideration of posthumous benefits like those of forgiveness gives some evidence for an afterlife.

Causal theory of content, religious experience, numinousness and naturalism

1. If naturalism (of the non-Aristotelian sort) is true, the causal theory of content is true. (It's the only decent naturalistically acceptable theory of content.)
2. The content of some religious experience involves the property of numinousness.
3. Numinousness is not a natural property and cannot be reduced solely to natural properties.
4. If the causal theory of content is true, and the content of an experience E involves a property P, then some experience is caused either by something's having P or by a combination of entities' having the properties that P reduces to.
5. If some experience is at least partly caused by something's having a non-natural property, then naturalism is false.
6. So, if naturalism is true, some experience is caused by something's being numinous or by a combination of entities' having the properties that numinousness reduces to. (1, 2 and 4)
7. So, if naturalism is true, some experience is at least partly caused by something's having a non-natural property. (3, 4)
8. So, if naturalism is true, naturalism is false. (5, 7)
9. So, naturalism is false. (8)

Another religious experience argument against naturalism

1. If something supernatural exists or if something has the causal power to produce something supernatural, then naturalism is false.
2. If it's causally possible for something supernatural to exist, then something supernatural exists or something has the causal power to produce something supernatural.
3. If p is causally possible, and p entails q, then q is causally possible.
4. For every natural* perceptual faculty in humans, it is causally possible for people to perceive veridically through it.
5. That someone perceives veridically through religious experience entails that there is something supernatural.
6. There is a natural* perceptual faculty of religious experience in humans.
7. So, naturalism is false.

Here, we can specify that a faculty is natural* provided that it is neither abnormal nor entirely dependent on culture.