Calvinism and the problem of sin

According to St Paul, we do not do evil that good might come of it. A plausible version of this principle is:

1. It is wrong to intend to produce an intrinsic evil, either as an end in itself or as a means (causal or constitutive) to an end.

Now, sin is an intrinsic evil. If (1) applies to God as well as to us, then a perfectly righteous God cannot intend anyone to sin. As far as (1) goes, it might be acceptable to permit someone to sin in order to bring a good out of it, but to cause someone to sin, even in order to bring a good out of it, is wrong. We might then formulate one of the distinctive views of Calvinism as that God intends sin in order to be glorified either by redeeming or by punishing the sinner. But then God violates (1).

Presumably, a standard Calvinist response will be that (1) applies to us but not to God. However, our ethics is supposed to be an ethics of love, and God, whether necessarily or by his contingent decision, always acts in love as well. (Some Calvinists think God doesn't love the reprobate. But the argument still applies insofar as on Calvinist views it seems that God intends the elect—whom he loves—to sin, in order that he be able to redeem them.) And principle (1) seems to be at such a high level of generality that if it follows from the duty to love in our case, it is likely to follow from that duty in the case of God, as well.

I think the Calvinist should deny that God intends sin. Instead, the Calvinist should give some sort of a Double Effect story on which God causes something that entails the existence of sin, but which is distinct from the sin and good, and to which the sin is not a means. Maybe instead of willing Sally to punch George, which was evil, God can intend Sally to swing her fist forward, which is not in itself an evil, but which, along with the other things God has willed, entails that Sally is punched by George. Then one ends up denying that God intends people to sin for the sake of his glory, instead asserting that for the sake of his glory he permits them to sin, while he (God) wills something that entails their sinning. Whether it is possible for a Calvinist to walk this fine line is not clear.

Another notion of supervenience

Say that the A-facts cs-supervene on the B-facts iff a computer simulation of the B-facts, based on the actual world's B-laws and initial conditions for B-facts, would necessarily also correctly simulate the A-facts.

Thesis: A-facts are problematic for physicalists iff A-facts do not cs-supervene on physical facts.

Corollary: There is a debate in the philosophy of biology on whether the apparent irreducibility of biological facts to physical facts is a problem for physicalism, and if not, why not. If the Thesis is true, then as long as biological facts cs-supervene on physical facts, then whether or not there is a reduction, biology does not present a problem for physicalism. I suspect that typical non-reductionist philosophers of biology would accept the cs-supervenience of the biological on the physical—for instance, they would agree that simulated natural selection would show up in the physical simulations—and hence their non-reductionism is not a problem for physicalism.

What about my argument against reductionism in biology, based on the instantaneousness of coming-into-being and the gradualness of chemical processes? I guess in the present setting the thing to say is that we wouldn't be able to tell from the computer simulation exactly when a new animal came into existence—just as we can't tell from empirical observation of the world—and so the computer simulation would not be simulating the coming into existence of animals.

A problem with the notion of cs-supervenience is the imprecision of the notion of "simulating".

Disjunction introduction

The following argument is valid by disjunction introduction:

1. I won't play the lottery.
2. I won't play the lottery or I'll win the lottery.

But the conclusion sounds wrong.

Now, here is a hypothesis. The wrong-sound of the or-sentences corresponds precisely to the appearance of falsity in the conditional:

3. If I play the lottery, I'll win the lottery.

Now if the indicative conditional is the material conditional, then (3) is true assuming I won't play the lottery. This is one of the standard objections to taking the indicative conditional to be a material conditional. But I think this objection is weakened if (2) also sounds wrong, since (2) is the material conditional analysis of (3). If (3) sounds wrong, and (2) is the analysis of (3), then if one of them sounds wrong, so should the other. And that's what we observe.

The following would be an interesting test. Consider this argument:

4. I won't play the lottery.
5. So: I won't play the lottery or I'll win the lottery.
6. So: If I do play the lottery, I'll win the lottery.

Where would ordinary folks situate the apparent fallacy? Would it be at step (5) or at step (6)? I don't know. But if only at (5), then the material conditional analysis of indicative conditionals is not challenged by the apparent wrongness of (3).

On the other hand, (3) sounds false to ordinary folks (unless the lottery is rigged!), but I don't know if (2) sounds false to ordinary folks.

An argument against physicalism in biology

1. (Premise) Every biologically significant purely physical event happens at the same time as a chemically significant purely physical event.
2. (Premise) No chemically significant purely physical event takes less than 10⁻²² seconds (it takes 1.7x10⁻²⁰ seconds for light to cross the Bohr radius of the hydrogen atom).
3. (Premise) The coming into existence of an animal is a biologically significant event.
4. (Premise) There is no vague existence at a time.
5. (Premise) If there is no vague existence at a time, transitions between non-existence and existence are instantaneous.
6. (Premise) Instantaneous events take less than 10⁻²² seconds.
7. Transitions between existence and non-existence do not happen at the same time as a chemically significant purely physical event. (2, 4, 5 and 6)
8. If the coming into existence of an animal is a purely physical event, it happens at the same time as a chemically significant purely physical event. (1 and 3)
9. The coming into existence of an animal is not a purely physical event. (7 and 8)

Epistemic and moral reasons

Suppose that I have epistemic reason not to believe p, but a failure to believe p will likely lead to a non-epistemic bad to another. Maybe I have epistemic reason not to believe that George is innocent, but a failure to believe that George is innocent will likely lead me to treat him unfairly. On what grounds should I decide whether to believe p? Not epistemic—for epistemic rationality is blind to non-epistemic bads, and hence either will automatically tell me to go with the epistemic good, which is not the right answer since epistemic goods do not outweigh all others, or will be unable to weigh the epistemic and non-epistemic goods and bads. On the other hand, morality tells me to pursue the good, and that includes the epistemic good. Morality, thus, is able to weigh both the epistemic and the non-epistemic goods and bads. Sometimes it will say to go for the epistemic good at the expense of the non-epistemic, and sometimes it will say to go for the non-epistemic good at the expense of the epistemic. It seems, thus, that the question of what should I do simpliciter in a case like this is a question of moral normativity.

Insofar, then, as morality can weigh epistemic goods, and morality is what delivers the answer of what I should do simpliciter, it seems that epistemic normativity tells me what I should or should not do considering a subset of the actual reasons—the subset of epistemically relevant ones. This subset is considered, however, by morality, and weighed against other reasons. As as the epistemic reasons are weighed by morality, they also constitute moral reasons. Moreover, since the answer to the question of what I should do simpliciter comes from morality, it does not appear that the epistemic reasons have any independent normative force: epistemic reasons are simply a special kind of moral reasons. There are many kinds of moral reasons. For instance, there are reasons relating to aretaic goods to self; reasons relating to non-aretaic goods to other persons; reasons relating to goods to non-persons; etc. And there are reasons relating to epistemic goods.

What I said about epistemic goods applies to prudential ones. (In fact, I now think that C. S. Lewis makes a similar argument early in Mere Christianity when he talks of how morality judges between instincts.)

One might think that what I said only applies in the case where the epistemic reasons have moral implications. But even if they didn't, the fact that they don't would be a matter for a moral judgment, and hence the reasons would not escape from the purview of moral normativity. And, anyway, since epistemic goods are goods—if they weren't, they wouldn't be worth pursuing—and since morality tells us to pursue the goods, there are always moral implications.

Why frequentism about probabilities is wrong

It would be surprising if the following argument were not already known. But I rather like the argument, so I'll post it. According to frequentism as I shall use the term, the probabilities of a random event are to be read off from actual infinite-run frequencies. Here is a disproof of frequentism.

1. It is logically possible to have a physically-realized random variable (a) whose probability measure is atom-free, i.e., has the probability that the measure of a singleton set is zero—uniform and Gaussian measures on continua have this property, and (b) the random variable is physically realized on a countable infinity of independent occasions in the history of the universe.
2. If frequentism is true, (1) is false.
3. Therefore, frequentism is not true.

Argument for (1): Scientists all the time model phenomena with atom-free measures. Consider, for instance, decay times of radioactive substances, or various Gaussian models. These models may not be entirely accurate, but it would be an amazing result if these models for basic philosophical reasons could not be right. (Besides, and this is more of an ad hominem, frequentism tends to be liked by folks of a reductionistic sort, and these folks tend to be in awe of science and unwilling to dictate to scientists what theories are and are not acceptable.)

Argument for (2): Suppose that in some world w we have physically realized a countable infinity of independent copies, X₁, X₂, ..., of an atom-free random variable X (a random variable whose associated measure is atom-free, i.e., a random variable such that the probability of its having any particular value is always zero). There are two versions of frequentism that I shall address: on one, we look at actual frequencies in an infinite run, and on the other, we consider the subjunctive conditional were the experiment repeated countably infinitely many times, what would the frequencies be? In w, however, the two come to the same thing in respect of X, because in w the experiment already is repeated countably infinitely many times. Let x₁,x₂,... be the outcomes of X₁,X₂,.... Let B be any Borel-measurable set in the space where X takes its values (e.g., a Borel susbet of the reals). Then,

3. If frequentism of either variety is true, then in w we have: P(X is in B) = lim N(B,n)/n

where the limit is taken as n goes to infinity, and N(B,n) is the number of values of x₁,...,x_n that are in the set of B. Now, let B be the set {x₁,x₂,...}. Then:

4. lim N(B,n)/n = 1

since all of the x_i are in B. Therefore:

5. If frequentism of either variety is true, then in w we have: P(X is in B)=1.

But the following is a theorem:

6. If B is any countable set and X is a variable whose distribution is atom-free, then P(X is in B)=0.

(This follows from countable additivity and being atom-free—just break B up into countably many singletons, note that each singleton has zero measure, and hence so does their union.) Therefore, if frequentism is true, the scenario that defined w cannot be possible.

The Cogito

Is Descartes' Cogito argument indubitably sound? Descartes tells us, basically:

1. I think.
2. Therefore, I am.

But I can doubt the proposition expressed by (1). For the proposition expressed by (1) is that Descartes exists, and I can doubt that. Now what I can't doubt is that I think. So the Meditations are different from a traditional philosophy book in that they do not give an argument that is supposed to convince the reader. Rather, they give an argument of which a homophone is supposed to convince the reader—the reader is expected to say to herself: "I think. Therefore, I am."

Instead of providing us with an argument against scepticism, Descartes shows us by example how we should argue our way out. In fact, there is no argument that he could give that we could use. Suppose Descartes said:

3. You think.
4. Therefore, you are.

But I could doubt that the person Descartes refers to with "you" exists. For instance, I could doubt that Descartes meant for me to read his book. Likewise, if the Descartes said "the reader", since I could doubt that I am reading, as I could simply be dreaming that I am reading; besides "the reader" is not of unambiguous reference.

I suppose this is why Descartes called these thoughts meditations.

Abortion and destruction of animal life

Say that an entity has "very high value" provided that it is wrong to destroy that entity for the sake of any good less than that of the saving of a typical life of a human person. (This is rough and heuristic, because incommensurability is very problematic here.) Human persons, then, have very high value. If fetuses have very high value as well, then it follows that typical abortions are wrong, but it does not follow that abortions done to save the mother's life are wrong. (My view is that they are wrong, because fetuses not only have very high value but are deserving of the same moral respect as adults. But I shan't be defending that here.)

One might try to argue that fetuses lack very high value on the basis of the following argument:

1. Only things that exhibit P have very high value.
2. Fetuses lack P.
3. Therefore, fetuses lack very high value.

Here, P is some "personal feature", like personhood, self-awareness or valuing one's existence.

But with these ways of filling in P, the argument is unsound because (1) is false for such P. Here is one example (taking "things" widely to include mereological sums or other collectives): the genus Equus, which currently comprises horses, donkeys and zebras. No member of this genus has P, and the genus as a whole also lacks P. But it would be wrong to wipe out the genus for the sake of any good less than that of saving a typical human life. My evidence for this is that a lot of people will think that it would not be irrational to sacrifice one's life to save the genus from extiction, and that it would be problematic even to save one's life at the expense of the genus. Where something less than a human life is at stake, it seems that it would be wrong to destroy the genus.

If this example does not convince, let's up the numbers. Suppose there is a galaxy, containing a hundred thousand planets teeming with life. The intellectual level of the life in that galaxy does not individually exceed that of a horse, and there is no collective intelligence, either. Then that galaxy, with all its living contents, does not exhibit P. But, surely, it has very high value—it would be wrong to destroy it, with all its contents, for any good that is less than that of the saving of a typical human life. And it is not implausible that it would be wrong to engage in such vast destruction of animal life even to save a typical human life.

Indicative conditionals

Here is a proposal. It can't be right—it's too messy and ad hoc—but it matches my intuitions about how conditionals are used. (Some people will say that the latter fact is evidence against the theory!) The proposal is this:

1. (if p, then q) iff [P(q|p) is high and (q or not p)].

I don't know if P should be subjective or objective probability, what the background for P is, and what level of "high" counts. It may be that all these things are determined by context.

The proposal solves the main problem with simply identifying the indicative conditional with a high conditional probability: a high conditional probability does not support modus ponens. But (1) supports modus ponens, since it makes the indicative conditional be stronger than the material conditional.

The proposal handles the problem with the material conditional generated by false antecedents. It isn't true that if the Queen will invite me for dinner tonight I will go in my pajamas, because although the material conditional conjunct on the right hand side of (1) is true, P(I go in my pajamas | the Queen invites me) is low.

The problem of true consequents is handled in at least a lot of cases. It's true that I won't accept the Fields Medal this year, because I won't be offered it. But the following indicative conditional is false on the proposal: If I am justly offered the Fields Medal, I won't accept it. It is false because although the consequent is true, it has low probability conditionally on the antecedent.

The proposal is problematic in the case of impossible antecedents, at least if P is objective probability, but everybody knows per impossibile conditionals are tricky. It is also problematic in the case of zero-probability antecedents, but there one can at least hope that in the important cases one can give a limiting case interpretation of the conditional probability.

Another problem is with contrapositives. If it's true that if p, then q, we'd like it to be true that if ~q, then ~p. That works fine for the material conditional. The problem is with the high-probability conjunct in (1). We're going to have this problem when P(q|p) is high and the material conditional holds, but P(~p|~q) is not high. These cases are also going to be a problem for the no-truth-value conditional-probability view of indicatives.

Justification and love

Justification consists in God's forgiveness of our sins. What does this forgiveness consist in? At least partly in the taking away of the penalty. But what, most deeply, is the penalty? One thinks here of hell-fire. But while hell may contain fire (or it may contain great cold!), it is not constituted by fire, but by separation from God. Now, lack of charity—lack of the right kind of love for God—is at the heart of separation from God.

So: Divine forgiveness must consist, at least in part, in the removal of the penalty of separation from God, and the removal of our lack of charity. Therefore, the instilling of charity is at least partly constitutive of divine forgiveness. Hence, basic sanctification—the movement from lack of charity to the presence of charity—is not merely causally tied to justification, but is at least partly constitutive of justification. Moreover, this sanctification is not appropriate, and maybe not possible, apart from justification, since a just being is unlikely to waive punishment without forgiving.

Another argument against actual infinity

I am actually kind of suspicious that there is a subtle problem with my conditional probabilities in the following argument. It's a rather complex argument. Start with this observation. Suppose I know that there are N people in the universe, and ten minutes ago, a random process independently occurred to each of the N people, bestowing upon each the unobservable property Q, with probability p. How likely is it, given this information, that I have Q? The answer is obvious: p. But now suppose that I learn some additional information: I learn exactly how many people now have Q and how many don't. Presumably, approximately pN have Q and (1-p)N don't, but I now have better than an approximation—I have an exact number. Let's say, K people have Q. So now my probability that I have Q is K/N. Observe that p has now dropped out, because the information I have supercedes the information that involves p. For instance, maybe N=10, and in one world where p=1/4, three people have Q, and in another world where p=1/3, also three people have Q. If I am in the first world, my probability for having Q should be 3/10, and in the second, likewise.

The above illustrates a general principle: If I know the actual distribution at t₁ of the property Q in the population, my best estimated probability for having Q depends on that actual distribution, and not on the history of how the members of the population got to have Q. Different historical processes could produce the same distribution.

Now, suppose an actual infinity is possible, and so there are countably many people in the population. If this is possible, it is also possible that some process at t₀ independently bestowed Q on some of the people, with probability p strictly between 0 and 1. Suppose this is all I know. Then I ought to assign probability p to the claim that I have Q. But by the above principle, if I were to learn what the distribution of Q in the population is, I should use that distribution to estimate the probability of my having Q, instead of using information about p. But I do know the distribution of Q in the population is—even without actually observing anything. For I know that countably infinitely many members of the population have Q and countably infinitely many members of the population lack Q. (This works best if the persons are otherwise indiscernible, or differ in respect of properties that have no ordering or topology or significance to them.) At least, the probability that this is so is 1, and that's surely good enough for knowledge. But now here is the funny thing: this fact about the distribution is independent of p. Whatever the value of p is, the distribution would almost surely (i.e., with probability one) be infinitely many Qs and infinitely many non-Qs. So, by the above trumping principle, regardless of the value of p, I ought to assign the same probability (1/2? undefined?) to my having Q. But it is obvious that I ought to assign p. So we have a contradiction.

Here's a perhaps clearer way to run the argument. Two processes independently bestowed the properties Q and R, bestowing Q with probability 1/10 and R with probability 9/10, on all the members of an infinite population. I now know that in the population, there now are infinitely many Qs and infinitely many non-Qs. This is exactly the same distribution as the distribution of Rs. If my probabilities should depend on the distribution of Qs and Rs, as the trumping principle says they should, it follows I should assign the same probability to my having Q as to my having R. But plainly this is false—it is nine times as likely that I would have R than that I would have Q. Hence we have a contradiction.

Probability on infinite sets and the Kalaam argument

Suppose there is an infinite line of paving stones, labeled 1, 2, 3, ..., on each of which there is a blindfolded person. You are one of these persons. That's all you know. How likely is it you're on a number not divisible by ten? The obvious answer is: 9/10. But now I give you a bit more information. Yesterday, all the same people were standing on the paving stones, but differently arranged. At midnight, all the people were teleported, in such a way that the people who yesterday were standing on numbers divisible by ten are now standing on numbers not divisible by ten, and vice versa. Should you change your estimate of the likelihood you're on a number divisible by ten?

Suppose you stick to your current estimate. Then we can ask: How likely is it that you were yesterday on a number not divisible by ten? Exactly the same reasoning that led to your 9/10 answer now should give you a 9/10 answer to the back-dated question. But the two probabilities are inconsistent: you've assigned probability 9/10 to the proposition p₁ that yesterday you were on a number not divisible by ten and 9/10 to the proposition p₂ that today you are on a number not divisible by ten, even though p₁ holds if and only if p₂ does not (this violates finite additivity).

So you better not stick to your current estimate. You have two natural choices left. Switch to 1/2 or switch to 1/10. Switching to 1/2 is not reasonable. Let's imagine that today is the earlier day, and you have a button you can choose to press. If you press it, the big switch will happen—the folks on numbers divisible by ten will be swapped with the folks on numbers not divisible by ten. If you had switched to 1/2 in my earlier story, then if you press the button, you should also switch your probabilities to 1/2, while if you don't press the button, you should clearly stick with 9/10. But it's absurd that your decision whether to press the button or not should affect your probabilities (assume that there is no correlation between what decision you make and what number you're on).

Alright, so the right answer seems to be: switch to 1/10. But this means that the governing probabilities in infinite cases are those derived from the initial arrangements. Why should that be so?

Here is a suggestion. We assume that the initial arrangement came from some sort of a regular process, perhaps stochastic (where "regular" is understood in the same sense as "regularity" in discussions of laws). For instance, maybe God or a natural process brought about which squares the people go on by taking the people one by one, and assigning them to squares using some natural probability distribution, like probability 1/2 to 1, 1/4 to 2, 1/8 to 3, and so on, with the assignment being iterated until a vacant square is found (equivalently do it in one step: use this distribution but condition on vacancy). And, maybe, for most of the "regular" distributions, once enough people are laid down, we get about a 9/10 chance that the process will land you on a square not divisible by ten.

This assumes, however, that there is a process that puts people on squares. Suppose this assumption is false. Then there seems to be no reason to privilege the probability distribution from the first time the folks are put on squares. And our intuitions now lead to inconsistency: assigning 9/10 to p₁ and 9/10 to p₂.

Where has all this got us? I think there is an argument here that absurdity results from an actual, simultaneous infinity of uncaused objects. But if an actual infinity of objects is possible, and it is possible to have a contingent uncaused object, then it is very plausible (this is an ampliative inference) that it is possible to have an actual infinity of simultaneous contingent uncaused objects.

Therefore: either it is impossible to have an uncaused object or it is impossible to have an actual infinity of simultaneous contingent objects. But it is possible to have an actual infinity of simultaneous contingent objects if it is possible to have an infinite past. This follows by al-Ghazali's argument: just imagine at each past day a new immortal soul coming into existence, and observe that by now we'll have a simultaneous infinity of objects. So, it is either impossible to have an uncaused contingent object or it is impossible to have an infinite past. We thus have an argument for the disjunction of the premises of the Kalaam argument, which is kind of cool, since both of the premises of the argument have been disputed. Of course, it would be nicer to have an argument for their conjunction. But this is some progress. And it may be the further thought along these lines will yield more fruit.

Gradual variation of moral standing

For simplicity, suppose all utilities are commensurable.

No finite amount of utility justifies killing a being with moral standing. Take this to be stipulative of moral standing, and further take it as a substantive thesis that adult humans have moral standing. For any being x at a time t, let u(x,t) be the greatest (finite or infinite) number u with the property that if u' is any number smaller than u, then it is wrong to destroy x at t to produce u' units of utility. For instance, if the units of utility are average human lives, maybe u(adult dog, now) is 0.0001—it would be wrong to kill a dog to produce less than 0.0001 times the value of an average human life, but it would not be wrong to kill a dog to produce 0.00011 times that value. The exact calibration will be obviously controversial, and some people will say that the right number for a dog is 0.1 or 0.5 or even 1. We could call u(x,t) the "moral significance of x at t". Note that x has moral standing at t if and only if u(x,t) is infinite.

Now consider the following plausible assumptions:

1. No earthly critter changes at all significantly in its natural properties over a period of time in its life that does not exceed the Planck time (5.4x10⁻⁴⁴ seconds).
2. If x is an earthly critter, and u(x,t) changes very significantly over a period of time, then x changes at least somewhat significantly in its natural properties over that period.
3. If u(x,t₂)>100u(x,t₁)+100, where the units are average human life utilities, then u(x,t) has changed very significantly between t₁ and t₂.

These have the following logical consequence:

4. If x is an earthly critter that has moral standing at some time in its life, then x has moral standing at all times in its life.

Add two more premises:

5. I was once a fetus.
6. I now have moral standing.

Conclusion:

7. The fetus I have grown out of had moral standing.

This argument is based on this argument by Mike Almeida.

The prodigal son

I was yesterday struck by a dimension of the parable of the prodigal son that I hadn't noticed before: prayer. The younger son asks for two outrageous things—his inheritance ahead of time and being received back—and in both cases his requests are granted (in the latter case, he gets more than he asked for, since he is not asking for his sonship). The older son complains that he never got a young goat to eat with his friends. But it is clear that he never bothered to ask for one. After all, the father says: "all that is mine is yours." (The younger son reminds me a little of Mrs. Fidget in C. S. Lewis's Four Loves, who quietly works her fingers to the bone, suffering for others in ways that they don't want her to. Mrs. Fidget, too, would not bother asking you to come back by a certain time—she would just stay up and wait.)

Maybe the older son could try to complain: "But didn't you know that I wanted a goat?" However, a request is not just the expression of a desire: a request has normative effects that a mere desire does not.

Johnson's framework for theistic arguments

Occasionally, I've been offering theistic arguments that border on begging the question. Here, for instance, is one that's basically due to Kant, but transposed into an argument in a way that Kant would not approve of:

1. (Premise) We should be grateful for the wondrous universe.
2. (Premise) If something is not the product of agency, we should not be grateful for it.
3. Therefore, the wondrous universe is the product of agency.

The argument is indisputably valid.[note 1] Moreover, if theism is true, it is also sound, and I do take theism to be true. But soundness is, of course, not enough for a good argument. While premise (2) is pretty plausible (in the objective sense of "should"), it feels like premise (1) "begs the question".

Nonetheless, I think there could be something to (1)-(3). Dan Johnson, in the January 2009 issue of Faith and Philosophy has a fascinating little article on the ontological and cosmological arguments. He argues that a certain kind of circularity is not vicious. Suppose that I know p₁. I then infer p₂ from p₁ in such a way that I also know p₂. I then non-rationally (or irrationally) stop believing p₁, but as it happens, I continue to believe p₂. It will then often be the case that there will be a good argument from p₂ back to p₁ (perhaps given some auxiliary premises), and if I use that argument, I will be able to regain my knowledge of p₁. This is true even though there is a circularity: from p₁, to p₂, and back to p₁. Here is an uncontroversial example: I am told my hotel room is 314. I infer that my hotel room is the first three digits of pi. I then forget that my hotel room is 314, but continue to believe it is the first three digits of pi. I then infer that my hotel room is 314.

Johnson proposes that by the sensus divinitatis one may come to know that God exists (actually, throughout this, I can't remember if he talks of knowledge or justified belief). One may then infer from this various things, such as that possibly God exists. Then, one irrationally rejects the existence of God (it does not have to be a part of the theory that every rejection of the existence of God is irrational), but some of the things one inferred from that belief remain. And arguments like the S5 ontological argument then make it possible to recover the knowledge of the existence of God from the things that one had inferred from that belief. Johnson also applies this to the cosmological argument.

This same structure may be present in my Kantian argument. By the sensus divinitatis one comes to know that God exists (obviously this is not a Kantian idea!). One infers that the universe is such that we should be grateful for it. One then irrationally comes to be an atheist (again, there is need be no claim that every atheist is irrationally such), but one continues to believe that gratitude is an appropriate response to the universe. And if that belief is sufficiently deeply engrained, one can reason back from it to theism or at least to agency behind the universe.

Now let me move a little beyond the Johnson paper. I think it is not necessary for this structure that the initial knowledge of God's existence come from the sensus divinitatis. Any other way of having knowledge of God's existence will do—say, by argument or testimony. In fact, it is not even necessary for this structure that one oneself ever had the knowledge or even belief that God exists. Suppose, for instance, one's parents knew that God exists (in whatever way), and inferred from this that the universe is worthy of gratitude. They then instilled this belief in one, and did so in such a way as to be knowledge-transmitting. (Surely, value beliefs can be instilled in such a way.) But they did not instill the belief that God exists (maybe because they thought that the existence of God was something everybody should figure out for themselves). One then knows (1), and can infer (3).

This transmission can be mediated by the wider culture, too. Culture can transmit knowledge, whether scientific or normative, and arguments can work at a cultural level. It could be that a theistic culture where the existence of God was known grew into a culture where (1) was known. The knowledge of (1) can remain even if the culture non-rationally rejects the existence of God (as American culture has not done, and might or might not do in the future). And then the individual can acquire the knowledge of (1) from the culture (we don't need to attribute knowledge to the culture if we don't want to; we can just talk of knowledge had by individuals participating in the culture), and then infer (3).

I think there are probably many consequences of theism that are embedded in the culture, from which consequences one can infer back to theism. If the participants in the culture knew theism to be true when these consequences were derived, then it is perfectly legitimate to reason back from these consequences to theism.