Believing of God that he exists

One formulation of Schellenberg’s argument from hiddenness depends on the premise:

(4) If for any capable finite person S and time t, God is at t open to being in a personal relationship with S at t, then for any capable finite person S and time t, it is not the case that S is at t nonresistantly in a state of nonbelief in relation to the proposition that God exists.

Schellenberg argues that God is always open to personal relationships if he exists, and that there are people nonresistantly in a state of nonbelief to the proposition that God exists, and so God doesn’t exist.

I want to worry about a logical problem behind (4). Schellenberg attempts to derive (4) from a principle he calls Not Open that says, with some important provisos that won’t matter for this post, that “if a person A … is … in a state of nonbelief in relation to the proposition that B exists” but B could have gotten A to believe that B exists, “then it is not the case that B is … open … to having a personal relationship with A”.

It seems that Schellenberg gets (4) by substituting “God” for “B” in Not Open. But “the proposition that B exists” creates a hyperintensional context for “B”, and hence one cannot blithely substitute equals for equals, or even necessarily coextensive expressions, in Not Open.

Compare: If I have a personal relationship with Clark Kent, I then automatically have a personal relationship with Superman, even if I do not believe the proposition that Superman exists, because Superman and Clark Kent are in fact the same person. It is perhaps necessary for a personal relationship with Superman is that I believe of Superman that he exists, but I need not believe it of him under the description “Superman”.

So it seems to me that the only thing Schellenberg can get from Not Open is something like:

(4*) If for any capable finite person S and time t, God is at t open to being in a personal relationship with S at t, then for any capable finite person S and time t, it is not the case that S is at t nonresistantly in a state where he does not believe of God that he (or it) exists.

Now, to believe of x that it exists is to believe, for some y such that in fact y = x, that y exists.

But then all that’s needed to believe of God that he exists is to believe in the existence of something that is in fact coextensive with God. For instance, suppose an atheist believes that her mother is the being that loves her most. Then she presumably believes that the being that loves her most exists. In doing so, she believes of the being that loves her most that it exists. But in fact, assuming theism is true, the being that loves her most is God. So she believes of God that it (or he) exists.

At this point it is really hard to find non-controversial cases of the relevant kind of nonbelief that (4*) expresses. By “non-controversial”, I mean cases that do not presuppose the non-existence of God. For if God does in fact exist, he falls under many descriptions: “The being who loves me most”, “The existent being that Jean Vanier loves the most”, “The most powerful conscious being active on earth”, etc.

It is true that Schellenberg needs only one case. So even if it is true, on the assumption that God exists, that the typical atheist or agnostic believes of God that he exists, perhaps there are some people who don’t. But they will be hard to find—most atheists, I take it, think there is someone who loves them most (or loves them most in some particular respect), etc. I think the most plausible cases of examples are small children and the developmentally challenged. But those aren’t the cases Schellenberg’s argument focuses on, so I assume that’s not the line he would want to push.

The above shows that the doxastic prerequisite for a personal relationship with B is not just believing of B that it exists, since that’s too easy to get. What seems needed (at least if the whole doxastic line is to get off the ground—which I am not confident it does) is to believe of B that it exists and to believe it under a description sufficiently relevant to the relationship. For instance, suppose Alice falsely believes that her brother no longer exists, and suppose that not only does Alice’s brother still exist but he has been working out in secret and is now the fastest man alive. Alice believes that the fastest man alive exists, and mistakenly thinks he is Usain Bolt rather than her brother. So she does count as believing of her brother that he exists, but because she believes this under the description “the fastest man alive”, a description that she wrongly attaches to Bolt, her belief doesn’t help her have a relationship with her brother.

So probably (4*) should be revised to:

(4**) If for any capable finite person S and time t, God is at t open to being in a personal relationship with S at t, then for any capable finite person S and time t, it is not the case that S is at t nonresistantly in a state where he does not believe of God that he (or it) exists, under a description relevant to his personal relationship with God.

This doesn’t destroy the hiddenness argument. But it does make the hiddenness argument harder to defend, for one must find someone who does not believe in anything that would be coextensive with God if God exists under a description that would be relevant to a personal relationship with God. But there are, plausibly, many descriptions of God that would be so relevant.

A different move is to say that there can be descriptions D that in fact are descriptions precisely of x but some cases of believing that D exists are not cases of believing of x that it exists. Again, one will need to introduce some relevance criterion for the descriptions, though.

[Note added later: This was, of course, written before the revelations about Jean Vanier's abusiveness. I would certainly have chosen a different example if I were writing this post now.]

A fun unsound argument for dualism

Here’s a fun argument for dualism.

1. What is a part of the body is a matter of social convention.

2. Persons are explanatorily prior to social conventions.

3. So, probably, persons are not bodies.

I think (2) is undeniable. And (1) is a not uncommon view among people thinking about prostheses, implants, transplants and the like.

That said, I think (1) is just false.

Evil, omniscience, and other matters

If God exists, there are many evils that God doesn’t prevent, even though it seems that we would have been obligated to prevent them if we could.

A sceptical theist move is that God knows something about the situations that we don’t. For instance, it may seem to us that the evil is pointless, but God sees it as interwoven with greater goods.

An interesting response to this is that even if we knew about the greater goods, we would be obligated to prevent the evil. Say, Carl sees Alice about to torture Bob, and Carl somehow knows (maybe God told him) that one day Alice will repent of the evil in response to a beautiful offer of forgiveness from Bob. Then I am inclined to think Carl should still prevent Alice from torturing Bob, even if repentance and forgiveness are goods so great that it would have been better for both Alice and Bob if the torture happened.

Here is an interesting sceptical theist response to this response. Normally, we don’t know the future well enough to know that great goods would arise from our permitting an evil. Because of this, our moral obligations to prevent grave evils have a bias in them towards what is causally closer to us. Moreover, this bias in the obligations, although it is explained by the fact that normally we don’t know the future very well, is present even in the exceptional cases where we do know the future sufficiently well, as in the Carl, Alice and Bob case.

This move requires an ethical system where a moral rule that applies in all circumstances can be explained by its usefulness in normal circumstances. Rule utilitarianism is of course such an ethical system. Divine command theory is as well: God can be motivated to issue an exceptionless rule because of the fact that normally the rule is a good one and it might not be good for us to be trying to figure out whether a case at hand is an exception to the rule (this is something I learned from Steve Evans). And St. Thomas Aquinas in his argument against nonmarital sex holds that natural law is also like that (he argues that typically nonmarital sex is bad for the offspring, and concludes that it is wrong even in the exceptional cases where it’s not bad for the offspring, because, as he says, laws are made with regard to the typical case).

Historically, this approach tends to be used to derive or explain deontic prohibitions (e.g., Aquinas’ prohibition on nonmarital sex). But the move from typical beneficiality of a rule to its holding always does not require that the rule be a deontic prohibition. A rule that weights nearer causal consequences more heavily could just as easily be justified in such a way, even if the rule did not amount to a deontic prohibition.

Similarly, one might use typical facts about our relationships with those closer to us—that we know what is good for them better than for strangers, that they are more likely to accept our help, that the material benefits of our help enhance the relationship—to explain why helping those closer to us should be more heavily weighted in our moral calculus than helping strangers, even in those cases where the the typical facts do not obtain. Once again, this isn’t a deontic case.

One might even have such typical-case-justified rules in prudential reasoning (perhaps a bias towards the nearer future is not irrational after all) and maybe even in theoretical reasoning (perhaps we shouldn’t be perfect Bayesian agents after all, because that’s not in our nature, given that normally Bayesian reasoning is too hard for us).

Recognizing the finite

We have a simple procedure for recognizing finite sequences. We start at the beginning and go through the sequence one item at a time (e.g., by scanning with our eyes). If we reach the end, we are confident the sequence was finite. This procedure can be relied on if and only if there are no supertasks—i.e., if and only if it is impossible to have an infinite sequence of tasks started and completed.

How do we know that there are no supertasks? Either empirically or a priori. To know it empirically, we would have to know that the various tasks we’ve completed were finite. But how would we know of any tasks we’ve completed that it’s finite if not by the above procedure?

So we have to know it a priori.

And the only story I know of how we could do that is by a priori cognizing some anti-infinity principle like Causal Finitism.

I am not sure how strong the above argument is. It is a little too close to standard sceptical worries for comfort.

One Body book at 45% off with free shipping

Notre Dame University Press has a 45% off sale today (Cyber Monday) on all print books, with free shipping. This includes my One Body: An Essay in Christian Sexual Ethics, which should be down to $24.75. You need to use the promo code 14CYBER.

Religions as "faiths"

It is common in our culture to see religion as a matter of faith. Indeed, religions are sometimes even called “faiths”.

Here is a reason why one should be cautious with conceptualizing things in this way. Faith is a specifically Christian concept, with Christianity being centrally conceptualized as a matter of faith in Jesus Christ. To think about all religions in terms of faith is to presuppose that the Christian understanding of what is central to Christianity yields a correct way of understanding the life of other religions.

Either Christianity is or is not basically true.

If Christianity is basically true, then its self-understanding in terms of faith is likely correct. However, the truth of Christianity does not give one good reason to think other religions, with the possible exception of Judaism, would be rightly understood in terms of the concept of faith.

If Christianity is not basically true, then we should be cautious even about its own self-characterization. Self-understanding is an epistemic achievement, and if Christianity is not basically true, then we should not take it for granted that faith has the central role it is claimed to have. And we should certainly not expect that the self-characterization of a religion that is not true should also apply to other religions.

Do we have to know that seven is finite to know that three is finite?

Three is a finite number. How do we know this?

Here’s a proof that three is finite:

1. 0 is finite. (Axiom)

2. For all n, if n is finite, then n + 1 is finite. (Axiom)

3. 3=0+1+1+1. (Axiom)

4. So, 0+1 is finite. (By a and b)

5. So, 0+1+1 is finite. (By b and d)

6. So, 0+1+1+1 is finite. (By b and e)

7. So, 3 is finite. (By c and f)

Let’s assume we can answer the difficult question of how we know axioms (a) and (b), and allow that (c) is just true by definition.

I want to raise a different issue. To know that three is finite by means of the above argument, it seems we have to know that the argument is a proof.

One might think this is easy: a proof is a sequence of statements such that each non-axiomatic statement logically follows from the preceding ones, and it’s clear that (d)-(g) each follow from the previous by well-established rules of logic.

One could ask about how we know these rules of logic to be correct—but I won’t do that here. Instead, I want to note that it is false that every sequence of statements such that each non-axiomatic statement logically follows from the preceding ones is a proof. This is the case only for finite sequences of statements. The following infinite sequence of statements is not a proof, even though every statement follows from preceding ones: “…, so I am Napoleon, so I am Napoleon, so I am Napoleon.”

Very well, so to know that (a)-(g) is a proof, I need to know that (a)-(g) are only finitely many statements. OK, let’s count: (a)-(g) are seven statements. So it seems we have to know that seven is finite (or something just as hard to know) in order to use the proof to know that three is finite.

This, of course, would be paradoxical. For to use a proof analogous to (a)-(g) to show that seven is finite, we would need a proof of eleven steps, and so we would need to know that eleven is finite to know that the proof is a proof.

Maybe we can just see that seven is finite? But then we gain nothing by (a)-(g), since the knowledge-by-proof will depend on just seeing that seven is finite, and it would be simpler and more reliable just to directly see that three is finite.

It might be better to say that we can just see that the proof exhibited above, namely (a)-(g), is finite.

It seems that knowledge-by-proof in general depends on recognition of the finite. Or else on causal finitism.

Ways of being and quantifying

Pluralists about ways of being say that there are multiple ways to be (e.g., substance and accident, divine being and finite being, the ten categories, or maybe even some indefinitely extendible list) and there is no such thing as being apart from being according to one of the ways of being. Each way of being comes with its own quantifiers, and there is no overarching quantifier.

A part of the theory is that everything that exists exists in a way of being. But it seems we cannot state this in the theory, because the "everything" seems to be a quantifier transcending the quantifiers over the particular ways of being. (Merricks, for instance, makes this criticism.)

I think there is a simple solution. The pluralist can concede that there are overarching unrestricted quantifiers ∀ and ∃, but they are not fundamental. They are, instead, defined in terms of more fundamental way-of-being-restricted quantifiers in the system:

1. ∀xF(x) if and only if ∀_BWoBb∀_bxF(x)

2. ∃xF(x) if and only if ∃_BWoBb∃_bxF(x).

The idea here is that for each way of being b, there are ∀_b and ∃_b quantifiers. But, the pluralist can say, one of the ways of being is being a way of being (BWoB). So, to use Merricks’ example, to say that there are no unicorns at all, one can just say that no way of being b is such that a unicorn b-exists.

Note that being a way of a being is itself a way of being, and hence BWoB itself BWoB-exists.

The claim that everything that exists exists in a way of being can now be put as follows:

3. ∀x(x = x → ∃_BWoBb∃_by(x = y)).

Of course, (3) will be a theorem of the appropriate ways-of-being logic if we expand out "∀x" in accordance with (1). So (3) may seem trivial. But the objection of triviality seems exactly parallel to worrying that it is trivial on the JTB+ account of knowledge that if you know something, you believe it. Whether we have triviality depends on whether the account of generic existence or knowledge, respectively, is stipulative or meant to be a genuine account of a pre-theoretic notion. And nothing constrains the pluralist to making (1) and (2) be merely stipulative.

Suppose, however, your motivations for pluralism are theological: you don’t want to say that God and humans exist in the same way. You might then have the following further theological thought: Let G be a fundamental way of being that God is in. Then by transcendence, G has to be a category that is special to God, having only God in it. Moreover, by simplicity, G has to be God. Thus, the only way of being that God can be in is God. But this means there cannot be a fundamental category of ways of being that includes divine and non-divine ways of being.

However, note that even apart from theological considerations, the BWoB-quantifiers need not be fundamental. For instance, perhaps, among the ways of being there might be being an abstract object, and one could hold that ways of being are abstract objects. If so, then ∀_BWoBbG(b) could be defined as ∀_BAb(WoB(b)→G(b)), where BA is being abstract and WoB(x) says that x is a way of being.

Coming back to the theological considerations, one could suppose there is a fundamental category of being a finite way of being (BFWoB) and a fundamental category of being a divine way of being (BDWoB). By simplicity, BDWoB=God. And then we could define:

4. ∀_BWoBbF(b) if and only if ∀_BDWoBbF(b) and ∀_BFWoBbF(b).

5. ∃_BWoBbF(b) if and only if ∃_BDWoBbF(b) or ∃_BFWoBbF(b).

Note that we can rewrite ∀_BDWoBbF(b) and ∃_BDWoBbF(b) as just F(God).

Eucharist talk

I gave a Thomistic Institute talk on the Real Presence today. Here are the slides and here is the audio. I am particularly pleased with my argument against the purely symbolic view of the Eucharist.

Emergence and the epistemological gap

After reading O’Connor and Churchill’s piece on emergence, one of my very smart undergraduate students commented that it follows from such emergentist views that one could know the mental facts from the physical facts. Here I will argue for this and discuss an unhappy consequence for the causal emergentist.

The causal emergentist thinks that mental properties are not physical, but they causally emerge from complexes of physical properties of a physical entity.

So, now, suppose that physical entity e has a causal power C to produce mental property M when it has a complex P of physical properties. This causal power C will then either be a physical or a non-physical property of e. If it is a physical property of e, then by knowing the physical properties of e, one can know that e has the causal power to produce M. And that, in turn, means M is knowable from physical properties. On the other hand, if C is non-physical, then we do not have emergence of the mental from the physical: we have emergence of the mental from the physical and non-physical. So, if we have genuine emergence of the mental from the physical, then in knowing the physical, we will know the mental.

The unhappy consequence of this is that qualia-based epistemological gap arguments against physicalism apply against causal emergence, since we could suppose M is a quale, and then knowing all about C will include knowing all about M.

Causal emergence may fare a little better with respect to zombie-type arguments. If an entity has an exact duplicate of your physical properties, it will have an exact duplicate of the physically-based causal powers, and hence it will have the causal power to make mental properties emerge. However, it is logically possible that these mental properties will in fact fail to emerge, because it is logically possible that some external causal power blocks the causal powers of the duplicate from achieving their effects. One could even imagine a whole world that is an exact physical duplicate of this one but where nobody physical has mental powers, because some non-physical entity blocks the mental-emergence powers of all the physical beings. So I guess this does some justice to zombie intuitions. But note that if the possibility-of-zombies intuition is satisfied by a non-physical entity blocking mental powers, then a dispositional functionalist could do justice to the zombie intuition by imagining a world just like this one, but where a non-physical entity changes our dispositional properties in the way of Frankfurt’s neurosurgeon. And it’s not clear that that really does justice to the zombie intuition. Maybe.

The above argument against causal emergentism supposes that knowing a cause implies knowing the range of its effects. That is correct on causal powers views of causation. It is not true on Humean views of causation. So a causal emergentist could simply adopt a Humean view of causation. It is also not true on views on which causation depends on laws of nature extrinsic to the particular things in the world. But the causal powers view is the correct one. (And it is one that O’Connor and Churchill embrace.)

What if the emergence relation is not causal in nature? Then it is still a dispositional fact about our physical entity e that it comes to have mental property M when it comes to have a complex P of physical properties. This fact seems like it should be grounded in the properties of e. These properties had better be physical, because the motivation for the theory seems to be that our non-physical properties emerge from our physical ones. And now we still have the danger that by knowing these physical grounds, one can come to know the dispositional fact, and hence come to know M. Perhaps there is a way out of this danger.

Perhaps the best way out for the emergentist, causal or not, is to acknowledge a non-emergent non-physical property in each minded entity grounding the emergence dispositions.

Of course, none of this is a problem if one is unimpressed by qualia-based epistemological gap arguments.

Medical conscience exemptions

After listening to a talk by Christopher Kaczor, and the ensuing discussion, I want to offer a defense of a moderate position on the state not compelling healthcare professionals to violate their conscience, even when their conscience is unreasonably mistaken. I think a stronger position than the moderate position may be true, but I won’t be defending that.

This is the central insight:

1. It is a significant harm to an individual to violate their conscience, even when the conscience is irrationally mistaken.

One reason that (1) is true is the Socratic insight is that it is much better to suffer wrong than to do wrong, together with the Conscience Principle that to act against conscience is always wrong.

My argument will need something a bit more precise than (1). For convenience, I will stipulate that I use “grave” for normative considerations, goods, bads and harms whose importance is at least of the order of magnitude of the value of a human life. The coincidence that “grave” not only means very serious but also place of burial in English—even though the etymologies are quite different—should remind us of this. When you read the following, whenever you read “grave” and cognates, don’t just read “serious”, but also imagine a grave.

Then what I need is this:

2. It is a grave harm to a conscientious individual to gravely violate their conscience, even when that conscience is unreasonably mistaken.

(I suspect this is true even if one drops the “conscientious” and “gravely”, but I am only defending a moderate position.) The reasons for (2) are moral and psychological. The moral reasons are based on the aforementioned Socratic insight about the importance of avoiding wrongdoing. But there are also psychological reasons. A conscientious person identifies with their conscience in such a way that gravely violating this conscience is shattering to the individual’s identity. It is a kind of death. It is no coincidence that the Catholic tradition talks of some sins as “mortal”.

Next, here is another reasonable principle:

3. Normally, the state should not require a healthcare professional to provide care when the care is likely to come at a grave cost to the professional.

For instance, the state should not require a healthcare professional to donate her own kidney to save a patient. For a less extreme case that I will consider some variations of, neither should the state require a professional who has a severe bee allergy to pass through a cloud of bees to help a patient when allergy reaction drugs are unavailable and when other professionals lacking such an allergy are available.

In order for (3) to be useful in pracice, we need some way of getting rid of the “Normally” in it.

Notice that (3) is true even when the grave cost to the professional results from the professional’s irrationality. For instance, normally a healthcare professional who has a grave phobia of bees should not be required to pass through the cloud of bees, even if it is known that the professional would not be seriously physically harmed. In other words, that the cost results from irrationality does count as an abnormality in (3).

Under what abnormal conditions, then, may the state require the professional to offer care that comes at grave cost to the professional? This is clearly a necessary condition:

4. The need is grave.

But even if the need is grave, if someone else can offer the care for whom offering the care does not come at a grave cost, they should offer it instead. If the way to save a patient’s life is for one doctor to pass through a cloud of bees, and there is a doctor available who is not allergic to bee stings, then a doctor who is allergic should not be made to do it. Thus, we have this condition:

5. There is no way of meeting the need without someone being required to take on a likely grave cost.

We can combine these two conditions into a neater condition (which may also be a bit weaker than the conjunction of (4) and (5)):

6. If the care is not provided by this professional, a grave harm will likely result to someone.

This suggests some principle like this:

7. Unless failure of this professional to provide this instance of care will likely result in a grave harm, the state should not require a healthcare professional to provide care when the care is likely to come at a grave cost to the professional.

Now we go back to (2), the claim about the grave cost of violating conscience. Let us charitably assume that most medical professionals are conscientious, so that any given medical professional is likely to be conscientious. Then we get something like this:

8. Unless failure of this professional to provide this instance of care will likely result in a grave harm, the state should not require a healthcare professional to provide care that gravely violates their conscience, even when that conscience is unreasonably mistaken.

But this cannot be the whole story. For there are also conditions that render one incapable of doing central parts of one’s job. For instance, someone with a grave phobia of fires should not be allowed to be a fire fighter. And while a fire fighter with that grave phobia should not be made to fight a fire when someone else is available, if they had the phobia at the time of hiring, they should not have been hired in the first place. And if they hid this phobia at the time of hiring, they should be fired.

We have, however, a well-developed societal model for dealing with such conditions: the reasonable accommodations model of disability legislation like the Americans with Disabilities Act. It is reasonable to require an office building to put in a ramp for an employee in a wheelchair who is unable to walk; it would be unreasonable for a bank to have to hire a guard specially to watch a kleptomaniac teller. What is and is not a reasonable accommodation depends on the centrality of an aspect of a job, the costs to the employer, and so on.

So my moderate proposal says that we handle the worry that a particular conscientious objection renders a professional incapable of doing their job by analogy to the reasonable and unreasonable accommodations model, and qualify (8) by allowing in hiring or licensure the requirement that the accommodations for a conscientious restriction on practice would have be reasonable in ways analogous to reasonable disability accommodations. A healthcare professional who has only one hand could, I assume, be reasonably accommodated in a number of specialities, but likely not as a surgeon.

The disability case also should push us towards a less judgmental attitude towards a healthcare professional whose conscientious objections are unreasonably mistaken. That an employee became a paraplegic from unreasonable daredevil recreational activity does not render the employee uneligible for otherwise reasonable accommodations.

What about the worry about the rare cases where a healthcare professional has morally repugnant conscientious views that would require discriminatory care, such as refusing to care for patients of a particular race? Could one argue that if patients of that race are rare in a given area, then allowing a restriction of practice on the basis of race could be a reasonable accommodation? We might imagine an employee who has panic attacks triggered by a particular rare configuration of a client’s personal appearance, and that does seem like a case for reasonable accommodations, after all.

Here I think there is a different thing to be said. We want our healthcare professionals to have certain relevant moral virtues to a reasonable degree. Moral virtues go beyond obedience to conscience. Someone with a mistaken conscience may not be to blame, for the wrongs they do, but they may nonetheless lack certain virtues. The case of the conscientious racist is one of those. So it is not so much because the conscientious racist would refuse to care for patients of a particular race that they should not be a healthcare professional but it is because they fail to have the right kind of respect for the dignity of all human beings.

One may think that this consideration makes the account not very useful. After all, a pro-life individual is apt to be accused of not caring enough for women. Here I just think we need to be honest and reasonably charitable. Caring about the embryo and fetus has human dignity does not render it less likely that one cares about women. Compare this case: A vegan physician believes that all higher animal life is sacred, and hence refuses to prescribe medication whose production essentially involves serious suffering of higher animals. Even if such a physician’s actions might cause harm to patients who need such (hypothetical?) medication, the belief that all higher animal life is sacred is not evidence that the physician does not care about such patients–indeed, it seems to render it more likely that the physician thinks the patients’ lives to be sacred as well, and hence to be cared for. There may be specialties where accommodation is unreasonable, but the mere fact of the belief is not evidence of lack of relevant virtues.

Provability from finite and infinite theories

Let #s be the Goedel number of s. The following fact is useful for thinking about the foundations of mathematics:

Proposition. There is a finite fragment A of Peano Arithmetic such that if T is a recursively axiomatizable theory, then there is an arithmetical formula P_T(n) such that for all arithmetical sentences s, A → P_T(#s) is a theorem of FOL if and only if T proves s.

The Proposition allows us to replace the provability of a sentence from an infinite recursive theory by the provability of a sentence from a finite theory.

Sketch of Proof of Proposition. Let M be a Turing machine that given a sentence as an input goes through all possible proofs from T and halts if it arrives at one that is a proof of the given sentence.

We can encode a history of a halting (and hence finite) run of M as a natural number such that there will be a predicate H_M(m, n) and a finite fragment A of Peano Arithmetic independent of M (I expect that Robinson arithmetic will suffice) such that (a) m is a history of a halting run of M with input m if and only if H_M(m, n) and (b) for all m and n, A proves whether H_M(m, n).

Now, let P_T(n) be ∃mH_M(m, n). Then A proves P_T(#s) if and only if there is an m₀ such that A proves H_M(m₀, n). (If A proves P_T(#s), then because A is true, there is an m such that H_M(m, #s), and then A will prove H_M(m₀, #s). Conversely, if A proves H_M(m₀, #s), then it proves ∃mH_M(m, #s).) And so A proves P_T(#s) if and only if T proves s.

A bad idea in the foundations of mathematics

The relativity of FOL-validity is the fact that whether a sentence ϕ of First Order Logic is valid (equivalently, provable from no axioms beyond any axioms of FOL itself) sometimes depends on the axioms of set theory, once we encode validity arithmetically as per Goedel.

More concretely, if Zermelo-Fraenkel-Choice (ZFC) set theory is consistent, then there is an FOL formula ϕ that is FOL-provable according to some but not other models of ZFC. So which model of ZFC should real provability be relativized to?

Here is a putative solution that occurred to me today:

• Say that ϕ is really provable if and only if there is a model M of ZFC such that according to M, ϕ has a proof.

If this solution works, then the relativity of proof is quite innocent: it doesn’t matter in which model of ZFC our proofs live, because proofs in any ZFC model do the job for us.

It follows from incompleteness (cf. the link above) that real provability is strictly weaker than provability, assuming ZFC is true and consistent. Therefore, some really provable ϕ will fail to be valid, and hence there will be models of the falsity of ϕ. The idea that one can really prove a ϕ such that there is a model of the falsity of ϕ seems to me to show that my proposed notion of “really provable” is really confused.

Post-Goedelian mathematics as an empirical inquiry

Once one absorbs the lessons of the Goedel incompleteness theorems, a formalist view of mathematics as just about logical relationships such as provability becomes unsupportable (for me the strongest indication of this is the independence of logical validity). Platonism thereby becomes more plausible (but even Platonism is not unproblematic, because mathematical Platonism tends towards plenitude, and given plenitude it is difficult to identify which natural numbers we mean).

But there is another way to see post-Goedelian mathematics, as an empirical and even experimental inquiry into the question of what can be proved by beings like us. While the abstract notion of provability is subject to Goedelian concerns, the notion of provability by beings like us does not seem to be, because it is not mathematically formalizable.

We can mathematically formalize a necessary condition for something to be proved by us which we can call “stepwise validity”: each non-axiomatic step follows from the preceding steps by such-and-such formal rules. To say that something can be proved by beings like us, then, would be to say that beings like us can produce (in speech or writing or some other relevantly similar medium) a stepwise valid sequence of steps that starts with the axioms and ends with the conclusion. This is a question about our causal powers of linguistic production, and hence can be seen as empirical.

Perhaps the surest way to settle the question of provability by beings like us is for us to actually produce the stepwise valid sequence of steps, and check its stepwise validity. But in practice mathematicians usually don’t: they skip obvious steps in the sequence. In doing so, they are producing a meta-argument that makes it plausible that beings like us could produce the stepwise valid sequence if they really wanted to.

This might seem to lead to a non-realist view of mathematics. Whether it does so depends, however, on our epistemology. If in fact provability by beings like us tracks metaphysical necessity—i.e., if B is provable by beings like us from A₁, ..., A_n, then it is not possible to have A₁, ..., A_n without B—then by means of provability by beings like us we discover metaphysical necessities.

Nitpicking about the causal exclusion argument

Exclusion arguments against dualism, and sometimes against nonreductive physicalism, go something like this.

1. Every physical effect has a sufficient microphysical cause.

2. Some microphysical effects have non-overdetermined mental causes.

3. If an event E has two distinct causes A and B, with A sufficient, it is overdetermined.

4. So, some mental causes are identical to microphysical causes.

But (3) is just false as it stands. It neglects such cases of non-overdetermining distinct causes A and B as:

5. A is a sufficient cause of E and B is a proper part of A, or vice versa. (Example: E=window breaking; A=rock hitting window; B=front three quarters of rock hitting window.)

6. A is a sufficient cause of B and B is a sufficient cause of E, or vice versa, with these instances of sufficient causation being transitive. (Example: E=window breaking; A=Jones throwing rock at window; B=rock impacting window.)

7. B is an insufficient cause of A and A is a sufficient cause of B, with these instances of causation being transitive. (Example: E=window breaking; B=Jones throwing rock in general direction of window; A=rock impacting window.)

8. A and B are distinct fine-grained events which correspond to one coarse-grained event.

To take care of (6) and (7), we could replace “cause” with “immediate cause” in the argument. This would require the rejection of causation by a dense sequence of causes (e.g., the state of a Newtonian system at 3 pm is caused by its state at 2:30 pm, its state at 2:45 pm, at 2:52.5 pm, and so on, with no “immediate” cause). I defend such a rejection in my infinity book. But the price of taking on board the arguments in my infinity book is that one then has very good reason to accept the Kalaam argument, and hence to deny (1) (since the first physical state will then have a divine, and hence non-microphysical, cause).

We could take care of (5) and (8) by replacing “distinct” with “non-overlapping” in (3). But then the conclusion of the argument becomes much weaker, namely that some mental causes overlap microphysical causes. And that’s something that both the nonreductive physicalist and hylomorphic dualist can accept for different reasons: the nonreductive physicalist may hold that mental causes totally overlap with microphysical causes; the hylomorphist will say that the form is a part of both the mental cause and of the microphysical cause. Maybe we still have an argument against substance dualism, though.

Infinity, Causation, and Paradox

Amazon USA finally has it in stock.

Two kinds of functionalism

There are two kinds of functionalism about the mind.

One kind upholds the thesis that if two systems exhibit the same overall function, i.e., the same overall functional mapping between sequences of system inputs and sequences of system outputs, then they have the same mental states if any. Call this systemic functionalism.

The other kind says that mental properties depend not just on overall system function, but also on the functional properties of the internal states and/or subsystems of the system. Call this subsystemic functionalism. The subsystemic functionalist allows that two systems may have the same overall function, but because the internal architecture (whether software or hardware) that achieve this overall function are different, the mental states of the systems could be different.

Systemic functionalism allows for a greater degree of multiple realizability. If we have subsystemic functionalism, we might meet up with aliens who behave just like we do, but who nonetheless have no mental states or mental states very different from ours, because the algorithms that are used to implement the input-to-output mappings in them are sufficiently different.

If subsystemic functionalism is true, then it seems impossible for us to figure out what functional properties constitute mental states, except via self-experimentation.

For instance, we would want to know whether the functional properties that constitute mental states are neuronal-or-above or subneuronal. If they are neuronal-or-above, then replacing neurons with prostheses that have the same input-to-output mappings will preserve mental states. If they are subneuronal, such replacement will only preserve mental states if the prostheses not only have the same input-to-output mappings, but also are functionally isomorphic at the relevant (and unknown to us) subneuronal level.

But how could we figure out which is the case? Here is the obvious thing to try: Replace neurons with prostheses whose internal architecture does not have much functional resemblance to neurons but which have the same input-to-output mappings. But assuming standard physicalist claims about there not being “swervy” top-down causation (top-down causation that is unpredictable from the microphysical laws), we know ahead of the experiment that the subject will behave exactly as before. Yet if we have rejected systemic functionalism, sameness of behavior does not guarantee sameness of mental states, or any mental states at all. So doing the experiment seems pointless: we already know what we will find (assuming we know there is no swervy top-down causation), and it doesn’t answer our question.

Well, not quite. If I have the experiment done on me, then if I continue to have conscious states after complete neuronal prosthetic replacement, I will know (in a Cartesian way) that I have mental states, and get significant evidence that the relevant system level is neuronal-or-above. But I won’t be able to inform anybody of this. If I tell people: “I am still conscious”, if they have rejected systemic functionalism, they will just say: “Yeah, he/it would say that even if he/it weren’t, because we have preserved the systemic input-to-output mappings.” And there will be significant limits to what even I can know. While I could surely know that I am conscious, I doubt that I would be able to trust my memory to know that my conscious states haven’t changed their qualia.

So with self-experimentation, I could know tht the relevant system level is neuronal-or-above. Could I know even with self-experimentation that the relevant system level is subneuronal. That’s a tough one. At first sight, one might consider this: Replace neurons with prostheses gradually and have me observe whether my conscious experiences start to change. Maybe at some point I stop having smell qualia, because the neurons involved in smell have been replaced with subsystemically functionally non-isomorphic systems. Oddly, though, given the lack of swervy top-down causation, I would still report having smell qualia, and act as if I had them, and maybe even think, albeit mistakenly, that I have them. I am not sure what to make of this possibility. It’s weird indeed.

Moreover, a version of the above argument shows that there is no experiment that we could do that would persons other than at most the subject know whether systemic or subsystemic functionalism is true, assuming there is no swervy top-down causation.

Things become simpler in a way if we adopt systemic functionalism. It becomes easier to know when we have strong AI, when aliens are conscious, whether neural prostheses work or destroy thought, etc. The downside is that systemic functionalism is just behaviorism.

On the other hand, if there is swervy top-down causation, and this causation meshes in the right way with mental functioning, then we are once again in the experimental philosophy of mind business. For then neurons might function differently when in a living brain than what the microphysical laws predict. And we could put in prostheses that function outside the body just like neurons, and see if those also function in vivo just like neurons. If so, then the relevant functional level is probably neuronal-or-above; if not, it's probably subneuronal.

The centrality of the natural numbers

The more I think about the foundations of mathematics, the more wisdom I see in Kronecker’s famous saying: “God made the natural numbers; all else is the work of man.” There is something foundationally deep about the natural numbers. We see this in the way theories of natural numbers is equivalent (e.g., via Goedel encoding) to the theories of strings of symbols that are central to logic, and in the way that when we fix our model of natural numbers, we fix the foundational notion of provability.