The context problem for Lewisian functionalism

One problem for functionalism is the problem of defect. David Lewis, for instance, talks of a madman for whom pain is triggered by something other than damage and whose pain triggers something other than avoidance. Lewis’s functionalist solution is to define the function of a mental state in terms of the role it normally plays in the species.

Here is a problem with this. Suppose that in mammals pains is realized by C-fiber firing. But now take the C-fibers inside a living mammalian skull, disconnect their outputs and connect external electrodes to their inputs. Make the C-fibers fire. Since the C-fiber outputs are disconnected, causing them to fire does not cause any of the usual pain behaviors, the formation of memories of pain, etc. In fact, it seems very plausible that there is no pain at all. Yet according to Lewisian functionalism, there is pain, because it is the normal connections of the C-fibers that define their functional role.

This thought experiment shows that the physical realizers of mental states need to occur in their proper context. But this bumps up against Lewis’s madman, in whom the pain states, and presumably their physical realizers, do not occur in their proper context.

It seems that what the functionalist needs to say is that in order to realize a mental state, a physical state must occur in a sufficient approximation to its proper context. If it’s too far, as in the case of the C-fibers with severed outputs, there is no mental state. If it’s close enough, as in a moderate version of the madman case (I don’t know what to say about Lewis’s more extreme one), the mental state occurs.

But how is the line to be drawn?

Perhaps there is no problem. Pain is not in fact C-fiber firing. Perhaps enough of the brain needs to be involved in conscious states that one cannot plausibly remove the states from their normal functional context? Still, this is worth thinking about.

M-rays

Let me tell a story. Some neuroscientists detected a physically novel form of radiation, M-rays, that is emitted by brains of subjects who are thinking consciously. They discovered this much like X-rays were discovered, namely by finding something impinging on equipment in the lab in a way that could be explained by conventional physics. Further experiment showed that M-rays are not emitted by anything that clearly isn’t conscious. Moreover, the line between animals that emitted M-rays and animals that didn’t seemed to correspond to a noticeable difference in cognitive sophistication. Finally, in humans the M-rays turned out to be modulated in a way that has a natural one-to-one correspondence with the phenomenal states reported by the conscious subjects, so that the scientists eventually learned to discern from the M-rays what the subject’s conscious state is. (The CIA was very interested.)

In this case, it would be eminently reasonable for a physicalist to conclude that consciousness is the emission of M-rays.

This thought experiment shows that mysterians like Colin McGinn are mistaken in holding that no discovery we could make would solve the hard problem of consciousness.

But of course few physicalists actually expect to find a physically novel phenomenon in the brain.

Independence of FOL-validity

A sentence ϕ of a dialect of First Order Logic is FOL-valid if and only if ϕ is true in every non-empty model under every interpretation. By the Goedel Completeness Theorem, ϕ is valid if and only if ϕ is a theorem of FOL (i.e., has a proof from no axioms beyond any axioms of FOL). (Note: This does not use the Axiom of Choice since we are dealing with a single sentence.)

Here is a meta-logic fact that I think is not as widely known as it should be.

Proposition: Let T be any consistent recursive theory extending Zermelo-Fraenkel set theory. Then there is a sentence ϕ of a dialect of First Order Logic such that according to some models of T, ϕ is FOL-valid (and hence a theorem of FOL) and according to other models of T, ϕ is not FOL-valid (and hence not a theorem of FOL).

Note: The claim that ϕ is FOL-valid according to a model M is shorthand for the claim that a certain complex arithmetical claim involving the Goedel encoding of ϕ is true according to M.

The Proposition is yet another nail in the coffins of formalism and positivism. It tells us that the mere notion of FOL-theoremhood has Platonic commitments, in that it is only relative to a fixed family of universes of sets (or at least a fixed model of the natural numbers or a fixed non-recursive axiomatization) does it make unambiguous sense to predicate FOL-theoremhood and its lack. Likewise, the very notion of valid consequence, even of a finite axiom set, carries such Platonic commitments.

Proof of Proposition: Let G be a Rosser-tweaked Goedel sentence for T with G being Σ₁ (cf. remarks in Section 51.3 here). Then G is independent of T. In ZF, and hence in T, we can prove that there is a Turing machine Q that halts if and only if G holds. (Just make Q iterate over all natural numbers, halting if the number witnesses the existential quantifier at the front of the Σ₁ sentence G.) But one can construct an FOL-sentence ϕ such that one can prove in ZF that ϕ is FOL-valid if and only if Q halts (one can do this for any Turing machine Q, not just the one above). Hence, one can prove in T that ϕ is FOL-valid if and only if I holds.

Thus, in T it is provable that ϕ is FOL-valid if and only if G holds. But T is a consistent theory (otherwise one could formalize in T the proof of its inconsistency). Since G is independent of T, it follows that the FOL-validity of ϕ is as well.

Preventing suffering

Theodicies according to which sufferings make possible greater moral goods are often subjected to this objection: If so, why should we prevent sufferings?

I am not near to having a full answer to the question. But I think this is related to a question everyone, and not just the theist, needs to face up to. For everyone should accept Socrates’ great insight that moral excellence is much more important than avoiding suffering, and yet we should often prevent suffering that we think is apt to lead to the more important goods. I don’t know why. That’s right now one of the mysteries of the moral life for me. But it is as it is.

Famously, persons with disabilities tend to report higher life satisfaction than persons without disabilities. But we all know that accepting this data should not keep us from working to prevent disability-causing car accidents. While higher life satisfaction is not the same as moral excellence, the example is still instructive. Our reasons to prevent disability-causing car accidents do not require us to refute the empirical data suggesting that persons with disabilities lead more satisfying lives. I do not know why exactly we still have on balance reason to prevent such accidents, but it is clear to me that we do.

Mother Teresa thought that the West is suffering from a deep poverty of relationships, with both God and neighbor. Plausibly she was right. We probably are not in a position to know that affluence is a significant cause of this deep poverty, but we can be open to the real epistemic possibility that it is, and we can acknowledge the deep truth that the riches of relationship are far more important than physical goods, without this sapping our efforts to improve the material lot of the needy.

Or suppose you are witnessing Alice torturing Bob, and an oracle informed you that in ten years they will be reconciled, with Bob beautifully forgiving Alice and Alice deeply repenting, with the goods of the reconciliation being greater than the bads in the torture. I think I should still stop Alice.

A quick corollary of the above cases is that consequentialism is false. But there is a deep paradox here that cuts more deeply than consequentialism. I do not know how to resolve it.

Here are some stories, none of which are fully satisfying to me in their present state of development.

Perhaps it is better if humans have a special focus on the relief of suffering and improvement of material well-being of the patient. An opposite focus might lead to an unhealthy condescension.

Perhaps it has something to do with our embodied natures that a special focus on the bodily good of the other is a particularly fitting way for humans to express love for one another. While letting another suffer in the hope of greater on-balance happiness might be better for the patient, it could well be worse for the agent and the relationship. Maybe we should think of what Catholics call the “corporal works of mercy” as a kind of kiss, or maybe even something like a sacrament.

Perhaps there is something about respect for the autonomy of the other. Maybe others’ physical good is also our business while moral development is more their own business.

I think there is more. But the point I want to make is just that this is not a special question for theism and theodicy. It is a paradox that all morally sensitive people should see both sides of.

Coming back to theodicy, note that the above speculative considerations may not apply to God as the agent. (God cannot but condescend, being infinitely above us. God is not embodied, except in respect of the Incarnation. And we have no autonomy rights against God, as God is closer to us than we are to ourselves.)

Minecraft cake cake

Our youngest wanted a Minecraft cake for her birthday. We settled on a Minecraft cake block cake using this clever method. (But we froze the frosting overnight on parchment paper—maybe wax paper would have been better—and instead of candy melts just used more frozen colored frosting; we also used only two layers.) The brown bottom is cinnamon—we had three people apply it simultaneously.

Groups and roles

I’ve had a grad student, Nathan Mueller, do an independent study in social epistemology in the hope of learning from him about the area (and indeed, I have learned much from him), so I’ve been thinking about group stuff once a week (at least). Here’s something that hit me today during our meeting. There is an interesting disanalogy between individuals and groups. Each group is partly but centrally defined by a role, with different groups often having different defining roles. The American Philosophical Association has a role defined by joint philosophical engagement, while the Huaco Bowmen have a role defined by joint archery. But this is not the case for individuals. While individuals have roles, the only roles that it is very plausible to say that they are partly and centrally defined by are general roles that all human beings have, roles like human being or child of God.

This means that if we try to draw analogies between group and individual concepts such as belief or intention, we should be careful to draw the analogy between the group concept and the concept as it applies not just to an individual but to an individual-in-a-role. Thus, the analogy is not between, say, the APA believing some proposition and my believing some proposition, but between the APA believing some proposition and my believing that proposition qua father (or qua philosopher or qua mathematician).

If this is right, then it suggests an interesting research program: Study the attribution of mental properties to individuals-in-roles as a way of making progress on the attribution of analogous properties to groups. For instance, there are well-founded worries in the social epistemology literature about simple ways of moving from the belief of the members of the group to the belief of the group (e.g., attributing to the group any belief held by the majority of the members). These might be seen to parallel the obvious fact that one cannot move from my believing p to my believing p qua father (or qua mathematician). And perhaps if we better understand what one needs to add to my believing p to get that I believe p qua father, this addition will help us understand the group case.

(I should say, for completeness, that my claim that the only roles that human beings are partly and centrally defined by are general roles like human being is controversial. Our recent graduate Mengyao Yan in her very interesting dissertation argues that we are centrally defined by token roles like child of x. She may even be right about the specific case of descent-based roles like child of x, given essentiality of origins, but I do not think it is helpful to analyze the attribution of mental properties to us in general in terms of us having these roles.)

My experience of temporality

This morning I find myself feeling the force of presentism. I am finding it hard to see my four-dimensional worm theory as adequately explaining why my experience only includes what I am experiencing now, instead of the whole richness of my four-dimensional life. I am also finding it difficult to satisfactorily explain the sequentiality of my experiences: that I will have different experiences from those that I have now, some of which I dread and some of which I anticipate eagerly.

When I try to write down the thoughts that make me feel the force of presentism, the force of the thoughts is largely drained. After all, to be fair, when I wrote that I have am having trouble “explaining why my experience only includes what I am experiencing now”, shouldn’t I have written: “explaining why my present experience only includes what I am experiencing now”, a triviality? And that mysterious sequentiality, is that anything beyond the fact that some of my experiences are in the future of my present experience?

The first part of the mystery is due to the chopped up nature of my consciousness on a four-dimensional view. Instead of seeing my life as a whole, as God sees it, I see it in very short (but probably not instantaneous) pieces. It is puzzling how my consciousness can be so chopped up, and yet be all mine. But we have good reason to think that this phenomenon occurs outside of temporality. Split brain patients seem to have such chopped up consciousnesses. And if consciousness is an operation of the mind’s, then on orthodox Christology, the incarnate Christ, while one person, had (and still has) two consciousnesses.

Unfortunately, both the split brains and the Incarnation are mysterious phenomena, so they don’t do much to take away the feeling of mystery about the temporal chopping up of the consciousness of my four-dimensional life. But they do make me feel that there is no good argument for presentism here.

The second part of the mystery is due to the sequentiality of the experiences. As the split brain and Incarnation cases show, the sequentiality of experiences in different spheres of consciousness is not universal. The split brain patient has two non-sequential, simultaneous spheres of consciousness. Christ has his temporal sphere (or spheres, if we take the four-dimensional view) of consciousness and his divine atemporal sphere of consciousness. But seeing the contingency of the sequentiality does not remove the mystery in the sequentiality.

It makes me feel a little better when I recall that the presentist story about the sequentiality has its own problems. If my future experiences aren’t real—on presentism they are nothing but stuff in the scope of a modal “will” operator that doesn’t satisfy the T axiom—then what am I anticipating or dreading? It seems I am just here in the present, and when I think about this, it feels just as mysterious as on four-dimensionalism what makes the future impend. Of course, the presentist can give a reductive or non-reductive account of the asymmetry between past and future, but so can the four-dimensionalist.

So what remains of this morning’s presentist feelings? Mostly this worry: Time is mysterious and our theories of time—whether eternalist or presentist—do not do justice to its mysteriousness. This is like the thought that qualia are mysterious, but when we give particular theories of them—whether materialist or dualist—it feels like something is left out.

But what if I forget about standard four-dimensionalism and presentism, and just try to see what theory of time fits with my experiences? I then find myself pulled towards a view of time I had when I was around ten years old. Reality is four-dimensional, but we travel through it. Future sufferings I dread are there, ahead of me. But I am not just a temporal part among many: there is no future self suffering future pains and enjoying future pleasures. The past and future have physical reality but it’s all zombies. As for me, I am wholly here and now. And you are wholly here and now. We travel together through the four-dimensional reality.

But these future pains and pleasures, how can they be if they are not had by me or anyone else? They are like the persisting smile of the Cheshire cat. (I wasn’t worried about this when I was ten, because I was mainly imagining myself as traveling through events, and not philosophically thinking about my changing mental states. It wasn’t a theory, but a way of thinking.) Put that way, maybe it’s not so crazy. After all, the standard Catholic view of the Eucharist is that the accidents of bread and wine exist without anything having them. So perhaps my future and past pains and pleasures exist without anyone having them—but one day I will have them.

Even this strange theory, though, does not do justice to sequentiality. What makes it be the case that I am traveling towards the future rather than towards the past?

And what about Relativity Theory? Why don’t we get out of sync with one another if we travel fast enough relative to one another? Perhaps the twin who travels at near light speed comes back to earth and meets only zombies, not real selves? That seems absurd. Maybe though the internal flow of time doesn’t work like that.

I do not think this is an attractive theory. It is the theory that best fits most of my experience of temporality, and that is a real consideration in favor of it. But it doesn’t solve the puzzle of sequentiality. I think I will stick with four-dimensionalism. For now. (!)

Extended simples

There are at least two reasons to think we are simple:

1. It is difficult to explain how a non-simple thing can have a unity of consciousness.

2. There is David Barnett’s “pairs” argument.

But we are clearly extended.

So, we are extended simples.

So, there are extended simples.

(That said, while I am happy with the idea that we are extended simples, I am suspicious of both 1 and 2.)

Ownership and ontology

We can own dogs, trees, forests, cars, chairs, computers and cupcakes, but of these examples, only dogs and trees really exist. Many of the things we own do not really exist. This makes me sceptical of the idea that there are strong property rights independent of positive law.

You might stop me by saying that my ontology is simply too restrictive. Maybe forests, cars, chairs, computers and cupcakes all really exist. I doubt it, but the examples of non-existent things we can in principle own can be multiplied. It is just as reasonable to talk of owning the vacuum inside a flask as it is to talk of owning the cocoa inside a cup. In both cases, labor was needed to generate the “thing” owned, and there is a reasonable moral expectation of non-interference with respect to it. (I would be destroying your property if I beamed a gas into your vacuum flask.)

What does this have to do with scepticism of strong property rights independent of positive law? First, it becomes very difficult to draw a principled line between ownables and non-ownables. Second, once we recognize that we can own things that don’t exist, such as vacua, it becomes difficult to distinguish “things” we have created and own from other kinds of outcomes of our activity. It then becomes plausible that the relevant right is one that should apply to outcomes of activity without much regard for whether that outcome is a thing that exists, a “thing” that doesn’t exist, or some other kind of outcome, such as a mountain’s being enchanted. There seems to be some kind of a right not to have the intended outcome of one’s virtuous activity destroyed without good reason. But how good the reason has to be will vary widely from case to case, so it is unlikely that this kind of a right will ground a strong view of property rights independent of positive law.

But the difficult is not the impossible. For it may be that although it would be difficult to make the needed distinctions, these distinctions could be grounded in highly detailed facts encoded in our natures.

The unity of consciousness

Here’s a familiar kind of argument:

1. A spatial arrangement of ingredients of a mental life would not yield a unity of consciousness.

2. We have a unity of consciousness.

3. Our mental life is not constituted by a spatial arrangement of ingredients.

4. So, our minds are not spatially extended entities (and in particular they are not brains).

(The last step requires some additional premises about extension and mereology.)

But our unity of consciousness also includes ingredients that take time. We are aware of motion, and motion takes time. We consciously think temporally extended thoughts. If we take the argument (1)-(4) seriously, it looks like we should similarly conclude that our souls are not temporally extended entities.

This might be a reductio ad absurdum of the line of argument (1)-(4). For it seems that even the dualist will recognize the essentially temporally extended nature of many of our conscious states.

Or maybe it’s an argument for a Kantian view on which we have a noumenal self that is beyond space and time as the physicists conceive of them.

The five minute hypothesis

Let S(t) be the state our universe actually has t units of time after the Big Bang.

Let’s suppose that our universe begins at time 0, i.e., at the Big Bang. Then we can ask this question:

1. Why did the universe begin to exist in state S(0) rather than in one of the infinitely many states S(t) for t > 0?

If naturalism is true, the universe’s beginning as it does is probably a brute (unexplained) contingent fact. And while the state S(0) is rather different in physical arrangement from the states S(t) for t > 0, it does not seem different with respect to the likelihood of brutely coming into existence ex nihilo. So the question (1) is not easily to be dismissed, in the way that one might dismiss the question “Why is the third digit of the gravitational constant in SI units a 7 instead of some other digit?” by saying “Well, it had to something, and there is nothing special about 7.” For there is something special about S(0), namely that it is a singularity state that blocks retrodiction, but this something special does not seem relevant to the likelihood of brutely coming into existence—if only because brute coming into existence cannot be probabilistically quantified.

Theism, on the other hand, provides a satisfying answer to (1). First, as a warmup, we might have a good theistic story why the initial state isn’t S(t) for a very large t: perhaps the universe will no longer be habitable after t, and God has good reason to create a habitable universe. Second, we have a good reason why the initial state should be S(0): a singularity is a natural barrier to retrodiction, and so creating a universe whose past goes back to a singularity is creating a universe where the retrodictive knowledge of beings like us is maximized (cf. Robin Collins’ ideas on our world being optimized for science). And knowledge is good. So God has reason to create a world in initial state S(0).

Technical note: Perhaps there is no singularity state S(0), but instead there are states S(t) for t > 0. If so, then replace S(t) in the above argument by something facts about limits from above of S(t). Abstractly, we can form such units as disjunctions of shrinking conjunctions: let U(t)=⋁_δ>0⋀_0<ϵ<δS(t+ϵ), and say that the universe “begins to exist in state U(t)” provided that it doesn’t exist time t or earlier, but that U(t) correctly describes an opening interval of the universe’s existence. Then replace (1) with the question:

2. Why did the universe begin to exist in state U(0) rather than in one of the infinitely many states S(t) or U(t)?

Materiality revisited

I’ve long been puzzled by materiality.

Here’s a thought: What if materiality isn’t characterized by anything deeply metaphysical, but by a physical quality? Perhaps to be material just is to have something like inertia, or mass, or energy?

(I think that to have zero of some quality like mass is still to have mass. A mass of x is a determinate of the determinable mass even if x = 0. Photons have mass, while numbers don’t.)

Yet another infinite hat-guessing story

Suppose first a countably infinite line of blindfolded people standing on tiles numbered 0,1,2,…, with the ones on a tile whose number is divisible by 10 having a red hat, and the others having blue hats. Suppose you’re in the line, with no idea where, but apprised of the above. It seems you should reasonably think: “Probably my hat is blue.”

But then the blindfolded people are shuffled, without any changes of hats, so that now it is the tiles with numbers divisible by 10 that have the blue hatters and the others have the red hatters. Such mere shuffling shouldn’t change what you think. So after being informed of the shuffle, it seems you should still think: “Probably my hat is blue.” It is already puzzling, though, why the first arrangement defined the probabilities and not the second. (What does temporal order have to do with these probabilities?)

Now suppose you gather the nine people after you (in the tile order—even though you are blindfolded, I suppose you can tell which direction the tile number numbers increase) along with yourself into a group of ten. In any group of ten successive people on the line, there is exactly one blue hat and nine red hats. Yet each of the ten of you thinks: “Probably my hat is blue.” And by a reasonable closure, you each also think: “Probably the other nine all have red hats.” You talk about it. You argue about it. “No, I am probably the one with the blue hat!” “No, my hat is probably the blue one.” “No, you’re probably both wrong: It’s probably mine.” I submit there is no rational room for any resolution to the disagreement, and indeed no budging of probabilities, no matter how much you pool your data, no matter how completely you recognize your epistemic peerhood, no matter how you apply exactly the same reasonable principles of reasoning. For nothing you learn from the other people is evidentially relevant. This is paradoxical.

Two tenure track jobs at Baylor

We have two tenure-track jobs at Baylor. Both are open, but we have different preferred (but not required) specializations for them:

• Job 1: roughly, LEMM and its history

• Job 2: roughly, non-LEMM and its history

Yet another reason we need social epistemology

Consider forty rational people each individually keeping track of the ethnicities and virtue/vice of the people they interact with and hear about (admittedly, one wonders why a rational person would do that!). Even if there is no statistical connection—positive or negative—between being Polish and being morally vicious, random variation in samples means that we would expect two of the forty people to gain evidence that there is a statistically significant connection—positive or negative—between being Polish and being morally vicious at the p = 0.05 level. We would, further, intuitively expect that one in the forty would come to conclude on the basis of their individual data that there is a statistically significant negative connection between Polishness and vice and one that there is a statistically significant positive connection.

It seems to follow that for any particular ethnic or racial or other group, at the fairly standard p = 0.05 significance level, we would expect about one in forty rational people to have a rational racist-type view about any particular group’s virtue or vice (or any other qualities).

If this line of reasoning is correct, it seems that it is uncharitable to assume that a particular racist’s views are irrational. For there is a not insignificant chance that they are just one of the unlucky rational people who got spurious p = 0.05 level confirmation.

Of course, the prevalence of racism in the US appears to be far above the 1/40 number above. However, there is a multiplicity of groups one can be a racist about, and the 1/40 number is for any one particular group. With five groups, we would expect that approximately 5/40=1/8 (more precisely 1 − (39/40)⁵) of rational people to get p = 0.05 confirmation of a racist-type hypothesis about one of the groups. That’s still presumably significantly below the actual prevalence of racism.

But in any case this line of reasoning is not correct. For we are not individual data gatherers. We have access to other people’s data. The widespread agreement about the falsity of racist-type claims is also evidence, evidence that would not be undercut by a mere p = 0.05 level result of one’s individual study.

So, we need social epistemology to combat racism.

A simple reductive theory of consciousness

I think it is possible for one mind to have multiple spheres of consciousness. One kind of case is diachronic: there need be no unity of consciousness between my awareness at t₁ and my awareness at t₂. Split brain patients provide synchronic example. (I suppose in both cases one can question whether there is really only one mind, but I’ll assume so.)

What if, then, it turned out that we do not actually have any unconscious mental states? Perhaps what I call “unconscious mental states” are actually conscious states that exist in a sphere of consciousness other than the one connected to my linguistic productions. Maybe it is the sphere of consciousness connected to my linguistic productions that I identify as the “conscious I”, but both spheres are equally mine.

An advantage of such a view would be that we could then accept the following simple reductive account of consciousness:

• A conscious state just is a mental state.

Of course, this is only a partial reduction: the conscious is reduced to the mental. I am happy with that, as I doubt that the mental can be reduced to the non-mental. But it would be really cool if the mystery of the conscious could be reduced.

However, the above story still doesn’t fully solve the problem of consciousness. For it replaces the puzzle as to what makes some of my mental states conscious and others unconscious with the puzzle of what makes a plurality of mental states co-conscious, i.e., a part of the same sphere of consciousness. Perhaps this problem is more tractable than the problem of what makes a state conscious was, though?

Scepticism about culpability

I rarely take myself to know that someone is culpable for some particular wrongdoing. There are three main groups of exception:

1. my own wrongdoings, so many of which I know by introspection to be culpable

2. cases where others give me insight into their culpability through their testimony, their expressions of repentance, etc.

3. cases where divine revelation affirms or implies culpability (e.g., Adam and David).

In type 2 cases, I am also not all that confident, because unless I know a lot about the person, I will worry that they are being unfair to themselves.

I am amazed that a number of people have great confidence that various infamous malefactors are culpable for their grave injustices. Maybe they are, but it seems easier to believe in culpability in the case of more minor offenses than greater ones. For the greater the offense, the further the departure from rationality, and hence the more reason there is to worry about something like temporary or permanent insanity or just crazy beliefs.

I don’t doubt that most people culpably do many bad things, and even that most people on some occasion culpably do something really bad. But I am sceptical of my ability to know which of the really bad things people do they are culpable for.

The difficulty with all this is how it intersects with the penal system. Is there maybe a shallower kind of culpability that is easier to determine and that is sufficient for punishment? I don’t know.

Being mistaken about what you believe

Consider:

1. I don’t believe (1).

Add that I am opinionated on what I believe:

2. For each proposition p, I either believe that I believe p or believe that I do not believe p.

Finally, add:

3. My beliefs are closed under entailment.

Now I either believe (1) or not. If I do not believe (1), then I don’t believe that I don’t believe (1), by closure. But thus, by (2), I do believe that I do believe (1). Hence in this case:

4. I am mistaken about what I do or do not believe.

Now suppose I do believe (1). Then I believe that I don’t believe (1), by closure and by what (1) says. So, (4) is still true.

Thus, we have an argument that if I am opinionated on what I believe and my beliefs are closed under entailment, then I am mistaken as to what I believe.

(Again, we need some way of getting God out of this paradox. Maybe the fact that God’s knowledge is non-discursive helps.)

Socratic perfection is impossible

Socrates thought it was important that if you didn't know something, you knew you didn't know it. And he thought that it was important to know what followed from what. Say that an agent is Socratically perfect provided that (a) for every proposition p that she doesn't know, she knows that she doesn't know p, and (b) her knowledge is closed under entailment.

Suppose Sally is Socratically perfect and consider:

1. Sally doesn’t know the proposition expressed by (1).

If Sally knows the proposition expressed by (1), then (1) is true, and so Sally doesn’t know the proposition expressed by (1). Contradiction!

If Sally doesn’t know the proposition expressed by (1), then she knows that she doesn’t know it. But that she doesn’t know the proposition expressed by (1) just is the proposition expressed by (1). So Sally doesn’t know the proposition expressed by (1). So Sally knows the proposition expressed by (1). Contradiction!

So it seems it is impossible to have a Socratically perfect agent.

(Technical note: A careful reader will notice that I never used closure of Sally’s knowledge. That’s because (1) involves dubious self-reference, and to handle that rigorously, one needs to use Goedel’s diagonal lemma, and once one does that, the modified argument will use closure.)

But what about God? After all, God is Socratically perfect, since he knows all truths. Well, in the case of God, knowledge is equivalent to truth, so (1)-type sentences just are liar sentences, and so the problem above just is the liar paradox. Alternately, maybe the above argument works for discursive knowledge, while God’s knowledge is non-discursive.

Epistemic scores and consistency

Scoring rules measure the distance between a credence and the truth value, where true=1 and false=0. You want this distance to be as low as possible.

Here’s a fun paradox. Consider this sentence:

1. At t₁, my credence for (1) is less than 0.1.

(If you want more rigor, use Goedel’s diagonalization lemma to remove the self-reference.) It’s now a moment before t₁, and I am trying to figure out what credence I should assign to (1) at t₁. If I assign a credence less than 0.1, then (1) will be true, and the epistemic distance between 0.1 and 1 will be large on any reasonable scoring rule. So, I should assign a credence greater than or equal to 0.1. In that case, (1) will be false, and I want to minimize the epistemic distance between the credence and 0. I do that by letting the credence be exactly 0.1.

So, I should set my credence to be exactly 0.1 to optimize epistemic score. Suppose, however, that at t₁ I will remember with near-certainty that I was setting my credence to 0.1. Thus, at t₁ I will be in a position to know with near-certainty that my credence for (1) is not less than 0.1, and hence I will have evidence showing with near-certainty that (1) is false. And yet my credence for (1) will be 0.1. Thus, my credential state at t₁ will be probabilistically inconsistent.

Hence, there are times when optimizing epistemic score leads to inconsistency.

There are, of course, theorems on the books that optimizing epistemic score requires consistency. But the theorems do not apply to cases where the truth of the matter depends on your credence, as in (1).

Evidentialism, and self-defeating and self-guaranteeing beliefs

Consider this modified version of William James’ mountaineer case: The mountaineer’s survival depends on his jumping over a crevasse, and the mountaineer knows that he will succeed in jumping over the crevasse if he believes he will succeed, but doesn’t know that he will succeed as he doesn’t know whether he will come to believe that he will succeed.

James used his version of the case to argue that pragmatic reasons can legitimately override lack of epistemic reasons.

But what is interesting to me in my variant is the way it provides a counterexample to evidentialism. Evidentialists say that you epistemically should form your beliefs only on the basis of evidence. But notice that although the belief that he will succeed at the jump needs to be formed in the absence of evidence for its truth, as soon as it is formed, the belief itself becomes its own evidence to the point that it turns into knowledge. The belief is self-guaranteeing. So there seems to be nothing to criticize epistemically about the formation of the belief, even though the formation is independent of evidence. In fact, it seems, there is a good epistemic reason to believe, since by believing the mountaineer increases the stock of his knowledge.

Moreover, we can even make the case be one where the evidence on balance points against the proposition. Perhaps the mountaineer has attempted, in safer circumstances, to get himself to believe that he can make such a jump, and seven times out of ten he has failed at both self-induction of belief, and also at the jump. But in the remaining three times out of ten, he succeeded at both. So, then, the mountaineer has non-conclusive evidence that he won’t manage to believe that he will succeed (and that he won’t succeed). If he comes to believe that he will succeed, he comes to believe this against the evidence—but, still, in doing, he increases his stock of knowledge, since the belief, once believed, is self-guaranteeing.

(This phenomenon of self-guaranteeing belief reminds me of things that Kierkegaard says about faith, where faith itself is a miracle that hence is evidence for its truth.)

Interestingly, we might also be able to construct cases of well-evidenced but self-defeating beliefs. Consider a jeweler who has noticed that she is successful at cutting a diamond if and only if she believes she will be unsuccessful. Her theory is that belief in her success makes her insufficiently careful. Over time, she has learned to suspend judgment in her success, and hence to be successful. But now she reflects on her history, and she finds herself with evidence that he will be successful in cutting the next diamond. Yet if she believes on this evidence, this will render her overconfident, and hence render the belief false!

This is related to the examples in this paper on lying.

So perhaps what the evidentialist needs to say is that you epistemically may believe p if and only if the evidence says that if you believe p, p is true?

"The" natural numbers

Benacerraf famously pointed out that there are infinitely many isomorphic mathematical structures that could equally well be the referrent of “the natural numbers”. Mathematicians are generally not bothered by this underdetermination of the concept of “the natural numbers”, precisely because the different structures are isomorphic.

What is more worrying are the infinitely many elementarily inequivalent mathematical structures that, it seems, could count as “the natural numbers”. (This becomes particularly worrying given that we’ve learned from Goedel that these structures give rise to different notions of provability.)

(I suppose this is a kind of instance of the Kripke-Wittgenstein puzzles.)

In response, here is a start of a story. Those claims about the natural numbers that differ in truth value between models are vague. We can then understand this vagueness epistemically or in some more beefy way.

An attractive second step is to understand it epistemically, and then say that God connects us up with his preferred class of equivalent models of the naturals.

Color-remapping

I’ve often wondered what would happen if one wore color-remapping glasses. Would the brain adapt, and everything would soon start looking normal, much as it does when you wear vision-inverting prisms. Well, it turns out that there is an interesting two-subject (the investigators) study using LCD glasses linked to a camera and did a color-space rotation. They found interesting results, but found that over six days there was no adaptation: stop signs still looked blue, the sky still looked green, and broccoli was red. This is not conclusive, since six days might just not be long enough. I would love to see the results of a longer study.

The philosophical relevance, of course, is to inverted-spectrum thought experiments.

Panrepresentationalism

Panpsychists, as the term is commonly understood, think everything is conscious. An attractive but underexplored view is that everything nonderivatively represents. This was Leibniz's view, I suspect. One can add to this a reduction of experience to representation, but one does not have to.

When God doesn't act for some reason

Here’s an argument for a thesis that pushes one closer to omnirationality.

1. God is responsible for all contingent facts about his will.

2. No one is responsible for anything that isn’t an action (perhaps internal) done for a reason or the result of such an action.

3. If God doesn’t act on a reason R that he could have acted on, that’s a contingent fact about his will.

4. So, if God doesn’t act on a reason R, then either (a) God couldn’t have acted on R, or (b) God’s not acting on R itself has a reason S behind it.