Our sharp existence

This argument is fairly well trodden, but I still have to say that I find it quite compelling:

1. If physicalism is true, then there was no sharp time at which I came into existence.

2. There was a sharp time at which I came into existence.

3. So, physicalism is false.

Why think (1) is true? Well, if physicalism is true, there is nothing more to me than an arrangement of particles. And which exact arrangements count as sufficient for my existence seems quite vague. And why think (2) is true? Well, if there is no sharp time at which I came into existence, then there will be worlds where it is vague whether I ever exist at all. For instance, if it is vague whether I already existed by time t₁, then imagine a world just like ours up to t₁, but where immediately thereafter everything is annihilated. If it is vague whether I existed by time t₁ in our world, then it that world it will be vague whether I ever exist. But it can’t be vague whether I ever exist—vague existence is an impossibility.

Objection 1: There are many entities very much like me, each of which comes into existence at a sharp time, sharing most of their particles, and I am one of them. None of these entities is privileged, but as it happens I am only one of them. The entities differ in fine details of persistence and existence conditions.

Response: If none are privileged, then all these entities are persons. And so in my armchair there are many persons, and likewise wherever any human being is, there are many persons. Now, notice that there is more room for such “slight variation” when an individual is physically larger (i.e., has more particles). So it follows from the view that where there is a larger person, there are more persons. All the persons co-located with me have presumably the same experiences and the same rights (since none are privileged). So it follows that if you have a choice between benefiting a larger and a smaller person, you should benefit the larger. This sizeism is clearly absurd.

Objection 2: A Markosian-style view on which there are brute facts about composition can say that there is only entity where I am, and the other clouds of particles do not compose an entity.

Response: Yes, but while that counts as materialism, it doesn’t count as physicalism. It adds to the fundamental ontology something beyond what physical science talks about, namely entities that are brutely composed. Moreover, presumably persons are causes. So the story adds to physicalism additional causes.

Objection 3: Nobody can say that there was a sharp time at which I came into existence.

Response: It’s easy for the dualist to say it. I come into existence when my soul comes into existence, joined to some bit of matter. There is no vagueness as to when this happens, but of course the details are not empirically knowable.

A sharp world

Here are one way of believing in a totally sharp world:

1. Epistemicism: All meaningful sentences have a definite truth value, but sometimes it’s not accessible to us.

This has the implausible consequence that there is a fact of the matter whether, say, four rocks can make a heap, or about exactly how much money one needs to have to be filthy rich.

A way of escaping such consequences is:

2. Second-level epistemicism: For any meaningful sentence s, it is definitely true that s is definitely true, or s is definitely false, or s is definitely vague.

While this allows us to save the common-sense idea that there are people who are vaguely filthy rich, it still has the somewhat implausible consequence that it is always definite whether someone is definitely filthy rich, vaguely filthy rich, or definitely not filthy rich. I think it is easier to bite the bullet here. For while we can expect our intuitions about the meaning of first-order claims like “Sally is filthy rich” to be pretty reliable, our intuitions about the meaning of claims like “It’s vague that Sally is filthy rich” are less likely to be reliable.

Still, we can do justice to the second-level vagueness intuition by going for one of these:

3. nth level epistemicism: For any meaningful sentence s, and any sequence of D₁, ..., D_n − 1 of vagueness operators (from among "vaguely", "definitely" and "definitely not"), the sentence D₁...D_n − 1s is definitely true or definitely false.

(Say, with n = 3.)

4. Bounded-level epistemicism: for some finite n we have nth level epistemicism.

5. Finite-level epistemicism: For any meaningful sentence s, there is a finite n such that for any sequence of D₁, ..., D_n − 1 of vagueness operators, the sentence D₁...D_n − 1s is definitely true or definitely false.

The difference between finite-level and bounded-level epistemicism is that the finite-level option allows the level at which vagueness disapppears to vary from sentence to sentence, while on the bounded-level option, there is some level at which it always disappears.

I suspect that if we have finite-level epistemicism, then we have bounded-level epistemicism. For my feeling is that the level of vagueness of a sentence is definitely by something like the maximum level of vagueness of its basic predicates and names. Since there are only finitely many basic predicates and names in our languages, if each predicate and name has a finite level of vagueness, there will be a maximal finite level of vagueness for all our basic predicates and names, and hence for all our sentences. But I am not completely confident about this hand-wavy argument.

In any case, I find pretty plausible that we have bounded-level epistemicism for our languages, but we can extend the level if we so wish by careful stipulation of new predicates. And bounded-level epistemicism is, I think, enough to do justice to the idea that our world is really sharp.

Partially defined predicates

Is cutting one head off a two-headed person a case of beheading?

Examples like this are normally used as illustrations of vagueness. It’s natural to think of cases like this as ones where we have a predicate defined over a domain and being applied outside it. Thus, “is being beheaded” is defined over n-headed animals that are being deprived of all heads or of no heads.

I don’t like vagueness. So let’s put aside the vagueness option. What else can we say?

First, we could say that somehow there are deep facts about the language and/or the world that determine the extension of the predicate outside of the domain where we thought we had defined it. Thus, perhaps, n-headed people are beheaded when all heads are cut off, or when one head is cut off, or when the number of heads cut off is sufficient to kill. But I would rather not suppose a slew of facts about what words mean that are rather mysterious.

Second, we could deny that sentences using predicates outside of their domain lack truth value. But that leads to a non-classical logic. Let’s put that aside.

I want to consider two other options. The first, and simplest, is to take the predicates to never apply outside of their domain of definition. Thus,

1. False: Cutting one head off Dikefalos (who is two headed) is a beheading.

2. True: Cutting one head off Dikefalos is not a beheading

3. False: Cutting one head off Dikefalos is a non-beheading.

4. True: Cutting one head off Dikefalos is not a non-beheading.

(Since non-beheading is defined over the same domain as beheading). If a pre-scientific English-speaking people never encountered whales, then in their language:

5. False: Whales are fish.

6. True: Whales are not fish.

7. False: Whales are non-fish.

8. True: Whales are not non-fish.

The second approach is a way modeled after Russell’s account of definite descriptors: A sentence using a predicate includes the claim that the predicate is being used in its domain of definition and, thus, all of the eight sentences exhibited above are false.

I don’t like the Russellian way, because it is difficult to see how to naturally extend it to cases where the predicate is applied to a variable in the scope of a quantifier. On the other hand, the approach of taking the undefined predicates to be false is very straightforward:

9. False: Every marine mammal is a fish.

10: False: Every marine mammal is a non-fish.

This leads to a “very strict and nitpicky” way of taking language. I kind of like it.

The vagueness of a prioricity

Let L be the following property of a positive integer:

• being large or being greater than the number of carbon atoms in a water molecule.

Necessarily, every positive integer has L. But notice that 1 has L a posteriori, while 10¹⁰⁰ has L a priori. For 1 is not large, and it has L because it is greater than the number of carbon atoms in a water molecule, but the latter is an a posteriori fact. On the other hand, it’s a priori that 10¹⁰⁰ is large.

If n is a positive number such that it is vague whether it is large (e.g., maybe n = 50), it will be vague whether the fact that n has L is a priori or a posteriori. For the largeness of a number is, I assume, an a priori matter, and so it will be vaguely true that it is a priori true that n is large, and hence that n has L.

Hyperintensional vagueness

“Water” and “H₂O” don’t mean the same thing in ordinary English: it is not a priori that water is H₂O. But I suspect that when a chemist uses the word “water” in the right kind of professional context, they use it synonymously with “H₂O”. Suppose this is right. But what if the chemist uses the word with fellow chemists in an “ordinary” way, telling a colleague that the tea water has boiled?

Here is a possibility: we then have a case of merely hyperintensional vagueness. In cases of merely hyperintensional vagueness, there is vagueness as to what an utterance means, but this vagueness has no effect on truth value.

I suspect that hyperintensional vagueness is a common phenomenon. Likely some people use “triangle” to mean a polygon with three angles (as the etymology indicates) and some use it to mean a polygon with three sides. (We can capture the difference by noting that to the latter group it is trivial that triangles have three sides while for the former it is a not entirely trivial theorem.) But consider a child who inherits the word “triangle” from two parents, one of whom uses it in the angle way and the other uses it in a side way. This is surely not an unusual phenomenon: much of the semantics of our language is inherited from users around us, and these users often have hyperintensional (or worse!) differences in meaning.

Semantic determinacy and indeterminacy

There are arguments that our language is paradoxically indeterminate. For instance, Wittgenstein-Kripke arguments for underdetermination of rules by cases, Quine’s indeterminacy of translation arguments, or Putnam’s model-theoretic arguments.

There are also arguments that our language is paradoxically determinate. First order logic shows that there is a smallest number of grains of sand that’s still a heap.

In other words, there are cases where we want determinacy, and we find indeterminacy threatening, and cases where we want indeterminacy, and we find determinacy puzzling. I wonder if there is any relevant difference between these cases other than the fact that we have different intuitions about them.

If we are to go with our intuitions, we need to bite the bullet on, or refute, both sets of arguments, in their respective cases. But if we embrace determinacy everywhere or embrace indeterminacy everywhere, then it’s neater: we only need to bite the bullet on, or refute, one family of arguments.

I find embracing determinacy everywhere rather attractive.

Vagueness and moral obligation

It sure seems like there is vagueness in moral obligation. For instance, torture of the innocent is always wrong, making an innocent person’s life mildly unpleasant for a good cause is not always wrong, and in between we can run a Sorites sequence.

What view could a moral realist have about this? Here are four standard things that people say about a vague term “ϕ”.

1. Error theory: nothing is or could be ϕ; or maybe “ϕ” is nonsense.

2. Non-classical logic: there are cases where attributions of “ϕ” are neither true nor false.

3. Supervaluationism: there are a lot of decent candidates for the meaning of “ϕ”, and no one of them is the meaning.

4. Standard epistemicism: there are a lot of decent candidates for the meaning of “$”, and one of them is the meaning, but we don’t know which one, because we don’t know the true semantic theory and the details of our linguistic usage.

If “ϕ” is “moral obligation”, and we maintain moral realism, then (1) is out. I think (3) and (4) are only possible options if we have a watered-down moral realism. For on a robust moral realism, moral obligations really central to our lives, and nothing else could play the kind of central role in our lives that they do. On a robust moral realism, moral obligation is not one thing among many that just as well or almost as well fit our linguistic usage. Here is another way to put the point. On both (3) and (4), the question of what exact content “ϕ” has is a merely verbal question, like the question of how much hair someone can have and still be bald: we could decide to use “bald” differently, with no loss. But questions about moral obligation are not merely verbal in this way.

This means that given robust moral realism, of the standard views of vagueness all we have available is non-classical logic. But non-classical logic is just illogical (thumps table, hard)! :-)

So we need something else. If we deny (1)-(3), we have to say that ultimately “moral obligation” is sharp, but of course we can’t help but admit that there are Sorites sequences and we can’t tell where moral obligation begins and ends in them. But we cannot explain our ignorance in the semantic way of standard epistemicism. What we need is something like epistemicism, but where moral obligation facts are uniquely distinguished from other facts—they have this central overriding role in our lives—and yet there are moral facts that are likely beyond human ken. One might want to call this fifth view “non-standard epistemicism about vagueness” or “denial of vagueness”—whether we call it one or the other may just be a verbal question. :-)

In any case, I find it quite interesting that to save robust moral realism, we need either non-classical logic or something that we might call “denial of vagueness”.

Against digital phenomenology

Suppose a digital computer can have phenomenal states in virtue of its computational states. Now, in a digital computer, many possible physical states can realize one computational state. Typically, removing a single atom from a computer will not change the computational state, so both the physical state with the atom and the one without the atom realize the same computational state, and in particular they both have the same precise phenomenal state.

Now suppose a digital computer has a maximally precise phenomenal state M. We can suppose there is an atom we can remove that will not change the precise phenomenal state it is in. And then another. And so on. But then eventually we reach a point where any atom we remove will change the precise phenomenal state. For if we could continue arbitrarily long, eventually our computer would have no atoms, and then surely it wouldn’t have a phenomenal state.

So, we get a sequence of physical states, each differing from the previous by a single atom. For a number of initial states in the sequence, we have the phenomenal state M. But then eventually a single atom difference destroys M, replacing it by some other phenomenal state or by no phenomenal state at all.

The point at which M is destroyed cannot be vague. For while it might be vague whether one is seeing blue (rather than, say, purple) or whether one is having a pain (rather than, say, an itch), whether one has the precise phenomenal state M is not subject to vagueness. So there must be a sharp transition. Prior to the transition, we have M, and after it we don’t have M.

The exact physical point at which the transition happens, however, seems like it will have to be implausibly arbitrary.

This line of argument suggests to me that perhaps functionalists should require phenomenal states to depend on analog computational states, so that an arbitrarily small of the underlying physical state can still change the computational state and hence the phenomenal state.

From naturalism to divine command theory by way of vagueness

Plausibly:

1. There is a collection K of atoms forming an early stage of the developmental history of a particular human, where it is vague whether it is morally permissible to disperse the atoms in K.

Pro-choice readers might take K to be the collection of atoms in a late fetus or an early newborn. Pro-life readers might take K to be the collection of atoms at some early point during the fusion of gametes. (Note that (1) does not say that the atoms in K form a human.)

It is widely held that:

2. Vague matters always depend on semantic plasticity.

But:

3. Whether it is permissible to disperse the atoms in K does not depend on any semantic questions.

Claims (1)–(3) look incompatible. So we should abandon at least one of them. Personally, as a pro-life Aristotelian dualist, I am happy to abandon (1)—whether it is permissible to disperse the Ks depends on whether there is a human form uniting the Ks, and that’s not a vague matter. But apart from a Markosian-style brute fact view of composition—a view that seems implausible—it seems hard for a naturalist to deny (1).

Claim (2) is pretty plausible.

This means that if we accept (1), we should deny (3). But how to deny (3)? How could the question of whether it is permissible to disperse the atoms at some developmental stage depend on semantic questions? Surely it is a totally abhorrent idea that whether it is acceptable to disperse the atoms in K is determined by our linguistic performances?

Indeed! But it need not depend on the semantics of our linguistic performances. There is exactly one major moral theory that make that allows for a reasonable denial of (3): divine command theory. Commands are linguistic performances. Whether some act violates a command is in part a linguistic matter. Given divine command theory, semantic plasticity in the terms of the commands can ground vagueness in the obligations constituted by the commands. If your commander says: “Shave until you’re bald”, your obligation seems to be vague precisely because of the vagueness of language.

And that the semantics of God’s linguistic performances matters to the right to life of K is far from as objectionable as some claim on which our linguistic performances have the determining role. (Though our linguistic performances have some role to play, since they may help define the words that God is using if God speaks our language.) Thus, we can imagine that God says: “Thou shalt not murder.” But perhaps “murder” is vague. And the vagueness in this word then translates to vagueness as to whether the dispersal of the atoms in K is permissible.

If the above is right, then we have an argument for a very surprising thesis:

4. If physicalism (about us) is true, then divine command theory is the correct moral theory, and hence God exists.

This is yet another in a series of observations that I’ve been making over the years, that theistic naturalism has resources that its atheistic cousin lacks.

All that said, I think physicalism (even about us) is false, and divine command theory is not the correct moral theory. Though I do think God exists.

Vagueness about moral obligation

There is a single normative property that is normatively above all others, that overrides all others: moral obligation.

I think the above intuition entails that there cannot be any non-epistemic vagueness about moral obligation.

There are two main non-epistemic approaches to vagueness: deviant logic and supervaluationism. Deviant logic is logically unacceptable. :-) That leaves supervaluationism. But on supervaluationism, there would have to be many acceptable precisifications of our concept of moral obligation. Each such precisification would presumably be a normative property. But only a precisification that was normatively above all others could be an acceptable precisification of our concept of moral obligation. And there can only be one precisification above all others. So there can only be one acceptable precisification of moral obligation.

The above argument is too quick. The supervaluationist can say that in the claim “Moral obligation is above all other normative properties”, we have another candidates for vagueness: “above” (or “overrides”). Then we need to engage in coordinated precisification of “moral obligation”, as well as “above”. For each coordinated precisification, the aboveness claim will be true: “Moral obligation_i is above_i all other normative properties.”

I think, however, that once we allow for a variety of precisifications of “above”, we betray the intuition behind the aboveness thesis. That in some sense moral obligation is above personal convenience is not the bold and bracing intuition of the overridingness of morality. Thus, I think that if we are to be faithful to that intuition, we cannot allow for non-epistemic vagueness about moral obligation.

And this, in turn, greatly limits how much non-epistemic vagueness there can be. For instance, if there is no vagueness about permissibility, then it cannot be vague whether something is a person, since vagueness about personhood leads to vagueness about moral obligations of respect. Indeed, it is not clear that there can be any non-epistemic vagueness if there is no non-epistemic vagueness about moral obligation. Suppose I promise to become bald, and I have a small amount of hair. Then I am non-bald if and only if I am obligated to remove some hair.

Unicorns and error theory

Kripke famously argued that unicorns cannot exist. For “unicorn” would have to refer to a natural kind. But there are multiple non-actual natural kinds to which “unicorn” could equally well refer, since it’s easy to imagine worlds w₁ and w₂ in each of which there is a natural kind of animal that matches the paradigmatic descriptions of unicorns in our fiction, but where the single-horned equines of w₁ are a different natural kind (at the relevant taxonomic level) from the single-horned equines of w₂. The proposition p expressed by “There are unicorns” is true in one of the worlds but not the other, or in both, or in neither. Symmetry rules out its being true in one but not the other. It can’t be true in both, because then “unicorn” would refer to two natural kinds (at the relevant taxonomic level), while it arguably refers to one (at least if we index it to a sufficiently specific body of fictional work). So, the proposition must be true in neither world, and by the same token, there will be no world where it’s true.

It seems to me, however, that rather than saying that the proposition expressed by “There are unicorns” is impossible, we should say that “There are unicorns” fails to express a proposition. Here’s why. We could imagine Rowling enriching the Harry Potter stories by introducing a new species of animals, the monokeratines. Suppose she never gives us enough detail to tell the two species apart, so all the descriptions of “unicorns” in her stories apply to “monokeratines” and vice versa, but she is clear that they are different species (perhaps the story hinges on one of them being an endangered species and the other not).

Now, if “There are unicorns” in these (hypothetical) stories expresses a proposition, so does “There are monokeratines”. But if they express propositions, they express different propositions (neither entails the other, for instance). Thus, suppose “There are unicorns” expresses p while “There are monokeratines” expresses q. But no reason can be given for why it’s not the other way around—why “There are unicorns” doesn’t express q while “There are monokeratines” expresses p. In fact, the exact same reasoning why Kripke rejected the hypothesis that “There are unicorns” is true in one of w₁ and w₂ but not in the other applies here. Thus, we should reject the claim that either sentence expresses a proposition.

But if we do that, then we should likewise reject the claim that in the actual world, where Rowling doesn’t talk about monokeratines, “There are unicorns” expresses p (say). For it could equally well express q.

Maybe.

But maybe there is another way. One could say that “There are unicorns” is vague, and handle the vagueness in a supervaluationist way. There are infinitely many species u such that “There are unicorns” can be taken to be precisified into expressing the proposition that there are us. Thus, there is no one proposition expressed by the sentence, but there are infinitely many propositions for each of which it is vaguely true that the sentence expresses it.

This might be a good response to my old argument that error theorists should say that “Murder is wrong” is nonsense. Maybe error theorists can say that “Murder is wrong” has infinitely many precisifications, but each one is false, just as “There are unicorns” has infinitely many precisifications, but each one is false.

This suggests a view of fiction on which claims about fictional entities always suffer from vagueness.

An interesting thing is that on this approach, we need to distinguish between in-story and out-of-story vagueness. Suppose a Rowling has a character say “There are unicorns.” In-story, that statement is not vague. I.e., according to the story there is a specific species to which the word “unicorn” as spoken by the character definitely refers. But out-of-story, we have vagueness: there are infinitely many possible species the claim could be about.

This suggests that the error theorist who takes the vagueness way out is not home free. For it is a part of our usage of “(morally) wrong” that it refers fairly unambiguously to one important property. But the error theorist claims vagueness. If the statements about wrongness were made in a story, then the error theorist could handle this by distinguishing in-story and out-of-story vagueness. But this distinction is not available here.

A similar problem occurs for a real-world person who claims that there are unicorns. Maybe one could say that the person intends in saying “There are unicorns” to express a single specific proposition, but fails, and vaguely expresses each of an infinity of propositions, all of them false. If so, then a similar move would be available to the error theorist. But I am sceptical of this move. I wonder if it’s not better to just say that “There are unicorns” as said by someone who intended to express an existential claim about a single definite species is nonsense, but there is a neighboring sentence, such as “There is an extant species of single-horned equines”, that makes sense and is true.

A Biblical argument for epistemicism

1. If God knows the exact number of hairs we have on our head, then there is a definite number of hairs we have on our head.

2. If there is a definite number of hairs we have on our head, vagueness is at most epistemic.

3. God knows the exact number of hairs we have on our head. (Luke 12:7)

4. So, vagueness is at most epistemic.

Premise 2 is based on observing that the number of hairs we have on our heads involves similar kinds of vagueness to more paradigmatic cases of vagueness. Think here about these questions:

• What’s the cut-off between hairs on the head and hairs on the upper neck?

• How much keratin needs to come out of a hair follicle before that keratin counts as a hair?

• How far must the molecules of a hair separate from the molecules of the skin before the hair counts as no longer attached?

One might worry that Premise 3 relies on biblical data too literalistically. Jesus is emphasizing the impressiveness of God’s knowledge. Suppose that instead of God knowing the exact number of hairs on my head, God knew the exact vagueness profile for the hairs on my head. That would be even more impressive. I see some force in this objection, but it implies that epistemicism holds at the level of vagueness profiles, and it seems (but perhaps isn’t?) ad hoc to go for epistemicism there rather than everywhere.

On reflection, I think premise 1 might be the most questionable premise. Perhaps God’s knowledge definitely matches the number of hairs: for every natural number n, it’s definitely true that: God believes I have n hairs if and only if I have n hairs, but there is no natural number n such that God definitely believes I have n hairs. In other words, the vagueness profile concerning God’s beliefs exactly matches the vagueness profile in reality. I am sceptical of this solution. It doesn’t feel like knowledge to me if it’s got this sort of vagueness to it.

More on the problem of short pains

Consider these two very plausible theses:

1. Whether I have pain at time t does not depend on any future facts.

2. There is a length of time δt such that you cannot have a pain lasting no more than δt, but you can have a pain lasting 4δt.

Now imagine that I feel a pain that lasts from t₀ to t₂ = t₀ + 4δt. Let t₁ = t₀ + (1/2)δt. Suppose it is now t₁. Then I am in pain. But now I claim that this counterfactual is true:

3. Were it to be the case that I was to be annihilated at t₀ + (3/4)δt, I wouldn’t have felt any pain now.

Why? For if I were so annihilated, my pain could only have lasted (3/4)δt, which is too short for a pain by (2).

But by (3), whether I feel pain now depends on whether I will shortly be annihilated, contrary to (1).

Hence, we need to reject one of (1) and (2).

It is hard to reject (2). After all, imagine the sequence of times: a second, a quarter second, a sixteenth of a second, …, 2⁻⁴⁰ seconds. Clearly I can feel a pain that lasts a second. The last of these is less than a picosecond, and clearly I can’t feel a pain that short. So somewhere in that sequence I must reach a δt which is too short for a pain, but where 4δt isn’t too short for a pain.

I think denying (1) isn’t as bad as it may seem.

But perhaps a less counterintuitive move is to deny that phenomenal times and physical times are as closely correlated as they intuitively seem. Here is a possible story. Phenomenal times are discrete points while physical times are continuous (or discrete on a much finer timescale). You can feel a pain that is located at exactly one phenomenal time. The spacing between the phenomenal times corresponds to a fairly large (and non-uniform) spacing between physical times, say of the order of magnitude of a millisecond. So, you can feel a pain that is there one millisecond and gone the next, but you may feel it at exactly one point of phenomenal time.

As far as I can tell, it is not possible to run the annihilation argument while keeping careful track of the continuous physical and discrete phenomenal timelines. I guess this is a way of rejecting (2) by making it not make sense.

Here is a third way out of the argument. Imagine that what it is to have a pain at t is to have had some constitutive physical or spiritual process P have lasted some threshold period of time δt. On this view, before P lasted over a period of δt, there was no pain: pain only starts once P has lasted δt. We might now suppose that δt is something like a millisecond. Then it is possible to have a pain that lasts only a picosecond: for that, all we need is the underlying process P to have lasted δt plus a picosecond—and only the last picosecond of that process would have constituted a pain. But we no longer need to make the implausible claim that we can be aware of picosecond-scale stuff. For in paining, our awareness is the awareness of the underlying process P, and that process always needs to have taken something in the millisecond range for it to constitute a paining.

This way out of the argument also has the consequence that it is not possible to have a pain before completing the first δt of one’s existence. Pain is not a momentary property.

The third way out will not appeal to dualists who think phenomenal states are fundamental.

Hyperintensional vagueness

The typical examples of vagueness in the literature are ones where it is vague whether a subject has a property (e.g., vagueness) or whether a statement is true. But there is another kind of vagueness which we might call “hyperintensional vagueness”, which looks like it should be quite widespread. The easiest way to introduce this is in a supervaluationist context: a term has vagueness provided it has more than one precisification. But one possibility here is that all the precisifications of the term are intensionally the same. In that case, we can say that the term is merely hyperintensionally vague.

For instance, the English word “triangle” looks like it’s only hyperintensionally vague. It has two precisifications: a three-sided polygon and a three-angled polygon (the etymology favors the latter, but we cannot rely on etymology for semantics). Since necessarily all and only three-sided polygons are three-angled polygons, the two precisifications are intensionally the same.

Hyeprintensional vagueness doesn’t affect first-order logic or even modal logic, so it doesn’t get much talked about. But it does seem to be an interesting phenomenon that is even harder to get rid of than extensional or even intensional vagueness. Consider the vagueness in “bachelor”: it is extensionally vague whether a man who had his marriage annulled or the Pope is a bachelor. But even after we settle all the intensional vagueness by giving precise truth conditions for “x is a bachelor” such as “x is a never validly married, marriageable man”, there will still be hyperintensionally differing precisifications of “bachelor” such as:

• a marriageable man none of whose past marriages was valid

• a marriageable man none of whose past valid statuses was a marriage

• a human being none of whose past marriages was valid and who is a man.

This makes things even harder for epistemicists who have to uphold a fact of the matter as to the hyperintensionally correct precisification. Moreover, at this point epistemicists cannnot make use of the standard classical logic argument for epistemicism. For while that argument has much force against extensional vagueness, it has no force against hyperintensional vagueness. One could hold that there is no extensional or intensional vagueness but there is hyperintensional vagueness, but that sounds bad to me.

Vagueness and degrees of truth

Consider the non-bivalent logic solution to the problem of vagueness where we assign additional truth values between false and true. If the number of truth values is finite, then we immediately have a regress problem once we ask about the boundaries for the assignment of the finitely many truth values: for instance, if the truth values are False, 0.25, 0.50, 0.75 and True, then we will be able to ask where the boundary between “x is bald” having truth value 0.50 and having truth value 0.75 lies.

So, the number of truth values had better be infinite. But it seems to be worse than that. It seems there cannot be a set of truth values. Here is why. If x has any less hair than y, but neither is definitely bald or non-bald, then “x is bald” is more true than “y is bald”. But how much hair one has is quantified in our world with real numbers, say real numbers measuring something like a ratio between the volume of hair and the surface area of the scalp (the actual details will be horribly messy). But there will presumably be possible worlds with finer-grained distances than we have—distances measured using various hyperreals. Supposing that Alice is vaguely bald, there will be possible people y who are infinitesimally more or less bald than Alice. And as there is no set of all possible infinitesimals (because there is no set of all systems of hyperreal), there won’t be a set of all truth values.

Moreover, there will be vagueness as to comparisons between truth values. One way to be less bald is to have more hairs. Another way is to have longer hairs. And another is to have thicker hairs. And another is to have a more wrinkly scalp. Unless one adopts epistemicism, there are going to be many cases where it will be vague whether “x is bald” is more or less or equally or incommensurably true as “y is bald”.

We started with a simple problem: it is vague what is and isn’t bald. And the non-bivalent solution led us to a vast multiplication of such problems, and a vast system of truth values that cannot be contained in a set. This doesn’t seem like the best way to go.

Epistemicism and physicalism

1. There is a precise boundary for the application of “bald”.

2. If there is a precise boundary for the application of “bald”, that boundary is defined by a linguistic rule of infinite complexity.

3. If physicalism is true, then no linguistic rules have infinite complexity.

4. So, physicalism is not true.

The argument for (1) is classical logic. The argument for (2) depends on the many-species considerations at the end of my last post. And if (3) is true, then linguistic rules are defined by our practices, and our practices are finitary in nature.

Objection: We are analog beings, and every analog system has possible states of infinite complexity.

Response 1: Our computational states ignore small differences, so in practice we have only finite complexity.

Response 2: There is a cardinality limit on the complexity of states of analog systems (analog systems can only encode continuum-many states). But there is no cardinality limit on the number of humanoid species with hair, as there are possible such species in worlds whose spacetime is based on systems of hyperreals whose cardinality goes arbitrarily far beyond that of the reals.

The unknowability part of epistemicism about vagueness

One way to present epistemicism is to say that

1. vague concepts have precise boundaries, but

2. it is not possible for us to know these boundaries.

A theist should be suspicious of epistemicism thus formulated. For if there are precise boundaries, God knows them. And if God knows them, he can reveal them to us. So it is at least metaphysically possible for us to know them.

Perhaps the “possible” in (b) should be read as something stronger than metaphysical possibility. But whatever the modality in (b) is, it seems to imply:

3. none of us will ever know these boundaries.

But if epistemicism entails (c), then we don’t know epistemicism to be true. For if there are sharp boundaries, for all we know God will one day reveal them to a pious philosopher who prays really hard for an answer.

I think the best move would be to replace (b) with:

4. it is not possible for us to know these boundaries without reliance on the supernatural.

This is more plausible, but it seems hard to be all that confident about (d). Maybe there is some really elegant semantic theory that has yet to be discovered that yields the boundaries. Or maybe our mind has natural powers beyond those we know.

Let me try, however, to offer a bit of an argument for (d). Let’s imagine what the boundary between bald and non-bald would be like. As a first attempt, we might think it’s like this:

1. Necessarily, x is bald iff x has fewer than n hairs.

But there is no n for which (1) is true. For n would have to be at least two, since it is possible to be bald but have a hair. Now imagine Bill the Bald who has n − 1 hairs, and now imagine that these hairs grow in length until each one is so long that Bill can visibly and fully cover his scalp with them. At that point, Bill wouldn’t be bald, yet he would still have n − 1 hairs. So, the baldness boundary cannot be expressed numerically in terms of the number of hairs.

As a second attempt, we might hope for a total-length criterion.

2. Necessarily, x is bald iff the total length of x’s hairs is less than x centimeters.

But it is possible to have two people with the same total length of hairs, one of whom is bald and the other is not. For the thickness of hairs counts: if one just barely has the requisite total length but freakishly thin hairs, that won’t do. On the other hand, clearly x would have to be at least four centimeters, since a single ordinary hair of four centimeters is not enough to render one non-bald, but one could have a total hair length of four centimeters and yet be non-bald, if one has four hairs, each one centimeter long and 10 centimeters in diameter, covering one’s scalp with a thick keratinous layer.

So, we really should be measuring total volume, not length. But there are other problems. Shape probably matters. Suppose Helga has a single hair, of normal diameter, but it is freakishly rigid and long, long enough to provide the requisite volume, but immovably sticking up away from the scalp and providing no coverage. Moreover, whatever we are measuring has to be relative to the size of the scalp. A baby needs less hair to cease to be bald than an adult. But it’s not just relative to the size of the scalp, but also the shape of the scalp. If one has a very large surface area of scalp but that is solely due to many tiny wrinkles, one doesn’t need an amount of hair proportional to that large surface area. To a first approximation, what matters is the surface area of the upper part of the convex hull of one’s scalp. But even that’s not right if we imagine a scalp that has very large wrinkles.

So, in fact, we have good reason to think the real boundary wouldn’t be simply numerical. It would involve some function of hair shape, volume and rigidity, as well as of scalp shape and size. And if we think about cases, we realize that it will be a very complex function, and we are nowhere close to being able to state the function. Moreover, to be honest, there are likely to be other variables that matter.

At this point, we start to see the immense amount of complexity that would be involved in any plausible statement of the precise boundary of baldness, and that gives us positive reason to doubt that short of something supernatural we could know where the boundary lies.

But suppose our confidence has not yet been quashed. We still have other serious problems. What we are looking for is a perfectly precise necessary and sufficient condition for someone to be bald. In that definition, we cannot use other vague terms. That would be cheating. What the epistemicist meant by saying that we don’t know where the boundaries lie was that we do not know any transparently precise statements of the boundaries, statements not involving other vague terms. But “hair” itself is a vague term. Both hair and horns are made of keratin. Where does the boundary between hair and horns lie? Similarly, “scalp” is vague, too. And it’s only the volume of the part of the hairs sticking out of the scalp that counts—the size of the root is irrelevant. But “sticking out” is vague, as is obvious when we Google for microscopic photography of scalps. And which particles are in the hair or in the scalp is going to be vague. Next, any volume and surface area measurements suffer from vagueness even if we fix the particles, because for quantum reasons particles will have spread out wavefunctions. And then Relativity Theory comes in: volume and surface area depend on reference frame, and so we need a fully precise definition of the relevant reference frame.

Once we see all the complexity needed in giving a transparently precise statement of the boundary of baldness, it becomes very plausible that we can’t know it by natural means, just as it is very plausible that no human can know the first million digits of π by natural means.

And things get even worse. For humans are not the only things that can be bald. Klingons can be bald, too. Probably, though, only humanoid things are bald in the same sense of the word, but even when restricted to humanoid things, a precise statement of the boundary of baldness will have to apply to beings from an infinite number of possible species. And the norms of baldness will clearly be species-relative. Not to mention the difficulty of defining what hair and scalp are, once we are dealing with beings whose biochemistry is different from ours. It is now starting to look like a transparently precise statement of the boundary of baldness might actually have infinite complexity.

An argument that the moment of death is at most epistemically vague

Assume vagueness is not epistemic. This seems a safe statement:

1. If it is vaguely true that the world contains severe pain, then definitely the world contains pain.

But now take the common philosophical view that the moment of death is vague, except in the case of instant annihilation and the like. The following story seems logically possible:

2. Rover the dog definitely dies in severe pain, in the sense that it is definitely true that he is in severe pain for the last hours of his life all the way until death, which comes from his owner humanely putting him out of his misery. The moment of death is, however, vague. And definitely nothing other than Rover feels any pain that day, whether vaguely or definitely.

Suppose that t₁ is a time when it is vague whether Rover is still alive or already dead. Then:

3. Definitely, if Rover is alive at t₁, he is in severe pain at t₁. (By 2)

4. Definitely, if Rover is not alive at t₁, he is not in severe pain at t₁. (Uncontroversial)

5. It is vague whether Rover is alive at t₁. (By 2)

6. Therefore, it is vague whether Rover is in severe pain at t₁. (By 3-5)

7. Therefore, it is vague whether the world contains severe pain at t₁. (By 2 and 6, as 2 says that Rover is definitely the only candidate for pain)

8. Therefore, definitely the world contains pain at t₁. (By 1 and 7)

9. Therefore, definitely Rover is in pain at t₁. (By 2 and 8, as before)

10. Therefore, definitely Rover is alive at t₁. (Contradiction to 5!)

So, we cannot accept story 2. Therefore, if principle 1 is true, it is not possible for something with a vague moment of death to definitely die in severe pain, with death definitely being the only respite.

In other words, it is impossible for vagueness in the moment of death and vagueness in the cessation of severe pain to align perfectly. In real life, of course, they probably don’t align perfectly: unconsciousness may precede death, and it may be vague whether it does so or not. But it still seems possible for them to align perfectly, and to do so in a case where the moment of death is vague—assuming, of course, that moments of death are the sort of thing that can be vague. (For a special case of this argument, assume functionalism. We can imagine a being of such a sort that the same functioning constitutes it as existent as constitutes it as conscious, and then vagueness in what counts as functioning will translate into perfectly correlated vagueness in the moment of death and the cessation of severe pain.)

The conclusion I’d like to draw from this argument is that moments of death are not the sort of thing that can be non-epistemically vague.

Note that 1 is not plausible on an epistemic account of vagueness. For the intuition behind 1 depends on the idea that vague cases are borderline cases, and a borderline case of severe pain will be a definite case of pain, just as a borderline case of extreme tallness will be a definite case of tallness. But if vagueness is epistemic, then vague cases aren't borderline cases: they are just cases we can't judge about. And there is nothing absurd about the idea that we might not be able to judge whether there is severe pain happening and not able to judge whether there is any pain happening either.

A measure of sincerity

On a supervaluationist view of vagueness, a sentence such as “Bob is bald” corresponds to a large number of perfectly precise propositions, and is true (false) if and only if all of these propositions are true (false). This is plausible as far as it goes. But it seems to me to be very natural to add to this a story about degrees of truth. If Bob has one hair, and it’s 1 cm long, then “Bob is bald” is nearly true, even though some precisifications of “Bob is bald” (e.g., that Bob has no hairs at all, or that his total hair length is less than 0.1 cm) are false. Intuitively, the more precisifications are true, the truer the vague statement:

1. The degree of truth of a vague statement is the proportion of precisifications that are true.

But for technical reasons, (1) doesn’t work. First, there are infinitely many precisifications of “Bob is bald”, and most of the time the proportion of precisifications that are true will be ∞/∞. Moreover, not all precisifications are equally good. Let’s suppose we somehow reduce the precisifications to a finite number. Still, let’s ask this question: If Bob is an alligator is Bob bald? This seems vague, even though the precisifications of “Bob is bald” that require Bob to be the sort of thing that has hair seem rather better. But for any precisification that requires Bob to be a hairsute kind of thing, there is one that does not. And so if Bob is an alligator, he is bald according to exactly half of the precisifications, and hence by (1) it would be half-true that he is bald. And that seems too much: if Bob is an alligator, he is closer to being non-bald than bald.

A better approach seems to me to be this. A language assigns to each sentence s a set of precisifications and a measure m_s on this set with total measure 1 (i.e., technically a probability measure, but it does not represent chances or credences). The degree of truth of a sentence, then, is the measure of the subset of precisifications that are actually true.

Suppose now that we add to our story a probability measure P representing credences. Then we can form the interesting quantity E_P(m_s) where E_P is the expected value with respect to P. If s is non-vague, then E_P(m_s) is just our credence for s. Then E_P(m_s) is an interesting kind of “sincerity measure” (though it may not be a measure in the mathematical sense) that combines both how true a statement is and how sure we are of it. When E_P(m_s) is close to 1, then it is likely that s is nearly true, and when it is close to 0, then it is likely that s is nearly false. But when it is close to 1/2, there are lots of possibilities. Perhaps, s is nearly certain to be half-true, or maybe s is either nearly true or nearly false with probabilities close to 1/2, and so on.

This is not unlikely worked out, or refuted, in the literature. But it was fun to think about while procrastinating grading. Now time to grade.

Presentism and vagueness

If presentism is true, then vagueness about the exact moment of cessation of existence implies vagueness about existence: for if it is vague whether an object has ceased to exist at t, then at time t it was, is or will be vague whether the object exists. But it is plausible that there is vagueness about the exact moment of cessation of existence for typical organisms (horses, trees, etc.). On the other hand, vagueness about existence seems to be a more serious logical problem: it makes unrestricted quantifiers vague.

Of course, the eternalist will have a similar problem with vagueness about existence-at-t. But existence-at-t is not fundamental logical existence on eternalism, so perhaps the problem is less serious.