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.

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.

The sharpness of the Platonic realm

I feel an intellectual pull to a view that also repels me. The view is that all contingent vague truths are grounded in contingent definite truths and necessary vague truths. For instance, that Jim is bald might be grounded in a contingent definite truth about the areal density of hair on his scalp and a necessary vague truth that anyone with that areal density of hair is bald.

On this view, any vague differences between possible worlds are grounded in definite differences between possible worlds.

But the view also repels me. I have the Platonic intuition that the realm of necessary truth should be clean, unchanging, sharp and definite. Plato would be very surprised to think that fuzziness in the physical world is grounded in fuzziness in the Platonic realm.

Epistemicism, of course, nicely reconciles the Platonic intuition about necessary truths with the intellectual pull of the grounding claim. For it is no surprise that there be things in the Platonic realm that are not accessible to us. If vagueness is merely epistemic, then there is no difficulty about vagueness in the Platonic realm.

We are not alone among language users

This argument is valid:

1. All semantic truths are knowable to members of the community of language users.
2. There are semantic truths that are not knowable to human language users.
3. Therefore, there is at least one non-human language user.

There is some reason to accept (1) in light of the conventionality of language. Premise (2) is going to be quite controversial. I justify it by means of a standard argument for epistemicism. Consider Queen Elizabeth II. There are 88 statements of the form:

• Elizabeth was not old at age n but she was old at age n+1

where n ranges from 1 to 88. It's a straightforward matter of classical logic to show that if all 88 statements are false, then:

4. Elizabeth was old at age 1 or Elizabeth is not old at age 89.

But (4) is clearly false: Her Majesty is old now at age 89, and she surely wasn't old at age one. So, at least one of the 88 statements is false. This means that there is a sharp transition from being not old to being old. But it is clear that no matter what we find out about our behavior, biology and other relevant things, we can't know exactly where that transition lies. It seems very plausible that the relevant unknowable fact about the transition is a semantic fact. Hence, (2) is true.

The most plausible candidate for the non-human language user who is capable of knowing such semantic facts is God. God could institute the fundamental semantic facts of human language and thereby know them.

The argument from vagueness

Here's an argument inspired by Plantinga's argument from counterfactuals:

1. The meaning of a word is wholly determined by the decisions of language users.
2. The meaning of "bald" is not wholly determined by the decisions earthly language users.
3. Therefore, there is a non-earthly language user whose decisions at least partly determine the meaning of "bald".

The argument for (2) is this:

4. In any hypothetical sequence to whose last member "bald" does not apply and to whose first it does, there is a transition point in the sequence, i.e., a member to whom "bald" applies but to whose successor it does not.
5. The points in a hypothetical sequence at which "bald" does or does not apply are wholly determined by the meaning of "bald".
6. There are hypothetical sequences where the decisions of earthly language users do not determine the transition point.
7. So, (2) is true.

The argument for (6) is simply to exhibit a series of people, with someone completely bereft of hair on one end, and someone with a full head of hair on the other, with very slight transitions. Clearly our decisions and those of our ancestors do not determine where the transition point is. Claim (5) seems very plausible.

That leaves (4). But that's a matter of logic for any fixed sequence, as a standard argument for epistemicism points out. For suppose there is no transition point. Then:

• not ("bald" applies to x_n and "bald" does not apply to x_n+1)

is true for n=1,...,N where N is the number of items in the series. Given the fact that "bald" applies to x₁, we can then conclude by classical logic that "bald" applies to x₂, and to x₃, and so on up to x_N, contrary to assumption.

And the best candidate for the non-earthly language user is God. For any finite language user, say an alien who gave us language, would be in the same boat: its decisions would be insufficient to determine all meaning.