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.

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.

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.

Vagueness and grounding

1. If p is a precification of q and p is true, then p grounds q.
2. If q is vaguely true, then q has a true precification.
3. So, every proposition that is vaguely true is grounded.

If we could add the thesis that every grounded proposition is non-fundamental (which in another post I argued against), we could conclude that all vaguely true propositions are non-fundamental.  But even without that thesis, groundedness is evidence of non-fundamentality.  So vagueness is evidence of non-fundamentality.

The possibility of unicorns

Kripke argued that it is not possible for there to be unicorns. For "unicorn" is a natural kind term. But there are many possible natural kinds of single-horned equines that match our unicorn stories, and there is no possible natural kind that has significantly more right to count as the kind unicorn. So none of them count, and no possible world contains unicorns.

But there is another approach, through vagueness and supervaluationism. Let's say that the term "unicorn" is vague. It can be precisified as a full description of any one of the possible natural kinds of single-horned equines that match our unicorn stories.

Now consider the sentence that Kripke is unwilling to assert but which seems intuitively correct:

1. Possibly there is a unicorn.

We get to say (1) if we can correctly affirm:

2. Definitely, possibly there is a unicorn.

And we certainly do get to say this. For (2) holds if and only if:

3. For every precification U of "unicorn", possibly there is a U.

And assuming that all the precifications of "unicorn" are natural kinds that are possibly instantiated (we don't allow as a precification of "unicorn" something whose eyes are square circles, etc.), (3) is true. And so even if there is no world at which definitely there is a unicorn (though there might be—maybe that world is very rich and contains animals that fall under all possible precifications of "unicorn"; Blake McAllister suggests a multiverse), it is definite that there is a world at which there is a unicorn.