Tuesday, June 9, 2020

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.


Murali said...

I'm not sure what you mean by hyperintensional vagueness. All three definitions of bachelor seem exactly synonymous to me. Could you explain why they are different (apart from using slightly different words?)

Alexander R Pruss said...

They are logically equivalent, yes, but they do not *mean* the same thing. It is possible to believe that Smith is a bachelor1 without believing that Smith is a bachelor2, and vice versa.

Murali said...

I'm still not seeing how someone could think that none of Smith's previous marriages were valid and at the same time deny that none of his previous valid statuses was a marriage. After all, if one of Smith's previous valid statuses was a marraige, then one of his previous marriages is valid. To put it more strongly to have a previous valid status which is a marriage just is to have a previous valid marriage. One way this could perhaps happen is if the person did not know the meaning of status, but I doubt that's what you mean.

You might say that referring to the same object is not sufficient for synonymy since Hesperus and Phosphorous have different meanings even though they refer to the same object. Howevr, while Hesperus and Phosphorous are proper names, I don't think everyday words like "status" work the same way as proper names. If they did, two different words could never mean the same thing and that is implausible.

The other way this could happen is if the person was blatantly irrational and simultaneously believed and failed to believe the same proposition.

Alexander R Pruss said...

Indeed, one way this could happen is if one didn't know what "status" meant. One needs to have the concept of a status to grasp propositions involving the second definition but one doesn't need to have the concept of a status to grasp propositions involving the first definition. Consequently, propositions using the two definitions are different.

Or consider how easily students make mistakes in logic. Consider two definitions that even closer than my first two:
- a marriageable man none of whose past marriages was a valid status
- a marriageable man none of whose past valid statuses was a marriage.

Now consider logical renditions of "Jones is a bachelor" using the two definitions:
- Marriageable(jones) & Man(jones) & ∀x(Marriage(x) → ~(Status(x) & Valid(x)))
- Marriageable(jones) & Man(jones) & ∀x((Status(x) & Valid(x)) → ~Marriage(x)).

It seems quite easy to imagine a logic student who accepts one of these claims but not the other. This suggests that they express different propositions.

I suspect that it is relatively rare for two words to mean the same thing, even in terms of truth conditions. It is well-known that color words have different boundaries in different languages. I suspect the same is true of most other words. For instance, I expect that "It's snowing" in Canadian English and "Il neige" in French French will have subtly different truth conditions, with slight differences as one crosses, say, the boundary from snow to sleet.

Andrew Dabrowski said...

I'm confused about your use "intensional" and "extensional". I would have said that the two definitions you give of "triangle" are instensionally different but extensionally equivalent.

Alexander R Pruss said...

extensional difference between F and G: some F is not a G or some G is not an F
intensional difference between F and G: possibly (some F is not a G or some G is not an F)

There is an extensional difference between "male" and "French".
There is no extensional difference between "human less than 10 feet tall" and "human".
There is an intensional difference between "human lass than 10 feet tall" and "human" (you could have a human who is more than 10 feet tall, but we don't actually have one, as far as I know).
There is neither extensional nor intensional difference between "polygon with three sides" and "polygon with three corners", but there is a hyperintensional one.

Andrew Dabrowski said...

Thanks. I was taking "intensional" to refer to cognition.