## Wednesday, May 13, 2020

### 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.

Avraham said...

I heard there are logics with many values of truth.

William said...

I wonder if this could be turned into support for your theory that there are no actual infinite causal chains, since the process of serially deciding an infinite series of gradations of persons as bald/not bald would require an infinite series of actions (though perhaps not if done in parallel?)

Martin Cooke said...

This simple problem does have a simple solution though: add just one non-classical logical description to cover, more or less well, all the cases in between true and false, namely:

The imprecision of the description is deliberate. Your vast system resulted from an attempt to keep the precision of classical logic in a situation where it had failed to apply. To apply classical logic, you need precise descriptions to begin with, such as the precise number of hairs on the head.

If you are working with words like "bald," then you need to have a population all of whom, in the universe of discourse, will be either bald or else not bald. When you want to have a bigger universe of discourse, and you want to keep precision, you have to use something other than "bald" in your descriptions.

It is that simple, really. If you try to have your cake and eat it, then you will end up being disappointed, one way or another. For more details see my blog...

Alexander R Pruss said...

Well, you still generate something *like* an infinite number of truth values once you nest.

For there will be cases where it's true that it's in-between that it's bald, and there will be cases closer to the boundary where it's in-between that it's in-between that it's bald. So you will have infinitely many quasi-truth values like:
... true that true that true
... between that between that true
... between that true that between
... true that true that false
and so on.

And then even once you have the infinite sequence, you can ask whether the infinite sequence's application is true, between or false.