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.

T-schema and bivalence

Tarski's T-schema says that for any sentence "s":

1. "s" is true if and only if s.

Suppose that "s" is neither true nor false. Then the left hand side of (1) is false, but the right hand side is neither true nor false. It seems to me that a reasonable multivalent logic will not allow an "if and only if" sentence to be true when one side of it is false and the other side is not false. So, it seems that the T-schema requires bivalence.

It's odd that I never noticed this before.

Pushing language too far

Quantum Mechanics has borne much fruit. Is this fruit poisonous? Probably not. But is the total weight of the fruit borne by Quantum Mechanics even or odd when rounded to the nearest pound? Unlike the question whether the fruit is poisonous, the question about whether the weight is even or odd is just silly—it pushes the metaphor too far, in the sense that there is no natural meaning in the metaphor to be assigned to any answer to it.

The formation rules for meaningful metaphorical discourse do not have unrestricted compositionality. While "The fruit borne by Quantum Mechanics is sometimes bitter" is probably meaningful, "The fruit borne by Quantum Mechanics is sometimes elongated" has no meaning unless we choose to assign one to it. The latter sentence takes the metaphor too far.

A sign of metaphor being pushed too far is that classical logic has the appearance of failing. It appears to be neither true that the fruit of Quantum Mechanics is sometimes elongated nor that the fruit of Quantum Mechanics is never elongated, contrary to bivalence. To ask which is the case is to be silly. The same apparent failure of classical logic can occur when we make too involved inferences from a metaphorical claim, for instance when we conclude that Quantum Mechanics is partly made of carbon atoms, because only plants bear fruit and all plants are partly made of carbon atoms.

Here is a different kind of metaphor (I heard this metaphor—though I don't remember if it was identified as a metaphor—in discussion at INPC): The average plumber has 2.3 children. Let's press on. Stipulate, no doubt contrary to fact, that exactly half of all plumbers are male and exactly half of all plumbers are female. So, is the average plumber male? No. Is the average plumber female? No. Is the average plumber human? Certainly (supposing there are no alien plumbers). So, the average plumber is a human who is neither male nor female. Now, maybe there are such rare humans (this is a difficult question about the metaphysics of sex), but since by stipulation none of them are plumbers, surely the average plumber isn't one of them. Wondering about this bit of weirdness is, however, silly. It is taking the metaphor of the average plumber too far. Once we start saying that the average plumber is a human who is neither male nor female, we take our metaphor beyond the narrow region of the space of sentences where it makes sense.

Now, some metaphysicians, including me, think that in an important sense there are no tables or chairs. There are only particles or fields arranged tablewise or chairwise. It is a tough problem for these metaphysicians to defend their own use of ordinary language about tables and chairs—their saying things like "There are ten chairs in the room."

I think our ordinary language about artifacts has some things in common with metaphorical language. Take something like the question of how much of the wood of the table you can replace, and in how large chunks, while maintaining the same table? I think one can have a sense of discomfort with the question. After enough fast replacement of wood, one is tempted both to deny that one has the same table and to deny that one has a distinct table. The question seems to be a matter for our decision—much as it is a question for our decision whether we count last year's average plumber, with his/her/its 2.29 children, as the same individual as this year's average plumber with his/her/its 2.30. In the plumber case, the decision is a decision what to understand identity across time in the metaphor as standing for (maybe by saying that the average plumber is the same last year as this year we want to metaphorically signal that there was no en masse replacement of plumbers). And, I think, the ordinary folk think there is something a bit humorous and unserious about pressing the question whether after the replacement we have the same table, just as they would in the plumber case.

These things suggest that when we ask whether we have the same table, we can be pushing language too far, just as we sometimes do in metaphorical cases. And this, in turn, suggests that we should not take "There is a table here" at face value.

I will stop short of saying that our ordinary language of tables and chairs is literally metaphorical, that "their existence is metaphorical". Instead I'd like to say that our ordinary language of tables and chairs behaves in certain important respects metaphorically. Among these respects is that we should not expect arbitrary compositions of such language to be meaningful, and we should not expect to have classical logic hold on the surface level.

I actually think classical logic holds even in cases of metaphor. But it holds not at the linguistic level, but at the level of the propositions expressed by the metaphorical claims. "The average plumber has 2.3 children" expresses the same proposition as some sentence like "The average of the numbers of children had by plumbers is 2.3", and the latter sentence better reflects the logical structure of the proposition.