Truthteller's relative

The truthteller paradox is focused on the sentence:

1. This sentence is true.

There is no contradiction in taking (1) to be true, but neither is there a contradiction in taking (1) to be false. So where is the paradox? Well, one way to see the paradox is to note that there is no more reason to take (1) to be true than to be false or vice versa. Maybe there is a violation of the Principle of Sufficient Reason.

For technical reasons, I will take “This sentence” in sentences like (1) to be an abbreviation for a complex definite syntactic description that has the property that the only sentence that can satisfy the description is (1) is itself. (We can get such a syntactic description using the diagonal lemma, or just a bit of cleverness.)

But the fact that we don’t have a good reason to assign a specific truth value to (1) isn’t all there is to the paradox.

For consider this relative of the truthteller:

2. This sentence is true or 2+2=4.

There is no difficulty in assigning a truth value to (2) if it has one: it’s got to be true because 2+2=4. But nonetheless, (2) is not meaningful. When we try to unpack its meaning, that meaning keeps on fleeing. What does (2) say? Not just that 2+2=4. There is that first disjunct in it after all. That first disjunct depends for its truth value on (2) itself, in a viciously circular way.

But after all shouldn’t we just say that (2) is true? I don’t think so. Here is one reason to be suspicious of the truth of (2). If (2) is true, so is:

3. This sentence is true or there are stars.

But it seems that if (3) is meaningful, then it should should have a truth value in every possible world. But that would include the possible world where there are no stars. However, in that world, the sentence (3) functions like the truthteller sentence (1), to which we cannot assign a truth value. Thus (3) does not
have a sensible truth value assignment in worlds where there are no stars. But it is not the sort of sentence whose meaningfulness should vary between possible worlds. (It is important for this argument that the description that “This sentence” is an abbreviation for is syntactic, so that its referent should not vary between worlds.)

It might be tempting to take (2) to be basically an infinite disjunction of instances of “2+2=4”. But that’s not right. For by that token (3) would be basically an infinite disjunction of “there are stars”. But then (3) would be false in worlds where there are no stars, and that’s not clear.

If I am right, the fact that (1) wouldn’t have a preferred truth value is a symptom rather than the disease itself. For (2) would have a preferred truth value, but we have seen that it is not meaningful. This pushes me to think that the problem with (1) is the same as with (2) and (3): the attempt to bootstrap meaning in an infinite regress.

I don’t know how to make all this precise. I am just stating intuitions.

Some paradoxes of reference

Liar-like:

• one plus the biggest integer that can be expressed in English in fewer than fifty words
• one; two; three; one plus the biggest integer mentioned in this list
• one; two; three; one plus the last integer mentioned in this list
• one plus the last integer mentioned in this list; two; three; one plus the first integer mentioned in this list
• one plus this integer

Truthteller-like:

• one; two; three; the biggest integer mentioned in this list
• one; two; three; the last integer mentioned in this list
• the last integer mentioned in this list; two; three; the first integer mentioned in this list
• this integer
• the square of this integer

Grounding

It is normal to think that a disjunctive proposition that is true is grounded in each of its true disjuncts.

This may be false. Let p be the proposition that 2+2=4. Let q be the infinite disjunction p or (p or (p or ...)). Then q is its own second disjunct. Moreover, q is true. But surely what q is grounded in is not q itself but p.

For the same reason, it does not appear correct to say that a conjunction is always partly grounded in at least one of its conjuncts. For instance, take the infinite conjunction: p and (p and (p and ...)). The second conjunct is the conjunction itself, but it does not seem that this conjunction is even partly grounded in itself.

When I came up with the disjunction example today, I thought that I could weaken the disjunctive grounding principle to say that a disjunction is grounded in at least one of its disjuncts. But even that is not clear to me right now. For perhaps we could construct a complex infinite disjunction such that each disjunct is the disjunction itself. But I am less sure that such a disjunction really does exist.

On reflection, I wonder if my examples work. Maybe there is no disjunctive proposition p or (p or (p or ...)), but only the disjunctive proposition ...(((p or p) or p) or p).... In other words, the direction of the infinite nesting may matter. The difference is that the latter disjunctive proposition has a starting point: we take p and disjoin p to it infinitely often. The former one does not.

Another interesting case is: "2+2=4 or the proposition expressed by this sentence is true."

Liar and truthteller questions

Here are some fun questions:

1. Is the answer to this question negative?
2. Is the answer to this question positive?
3. What is the answer to this question?

The last one is due to my six-year-old.