This has turned into a week on the liar paradox. Here is an interesting variant on the liar paradox that includes no self-reference, either direct or indirect. There is a possible world which contains infinitely many sentence tokens s1,s2,.... (Perhaps they exist simultaneously, or perhaps they exist successively. If successively, they may exist successively in either direction—s1 before s2 before s3 ... or s1 after s2 after s3 ....) Now suppose that sn says:
sn+1 is false.
Now, in one sense there is no paradox here. We could just say that s1,s3,s5,... are true and s2,s4,s6,... are false. There is no contradiction. But that is ad hoc. There is no more reason for this truth-value assignment than for the opposite truth-value assignment. Truth is about reality. But the reality of the possible world containing these tokens is equally compatible with one truth-value assignment and with the other.
If I want to make the problem more pressing, I can suppose a doubly infinite sequence of sentence tokens: ...,s−3,s−2,s−1,s0,s1,s2,s3,.... Suppose that the tokens do not differ in any significant way (if the Principle of the Identity of Indiscernibles holds, they must differ in some way), because each one of them says:
The next token in the sequence is false.Then there can be no reason for distinguishing, say, the even-numbered ones for being true and the odd-numbered ones for being false, rather than the other way around.
I can also do this as a version of the truth-teller paradox. Just let sn say:
sn+1 is true.Then, if we're going to have a truth-value assignment, the same truth-value will be assigned to each sentence. But there is no more and no less reason to assign true to all of them than to assign false.
So not only must we be careful about self-reference, but also about regresses.
If affirming truth of a sentence is a way of taking up that sentence into one's own speech, as per one version of deflationism, then it is easy to see why we can't have self-reference, circular reference or infinite regress in respect of truth, since these processes fail to produce a finite well-formed sentence—they produce an infinite nonsensical sentence like "It is not the case that it is not the case that it is not the case that....". (Quantification would presumably be done in terms of possibly infinite conjunctions or disjunctions. But unfortunately, deflationism has trouble with quantification and modality. Thus, the claim that possibly George's favorite proposition is false, which claim is surely true, is troubling if we evaluate it by inserting George's favorite proposition in it—for that proposition might, as a matter of fact, be a necessary truth.) This book may be very much relevant, and may in fact already contain a full development of my inchoate ideas, and I've ordered it from interlibrary loan.
It's interesting, by the way, to note that some infinite sentences are nonsense, but others aren't. Thus,
- 2 is even and 4 is even and 6 is even and ...
- it is false that it is false that it is false that ...
Sorry, I'm rambling. This post is really just a bunch of notes for me to think about later.