Liar in three sentences

• No January 20, 2009, post by Alexander Pruss contains a true sentence colored blue.
• No January 20, 2009, post by Alexander Pruss contains a true sentence colored purple.
• No January 20, 2009, post by Alexander Pruss contains a true sentence colored green.

I assume that I will post nothing else in green, blue or purple on January 20, 2009.

What do we have to say about the above three bulleted sequences of text? First, I think that whatever we say about any one of them, we say about all of them—the situation is symmetric. One might try to distinguish between them in terms of the order in which I wrote them, but I could have written them all at once (with two hands and a foot) on the outside of a drum, with no inherent order. So, if we assign a truth value to any one of the three, we must assign the same one to all three; if we say any one carries a meaning, we must say that all three do, and the meanings have to be very similar.

So what can we do? If we say (1)-(3) are all true or that they are all false, we arrive at a contradiction, at least on the obvious interpretation of the texts. Assuming, as we ought, classical logic, no other truth value can be assigned. (And if we have a non-classical logic with a truth value V distinct from true and false, we will still run into trouble as long as we can make the inference from "s has truth value V" to "s is not true".)

Thus, none of the three has a truth value. Since every sentence that expresses a proposition has a truth value, none of the three claims expresses a proposition. But here is an oddity. If none of the three claims expresses a proposition, then I haven't posted anything true in green, blue or purple on January 20, 2009, since only something that expresses a proposition can be true. So it seems that if none of the three claims expresses a proposition, then all three claims are true, which is absurd.

But let's slow down. We have seen that we must say that no one of the three colored claims above can express a proposition that is true. Assuming I post nothing else in the relevant colors, it does indeed follow that:

• No January 20, 2009, post by Alexander Pruss contains a true sentence colored blue.
• No January 20, 2009, post by Alexander Pruss contains a true sentence colored purple.
• No January 20, 2009, post by Alexander Pruss contains a true sentence colored green.

It seems that from this we should infer the truth of each of the colored statements, and then we are in trouble. But here we are either confusing types and tokens, or else we are implicitly using the following principle:

1. If two written textual sequences are orthographically equivalent, they are semantically equivalent (so if one has a truth value, the other has the same truth value, and so on).

Two sequences are orthographically equivalent provided that they are equivalent according to the rules for making inscriptions in the language—two sequences in a language using the Latin alphabet are "orthographically equivalent" provided that they are tokens of the same sequences of letters and symbols (perhaps in different font, color, etc.)

Now, (1) is in fact false—the cases of homonymy and indexical sentences show this. So we need a weaker form of (1), restricted to "nice" sequences, i.e., ones lacking indexical expressions and where homonymy is not an issue. Let that be (1*). What our argument so far has shown, and I think there is really no way of disputing this within classical logic (and some non-classical ones as well) is that if (1*) is true, then each of the initial colored sequences is true. Since the latter leads to a contradiction, we must either deny (1*) or affirm that the colored statements are indexical or homonymous in a hidden way. Neither of these two moves is plausible. But either move is better than contradiction, and we must make at least one of the two moves on pain of contradiction.

I think what I said so far cannot really be disputed by a reasonable person (adopting a non-classical logic is unreasonable, since it is adopting an incorrect logic). What I will say now, however, will just be speculation. Let us take at face value the fact that the corresponding colored and black sequences lack homonymy and indexicality. It follows, I think, that the truth of a sequence of symbols, even in non-indexical cases, is not a function of the sequence type, but of the sequence token. We have the same sequence type in black being true and in green not being true. This suggests that semantic approaches that quickly leap from sequence token to sequence type are mistaken. It is the token that expresses a proposition, not mediated by the type's expression of anything.

This suggests that linguistics should be done in terms of tokens, not types.

There is still a puzzle as to why the green, purple and blue tokens fail to express propositions, while the corresponding black ones express true propositions. But a puzzle does not suffice for paradox. We would like to have a general account here. My suspicion is that a general account is not available, because language is a lossy encoding of thought, and has meaning derivatively from the meaning of the thought, and one cannot recover the thought algorithmically from its lossy encoding (but in many cases we can make a very good guess). And to have a thought is harder than people imagine it is. This is a lesson from Spinoza and the Tractatus. (It may even be the case that there is something true in the vicinity of Spinoza's radical idea that to have a belief that p, one must know that p. That would be a weird direction to have to move in.)

[I fixed one false to a not true, and some minor typos.]