I've got a cold and am running a fever. Posting on the interwebs is contra-indicated by such a condition. Nonetheless, in my addled state I'm haunted by the following.

To say that "

*P*is true" is neither true nor false (NT

*P*) is equivalent to saying that

*P*is neither true nor false (N

*P*).

To say that "

*P*is false" is neither true nor false (NF

*P*) is also equivalent to saying that

*P*is neither true nor false (if

*P*were false, then "

*P*is false" would be true; if

*P*were true, then "

*P*is false" would be false; so

*P*must be neither).

But what does it say about

*P*to say that "

*P*is neither true nor false" is neither true nor false (NN

*P*)? I can't make sense of this, and I'm sure I'm goofing something up. Here's an argument that the notion of NN

*P*is incoherent.

*P*is such that NN

*P*holds. By trivalence

*P*is true, false, or neither. Assume

*P*is neither (N

*P*). Then it is true that P is neither (TN

*P*), but this contradicts NN

*P*. So our assumption is wrong, and we know that from NN

*P*it follows that

*P*is not neither, which by trivalence entails that

*P*is either true or false.

But if

*P*is either true or false then we know that P is

**not**neither true nor false which contradicts the claim that NN

*P*.

I'm pretty sure that here last step is probably wrong, because asserting that something is not the case is the assertion that it is not true, which is consistent with it being neither.

But this still doesn't help me make sense of what is being affirmed about a sentence when we assert that the claim that it is neither true nor false is itself neither true nor false. Is it just to assert that *P* is either true or false?

I realize the above could be the fever. I won't be embarrassed if someone points out an obvious flaw in the above reasoning. I would be relieved. Especially if the pointing out also explained just what is going on with NN*P* (in particular, what this tells us about *P* in three valued logics).