Normally, "It is not true that F," equals, "It is true not that F," or even, "It is true that not F." I can't figure out how to carry this out with the way the truth predicate is used in the (strong) liar sentence:
- This sentence is not true → This sentence is true not...? Or This sentence is truly not that...? Truly not that what?
- This sentence is not true → This sentence is truly that not...? What is it such that "it is truly that not"...?
To be fair, my reaction to this is to want to dig-in on the idea that the argument for the liar paradox is equivocal,?! but I'd also like to see if I can avoid jumping to that conclusion again.
?!The idea again being that because "is (not) true" is in predicate position in the honest (liar) sentence, then the meaning of "true" is not the same as in, "It is (not) true that..." insofar as the truth-term is a sentential operator, there. If the "not" of the liar sentence doesn't commute over "true," this illustrates how the truth-term in predicate position is not the same in meaning as the truth-term in sentential-operator position, here.
"QED"... except I wonder if stipulative intent is enough to override this factoid (if intending to mean the same thing by "true" in both positions is enough to hold this meaning in place) and so if the revenge sentence, "This sentence is sententially false," is admissible. (One needs to consider that stipulative and ostensive definition seem like "inverses" or "reciprocals" of each other relative to the direction-of-fit parameter: a stipulative definition maps from a term to a possible meaning, an ostensive definition from a possible meaning to a term, and then both otherwise directly with their meanings. Unlike descriptive definitions, then, which don't directly relate their terms to their specific targets. But now if stipulation and ostension are "inverses," should, "This sentence is stipulated to be false," be hand-in-hand with, "This sentence is ostensibly false"? It seems, though, as if in the case of the liar sentence (or the honest one, for that matter), the distinction between stipulative and ostensive truth-by-definition is erased. Whether this moment of conflation is tantamount to the nullification of our ability to stabilize the definition of a pure sentence as a self-looping truth-term, remains to be seen.)
Sidebar: tantalizingly, Dalglish half-mentions the commutation question for T but doesn't, so far as I've read through that text, apply the question to the characterization of the liar sentence (using "Find:" for the text gave only 2 results for the word "commute").