i had been thinking about a similar idea. but these work fine as a beginning. a good hierarchy needs a lvl for approximate truth as well (like Newton’s laws), as well as actual truths. but also perhaps a dimension for the relevance of the information conveyed. a sentence can express a true proposition without that proposition being relevant the making of just about any real life decision. for instance, the true proposition expressed by “42489054329479823423 is larger than 37828234747″ will in all likelihood never, ever be relevant for any decision. also one can cote that the relevance dimension only begins when there is actually some information conveyed, that is, it doesnt work before level 2 and beyond, as those below are meaningless pieces of language.
and things that are inconsistent can also be very useful, so its not clear how the falseness, approximate truth, and truth related to usefulness. but i think that they closer it is the truth, the more likely that it is useful. naive set theory is fine for working with many proofs, even if it is an inconsistent system.