A rigorous, set-theoretic definition of truth and meaning, due entirely to yours truly.
A ‘truth’ is a true proposition, a proposition being what is affirmed, denied, believed, disbelieved, etc., examples being snow is white, grass is green, and the like.
What is a proposition?
A proposition is a class of properties---of characteristics, in other words---and for a proposition to be true is for the members of the corresponding property-set to be instantiated.
Let k be the smallest set containing the following three properties.
P1. The property of being identical with John P2. The property of smoking, and
P3. The property of being a thing x such that x is identical with John and such that x smokes.
John smokes if, and only if, these three properties are jointly instantiated. In other words, if they are instantiated, then John smokes; and if John smokes they are instantiated.
Therefore, the proposition John smokes is to be identified with k and that proposition’s being
true is to be identified with those three properties’ being jointly instantiated.
To be sure, if P3 is instantatiated, then so are P1 and P2; so it might seem that John smokes could be identified with P3, and its being true with P3’s being instantiated, it being unnecessary to identify that proposition with k, since k contains P1 and P2.
But there are two conditions that a viable theory of truth must satisfy. First, it must account for any given proposition’s truth-conditions. In other words, for any given proposition T, if T is true under such and such circumstances, then a viable theory of truth must account for that fact; and if T is false under thus and such circumstances, a viable theory of truth must account for that fact as well.
At the same time, a viable theory of truth must also do justice to the internal, structural properties of propositions. It must account for their decompositional properties, in other words. It is a datum that John smokes has a discrete part corresponding to John, a second discrete part corresponding to smokes, and a third part, this being an improper part, corresponding to the fusion of the two.
Our theory is consistent with these facts about the decomposition of John Smokes. In general, our theory, unlike the other theory considered, is consistent with the decompositional properties of propositions.