Previously I discussed two kinds of claims, those that attempt to accurately reflect some kind of subject matter and those that are pure formalisms, which are not judged entirely according to some set of rules and not by how well they reflect some subject matter. Which of these categories do our statements about “truth” fall into? The answer isn’t immediately obvious. In a sense the pure formalisms are in a certain way conventional, there is nothing that forces us to structure a pure formalism in a certain way. And truth certainly doesn’t seem conventional. But, on the other hand, it certainly doesn’t seem like truth is itself an object, such that we can match our claims about truth up against it and thus test them, which is necessary if we are to consider statements about truth to have some subject matter.
Part of the problem with placing truth into one of these categories (or, alternatively, discovering that it needs some third category of its own to fit in) is that questions about truth often end up tangled with questions about reference. Reference can be defined as a set of transformation rules that put the statements in some language or representation into an equivalence with the subject matter itself (or possible arrangements of the subject matter). Obviously this is a “third person” account of reference, in the sense that this is an idealized (theoretical) conception of reference since no one can directly get at the subject matter itself to contemplate the theoretical equivalences themselves. Still, we suppose that the equivalences could be recognized as holding (or failing to hold, when the subject matter is not actually as the statements or representation portrays it as), it’s just that we don’t know whether they actually do or not. As such reference can be thought of as both a pure formalism and as describing a kind of subject matter, depending on the context. As a pure formalism the transformation rules for reference can be defined in an essentially arbitrary manner and no one can say that they are “wrong” because when employed in this way the rules of reference are being defined by stipulation in order to give the statements that they are to be subsequently applied to an unambiguous interpretation. Or rules of reference can be given that are supposed to reflect the actual rules that we use in context with some language (a mental construct), and in this context they can be compared to their subject matter, by seeing whether actual users of the language agree that they are referring to what the proposed rules say that they are when all the facts are revealed to them. (In this way rules of reference are much like the rules of arithmetic which by themselves are purely formal, but cease to be when they are given an interpretation in terms of counting and combining groups of objects.)
With that said we can now consider how we would investigate a sentence such as “X is true”. We might first consider testing this claim by investigating whether X is actually the case, reasoning that if X is not the case then our theory of truth has failed by labeling a false sentence true. But that surely can’t be how we investigate truth because that is how we investigate the claims made by the sentence X. For example, if X is “the moon revolves around the earth” checking the accuracy of that statement is to investigate whether our claims about the moon and the earth are right, not to check whether our claims about truth are right. So we might instead consider investigating whether “X is true” would be endorsed by most competent speakers of the language, if they knew all the facts about the subject matter that X makes claims about. The problem here though is that this is now an investigation concerning whether the rules of reference we are using are the ones that people actually use to interpret sentences. If the people we consult use the same rules of reference that we did to arrive at the claim X and they agree with us about the facts regarding the subject matter then they will agree with the claim that X is true. But if they use rules of reference that lead to a different interpretation of “X” then they will claim that we are wrong when we assert that X is true, and that X is instead false. But, if we stipulate both the facts about the subject matter and the rules of reference then there is nothing left to investigate, the facts about the subject matter and the rules of reference completely determine which statements are true and false.
Thus we come to a crossroads. I certainly don’t think that there is a “correct” definition of truth because I think that the idea that there are correct definitions is absurd. Certainly there are conventional definitions, but how a word is defined conventionally may not be very useful. Given the above investigations we have a number of options in front of us as to how we would like to define truth, and which definition we choose to adopt determines what category truth falls in. First we have the completely minimal definition of truth. Given that the rules of reference and the subject matter itself seem to completely determine what is and isn’t true we may be moved to discard truth as signifying anything. Truth may be considered simply a linguistic device, used to call attention to the fact that a claim holds (rather than the content, which is what we usually focus on). Such an interpretation of truth makes it neither a claims about some subject matter nor a pure formalism, because it isn’t a claim at all, and hence cannot be classified in this way.
Alternatively, given that the truth of a statement is completely determined only by a combination of both facts about the subject matter and facts about the rules of reference we may argue that it is a term for referring to both of these sets of facts at the same time. To assert that a statement is true then would be to assert both that the rules of reference being used are the correct ones (the ones used by most speakers) and that the subject matter is as claimed. If this is our understanding of truth then it is not a pure formalism because it has a subject matter, albeit a very unusual one, a combination of the rules of reference actually used and the subject matter under discussion (I described this as unusual because usually the subject matter of a claim is a clearly defined domain, not a complex).
Finally, we might identify truth with the subject matter itself, but in a completely general way. Consider one definition of what is true, namely that it is everything that is the case. Truth then designates a subject matter in essentially the same way that “biology” designates a subject matter, except that in the case of truth the subject matter is everything. To say that a statement is true under this interpretation is to say that its subject matter is something that is the case (and saying that it is false would be to say that its subject matter is that which is not the case, or that it has no proper reflection in the subject matter). Since truth under this definition designates a category it would be a pure formalism. Just as the statements of biology have a subject matter while “biology” itself is an abstract classification so the true statements have subject matter while there is no actually existing category that truth can be matched up to.
Personally I lean towards the first alternative, since I would rather deal with fewer terms than more, but none of the definitions given here is “ruled out”, nor can they be so long as they are internally consistent.
Final note: Obviously truth as we are considering it here is truth as is applied to statements which are themselves supposed to have some subject matter. Truth in the context of statements that are purely formal has a different meaning, and is much easier to analyze. To say that a purely formal statement is true is simply to say that it conforms to the rules that define the formalism. If you hold that logic is purely formal, and not properly about formalisms, then such statements are themselves pure formalisms. On the other hand if you hold that logic can legitimately have as its subject matter pure formalisms then such claims about truth also have pure formalisms as their subject matter.