Part 3: Philosophical Implications
I think my approach has the advantage that it can explain why necessary truths don't have everything as a truth-maker. Granted, "The Earth has exactly one moon --> the Earth has exactly one moon" is true no matter what, but it does not therefore have everything as a truth-maker. By my definitions, a truth-maker for that material conditional is either a false-maker for its antecedent or a truth-maker for its consequent. As the antecedent and consequent are the same in this case, and as it is in fact true, every truth-maker for "the Earth has exactly one moon" will be a truth-maker for our conditional, and nothing else will. Truth-making, on my account, is not trivial for necessary truths, not even for paradigm cases of tautologies. A different instance of the Law of Identity, say "Sacramento is the capitol of California --> Sacramento is the capitol of California", will have (a) different truth-maker(s) from the previous instance. We can call such sentences 'analytic' if we like, in the sense that their meaning fixes their truth value--they could not have the same meaning, but a different truth value--but on this view it would be wrong to say that they are true solely in virtue of meaning. They have truth-makers, and they are not trivially made true by everything, nor will even different logical truths of the same form necessarily have the same truth-makers.
Another advantage of my account is that it enables one to dispense with a primitive, cross-categorical relation between objects (including sets) and truths. We can, if we wish, instead define necessitation in terms of truth-making (which itself is defined in terms of reference-making): An object x necessitates p iff x is a truth-maker for p. I count this as an advantage because we can now explain why an object necessitates the truth of p via reference-making, which itself can be explained via meaning, which itself
can be explained via the use and/or causal history of terms. On an account like that of D. M. Armstrong, it would appear to be brute that an object necessitates a truth-bearer. Brute facts are not always a bad thing--"Explanation comes to an end somewhere", as Wittgenstein said. But, first, it seems odd that there should not be an explanation for the obtaining of the necessitation relation, when we are trying to account for how the truth of truth-bearers is grounded in reality. To explain it in terms of necessitation by objects, and then offer no explanation as to why objects necessitate the truths they do, seems little better than taking truth to be a primitive property which just happens to attach to some truth-bearers and not others. Second, it also seems odd that an account of why truth-bearers are true would say nothing about how their truth depends on the reference and structure of their components. My account is designed to do exactly that, and in virtue of doing that it can explain why objects make true the truth bearers they do--which, in my opinion, is perhaps the only kind of explanation of this that can be had, and perhaps also the only kind of explanation of it that we should desire.
My account is neutral with respect to the existence of facts or states of affairs. Objects and sets are the only entities that it posits as truth-makers. If the relation of reference-making that holds between nouns or predicates and objects or sets counts as a fact, or a state of affairs, then of course my account will not work without such entities; but given them, it needs no others. All other predicational, relational, truth-functional and quantificational sentences can be accounted for in terms of my definitions. And if the reference-making relation does not count as a fact or state of affairs, then my needs none of them at all.
My account is also neutral with respect to the existence of properties, including relations, and for much the same reasons as just stated. If reference-making counts as a genuine relation, then of course my account needs genuine relations to work, but it will need no others for the purposes of explaining truth-making. One may need to posit genuine properties for other reasons, but they are not items that my account is committed to. My account, then, is metaphysically chaste: If it requires any potentially dubious entities, it requires only the least amount necessary to achieve its purposes. I hope it should thus be acceptable to philosophers of a variety of metaphysical persuasions.
For some final thoughts on truth-makers for negative truths, please see the Appendix.