Nice. As you imply, no theory is, in itself, true, though there are truths about the theory. And so, there can be no theory of truth.
It would be more precise to say that there can be no theory the does not rely on the truth of its axioms. Logic is the theory of truth. You certainly must have axioms, which are unprovable. Assuming your axioms are all true, then anything you prove based on them will also be true. The truth of your axioms is verifiable, via the consistency of the theorems that you can prove using them. If extensionality (∀A∀B(∀X(X∈A ⇔ X∈B) ⇒ A=B) has not been contradicted by the entirety of mathematics, it’s a good bet that it’s true. It’s just not provable by mathematics.
Ethics are of course not logic, and trying to apply the same sort of rigor to them as one does to formal logic or mathematics is difficult. Stefan Molyneux does a good job, and I find no flaws in his ethical statements, but some of his arguments for them are appeals to authority (with the abstract “science” being the authority), IMO.
The question of truth is an interesting one. Can you know the truth of the axioms you are building your case on? For example, here are the axioms of classical logic:
- The Law of Identity: A = A. Or: anything is itself. Or: if a proposition is true, then it is true.
- The Law of the Excluded Middle: ∀B, B=A ∨ B<>A. Or: anything is either A or not A. Or: a proposition is either true or false.
- The Law of Noncontradiction: ¬(∃B | B=A ^ B<>A). Or: nothing can be both A and not A. Or: a proposition can not be both true and false.
These are axioms of the self evident variety. If you accept them to be true, you can begin reasoning logically based on them, and they form the basis of a mountain of logical reasoning. For example, using classical logic, you can prove that equality is transitive. I.e.:
Now, if you are a Christian, you will likely believe the Athanasian Creed. This states that:
- The Father is God
- The Son is God
- The Father is not the Son
But the transitivity of equality is contradicted by the Athanasian Creed, since (F=G)^(G=S)⇒F=S is, according to the creed, not true. Therefore, either at least one of the axioms of classical logic is false, or the doctrine of the trinity is false. Christians tend to go with the first alternative, essentially saying that logic does not apply to God. This makes debating the existence of God (see the PBS Crash Course Philosophy on the Problem of Evil) rather pointless. If I want to have both God and logic, then the Athanasian Creed has got to go. That’s why many protestant sects (for example, the United Church) disavow it.