As I see it, the concept of truth can refer to two different ideas, which I will refer to as syntactic truth and semantic truth.
By syntactic truth, I mean that a statement in a formal system is true if it is consistent with the axioms and the production rules of the formal system. You can also talk about whether the production rules are consistent with each other. So when you ask whether logic is true, you are in this case asking whether the rules of logic are consistent, which can be proved. The problem is that every formal system will have undecidable statements, meaning statements that are true (i.e., consistent), but whose truth cannot be proved within that formal system. So logic can take you only so far.
In semantic truth, we are talking about whether a formal system statement decodes into some verifiable observable event in a natural system. Specifically, if we encode a first observable event into some first formal statement, manipulate the formal statement with the production rules into a second statement, and decode the second statement into a second observable event, does this correlate with what we observe? Did we observe the second event in response to the first event? In other words, did the formal system predict the second event correctly? In yet other words, to ask whether the formal system is true is to ask whether the formal system captures the entailment structure of the natural system. This is much harder, because we may not be able to tell whether the failure of a prediction arises out of faulty encoding, a faulty the formal system, a faulty decoding, or faulty observation.
When you say "So naturalism might be true, but it couldn't be logically affirmed, just like my argument," you may be referring to either of these issues. Ghosts of Godel and whatnot.
There are my thoughts at 3:30 in the morning :)
“The welfare of the people in particular has always been the alibi of tyrants.” — Albert Camus
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