Here are a couple of claims I'd affirm, albeit not too confidently, if someone had asked:
If N is a proof of a statement about objects in a domain D, then N is a derivation in a language interpreted as referring to D
Every semantic rule about a language L, as expressed in a metalanguage L-prime (i.e. every rule in L-prime concerning what symbols in L refer to what objects in L's domain) can be effectively considered as a syntactic rule in a meta-metalanguage L-prime-prime (i.e. as a rule concerning which symbols in L-prime may appear in what order in to satisfy requirements of well-formedness and valid derivation).
Are the above two claims trivially true, true but not obvious, controversial, false but could be defended by serious people, trivially false, or something else (for example, in need of clarification)?
I am (obviously) not a logician so it is more than possible that the two claims, as stated, need cleaning up or clarification. If this is needed, any help you are willing to offer along those lines is also appreciated. I've already had feedback from a logician acquaintance of mine who says the first claim is reasonable if taken informally, but that since proof and derivation are proof-theoretical predicates, "reference" and "object" don't strictly apply. But I have no idea what that means, and I don't want to pester the guy!
Anyway, if you're wondering where this all comes from, it's basically a kind of bet-settling question. I was told elsewhere, for saying a natural-english version of the above, that I was just ridiculously off my rocker, and was treated with a very dismissive attitude as a result. But...... I don't think these claims are that crazy. Are they? (I actually thought they were right..... but even if that's wrong, are they actually crazy?