They will for instance equally hold for certain notions of
provability.1 Now it is true that various notions of
provability will have their own distinctive axioms which distinguish them from metaphysical necessity.
It extends the guard computation mechanism from a simple one-way unification solving problem to a more general
provability check of conditions in the guard part under a given set of constraints using the ash operation.
Godel's second theorem is included because it involves an iteration of deviant coextensive
provability operations that satisfy different principles in Peano arithmetic, relevant to the mechanistic arguments in the philosophy of mind.
(such as correspondence to reality,
provability, etc.), which is supposed to specify what truth is.
In other words, in the case of empirical discourse the perfect analogue of the antirealist notion of constructive
provability is the notion of constructive falsifiability.
To motivate this, McGee points out that D is a notion of
provability, and that, surprising as it is, analogous principles (though they may be true) are not provable.
"I assumed [Carnap writes] that he had in mind a syntactical definition of logical truth or
provability. I was surprised when he said that he meant truth in the customary sense, including contingent factual truth.
For example if we add to first-order Peano arithmetic the statement that Peano arithmetic is inconsistent, the resulting
provability logic fails to count as modal by Koslow's criterion.
First, Dummett believes that an intuitionistic
provability semantics already is his kind of theory of meaning for restricted cases.
Boolos, George 1993: The Logic of
Provability. Cambridge: Cambridge University Press.
We know, for example, that if the system is consistent there is an undecidable sentence, and we know this in addition to the
provability of the sentence which expresses that conditional in the formal system: the impoverished "omniscience" that Grim outlines would not know the metamathematical propositions but only, if it made sense to speak so, the formal analogues of them.
They discuss Cantor and infinity, axiomatic set theory, elementary number theory, computability and
provability, Hilbert's tenth problem and applications of it, and Hilbert's tenth problem over number fields.