Could someone help me understand what is McGee's argument on restriction of consistent instances of T-schema about please?
This is gonna be messy so please bear with me.
Halbach and Holsten's "Norms for Theories of Reflexive Truth" argues that one of the desiderata of a (formal) theory of truth should be that Tr('ϕ') and ϕ (I am using 'ϕ' to mean the Godel number of ϕ here, since Mathjax doesn't work here) can be substituted salva demonstrabilitate. (I take it this means that given we can prove that ϕ is true, the two are interchangeable)
Halbach and Holsten (H&H) then talk about how we can restrict the class of instances of the T-schema.
One option is to restrict it to consistent instances, but H&H cite McGee's paper "Maximal consistent sets of instances of Tarski's Schema T" and argue that this does not suffice, and I do not understand why McGee's argument shows that.
H&H argue that McGee shows that with the diagonal lemma every sentence is equivalent to a T-schema. Thus any sentence independent from the base theory can be decided using a consistent instance of the T-schema. (Base theory refers to Peano arithmetic, or PA)
As far as I know, the diagonal lemma states that if theory T satisfies certain properties, and φ(x) is a formula with x being a free variable, then there is a sentence γ such that T⊢γ⟺φ('γ')$. But I don't see how applying this lemma gives us B_ϕ⟺(ϕ⟺ Tr('B_ϕ')).
I also don't understand what "any sentence independent from the base theory can be decided using a consistent instance of the T-schema" means, and why it shows that restricting to consistent instances does not work.
I apologise in advance of the messiness of this question, but I am really confused and I've tried my best to state it as best as I could.
H&H's argument. (Base theory refers to Peano arithmetic, or PA)
Some preliminaries from McGee's paper, the relevant part is on what S and R mean.
The relevant result from McGee's paper, although I suspect only the first half is relevant. (Theory S refers to some consistent arithmetical theory which entails the axioms of Robinson's R)