McGee's argument on restriction of consistent instances of T-schema

Question

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.

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H&H's argument. (Base theory refers to Peano arithmetic, or PA)

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Some preliminaries from McGee's paper, the relevant part is on what S and R mean.

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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)

Answer 1

The first question was answered in the comments. The short answer to the second is that they assume that the reader can figure out how the truth about there being many incompatible maximal consistent ways to extend PA with T-bivalences follows once it has been pointed out that PA proves that for every φ in L_T, there is an equivalent T-bivalence_φ, plus they give the reference that explains it in detail.

What's going on in this passage more generally:

The authors are discussing the problem faced by those wanting to develop an untyped truth theory, namely, that the theory containing PA + all instances of the T-bivalence without restriction, is inconsistent.

One proposed way to answer this (by Paul Horwich) is to include as many T-bivalence instances as possible retaining consistency (i.e. "maximal consistent" set). However the authors refer to McGee's paper that shows, for one, that there are great many such sets that are maximal consistent (with PA) but mutually incompatible. So the maximal consistency answer as such doesn't give us enough guidance.

(They also note that Cieśliński showed that restriction to a maximal conservative set suffers from similar problems.)

They point to the key fact contained in the proof that illustrates some of the problems in deciding which consistent T-bivalences to allow. As said, applying the diagonal lemma they show that for any sentence φ of L_T, PA proves that φ and some T-bivalence instance are equivalent (call this instance T-bivalence_φ).

I assume that they are trying to illustrate that while for some cases of φ, the question whether or not to include the respective T-bivalence_φ is unproblematic (e.g. when φ is inconsistent with PA, so is the T-bivalence_φ, and it should therefore not be included), in other cases the consistency criterion doesn't give us the answer.

In particular they point out that in cases where φ is independent from PA (i.e. PA doesn't prove either φ or not-φ, i.e. doesn't "decide" φ), we get a consistent extension of PA with PA+φ and with PA+not-φ, and so we also get a consistent extension by PA+T-bivalence_φ and with PA+T-bivalence_not-φ. Which of these should we choose? The section 4.1 of the SEP article that discusses this issue mentions a drastic case, a false arithmetical statement φ that is independent of PA, so there are sets of T-bivalences consistent with PA that prove false arithmetical statements.