It is well known that adding a truth predicate to arithmetic in the most natural way leads to a contradiction.
Suppose as usual that we add a one place relation T to the language of arithmetic, and define some system of Godel numbering $\ulcorner \cdot \urcorner$ for this expanded language. Given a set of axioms A in this language (e.g., PA), we extend the usual provability relation $A \vdash \phi$ by demanding that it satisfy the following:
R1. if $A \vdash \phi$, then $A \vdash T(\ulcorner \phi \urcorner)$
R2. if $A \vdash T(\ulcorner \phi \urcorner)$, then $A \vdash \phi$
R3. if $A \vdash \neg \phi$, then $A \vdash \neg T(\ulcorner \phi \urcorner)$
R4. if $A \vdash \neg T(\ulcorner \phi \urcorner)$, then $A \vdash \neg \phi$
• We are not adding any new axioms (in particular, we are not adding any Tarski biconditionals $\phi \leftrightarrow T(\ulcorner \phi \urcorner)$ to A), or any other rules of inference of any sort related to $T$. All we are adding to A are the above rules.
• Requirements R1-R4 are different from adding rules of inference. Obviously, adding rules of inference that tell that us we can go from $\phi$ to $T(\ulcorner \phi \urcorner)$ and back, and $\neg \phi$ to $\neg T(\ulcorner \phi \urcorner)$ and back, would be too strong and lead to contradictions in well known ways if A is sufficiently strong. Loosely speaking, with R1-R4, we can't reason hypothetically about truth, though we can talk about the truth or falsity of something when we've actually established its truth or falsity.
Question: Given the above requirements, do we nevertheless have $PA \vdash \bot$ ?