Full Idea
The Naïve Theory of Truth collects all the Tarski bi-conditionals of a language and takes them as axioms. But no consistent theory extending Peano arithmetic can prove all of them. It is inconsistent, and even formalises the liar paradox.
Clarification
The bi-conditional is 'the sentence-is-true is equivalent to the sentence'
Gist of Idea
The Naïve Theory takes the bi-conditionals as axioms, but it is inconsistent, and allows the Liar
Source
Leon Horsten (The Tarskian Turn [2011], 03.5.2)
Book Reference
Horsten,Leon: 'The Tarskian Turn' [MIT 2011], p.38
A Reaction
[compressed] This looks to me like the account of truth that Davidson was working with, since he just seemed to be compiling bi-conditionals for tricky cases. (Wrong! He championed the Compositional Theory, Horsten p.71)