Is mathematical logic equally expressive if we to expunge its language of the notion of "TRUTH"; and replace the outputs True, False, Independent Of, Unprovable In etc., with only CONTRADICTORY?
For a statement $s$ which is currently thought of as FALSE we can still express as much by stating:
"Some theory + $s$ is contradictory."
Meanwhile TRUTH of $s$ according to some theory $t_0$ would be satisfactorily expressed by something like:
"'$t_0$ + $s$ is contradictory' is itself contradictory.
But I'm unclear on how to correctly state this with reference some appropriate model $t_1$ of $t_0$ which can "see" whether $s$ is true, independent, unprovable etc.
What are these statements ($s$ is TRUE, $s$ is Independent of $t_0$, etc.)?
A little background
The distillation to a single output more clearly expresses the indefinable status of "truth" while at the same time clarifying some areas of paradox and difficulty. I'd like to rewrite the modified liar paradox in this form which I expect to look something like "this statement is contradictory in any theory in which it is provable", giving a clearer rendition of why theories do not fall to the Modified Liar Paradox. I want to achieve clarity with respect to such statements which are contradictory irrespective of the model in which they're considered.
I then want to begin understanding of Godel's incompleteness theorems, razoring off "contradictory statements" from any proof of incompleteness in order to establish that the theorems extend to statements which are not "contradictory irrespective of any model/theory in which they're provable" to see what the remaining statements look like.