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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.

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    $\begingroup$ My vote: no, such expungement doesn't make sense. $\endgroup$ – GEdgar Nov 8 '17 at 15:12
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    $\begingroup$ In propositional logic, and i think that in first order too, you can define contradiction with true and false, or 1, 0 (think them as you wish). But the converse is also the case, since something contradictory is always "false" or 0, and its negation is always "true" or 1. Your notion seems to appeal to a modal prefix, but at first glance it would be somehow equivalent to the negation. Give it a try. $\endgroup$ – edgar alonso Nov 8 '17 at 15:25
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    $\begingroup$ Robert, you might want to stick to poetry (rather than the genre of the novel). $\endgroup$ – Mikhail Katz Nov 8 '17 at 15:48
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    $\begingroup$ @MikhailKatz I see a future for you in comedy. $\endgroup$ – user334732 Nov 8 '17 at 16:03
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    $\begingroup$ See this. $\endgroup$ – Mikhail Katz Nov 8 '17 at 16:04

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