You have a surface familiarity with this issue you are entirely too sure of. There are many, many applications in the real world that are not wrong or mistaken, that nonetheless, cannot resolve because, stated as formal math instead of programs (which is doable if their grammars are chomsky-normal), they are contradictions, in that an attempt to return their truth values to the operating program result in endless loop hangups. A contradiction does not return a consistent truth value--that is why it is a contradiction. Whether it makes sense or not depends on what you are doing, and what your domain of discourse is. Contradictions occur when a domain of discourse which can be rendered as a venn diagram whose sets contain all the elements under discussion contains elements whose truth value is different for different, supposedly valid operators in the domain. Nothing forces you to be confined to said domain. If you are outside the domain, contradictions can have useful meaning, like "this domain is invalid" for example.
Your attempt at justifying contradictions is like performing neurosurgery to remove a hangnail--it's both overly complex, and misapplied.
We are talking about simple truth values. We need no more than a simple predicate calculus. A contradiction always has a truth value of false. It occurs with a conjunction of statements that cannot both be true. It is a fundamental of all rational logic. "A and not A" is false in all rational set theory, grammars, automatons--even Chomsky's (despite his political absurdities he apparently does not deny the law of noncontradiction, at least in his linguistic theories). That you have construed ways in which you think a contradiction is at times NOT false shows your misapplication of the fundamentals upon which the theories you think you understand are founded.
Besides. I don't fall for diversions.