How can statements contradict when contradictions do not exist?
A contradiction, by definition according to "Discrete Mathematics and Its Applications" by Kenneth H. Rosen, is a compound proposition that is always false. This means that contradictions are always false. Additional sources, such as Merriam-Webster’s Online Dictionary, assert a contradiction is "a proposition, statement, or phrase that asserts or implies both the truth and falsity of something." Since propositions are either true or false, no proposition can be both true and false. Thus, even by this definition, contradictions can never exist (are always false).
Since contradictions are always false, how can we say two statements contradict? Nothing is contradictory! Nothing can contradict!
Do not answer with IF or WHEN two statements contradict, then…, because that’s impossible!
You are confusing (or at least conflating) a mathematical definition (Rosen) with a colloquial definition (Merriam-Webster). That’s not likely to lead you anywhere useful.
I also don’t recommend a dictionary for mathematical definitions of "ring," "field," "space," "vector," "set," "paradoxical," or any other mathematical terms that have common English meanings.
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