# In logic, is there a name for a secondary contradiction that arises from operating (unkowingly) under a first contradiction?

Say I assume predicate A and B.

Say I show that A AND B is a contradiction.

If I then apply the law of excluded middle to say NOT A, this is a fallacy, no? Does it have a particular name?

• Anything follows from a contradiction by the law of explosion, so, as described, this is not a fallacy. (A ∧ B) ∧ ¬ (A ∧ B) → ¬ A is not very useful, but valid. Did you mean something else? I do not see where the law of excluded middle is supposed to be applied. Aug 31 at 12:01
• In so-called Classical Logic (the "mainstream logic used in mathematics) a contradiction is a fatal wound for the "system" and produces its bankruptcy. So, there are no different levels of inconsistency... Aug 31 at 12:25
• I don't believe it has a formal name, I propose we refer to it as a variant of the Cherry Pick Fallacy, since it involves a situation where we have ¬ A ∨ ¬ B and we concluded ( falsely) ¬ A. Also, for clarity: Predicates are always conjoined, formally. So taking A, and taking B, two "seperate" predicates, is really taking A ∧ B as a predicate. We always take the conjunction of every predicate when we do a proof. Aug 31 at 15:33