Negation introduction
This is a logical inference that basically says "if a given antecedent implies both the consequent and it's compliment, then the antecedent is a contradiction" [1]
It can be written as: (P → Q) ∧ (P → ¬Q) ↔ ¬P
A prose example of its use is found whenever you realize someone is attempting to bullshit you by proving two contradictory statements from the same fact. Intuitively you consider their point to be false because it doesn't make sense that a single argument can both prove and disprove the same thing. For example, if someone were to say to you "When the phone rings I get so excited" and then later say "I get annoyed when people call me", your only sensible conclusion to this contradictory information about telephones ringing would be that your friend is making false statements. Yes, phones ring and people may or may not get excited about it but, all you know in this instance is falsehood.
(phone ringing) → excitement ∧ (phone ringing) → annoyance ↔ (doughnuts, no seriously, anything can be "proven" from contradiction. Since contradiction basically states that true is equal to false, it's safe to admit that nothing has meaning and just start making things up. Of course it makes sense to say that P is false because then the contradictory implications can be discarded without further thought)