Are there any cases where b and C are real world statements where b⊢C∧¬b⊢C∧b⇒C∧¬b⇒C where b and C are not tautologies? It may seem like a silly question, but after searching hard and deep, I couldn't find an answer! Please help me with this question.
Assume classical propositional calculus.
If C is a tautology, then ⊨ C, and thus B ⊢ C for B whatever, and thus ⊢ B → C.
If not, if we have both B ⊢ C and ¬B ⊢ C we apply the Deduction Theorem and we get :
⊢ ¬B → C and ⊢ B → C.
But if C is not a tautology, this contradicts the soundness of the calculus.