Show that proposition C is a logical consequence of the formula A and (A -> (B or C)) and (B -> not A) using truth tables.

GATE 1993 · Discrete Mathematics · Propositional Logic · medium

Answer: On every interpretation making the formula True we get C = True, so C is a logical consequence of A and (A -> (B or C)) and (B -> not A).

1. The conjunction forces A True and B False: For the formula F = A and (A -> (B or C)) and (B -> not A) to be True, every conjunct is True. So A = True. In B -> not A, not A = False; an implication with a False consequent forces a False antecedent, so B = False.
2. A -> (B or C) then forces C True: Since A = True and A -> (B or C) is True, the consequent B or C is True. As B = False, the disjunction reduces to C, so C = True. Thus the only model of F is A=True, B=False, C=True, and there C is True.