Solution to Is (p q) > I(p>q)> q] a tautology? Why or why not? - Sikademy
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Archangel Macsika

Is (p q) > I(p>q)> q] a tautology? Why or why not?

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Let us consider the formula (pq) \to [(p\to q)\to q]. Let us show that this formula is a tautology using the method by contradiction. Suppose that the formula is not tautology. Then there exists (p_0,q_0) such that |(p_0q_0) \to [(p_0\to q_0)\to q_0]|=F. It follows from definition of implication that |p_0q_0|=T, |(p_0\to q_0)\to q_0|=F. It follows from definition of conjunction that |p_0|=|q_0|=T.

On the other hand, we have |(p_0\to q_0)\to q_0|=(T\to T)\to T=T\to T=T\ne F. This contradiction proves that the formula (pq) \to [(p\to q)\to q] is a tautology.

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