Classical propositional logic


THEOREM (Principle of explosion)    $$\bot \vdash P$$

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LEMMA    \(P \vee Q \vdash \neg P \Rightarrow Q\)

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LEMMA    \(\neg P \Rightarrow Q \vdash P \vee Q\)

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THEOREM    \(P \vee Q \dashv\vdash \neg P \Rightarrow Q\)

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THEOREM    \(P \Rightarrow Q \vdash \neg P \vee Q\)

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THEOREM    \(\neg P \vee Q \vdash P \Rightarrow Q\)

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THEOREM    \(\neg P \Rightarrow \neg Q \vdash Q \Rightarrow P\)

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LEMMA    \(P \Rightarrow \neg Q \vdash Q \Rightarrow \neg P\)

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LEMMA    \(Q \Rightarrow \neg P \vdash P \Rightarrow \neg Q\)

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