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I could easily prove

$\neg P \lor Q$ entails $P \rightarrow Q$.

It is well known that

$P \rightarrow Q$ entails $\neg P \lor Q$

but I couldn't find a way to prove it.

Although there is the same question; How to prove that $P \rightarrow Q$ is equivalent with $\neg P \lor Q $?; it's a little bit confusing and I need to see step by step solution.

Could you show me the way by using Natural Deduction?

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Cnqt
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1 Answers1

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Assume : $P \rightarrow Q$ --- premise

1) $\lnot (\lnot P \lor Q)$ --- assumed [a]

2) $\lnot P$ --- assumed [b]

3) $\lnot P \lor Q$ --- from 2) by $\lor$I

4) $\bot$ --- from 1) and 3) by $\lnot$E (or $\rightarrow$E)

5) $P$ --- from 2) and 4) by Double Negation, discharging [b]

6) $Q$ --- from premise and 5) by $\rightarrow$E

7) $\lnot P \lor Q$ --- from 6) by $\lor$I

8) $\bot$ --- from 1) and 7) by $\lnot$E (or $\rightarrow$E)

9) $\lnot P \lor Q$ --- from 1) and 8) by Double Negation, discharging [a]

Thus :

$P \rightarrow Q \vdash \lnot P \lor Q$.

Mauro ALLEGRANZA
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  • thank you mauro, but i couldn't understand the implication elimination in contradiction lines(4,8). which implication we eliminate? – Cnqt Nov 02 '14 at 17:29
  • @Cnqt - perhaps you know the rule $\lnot$E : from $\varphi, \lnot \varphi$, infer $\bot$; I've applied this rule in the form : $\lnot \varphi := \varphi \rightarrow \bot$. Assuming this abbreviation, the above rule is from $\varphi, \varphi \rightarrow \bot$, infer $\bot$, taht is simply an application of $\rightarrow$E. – Mauro ALLEGRANZA Nov 02 '14 at 17:57
  • @MauroALLEGRANZA. Your answer is great. Very easy to follow. Going on a bit of a tangent. Is there a way to quickly determine if a conclusion does not entail its premises? – Peter Chaula Sep 02 '21 at 08:20
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    @peter - " if a conclusion is not entailed by its premises"... Yes, for propositional logic: truth table. – Mauro ALLEGRANZA Sep 02 '21 at 08:23
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    More shortly, try to define a truth assignment that falsify the conclusion and "see what happen" with the premises. If you find a contradiction, then there is entailment; if not, you have succeeded to find a counter-example. – Mauro ALLEGRANZA Sep 02 '21 at 08:24
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    @peter - apply it to the example above. In order to falsify the conclusion we need an assignment such that P is True and Q is False. – Mauro ALLEGRANZA Sep 02 '21 at 08:26
  • @MauroALLEGRANZA, that makes sense. Thank you. I suppose, for simple problems like these, I can determine validity by inspection? In other words, without working out the full truth table – Peter Chaula Sep 02 '21 at 08:31
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    @peter - exactly. A simple proof procedure based on this approach is that of Truth tree. – Mauro ALLEGRANZA Sep 02 '21 at 08:34