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According to wikipedia, when we compute the CNF of propositional logic, we substitute $P \rightarrow Q$ with $\lnot P \vee Q$.

However, it seems to that when $P$ is False and $Q$ is True, then $P \rightarrow Q$ is not true but $\lnot P \vee Q$ is true, so those two don't seem equivalent.

gust
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3 Answers3

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No. $P\to Q$ is true when $P$ is false and $Q$ is true.

$P\to Q$ is only falsified when $P$ is true and $Q$ is false.

$\def\top{\mathsf T}\def\bot{\mathcal F}$

$$\begin{array}{|c|c|}\hline P & Q & P\to Q& \neg P\lor Q\\\hline \top &\top & \top & \top\\\hdashline \top &\bot&\bot&\bot\\\hdashline\bot&\top&\top&\top\\\hdashline\bot&\bot&\top&\top\\\hline\end{array}$$

Graham Kemp
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  • In other words, the only way to falsify an implication is to find a case where the conclusion is false but the hypothesis is true. – Lubin Sep 16 '20 at 22:48
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    @Lubin, exactly. $P\to Q$ makes no claim about the value of $Q$ when $P$ is false. – Graham Kemp Sep 16 '20 at 22:53
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This is so-called material implication. The standard truth-table for $P\rightarrow Q$ has F when $P$ is true and $Q$ is false, and T for the other three cases. That's the same as the truth table for $\neg P\vee Q$.

This often seems unintuitive to beginners. On the other hand, it's hard to justify any other truth-table. The most natural alternative is to say that if $P$ is false, then $P\rightarrow Q$ is meaningless, or depends on exactly what $P$ and $Q$ are. But this causes technical problems.

A trace of the material conditional can be found in an English sentence like, "If that horse wins the race, then I'm a monkey's uncle." The speaker expects the antecedent to be false, so he wants the whole sentence to be an instance of F$\rightarrow$F, which is T according to the standard truth-table.

Another example, cribbed from one of the references below: "If the train is on time, then I'll be home by 10." Now suppose the train is late, but the speaker manages to make it home by 10 anyway. So we have an instance of F$\rightarrow$T. Do you want to say that the speaker's statement was false? That seems perverse. The only reasonable choices, it would seem, are either T or to say that the implication doesn't have a truth value at all.

You'll find more in Wikipedia ("Material Conditional: Philosophical Problems"), and a much more extensive discussion at the Stanford Encyclopedia of Philosophy ("Indicative Conditionals" and "Relevance Logic").

Michael Weiss
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  • Thank you for the non-patronizing, detailed answer. – gust Sep 17 '20 at 02:53
  • You’re welcome. As you will see from the references, some very knowledgeable and insightful people have had their qualms about the material conditional. – Michael Weiss Sep 17 '20 at 03:38
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According wikipedia:

The compound p → q is false if and only if p is true and q is false

Hence, $P \rightarrow Q$ is true in your case

trancendentalFork
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