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P | Q | P->Q
T | T | T (That makes sense)
T | F | F (That makes sense)
F | T | T (Why the anwser is T?)
F | F | T (Why the anwser is T?)

Sworduy
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  • The idea is that if you begin with a false statement you can prove anything, either true or false. – fonfonx Aug 07 '17 at 04:24
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    Compare https://math.stackexchange.com/questions/70736/in-classical-logic-why-is-p-rightarrow-q-true-if-p-is-false-and-q-is-tr – Martin R Aug 07 '17 at 04:27
  • Another answer: https://math.stackexchange.com/questions/2208612/logical-conditional-truth-table-rationale – Bram28 Aug 07 '17 at 14:38

3 Answers3

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$(P\implies Q)$ merely says that we cannot have $P$ true and $Q$ false. In other words, $(P\implies Q)$ is equivalent to $\sim (P\land \sim Q).$

DanielWainfleet
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The implication/conditional: "$P\to Q$" means "If $P$, then $Q$" .

It is a guarantee that $Q$ is true whenever $P$ is true.

When $P$ is false, we might say the guarantee is counterfactual, but it is not falsified.   If the guarantee is not false, what is it?

The guarantee is only considered falsified when $P$ is true while $Q$ is false.

So we consider the implication to be true when either $P$ is false or $Q$ is true.   That is $\neg P\vee Q$.

Graham Kemp
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Suppose I tell you "If it rains in the evening then I will meet you." Then there are four possibilities (Look at the truth table). The only instance my statement is false is when it actually rains and I don't meet you. But my statement will still be true when it doesn't rain in the evening but I come and meet you, and also when it doesn't rain in the evening and I don't come and meet you. All I said was that if it rains in the evening then I will meet you.

Janitha357
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