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According to the truth table, If $P$ is false,then $P->Q$ is true.

if pigs fly, then $1+1=3$. Why is this implication true? How do you prove it?

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  • $\begingroup$ You don't prove it, it's a definition. $\endgroup$
    – Tyler
    Mar 4, 2014 at 23:54
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    $\begingroup$ Who says pigs don't fly? $\endgroup$ Mar 4, 2014 at 23:55
  • $\begingroup$ maybe look at this $\endgroup$
    – Tyler
    Mar 4, 2014 at 23:59
  • $\begingroup$ You are confusing a priori false, such as $1\ne 1$, with a posteriori false, such as "the capital of mexico is london". Pigs not flying is not a priori false unless you say it is, and you haven't. en.wikipedia.org/wiki/A_priori_and_a_posteriori $\endgroup$
    – DanielV
    Mar 5, 2014 at 0:27
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    $\begingroup$ My professor always said: "If the moon consists of green cheese then I am the imperator of China. But the moon is not made of green chees. So I didn't claim anything.". Somehow this made it clear for me. Maybe it will help you aswell :) $\endgroup$
    – Luca
    Mar 5, 2014 at 0:47

4 Answers 4

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Suppose you say that "If it's raining, then the ground is wet."

Then someone responds: "But the ground is dry."

Your response would be: "So what? It's not raining, so my statement is still valid!"

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We know "pigs can't fly" is true, and by the law of the excluded middle, only one of the statements of { "pigs can't fly" , "pigs can fly"} is true.

But if we now suppose "pigs can fly" is true, then two of the statements of { "pigs can't fly" , "pigs can fly"} are true. But we've already shown only one is true, hence $1=2$. Adding one to both sides gives $1+1=2+1=3$.

QED.

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  • $\begingroup$ This really made my day. :) $\endgroup$
    – Mann
    Jun 18, 2016 at 9:20
  • $\begingroup$ But if "pigs can fly" is true, then wouldn't "pigs can't fly" be false? $\endgroup$
    – Hrit Roy
    Nov 29, 2017 at 21:07
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In the style of Bertrand Russell and the Pope:

Assume we have a set of pigs $S$. Two can't fly and one can. How many pigs are in set $S$? Well $|S| = 2 + 1 = 3$.

Let's $S_F$ be the number of pigs in $S$ that can fly. Let $S_{\lnot F}$ be the number of pigs in $S$ that can't fly. Since pigs can't fly, $|S_F| = 0$. And we are given that $S_{\lnot F} = 2$. So $|S| = |S_F| + |S_{\lnot F}| = 0 + 2 = 1 + 1$.

So $3 = 1 + 1$.

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  • $\begingroup$ How come |𝑆𝐹| = 0 ? It should be 1 as we have assumed that one pig in S can fly. @DanielV $\endgroup$ Sep 14, 2019 at 14:04
  • $\begingroup$ We also assumed that pigs can't fly. $\endgroup$
    – DanielV
    Sep 14, 2019 at 16:04
  • $\begingroup$ Can we assume this ? $\endgroup$ Sep 15, 2019 at 18:52
  • $\begingroup$ @AbhinavArya That's the whole point of the question, to assume something that is false. (And for it to be false, you must also be assuming the opposite, so there are 2 assumptions.) $\endgroup$
    – DanielV
    Sep 15, 2019 at 20:16
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It's a convention, so you can say eg. for all $x\in\mathbb R$ the following is true:

$$x\geq0\Longrightarrow x^2\geq0$$

This would be false if you didn't define it that way, because $-1<0$ even though $(-1)^2>0$.

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