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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?

Gavin Z.
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  • You don't prove it, it's a definition. – Tyler Mar 04 '14 at 23:54
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    Who says pigs don't fly? – Daniel Fischer Mar 04 '14 at 23:55
  • maybe look at this – Tyler Mar 04 '14 at 23:59
  • 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. http://en.wikipedia.org/wiki/A_priori_and_a_posteriori – DanielV Mar 05 '14 at 00:27
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    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 :) – Luca Mar 05 '14 at 00:47

4 Answers4

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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!"

Jair Taylor
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  • I don't think many people are confused that false can imply false. So more importantly: false can also imply true, and the ground can be wet even if it didn't rain. Neighbors could have washed their car, for example. Your description could mislead people into thinking that "it's raining $\Leftrightarrow$ the ground is wet". – Eric Duminil Apr 23 '23 at 13:33
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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.

Neil W
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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 $3 = 1 + 1$.

DanielV
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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 $(-1)^2>0$.

user2345215
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