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Let $P$ = "Pigs can fly" and $Q$ = "I'm king".

Apparently, there's a rule stating that $P \implies Q$ is true, if $P$ is false.

In this example, $P$ is indeed false, because pigs cannot fly. But how does this make the implication true?

The way I see it, pigs learning to fly will not cause me to be crowned king.

What am I missing here?

Any help appreciated?


marked as duplicate by Zev Chonoles, Joffan, Community May 25 '15 at 18:37

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  • $\begingroup$ this is the principle of explosion. It's unintuitive $\endgroup$ – man and laptop May 25 '15 at 18:23
  • $\begingroup$ Well they say that one could deduce anything based on false knowledge $\endgroup$ – alkabary May 25 '15 at 18:23
  • $\begingroup$ if $1=2$ then $2 =3$, you see now how it might work ? $\endgroup$ – alkabary May 25 '15 at 18:24
  • 4
    $\begingroup$ This is a duplicate of many previous questions... $\endgroup$ – anon May 25 '15 at 18:25
  • $\begingroup$ In another system of logic-of your own making-it sure can be true,even a truism.. $\endgroup$ – MathematicianByMistake May 25 '15 at 18:26

One way you could interpret your implication would be "every time a pig has been able to fly, I have been king." In order to show this was not true, you would have to demonstrate a time when (a) pigs have flown ($P$ is true), and (b) you have not been king ($Q$ is false). But, $P$ is never true, so you can't do this. Thus, the implication is valid.

  • $\begingroup$ I think this slightly misstates the role of "If pigs could fly", which is presented in the idiom as an absolute falsity. In particular the listener is not being invited to consider that there might be any occasion when it's true. However this is probably an English language point rather than a mathematical logic point. $\endgroup$ – Joffan May 25 '15 at 18:33

Taking this back to natural language, this says "Assume pigs fly. Since pigs fly, I am king". It reads completely ridiculus because it is, but it doesn't make it wrong. You're starting from a false assumption.

For example, if you wanted to discover the properties of a hypothetical object, but didn't know one exists or not. You'd start with "If [OBJECT] exists, we should expect to see..." Which would have reaonable conclusions but without knowing the thing exists to begin with! That make the statement "If [OBJECT], then ..." a true statement.


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