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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?

Alec
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  • this is the principle of explosion. It's unintuitive – wlad May 25 '15 at 18:23
  • Well they say that one could deduce anything based on false knowledge – alkabary May 25 '15 at 18:23
  • if $1=2$ then $2 =3$, you see now how it might work ? – alkabary May 25 '15 at 18:24
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    This is a duplicate of many previous questions... – anon May 25 '15 at 18:25
  • In another system of logic-of your own making-it sure can be true,even a truism.. – MathematicianByMistake May 25 '15 at 18:26
  • @alkabary - No, because if 1=2, by mathematical operations, we can preserve the "true-ness" of the equation by adding 1 to both sides, which gives us 2=3. But in the original example, I don't see such an inference. – Alec May 25 '15 at 18:29
  • it might be more intuitive to think about it in disjunctive form, one of these two statements must be true: either 'pigs can't fly' or 'I'm king'. – Set May 25 '15 at 18:32
  • @Alec not over the integers. We can derive a bunch of fantastic things like that $1 < 1$. In fact in classical logic you can derive everything from it. – wlad May 25 '15 at 18:32

2 Answers2

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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.

  • 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. – Joffan May 25 '15 at 18:33
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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.

ZMoltion
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  • Your answer is not wrong, but your wording could be distilled to strengthen your answer. Do away with "hypothetical", it suggests the object may exist, creating potential doubt about the falsity of P. Whether or not your conclusion is "reasonable" has no bearing on the truth of the implication, so I would not use it. Ultimately, it is the absolute falsity of P that does not allow an alternative observation to the stated Q, so we cannot not disprove the implication. The implication remains true on this reasoning alone. – spacedustpi Jan 25 '20 at 13:08