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Suppose I have

$p, q$ as variables and lets say

$p$ is false and $q$ is false, then why and how is $p \implies q$ true?

ayylmao
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  • If you like physics to me an example could be classic physics imply $p=mv$ which is true but classic physics is false and the formula is false (the relativistic one is correct) – kingW3 May 01 '17 at 18:40
  • Physicists don't run around saying classical physics is false, they say it's the classical limit of QM/GR And thus a very useful approximation. – Triatticus May 01 '17 at 19:21

6 Answers6

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"If I have a unicorn in my closet, then he is hungry." is a true statement. Every single unicorn in my closet (none of them) hasn't been fed in a long time.

Implications $p\to q$ are true when $p$ are false; this is called "vacuously true" and is notoriously difficult for students to accept since it conflicts with their intuition.

vadim123
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Think of $p \implies q$ as an insurance policy. $p$ = an accident has occurred. $q$ = the insurance company has made a payout. The only time when the insurance company breaches the contract is when $p$ is true, but $q$ is false.

Now, $p$ = false and $q$ = false would mean that no accident occurred and no payout was made; i.e., the insurance company is still in the clear.

Also, it is helpful to know that $p \implies q$ is logically equivalent to $\neg p \vee q$.

avs
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It is true because $A\implies B$ means "whenever $A$ is true, then $B$ is also true".

So to check if $A\implies B$ is true, it is equivalent to check that whenever $B$ is false, then $A$ is also false (it is called the contraposition).

Then in your case, since $q$ is always false, and $p$ is always false, $p\implies q$ is always true.

E. Joseph
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It $p\implies q$ is not true, then it must be false. So the contrapositive $$\sim q\implies\sim p$$ would have to be false, too. But this is a "true$\implies$true" implication!

Martin Argerami
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We have $$p\implies q $$

$$\iff $$ $$\lnot \;p \lor q $$

thus

$p$ false $\implies \lnot\; p $ true

$\implies \lnot\; p \lor q $ true.

To check if $p\implies q $ is false,

$p $ must be true. if $p $ is false, you can say nothing about $q $.

$$1=3\implies 4=7$$ $$1=3\implies 4=4. $$

hamam_Abdallah
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In a strictly formal sense, one can say that given a logically false statement, any other statement is immediately true. This seems strange, but look at the following example. Take some statement $P$ like "Pigs can fly" which is false from our everyday experience. So we also know that $\neg P$ "Pigs cannot fly" is true. Now take your favorite absurd statment $Q$ like "unicorns exist" and do the following:

  1. As "Pigs can fly" is assumed to be true, we can say that the statement "Pigs can fly or Unicorns exist" is true as well, formally $P\vee Q$. We just added another statement that does not influence the truth value in any way.
  2. Now we have $P\vee Q$, but also we have $\neg P$. So we have the two options $P$ and $Q$, and from $P\vee Q$ we know that at least one must be true. But from $\neg P$ we know, that it cannot be $P$. Hence it must be $Q$ and this shows that unicorns exist.

You see that from a wrong statment I can conclude a contradiction and from this anything I want. Further, if $p$ is false, then $p\Rightarrow q$ is true because it does not matter what $q$ is because you can derive it anyway.

Just remember: from wrong assumption, anything can be derived. Hence anything must be assumed to be true because we cannot distinguish truth from falsehood.

The text above addresses the explanation from an intuitive point of view. If you are only interested in the formal background, then remember that $p\Rightarrow q$ is the same as $\neg p\vee q$, hence a false $p$ will make the whole statement true.

M. Winter
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