1
$\begingroup$

I just started doing logic and everything is going fine. But we just got to Material Implication and I don't really understand it. From for example $q$, we can get $q \lor \lnot p$ by Disjunction, and from there, using Material Implication, we can get $p \to q$.

The thing I don't understand is how is this valid. Let's say

  • $p$ is "it's raining", and
  • $q$ is "the floor is wet",

and $p \to q$ holds. But just because the floor is wet, doesn't mean that it's raining. So how can we conclude that $p \to q$ just from $q$? I'm sorry if it's confusing.

$\endgroup$
4
$\begingroup$

If you know $Q$ holds, then $P\to Q$ says nothing about $P$.

A perhaps better way of thinking about it is the following. Let's say that given $R$ you could prove $Q$. Then clearly if I give you both $R$ and $P$, you could still prove $Q$ by simply ignoring $P$. Since you're not actually "using" $P$, it doesn't matter whether it is true or not.

$\endgroup$
  • $\begingroup$ Thank you, well explained $\endgroup$ – George S Feb 9 '18 at 7:35
1
$\begingroup$

In your example, if the floor is wet, you are right that we cannot conclude it is raining. Neither can we conclude that rain will eventually cause the floor to be wet. Given that the floor is wet, we can conclude, however, that the implication "if it is raining then the floor is wet" is true.

Yes, it's a bit counter-intuitive, but, in general, if $P$ and $Q$ are logical propositions that are unambiguously either true or false in the moment, then we can easily prove that $Q\implies [P \implies Q]$.

In words, that which is true follows from anything, be it true or false. Similarly, anything follows from a falsehood.

See my answer at: Arguments pro material implication

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.