# Understanding of Material Implication

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.

• Could you resist verbalising $p\to q$ as "$p$ implies $q$"? In natural language, "implies" generally connotes some causality; for the material implication causality is immaterial. – Lord Shark the Unknown Feb 9 '18 at 7:21
• Yeah that makes sense, but it's just a bit hard to understand in general. Thank you – George S Feb 9 '18 at 7:32
• If you know that $q$ holds, you can for sure assert $p \to q$, because a conditional with a true consequent is always true, irrespective of the truth-value of the antecedent. There is no reaso to assume in addition that $p$ must be the "cause" or "source" of $q$. – Mauro ALLEGRANZA Feb 9 '18 at 7:36
• – Mauro ALLEGRANZA Feb 9 '18 at 7:37
• And see also the post why is $p \rightarrow q$ true if $p$ is false and $q$ is true ?. – Mauro ALLEGRANZA Feb 9 '18 at 7:41

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.

• Thank you, well explained – George S Feb 9 '18 at 7:35

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