Understanding of Material Implication

Question

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.

Answer 1

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.

Answer 2

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

Answer 3

Like many, I am struggling with this. In a previous answer, the author writes:

"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."

The first sentence clearly states that a wet floor does not mean it is raining. The second states that rain does not mean the floor is wet. The third sentence that the floor is wet leads to the conclusion that the implication is true. It seems to conflict with the first two sentences. At best, it seems that the only conclusion to be drawn from the wet floor is that the floor is wet. This is my interpretation and may have very little to do with what the author meant. It only reflects my issues with translating from English to logic.

Given the vagaries of the English language, the only reasonable way to evaluate an If-Then, implication, or what ever title the sentence has as you are reading it is by what amounts to definition as show in a truth table.

The rules, or definition say that if Q is true or if P is false, then $P \implies Q$ is true.