If it rains, John is sick. It didn't rain. $\vdash$ John wasn't sick. Is this valid?

Question

If it rains, John is sick. It didn't rain. $\vdash$ John wasn't sick.

I would say that this is false since the weather isn't directly influencing John's health. Am I right or wrong? Should I use a specific strategy in order to solve this statement?

Answer 1

The argument is an example of the fallacy of negating the antecedent (and as such, the argument is not valid).

Let $p$ denote "It rains."
Let $q$ denote "John is sick."
So $\lnot p$ denotes "It doesn't rain."
And $\lnot q$ denotes "John is not sick."

Then we have an invalid argument: $$p\rightarrow q$$ $$\lnot p$$ $$\therefore \lnot q$$

From $p \rightarrow q$, it does not follow logically that $\lnot p \rightarrow \lnot q$.

What is true is $p \rightarrow q \equiv \lnot q \rightarrow \lnot p$ (contraposition).
"If it rains, John is sick" is thus equivalent to "If John is not sick, then it isn't raining.

If we are given $p\rightarrow q$, $\lnot q$, we then can affirm therefore $\lnot p$ by modus tollens. This would be the argument $$p \rightarrow q$$ $$\lnot q$$ $$\therefore\;\;\lnot p$$

Answer 2

Let your statements be denoted as follows:

• $P : $ It is raining.
• $Q : $ John is sick.

You are given that $P\to Q$. Then you are asked to evaluate the validity of $\neg P\to\neg Q$, the inverse of $P\to Q$. You can see that $P\to Q\not\equiv\neg P\to\neg Q$ (by truth table or otherwise); thus, the deduction that John wasn't sick because it wasn't raining is in invalid deduction. However, if you were asked to evaluate the validity of $\neg Q\to\neg P$, then you could say this is valid because $\neg Q\to\neg P$ is the contrapositive of $P\to Q$, and we know $P\to Q\equiv\neg Q\to\neg P$.

Answer 3

Is John sick ONLY when it rains? Or might he also be sick in other circumstances? The statement "If it rains, John is sick" means only that it is never the case that it is both raining and John is not sick.

$P\implies Q \equiv \neg[P\land \neg Q]$

From "It is not raining," we cannot infer the John is not sick. Neither can we infer that he is sick.