# Are P and Q assumed to be independent?

Suppose P and Q are each true. Classify the following statement as True or False (choose one): If (not P) then (not Q).

I put False, the answer was TRUE

I assumed that P and Q were independent events, as it was never mentioned that they had anything to do with each other. Therefore, P has no influence on Q, so P can be false without affecting Q and vice versa.

Therefore, it is not necessarily true that negating P negates Q too.

Where did I screw up?

• Material implication does not equal causal implication. – Hayden Sep 25 '17 at 3:12
• The implication "False $\implies X$" where $X$ can either be True or False is always (vacuously) true. – Prasun Biswas Sep 25 '17 at 3:15
• $P \land Q \land \neg P$ is always false. And from a falsehood, anything at all, including $\neg Q$, can follow. It is commonly used method of proof. See my answer to a related question at math.stackexchange.com/questions/1551320/… – Dan Christensen Sep 25 '17 at 13:09

## 4 Answers

The problem opens with

Suppose P and Q are each true.

You then think about the possibility that

P can be false.

But P can't be false - the assumption is that P is true. The mistake you're making is thinking in terms of possibility, e.g. that P could be true even though it isn't, and then interpreting "If not P, then not Q" as a statement about possibilities. But that's not the sense being treated here. P is true, Q is true; so "not P" and "not Q" are both false. The question, then, is: how do we think about "false implies false"? (Or even worse, "false implies true"?)

Ultimately this comes down to how we define implication in propositional logic, but the point is that "false implies ---" is vacuously true; it might be easier to first think about why a statement like "Every purple flying elephant is the king of France" could be considered true.

One important takeaway here is that we're not thinking of "implies" in terms of causality or possibility. If you want to talk about such things, we have to go beyond propositional logic - modal logic is a good place to set up shop.

Any if-then statement beginning with "If" and then something that has a value of False (not P in this case) is what's called "vacuously true". In some sense, it doesn't matter whether or not the implication would have held if that false statement after the if were true, because it isn't.

Your reasoning is fine ... but it's just that within the context of logic, any kind of 'if .. then ..' statement is analyzed using the mathematically defined material conditional ... and that can lead to some unexpected results since thate material conditional does not always quite match the English conditional.

So, you didn't "screw up" ... in fact, you were right to question this very result, but there are also excellent reasons for analyzing 'if .. then..' using the material conditional, so you better get used to the material conditional in the context of logic.

If you want to know more about this very issue, I suggest you look up "paradoxes of material implication"

You need to go back to the definition of the implication :

$$\begin{matrix} P & Q & P \Rightarrow Q \\ true & true & true\\ true & false & false\\ false & true & true\\ false & false & true \\ \end{matrix}$$

Now let's adapt it to the proposition ' If (not P) then (not Q)', i.e. $not(P) \Rightarrow not(Q)$:

$$\begin{matrix} P & Q & not(P) & not(Q) & not(P) \Rightarrow not(Q) \\ true & true & false & false & true\\ true & false & false & true & true\\ false & true & true & false & false\\ false & false & true & true & true\\ \end{matrix}$$

You are in the case $P=true$ and $Q=true$, so $(not(P) \Rightarrow not(Q))=true$.