Intuitive reason of why the negation of $P \Rightarrow Q$ is $P \land \neg Q$ instead of $P \Rightarrow \neg Q$

Question

I'm trying to figure out in an intuitive way the reason why the negation of $P\Rightarrow Q$ is $P \land \neg Q$ and not $P \Rightarrow \neg Q$; I'm trying to figure out with phrases like "if today is sunny then this afternoon I'll go to the sea", using the correct negation I get "today is sunny and this afternoon I will not go to the sea", but to me it seems the same of "if today is sunny then this afternoon I will not go to the sea".

Can I have an help? Thanks.

Answer 1

Here's the example I give to my students: there's a rule that says that "If a traffic light is red, then you must stop". Recall that this is what implication means: $P \implies Q$ means "if $P$, then $Q$".

To break that rule, you must run a red light. There's absolutely no way to break that specific rule when the light is not red, no matter how rebellious you feel.

So to make $P \implies Q$ false, you have to be in a situation where $P$ is true and, nevertheless, $Q$ is false, which is exactly the same as saying that $P \wedge \neg Q$ is true.

I try to make my students remember that rule by using the "slogan":

You cannot run a green light.

Answer 2

Like half intuitive: $P \implies Q$ is equivalent to $\neg P \lor Q$ because the implication is true when

• the premise $P$ is false: Then you can imply everything and whatever $Q$ is, $P \implies Q$ is true. In other words: A false statement can imply a false statement and a true statement.

• the conclusion $Q$ is true: Both a true and a false statement can imply a true statement. So when the conclusion is true, the implication is true - whatever the premise $P$ is.

So the negation has to be $\neg(\neg P \lor Q)$ and that is $P \land \neg Q$.

Your real world example:

today is sunny $\implies$ you go to the sea

If this implication is true, what are the cases which can occur?

• It is sunny and you go to the sea. Yeah. That's the trivial case.

• It isn't sunny and you don't go to the sea. That's possible. We only made a statement about the sunny situation.

• It is sunny and you don't go to the sea. NO! We said, if it is sunny (and that is the case), you go to the sea.

• It isn't sunny and you go to the sea. That is the thing: That's possible. We said you will you to the sea if it is sunny. But we didn't say something about what you do when it is not sunny. If it's sunny, you go, if it is not - nobody knows.

If it is sunny, I know that you went to the sea. But when you are at the sea, I don't know if it is sunny (that means I cannot conclude that it is sunny when I see you at the sea).

What is now the negation of the implication? That is exactly the third statement. We said, always when the sun shines, you go to the sea. So the correct negation is the case that the sun shines but the implication isn't fulfilled - you don't go to the sea.

Let $P$ be it is sunny and let $Q$ be you go to the sea, then $$ \neg(P \implies Q) \iff P \land \neg Q $$ as we saw.

That is not the same as $P \implies \neg Q$ which means always when it is sunny, you do not go to the sea. We only considered the negation of the one implication (not always when it is sunny, you go to the sea) and didn't make a statement about all non-sunny days (always when it is sunny, you don't go to the sea is something different!).

The negation sais: There is one situation where it is sunny (that's $P$) but you do not go to the sea (that's $\neg Q$). So it is not the case that $P \implies Q$.

Answer 3

Here is my favourite example to introduce a material (logical) implications:

If it is raining, then it is cloudy.

$Rain \implies Cloudy$

This does not mean that rain causes cloudiness. It means only that, at the moment (usually the present), is not both (a) raining and (b) not cloudy.

$Rain \implies Cloudy \space \space \equiv \space \space \neg [Rain \land \neg Cloudy]$

From your examples:

1. $P \implies Q \space \space \equiv \space \space \neg [P \land \neg Q].~~$ So, the negation would be $P \land \neg Q$
2. To fit into the above framework, you sailing example would have be recast as follows: If it is a sunny afternoon, then I am sailing: $$SunnyAfternoon \implies Sailing$$ Its negation would be: $$SunnyAfternoon \land \neg Sailing$$This implication will be false if and only if it is a sunny afternoon and I am not sailing.

Answer 4

A very intuitive reason:

The best way to prove an implication is false is to find a counterexample, i.e. an example where the premise $P$ is true and yet, the conclusion $Q$ is false. Formally this translates precisely as $\;P\wedge \neg Q$.

Answer 5

Another consideration. We quite often establish that a proposition $P$ is false by arguing like this (here using $\to$ to abbreviate an ordinary-language conditional):

1. Mumble, mumble, mumble; so $P \to Q$.
2. More mumble, mumble, mumble; so $P \to \neg Q$.
3. Ahah! Hold on!! The truth of $P$ would give us both $Q$ and $\neg Q$ and that's impossible -- so $\neg P$!!!

Now if this sort of argument is indeed to show the truth of $\neg P$, we need both the premisses $P \to Q$ and $P \to \neg Q$ to be true together. They couldn't be true together if one really were the negation of the other. So in general, $P \to Q$ can't rule out the simultaneous truth of $P \to \neg Q$.

Answer 6

1. A sentence and its negation, by definition, form an inconsistent pair; that is, they have opposite truth values in all interpretations.

2. When condition $\boldsymbol P$ is satisfied, the conditionals $$P\implies Q\tag1$$ and $$P\implies ¬Q\tag2$$ certainly have opposite truth values.

   This is because a conditional sentence $(P{\implies}R)$ is equivalent to its consequent $(R)$ when its premise $(P)$ is true.

3. When folks are mistakenly thinking that the sentences $(1)$ and $(2)$ contradict each other, they are fact implicitly assuming that premise $P$ holds; that is, they are forgetting that $P$ need not hold, in which case sentences $(1)$ and $(2)$ actually have the same truth value.

4. $$P\land\lnot Q\tag{1n}$$ On the other hand, sentences $(1)$ and $(1n)$ are correct negations of each other, as they have opposite truth values regardless of interpretation.

5. To be clear: sentences $(1)$ and $(2)$ do not have opposite truth values in an interpretation in which $P$ is false, for example, when $P$ means “cats bark”.