# Clarification on necessity

I am currently just being introduced to the concept of logic in mathematics and my lecturer was talking about "if...then" statements.

For example, for some statements $$P$$ and $$Q$$, "if $$P$$ then $$Q$$".

I think sufficiency is quite intuitive, but why do we say that $$Q$$ is a necessary condition for $$P$$? Wikipedia provides one reason to be that it is impossible to have $$P$$ without $$Q$$, but why can't I have $$P$$ standalone? I mean, it is a statement in itself after all, right? Would it not make sense if I just wrote a $$P$$ down as a sentence? Why does $$Q$$ have to exist for $$P$$ to exist?

Perhaps if I can be provided with a detailed explanation, as well as an intuitive one, that will be nice :)

• If $Q$ is false , then $P$ cannot be true , since then the implication $P\implies Q$ would be false. So, $P$ can only be true, if $Q$ is true, in other words, $Q$ must necessarily be true, or in short, is necessary. – Peter Jan 25 '20 at 8:53
• @Peter thank you so much :) – Ethan Mark Jan 25 '20 at 9:37

If $$\underbrace {one\space is\space the\space strongest\space man\space alive}_P$$ $$\underbrace{then}_{\implies}$$ $$\underbrace{one\space is\space strong.}_Q$$

Because one can't be the strongest if one isn't strong.

Imagine these two statements:

1. There exists a strongest man in the world.
2. There exists no man that is strong.

I hope this makes the contradiction obvious.

And also that argument from Peter in the comments, very important thing.

• Thank you for your insight! This makes sense! – Ethan Mark Jan 25 '20 at 9:37

We are told $$P\implies Q$$.

There are $$4$$ hypothetical possibilities.

1. $$P$$ is not true and $$Q$$ is not true.

This could happen.

1. $$P$$ is not true and $$Q$$ is true.

The could happen.

1. $$P$$ is true and $$Q$$ is not true.

This can not happen. We we told $$P\implies Q$$ so if we have $$P$$ we must have $$Q$$ be true. This is just not possible.

1. $$P$$ is tru and $$Q$$ is true.

Now in every single case where $$P$$ is true we had to have $$Q$$ also be true. So.... $$Q$$ is necessary for $$P$$.

That's all.

.....

but why can't I have P standalone?

Well, what does $$Q$$ do, while we have $$P$$ stand by itself? We can ignore $$Q$$ and not look at $$Q$$ but and we can choose to not invite $$Q$$ to our party, but we can't make $$Q$$ ... disappear. Now matter what we do with $$P$$, $$Q$$ is still going to exist and either be true or false even if you don't look at it.

So we have $$P$$ at our party and there it is... being true. That's fine but someone might wonder "Hmm, I wonder what $$Q$$ is doing" and everyone else can ignore $$Q$$ but, $$Q$$ will be sitting on the other side of the hill doing it's $$Q$$ thing of being true or false. So someone wonders over to check on $$Q$$ and what does she see?

Well, $$P\implies Q$$ so, lo and behold, $$Q$$ is being true. It doesn't matter if we look at $$Q$$. So long as $$P\implies Q$$ then if we ever have $$P$$, it will always be that $$Q$$ is true, whether we look at it or not.

And since every time $$P$$ is true, $$Q$$ is true, we can't have $$P$$ be true without $$Q$$ being true (even if we stick $$Q$$ is a little cottage three thousand miles away and pass a law that no-one is allowed to check in on him-- $$Q$$ will just sit there being true all by himself unnoticed and unloved), so $$Q$$ is necessary for $$P$$ to be true.