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 :)

  • 2
    $\begingroup$ 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. $\endgroup$ – Peter Jan 25 '20 at 8:53
  • $\begingroup$ @Peter thank you so much :) $\endgroup$ – 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.

  • $\begingroup$ Thank you for your insight! This makes sense! $\endgroup$ – 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.


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