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Logic Makes Me Die Inside

Can I logically valid argument have true premises and a false conclusion? Let me put this another way ...
Do you use facebook? I’m not on it. I already have a friend, so it didn't seem necessary.
But I was very proud to learn (from a TA) that one year some of my students set up a facebook group for logic. How cool is that? You love logic so much that you have a facebook group to celebrate. I'm like the Burial of logic lecturers ...
Or not. But let's have a closer look at the argument.
One premise, one conclusion.
The funny thing about this argument is not just that there's a counterexample to it, ...
... but that the actual world is a counterexample to it.
So if you think about it, the facebook group was actually for people who love logic after all.
Earlier I asked, Can a logically valid argument have true premises and a false conclusion?
Please think about this question before writing about logic.
So what was logic making me die inside really about? I think it was probably intended to be a response to the logical validity of two arguments, this one ...

P ∧ ¬P

... and this one.
If you are really good, you'll be able to prove both arguments by the end of the course.
In fact, you could prove the argument on your left by the end of today.
In fact, let's see how to do that right away (in unit 270).


P ∨ ¬P