[music]
If you need a simple introduction to truth-functional symbolic logic,
click here instead. This video is about two difficult translations that
tend to cause headaches for most logic students. Number one:
"both not" versus "not both". In English, word order matters.
"Both not" is entirely different from "not both".
It's easy to see with an example: Alex and Betty are both not going to the
party. Think about it. Who's going to the party?
Not Alex. And not Betty. Why? Because we just said they're both not
going. So we translate this to "Not-A
and not-B" because Alex is not going and Betty is not going. They're both
not going to the party. But notice "both not" is very different from "not both".
Consider: Carla and David are not both going to the party. What does this mean?
Well, it's not clear whether Carla is going and we're not sure whether David
is going but one thing is certain... both of them
together is something you will not see at the
party. Why? Because they're not both going.
So how does this translate to symbolic logic?
Like this. Here in the parentheses it says,
"Both C and D" but the tilde out here negates whatever
is in the parentheses. Since this says "both", this
says "not both". So this is the correct translation for,
"Carla and David are not both going to the party."
Let's compare. Here is the "both not" construction and here is the "not both"
construction. Quick practice: At the birthday party
David says, "Mom, can I have cake and ice cream?" and
apparently being the meanest mom in the world,
she says, "No." If C stands for "You can have cake"
and I stands for "You can have ice cream," how would you express
mom's answer in symbolic logic? Hit pause and give it a try.
Well, David asked for cake and ice cream. Let's start there. This says
"You can have cake and you can have ice cream,"
but mom said David can not have cake and ice cream...
so do we put the negation here? This says "You cannot have cake and
you can have ice cream." But that's not what mom said.
So where did we go wrong? We forgot to add parentheses.
Enclose "You can have cake and ice cream,"
and then negate it. This says "Here is something that is not true: you
can have cake and ice cream." Alternatively, you could
understand it like this" "Cake and ice cream are something you're
not getting." In other words, David asked for both
and mom denied the request. So her answer was,
"Not both". This is the correct translation. But don't feel too bad for David. She
said he can not have both but she didn't say that cake and
ice cream are both things he's not having. There is
a difference between "not both" and "both not". In other words, she didn't
say "You're not having cake and you're not
having ice cream." This is the "both not" construction and
lucky for David, that's not what mom said. She just said
that both together is something that ain't
happening. But we can hold out hope for David that
she lets him have just ice cream... or just cake. Number two:
"Not either" versus "iEither not". Let's say David isn't about to give up
yet, so he changes up the question, "Okay then, mom, can i have either cake or
ice cream?" And despite being at a birthday party,
let's say mom bizarrely says, "No." Hit pause and try to express mom's
answer in symbolic logic. How would it go? David asked for either
cake or ice cream, but mom said no, so we need a tilde.
Where does it go? Right here. "You're not having cake or
ice cream." Or think of it this way, "You can have cake or ice cream...
just kidding!" But no matter how you think of it,
the result is the same: poor David is not having ice cream
and not having cake. He's getting neither of them.
Did you catch that? We just stumbled on something important.
The "not either" construction--- sometimes we just say "neither"---is logically equivalent to the both not construction.
In other words, "neither" and "both not" mean
the same thing, logically speaking. Quick example: is either Alex or Carla
going to the party? No. Not either of them. That is,
neither of them is going. Which is logically the same as saying,
Alex is not going and Carla is not going. They're both not going. So that's the
"neither" or "not either" form, which is very different from "either not."
Back to David. He's asking for cake and ice cream again.
But this time mom says, "No. You have to either
not have cake or not have ice cream." In symbolic logic, either not cake or not
ice cream. Did you spot the equivalence? The "either
not" construction is logically equivalent to
the "not both" construction. It's easy to see why. If you have to
either not have one or not have the other,
then obviously... you're not having both! Let's close with a rapid fire review.
Neither Betty nor Carla is going to the party.
Either Alex is not going to the party or David is not going.
Both Betty and Carla are not going to the party.
Alex and David are not both going so you might see one at the party
or you might see the other... but not both. I hope you found this video helpful and
entertaining. If not, then I was either not helpful or
not entertaining. Fair enough---quality videos are hard to
make! But if I was neither helpful nor entertaining...well,
my apologies for the flop.
[music]