# Implication

One of the hazards of a specialized technical training is suffering through bad representations of it in pop culture.   I know scientists who pull their hair at the ability of TV’s CSIs to instantly extract an entire genetic sequence from a drop of blood and the USB port on their laptop and computer scientists who gouge their eyes out at the elaborate GUIs on the screens of every TV hacker.

As a mathematician, I’m usually spared this, since college-level mathematics isn’t usually fodder for prime-time popularity, Numb3rs being the exception to the rule, I suppose. That being said, however, I’m constantly bombarded by one particular example of bad logic.   It can, for example, be found in the Liza Minelli song Maybe This Time, by which, as a consequence of its being covered on the Queen B/Ladybug fave Glee, I am constantly pummeled whilst in the Queen B’s car.   The relevant bit is as follows:

(I’ll be) Not a loser
Like last time
And the time before.
Everybody loves a winner,
So nobody loves me….

Did you catch it?

The statement everybody loves a winner is more or less equivalent to if X is a winner, then Y loves X for all Y.   This is in turn equivalent to the statement that If there exists someone who doesn’t love X, then X is not a winner, and from which we can conclude that if nobody loves me then I am a loser (or, at the very least, I am not a winner.)   However, it does not follow from any of these statements that If X is not a winner then Someone doesn’t love X.   That is, the logical error made is the following untrue conclusion:

$big[P implies Qbig] qquad mbox{therefore} qquad big[neg P implies neg Qbig]$

With less symbols, this is the logical fallacy of assuming that, say, “If X is a Labrador, then X is a dog,” and then using that to conclude that “Since Doug is not a Labrador, Doug is therefore not a dog,” which need not be true.   (Doug could be a Great Dane, for example.)

Another obvious example of this error can be found in Phineas and Ferb: Summer Belongs to You, during which our protagonists are arguing with Buford the Bully about making the most out of summer:

Phineas: In this ship, we’ll travel around the world as the sun does, making this, the longest day of summer, even longer.   That’ll give us twenty-four hours of continuous daylight.   Add that to the fifteen hours we would have had anyway, and that’s nearly a forty-hour day.

Buford: It can be done.   There’s only twenty-four hours in a day, and that’s that.

Phineas: Well, yes and no.   You see Buford, if you define the day by the passage of the sun and we follow the sun by traveling around the world…

Buford: Nah, nah, nah.   Don’t try and confuse me with your sorcerer’s ways.   There’s nothing I’ve ever seen that would make me believe you could pull this off.   Except for that time machine thing… Oh! and the roller coaster…   But other than that, nothin’!   Oh!   and that time you played the song when platypus came back… Ah, man, nature just bends to your will, doesn’t it?

Phineas: Yeah.   Nothing’s impossible if you believe you can do it.

Buford: Well, I don’t believe, and therefore it’s impossible.

For such a frequently nerd-centric show, I can’t believe that go that wrong.   (As evidence of   its penchant for nerdery, consider the following bit from the very same same episode:

Candace: You know when you know someone, and you see they have another, like,   life away from you, and it fells weird.

Phineas: Like when-you-see-your-teacher-at-the-grocery-store weird, or like when-someone-you’ve-known-for-a-long-time-starts-wearing-a-cowboy-hat weird?

Candace: Uh… the first thing weird.

Phineas: That’s good, ’cause I was thinking of getting a cowboy hat…. Candace, like the song says, you have to believe you can.

Bajeet: I think Candace may have missed the song.

Buford: Yeah, man, she was in the house.

Phineas: You missed Clay Aiken and Chaka Khan?

Candace: Uh… apparently.

Phineas: That’s too bad.   But the gist of it was that you’ve got believe in yourself.

Candace: That’s easy for you, Phineas.   Look at all the things you’ve done.   Summer belongs to you.

Phineas: Summer doesn’t belong to me, it belongs to everyone.   You have to believe that.

Candace: Well, my summer’s been a series of failures.   I have a hard time believing in anything.

Phineas: Well, you got on this plane… That means you believed in us.

Ferb: And we believe in you.

Phineas: And therefor through the Transitive Property of belief, you do believe in yourself!

Hurray for binary relations!)

Of course, not everybody gets this wrong, and thankfully I can look to an even more nerd-centric cartoon for logical redemption: Futurama.   In the episode In a Gadda de Leela, we find Leela trapped under a fallen log:

Leela: Zapp!   Wake up!   We crashed on an uncharted planet.   When I woke up, I was pinned under this tree. Can you help me?

Zapp: If anyone can move it, I can.

[ Grunts and strains, but the tree doesn’t budge. ]

Zapp: No one can move it.

Then again, what else would you expect from a show that actually proved a new result in combinatorial algebra as a sight gag for an episode?

Perhaps there’s hope for pop culture yet.

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