# Entertaining examples of multiply quantified statements

I am teaching a discrete math course, and doing multiply quantified statements. All the book examples are sober and forgettable:

Every real number has a reciprocal.

For all triangles x [in a "Tarski world" with a bunch of shapes of different colors], there is a square y such that x and y have the same color.

There is no smallest positive real number.

And so on. These are good, but I would like to supplement these with examples from pop culture, song lyrics, contemporary politics, etc. -- especially anything amusing, complicated, and/or which drastically changes meaning when you change the order of the quantifiers. Something the students might remember the day after the lecture.

Usually I am good at this kind of thing but for now I am drawing a blank. Any ideas?

• Zero does not have a reciprocal. Feb 11 '15 at 7:17

For single quantifiers, there is the standard -- jazz standard -- example:

• Everybody loves my baby.
• My baby loves nobody but me.

For multiple quantifiers, a classic is:

• You can fool all of the people some of the time
• You can fool some of the people all of the time
• But you can't fool all the people all the time.

You could have students draw up the quantifications for the following, taken from Alice's Adventures in Wonderland.

‘Then you should say what you mean,’ the March Hare went on.

‘I do,’ Alice hastily replied; ‘at least--at least I mean what I say--that’s the same thing, you know.’

‘Not the same thing a bit!’ said the Hatter. ‘You might just as well say that “I see what I eat” is the same thing as “I eat what I see”!’

‘You might just as well say,’ added the March Hare, ‘that “I like what I get” is the same thing as “I get what I like”!’

‘You might just as well say,’ added the Dormouse, who seemed to be talking in his sleep, ‘that “I breathe when I sleep” is the same thing as “I sleep when I breathe”!’

Or the following, from Through the Looking Glass.

‘Who did you pass on the road?’ the King went on, holding out his hand to the Messenger for some more hay.

‘Nobody,’ said the Messenger.

‘Quite right,’ said the King: ‘this young lady saw him too. So of course Nobody walks slower than you.

‘I do my best,’ the Messenger said in a sulky tone. ‘I’m sure nobody walks much faster than I do!’

‘He can’t do that,’ said the King, ‘or else he’d have been here first. However, now you’ve got your breath, you may tell us what’s happened in the town.’

• I approve of using Alice! (Second one is from Through the Looking Glass though.) Feb 9 '15 at 8:33
• Oops, you are right. I glanced too quickly at the heading. I'll edit my answer. Thanks for the catch! Feb 9 '15 at 9:24

There's a lovely bit from Catch-22 that I'm quite fond of (I'll boldface the relevant part, but you really need the full passage to appreciate how wonderfully Carollian it is):

"All right," said the colonel. "Just what the hell did you mean?"

"I didn't say you couldn't punish me, sir."

"When what, sir?"

"Now you're asking me questions again."

"I'm sorry, sir. I'm afraid I don't understand your question."

"When didn't you say we couldn't punish you? Don't you understand my question?"

"No, sir. I don't understand."

"You've just told us that. Now suppose you answer my question."

"But how can I answer it?"

"That's another question you're asking me."

"I'm sorry, sir. But I don't know how to answer it. I never said you couldn't punish me."

"Now you're telling us when you did say it. I'm asking you to tell us when you didn't say it."

Clevinger took a deep breath. "I always didn't say you couldn't punish me, sir."

"That's much better, Mr. Clevinger, even though it is a barefaced lie. Last night in the latrine. Didn't you whisper that we couldn't punish you to that other dirty son of a bitch we don't like? What's his name?"

"Yossarian, sir," Lieutenant Scheisskopf said.

"Yes, Yossarian. That's right. Yossarian. Yossarian? Is that his name? Yossarian? What the hell kind of a name is Yossarian ?"

Lieutenant Scheisskopf had the facts at his finger tips. "It's Yossarian's name, sir," he explained.

Everybody doesn't like something. But nobody doesn't like Sara Lee.

and

There isn't a car on the road that shouldn't be using it.

Another song: Happy Together, by The Turtles

The lines: I can't see me lovin' nobody but you For all my life.

And: The only one for me is you And you for me.

(The fact that he says the same thing twice, instead of changing direction, always bugged me.)

From "Maybe This Time" from the musical Cabaret: "Everybody loves a winner/So nobody loved me." (This exhibits, of course, a very common mistake in quantifier manipulation, which bothers me every time I hear the song.)

Raymond Smullyan gave a nice example in one of his books (I forget now which): Two restaurants have signs on their doors. One says

"Good food is not cheap."

The other says

"Cheap food is not good."

The two statements are equivalent, but one makes you think of delicacies, while the other about disgusting greasy bites...

• Funny! I think the apparent dissonance or paradox between the two interpretations/connotations is resolved (and the humor ruined) by noting that they are just two reflections or facets of one statement of nonexistence: that there is no food that is cheap and good. This symmetric version feels more neutral yet says the same thing. Nov 19 '15 at 22:55
• Incidentally, I also find that a similar statement (namely, that no function is simultaneously entire, bounded, and non-constant) of Liouville's theorem from complex analysis helps prevent moments such as this. Typically it's rendered in the form $(P \wedge Q) \Longrightarrow \neg R$ for whatever reason, whereas $\neg (P \wedge Q \wedge R)$ seems less arbitrary at least to me. Nov 19 '15 at 22:59

This one might take some extra massaging to make it "mathy" (re: "half" and "part" and "some"), but I like it as a fun twist on Abe Lincoln's classic (which @MattF used in his answer):

Half of the people can be part right all of the time

Some of the people can be all right part of the time

But all of the people can't be all right all of the time

I think Abraham Lincoln said that

Bob Dylan, in "Talkin' World War III Blues", from The Freewheelin' Bob Dylan

• "Some" and "all" are good, but there is no compact way to formalize "half of" with these sorts of quantifiers.
– user173
Feb 9 '15 at 15:41

Phew!... I got first to enter this answer:

The "Who's on First?" routine by Abbott & Costello.

Can anybody find me somebody to love?

• Those are good quantifiers! I think the question is whether (exists x)(exists y)(x can find y and I can love y), so it's a pity that the quantifier order is irrelevant.
– user173
Feb 12 '15 at 4:14
• The system accused this post as being low-quality. No love from the system. Feb 12 '15 at 16:53