12

In the movie Mystery Men, there is this scene:

Captain Amazing (good guy): I knew you couldn't change.
Casanova Frankenstein (bad guy): I knew you'd know that.
Captain Amazing: Oh, I know. And I knew you'd know I'd know you knew.
Casanova Frankenstein: But I didn't. I only knew that you'd know that I knew. Did you know that?
Captain Amazing: (clears throat) ... Of course.
[source]

Amusing.

But would be the propositional logic of each sentence? Would I be correct in intuitively sensing that Casanova Frankenstein forced Captain Amazing into accepting a contradictory dead-end?

(It's been years since I once knew discrete maths.)

EDIT: Added the missing word "accepting", to better describe my initial suspicion; sorry about that.

LateralFractal
  • 315

2 Answers2

2

Let's try to simplify it a little. Ignoring the difference between "know" and "knew", using A and F for Amazing and Frankenstein instead of "I" and "you", and using P for the original proposition "F can't change" here is how I interpret the relevant parts of the third and fourth lines, namely

Amazing: "...I knew you'd know I'd know you knew" and

Frankenstein: "I didn't. I only knew that you'd know that I knew."

In both of these quotations I think that the "knew" and the end means "knew A knows P."

Writing the quotation from Amazing as "...I knew Q" where Q is the proposition "you'd know I'd know you knew" I interpret Frankenstein's saying "I didn't" as a denial of Q. So here is how I understand the exchange:

Amazing: "A knows F knows A knows F knows A knows P"

Frankenstein: "F knows A knows F knows A knows P" is false, but only "F knows A knows F knows A knows P" is true.

So in this interpretation it is Frankenstein, not Amazing, who contradicts himself.

By the way, you might be interested in the notion of (logical) common knowledge and this puzzle in particular.

Trevor Wilson
  • 17,236
1

It is Amazing who contradicts himself, but he gets himself into it by not being careful. He tries to jump a step ahead, but Frankenstein denies his presumed intervening move, and says he didn't know what Amazing just assumed he did.

Let $A$ and $F$ represent Amazing and Frankenstein, let $\supset$ represent knowledge, and let $Q$ represent the statement "[Frankenstein] couldn't change". Parts in parentheses are unstated in the dialogue.

Amazing: $A \supset Q$

Frankenstein: $F \supset A \supset Q$

Amazing: $A \supset F \supset A \supset Q$ and $A \supset F \supset A \supset F \supset (A \supset Q)$

Frankenstein: $F \not\supset A \supset F \supset (A \supset Q).$

Then Frankenstein asks, "did you know that?", when Amazing has just claimed to have known the contradiction of that. Therefore, when Amazing replies with, "of course," he contradicts himself.

G Tony Jacobs
  • 32,144
  • Oh, I see, if the "did you know that?" applies to the whole line, then Amazing contradicts himself in the reply. But aren't you ignoring the part where Frankenstein contradicts himself first, right before asking "did you know that"? – Trevor Wilson Sep 07 '13 at 02:15
  • 1
    Darn it, you're right. When Frankenstein says "I didn't", and then goes on to say that he did. Then he asks Amazing to agree with the direct contradiction he just stated, and Amazing does. Huh. – G Tony Jacobs Sep 07 '13 at 02:19
  • I guess bad guys are allowed to lie but good guys aren't :-) – Trevor Wilson Sep 07 '13 at 02:24