No account? Create an account
 Buffalo proof - Ben FrantzDale [entries|archive|friends|userinfo]
Ben FrantzDale

 [ userinfo | livejournal userinfo ] [ archive | journal archive ]

Buffalo proof [Feb. 6th, 2005|01:21 pm]
Ben FrantzDale
 [ Tags | english, language, logic, nerdy ] [ music | Laguardia, Sleepover (116 Overture) ]

Continued from my previous post, a proof of the infinite number buffalo sentences:

Since I can't diagram sentences easily, let's use this notation: subject[modifier].verb(direct object). For brevity, let B be the animal, and b be the verb.

So we have:

1. .b() // Imperative: Hey you, bamboozle someone!
2. B.b() // Declarative: There are bison who bamboozle.
3. B.b(B) // There are bison who bamboozle.
4. B[B.b(~)].b() // Bison whom other bison have bamboozled tend to bamboozle. (Let the ~ mean that the thing B.b(~) is modifying is the argument of b(). That is, people[people.know(~)] is equivalent to “People whom people know”
5. B[B.b(~)].b(B) // Bison whom other bison have bamboozled tend to bamboozle bison.
6. B[B[B.b(~)].b(~)].b() // etc.
7. B[B.b(~)].b(B[B.b(~)]) // etc.

Now there are three transformations I see in use here:

(i) . ⇒ B. // This is only useful once; it gets you out of imperative but isn't useful after that.
(ii) b() ⇒ b(B) // This is also only useful once, because only the top-level b has an empty argument, the others all act on their parent.
(iii) B ⇒ B[B.b(~)]

So (ii) adds one word and leaves you with a new B leaf; (iii) adds two Bs and also leaves you with a new B leaf. As base cases, consider the sentences of length two and three above. Both have B leaves. Applying (iii) to #2 gives #4. Applying (iii) to #3 gives #5. Applying (iii) to #4 gives #6. Because (iii) preserves B leaves, ∃ sentence of length n ⇒ ∃ sentence of length n+2 ∀ n ∈ { 2, 3, 4, … }. But by example ∃ sentence of length 1 and 2.

∴ ∃ sentence of length nn ∈ { 1, 2, 3, 4, … }.

However, because #3 has two B leaves, #5 could be either B[B.b(~)].b(B) or B.b(B[B.b(~)]), so the parsing is not unique. Hence, as if it's any surprise, sentences of length five and up are ambiguous.

 From: 2005-02-06 09:41 pm (UTC) (Link)
Ben, you are a horrendous geek. I love it!!!
 From: 2005-02-06 09:45 pm (UTC) (Link)
d00d
 From: 2005-02-08 08:19 pm (UTC) (Link)
If I'm reading your notation correctly, the number 6 you have is congruent to:

"The cheese the rat the cat chased ate was moldy."

That's probably why you had trouble parsing it. This sentence is at the boundary of how many things most normal humans can hold in relation to each other in a single thought.