# Formalize “Buffalo buffalo buffalo buffalo buffalo”

Source: p 145. Sweet Reason: A Field Guide to Modern Logic (2010 2 ed) by Henle, Garfield, Tymoczko.

"Buffalo" sentences (pp. 73, 101, and 139) can be formalized if we establish two conventions. First, that the noun "buffalo" should be interpreted as "all buffalo" (or "all bison"). Second, if there is no object to the verb "buffalo", then we interpret it as "buffalo some buffalo" (or "intimidate some bison"). So "Buffalo buffalo" is "All bison intimidate some bison". Let Bx mean that x is a buffalo (or bison). Let Ixy mean that x buffalos (or intimidates) y. Then we can express "Buffalo buffalo" as ∀x ( Bx ⟹ ∃y(By ∧ Ixy) ).

1. Formalize "Buffalo buffalo buffalo buffalo buffalo" where we mean "All bison intimidate all bison, that all bison intimidate".

[ p 360 : ] 17. ∀x ( Bx ⟹ ∀y ( ( By ∧ ∀z (Bz ⟹ Izx) ) ⟹ Ixy) )

The following was my attempt; I added colour and changed the brackets to ease reading.

Why should the red y instead be x (per the above answer)? I wrote the red y because the green restrictive relative clause modifies bison2, and not bison1.

• You are right, this is an error, the original referent (x) is the subject and the pronoun refers back to the direct object (y) of the first verb, so in no place is the subject (x) the direct object of the verb. In any good reformulation, there would never be and I whose second argument was x. – jobermark Aug 6 '16 at 15:17
• That said, this is still poorly constructed. Where does the existential quantifier from the stated interpretation of "Buffalo buffalo" go when we start conflating sets? "Children eat food mothers prepare" means "For child c, for food f if c eats f there exists mother m who preprepares food f". Where is this final existential in the case in question? – jobermark Aug 6 '16 at 15:24
• intimidate has two senses, one transitive and the other deponent. If people severely intimidate you in the transitive sense, that is mass oppression. If people severely intimidate you in the deponent sense, you have a social anxiety disorder. The book's answer only makes useful sense by conflating the two incompatible meanings onto a single predicate. – jobermark Aug 6 '16 at 15:43
• I'm pretty sure this question is off-topic, being about semantics. It belongs on the linguistics SE – commando Aug 6 '16 at 15:59
• Translations in and out of logical formalism do traditionally belong here, rather than in math or language SE's, even if the difficulties are caused by mathematical or language problems rather than logical ones. (How often do we discuss misinterpretations of Godel, Quantum Theory, or Latin terms in Augustine?) – jobermark Aug 8 '16 at 17:40