A textbook "Language, Proof and Logic" by John Barwise and John
Etchemendy (in collaboration with Gerard Allwein, Dave Barker-Plummer
and Albert Liu)|
(was) available online at
devotes several chapters to translation from English into the language
of the first-order predicate logic. The book explains in great detail
peculiarities of translating 'each' and 'some' sentences. Sentences
such as "more than a half of the students liked the class" cannot be
expressed with the traditional quantifiers. The book explains why
(the determiner "more than a half" is irreducible) and what to do
about it. The book also examines the differences between the logical
implication and a conversational implicature.
Incidentally, the book discusses the ambiguity of a sentence like
Every minute a man is mugged in New York City.
There are two translations:
a weak reading:
all x. (Minute(x) -> exists y. (Man(y) & MuggedDuring(y, x)))
a strong reading:
exists y.( Man(y) & all x.(Minute(x) -> MuggedDuring(y,x)))
The weak reading says that every minute some man is robbed -- not
necessarily the same man. The second reading specifically says that
there is an unfortunate man who is being mugged minute after minute.
The readings are called 'weak' and 'strong' because the latter implies
the former, but not vice versa.
In the above case, the weak reading is more natural (and the one that
first comes to mind). The book points out the following "Saturday
Night Live" joke that brings up the strong reading:
"Every minute a man is mugged in New York City. We are going to
interview him tonight."
The gist of the joke is this ambiguity and the low likelihood of the strong
More interesting examples (of so-called "Donkey sentences")
"Every cube in back of a dodecahedron is also smaller than it."
The 'it' is especially troublesome. The translation therefore looks
nothing like the original sentence:
all x.(Dodec(x) -> all y.((Cube(y) & BackOf(x,y)) -> Smaller(y,x)))
"If you always do right, you will gratify some people and astonish
all x.(RightThing(x) -> DoThing(you,x)) ->
exists y. (Person(y) & Gratify(you,y)) &
all y.(Person(y) -> Gratify(you,y) || Astonish(you,y)))