will then take care of the rest. The category-to-type function type is de-
fined as follows. type(txt) = type(s) = s(st), type(n) = π and type(cn)
= π(s(st)), while type(a / b) = type(b \ a) = (type(b), type(a)) in ac-
cordance with our previous requirement. It is handy to abbreviate a type of
the form α1 (. . . (αn (s (st)). . . ) as [α1 . . . αn], so that the type of a sentence
now becomes [ ] (a box!), the type of a common noun [π] and so on.
|
expr. |
categories |
type |
translation |
|
an |
(s/(n\s))/cn ((s/n)\s)/cn |
[[π][π]] |
λPλP0([un | ]; P0(un); P(un)) |
|
non |
(s/(n\s))/cn ((s/n)\s)/cn |
[[π][π]] |
λPλP0[ | not([un | ]; P 0(un); P(un))] |
|
everyn |
(s/(n\s))/cn ((s/n)\s)/cn |
[[π][π]] |
λPλP 0[ | ([un | ]; P 0(un)) ⇒ P (Un)] |
|
Maryn |
s/(n\s) |
[[π]] |
λP([un | un is mary] ; P(un)) |
|
hen |
s/(n\s) |
[[π]] |
λP(P(un)) |
|
himn |
(s/n)\s |
[[π]] |
λP(P(un)) |
|
who |
(cn\cn)/(n\s) |
[[π][π]π] |
λP0λPλv(P(v) ; P0(v)) |
|
man |
cn |
[π] |
λv[ | man v] |
|
stinks |
n\s |
[π] |
λv[ | stinks v] |
|
adores |
(n\s)/n |
[ππ] |
λv0λv [ | v adores v0] |
|
if |
(s/s)/s |
[[][]] |
λpq [ | p ⇒ q] |
|
. |
s\(txt/s) |
[[][]] |
λpq (p ; q) |
|
and |
s\(s/s) |
[[][]] |
λpq (p ; q) |
|
or |
s\(s/s) |
[[][]]________ |
λpq [ | p or q] |
Table 1: The Lexicon
In Table 1 the lexicon for a limited fragment of English is given. The sen-
tences in this fragment are indexed as in Barwise [1987]: possible antecedents
with superscripts, anaphors with subscripts. The second column assigns one
or two categories to each word in the first column, the third column lists
the types that correspond to these categories according to the function type
and the last column gives each word a translation of this type. Here P is a
variable of type [π], p and q are variables of type [ ], and v is a variable of