Notes on computational linguistics.pdf - UCLA Department of ...

Stabler - Lx 185/209 2003
12.5 Binary relations among properties of things
Now we turn to binary quantifiers. Most English quantifiers are binary. In fact, everything, everyone, something,
someone, nothing, noone, are obviously complexes built from the binary quantifiers every, some, no and a noun
thing, one that specifies what “sort” of thing we are talking about. A binary quantifier is a binary relation
among properties of things. Unfortunately, there are too many too diagram easily, because in a universe of
n things, there are 2n properties of things, and so 2n × 2n = 22n pairs of properties, and 222n sets of pairs of
properties of things. So in a universe of 2 things, there are 4 properties, 16 pairs of properties, and 65536 sets
of pairs of properties. We can consider some examples though:
[[every]] ={〈p, q〉| p ⊆ q}
[[some]] ={〈p, q〉| (p ∩ q) = ∅}
[[no]] ={〈p, q〉| (p ∩ q) =∅}
[[exactly N]] ={〈p, q〉| |p ∩ q| =N} for any N ∈ N
[[at least N]] ={〈p, q〉| |p ∩ q| ≥N} for any N ∈ N
[[at most N]] ={〈p, q〉| |p ∩ q| ≤N} for any N ∈ N
[[all but N]] ={〈p, q〉| |p − q| =N} for any N ∈ N
[[between N and M]] ={〈p, q〉| N ≤|p ∩ q| ≤M} for any N,M ∈ N
[[most]] ={〈p, q〉| |p − q| > |p ∩ b|}
[[the N]] ={〈p, q〉| |p − q| =0and|p ∩ q| =N} for any N ∈ N
For any binary quantifier Q we use ↑Q to indicate that Q is (monotone) increasing in its first argument,
which means that whenever 〈p, q〉 ∈Q and r ⊇ p then 〈r,q〉∈Q. Examplesaresome and at least N.
For any binary quantifier Q we use Q↑ to indicate that QQis (monotone) increasing in its second argument
iff whenever 〈p, q〉 ∈Q and r ⊇ q then 〈p, r〉 ∈Q. Examplesareevery, most, at least N, the, infinitely many,….
For any binary quantifier Q we use ↓Q to indicate that Q is (monotone) decreasing in its first argument,
which means that whenever 〈p, q〉 ∈Q and r ⊆ p then 〈r,q〉∈Q. Examplesareevery, no, all, at most N,…
For any binary quantifier Q we use Q↓ to indicate that QQis (monotone) decreasing in its second argument
iff whenever 〈p, q〉 ∈Q and r ⊆ q then 〈p, r〉 ∈Q. Examplesareno, few, fewer than N, at most N,….
Since every is decreasing in its first argument and increasing in its second argument, we sometimes write
↓every↑. Similarly, ↓no↓, and↑some↓.
13 Correction: quantifiers as functionals
14 A first inference relation
We will define our inferences over derivations, where these are represented by the lexical items in those derivations,
in order. Recall that this means that a sentence like
Every student sings
is represented by lexical items like this (ignoring empty categories, tense, movements, for the moment):
sings every student.
If we parenthesize the pairs combined by merge, we have:
(sings (every student)).
The predicate of the sentence selects the subject DP, and the D inside the subject selects the noun. Thinking
of the quantifier as a relation between the properties [student] and [sing], we see that the predicate [sing] is,
in effect, the second argument of the quantifier.
237