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

Stabler - Lx 185/209 2003
A first, basic account of what some of the semantic features is usually given roughly as follows.
1. First, we let names donote in some set of individuals e, the set of all the things you could talk about.
We can add this information to our names like this:
Titus::D -k::e
2. Simple intransitive predicates and adjectives can be taken as representing properties, and we can begin by
thinking of these as sets, as the function from individuals to truth values, e → t, that maps an individual
to true iff it has the property.
laugh::V::e → t happy::A::e → t
Simple transitive verbs and adjectives take two arguments:
praise::=D +k V::e → e → t proud::=p A::e → e → t
3. For the moment, we can dodge the issue of providing an adquequate account of tense T by simply interpreting
each lexical item in this as the identity function. We will use id to refer to the identity function, so
as not to confuse it with the symbol we are using for selection.
-s::=v T::id
We will do the same thing for all elements in in the functional categories Num, Be, Have, C, a, v, and p.
Then we can interpret simple intransitive sentences. First, writing + for the semantic combinations we
want to make
[[C(-s(be(a(mortal(Titus)))))]] = id + (id + (id + (id + (e → t :mortal+ (e :Titus))))).
Now suppose that we let the semantic combinations be forward or backward application 50 In this case,
forward application suffices:
[[C(-s(be(a(mortal(Titus)))))]] = id + id + id + id + e → t :mortal(e :Titus)))))
= e → t :mortal(e :Titus)
= t :mortal(Titus)
4. While a sentence like Titus be -s mortal entails that something is mortal, a sentence like no king be -s mortal
obviously does not.
In general, the entailment holds when the subject of the intransitive has type e, but may not hold when it is
a quantifier, which we will say is a function from properties to truth values, a function of type (e → t) → t.
To get this result, we will say that a determiner has type (e → t) → (e → t) → t. Then the determination of
semantic values begins as follows:
[[C(−s(be(a(mortal(no(Num(king)))))))]]
= id + id + id + id + e → t :mortal+ (e → t) → (e → t) → t :no+ (id + ((e → t) :king))
= e → t :mortal+ (e → t) → (e → t) → t :no((e → t) :king))
= t :no(king)(mortal)
50 An alternative is to use forward application and Curry’s lifting combinator C∗ which is now more often called T for “type raising”
(Steedman, 2000; Smullyan, 1985; Curry and Feys, 1958; Rosser, 1935).
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