Categorial Grammar and Discourse

type π.

Let us see how this immediately provides us with a semantics. We have
seen before that our Lambek analysis of (1) provides us with a semantic
recipe that is reprinted as (2) below. If we substitute the translation of a¹ ,
λP⁰λP ([u1 | ] ; P⁰(u1 ) ; P(u1 )) for D in the succedent of (2) and substitute
λv [ | man v] for P, we get a lambda term that after a few conversions reduces
to (3). This can be reduced somewhat further, for now the merging lemma
applies, and we get (4). Proceeding further in this way, we obtain (5), the
desired translation of (1).

(1) A¹ man adores a² woman

(2) D,P,R,D0,P0' D(P)(λv.D⁰(P⁰)(λv⁰.R(v⁰)(v)))

(3) [u₁ | ] ; [ | man u₁] ; D⁰(P⁰)(λv⁰.R(v⁰)(u₁))

(4) [u₁ | man u₁] ; D⁰(P⁰)(λv⁰.R(v⁰)(u₁))

(5) [u₁ u₂ | man u₁ , woman u₂, u₁ adores u₂]

(6) Every¹ man adores a² woman

(7) [ | [u1 | man u1 ] ⇒ [u2 | woman u2 , u1 adores u2]]

(8) D,P,R,D⁰,P0' D0(P0)(λv0.D(P)(λv.R(v0)(v)))

(9) [u2 | woman u2 , [u1 | man u1 ] ⇒ [ | u1 adores u2]]

(10) A¹ man adores a² woman. She₂ abhors him₁

(11) [u₁ u₂ | man u₁ , woman u₂, u₁ adores u₂ , u₂ abhors u₁]

(12) If a¹ man bores a² woman she₂ ignores him₁

(13) [ | [u1 u2 | man u1 , woman u2, u1 bores u2] ⇒ [ | u2 ignores u1]]

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