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]]
10
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