PLIN3004/PLING218 Advanced Semantic Theory
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Lambda-Notation
Assignment 3
(adopted from Heim & Kratzer 1998: p. 39)
Describe the following functions in words. E.g., rλ x : x is an individual in M. 1 iff x smokes in Ms
is the function that maps any individual in M to 1 if he or she smokes in M, and to 0 if he or
she doesn’t smoke in M.1
i)
λ x P N. 1 iff x ą 3 and x ă 7
ii)
λ x : x is a peson. x’s biological father
iii) λ X : X is a set of individuals. t y | y is a man and y R X u
iv)
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λ X : X P ℘pDq. rλ y : y is a man. 1 iff y R Xs
λ-conversion
Simplify the following descriptions by λ-conversion.
i) rλ x P N. x ˆ 5sp12q
ii)
rλz P N. rλ y P N. rλ x P N. px ` yq ˆ zsp8qssp5q
iii) rλz P N. rλ y P N. rλ x P N. 1 iff x ą 3 and x ă 7spyqspzqs
iv) rλ x P D rλ y P D. 1 iff y likes xsspAnnq
v)
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rλ x P D. rλ y P D. 1 iff y likes xspAnnqs
Computation
Assume a model M such that vJohnw M “ j where j is some individual who didn’t leave in M.
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Also, for any model M 1 , vleftw M “ rλ x P D. 1 iff x left in M 1 s. Then, we can compute the
denotation of the sentence ‘John left’ in M as follows:
<M
4
S
>
6
>
6
6 DP VP >
=
5
John left
4
< M ¨4
<M ˛
DP
VP
= ˚
= ‹
“5
(by Branching Node Rule)
‚
˝5
John
left
¨4
<M ˛
˚ DP = ‹
“ vleftw M ˝5
(by Non-Branching Node Rule)
‚
John
“ vleftw M pvJohnw M q
“ vleftw M p jq
“ rλ x P D. 1 iff x left in Msp jq
“0
(by Non-Branching Node Rule)
(because vJohnw M “ j)
(Lexicon)
(λ-conversion)
1N is the set of natural numbers, and x R S means x is not a member of S. D is the set of all individuals in a
give model.
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Compute the denotation of “Mary smokes” in M. Assume that vMaryw M “ m, an individual
1
who smokes in M, and for any model M 1 , vsmokesw M “ rλ x P D. 1 iff x smokes in M 1 s. You
may use the bracket notation instead of tree diagrams.
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