Assignment 4
PLIN3004/PLING218 Advanced Semantic Theory
1 Semantic Types
(Heim & Kratzer 1998: p.40)
Replace “?” in each of the following statements (you can use the shorthand et :“ xe, ty)
e.g. rλ f P Dxe,t y .rλ x P De . 1 iff f pxq “ 1 and x is grayss P D? — xxe, ty, xe, tyy (or xet, ety)
i)
rλ f P Dxe,xe,t yy .rλ x P De . 1 iff f pxqpAnnq “ 1ss P D?
ii)
rλ y P De .rλ f P Dxe,t y .rλ x P De . 1 iff f pxq “ 1 and x is in ysss P D?
iii) rλ f P Dxe,t y . 1 iff there is some x P De such that f pxq “ 1s P D?
iv) rλ f P Dxe,t y . Marys P D?
v)
rλ f P Dxe,t y .rλg P Dxe,t y . 1 iff there is no x P De such that f pxq “ 1 and gpxq “ 1ss P
D?
2 Computation
Assume the following lexical entries.
vJohnw M1 “ j
vMaryw M1 “ m
vkissedw M1 “ rλ y P De . rλ x P De . 1 iff x kissed y in M1 ss
Using the two compositional rules, the Branching Node Rule and Non-Branching Node Rule,
compute the denotation of (1) in model M1 . Do not omit any step, i.e. in each step you may
perform only one replacement. You may do the computation top-down or bottom-up.
S
(1)
VP
NP
NP
John kissed
Mary
3 (Optional) ‘If’
In the lecture, we analyzed the denotations of ‘and’ and ‘or’ based on the meanings of logical
connectives ^ and _, respectively. In this optional exercise, you will develop a similar analysis
for ‘if’, based on logical connective Ñ.
Firstly, let us assume the following syntax.
(2)
S
S
Cond
if
DP
S
DP
VP
VP Mary smoked
John left
1
In order to generate this sentence, we need the following syntactic rules:
(3)
a.
b.
S Ñ Cond S
Cond Ñ if S
a) The first thing to do is to figure out the semantic type of ‘if’. Annotate each constituent in
the following tree with its semantic type. Don’t leave out the semantic types of the terminal
nodes.
???
(4)
???
???
???
???
if???
???
???
???
Mary??? smoked???
John??? left???
b) Recall that the logical connective Ñ has the following meaning.
p Ñ q iff p “ 0 or q “ 1
Based on this, formulate the denotation of ‘if’.
For any model M, vifw M = ???
2