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6AANB033
Section A
1. (a) What do the basic types e and t represent?
(b) What is the semantic type of a transitive verb?
(c) Give a recursive definition of the domain of interpretation of types.
(d) What is the domain of the type (e → t) → (e → t)?
(e) Give an example of a word which would be naturally modeled with
a constant of type (e → t) → (e → t) and describe the denotation of
that constant.
(f) Assign types to the following words:
i.
ii.
iii.
iv.
v.
vi.
John
every
cat
cuddles
violently
student
(g) Using the type assignments given in (f), show that the sentence “John
violently cuddles every cat.” has type t.
(h) The notion of semantic consequence is defined in type theory, but
not the notion of syntactic consequence. Explain this absence.
2. (a) Define α, β and η reduction.
(b) When is α-reduction necessary? Give an example of an expression
which requires α-reduction to be normalized.
(c) Normalise these expressions, showing each step that you take.
i. (λP.P (likes))(λQ.Q(j))
ii. (λx.λy(likes)(y)(x))(john)
(d) Define [[λx.α]] with respect to a model M and variable assignment
g.
(e) Argue that β-reduction is sound. Sketch a proof.
3. (a) What is the role of intensional types in formal semantics?
(b) Give a definition of the set of types used in intensional logic.
(c) Define the syntax and semantics of the intensional operators
∨
.
∧
and
(d) Give an example of sentences where the intensional operators
∨
are used.
∧
and
(e) Are
∧∨
α and
∨∧
α always equivalent to α? Explain your answers.
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TURN OVER
6AANB033
Section B
4. What are the advantages and limitations of representing the meaning of
a proposition as a function from possible worlds to truth values?
5. Discuss the limitations of First-Order Logic as a representation language
for natural languages, and how they can be overcome using higher-order
logic.
6. Discuss the role of meaning postulates in Montague semantics.
7. How is a uniform treatment of noun-phrases obtained in Montague semantics? Explain how this could be applied to the phrase “John and every
dog”.
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END OF PAPER