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SECTION A
6AANB033
1. (a) Assign appropriately typed lambda terms (that is elements of con) to the
following words:
‘John’, ‘Miriam’, ‘every’, ‘cat’, ‘black’, ‘believes that’ (considered as one
word), and ‘adores’.
(b) Now give the terms that correspond to Every black cat adores John and
John believes that every cat adores mary and show that they are of type
‘t’.
(c) If α is the term you have assigned to ‘adore’, then define a term corresponding to ‘be adored by’ using only α as a constant. Now give terms
γ, γ 0 that correspond to ‘John adores Miriam’ and ‘Miriam is adored by
John’ respectively, such that ` γ ⇒ γ 0 .
2. Let α ∈ con(e→(e→t)) and β = λx.(λy.(α(x))(y)) be terms of the simply typed
lambda calculus. Prove the following statements
(a) α ≡ β
(b) ` α ⇒ β
(c) ` (β(y))(x) ⇒ (α(y))(x).
3. (a) Define α- β- and η-reduction for the simply typed lambda calculus and
motivate the variable conditions.
(b) The proof of the soundness of β-reduction in the simply typed lambda
calculus is based on the lemma (in the notation of Carpenter): Let β be
free for x in α, then
θ[x:=[[β]]θM ]
[[α]]M
= [[α[x 7→ β]]]θM .
Assuming this already holds for γ and its subterms prove this lemma for
abstraction terms α = λz.γ (in the proof you may use the lemma: if x
θ[x:=a]
does not occur free in α, then [[α]]M
= [[α]]θM ).
TURN OVER
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6AANB033
SECTION B
4. (a) Describe how the typing system of the Intensional Theory of Types extends that of the Simply Typed Lambda Calculus.
(b) The intensional typing system allows the definition of intension- and
extension- operators ∧ α, ∨ α applied to a term α. Give the definition of
their denotations and argue that ∨∧ α and α are logically equivalent.
(c) Explain how formalisation in the language of the intensional lambda calculus may block the conclusion “90o Celsius is rising” from the premisses
“The temperature is 90o Celsius” and “The temperature is rising”.
5. (a) What are Type(1) and Type(1,1) generalized quantifiers?
(b) Define formally transitive generalised quantifiers and symmetric ones and
give some natural language examples
(c) Give and motivate the left- and right-monotonicity properties of ‘at least
three’ as a Type(1,1) quantifier. Is this quantifier reflexive, is it symmetric,
is it transitive?
(d) Define Conservativity for generalized quantifiers and define when a quantifier is quantitative. Discuss these principles as constraints on natural
language quantifiers.
6. Consider the ‘text’: “If a teenager owns a car, he crashes it.” Discuss the
problem with the formalisation of this sentence in first-order predicate logic,
and show how a DRT formalisation addresses this problem.
END OF PAPER
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