(1) John is a smoker (2) John smokes
⟦a⟧M = [λf∈D⟨e,t⟩. f] ⟦is⟧M = [λf∈D⟨e,t⟩. f]
Mary is a student of linguistics
(we assume linguistics denotes an abstract individual)
NB: et = ⟨e,t⟩
Mary is a student of lingusitics Mary is a student
Mary ate pizza Mary ate
Intransitive
Transitive
Semantically vacuous items denote identity functions.
John is blond John is fond of Mary
⟦blond⟧M = [λx∈De 1 iff x is blond in M]
⟦blond boy⟧M = [λx∈De. 1 iff x is a blond boy in M]
⟦blond⟧M(x) = 1 iff x is blond in M
⟦boy⟧M(x) = 1 iff x is a boy in M
To distinguish this rule from our earlier rule for branching nodes, let's call the latter Functional Application from now on.
For each branching node, either one of these rules will apply, and which one to apply depends on the semantic types of the daughters B and C.
Predicate Modification For any model M, if A is a branching node with B and C as its daughters, and both ⟦B⟧M and ⟦C⟧M are functions of type ⟨e,t⟩, then ⟦A⟧M = [λx∈De. 1 iff ⟦B⟧M(x)=1 and ⟦C⟧M(x)=1]
Cf. {x | x is a blond boy in M} = {x | x is blond in M} ∩ {x | x is a boy in M} (More on the relation between functions and sets in Lecture 7)]