assert that he0 likes Mary
(t/e)/t t, 4
he0 like Mary
t/(t/e) t/e, 5
like Mary
(t/e)/(t/(t/e)) t/(t/e)
Every man can be analysed further:
every man
t/(t/e), 6
every man
(t/(t/e))/(t//e) t//e
(b) Translate the sentence in (a) into an expression of intensional logic, and show that the translation algorithm for mapping categories of English into types of intensional logic has been adhered to.
every man = P "x [man'(x) (r) P(x)]
he0 asserts that he0 likes Mary = PP{x0} (_assert-that' (_PP{x0} (_like' (_m*))))
Substitute every man into he0:
P "x [man'(x) (r) P(x)] _x0 [PP{x0} (_assert-that' (_PP{x0} (_like' (_m*))))]
This translation is easier to follow in the form of a tree as demonstrated on the next page.
every man asserts that he likes Mary
P "x [man'(x) (r) P(x)] _x0 [PP{x0} (_assert-that' (_PP{x0} (_like' (_m*))))]
every man he0 asserts that he0 likes Mary
P "x [man'(x) (r) P(x)] PP{x0} (_assert-that' (_PP{x0} (_like' (_m*))))
he0 asserts that he0 likes Mary
PP{x0} assert-that' (_PP{x0} (_like' (_m*)))
assert that he0 likes Mary
assert-that' PP{x0} (_like' (_m*))
he0 like Mary
PP{x0} like' (_m*)
like Mary
like' m*
(c) Simplify (as far as possible) the intensional logic expression derived in (b).
P "x [man'(x) (r) P(x)] _x0 [PP{x0} (_assert-that' (_PP{x0} (_like' (_m*))))]
m* is the superstar definition of P[P{_m}]
Abstraction application gives:
Þ P "x [man'(x) (r) P(x)] _x0 [PP{x0} (_assert-that' (_like' (_m*) {x0}))]
Abstraction application gives:
Þ P "x [man'(x) (r) P(x)] _x0 [_assert-that' (_like' (_m*) {x0}){x0}]
Brace convention and down-up cancellation gives:
Þ P "x [man'(x) (r) P(x)] _x0 [assert-that' (_like' (_m*) (x0)) (x0)]
Abstraction application gives:
Þ "x [man'(x) (r) _x0 [assert-that' (_like' (_m*) (x0)) (x0)] {x}]
Brace convention and up-down cancellation gives:
Þ "x [man'(x) (r) x0 [assert-that' (like' (_m*) (x0)) (x0)] (x)]
gives:
Þ "x [man'(x) (r) [assert-that' (like' (_m*) (x0)) (x0)] ]
Application of the notation for two-place relations gives:
Þ "x [man'(x) (r) [assert-that' (like' (x0 ,_m*) ) (x0)] ]