SEMANTICS

Answers to assignment #9

Problem 4-2
If ^ was included in ME<t, <t, t>>, and α and β were members of ME<t>, then by rule C2 ^( α) would a member of ME<t, t>, and ^(α)(β) would be a member of Me<t>. Both expressions are ill-formed in the language under consideration.

Problem 4-3
i. ∀v0,<e, t>[c0,<<e, t>, t>(v0,<e, t>) → v0, e[v0, <e, t>(v0, e)]]
All sets of individuals which are members of the set of sets of individuals denoted by c0, <<e, t>, t> have at least one member.

ii. v0, e∀v0, <e, t>[c0, <<e, t>, t>(v0, <e, t>) → [v0, <e, t>(v0, e)]]
Some individual is a member of each set of individuals that is a member of the set of sets of individuals denoted by c0, <<e, t>, t>.

iii. !v0, e∀v0, <e, t>[c0, <<e, t>, t>(v0, <e, t>) <-> [v0, <e, t>(v0, e)]]
There is exactly one e in A which belongs to all and only those sets of individuals which are in the set of sets of individuals denoted by c0, <<e, t>, t>.
[What is the difference between a predicate and a set of individuals? Must an individual be a person?]

Problem 4-4
Assume that the two individuals are a and b.
D<e, t> is the set of all possible denotations of type <e, t>. Its members are:
a → 1		a → 1		a → 0		a → 0
b → 1		b → 0		b → 1		b → 0

Therefore, there are 4 members in set D<e, t>.

a → 1			a → 0
	b → 1			b → 0

a → 1			a → 0
h =		b → 0			b → 1

a → 0			a → 1
		b → 1			b → 0

a → 0			a → 1
		b → 0			b → 1

Problem 4-5
[[c12,<<e,<e,t>>,<e,<e,t>>>]]M, g is that function h from D<e, <e, t>> into D<e, <e, t>> such that for any k in D<e,<e, t>> and any individuals e and f in A, 
[h(k)](e)(f) = 1 iff k(e)(f) = 0, and [h(k)](e)(f) = 0 iff k(e)(f) = 1.