Semantics
Answers to assignment #10

Problem 4-6
i.	2nn  = 233  = 512
ii.	 
			a		1
	[[Ro(K)]] =	b	
			c		0

     			a		1
	[[Ro(L)]] =	b	
			c		0

     			a		1
	[[Rs(K)]] =	b	
			c		0

 			a		1
	[[Rs(L)]] =	b	
			c		0

iii.	[[ [Ro(K)](j) ]] = 1
[[ [Rs(L)](m) ]] = 1

Problem 4-7
[[Rr]]M,g is that function h from D<e, <e, t>> into D<e, t> such that for all functions k in D<e, <e, t>> and all individuals e in A, [h(k)](e) = 1 iff k(e)(e) =1.

Problem 4-8
•.	lx[2x+3 = 13](5)
2(5)+3 = 13; True.

•.	ly[2x+3y=15](5)
2x+3(5) = 15; No truth value.

•.	lx[4x2-2x+5=140](7)
4(7)2-2(7)+5=140; False.

•.	lx[∀y[y<7 → x>(2y-x)]](11)
∀y[y<7 → 11>(2y-11)]; True.

Problem 4-9

In the model:
			a		1
	F(B) = 	b
			c		0
and,
	F(L)  =  {<a,c>, <b,a>, <c,a>, <c,c>}  
lx[B(x) ↔ L(j)(x)] is that function f in D<e,t> which maps each e in A to the denotation of [B(e)↔L(j)(e)]. In the given model for all e in A, B(e) and L(j)(e) have the same denotation, that is, for all e in A, [B(e) ↔ L(j)(e)] denotes 1. Therefore, f maps every member of A to 1.

Problem 4-10
i.	The function that maps each person into his or her height on April 2, 1980
ii.	The function that maps each day into Mickey Mantle's height on that day
iii.	The function that maps each day into Mickey Mantle's height on that day
iv.	This is an ill-formed expression.
v.	The function that maps each person into his or her height on April 2, 1980
vi.	The function that maps each person into a function which for each day yields the height of that person on that day
vii. The function that maps each day into a function which for each person yields the height of that person on that day.

Problem 4-11
The function that maps each function from integers into integers into the value obtained when that function is applied at the argument 2 and then 1 is added to the result.

Problem 4-12
The following is a notational variant of the formula under consideration:
lP[lx[yP(y)(x)]]
It denotes that function from D<e,<e,t>> into D<e,t> which maps each function f in its domain into that function h in its range such that for each e in A h(e) =1 just in case for some e’ in A f(e’)(e) = 1. In other words, the semantic value of the formula is a function that maps the denotation of a binary predicate into that of a unary predicate such that the latter is true of any e in A if and only if for some e’ in A, the former is true of <e,e’>. Thus, lv0,<e,<e,t>>[lx[y v0,<e,<e,t>>(y)(x)]] is the same as [[Ro]]M, g.
[This proof relies heavily on your intuitive understanding of Ro. The answer given in DWP is better as a formal demonstration of the equivalence of the two formulae.]

Problem 4-13
Rs = lP[lx[yP(x)(y)]]
Rr = lP[lx[P(x)(x)]]