245: PHL Second Practice Test Answers
A1: a advertises.
D1: a has diamonds on it.
G1: a is a restaurant
M2: a is getting married to b.
B1:
E1:
H1:
N2:
a0: Tiffanys
d1: the fianc of a
b0: Brian
e1: the sister of a
1.
a is a business
a is an engagement ring.
a stays in business.
a is more expensive than b.
C1:
F1:
L2:
O3:
a is a store.
a is a person.
a is displayed at b.
a buys b from c (c sells b to a).
c0: Carol
b2: the best man at the wedding of a and b.
Use the above to symbolize the following sentences (symbolize without iota except for the
questions indicated):
(a) Although stores and restaurants are businesses, not all businesses advertise.
x(Cx Gx Bx) ~x(Bx Ax) OR x(Cx Bx) x(Gx Bx) x(Bx ~Ax)
(b) Assuming that people dont buy anything from businesses that dont advertise, restaurants stay
in business only if they do.
x(Bx ~Ax ~yz(Fy O(yzx))) x(Gx (Hx Ax))
or ~x (Bx ~Ax yz(Fy O(yzx))) x(Gx Hx Ax)
(c) In order for Carols fianc to get married to Carol, it is necessary that he buys her an
engagement ring from Tiffanys.
M(d(c)c) x(Ex O(d(c)xa))
(d) Everyone buys things from stores, but no store sells things to everyone.
x(Fx yz(Cz O(xyz))) ~x(Cx y(Fy zO(yzx))
OR second conjunct: x(Cx y (Fy ~z O(yzx)))
(e) Brian is the only person that buys an engagement ring from Tiffanys.
x(Fx (y(Ey O(xya)) x=b))
(f) If anybody buys an engagement ring from a store then he/she is getting married to somebody.
x(Fx y(Ey z(Cz O(xyz))) w(Fw M(xw))
(g) The best man at Carol and Brians wedding is the Carols sisters fianc.
a(bc)=d(e(c))
1
(h) Give an idiomatic English translation of:
xy ((Cy z(L(zy) Dz) x=y) z(Cz ~wv(Fw Dv O(zvw)) N(xz))).
The store that only displays things with diamonds on it is more expensive than any store that
doesnt sell things with diamonds to anybody.
Or there is exactly one store that only displays things with diamonds on it, and it is more
expensive than every store that sells nothing with diamonds on it to anyone.
(i) Disambiguate this ambiguous sentence by providing two symbolizations. For each, provide an
English sentence that makes the meaning clear.
Somebody buys everything from some store.
x(Fx y(Cy zO(xzy))) There is a person who buys everything from some one store.
x(Fx zy(Cy O(xzy)))
There is a person who buys everything from stores.
z x(Fx y(Cy O(xzy))) Everything is such that somebody buys it from some store or
another.
x(Fx y(Cy zw(Fw O(wzy) x=w))
There is some one person and some one store, and that
person buys everything that the store sells to anyone.
3. Use a full truth table to determine whether the following is a tautology, a contradiction or a
contingent sentence. State which it is and briefly explain how you know.
P ((Q R) ~(P Q))
PQR
P
T
T
T
T
F
F
F
F
T
T
T
T
F
F
F
F
T
T
F
F
T
T
F
F
T
F
T
F
T
F
T
F
F
F
T
F
T
T
T
T
((Q
TF
FF
TT
FF
TT
FT
TF
FF
~
T
T
T
T
F
T
F
F
T
T
T
T
F
T
F
F
R) (P
FTT
FTT
TT F
TT F
TF F
TF F
FF T
FF T
Q))
T
T
F
F
T
T
F
F
false on this TVA
true on this TVA
This is a contingent sentence since some lines below the main connective are true and some false.
2
4. Use a shortened truth-table of one line to show that the following argument is INVALID.
P (Q R). ~Q (~S P). P R.
P
Q
R
S
T
T
F
T
P (Q R) . ~ (Q (~ S P) P R
TT
T
TF
TT
F
F
TFT
TF
F
The premises are both true, but the conclusion is false. Therefore, the argument is INVALID.
5. Show that the following arguments are valid:
a) x(Hx yM(xa(y))). xy(Lx ~M(xy)). y(Ly Hy) zFz
b) ~x(Fx ~Gx) yx(Gx M(yx)). xy(M(xy) L(yx)). x(Fx yL(xy)).
c)
x(yM(yx) (z(~Kz F(xz)) wG(xwx))).
x(Kx ~Hx). xy(F(xy) F(yx))
xy(L(xy) M(yx)).
(yL(ay) z(Hz F(az))) zyG(ayz).
3
P roblem: PT2 5 a)
PT2 5 a): x (Hx yM (xa(y))). x y( Lx M( xy)). y( Ly Hy) zFz
1
Show y( Ly Hy) zFz
"show conc"
2
y( Ly Hy)
ass cd
3
Show zFz
"show cons"
4
zFz
ass id
5
Li Hi
2 ei
6
Hi yM (ia(y))
pr1 ui
7
Hi
5 sr
8
yM (ia(y))
6 7 mp
9
M( ia (n ))
8 ei
10
y( Li M( iy))
pr2 ui
11
Li M( ia (n ))
10 ui
12
Li
5 sl
13
M( ia (n ))
11 12 mp
uniform derivation
14
13 9 id
15
3 cd
16
uniform derivation
P roblem: PT2 5 b)
PT2 5 b) : x (Fx Gx) y x (Gx M(y x)). x y( M(x y) L(y x)). x (Fx
yL (xy))
1
2
Show x (Fx yL (xy))
Show Fx yL (xy)
"show conc"
"show inst"
3
Fx
ass cd
4
x (Fx Gx)
pr1 sl
5
y x (Gx M(yx ))
pr1 sr
6
x (Gx M( ix ))
5 ei
7
x (Fx Gx)
4 qn
8
Gx M( ix )
6 ui
9
(Fx Gx)
7 ui
10
Fx Gx
9 dm
11
Gx
3 dn 10 mtp
12
M( ix )
11 dn 8 mp
13
y( M(iy) L(y i))
pr2 ui
14
M( ix ) L(xi)
13 ui
15
L(xi)
14 bc 12 mp
16
yL (xy)
15 eg
17
16 cd
18
2 ud
19
uniform derivation
P roblem: PT2 5 c)
PT2 5c): x ( yM (yx ) ( z( Kz F(xz)) wG(xwx ))) . x y( L(x y) M(y x)).
x (Kx Hx ). x y( F(x y) F(y x)) ( yL (ay) z(Hz F(az)))
z yG( ay z)
1
Show yL (ay) z(Hz F(az)) z yG( ay z)
"show conc"
2
yL (ay) z(Hz F(az))
ass cd
3
yL (ay)
2 sl
4
z(Hz F(az))
2 sr
5
L(ai)
3 ei
6
Hn F(an)
4 ei
7
Hn
6 sl
8
F(an)
6 sr
9
y( L(a y) M(ya ))
pr2 ui
10
L(ai) M( ia )
9 ui
11
M( ia )
5 10 mp
12
yM (ya )
11 eg
13
yM (ya ) ( z( Kz F(az)) wG(awa ))
pr1 ui
14
z( Kz F(az)) wG(awa )
12 13 mp
15
Kn Hn
pr3 ui
16
Kn
7 dn 15 mt
17
Kn F(an)
16 8 adj
18
z( Kz F(az))
17 eg
19
wG(awa )
18 14 mp
20
G(aoa)
19 ei
21
yG( ay a)
20 eg
22
z yG( ay z)
21 eg
23
24
22 cd
uniform derivation