Chap.6
Exercise IV (page 146)
1. non-propositional (It is perhaps possible to interpret this sentence as “A Catholic priest married John to Amy, and a Catholic priest married Mary to Peter.” However, the most natural reading is “A Catholic priest married John and Mary to each other.” That is non-propositional.)
1. propositional (If Fred had pie with ice cream on top or on the side, he still had pie, and he also had ice cream, so the propositional conjunction is true.)
1. non-propositional (this sentence means that the winning presidential candidate rarely loses both states; it’s not saying that the winning candidate rarely loses New York, and rarely loses California.)
1. propositional
1. propositional
1. non-propositional (We cannot infer that someone who speaks French is bilingual and someone who speaks English is bilingual.)
1. propositional
1. If the claim is that Miranda and Nick cooked dinner together, then it would be non-propositional. However, if the sentence means that both Miranda and Nick cooked dinner, so they ended up with two separate dinners, then it would be propositional.
1. non-propositional (We cannot infer that I doubt John is poor and I doubt John is happy, because I doubt that John is poor and happy if I doubt that he is happy, but I do not doubt that he is poor.)
Exercise V (page 149)
1. Valid by virtue of propositional form
1. Not valid by virtue of propositional form (One could imagine a law requiring that in order to own a house one must also own a piece of land, but then the argument would be valid by virtue of that law instead of being valid by virtue of its form.)
1. Valid by virtue of propositional form
1. Valid by virtue of propositional form
1. Valid by virtue of propositional form
1. Not valid by virtue of propositional form
(This problem is an interesting one as the argument is technically formally valid. However, using the system of Propositional Logic – that we’re using in this class – you would not be able to see that this is formally valid. Knowing that Bess tied with Katie does entail that Bess tied with someone. But Propositional Logic can only treat the premises as one proposition (like “P) and the conclusion as another (like “Q”). That’s not formally valid. If you want to see the complicated Predicate Logic version, I’m happy to demonstrate it for you! Just let me know)
Exercise XII (page 156)
1. T
1. T
1. T
1. F
1. T
1. T
1. T
1. F
1. F
1. F
1. T
1. T
1. T
1. T
1. T
1. T
Exercise XV (page 159-60)
Again, parentheses after the word “Invalid” indicate a row where an invalid argument has true premises but a false conclusion.
(1) Invalid (TT)
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|
|
Pr. |
Pr. |
Conc. |
|
|
p |
q |
(~ p q) |
p |
~ q |
|
|
T |
T |
T |
T |
F |
Invalid |
|
T |
F |
F |
T |
T |
|
|
F |
T |
T |
F |
F |
|
|
F |
F |
T |
F |
T |
|
(2) Valid
|
|
|
Pr. |
Conc. |
|
|
p |
q |
~( p q) |
~ q |
|
|
T |
T |
F |
F |
|
|
T |
F |
F |
T |
|
|
F |
T |
F |
F |
|
|
F |
F |
T |
T |
OK |
(3) Valid (Surprisingly, because the premises are never both true.)
|
|
|
Pr. |
Pr. |
Conc. |
|
|
p |
q |
~( p q) |
p |
q |
|
|
T |
T |
F |
T |
T |
|
|
T |
F |
F |
T |
F |
|
|
F |
T |
F |
F |
T |
|
|
F |
F |
T |
F |
F |
|
(4) Valid (for the same reason as (3))
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|
|
|
Pr. |
Pr. |
Conc. |
|
|
p |
q |
r |
~( p q) |
p |
r |
|
|
T |
T |
T |
F |
T |
T |
|
|
T |
T |
F |
F |
T |
F |
|
|
T |
F |
T |
F |
T |
T |
|
|
T |
F |
F |
F |
T |
F |
|
|
F |
T |
T |
F |
F |
T |
|
|
F |
T |
F |
F |
F |
F |
|
|
F |
F |
T |
T |
F |
T |
|
|
F |
F |
F |
T |
F |
F |
|
(5) Valid
|
|
|
Pr. |
Pr. |
Conc. |
|
|
p |
q |
~( p & q) |
q |
~ p |
|
|
T |
T |
F |
T |
F |
|
|
T |
F |
T |
F |
F |
|
|
F |
T |
T |
T |
T |
OK |
|
F |
F |
T |
F |
T |
|
(6) Invalid (FF)
|
|
|
Pr. |
Pr. |
Conc. |
|
|
p |
q |
~( p & q) |
~ q |
p |
|
|
T |
T |
F |
F |
T |
|
|
T |
F |
T |
T |
T |
OK |
|
F |
T |
T |
F |
F |
|
|
F |
F |
T |
T |
F |
Invalid |
(7) Valid
|
|
|
|
Pr. |
Conc. |
|
|
p |
q |
r |
( p & q) ( p & r) |
p & ( q r) |
|
|
T |
T |
T |
T |
T |
OK |
|
T |
T |
F |
T |
T |
OK |
|
T |
F |
T |
T |
T |
OK |
|
T |
F |
F |
F |
F |
|
|
F |
T |
T |
F |
F |
|
|
F |
T |
F |
F |
F |
|
|
F |
F |
T |
F |
F |
|
|
F |
F |
F |
F |
F |
|
(8) Invalid (TFF, FTT)
|
|
|
|
Pr. |
Conc. |
|
|
p |
q |
r |
( p q) & ( p r) |
p & ( q r) |
|
|
T |
T |
T |
T |
T |
OK |
|
T |
T |
F |
T |
T |
OK |
|
T |
F |
T |
T |
T |
OK |
|
T |
F |
F |
T |
F |
Invalid |
|
F |
T |
T |
T |
F |
Invalid |
|
F |
T |
F |
F |
F |
|
|
F |
F |
T |
F |
F |
|
|
F |
F |
F |
F |
F |
|
(9) Valid
|
|
|
|
Pr. |
Conc. |
|
|
p |
q |
r |
p& q |
( p r) & ( q r) |
|
|
T |
T |
T |
T |
T |
OK |
|
T |
T |
F |
T |
T |
OK |
|
T |
F |
T |
F |
T |
|
|
T |
F |
F |
F |
F |
|
|
F |
T |
T |
F |
T |
|
|
F |
T |
F |
F |
F |
|
|
F |
F |
T |
F |
T |
|
|
F |
F |
F |
F |
F |
|
(10) Invalid (TTF, TFF, FTF)
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|
|
|
Pr. |
Conc. |
|
|
p |
q |
r |
p q |
( p & r) ( q & r) |
|
|
T |
T |
T |
T |
T |
OK |
|
T |
T |
F |
T |
F |
Invalid |
|
T |
F |
T |
T |
T |
OK |
|
T |
F |
F |
T |
F |
Invalid |
|
F |
T |
T |
T |
T |
OK |
|
F |
T |
F |
T |
F |
Invalid |
|
F |
F |
T |
F |
F |
|
|
F |
F |
F |
F |
F |
|
Exercise XXIV (page 168-9)
(5) Valid
|
P |
Q |
R |
P Q |
Q R |
~R |
~P |
|
T |
T |
T |
T |
T |
F |
F |
|
T |
T |
F |
T |
F |
T |
F |
|
T |
F |
T |
F |
T |
F |
F |
|
T |
F |
F |
F |
T |
T |
F |
|
F |
T |
T |
T |
T |
F |
T |
|
F |
T |
F |
T |
F |
T |
T |
|
F |
F |
T |
T |
T |
F |
T |
|
F |
F |
F |
T |
T |
T |
T |
(7) Valid
|
P |
Q |
R |
P vQ |
P Q |
Q R |
R |
|
T |
T |
T |
T |
T |
T |
T |
|
T |
T |
F |
T |
T |
F |
F |
|
T |
F |
T |
T |
F |
T |
T |
|
T |
F |
F |
T |
F |
T |
F |
|
F |
T |
T |
T |
T |
T |
T |
|
F |
T |
F |
T |
T |
F |
F |
|
F |
F |
T |
F |
T |
T |
T |
|
F |
F |
F |
F |
T |
T |
F |
(13) Valid
|
P |
Q |
R |
P Q R) |
Q |
~R |
~P |
|
T |
T |
T |
T |
T |
F |
F |
|
T |
T |
F |
F |
T |
T |
F |
|
T |
F |
T |
T |
F |
F |
F |
|
T |
F |
F |
T |
F |
T |
F |
|
F |
T |
T |
T |
T |
F |
T |
|
F |
T |
F |
T |
T |
T |
T |
|
F |
F |
T |
T |
F |
F |
T |
|
F |
F |
F |
T |
F |
T |
T |
(18) Valid
|
P |
Q |
(P v Q) (P & Q) |
(P Q) & (Q P) |
|
T |
T |
T |
T |
|
T |
F |
F |
F |
|
F |
T |
F |
F |
|
F |
F |
T |
T |
Exercise XXVIII (page 176)
(6) M L
~L
————
~M Valid
|
M |
L |
M L |
~L |
~M |
|
T |
T |
T |
F |
F |
|
T |
F |
F |
T |
F |
|
F |
T |
T |
F |
T |
|
F |
F |
T |
T |
T |
(7) PH
H
————
P Invalid
This argument might seem valid if we imagine contexts where it is conversationally implied that John will play if the situation is hopeless.
|
P |
H |
PH |
H |
P |
|
T |
T |
T |
T |
T |
|
T |
F |
F |
F |
T |
|
F |
T |
T |
T |
F |
|
F |
F |
T |
F |
F |
(8) B & L
L F
————
F Valid
|
B |
L |
F |
B & L |
L F |
F |
|
T |
T |
T |
T |
T |
T |
|
T |
T |
F |
T |
F |
F |
|
T |
F |
T |
F |
T |
T |
|
T |
F |
F |
F |
T |
F |
|
F |
T |
T |
F |
T |
T |
|
F |
T |
F |
F |
F |
F |
|
F |
F |
T |
F |
T |
T |
|
F |
F |
F |
F |
T |
F |
(9) (A v C) & ~(A & C)
(C v G) & ~(C & G)
————
(A v G) & ~(A & G) Invalid
|
A |
C |
G |
(A v C) & ~(A & C) |
(C v G) & ~(C & G) |
(A v G) & ~(A & G) |
|
T |
T |
T |
F |
F |
F |
|
T |
T |
F |
F |
T |
T |
|
T |
F |
T |
T |
T |
F |
|
T |
F |
F |
T |
F |
T |
|
F |
T |
T |
T |
F |
T |
|
F |
T |
F |
T |
T |
F |
|
F |
F |
T |
F |
T |
T |
|
F |
F |
F |
F |
F |
F |
(10) D O
————————
(D & ~O) W Valid
|
D |
O |
W |
D O |
(D & ~O) W |
|
T |
T |
T |
T |
T |
|
T |
T |
F |
T |
T |
|
T |
F |
T |
F |
T |
|
T |
F |
F |
F |
F |
|
F |
T |
T |
T |
T |
|
F |
T |
F |
T |
T |
|
F |
F |
T |
T |
T |
|
F |
F |
F |
T |
T |
(11) (J B) & ~(J & B)
B S
J N
————
N S Valid
|
J |
B |
S |
N |
(J B) & ~(J & B) |
B S |
J N |
N S |
|
T |
T |
T |
T |
F |
T |
T |
T |
|
T |
T |
T |
F |
F |
T |
F |
T |
|
T |
T |
F |
T |
F |
F |
T |
T |
|
T |
T |
F |
F |
F |
F |
F |
F |
|
T |
F |
T |
T |
T |
T |
T |
T |
|
T |
F |
T |
F |
T |
T |
F |
T |
|
T |
F |
F |
T |
T |
T |
T |
T |
|
T |
F |
F |
F |
T |
T |
F |
F |
|
F |
T |
T |
T |
T |
T |
T |
T |
|
F |
T |
T |
F |
T |
T |
T |
T |
|
F |
T |
F |
T |
T |
F |
T |
T |
|
F |
T |
F |
F |
T |
F |
T |
F |
|
F |
F |
T |
T |
F |
T |
T |
T |
|
F |
F |
T |
F |
F |
T |
T |
T |
|
F |
F |
F |
T |
F |
T |
T |
T |
|
F |
F |
F |
F |
F |
T |
T |
F |
(12) F M
(F M) S
————
S Valid
|
F |
M |
S |
F M |
(F M) S |
S |
|
T |
T |
T |
T |
T |
T |
|
T |
T |
F |
T |
F |
F |
|
T |
F |
T |
T |
T |
T |
|
T |
F |
F |
T |
F |
F |
|
F |
T |
T |
T |
T |
T |
|
F |
T |
F |
T |
F |
F |
|
F |
F |
T |
F |
T |
T |
|
F |
F |
F |
F |
T |
F |