+1(978)310-4246 credencewriters@gmail.com
  

1. Translate the following into GPLI:

Everyone apart from Alice likes Sydney and at least one other city.

[Note: regard this proposition as stating nothing about what Alice does, or does not,

like.]

2. For the following proposition, describe

(i) a model on which it is true (and explain

why it is true on this model), and

(ii) a model on which it is false (and explain why it is

false on this model).

If there is no model of one of these types, explain why.

∀

x

∀

y

(

Rxy

→

(

x

=

y

∧

P x

∧

Qy

))

∧¬∃

x

(

P x

∧

Qx

)

LOGIC
The Drill
Nicholas J.J. Smith and John Cusbert
Copyright c 2012 by Nicholas Jeremy Josef Smith and John Cusbert
All rights reserved
Cover photograph: Alser Straße, Vienna. c Nicholas J.J. Smith
30 May 2012: first published.
29 November 2014: corrections.
16 May 2017: corrections.
Preface
The first part of this volume contains all the exercise questions that appear in Logic: The Laws of Truth by Nicholas J.J. Smith (Princeton University Press, 2012). The second part contains answers to almost all of these
exercises. Both the questions and the answers are a collaborative effort
between Nicholas J.J. Smith and John Cusbert.
One obvious use of this work is as a solutions manual for readers of Logic:
The Laws of Truth—but it should also be of use to readers of other logic
books. Students of logic need a large number of worked examples and
exercise problems with solutions: the more the better. This volume should
help to meet that need.
After each question, a cross-reference of the form ‘[A p.x]’ appears. This
indicates the page on which the answer to that question can be found. You
can click on the cross-reference to be taken directly to the answer. Each
answer then contains a cross-reference of the form ‘[Q p.x]’ which leads
back to the corresponding question. Other blue items are also links: for
example, clicking on an entry in the Contents pages takes you directly to
the relevant section, and at the end of each exercise set and each answer
set there is a link back to the Contents.
If you find any errors—or have any other comments or suggestions—
please email us at:
logicthedrill@gmail.com
The latest version of this work can be found at:
http://njjsmith.com/philosophy/lawsoftruth/
Any significant revisions (e.g. corrections or additions to the exercises or
answers) will be documented on the copyright page.
iii
Contents
Preface
iii
Questions
2
1. Propositions and Arguments
2
Exercises 1.2.1
2
Exercises 1.3.1
3
Exercises 1.4.1
3
Exercises 1.5.1
4
Exercises 1.6.1.1
4
Exercises 1.6.2.1
5
Exercises 1.6.4.1
5
Exercises 1.6.6
6
2. The Language of Propositional Logic
7
Exercises 2.3.3
7
Exercises 2.3.5
8
Exercises 2.3.8
8
iv
Exercises 2.5.1
10
Exercises 2.5.3.1
11
Exercises 2.5.4.1
11
Exercises 2.5.5.1
11
3. Semantics of Propositional Logic
13
Exercises 3.2.1
13
Exercises 3.3.1
14
Exercises 3.4.1
14
Exercises 3.5.1
15
4. Uses of Truth Tables
16
Exercises 4.1.2
16
Exercises 4.2.1
17
Exercises 4.3.1
18
Exercises 4.4.1
18
5. Logical Form
20
Exercises 5.1.1
20
Exercises 5.2.1
20
Exercises 5.3.1
21
Exercises 5.4.1
22
Exercises 5.5.1
23
v
6. Connectives: Translation and Adequacy
24
Exercises 6.5.1
24
Exercises 6.6.3
25
7. Trees for Propositional Logic
27
Exercises 7.2.1.1
27
Exercises 7.2.2.1
27
Exercises 7.2.3.1
28
Exercises 7.3.1.1
28
Exercises 7.3.2.1
29
Exercises 7.3.3.1
30
Exercises 7.3.4.1
30
Exercises 7.3.5.1
31
8. The Language of Monadic Predicate Logic
32
Exercises 8.2.1
32
Exercises 8.3.2
33
Exercises 8.3.5
34
Exercises 8.4.3.1
35
Exercises 8.4.5.1
36
9. Semantics of Monadic Predicate Logic
38
Exercises 9.1.1
38
Exercises 9.2.1
39
vi
Exercises 9.3.1
39
Exercises 9.4.3
40
Exercises 9.5.1
43
10. Trees for Monadic Predicate Logic
44
Exercises 10.2.2
44
Exercises 10.3.8
45
11. Models, Propositions, and Ways the World Could Be
47
12. General Predicate Logic
48
Exercises 12.1.3.1
48
Exercises 12.1.6
49
Exercises 12.1.9
50
Exercises 12.2.2
52
Exercises 12.3.1
54
Exercises 12.4.1
57
Exercises 12.5.4
57
13. Identity
58
Exercises 13.2.2
58
Exercises 13.3.1
60
Exercises 13.4.3
61
Exercises 13.5.1
63
vii
Exercises 13.6.1.1
64
Exercises 13.6.2.1
65
Exercises 13.6.3.1
66
Exercises 13.7.4
66
14. Metatheory
70
Exercises 14.1.1.1
70
Exercises 14.1.2.1
71
Exercises 14.1.3.1
71
15. Other Methods of Proof
72
Exercises 15.1.5
72
Exercises 15.2.3
74
Exercises 15.3.3
75
16. Set Theory
76
Answers
78
1. Propositions and Arguments
78
Answers 1.2.1
78
Answers 1.3.1
78
Answers 1.4.1
79
Answers 1.5.1
79
viii
Answers 1.6.1.1
80
Answers 1.6.2.1
80
Answers 1.6.4.1
80
Answers 1.6.6
81
2. The Language of Propositional Logic
84
Answers 2.3.3
84
Answers 2.3.5
84
Answers 2.3.8
85
Answers 2.5.1
87
Answers 2.5.3.1
89
Answers 2.5.4.1
90
Answers 2.5.5.1
90
3. Semantics of Propositional Logic
92
Answers 3.2.1
92
Answers 3.3.1
93
Answers 3.4.1
95
Answers 3.5.1
98
4. Uses of Truth Tables
99
Answers 4.1.2
99
Answers 4.2.1
102
Answers 4.3.1
105
ix
Answers 4.4.1
109
5. Logical Form
112
Answers 5.1.1
112
Answers 5.2.1
113
Answers 5.3.1
114
Answers 5.4.1
116
Answers 5.5.1
117
6. Connectives: Translation and Adequacy
118
Answers 6.5.1
118
Answers 6.6.3
128
7. Trees for Propositional Logic
134
Answers 7.2.1.1
134
Answers 7.2.2.1
135
Answers 7.2.3.1
136
Answers 7.3.1.1
137
Answers 7.3.2.1
141
Answers 7.3.3.1
144
Answers 7.3.4.1
148
Answers 7.3.5.1
151
8. The Language of Monadic Predicate Logic
156
x
Answers 8.2.1
156
Answers 8.3.2
158
Answers 8.3.5
159
Answers 8.4.3.1
161
Answers 8.4.5.1
163
9. Semantics of Monadic Predicate Logic
165
Answers 9.1.1
165
Answers 9.2.1
165
Answers 9.3.1
166
Answers 9.4.3
167
Answers 9.5.1
171
10. Trees for Monadic Predicate Logic
172
Answers 10.2.2
172
Answers 10.3.8
180
11. Models, Propositions, and Ways the World Could Be
190
12. General Predicate Logic
191
Answers 12.1.3.1
191
Answers 12.1.6
192
Answers 12.1.9
193
Answers 12.2.2
196
xi
Answers 12.3.1
200
Answers 12.4.1
217
Answers 12.5.4
220
13. Identity
221
Answers 13.2.2
221
Answers 13.3.1
223
Answers 13.4.3
226
Answers 13.5.1
244
Answers 13.6.1.1
258
Answers 13.6.2.1
259
Answers 13.6.3.1
260
Answers 13.7.4
261
14. Metatheory
265
Answers 14.1.1.1
265
Answers 14.1.2.1
265
Answers 14.1.3.1
267
15. Other Methods of Proof
269
Answers 15.1.5
269
Answers 15.2.3
284
Answers 15.3.3
303
xii
16. Set Theory
304
xiii
Questions
1
Chapter 1
Propositions and Arguments
Exercises 1.2.1
Classify the following as propositions or nonpropositions.
1. Los Angeles is a long way from New York.
[A p.78]
2. Let’s go to Los Angeles!
[A p.78]
3. Los Angeles, whoopee!
[A p.78]
4. Would that Los Angeles were not so far away.
[A p.78]
5. I really wish Los Angeles were nearer to New York.
[A p.78]
6. I think we should go to Los Angeles.
[A p.78]
7. I hate Los Angeles.
[A p.78]
8. Los Angeles is great!
[A p.78]
9. If only Los Angeles were closer.
[A p.78]
10. Go to Los Angeles!
[A p.78]
[Contents]
2
Exercises 1.3.1
Represent the following lines of reasoning as arguments.
1. If the stock market crashes, thousands of experienced investors will
lose a lot of money. So the stock market won’t crash.
[A p.78]
2. Diamond is harder than topaz, topaz is harder than quartz, quartz is
harder than calcite, and calcite is harder than talc, therefore diamond
is harder than talc.
[A p.79]
3. Any friend of yours is a friend of mine; and you’re friends with everyone on the volleyball team. Hence, if Sally’s on the volleyball
team, she’s a friend of mine.
[A p.79]
4. When a politician engages in shady business dealings, it ends up on
page one of the newspapers. No South Australian senator has ever
appeared on page one of a newspaper. Thus, no South Australian
senator engages in shady business dealings.
[A p.79]
[Contents]
Exercises 1.4.1
State whether each of the following arguments is valid or invalid.
1.
All dogs are mammals.
All mammals are animals.
All dogs are animals.
2.
[A p.79]
All dogs are mammals.
All dogs are animals.
All mammals are animals.
3.
[A p.79]
All dogs are mammals.
No fish are mammals.
No fish are dogs.
[A p.79]
3
4.
All fish are mammals.
All mammals are robots.
All fish are robots.
[A p.79]
[Contents]
Exercises 1.5.1
1. Which of the arguments in Exercise 1.4.1 are sound?
[A p.79]
2. Find an argument in Exercise 1.4.1 that has all true premises and a
true conclusion but is not valid and hence not sound.
[A p.79]
3. Find an argument in Exercise 1.4.1 that has false premises and a false
conclusion but is valid.
[A p.79]
[Contents]
Exercises 1.6.1.1
1. What is the negand of:
(i) Bob is not a good student
[A p.80]
(ii) I haven’t decided not to go to the party.
[A p.80]
(iii) Mars isn’t the closest planet to the sun.
[A p.80]
(iv) It is not the case that Alice is late.
[A p.80]
(v) I don’t like scrambled eggs.
[A p.80]
(vi) Scrambled eggs aren’t good for you.
[A p.80]
2. If a proposition is true, its double negation is. . . ?
[A p.80]
3. If a proposition’s double negation is false, the proposition is. . . ?
[A p.80]
[Contents]
4
Exercises 1.6.2.1
What are the conjuncts of the following propositions?
1. The sun is shining, and I am happy.
[A p.80]
2. Maisie and Rosie are my friends.
[A p.80]
3. Sailing is fun, and snowboarding is too.
[A p.80]
4. We watched the movie and ate popcorn.
[A p.80]
5. Sue does not want the red bicycle, and she does not like the blue one.
[A p.80]
6. The road to the campsite is long and uneven.
[A p.80]
[Contents]
Exercises 1.6.4.1
What are the (a) antecedents and (b) consequents of the following propositions?
1. If that’s pistachio ice cream, it doesn’t taste the way it should.
[A p.80]
2. That tastes the way it should only if it isn’t pistachio ice cream.
[A p.80]
3. If that is supposed to taste that way, then it isn’t pistachio ice cream.
[A p.81]
4. If you pressed the red button, then your cup contains coffee.
[A p.81]
5. Your cup does not contain coffee if you pressed the green button.
[A p.81]
6. Your cup contains hot chocolate only if you pressed the green button.
[A p.81]
[Contents]
5
Exercises 1.6.6
State what sort of compound proposition each of the following is, and
identify its components. Do the same for the components.
1. If it is sunny and windy tomorrow, we shall go sailing or kite flying.
[A p.81]
2. If it rains or snows tomorrow, we shall not go sailing or kite flying.
[A p.81]
3. Either he’ll stay here and we’ll come back and collect him later, or
he’ll come with us and we’ll all come back together.
[A p.81]
4. Jane is a talented painter and a wonderful sculptor, and if she remains interested in art, her work will one day be of the highest quality.
[A p.81]
5. It’s not the case that the unemployment rate will both increase and
decrease in the next quarter.
[A p.82]
6. Your sunburn will get worse and become painful if you don’t stop
swimming during the daytime.
[A p.82]
7. Either Steven won’t get the job, or I’ll leave and all my clients will
leave.
[A p.82]
8. The Tigers will not lose if and only if both Thompson and Thomson
get injured.
[A p.82]
9. Fido will wag his tail if you give him dinner at 6 this evening, and if
you don’t, then he will bark.
[A p.82]
10. It will rain or snow today—or else it won’t.
[A p.83]
[Contents]
6
Chapter 2
The Language of Propositional
Logic
Exercises 2.3.3
Using the glossary:
A: Aristotle was a philosopher
B: Paper burns
F: Fire is hot
translate the following from PL into English.
1. ¬ A
[A p.84]
2. ( A ^ B)
[A p.84]
3. ( A ^ ¬ B)
[A p.84]
4. (¬ F ^ ¬ B)
[A p.84]
5. ¬( F ^ B)
[A p.84]
[Contents]
7
Exercises 2.3.5
Using the glossary of Exercise 2.3.3, translate the following from PL into
English.
1. (( A ^ B) _ F )
[A p.84]
2. (¬ A _ ¬ B)
[A p.84]
3. (( A _ B) ^ ¬( A ^ B))
[A p.84]
4. ¬( A _ F )
[A p.84]
5. ( A ^ ( B _ F ))
[A p.85]
[Contents]
Exercises 2.3.8
1. Using the glossary:
B: The sky is blue
G: Grass is green
R: Roses are red
W: Snow is white
Y: Bananas are yellow
translate the following from PL into English.
(i) (W ! B)
[A p.85]
(ii) (W $ (W ^ ¬ R))
[A p.85]
(iv) (( R _ W ) ! ( R ^ ¬W ))
[A p.85]
(vi) ( G _ (W ! R))
[A p.85]
(iii) ¬( R ! ¬W )
[A p.85]
(v) ((W ^ W ) _ ( R ^ ¬ B))
[A p.85]
(vii) ((Y $ Y ) ^ (¬Y $ ¬Y ))
[A p.85]
(viii) (( B ! W ) ! (¬W ! ¬ B))
(ix) ((( R ^ W ) ^ B) ! (Y _ G ))
8
[A p.85]
[A p.85]
(x) ¬(¬ R ^ (¬W _ G ))
[A p.85]
2. Translate the following from English into PL.
(i) Only if the sky is blue is snow white.
[A p.86]
(ii) The sky is blue if, and only if, snow is white and roses are
not red.
[A p.86]
(iii) It’s not true that if roses are red, then snow is not white.
[A p.86]
(iv) If snow and roses are red, then roses are red and/or snow isn’t.
[A p.86]
(v) Jim is tall if and only if Maisy is, and Maisy is tall only if Nora
is not.
[A p.86]
(vi) Jim is tall only if Nora or Maisy is.
[A p.86]
(vii) If Jim is tall, then either Maisy is tall or Nora isn’t.
[A p.86]
(viii) Either snow is white and Maisy is tall, or snow is white and she
isn’t.
[A p.86]
(ix) If Jim is tall and Jim is not tall, then the sky both is and is not
blue.
[A p.86]
(x) If Maisy is tall and the sky is blue, then Jim is tall and the sky is
not blue.
[A p.86]
3. Translate the following from English into PL.
(i) If it is snowing, we are not kite flying.
[A p.87]
(ii) If it is sunny and it is windy, then we are sailing or kite flying.
[A p.87]
(iii) Only if it is windy are we kite flying, and only if it is windy are
we sailing.
[A p.87]
(iv) We are sailing or kite flying—or skiing.
[A p.87]
(v) If—and only if—it is windy, we are sailing.
[A p.87]
(vi) We are skiing only if it is windy or snowing.
[A p.87]
(vii) We are skiing only if it is both windy and snowing.
[A p.87]
(viii) If it is sunny, then if it is windy, we are kite flying.
[A p.87]
(ix) We are sailing only if it is sunny, windy, and not snowing.
[A p.87]
9
(x) If it is sunny and windy, we’re sailing, and if it is snowing and
not windy, we’re skiing.
[A p.87]
[Contents]
Exercises 2.5.1
1. State whether each of the following is a wff of PL.
(i) (( A ! B))
[A p.87]
(ii) ( A !! B)
[A p.87]
(iv) A ! (( A ! A))
[A p.87]
(vi) ( A _ ( A _ ( A _ ( A _ ( A _ ( A _ ( A _ A))))))
[A p.87]
(iii) ( A ! ( A ! A))
[A p.87]
(v) (( A ^ B)^) A
[A p.87]
(vii) (( AA _ ^ BC ))
[A p.87]
(viii) (( A _ A) ^ BC ))
[A p.87]
(ix) ABC
[A p.87]
(x) (( A _ A) ^ (( A _ A) ^ (( A _ A) ^ A)))
[A p.87]
2. Give recursive definitions of the following.
(i) The set of all odd numbers.
[A p.88]
(ii) The set of all numbers divisible by five.
[A p.88]
(iii) The set of all “words” (finite strings of letters) that use only (but
not necessarily both of) the letters a and b.
[A p.88]
(iv) The set containing all of Bob’s ancestors.
[A p.88]
(v) The set of all cackles: hah hah hah, hah hah hah hah, hah hah
hah hah hah, and so on.
[A p.88]
[Contents]
10
Exercises 2.5.3.1
Write out a construction for each of the following wffs, and state the main
connective.
1. (¬ P _ ( Q ^ R))
[A p.89]
2. ¬( P ^ ( Q _ R))
[A p.89]
3. ((¬ P ^ ¬ Q) _ ¬ R)
[A p.89]
4. (( P ! Q) _ ( R ! S))
[A p.89]
5. ((( P $ Q) $ R) $ S)
[A p.89]
6. ((¬ P ^ ¬¬ P) ! ( P ^ ¬ P))
[A p.90]
[Contents]
Exercises 2.5.4.1
1. For each of the remaining orderings (2–6) of the connectives !, ^,
and _ given in §2.5.4, state which disambiguation (1–5) results from
restoring parentheses to our original expression in this order.
[A p.90]
[Contents]
Exercises 2.5.5.1
1. Write the following in the notation of this book:
(i) _ ¬ P ^ QR
[A p.90]
(iii) ^ ¬ _ PQR
[A p.90]
(ii) ¬ ^ _ PQR
[A p.90]
(iv) _ ^ ¬ P¬ Q¬ R
[A p.90]
(v) $$$ PQRS
[A p.90]
2. Write the following in Polish notation:
(i) ¬( P ^ ( Q _ R))
[A p.91]
11
(ii) ([ P ! ( Q _ R)] ! S)
[A p.91]
(iv) ( P ! [( Q _ R) ! S])
[A p.91]
(iii) [( P ! Q) _ ( R ! S)]
[A p.91]
(v) [(¬ P ^ ¬¬ P) ! ( P ^ ¬ P)]
12
[A p.91]
[Contents]
Chapter 3
Semantics of Propositional Logic
Exercises 3.2.1
Determine the truth values of the following wffs, given the truth values
for their basic components, which are written under those components.
1.
(¬ P ^ ( Q _ R))
T
T
F
[A p.92]
2.
¬ ( P _ ( Q ! R))
T
T
F
[A p.92]
3.
(¬ ¬ P ^ ( Q ! ( R _ P)))
F
T
T
F
[A p.92]
4.
(¬ ¬ P ^ ( Q ! ( R _ P)))
T
F
F
T
[A p.92]
5.
(( P _ Q) ! ( P _ P))
F
T
F
F
[A p.92]
6.
(( P _ Q) ! ( P _ P))
T
F
T
T
[A p.93]
7.
( P ! ( Q ! ( R ! S)))
T
T
T
F
[A p.93]
8.
( P ! ( Q ! ( R ! S)))
F
T
F
T
[A p.93]
9.
¬ (((¬ P $ P) $ Q) ! R)
F
F
F
F
[A p.93]
13
10.
¬ (((¬ P $ P) $ Q) ! R)
T
T
T
T
[A p.93]
[Contents]
Exercises 3.3.1
Draw up truth tables for the following propositions.
1. (( P ^ Q) _ P)
[A p.93]
2. ( P ^ ( P _ P))
[A p.94]
3. ¬(¬ P ^ ¬ Q)
[A p.94]
4. ( Q ! ( Q ^ ¬ Q))
[A p.94]
5. ( P ! ( Q ! R))
[A p.94]
6. (( P _ Q) $ ( P ^ Q))
[A p.94]
7. ¬(( P ^ Q) $ Q)
[A p.94]
8. ((( P ! ¬ P) ! ¬ P) ! ¬ P)
[A p.94]
9. ¬( P ^ ( Q ^ R))
[A p.95]
10. ((¬ R _ S) ^ (S _ ¬ T ))
[A p.95]
[Contents]
Exercises 3.4.1
Draw up a joint truth table for each of the following groups of propositions.
1. ( P ! Q) and ( Q ! P)
[A p.95]
2. ¬( P $ Q) and (( P _ Q) ^ ¬( P ^ Q))
[A p.95]
3. ¬( P ^ ¬ Q) and ¬ Q
[A p.95]
4. (( P ! Q) ^ R) and ( P _ ( Q _ R))
[A p.96]
5. (( P ^ Q) ^ (¬ R ^ ¬S)) and (( P _ ( R ! Q)) ^ S)
[A p.96]
14
6. ( P ^ ¬ P) and ( Q ^ ¬ Q)
[A p.96]
7. ( P _ ( Q $ R)) and (( Q ! P) ^ Q)
[A p.96]
8. ¬(( P ^ Q) ^ R) and (( P ! Q) $ ( P ! R))
[A p.97]
9. ( P _ Q), ¬ P and ( Q _ Q)
[A p.97]
10. ( P ! ( Q ! ( R ! S))) and ¬S
[A p.97]
[Contents]
Exercises 3.5.1
1. Can the meaning of any of our two-place connectives (^, _, !, $)
be specified as the truth function f 22 defined in Figure 3.2? [A p.98]
2. Define truth functions f 42 and f 52 such that the meanings of ^ and !
(respectively) can be specified as these truth functions.
[A p.98]
3. Suppose we introduce a new one-place connective ? and specify its
meaning as the truth function f 11 defined in Figure 3.2. What is the
truth value of ? A when A is T?
[A p.98]
4. What truth values do you need to know to determine the truth value
of ?( A ! B)?
(i) The truth values of A and B.
(ii) The truth value of A but not of B.
(iii) The truth value of B but not of A.
(iv) None.
[A p.98]
5. Which of our connectives could have its meaning specified as the
two-place function g( x, y) defined as follows?
g( x, y) = f 32 ( f 21 ( x ), y)
[A p.98]
[Contents]
15
Chapter 4
Uses of Truth Tables
Exercises 4.1.2
Use truth tables to determine whether each of the following arguments is
valid. For any argument that is not valid, give a counterexample.
1. A _ B
A!C
) (B ! C) ! C
[A p.99]
2. ¬ A
) ¬(( A ! B) ^ ( B ! C )) _ C
3. ( A ^ ¬ B) ! C
¬C
B
) ¬A
[A p.99]
[A p.100]
4. ( A ^ B) $ C
A
) C!B
[A p.100]
5. (¬ A ^ ¬ B) $ ¬C
¬( A _ B)
) C ! ¬C
[A p.100]
6. A _ B
¬A _ C
B!C
) C
[A p.101]
16
7. ¬( A _ B) $ ¬C
¬ A ^ ¬B
) C ^ ¬C
[A p.101]
8. ¬( A ^ B) ! (C _ A)
¬ A _ ¬B
A
) ¬(C _ ¬C )
[A p.101]
9. A ! ( B ^ C )
B $ ¬C
) ¬A
[A p.102]
10. A ! B
B!C
¬C
) ¬A
[A p.102]
[Contents]
Exercises 4.2.1
Write out truth tables for the following propositions, and state whether
each is a tautology, a contradiction, or neither.
1. (( P _ Q) ! P)
[A p.102]
2. (¬ P ^ ( Q _ R))
[A p.103]
3. ((¬ P _ Q) $ ( P ^ ¬ Q))
[A p.103]
4. ( P ! ( Q ! ( R ! P)))
[A p.103]
5. ( P ! (( P ! Q) ! Q))
[A p.103]
6. ( P ! (( Q ! P) ! Q))
[A p.104]
7. (( P ! Q) _ ¬( Q ^ ¬ Q))
[A p.104]
8. (( P ! Q) _ ¬( Q ^ ¬ P))
[A p.104]
9. (( P ^ Q) $ ( Q $ P))
[A p.104]
10. ¬(( P ^ Q) ! ( Q $ P))
[A p.104]
[Contents]
17
Exercises 4.3.1
Write out joint truth tables for the following pairs of propositions, and
state in each case whether the two propositions are (a) jointly satisfiable,
(b) equivalent, (c) contradictory, (d) contraries.
1. ( P ! Q) and ¬( P ^ ¬ Q)
[A p.105]
2. ( P ^ Q) and ( P ^ ¬ Q)
[A p.105]
3. ¬( P $ Q) and ¬( P ! Q) _ ¬( P _ ¬ Q)
[A p.105]
4. ( P ! ( Q ! R)) and (( P ! Q) ! R)
[A p.106]
5. ( P ^ ( Q ^ ¬ Q)) and ¬( Q ! ¬( R ^ ¬ Q))
[A p.106]
6. ( P ^ ¬ P) and ( R _ ¬ R)
[A p.107]
7. ( P ^ ¬ P) and ¬( Q ! Q)
[A p.107]
8. (( P ! Q) ! R) and ¬( P _ ¬( Q ^ ¬ R))
[A p.108]
9. ( P $ Q) and (( P ^ Q) _ (¬ P ^ ¬ Q))
[A p.108]
10. ( P $ Q) and (( P ^ Q) _ (¬ P ^ ¬ Q))
[A p.109]
[Contents]
Exercises 4.4.1
Write out a joint truth table for the propositions in each of the following
sets, and state whether each set is satisfiable.
1. {( P _ Q), ¬( P ^ Q), P}
[A p.109]
2. {¬( P ! Q), ( P $ Q), (¬ P _ Q)}
[A p.109]
3. {( P ! ¬ P), ( P _ ¬ P), (¬ P ! P)}
[A p.109]
4. {(( P _ Q) _ R), (¬ P ! ¬ Q), (¬ Q ! ¬ R), ¬ P}
[A p.110]
5. {( P $ Q), ( Q _ R), ( R ! P)}
[A p.110]
6. {(¬ P ! ¬ Q), ( P $ Q), P}
[A p.110]
7. {¬ P, ( P ! ( P ! P)), (¬ P $ P)}
[A p.110]
18
8. {( P _ ¬ Q), ( P ! R), ¬ R, (¬ R ! Q)}
[A p.111]
9. {¬ R, ¬ P, (( Q ! ¬ Q) ! R)}
[A p.111]
10. {(¬ P _ ¬ Q), ¬( P ^ ¬ Q), ( P _ ¬ Q), ¬(¬ P ^ ¬ Q)}
[A p.111]
[Contents]
19
Chapter 5
Logical Form
Exercises 5.1.1
For each of the following propositions, give three correct answers to the
question “what is the form of this proposition?”
1. ¬( R ! ( R ! Q))
[A p.112]
2. ( R _ P) ! ( R _ P)
[A p.112]
3. P ^ (¬ P ! Q)
[A p.112]
4. ((¬ P _ Q) ^ P) $ R
[A p.112]
[Contents]
Exercises 5.2.1
1. The following propositions all have three logical forms in common.
State what the three forms are, and in each case, show what replacements of variables by propositions are required to obtain the three
propositions from the form.
(i) ¬¬C
(ii) ¬¬( A ^ B)
(iii) ¬¬(C ^ ¬ D )
[A p.113]
20
2. State whether the given propositions are instances of the given form.
If so, show what replacements of variables by propositions are required to obtain the proposition from the form.
(i) Form: ¬(a ! b)
Propositions:
(a) ¬( P ! Q)
(b) ¬( R ! Q)
(c) ¬( R ! ( R ! Q))
[A p.113]
[A p.113]
[A p.113]
(a) ¬( P ! ( P ! Q))
(b) ¬( P ! ( P ! P))
(c) ¬( P ! ( Q ! P))
[A p.113]
[A p.113]
[A p.113]
(a) (¬ P _ Q) ! (¬ P ^ Q)
(b) ( P _ ¬ P) ! ( P ^ ¬ P)
(c) ¬( R _ S) ! ¬( R ^ S)
[A p.113]
[A p.113]
[A p.113]
(a) ( P _ Q) _ ( Q _ ( P _ Q))
(b) Q _ (¬ Q _ ( Q ^ Q))
(c) ¬ P _ (¬¬ P _ ¬ P)
[A p.113]
[A p.113]
[A p.113]
(ii) Form: ¬(a ! (a ! b))
Propositions:
(iii) Form: (a _ b) ! (a ^ b)
Propositions:
(iv) Form: a _ (¬ b _ a)
Propositions:
[Contents]
Exercises 5.3.1
For each of the following arguments, give four correct answers to the question “what is the form of this argument?” For each form, show what replacements of variables by propositions are required to obtain the argument from the form.
1. ¬( R ! ( R ! Q))
) R _ ( R ! Q)
[A p.114]
21
2. ( P ^ Q) ! Q
¬Q
) ¬( P ^ Q)
[A p.114]
3. ¬ Q ! ( R ! S)
¬Q
) R!S
[A p.115]
4. ( P ! ¬ Q) _ (¬ Q ! P)
¬(¬ Q ! P)
) P ! ¬Q
[A p.115]
[Contents]
Exercises 5.4.1
For each of the following arguments, (i) show that it is an instance of the
form:
a
a!b
) b
by stating what substitutions of propositions for variables have to be made
to otbain the argument from the form, and (ii) show by producing a truth
table for the argument that it is valid.
1. P
P!Q
) Q
[A p.116]
2. ( A ^ B)
( A ^ B) ! ( B _ C )
) (B _ C)
[A p.116]
3. ( A _ ¬ A)
( A _ ¬ A) ! ( A ^ ¬ A)
) ( A ^ ¬ A)
[A p.116]
4. ( P ! ¬ P)
( P ! ¬ P) ! ( P ! ( Q ^ ¬ R))
) ( P ! ( Q ^ ¬ R))
[A p.116]
[Contents]
22
Exercises 5.5.1
1.
(i) Show by producing a truth table for the following argument
form that it is invalid:
a
) b
[A p.117]
(ii) Give an instance of the above argument form that is valid; show
that it is valid by producing a truth table for the argument.
[A p.117]
2. While it is not true in general that every instance of an invalid argument form is an invalid argument, there are some invalid argument
forms whose instances are always invalid arguments. Give an example of such an argument form.
[A p.117]
[Contents]
23
Chapter 6
Connectives: Translation and
Adequacy
Exercises 6.5.1
Translate the following arguments into PL and then assess them for validity (you may use shortcuts in your truth tables).
1. Bob is happy if and only if it is raining. Either it is raining or the sun
is shining. So Bob is happy only if the sun is not shining. [A p.118]
2. If I have neither money nor a card, I shall walk. If I walk, I shall get
tired or have a rest. So if I have a rest, I have money.
[A p.119]
3. Maisy is upset only if there is thunder. If there is thunder, then there
is lightning. Therefore, either Maisy is not upset, or there is lightning.
[A p.121]
4. The car started only if you turned the key and pressed the accelerator. If you turned the key but did not press the accelerator, then the
car did not start. The car did not start—so either you pressed the accelerator but did not turn the key, or you neither turned the key nor
pressed the accelerator.
[A p.122]
5. Either Maisy isn’t barking, or there is a robber outside. If there is a
robber outside and Maisy is not barking, then she is either asleep or
depressed. Maisy is neither asleep nor depressed. Hence Maisy is
barking if and only if there is a robber outside.
[A p.123]
24
6. If it isn’t sunny, then either it is too windy or we are sailing. We are
having fun if we are sailing. It is not sunny and it isn’t too windy
either—hence we are having fun.
[A p.124]
7. Either you came through Singleton and Maitland, or you came through
Newcastle. You didn’t come through either Singleton or Maitland—
you came through Cessnock. Therefore, you came through both
Newcastle and Cessnock.
[A p.125]
8. We shall have lobster for lunch, provided that the shop is open. Either the shop will be open, or it is Sunday. If it is Sunday, we shall go
to a restaurant and have lobster for lunch. So we shall have lobster
for lunch.
[A p.126]
9. Catch Billy a fish, and you will feed him for a day. Teach him to fish,
and you’ll feed him for life. So either you won’t feed Billy for life, or
you will teach him to fish.
[A p.127]
10. I’ll be happy if the Tigers win. Moreover, they will win—or else they
won’t. However, assuming they don’t, it will be a draw. Therefore,
if it’s not a draw, and they don’t win, I’ll be happy.
[A p.128]
[Contents]
Exercises 6.6.3
1. State whether each of the following is a functionally complete set of
connectives. Justify your answers.
(i) {!, ¬}
[A p.128]
(ii) {$, Y}
[A p.129]
(iv) {!, ^}
[A p.130]
(vi) {_, ≠4 }
[A p.131]
(iii) {≠15 } (The connective ≠15 is often symbolized by #;
another common symbol for this connective is NOR.) [A p.130]
(v) {¬, ≠12 }
[A p.131]
2. Give the truth table for each of the following propositions.
(i) B ≠14 A
[A p.131]
25
(ii) ( A ≠11 B) ≠15 B
[A p.131]
(iii) ¬( A _ ( A ≠6 B))
[A p.131]
(iv) A $ ( A ≠3 ¬ B)
[A p.131]
(v) ( A ≠12 B) Y ( B ≠12 A)
[A p.132]
(vi) ( A ≠12 B) Y ( B ≠16 A)
[A p.132]
3. Consider the three-place connectives ] and , whose truth tables are
as follows:
a b
T T
T T
T F
T F
F T
F T
F F
F F
g
T
F
T
F
T
F
T
F
](a, b, g)
T
F
T
T
T
F
T
T
(a, b, g)
F
F
T
T
T
T
F
F
(i) Define ] using only (but not necessarily all of) the connectives
_, ^, and ¬.
[A p.132]
(ii) Do the same for .
[A p.132]
4. State a proposition involving only the connectives ¬ and ^ that is
equivalent to the given proposition.
(i) ¬( A ! B)
[A p.132]
(ii) ¬( A _ B)
[A p.132]
(iv) ¬(¬ A _ B)
[A p.132]
(vi) ( A ! B) _ ( B ! A)
[A p.132]
(iii) ¬ A _ ¬ B
5.
[A p.132]
(v) A $ B
[A p.132]
(i) What is the dual of ¨1 ?
[A p.132]
(ii) What is the dual of !?
[A p.133]
(iii) Which one-place connectives are their own duals?
[A p.133]
(iv) Which two-place connectives are their own duals?
[A p.133]
[Contents]
26
Chapter 7
Trees for Propositional Logic
Exercises 7.2.1.1
Apply the appropriate tree rule to each of the following propositions.
1. (¬ A _ ¬ B)
[A p.134]
2. (¬ A ! B)
[A p.134]
3. (( A ! B) ^ B)
[A p.134]
4. (( A $ B) $ B)
[A p.134]
5. ¬( A $ ¬¬ A)
[A p.134]
6. ¬(¬ A _ B)
[A p.135]
[Contents]
Exercises 7.2.2.1
Construct finished trees for each of the following propositions.
1. (( A ! B) ! B)
[A p.135]
2. (( A ! B) _ ( B ! A))
[A p.135]
3. ¬(¬ A ! ( A _ B))
[A p.135]
4. ¬¬(( A ^ B) _ ( A ^ ¬ B))
[A p.135]
[Contents]
27
Exercises 7.2.3.1
Construct finished trees for each of the following propositions; close paths
as appropriate.
1. ¬( A ! ( B ! A))
[A p.136]
2. (( A ! B) _ (¬ A _ B))
[A p.136]
3. ¬(( A ! B) _ (¬ A _ B))
[A p.136]
4. ¬¬¬( A _ B)
[A p.136]
5. ¬( A ^ ¬ A)
[A p.136]
6. ¬(¬( A ^ B) $ (¬ A _ ¬ B))
[A p.137]
[Contents]
Exercises 7.3.1.1
Using trees, determine whether the following arguments are valid. For
any arguments that are invalid, give a counterexample.
1.
A
) ( A _ B)
2.
( A _ B)
) B
3.
( A _ B)
( A ! C)
(B ! D)
) (C _ D )
4.
5.
[A p.137]
[A p.137]
[A p.138]
(( A _ ¬ B) ! C )
(B ! ¬D)
D
) C
[A p.138]
B
( A ! B)
) A
[A p.138]
28
6.
7.
8.
9.
10.
A
( A ! B)
) B
[A p.138]
( A _ ( B ^ C ))
( A ! B)
(B $ D)
) (B ^ D)
[A p.139]
¬(¬ A ! B)
¬(C $ A)
( A _ C)
¬(C ! B)
) ¬( A ! B)
[A p.139]
( A $ B)
(B ! C)
(¬ B ! ¬C )
( A _ ( B ^ ¬ B))
) C
[A p.140]
( A ! B)
(B ! C)
(C ! D )
( D ! E)
) ¬( A ^ ¬ E)
[A p.140]
[Contents]
Exercises 7.3.2.1
1. Using trees, test whether the following propositions are contradictions. For any proposition that is satisfiable, read off from an open
path a scenario in which the proposition is true.
(i) A ^ ¬ A
[A p.141]
(ii) ( A _ B) ^ ¬( A _ B)
[A p.141]
(iv) ( A ! ¬( A _ B)) ^ ¬(¬( A _ B) _ B)
[A p.141]
(vi) ( A $ ¬ A) _ ( A ! ¬( B _ C ))
[A p.142]
(iii) ( A ! B) ^ ¬( A _ B)
[A p.141]
(v) ¬((¬ B _ C ) $ ( B ! C ))
29
[A p.142]
2. Using trees, test whether the following sets of propositions are satisfiable. For any set that is satisfiable, read off from an open path a
scenario in which all the propositions in the set are true.
(i) {( A _ B), ¬ B, ( A ! B)}
[A p.142]
(ii) {( A _ B), ( B _ C ), ¬( A _ C )}
[A p.142]
(iii) {¬(¬ A ! B), ¬(C $ A), ( A _ C ), ¬(C ! B), ( A ! B)}
[A p.143]
(iv) {( A $ B), ¬( A ! C ), (C ! A), ( A ^ B) _ ( A ^ C )} [A p.143]
[Contents]
Exercises 7.3.3.1
Test whether the following pairs of propositions are contraries, contradictories, or jointly satisfiable.
1. (¬ A ! B) and ( B ! A)
[A p.144]
2. ( A ! B) and ¬( A ! ( A ! B))
[A p.144]
3. ¬( A $ ¬ B) and ¬( A _ ¬ B)
[A p.145]
4. ¬( A _ ¬ B) and (¬ A ! ¬ B)
[A p.146]
5. (¬ A ^ ( A ! B)) and ¬(¬ A ! ( A ! B))
[A p.147]
6. (( A ! B) $ B) and ¬( A ! B)
[A p.147]
[Contents]
Exercises 7.3.4.1
Test whether the following propositions are tautologies. (Remember to
restore outermost parentheses before adding the negation symbol at the
front—recall §2.5.4.) For any proposition that is not a tautology, read off
from your tree a scenario in which it is false.
1. A ! ( B ! A)
[A p.148]
2. A ! ( A ! B)
[A p.148]
30
3. (( A ^ B) _ ¬( A ! B)) ! (C ! A)
[A p.148]
4. ( A ^ ( B _ C )) $ (( A ^ B) _ ( A ^ C ))
[A p.149]
5. ¬ A _ ¬( A ^ B)
[A p.149]
6. A _ (¬ A ^ ¬ B)
[A p.149]
7. ( A ! B) _ ( A ^ ¬ B)
[A p.150]
8. ( B ^ ¬ A) $ ( A $ B)
[A p.150]
9. ( A _ ( B _ C )) $ (( A _ B) _ C )
[A p.150]
10. ( A ^ ( B _ C )) $ (( A _ B) ^ C )
[A p.151]
[Contents]
Exercises 7.3.5.1
Test whether the following are equivalent. Where the two propositions
are not equivalent, read off from your tree a scenario in which they have
different truth values.
1. P and ( P ^ P)
[A p.151]
2. ( P ! ( Q _ ¬ Q)) and ( R ! R)
[A p.151]
3. ¬( A _ B) and (¬ A ^ ¬ B)
[A p.152]
4. ¬( A _ B) and (¬ A _ ¬ B)
[A p.152]
5. ¬( A ^ B) and (¬ A ^ ¬ B)
[A p.152]
6. ¬( A ^ B) and (¬ A _ ¬ B)
[A p.153]
7. A and (( A ^ B) _ ( A ^ ¬ B))
[A p.153]
8. ¬( P $ Q) and (( P ^ ¬ Q) _ (¬ P ^ Q))
[A p.154]
9. (( P ^ Q) ! R) and ( P ! (¬ Q _ R))
[A p.154]
10. ¬( P $ Q) and ( Q ^ ¬ P)
[A p.155]
[Contents]
31
Chapter 8
The Language of Monadic
Predicate Logic
Exercises 8.2.1
Translate the following propositions from English into MPL:
1. The Pacific Ocean is beautiful.
[A p.156]
2. New York is heavily populated.
[A p.156]
3. Mary is nice.
[A p.156]
4. John is grumpy.
[A p.157]
5. Seven is a prime number.
[A p.157]
6. Pluto is a planet.
[A p.157]
7. Bill and Ben are gardeners
[A p.157]
8. If Mary is sailing or Jenny is kite flying, then Bill and Ben are grumpy.
[A p.157]
9. Mary is neither sailing nor kite flying.
[A p.157]
10. Only if Mary is sailing is Jenny kite flying.
[A p.157]
11. John is sailing or kite flying but not both.
[A p.157]
12. If Mary isn’t sailing, then unless he’s kite flying, John is sailing.
[A p.157]
32
13. Jenny is sailing only if both Mary and John are.
[A p.157]
14. Jenny is sailing if either John or Mary is.
[A p.157]
15. If—and only if—Mary is sailing, Jenny is kite flying.
[A p.157]
16. If Steve is winning, Mary isn’t happy.
[A p.157]
17. Two is prime, but it is also even.
[A p.157]
18. Canberra is small—but it’s not tiny, and it’s a capital city. [A p.157]
19. If Rover is kite flying, then two isn’t prime.
[A p.157]
20. Mary is happy if and only if Jenny isn’t.
[A p.157]
[Contents]
Exercises 8.3.2
Translate the following from English into MPL.
1. If Independence Hall is red, then something is red.
[A p.158]
2. If everything is red, then Independence Hall is red.
[A p.158]
3. Nothing is both green and red.
[A p.158]
4. It is not true that nothing is both green and red.
[A p.158]
5. Red things aren’t green.
[A p.158]
6. All red things are heavy or expensive.
[A p.158]
7. All red things that are not heavy are expensive.
[A p.158]
8. All red things are heavy, but some green things aren’t.
[A p.158]
9. All red things are heavy, but not all heavy things are red. [A p.158]
10. Some red things are heavy, and furthermore some green things are
heavy too.
[A p.158]
11. Some red things are not heavy, and some heavy things are not red.
[A p.158]
33
12. If Kermit is green and red, then it is not true that nothing is both
green and red.
[A p.159]
13. Oscar’s piano is heavy, but it is neither red nor expensive. [A p.159]
14. If Spondulix is heavy and expensive, and all expensive things are red
and all heavy things are green, then Spondulix is red and green.1
[A p.159]
15. If Kermit is heavy, then something is green and heavy.
[A p.159]
16. If everything is fun, then nothing is worthwhile.
[A p.159]
17. Some things are fun and some things are worthwhile, but nothing is
both.
[A p.159]
18. Nothing is probable unless something is certain.
[A p.159]
19. Some things are probable and some aren’t, but nothing is certain.
[A p.159]
20. If something is certain, then it’s probable.
[A p.159]
[Contents]
Exercises 8.3.5
Translate the following propositions from English into MPL.
1. Everyone is happy.
[A p.159]
2. Someone is sad.
[A p.159]
3. No one is both happy and sad.
[A p.159]
4. If someone is sad, then not everyone is happy.
[A p.159]
5. No one who isn’t happy is laughing.
[A p.160]
6. If Gary is laughing, then someone is happy.
[A p.160]
7. Whoever is laughing is happy.
[A p.160]
1 “Spondulix”
is the name of a famous gold nugget, found in 1872.
34
8. Everyone is laughing if Gary is.
[A p.160]
9. Someone is sad, but not everyone and not Gary.
[A p.160]
10. Gary isn’t happy unless everyone is sad.
[A p.160]
11. All leaves are brown and the sky is gray.
[A p.160]
12. Some but not all leaves are brown.
[A p.160]
13. Only leaves are brown.
[A p.160]
14. Only brown leaves can stay.
[A p.160]
15. Everyone is in trouble unless Gary is happy.
[A p.160]
16. Everyone who works at this company is in trouble unless Gary is
happy.
[A p.160]
17. If Stephanie is telling the truth, then someone is lying.
[A p.160]
18. If no one is lying, then Stephanie is telling the truth.
[A p.160]
19. Either Stephanie is lying, or no-one’s telling the truth and everyone
is in trouble.
[A p.160]
20. If Gary is lying, then not everyone in this room is telling the truth.
[A p.160]
[Contents]
Exercises 8.4.3.1
Write out a construction for each of the following wffs, and state the main
operator.
1. 8 x ( Fx ! Gx )
[A p.161]
2. 8 x ¬ Gx
[A p.161]
3. ¬9 x ( Fx ^ Gx )
[A p.161]
4. ( Fa ^ ¬9 x ¬ Fx )
[A p.161]
5. 8 x ( Fx ^ 9y( Gx ! Gy))
[A p.162]
35
6. (8 x ( Fx ! Gx ) ^ Fa)
[A p.162]
7. ((¬ Fa ^ ¬ Fb) ! 8 x ¬ Fx )
[A p.162]
8. 8 x 8y(( Fx ^ Fy) ! Gx )
[A p.163]
9. 8 x ( Fx ! 8yFy)
[A p.163]
10. (8 xFx ! 8yFy)
[A p.163]
[Contents]
Exercises 8.4.5.1
Identify any free variables in the following formulas. State whether each
formula is open or closed.
1. Tx ^ Fx
[A p.163]
2. Tx ^ Ty
[A p.163]
3. 9 xTx ^ 9 xFx
[A p.163]
4. 9 xTx ^ 8yFx
[A p.163]
5. 9 xTx ^ Fx
[A p.164]
6. 9 x ( Tx ^ Fx )
[A p.164]
7. 8y9 xTy
[A p.164]
8. 9 x (8 xTx ! 9yFx )
[A p.164]
9. 9y8 xTx ! 9yFx
[A p.164]
10. 8 x (9 xTx ^ Fx )
[A p.164]
11. 8 x 9 xTx ^ Fx
[A p.164]
12. 9 xTy
[A p.164]
13. 8 xTx ! 9 xFx
[A p.164]
14. 9 x 8y( Tx _ Fy)
[A p.164]
15. 8 xFx ^ Gx
[A p.164]
36
16. 8 x 8yFx ! Gy
[A p.164]
17. 8 x 8y( Fx ! 8 xGy)
[A p.164]
18. 9yGb ^ Gc
[A p.164]
19. 9yGy ^ 8 x ( Fx ! Gy)
[A p.164]
20. 8 x (( Fx ! 9 xGx ) ^ Gx )
[A p.164]
[Contents]
37
Chapter 9
Semantics of Monadic Predicate
Logic
Exercises 9.1.1
For each of the propositions:
(i) Pa
(ii) 9 xPx
(iii) 8 xPx
state whether it is true or false on each of the following models.
1. Domain: {1, 2, 3, . . . }2
Referent of a: 1
Extension of P: {1, 3, 5, . . . }3
[A p.165]
2. Domain: {1, 2, 3, . . . }
Referent of a: 1
Extension of P: {2, 4, 6, . . . }4
[A p.165]
3. Domain: {1, 2, 3, . . . }
Referent of a: 2
Extension of P: {1, 3, 5, . . . }
[A p.165]
4. Domain: {1, 2, 3, . . . }
Referent of a: 2
Extension of P: {2, 4, 6, . . . }
[A p.165]
2 That
is, the set of positive integers.
is, the set of odd numbers.
4 That is, the set of even numbers.
3 That
38
5. Domain: {1, 2, 3, . . . }
Referent of a: 1
Extension of P: {1, 2, 3, . . . }
[A p.165]
6. Domain: {1, 2, 3, . . . }
Referent of a: 2
Extension of P: ∆
[A p.165]
[Contents]
Exercises 9.2.1
State whether each of the following propositions is true or false in each of
the six models given in Exercises 9.1.1.
(i) (¬ Pa ^ ¬ Pa)
(ii) (¬ Pa ! Pa)
(iii) ( Pa $ 9 xPx )
(iv) (9 xPx _ ¬ Pa)
(v) ¬(8 xPx ^ ¬9 xPx )
[Answers p.165]
[Contents]
Exercises 9.3.1
1. If a( x ) is ( Fx ^ Ga), what is
(i) a( a/x )
[A p.166]
(ii) a(b/x )
[A p.166]
2. If a( x ) is 8y( Fx ! Gy), what is
(i) a( a/x )
[A p.166]
(ii) a(b/x )
[A p.166]
3. If a( x ) is 8 x ( Fx ! Gx ) ^ Fx, what is
(i) a( a/x )
[A p.166]
39
(ii) a(b/x )
[A p.166]
4. If a( x ) is 8 x ( Fx ^ Ga), what is
(i) a( a/x )
[A p.166]
(ii) a(b/x )
[A p.166]
5. If a(y) is 9 x ( Gx ! Gy), what is
(i) a( a/y)
[A p.166]
(ii) a(b/y)
[A p.166]
6. If a( x ) is 9y(8 x ( Fx ! Fy) _ Fx ), what is
(i) a( a/x )
[A p.166]
(ii) a(b/x )
[A p.166]
[Contents]
Exercises 9.4.3
1. Here is a model:
Domain: {1, 2, 3, 4}
Extensions: E: {2, 4} O: {1, 3}
State whether each of the following propositions is true or false in
this model.
(i) 8 xEx
[A p.167]
(iii) 9 xEx
[A p.167]
(ii) 8 x ( Ex _ Ox )
[A p.167]
(iv) 9 x ( Ex ^ Ox )
[A p.167]
(vi) 8 xEx _ 9 x ¬ Ex
[A p.167]
(v) 8 x (¬ Ex ! Ox )
[A p.167]
2. State whether the given proposition is true or false in the given models.
(i) 8 x ( Px _ Rx )
(a) Domain: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Extensions: P: {1, 2, 3} R: {5, 6, 7, 8, 9, 10}
40
[A p.167]
(b) Domain: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Extensions: P: {1, 2, 3, 4} R: {4, 5, 6, 7, 8, 9, 10}
[A p.167]
(ii) 9 x (¬ Px $ ( Qx ^ ¬ Rx ))
(a) Domain: {1, 2, 3, . . .}
Extensions: P: {2, 4, 6, . . .}
Q: {1, 3, 5, . . .}
R: {2, 4, 6, . . .}
[A p.167]
(b) Domain: {1, 2, 3, . . .}
Extensions: P: {2, 4, 6, . . .}
Q: {2, 4, 6, . . .}
R: {1, 3, 5, . . .}
[A p.167]
(iii) 9 xPx ^ Ra
(a) Domain: {1, 2, 3, . . .}
Referent of a: 7
Extensions: P: {2, 3, 5, 7, 11, . . .}5 R: {1, 3, 5, . . .} [A p.167]
(b) Domain: {Alice, Ben, Carol, Dave}
Referent of a: Alice
Extensions: P: {Alice, Ben} R: {Carol, Dave} [A p.167]
3. Here is a model:
Domain: {Bill, Ben, Alison, Rachel}
Referents: a: Alison r: Rachel
Extensions: M: {Bill, Ben} F: {Alison, Rachel}
J: {Bill, Alison} S: {Ben, Rachel}
State whether each of the following propositions is true or false in
this model.
(i) ( Ma ^ Fr ) ! 9 x ( Mx ^ Fx )
[A p.167]
(ii) 8 x 8y( Mx ! My)
[A p.167]
(iv) 8 xMx ! 8 xJx
[A p.167]
(vi) 9 x ( Fx ^ Sx ) ! 8 x ( Fx ! Sx )
[A p.167]
(iii) (¬ Ma _ ¬ Jr ) ! 9 x 9y( Mx ^ Fy)
(v) 9 x 9y( Mx ^ Fy ^ Sr )
5 That
[A p.167]
[A p.167]
is, the set of prime numbers.
41
4. For each of the following propositions, describe (a) a model in which
it is true, and (b) a model in which it is false. If there is no model of
one of these types, explain why.
(i) 8 x ( Fx ! Gx )
[A p.167]
(ii) 8 xFx ^ ¬ Fa
[A p.167]
(iv) 9 x ( Fx ^ Gx )
[A p.168]
(vi) 9 xFx ^ 9 xGx
[A p.168]
(viii) 9 x ( Fx ^ ¬ Fx )
[A p.168]
(x) 9 x ( Fx ! Fx )
[A p.169]
(xii) 9 xFx ! 8 xGx
[A p.169]
(xiv) 8 x ( Fx ! Fa)
[A p.169]
(xvi) 8 x ( Fx _ Gx )
[A p.170]
(xviii) 8 x ( Fx ^ ¬ Fx )
[A p.170]
(iii) 9 xFx ^ ¬ Fa
[A p.168]
(v) 8 x ( Fx ! Fx )
[A p.168]
(vii) 8 xFx ! 9 xFx
[A p.168]
(ix) 9 xFx ^ 9 x ¬ Fx
[A p.168]
(xi) 9 xFx ! 9 xGx
[A p.169]
(xiii) 8 xFx ! Fa
[A p.169]
(xv) Fa ! Fb
[A p.169]
(xvii) 9 x ( Fx _ Gx )
[A p.170]
(xix) 8 x 9y( Fx ! Gy)
[A p.170]
(xx) 8 x ( Fx ! 9yGy)
5.
[A p.170]
(i) Is 8 x ( Fx ! Gx ) true or false in a model in which the extension
of F is the empty set?
[A p.170]
(ii) Is 9 x ( Fx ^ Gx ) true in every model in which 8 x ( Fx ! Gx ) is
true?
[A p.170]
[Contents]
42
Exercises 9.5.1
For each of the following arguments, either produce a countermodel (thereby
showing that the argument is invalid) or explain why there cannot be a
countermodel (in which case the argument is valid).
1. 9 xFx ^ 9 xGx
) 9 x ( Fx ^ Gx )
[A p.171]
2. 9 x ( Fx ^ Gx )
) 9 xFx ^ 9 xGx
[A p.171]
3. 8 x ( Fx _ Gx )
¬8 xFx
) 8 xGx
[A p.171]
4. 8 x ( Fx ! Gx )
8 x ( Gx ! Hx )
) 8 x ( Fx ! Hx )
[A p.171]
5. 8 x ( Fx ! Gx )
8 x ( Gx ! Hx )
) 8 x ( Hx ! Fx )
[A p.171]
[Contents]
43
Chapter 10
Trees for Monadic Predicate Logic
Exercises 10.2.2
1. Using trees, determine whether the following propositions are logical truths. For any proposition that is not a logical truth, read off
from your tree a model in which it is false.
(i) Fa ! 9 xFx
[A p.172]
(ii) 9 xFx ! ¬8 x ¬ Fx
[A p.172]
(iv) 8 xFx ! 9 xFx
[A p.173]
(vi) 9 xFx ^ 9 x ¬ Fx
[A p.174]
(iii) 8 x (( Fx ^ ¬ Gx ) ! 9 xGx )
[A p.173]
(v) ( Fa ^ ( Fb ^ Fc)) ! 8 xFx
[A p.173]
(vii) 9 x ( Fx ! 8yFy)
[A p.174]
(viii) 8 x ( Fx ! Gx ) ! ( Fa ! Ga)
(ix) ¬8 x ( Fx ^ Gx ) $ 9 x ¬( Fx ^ Gx )
(x) ¬9 x ( Fx ^ Gx ) $ 8 x (¬ Fx ^ ¬ Gx )
[A p.174]
[A p.175]
[A p.175]
2. Using trees, determine whether the following arguments are valid.
For any argument that is not valid, read off from your tree a model
in which the premises are true and the conclusion false.
(i) 9 xFx ^ 9 xGx
) 9 x ( Fx ^ Gx )
[A p.176]
(ii) 9 x 8y( Fx ! Gy)
) 8y9 x ( Fx ! Gy)
[A p.176]
44
(iii) Fa ! 8 xGx
) 8 x ( Fa ! Gx )
[A p.177]
(iv) Fa ! 8 xGx
) 9 x ( Fa ! Gx )
[A p.177]
(v) 8 x ( Fx _ Gx )
¬8 xFx
) 8 xGx
[A p.177]
(vi) 9 x ( Fx ^ Gx )
) 9 xFx ^ 9 xGx
[A p.178]
(vii) 8 x ( Fx ! Gx )
Fa
) Ga
[A p.178]
(viii) ¬8 x ( Fx _ Gx )
) 9 x (¬ Fx ^ ¬ Gx )
[A p.178]
(ix) 8 x ( Fx ! Gx )
8 x ( Gx ! Hx )
) ¬9 x (¬ Fx ^ Hx )
[A p.179]
(x) 8 x ( Fx _ Gx )
) ¬9 x ( Fx ^ Gx )
[A p.179]
[Contents]
Exercises 10.3.8
Translate the following arguments into MPL, and then test for validity using trees. For any argument that is not valid, read off from your tree a
model in which the premise(s) are true and the conclusion false.
1. All dogs are mammals. All mammals are animals. Therefore, all
dogs are animals.
[A p.180]
2. If everything is frozen, then everything is cold. So everything frozen
is cold.
[A p.181]
3. If a thing is conscious, then either there is a divine being, or that thing
has a sonic screwdriver. Nothing has a sonic screwdriver. Thus, not
everything is conscious.
[A p.182]
4. All cows are scientists, no scientist can fly, so no cow can fly.
[A p.183]
45
5. Someone here is not smoking. Therefore, not everyone here is smoking.
[A p.184]
6. If Superman rocks up, all cowards will shake. Catwoman is not a
coward. So Catwoman will not shake.
[A p.185]
7. Each car is either red or blue. All the red cars are defective, but some
of the blue cars aren’t. Thus, there are some defective cars and some
nondefective cars.
[A p.186]
8. For each thing, it swims only if there is a fish. Therefore, some things
don’t swim.
[A p.187]
9. All robots built before 1970 run on kerosene. Autovac 23E was built
before 1970, but it doesn’t run on kerosene. So it’s not a robot.
[A p.188]
10. Everyone who is tall is either an athlete or an intellectual. Some people are athletes and intellectuals, but none of them is tall. Graham is a
person. Therefore, if he’s an athlete, then either he’s not an intellectual, or he isn’t tall.
[A p.189]
[Contents]
46
Chapter 11
Models, Propositions, and Ways
the World Could Be
There are no exercises for chapter 11.
47
[Contents]
Chapter 12
General Predicate Logic
Exercises 12.1.3.1
State whether each of the following is a wff of GPL.
1. 8 xF1 y
[A p.191]
2. 8 x 9yF1 y
[A p.191]
3. 8 xR2 xy
[A p.191]
4. 8 x 9 xR2 yy
[A p.191]
5. R2 x
[A p.191]
6. 8 xR2 x
[A p.191]
7. 8 x ( F1 x ! R2 x )
[A p.191]
8. 8 x 9y( F1 x ! R2 xy)
[A p.191]
9. 8 x 9y( F1 xy ! R2 y)
[A p.191]
10. 8 x 9y8 x 9yR2 xy
[A p.191]
[Contents]
48
Exercises 12.1.6
Translate the following into GPL.
1. Bill heard Alice.
[A p.192]
2. Bill did not hear Alice.
[A p.192]
3. Bill heard Alice, but Alice did not hear Bill.
[A p.192]
4. If Bill heard Alice, then Alice heard Bill.
[A p.192]
5. Bill heard Alice if and only if Alice heard Alice.
[A p.192]
6. Bill heard Alice, or Alice heard Bill.
[A p.192]
7. Clare is taller than Dave, but she’s not taller than Edward. [A p.192]
8. Mary prefers Alice to Clare.
[A p.192]
9. Mary doesn’t prefer Dave to Clare; nor does she prefer Clare to Dave.
[A p.192]
10. Edward is taller than Clare, but he’s not tall.
[A p.192]
11. The Eiffel tower is taller than both Clare and Dave.
[A p.192]
12. If Dave is taller than the Eiffel tower, then he’s tall.
[A p.192]
13. Although the Eiffel tower is taller, Clare prefers Dave.
[A p.193]
14. If Alice is taller than Dave, then he prefers himself to her. [A p.193]
15. Dave prefers Edward to Clare only if Edward is taller than the Eiffel
tower.
[A p.193]
16. Dave prefers Edward to Clare only if she’s not tall.
[A p.193]
17. Mary has read Fiesta, and she likes it.
[A p.193]
18. Dave doesn’t like Fiesta, but he hasn’t read it.
[A p.193]
19. If Dave doesn’t like The Bell Jar, then he hasn’t read it.
[A p.193]
20. Dave prefers The Bell Jar to Fiesta, even though he hasn’t read either.
[A p.193]
[Contents]
49
Exercises 12.1.9
Translate the following into GPL.
1.
(i) Something is bigger than everything.
[A p.193]
(ii) Something is such that everything is bigger than it.
[A p.193]
(iii) If Alice is bigger than Bill, then something is bigger than Bill.
[A p.193]
(iv) If everything is bigger than Bill, then Alice is bigger than Bill.
[A p.193]
(v) If something is bigger than everything, then something is bigger
than itself.
[A p.193]
(vi) If Alice is bigger than Bill and Bill is bigger than Alice, then
everything is bigger than itself.
[A p.193]
(vii) There is something that is bigger than anything that Alice is bigger than.
[A p.193]
(viii) Anything that is bigger than Alice is bigger than everything that
Alice is bigger than.
[A p.193]
(ix) Every room contains at least one chair.
[A p.193]
(x) In some rooms some of the chairs are broken; in some rooms all
of the chairs are broken; in no room is every chair unbroken.
[A p.194]
2.
(i) Every person owns a dog.
[A p.194]
(ii) For every dog, there is a person who owns that dog. [A p.194]
(iii) There is a beagle that owns a chihuahua.
[A p.194]
(iv) No beagle owns itself.
[A p.194]
(v) No chihuahua is bigger than any beagle.
[A p.194]
(vi) Some chihuahuas are bigger than some beagles.
[A p.194]
(vii) Some dogs are happier than any person.
[A p.194]
(viii) People who own dogs are happier than those who don’t.
[A p.194]
(ix) The bigger the dog, the happier it is.
[A p.194]
(x) There is a beagle that is bigger than every chihuahua and smaller
than every person.
[A p.194]
50
3.
(i) Alice is a timid dog, and some cats are bigger than her.
[A p.195]
(ii) Every dog that is bigger than Alice is bigger than Bill. [A p.195]
(iii) Bill is a timid cat, and every dog is bigger than him.
[A p.195]
(iv) Every timid dog growls at some gray cat.
[A p.195]
(v) Every dog growls at every timid cat.
[A p.195]
(vi) Some timid dog growls at every gray cat.
[A p.195]
(vii) No timid dog growls at any gray cat.
[A p.195]
(viii) Alice wants to buy something from Woolworths, but Bill doesn’t.
[A p.195]
(ix) Alice wants to buy something from Woolworths that Bill doesn’t.
[A p.195]
(x) Bill growls at anything that Alice wants to buy from Woolworths.
[A p.195]
4.
(i) Dave admires everyone.
[A p.196]
(ii) No one admires Dave.
[A p.196]
(iii) Dave doesn’t admire himself.
[A p.196]
(iv) No one admires
himself.6
[A p.196]
(v) Dave admires anyone who doesn’t admire himself.7 [A p.196]
(vi) Every self-admiring person admires Dave.
[A p.196]
(vii) Frank admires Elvis but he prefers the Rolling Stones. [A p.196]
(viii) Frank prefers any song recorded by the Rolling Stones to any
song recorded by Elvis.
[A p.196]
(ix) The Rolling Stones recorded a top-twenty song, but Elvis didn’t.
[A p.196]
(x) Elvis prefers any top-twenty song that the Rolling Stones recorded
to any song that he himself recorded.
[A p.196]
[Contents]
6 Read
“himself” here as gender-neutral—that is, the claim is that no one self-admires.
“himself” here as gender-neutral—that is, the claim is that Dave admires anyone who doesn’t self-admire.
7 Read
51
Exercises 12.2.2
1. Here is a model:
Domain:
Referents:
Extensions: E:
P:
L:
{1, 2, 3, . . .}
a: 1 b: 2 c: 3
{2, 4, 6, . . .}
{2, 3, 5, 7, 11, . . .}8
{h1, 2i, h1, 3i, h1, 4i, . . . , h2, 3i, h2, 4i, . . . , h3, 4i, . . .} 9
State whether each of the following propositions is true or false in
this model.
(i) Lba
[A p.196]
(ii) Lab _ Lba
[A p.196]
(iii) Laa
[A p.196]
(iv) 9 xLxb
[A p.196]
(vi) 9 xLxx
[A p.197]
(viii) 8 x 9yLyx
[A p.197]
(v) 9 xLxa
[A p.196]
(vii) 8 x 9yLxy
[A p.197]
(ix) 9 x ( Px ^ Lxb)
[A p.197]
(xi) 8 x 9y( Ey ^ Lxy)
[A p.197]
(x) 9 x ( Px ^ Lcx )
[A p.197]
(xii) 8 x 9y( Py ^ Lxy)
[A p.197]
(xiv) 8 x (( Lax ^ Lxc) ! Ex )
[A p.197]
(xvi) 9 x 9y9z( Ex ^ Py ^ Ez ^ Pz ^ Lxy ^ Lyz)
[A p.197]
(xiii) 8 x ( Lcx ! Ex )
[A p.197]
(xv) 8 x 8y( Lxy _ Lyx )
[A p.197]
(xvii) 9 x 9y9z( Lxy ^ Lyz ^ Lzx )
(xviii) 8 x 8y8z(( Lxy ^ Lyz) ! Lxz)
8 That
[A p.197]
[A p.197]
is, the set of prime numbers.
is, the set of all pairs h x, yi such that x is less than y. A more compact way of
writing this set is {h x, yi : x < y}. See §16.1 for an explanation of this kind of notation for sets. 9 That 52 2. Here is a model: Domain: Referents: Extensions: F: G: R: S: {1, 2, 3} a: 1 b: 2 c: 3 {1, 2} {2, 3} {h1, 2i, h2, 1i, h2, 3i} {h1, 2, 3i} State whether each of the following propositions is true or false in this model. (i) 8 x 8y( Rxy ! Ryx ) [A p.197] (ii) 8 x 8y( Ryx ! Rxy) [A p.197] (iv) 8 x ( Fx ! 9y( Gy ^ Rxy)) [A p.197] (vi) 9 x 9ySxay [A p.197] (iii) 8 x 9y( Gy ^ Rxy) [A p.197] (v) 9 x 9y9zSxyz [A p.197] (vii) 9 x 9ySxby [A p.197] (viii) 9 xSxxx [A p.197] (ix) 9 x 9y( Fx ^ Fy ^ Sxby) [A p.197] (x) 9 x 9y( Fx ^ Gy ^ Sxby) [A p.197] 3. Here is a model: Domain: Referents: {Alice, Bob, Carol, Dave, Edwina, Frank} a:Alice b: Bob c: Carol d: Dave e: Edwina f : Frank Extensions: M: {Bob, Dave, Frank} F: {Alice, Carol, Edwina} L: {hAlice, Caroli, hAlice, Davei, hAlice, Alicei, hDave, Caroli, hEdwina, Davei, hFrank, Bobi} S: {hAlice, Bobi, hAlice, Davei, hBob, Alicei, hBob, Davei hDave, Bobi, hDave, Alicei} State whether each of the following propositions is true or false in this model. (i) 8 x 8y( Lxy ! Lyx ) [A p.197] (ii) 9 xLxx [A p.198] (iii) ¬9 xSxx [A p.198] 53 (iv) 8 x 8y(Sxy ! Syx ) [A p.198] (vi) 8 x ( Mx ! 9yLyx ) [A p.198] (v) 8 x 8y8z((Sxy ^ Syz) ! Sxz) [A p.198] (vii) 8 x ( Fx ! 9yLyx ) [A p.198] (ix) 9 x 9y( Lax ^ Lyb) [A p.198] (viii) 8 x ( Fx ! 9yLxy) [A p.198] (x) 8 x (( Lxd _ Ldx ) _ Mx ) [A p.198] 4. For each of the following propositions, describe (a) a model in which it is true and (b) a model in which it is false. If there is no model of one of these types, explain why. (i) 8 xFxx [A p.198] (ii) 8 x 8y( Fxy ! Fyx ) [A p.198] (iv) 9 x 8yFxy [A p.198] (vi) 9 x 9yFxy [A p.198] (iii) 8 x 8y( Fxy $ Fyx ) [A p.198] (v) 8 x 9yFxy [A p.198] (vii) 8 x 8yFxy [A p.199] (viii) 9 x 9yFxy ^ ¬ Faa [A p.199] (ix) 8 x 8yFxy ^ ¬ Faa [A p.199] (x) 8 x 8y( Fxy $ Fyx ) ^ Fab ^ ¬ Fba [A p.199] [Contents] Exercises 12.3.1 1. Using trees, determine whether the following propositions are logical truths. For any proposition that is not a logical truth, read off from your tree a model in which it is false. (i) 8 x ( Rxx ! 9yRxy) [A p.200] (iii) 8 xRax ! 8 x 9yRyx [A p.200] (ii) 8 x (9yRxy ! 9zRzx ) [A p.200] (iv) 8 x 9y9zRyxz ! 9 x 9yRxay [A p.201] 54 (v) ¬8 x 9yRxy [A p.201] (vi) 8 x 8y8z(( Rxy ^ Ryz) ! Rxz) [A p.202] (vii) 9 x 8yRxy ! 8 x 9yRxy [A p.202] (ix) 9 x 8yRxy ! 9 x 9yRxy [A p.203] (viii) 9y8 xRxy ! 8 x 9yRxy [A p.203] (x) 8 x 8y9zRxyz _ 8 x 8y8z¬ Rxyz [A p.204] 2. Using trees, determine whether the following arguments are valid. For any argument that is not valid, read off from your tree a model in which the premises are true and the conclusion false. (i) 8 x 8y8z(( Rxy ^ Ryz) ! Rxz) Rab Rba ) 9 xRxx (ii) 8 xFxa ! 9 xFax 9 xFxa ) 9 xFax [A p.204] [A p.205] (iii) 9 x 9y9z( Rxy ^ Rzy) ) 9 xRxx [A p.205] (iv) 8 x 8y( Rxy ! Ryx ) 9 xRxa ) 9 xRax [A p.206] (v) 8 x 8y(¬ Rxy ! Ryx ) ) 8 x 9yRyx [A p.206] (vi) 8 x 8y( Rxy ! ( Fx ^ Gy)) ) ¬9 xRxx [A p.206] (vii) 8 x ( Fx ! (8yRxy _ ¬9yRxy)) Fa ¬ Rab ) ¬ Raa (viii) 8 x 8y(9z( Rzx ^ Rzy) ! Rxy) 8 xRax ) 8 x 8yRxy (ix) 8 x 9yRxy ) 9 xRxb [A p.207] [A p.207] [A p.208] 55 (x) 9 x 8y( Fy ! Rxy) 9 x 8y¬ Ryx ) 9 x ¬ Fx [A p.208] 3. Translate the following arguments into GPL and then test for validity using trees. For any argument that is not valid, read off from your tree a model in which the premises are true and the conclusion false. (i) Alice is older than Bill, and Bill is older than Carol, so Alice must be older than Carol. [A p.209] (ii) Alice is older than Bill. Bill is older than Carol. Anything older than something is older than everything that that something is older than. It follows that Alice is older than Carol. [A p.209] (iii) I trust everything you trust. You trust all bankers. Dave is a banker. Thus, I trust Dave. [A p.210] (iv) Everybody loves somebody, so everybody is loved by somebody. [A p.211] (v) Nancy is a restaurateur. She can afford to feed all and only those restaurateurs who can’t afford to feed themselves. So Nancy is very wealthy. [A p.212] (vi) Everything in Paris is more beautiful than anything in Canberra. The Eiffel tower is in Paris, and Lake Burley Griffin is in Canberra. Therefore, the Eiffel tower is more beautiful than Lake Burley Griffin. [A p.213] (vii) Politicians only talk to politicians. No journalist is a politician. So no politician talks to any journalist. [A p.214] (viii) There is no object that is smaller than all objects; therefore, there is no object such that every object is smaller than it. [A p.215] (ix) Either a movie isn’t commercially successful or both Margaret and David like it. There aren’t any French movies that Margaret and David both like. So there aren’t any commercially successful French movies. [A p.216] (x) There’s something that causes everything. Thus, there’s nothing that is caused by everything. [A p.217] [Contents] 56 Exercises 12.4.1 For each of the following arguments, first translate into GPL and show that the argument is invalid using a tree. Then formulate suitable postulates and show, using a tree, that the argument with these postulates added as extra premises is valid. 1. Roger will eat any food; therefore, Roger will eat that egg. [A p.217] 2. Bill weighs 180 pounds. Ben weighs 170 pounds. So Bill is heavier than Ben. [A p.218] 3. John ran 5 miles; Nancy ran 10 miles; hence, Nancy ran farther than John. [A p.218] 4. Sophie enjoys every novel by Thomas Mann, so she enjoys Buddenbrooks. [A p.219] 5. Chris enjoys novels and nothing else; therefore, he does not enjoy anything by Borges. [A p.219] [Contents] Exercises 12.5.4 For each of the following wffs, find an equivalent wff in prenex normal form. 1. (8 xPx _ 8 xQx ) [A p.220] 2. (9 xPx _ 9 xQx ) [A p.220] 3. (8 xPx ! 8 xPx ) [A p.220] 4. (8 xPx $ 8 xPx ) [A p.220] 5. ¬8 x (Sx ^ (9yTy ! 9zUxz)) [A p.220] [Contents] 57 Chapter 13 Identity Exercises 13.2.2 Translate the following into GPLI. 1. Chris is larger than everything (except himself). [A p.221] 2. All dogs are beagles—except Chris, who is a chihuahua. [A p.221] 3. Ben is happy if he has any dog other than Chris by his side. [A p.221] 4. Chris is happy if he is by anyone’s side but Jonathan’s. [A p.221] 5. Jonathan is larger than any dog. [A p.222] 6. Everything that Mary wants is owned by someone else. [A p.222] 7. Mary owns something that someone else wants. [A p.222] 8. Mary owns something she doesn’t want. [A p.222] 9. If Mary owns a beagle, then no one else does. [A p.222] 10. No one other than Mary owns anything that Mary wants. [A p.222] 11. Everyone prefers Seinfeld to Family Guy. [A p.222] 12. Seinfeld is Adam’s most preferred television show. [A p.222] 13. Family Guy is Adam’s least preferred television show. [A p.222] 58 14. Jonathon watches Family Guy, but he doesn’t watch any other television shows. [A p.222] 15. Jonathon is the only person who watches Family Guy. [A p.222] 16. Diane is the tallest woman. [A p.222] 17. Edward is the only man who is taller than Diane. [A p.222] 18. Diane isn’t the only woman Edward is taller than. [A p.222] 19. No one whom Diane’s taller than is taller than Edward. [A p.222] 20. Edward and Diane aren’t the only people. [A p.222] 21. You’re the only one who knows Ben. [A p.222] 22. I know people other than Ben. [A p.222] 23. Everyone Ben knows (not including Chris and me) is happy. [A p.222] 24. The only happy person I know is Ben. [A p.222] 25. Ben is the tallest happy person I know. [A p.222] 26. Jindabyne is the coldest town between Sydney and Melbourne. [A p.222] 27. There’s a colder town than Canberra between Sydney and Melbourne. [A p.222] 28. For every town except Jindabyne, there is a colder town. [A p.222] 29. No town between Sydney and Melbourne is larger than Canberra or colder than Jindabyne. [A p.222] 30. Jindabyne is my most preferred town between Sydney and Melbourne. [A p.223] [Contents] 59 Exercises 13.3.1 1. Here is a model: Domain: Referents: Extensions: F: R: {Clark, Bruce, Peter} a: Clark b: Clark e: Peter f : Peter {Bruce, Peter} {hClark, Brucei, hClark, Peteri, hBruce, Brucei, hPeter, Peteri} State whether each of the following propositions is true or false in this model. (i) 8 x (¬ Fx ! x = a) [A p.223] (ii) 8 x ( x = a ! 8yRxy) [A p.223] (iv) 8 x ( x 6= b ! Rax ) [A p.223] (vi) 9 x ( x 6= e ^ Rxx ) [A p.223] (iii) 9 x ( x 6= f ^ F f ^ Rx f ) [A p.223] (v) 9 x ( x 6= a ^ 8y( Fy ! Rxy)) 2. Here is a model: Domain: Referents: Extensions: F: G: R: [A p.223] {1, 2, 3, . . .} a: 1 b: 1 c: 2 e: 4 {1, 2, 3} {1, 3, 5, . . .} {h1, 2i, h2, 3i, h3, 4i, h4, 5i, . . .} State whether each of the following propositions is true or false in this model. (i) 9 x ( Rax ^ ¬ Rbx ) [A p.223] (ii) 8 x (( Fx ^ ¬ Gx ) ! x = c) [A p.223] (iv) 8 x ( Gx ! 9y9z( Rxy ^ Ryz ^ Gz)) [A p.223] (vi) 9 x (¬ Fx ^ x 6= e ^ 9y( Fy ^ Ryx )) [A p.223] (iii) 8 x ( x 6= a ! 9yRyx ) (v) 8 x (( x = a _ x = b) ! x 6= c) [A p.223] [A p.223] 3. For each of the following propositions, describe (a) a model in which it is true and (b) a model in which it is false. If there is no model of one of these types, explain why. 60 (i) 8 x ( Fx ! x = a) [A p.223] (ii) 9 x ( x = a ^ x = b) [A p.223] (iv) 8 x 8y( Rxy ! x = y) [A p.224] (vi) 9 x ( x = a ^ a 6= x ) [A p.224] (iii) 9 x 8y( x 6= y ! Rxy) [A p.223] (v) 8 x 8y( x 6= y ! 9zRxyz) [A p.224] (vii) 8 x 8y(( Fx ^ Fy) ! x = y) [A p.224] (viii) 9 x ( Fx ^ 8y( Gy ! x = y)) [A p.224] (x) 8 x (( Fx ^ Rax ) ! x 6= a) [A p.224] (ix) 8 x ( Fx ! 9y( x 6= y ^ Rxy)) [A p.224] (xi) 9 x 9y9z( x 6= y ^ y 6= z ^ x 6= z ^ Rxyz) [A p.224] (xii) 8 x 8y8z( Rxyz ! ( x 6= y ^ y 6= z ^ x 6= z)) [A p.224] (xiv) 9 x ( Fx ^ 8y(( Fy ^ x 6= y) ! Rxy)) [A p.225] (xvi) 8 x ( Rxx ! 8y( x = y ! Rxy)) [A p.225] (xiii) 8 x 8y( x 6= y ! ( Fx _ Fy)) (xv) 8 x 8y8z( Rxyz ! ( Rxxx ^ Ryyy ^ Rzzz)) (xvii) ( Fa ^ Fb) ^ 8 x 8y(( Fx ^ Fy) ! x = y) (xviii) 9 x 9y( Fx ^ Fy ^ 8z[ Fz ! ( x = z _ y = z)]) [A p.225] [A p.225] [A p.225] [A p.225] [Contents] Exercises 13.4.3 1. Using trees, determine whether the following sets of propositions are satisfiable. For any set that is satisfiable, read off from your tree a model in which all propositions in the set are true. (i) { Rab ! ¬ Rba, Rab, a = b} [A p.226] (iii) {8 x ( Fx ! x = a), Fa, a 6= b} [A p.227] (ii) { Rab, ¬ Rbc, a = b} [A p.226] (iv) {8 x ( Fx ! Gx ), 9 xFx, ¬ Ga, a = b} [A p.227] (vi) {9 x 8y( Fy ! x = y), Fa, Fb} [A p.228] (v) {8 x ( x 6= a ! Rax ), 8 x ¬ Rxb, a 6= b} 61 [A p.228] (vii) {8 x 8y( Rxy ! x = y), Rab, a 6= b} (viii) {8 x (( Fx ^ Rxa) ! x 6= a), Fb ^ Rba, a = b} (ix) {9 x 9y9zRxyz, 8 x ( x = x ! x = a)} (x) {8 x ¬ Rxx, 8 x 8yx = y, 9 xRax } [A p.229] [A p.229] [A p.230] [A p.231] 2. Using trees, determine whether the following arguments are valid. For any argument that is not valid, read off from your tree a model in which the premises are true and the conclusion false. (i) 9 xFx 9yGy 8 x 8yx = y ) 9 x ( Fx ^ Gx ) [A p.231] (ii) 9 x 9y( Fx ^ Gy ^ 8z(z = x _ z = y)) ) 9 x ( Fx ^ Gx ) (iii) Rab ) 8 x 8y8z((( Rxy ^ Ryz) ^ x = z) ! Ryy) (iv) 8 x 8y( Rxy ! Ryx ) 9 x ( Rax ^ x 6= b) ) 9 x ( Rxa ^ x 6= b) [A p.232] [A p.233] [A p.233] (v) 8 x 8yx = y ) 8 x 8y( Rxy ! Ryx ) [A p.234] (vi) 8 x 8y8z(( Rxy ^ Rxz) ! y = z) Rab ^ Rcd b 6= d ) a 6= c (vii) 9 x 9y( Rxy ^ x = y) ) ¬8 xRxx [A p.234] [A p.235] (viii) 8 x ( x = a _ x = b) ) 8 xx = a [A p.235] (ix) 8 xRax ¬8 x 8yx = y ) 9 x 9y9z( Rxy ^ Rxz ^ y 6= z) (x) 8 xx = a ) 8 xx = b [A p.236] [A p.236] 62 3. Translate the following propositions into GPLI and then test whether they are logical truths using trees. For any proposition that is not a logical truth, read off from your tree a model in which it is false. (i) If Stan is the only firefighter, then nothing else is a firefighter. [A p.237] (ii) If Julius Caesar is left-handed but Lewis Carroll isn’t, then Lewis Carroll isn’t Julius Caesar. [A p.237] (iii) If the sun is warming all and only things other than itself, then the sun is warming Apollo. [A p.238] (iv) If Kevin Bacon isn’t Kevin Bacon, then he’s Michael J. Fox. [A p.238] (v) If no one who isn’t Twain is a witty author, and Clemens is an author, then Clemens is not witty. [A p.239] (vi) No spy trusts any other spy. [A p.240] (vii) Either everything is identical to this ant, or nothing is. [A p.240] (viii) If Doug is afraid of everything but Santa Claus, then either he’s afraid of himself, or else he’s Santa Claus. [A p.241] (ix) If Mark respects Samuel and only Samuel, then Mark doesn’t respect himself. [A p.242] (x) Either I am a physical body, or I am identical to something that’s not a physical body. [A p.243] [Contents] Exercises 13.5.1 1. Translate the following propositions into GPLI and then test whether they are logical truths using trees. For any proposition that is not a logical truth, read off from your tree a model in which it is false. (i) There are at most two gremlins. [A p.244] (ii) There are at least three Beatles. [A p.245] (iii) There is exactly one thing that is identical to Kevin Bacon. [A p.246] (iv) If there are at least two oceans, then there is an ocean. [A p.247] 63 (v) Take any two distinct dogs, the first of which is larger than the second; then the second is not larger than the first. [A p.247] (vi) If there is exactly one apple, then there is at least one apple. [A p.248] (vii) It’s not the case both that there are at least two apples and that there is at most one apple. [A p.248] (viii) Either there are no snakes, or there are at least two snakes. [A p.249] 2. Translate the following arguments into GPLI and then test for validity using trees. For any argument that is not valid, read off from your tree a model in which the premises are true and the conclusion false. (i) There are at least three things in the room. It follows that there are at least two things in the room. [A p.250] (ii) There are at least two bears in Canada, so there are at most two bears in Canada. [A p.251] (iii) There is at most one barber. So either every barber cuts his own hair, or no barber cuts any barber’s hair. [A p.252] (iv) There are at most two things. If you pick a first thing and then pick a second thing (which may or may not be a different object from the first thing), then one of them is heavier than the other. So everything is either the heaviest or the lightest thing. [A p.253] (v) Some football players are athletes. Some golfers are athletes. Thus, there are at least two athletes. [A p.254] (vi) Everything is a part of itself. So everything has at least two parts. [A p.255] (vii) There are at least two things that are identical to the Eiffel tower. Therefore, there is no Eiffel tower. [A p.256] (viii) I’m afraid of Jemima and the chief of police. So either Jemima is the chief of police, or I’m afraid of at least two things. [A p.257] [Contents] Exercises 13.6.1.1 Translate the following into GPLI, using Russell’s approach to definite descriptions. 64 1. Joseph Conrad is the author of The Shadow Line. [A p.258] 2. The author of The Shadow Line authored Lord Jim. [A p.258] 3. The author of The Shadow Line is the author of Lord Jim. [A p.258] 4. Vance reads everything authored by the author of Lord Jim. [A p.258] 5. Joseph Conrad authored The Inheritors, but it’s not the case that he is the author of The Inheritors. [A p.258] 6. The author of The Shadow Line is taller than any author of Lord Jim. [A p.258] 7. There is something taller than the author of The Shadow Line. [A p.258] 8. The author of The Shadow Line is taller than Joseph Conrad, who is taller than the author of Lord Jim. [A p.258] 9. The father of the author of The Shadow Line is taller than Joseph Conrad. [A p.258] 10. The father of the author of The Shadow Line is taller than the author of The Shadow Line. [A p.258] [Contents] Exercises 13.6.2.1 Translate the claims in Exercises 13.6.1.1 into GPLID, using the definite description operator to translate definite descriptions. [A p.259] [Contents] 65 Exercises 13.6.3.1 Translate the claims in Exercises 13.6.1.1 into GPLI, treating definite descriptions as names and stating appropriate uniqueness assumptions as postulates. [A p.260] [Contents] Exercises 13.7.4 1. Translate the following into GPLIF. (i) 2 + 2 = 4 [A p.261] (ii) 2 ⇥ 2 = 4 [A p.261] (iii) 2 + 2 = 2 ⇥ 2 [A p.261] (iv) 22 = 2 ⇥ 2 [A p.261] (v) ( x + y)2 = ( x + y)( x + y) [A p.261] (vi) ( x + y)2 = x2 + 2xy + y2 [A p.262] (vii) Whether x is even or odd, 2x is even. [A p.262] (viii) Tripling an odd number results in an odd number; tripling an even number results in an even number. [A p.262] (ix) 5x < 6x [A p.262] (x) If x < y, then 3x < 4y [A p.262] 66 2. Here is a model: Domain: Referents: Extensions: Values of function symbols: F: M: S: f: m: s: {Alison, Bruce, Calvin, Delilah} a: Alison b: Bruce c: Calvin d: Delilah {Alison, Delilah} {Bruce, Calvin} {hAlison, Brucei, hAlison, Calvini, hAlison,Delilahi,hBruce, Calvini, hBruce, Delilahi,hCalvin, Delilahi} {hAlison, Brucei,hBruce, Calvini, hCalvin, Brucei, hDelilah, Calvini} {hAlison, Delilahi, hBruce, Alisoni, {hCalvin, Delilahi, hDelilah, Alisoni} {hAlison, Alison, Brucei, hAlison, Bruce, Calvini, hAlison, Calvin, Delilahi, hAlison, Delilah, Alisoni, hBruce, Alison, Calvini, hBruce, Bruce, Calvini, hBruce, Calvin, Delilahi, hBruce, Delilah, Alisoni, hCalvin, Alison, Delilahi, hCalvin, Bruce,Delilahi, hCalvin, Calvin, Delilahi, hCalvin, Delilah, Alisoni, hDelilah, Alison, Alisoni, hDelilah, Bruce, Alisoni, hDelilah, Calvin, Alisoni, hDelilah, Delilah, Alisoni} State whether each of the following propositions is true or false in this model. (i) 8 xM f ( x ) [A p.262] (ii) 9 xMm( x ) [A p.262] (iii) s(c, b) = d [A p.262] (iv) s( a, a) = f (c) [A p.262] (v) F f (b) ! M f (b) [A p.262] (vi) 8 x 8y9z8w(s( x, y) = w $ w = z) (vii) 9 x 9y9z9w(s( x, y) = z ^ s( x, y) = w ^ z 6= w) 67 [A p.262] [A p.262] (viii) s(s(b, a), s(d, a)) = s(b, c) [A p.262] (ix) 9 x 9ys( x, y) = m(y) [A p.262] (x) 8 x 9ys(y, x ) = x 3. Here is a model: Domain: Referents: Extensions: Values of function symbols: [A p.262] {1, 2, 3, . . .} a1 : 1 a2 : 2 a3 : 3 . . . E: {2, 4, 6, . . .} O: {1, 2, 3, . . .} L: {h x, yi : x < y} 10 q: s p {h x, yi : y = x2 } 11 {h x, y, zi : z = x + y} 12 {h x, y, zi : z = x ⇥ y} 13 State whether each of the following propositions is true or false in this model. (i) s( a2 , a2 ) = a5 [A p.262] (ii) p( a2 , a2 ) = a3 [A p.262] (iii) s( a2 , a2 ) = p( a2 , a2 ) [A p.262] (iv) q( a2 ) = p( a1 , a2 ) [A p.262] (v) 8 x 8yq(s( x, y)) = p(s( x, y), s( x, y)) [A p.262] (vi) 8 x 8yq(s( x, y)) = s(s(q( x ), p( a2 , p( x, y))), q(y)) (vii) 8 xEp( a2 , x ) (viii) 8 x ((Ox ! Op( a3 , x )) ^ ( Ex ! Ep( a3 , x ))) (ix) 9 xLp( a5 , x ) p( a5 , x ) (x) 8 x 8y( Lyx ! Lp( a3 , x ) p( a4 , y)) 10 That [A p.262] [A p.262] [A p.262] [A p.262] [A p.262] is, {h1, 2i, h1, 3i, h2, 3i, h1, 4i, h2, 4i, h3, 4i, h1, 5i, h2, 5i, h3, 5i, h4, 5i, . . .}. is, {h1, 1i, h2, 4i, h3, 9i, h4, 16i, . . .}. 12 That is, {h1, 1, 2i, h2, 1, 3i, h2, 2, 4i, h1, 2, 3i, h3, 1, 4i, h3, 2, 5i, h3, 3, 6i, h2, 3, 5i, h1, 3, 4i, h4, 1, 5i, . . . }. 13 That is, {h1, 1, 1i, h2, 1, 2i, h2, 2, 4i, h1, 2, 2i, h3, 1, 3i, h3, 2, 6i, h3, 3, 9i, h2, 3, 6i, h1, 3, 3i, h4, 1, 4i, . . . }. 11 That 68 4. For each of the following propositions, describe (a) a model in which it is true and (b) a model in which it is false. If there is no model of one of these types, explain why. (i) f ( a) = f (b) [A p.263] (ii) f ( a) 6= f (b) [A p.263] (iv) 8 x 9y f ( x ) = y [A p.263] (vi) 8 x 8ys( x, y) = s(y, x ) [A p.263] (iii) f ( a) 6= f ( a) [A p.263] (v) 9 x 8y f ( x ) = y [A p.263] (vii) 8 x 8y f (s( x, y)) = s( f ( x ), f (y)) (viii) 9 x 9ys( x, y) = f ( x ) ! 9 x 9ys( x, y) = f (y) (ix) 9 x 9ys( x, y) = f ( x ) ! 9 x 9y f (s( x, y)) = f ( x ) (x) 8 x 8y9z8w(s( x, y) = w $ w = z) 69 [A p.263] [A p.264] [A p.264] [A p.264] [Contents] Chapter 14 Metatheory Exercises 14.1.1.1 What is the complexity of each of the following wffs? 1. Fa [A p.265] 2. ( Hx ! 8 x ( Fx ! Gx )) [A p.265] 3. 8 xx = x [A p.265] 4. 8 x 9y¬ Rxy [A p.265] 5. ¬8 xa 6= x [A p.265] 6. 8 x ( Fx ! 9yRxy) [A p.265] 7. (8 xa = x ^ ¬9 xa 6= x ) [A p.265] 8. ( Fa ^ ( Fa ^ ( Fa ^ ( Fa ^ ( Fa ^ Fa))))) [A p.265] 9. 8 x ( Fx ! 8 x ( Fx ! 8 x ( Fx ! 8 x ( Fx ! 8 x ( Fx ! Fx ))))) [A p.265] 10. (((¬9 x (¬ Fx _ Gx ) ^ a 6= b) ! ¬ Fa) _ (¬9 x (¬ Fx _ Gx ) _ ¬ Fa)) [A p.265] [Contents] 70 Exercises 14.1.2.1 In §10.1 we showed that the tree rules for (negated and unnegated) disjunction and the quantifiers are truth-preserving (in the precise sense spelled out in §14.1.2), and in §13.4 we showed that the tree rule SI is truth-preserving. Complete the soundness proof by showing that the remaining tree rules are truth-preserving: 1. Unnegated conjunction. [A p.265] 2. Negated conjunction. [A p.266] 3. Unnegated conditional. [A p.266] 4. Negated conditional. [A p.266] 5. Unnegated biconditional. [A p.266] 6. Negated biconditional. [A p.267] 7. Negated negation. [A p.267] [Contents] Exercises 14.1.3.1 Fill in the remaining cases in step (III) of the completeness proof. 1. g is of the form ¬a, and a’s main operator is conjunction. [A p.267] 2. g is of the form ¬a, and a’s main operator is the conditional. [A p.267] 3. g is of the form ¬a, and a’s main operator is the biconditional. [A p.268] 4. g is of the form ¬a, and a’s main operator is the existential quantifier. [A p.268] 5. g’s main operator is the biconditional. [A p.268] [Contents] 71 Chapter 15 Other Methods of Proof Exercises 15.1.5 1. Show the following in A1 by producing formal proofs. (i) ¬ P ! Q, ¬ P ` Q [A p.269] (ii) P ` ¬ Q ! P [A p.269] (iv) ` P ! P [A p.269] (vi) P, ¬ P ` Q [A p.270] (iii) ¬ Q ` (¬ P ! Q) ! P [A p.269] (v) ¬( P ! ¬ Q) ` Q [A p.270] (vii) P ^ Q ` ( P ! ¬ Q) ! ¬( P ! ¬ Q) [A p.270] 2. Show the following in A1 by producing formal or informal proofs. (i) ` ¬( P ! ¬ Q) ! Q [A p.270] (ii) ` P ! ( P _ Q) [A p.270] (iv) ` ( P ! Q) ! (¬ Q ! ¬ P) [A p.271] (vi) P ! Q, ¬ Q ! P ` Q [A p.272] (iii) ` (( P ! Q) ! ( P ! R)) ! ( P ! ( Q ! R)) (v) P ! Q, P ! ¬ Q ` ¬ P (vii) ` ( P ! ( Q ! R)) ! ( Q ! ( P ! R)) [A p.271] [A p.271] [A p.272] 3. Show the following in A2 by producing formal or informal proofs. (i) ` P ! ¬¬ P [A p.272] 72 (ii) P ! ¬ P ` ¬ P [A p.273] (iv) ` ¬ Q ! ( Q ! P) [A p.273] (vi) ¬ Q ` ( P _ Q) ! P [A p.274] (viii) ¬( P _ Q) ` ¬ P ^ ¬ Q [A p.275] (iii) P ! Q ` ¬ Q ! ¬ P [A p.273] (v) P ^ Q ` P ! Q [A p.273] (vii) ¬ P ^ ¬ Q ` ¬( P _ Q) [A p.274] (ix) ` ¬( P ^ ¬ P) [A p.275] (xi) ` P $ P [A p.276] (x) ` ( P ^ ¬ P) ! Q [A p.276] (xii) ` P ! (¬ P ! Q) [A p.277] 4. Show the following in A18= by producing formal or informal proofs. (i) 8 x ( Fx ! Gx ), Fa ` Ga [A p.277] (ii) 8 xFx ` 8 x ( Gx _ Fx ) [A p.277] (iv) 9 xFx ! ¬ Ga ` Ga ! 8 x ¬ Fx [A p.278] (vi) Fa, a = b ` Fb [A p.280] (iii) 8 x 8y( Rxy ! Ryx ), Rab ` Rba (v) ` Fa ! 9 xFx (vii) 8 x 8yx = y ` a = b [A p.277] [A p.279] [A p.280] (viii) a = b, a = c ` c = b [A p.280] (x) Fa, ¬ Fb ` ¬ a = b [A p.281] (ix) ` a = b ! b = a [A p.281] (xi) ¬b = a, 8 x (¬ Fx ! x = a) ` Fb [A p.283] (xii) ` 8 xFx ! 8yFy [A p.283] 5. Explain why the original unrestricted deduction theorem does not hold in A18= and why the restricted version stated at the end of §15.1.1.1 does hold. [A p.283] [Contents] 73 Exercises 15.2.3 1. Show the following in N1 . (i) ` (¬ P ! P) ! P [A p.284] (ii) A ! C, B ! C, A _ B ` C [A p.284] (iv) ¬( A _ B) ` ¬ A ^ ¬ B [A p.285] (vi) A ! B, B ! C ` A ! C [A p.285] (iii) ` ¬¬ P ! P [A p.284] (v) A, ¬ A ` B [A p.285] (vii) P ! Q ` ¬ Q ! ¬ P [A p.286] (viii) A _ B, ¬ A ` B [A p.286] (ix) P ! R, Q ! R, P _ Q ` R (x) P ! Q ` ¬( P ^ ¬ Q) [A p.287] [A p.287] 2. Establish each of the following in each of the systems N2 through N5 . (i) ` A _ ¬ A [A p.288] (ii) A ^ ¬ A ` B [A p.289] (iv) ` ¬( A ^ ¬ A) [A p.291] (iii) ` ¬¬ A ! A [A p.289] 3. Show the following in N189= . (i) ` 8 x ( Fx ! Fx ) [A p.291] (ii) 9 x ( Fx ^ Gx ) ` 9 xFx ^ 9 xGx [A p.291] (iv) 8 x ( Fx ! x = a) ` Fb ! a = b [A p.292] (vi) ` 8 xRxx ! 8 x 9yRxy [A p.293] (iii) 8 x ( Fx ! Gx ), ¬9 xGx ` ¬9 xFx (v) 8 x 8yx = y, Raa ` 8 x 8yRxy (vii) ` 9 xFx ! ¬8 x ¬ Fx [A p.292] [A p.293] [A p.294] (viii) ¬9 xFx ` 8 x ¬ Fx [A p.294] (ix) 8 xx = a ` b = c [A p.294] (x) ` 8 x 8y(( Fx ^ ¬ Fy) ! ¬ x = y) 74 [A p.295] 4. (i) Reformulate the rules of system N1 in list style. Re-present your answers to Question 1 above as proofs in the list style. [A p.295] (ii) Reformulate the rules of system N1 in stack style. Re-present your answers to Question 1 above as proofs in the stack style. [A p.299] 5. State natural deduction rules (i.e., introduction and elimination rules) for $. [A p.302] [Contents] Exercises 15.3.3 1. Define the following notions in terms of sequents. (i) The proposition a is: (a) a contradiction (b) satisfiable [A p.303] [A p.303] (ii) Propositions a and b are: (a) jointly satisfiable (b) equivalent [A p.303] [A p.303] 2. Redo some of Exercise 7.3.1.1 and Exercise 7.3.2.1 using the sequent calculus S1 instead of trees. [A p.303] 3. Redo some of Exercise 10.2.2, Exercise 12.3.1 and Exercise 13.4.3 using the sequent calculus S189= instead of trees. [A p.303] 4. State sequent rules (i.e., left and right introduction rules) for $. [A p.303] 5. State a (new) tree rule that is the analogue of Cut. [A p.303] [Contents] 75 Chapter 16 Set Theory There are no exercises for chapter 16. 76 [Contents] Answers 77 Chapter 1 Propositions and Arguments Answers 1.2.1 1. Proposition [Q p.2] 2. Non-proposition (Exhortation) [Q p.2] 3. Non-proposition (Exclamation) [Q p.2] 4. Non-proposition (Wish) [Q p.2] 5. Proposition (Not a wish: the speaker is making a statement about what she wishes.) [Q p.2] 6. Proposition [Q p.2] 7. Proposition [Q p.2] 8. Proposition [Q p.2] 9. Non-proposition (Wish) [Q p.2] 10. Non-proposition (Command) [Q p.2] [Contents] Answers 1.3.1 1. If the stock market crashes, thousands of experienced investors will lose a lot of money. The stock market won’t crash. 78 [Q p.3] 2. Diamond is harder than topaz. Topaz is harder than quartz. Quartz is harder than calcite. Calcite is harder than talc. Diamond is harder than talc. 3. [Q p.3] Any friend of yours is a friend of mine. You’re friends with everyone on the volleyball team. If Sally’s on the volleyball team, she’s a friend of mine. [Q p.3] 4. When a politician engages in shady business dealings, it ends up on page one of the newspapers No South Australian senator has ever appeared on page one of a newspaper. No South Australian senator engages in shady business dealings. [Q p.3] [Contents] Answers 1.4.1 1. Valid. [Q p.3] 2. Invalid. [Q p.3] 3. Valid. [Q p.3] 4. Valid. [Q p.4] [Contents] Answers 1.5.1 1. Arguments 1 and 3. [Q p.4] 2. Argument 2. [Q p.4] 3. Argument 4. [Q p.4] [Contents] 79 Answers 1.6.1.1 1. (i) Bob is a good student [Q p.4] (ii) I have decided not to go to the party. [Q p.4] (iii) Mars is the closest planet to the sun. [Q p.4] (iv) Alice is late. [Q p.4] (v) I like scrambled eggs. [Q p.4] (vi) Scrambled eggs are good for you. [Q p.4] 2. True. [Q p.4] 3. False. [Q p.4] [Contents] Answers 1.6.2.1 1. The sun is shining. I am happy. [Q p.5] 2. Maisie is my friend. Rosie is my friend. [Q p.5] 3. Sailing is fun. Snowboarding is fun. [Q p.5] 4. We watched the movie. We ate popcorn. [Q p.5] 5. Sue does not want the red bicycle. Sue does not like the blue bicycle. [Q p.5] 6. The road to the campsite is long. The road to the campsite is uneven. [Q p.5] [Contents] Answers 1.6.4.1 1. (a) That’s pistachio ice cream. (b) That doesn’t taste the way it should. [Q p.5] 2. (a) That tastes the way it should. (b) That isn’t pistachio ice cream. 80 [Q p.5] 3. (a) That is supposed to taste that way. (b) That isn’t pistachio ice cream. [Q p.5] 4. (a) You pressed the red button. (b) Your cup contains coffee. [Q p.5] 5. (a) You pressed the green button. (b) Your cup does not contain coffee. [Q p.5] 6. (a) Your cup contains hot chocolate. (b) You pressed the green button. [Q p.5] [Contents] Answers 1.6.6 1. This is a conditional with antecedent ‘It will be sunny and windy tomorrow’ and consequent ‘We shall go sailing or kite flying tomorrow’. The antecedent is a conjunction with conjuncts ‘It will be sunny tomorrow’ and ‘It will be windy tomorrow’. The consequent is a disjunction with disjuncts ‘We shall go sailing tomorrow’ and ‘We shall go kite flying tomorrow’. [Q p.6] 2. This is a conditional with antecedent ‘It will rain or snow tomorrow’ and consequent ‘We shall not go sailing or kite flying tomorrow’. The antecedent is a disjunction with disjuncts ‘It will rain tomorrow’ and ‘It will snow tomorrow’. The consequent is a negation with negand ‘We shall go sailing or kite flying tomorrow’. The negand, as mentioned in answer to the previous question, is a disjunction with disjuncts ‘We shall go sailing tomorrow’ and ‘We shall go kite flying tomorrow’. [Q p.6] 3. This is a disjunction with disjuncts ‘He’ll stay here and we’ll come back and collect him later’ and ‘He’ll come with us and we’ll all come back together’. The first of these disjuncts is a conjunction with conjuncts ‘He’ll stay here’ and ‘We’ll come back and collect him later’; the second of the disjuncts is also a conjunction, with conjuncts ‘He’ll come with us’ and ‘We’ll all come back together’. [Q p.6] 4. This is a conjunction with conjuncts ‘Jane is a talented painter and a wonderful sculptor’ and ‘If she remains interested in art, her work 81 will one day be of the highest quality.’ The first of these conjuncts is itself a conjunction, with conjuncts ‘Jane is a talented painter’ and ‘Jane is a wonderful sculptor’; the second conjunct is a conditional, with antecedent ‘Jane remains interested in art’ and consequent ‘Jane’s work will one day be of the highest quality’. [Q p.6] 5. This is a negation with negand ‘The unemployment rate will both increase and decrease in the next quarter’. The negand is a conjunction with conjuncts ‘The unemployment rate will increase in the next quarter’ and ‘The unemployment rate will decrease in the next quarter’. [Q p.6] 6. This is a conditional with antecedent ‘You don’t stop swimming during the daytime’ and consequent ‘Your sunburn will get worse and become painful’. The antecedent is a negation with negand ‘You stop swimming during the daytime’; the consequent is a conjunction with conjuncts ‘Your sunburn will get worse’ and ‘Your sunburn will become painful’. [Q p.6] 7. This is a disjunction with disjuncts ‘Steven won’t get the job’ and ‘I’ll leave and all my clients will leave’. The first of these disjuncts is a negation with negand ‘Steven will get the job’; the second disjunct is a conjunction with conjuncts ‘I’ll leave’ and ‘All my clients will leave’. [Q p.6] 8. This is a biconditional with components ‘The Tigers will not lose’ and ‘Both Thompson and Thomson will get injured’. The first is a negation with negand ‘The Tigers will lose’; the second is a conjunction with conjuncts ‘Thompson will get injured’ and ‘Thomson will get injured’. [Q p.6] 9. This is a conjunction with conjuncts ‘Fido will wag his tail if you give him dinner at 6 this evening’ and ‘Fido will bark if you do not give him dinner at 6 this evening’. The first of these conjuncts is a conditional with antecedent ‘You will give Fido dinner at 6 this evening’ and consequent ‘Fido will wag his tail [at 6 this evening]’; the second conjunct is a conditional with antecedent ‘You do not give Fido dinner at 6 this evening’ and consequent ‘Fido will bark [at 6 this evening]’. Finally, the antecedent of this last conditional is a negation with negand ‘You give Fido dinner at 6 this evening’. [Q p.6] 82 10. This is a disjunction with disjuncts ‘It will rain or snow today’ and ‘It will not rain or snow today’. The first of these disjuncts is itself a disjunction, with disjuncts ‘It will rain today’ and ‘It will snow today’. The second of these disjuncts is a negation with negand ‘It will rain or snow today’. The latter, as already mentioned, is a disjunction, with disjuncts ‘It will rain today’ and ‘It will snow today’. [Q p.6] [Contents] 83 Chapter 2 The Language of Propositional Logic Answers 2.3.3 1. Aristotle was not a philosopher. [Q p.7] 2. Aristotle was a philosopher and paper burns. [Q p.7] 3. Aristotle was a philosopher and paper doesn’t burn. [Q p.7] 4. Fire is not hot and paper does not burn. [Q p.7] 5. It’s not true both that fire is hot and that paper burns. [Q p.7] [Contents] Answers 2.3.5 1. Either Aristotle was a philosopher and paper burns, or fire is hot. [Q p.8] 2. Either Aristotle wasn’t a philosopher, or paper doesn’t burn. [Q p.8] 3. Aristotle was a philosopher or paper burns—but not both. [Q p.8] 4. It’s not the case either that Aristotle was a philosopher or that fire is hot. [Q p.8] 84 5. Aristotle was a philosopher, and either paper burns or fire is hot. [Q p.8] [Contents] Answers 2.3.8 1. (i) If snow is white, then the sky is blue. [Q p.8] (ii) Snow is white if and only if both snow is white and roses are not red. [Q p.8] (iii) It’s not the case that if roses are red then snow is not white. [Q p.8] (iv) If roses are red or snow is white, then roses are red and snow is not white. [Q p.8] (v) Either snow is white and snow is white, or roses are red and the sky is not blue. [Q p.8] (vi) Either grass is green, or if snow is white then roses are red. [Q p.8] (vii) Bananas are yellow if and only if they’re yellow; and they’re not if and only if they’re not. [Q p.8] (viii) If, if the sky is blue then snow is white, then if snow isn’t white then the sky isn’t blue. [Q p.8] (ix) If roses are red, snow is white and the sky is blue, then either bananas are yellow or grass is green. [Q p.8] (x) It’s not the case both that roses aren’t red and that either snow isn’t white or grass is green. [Q p.9] 85 2. Glossary: B: The sky is blue E: Snow is red J: Jim is tall M: Maisy is tall N: Nora is tall R: Roses are red W: Snow is white (i) (W ! B) [Q p.9] (ii) ( B $ (W ^ ¬ R)) [Q p.9] (iv) (( E ^ R) ! ( R _ ¬ E)) [Q p.9] (vi) ( J ! ( N _ M )) [Q p.9] (iii) ¬( R ! ¬W ) [Q p.9] (v) (( J $ M) ^ ( M ! ¬ N )) [Q p.9] (vii) ( J ! ( M _ ¬ N )) [Q p.9] (viii) ((W ^ M ) _ (W ^ ¬ M)) [Q p.9] (x) (( M ^ B) ! ( J ^ ¬ B)) [Q p.9] (ix) (( J ^ ¬ J ) ! ( B ^ ¬ B)) [Q p.9] 86 3. Glossary: G: We are skiing K: We are kite flying L: We are sailing S: It is snowing U: It is sunny W: It is windy (i) (S ! ¬K ) [Q p.9] (ii) ((U ^ W ) ! ( L _ K )) [Q p.9] (iv) (( L _ K ) _ G ) [Q p.9] (vi) ( G ! (W _ S)) [Q p.9] (viii) (U ! (W ! K )) [Q p.9] (iii) ((K ! W ) ^ ( L ! W )) [Q p.9] (v) (W $ L) [Q p.9] (vii) ( G ! (W ^ S)) [Q p.9] (ix) ( L ! ((U ^ W ) ^ ¬S)) [Q p.9] (x) (((U ^ W ) ! L) ^ ((S ^ ¬W ) ! G )) [Q p.10] [Contents] Answers 2.5.1 1. (i) No [Q p.10] (ii) No [Q p.10] (iii) Yes [Q p.10] (iv) No [Q p.10] (v) No [Q p.10] (vi) No [Q p.10] (vii) No [Q p.10] (viii) No [Q p.10] (ix) No [Q p.10] (x) Yes [Q p.10] 87 2. (i) (a) 1 is an odd number. (b) If x is an odd number then so is x + 2. (c) Nothing else is an odd number. Note: We are assuming here that ‘number’ means ‘positive integer’. If it is taken to mean ‘integer’ (i.e. positive, negative or zero) then the answer is: (a) 1 is an odd number. (b) If x is an odd number then so are x + 2 and x (c) Nothing else is an odd number. 2. [Q p.10] (ii) (a) 5 is divisible by five. (b) If x is divisible by five then so is x + 5. (c) Nothing else is divisible by five. Note: We are assuming here that ‘number’ means ‘positive integer’. If it is taken to mean ‘integer’ (i.e. positive, negative or zero) then the answer is: (a) 5 is divisible by five. (b) If x is divisible by five then so are x + 5 and x (c) Nothing else is divisible by five. (iii) (a) a is such a word; b is such a word. (b) If x is such a word then so are xa and xb. (c) Nothing else is such a word. 5. [Q p.10] [Q p.10] (iv) (a) Bob’s mother is in the set; Bob’s father is in the set. (b) If x is in the set then so are x’s mother and x’s father. (c) Nothing else is in the set. [Q p.10] (v) (a) hah hah hah is a cackle. (b) If x is a cackle then so is x hah. (c) Nothing else is a cackle. [Q p.10] [Contents] 88 Answers 2.5.3.1 1. 1. 2. 3. 4. 5. 6. P Q R ¬P ( Q ^ R) (¬ P _ ( Q ^ R)) / (2i) / (2i) / (2i) 1 / (2ii¬) 2, 3 / (2ii^) 4, 5 / (2ii_) P Q R ( Q _ R) ( P ^ ( Q _ R)) ¬( P ^ ( Q _ R)) / (2i) / (2i) / (2i) 2, 3 / (2ii_) 1, 4 / (2ii^) 5 / (2ii¬) Main connective is _. 2. 1. 2. 3. 4. 5. 6. Main connective is ¬. 3. 1. 2. 3. 4. 5. 6. 7. 8. P Q R ¬P ¬Q ¬R (¬ P ^ ¬ Q) ((¬ P ^ ¬ Q) _ ¬ R) Main connective is _. 4. 1. 2. 3. 4. 5. 6. 7. / (2i) / (2i) / (2i) 1 / (2ii¬) 2 / (2ii¬) 3 / (2ii¬) 4, 5 / (2ii^) 6, 7 / (2ii_) P Q R S ( P ! Q) ( R ! S) (( P ! Q) _ ( R ! S)) Main connective is _. / (2i) / (2i) / (2i) / (2i) 1, 2 / (2ii!) 3, 4 / (2ii!) 5, 6 / (2ii_) 89 [Q p.11] [Q p.11] [Q p.11] [Q p.11] 5. 1. 2. 3. 4. 5. 6. 7. P Q R S ( P $ Q) (( P $ Q) $ R) ((( P $ Q) $ R) $ S) Main connective is $. 6. 1. 2. 3. 4. 5. 6. / (2i) / (2i) / (2i) / (2i) 1, 2 / (2ii$) 3, 5 / (2ii$) 4, 6 / (2ii$) P ¬P ¬¬ P (¬ P ^ ¬¬ P) ( P ^ ¬ P) ((¬ P ^ ¬¬ P) ! ( P ^ ¬ P)) Main connective is !. / (2i) 1 / (2ii¬) 2 / (2ii¬) 2, 3 / (2ii^) 1, 2 / (2ii^) 4, 5 / (2ii!) [Q p.11] [Q p.11] [Contents] Answers 2.5.4.1 1. ordering 2 3 4 5 6 disambiguation 2 3 4 2 5 [Q p.11] [Contents] Answers 2.5.5.1 1. (i) (¬ P _ ( Q ^ R)) [Q p.11] (iii) (¬( P _ Q) ^ R) [Q p.11] (ii) ¬(( P _ Q) ^ R) [Q p.11] (iv) ((¬ P ^ ¬ Q) _ ¬ R) [Q p.11] (v) ([( P $ Q) $ R] $ S) [Q p.11] 90 2. (i) ¬ ^ P _ QR [Q p.11] (ii) !! P _ QRS [Q p.12] (iv) ! P ! _ QRS [Q p.12] (iii) _ ! PQ ! RS [Q p.12] (v) ! ^¬ P¬¬ P ^ P¬ P [Q p.12] [Contents] 91 Chapter 3 Semantics of Propositional Logic Answers 3.2.1 1. 2. 3. ^ ( Q _ R)) T F T F [Q p.13] ¬ ( P _ ( Q ! R)) phase 0: T T F phase 1: F phase 2: T phase 3: F [Q p.13] phase 0: phase 1: phase 2: (¬ P T F (¬ ¬ P F T F ^ ( Q ! ( R _ P))) T T F T T F [Q p.13] 4. (¬ ¬ P phase 0: T phase 1: F phase 2: T phase 3: ^ ( Q ! ( R _ P))) F F T T T T [Q p.13] 5. (( P _ Q) ! ( P _ P)) phase 0: F T F F phase 1: T F phase 2: F phase 0: phase 1: phase 2: phase 3: 92 [Q p.13] 6. (( P _ Q) ! ( P _ P)) phase 0: T F T T phase 1: T T phase 2: T [Q p.13] 7. ( P ! ( Q ! ( R ! S))) phase 0: T T T F phase 1: F phase 2: F phase 3: F [Q p.13] ( P ! ( Q ! ( R ! S))) F T F T T T T [Q p.13] 8. phase 0: phase 1: phase 2: phase 3: 9. ¬ (((¬ P $ P) $ Q) ! R) phase 0: F F F F phase 1: T phase 2: F phase 3: T phase 4: F phase 5: T [Q p.13] 10. ¬ (((¬ P $ P) $ Q) ! R) phase 0: T T T T phase 1: F phase 2: F phase 3: F phase 4: T phase 5: F [Q p.14] [Contents] Answers 3.3.1 1. P T T F F Q T F T F (( P ^ Q) _ T / T F / T F / F F / F [Q p.14] P) 93 2. P T F ( P ^ (P _ T T / F / F P)) [Q p.14] 3. P T T F F Q T F T F ¬(¬P ^ ¬Q) T F / F / F / T F / F / T / T T / F / F / F T / T / T / [Q p.14] 4. Q T F ( Q ! (Q ^ ¬Q)) F / / F F T F / T / [Q p.14] 5. P T T T T F F F F Q T T F F T T F F R T F T F T F T F 6. P T T F F Q T F T F ((P 7. P T T F F Q T F T F ¬(( P ^ Q) $ Q) F T / T / F F / T / T F / / F F F / T / 8. P T F ((( P ! ¬ P) ! ¬ P) ! ¬ P) F / F / T / / F F / F T / T / T / T / T T / (P ! (Q ! T T / F / F T T / T T / T T / T F / T T / T T / R)) _ Q) $ ( P ^ Q)) T / T T / T / F F / T / F F / F / T F / 94 [Q p.14] [Q p.14] [Q p.14] [Q p.14] 9. P T T T T F F F F Q T T F F T T F F R T F T F T F T F ¬( P F T T T T T T T ^ T / F / F / F / F / F / F / F / 10. R T T T T F F F F S T T F F T T F F T T F T F T F T F ((¬ R F / F / F / F / T / T / T / T / _ S) T / T / / F / F T / T / T / T / [Q p.14] ( Q ^ R)) T / F / F / F / T / F / F / F / ^ (S T T F F T T F T _ T / T / F / T / T / T / / F T / ¬ T )) F / T / / F T / F / T / F / T / [Q p.14] [Contents] Answers 3.4.1 1. P T T F F Q T F T F ( P ! Q) T F T T 2. P T T F F Q T F T F ¬ ( P $ Q) F T / T F / T F / F T / 3. P T T F F Q T F T F ¬( P ^ ¬ Q) T F / F / F T / T / T F / F / T F / T / [Q p.14] ( Q ! P) T T F T (( P _ Q) ^ ¬ ( P ^ Q)) T / F / F T / T / T T / F / T / T T / F / F / F T / / F [Q p.14] [Q p.14] ¬Q F T F T 95 4. P T T T T F F F F Q T T F F T T F F 5. P T T T T T T T T F F F F F F F F Q T T T T F F F F T T T T F F F F R T F T F T F T F R T T F F T T F F T T F F T T F F (( P ! Q) T / T / F / F / T / T / T / T / S T F T F T F T F T F T F T F T F ((P ^ Q) T / T / T / T / F / F / F / F / F / F / F / F / F / F / F / F / ^ F F F T F F F F F F F F F F F F ( P _ ( Q _ R)) [Q p.14] T T / T T / T T / T F / T T / T T / T T / F F / ^ R)) T F F F T F T F (¬ R F / F / T / T / / F / F T / T / / F / F T / T / / F / F T / T / ^ F / F / F / T / / F / F / F T / / F / F / F T / / F / F / F T / ¬S)) F / T / / F T / F / T / F / T / F / T / F / T / F / T / F / T / (( P _ T / T / T / T / T / T / T / T / T / T / T / T / F / F / T / T / (R ! T / T / T / T / / F / F T / T / T / T / T / T / / F / F T / T / Q)) ^ S) T F T F T F T F T F T F F F T F [Q p.14] 6. P T T F F Q T F T F (P ^ F F F F 7. P T T T T F F F F Q T T F F T T F F R T F T F T F T F ¬ P) F / F / T / T / (P _ T T T T T F F T (Q ^ F F F F [Q p.15] ¬ Q) F / T / F / T / ( Q $ R)) T / F / F / T / T / F / F / T / 96 (( Q ! P) T / T / T / T / F / F / T / T / ^ T T F F F F F F Q) [Q p.15] 8. P T T T T F F F F Q T T F F T T F F R T F T F T F T F ¬ (( P ^ Q) F T / T T / T / F T / F T / F T / F T / F T / F ^ T / / F / F / F / F / F / F / F R) (( P ! Q) $ T / T T / F F / F F / T T / T T / T T / T T / T ( P ! R)) T / F / T / F / T / T / T / T / [Q p.15] 9. P T T F F Q T F T F ( P _ Q) T T T F 10. P T T T T T T T T F F F F F F F F Q T T T T F F F F T T T T F F F F R T T F F T T F F T T F F T T F F S T F T F T F T F T F T F T F T F ¬P F F T T [Q p.15] ( Q _ Q) T F T F (P ! (Q T F T T T T T T T T T T T T T T ! T / F / T / T / T / T / T / T / T / F / T / T / T / T / T / T / ( R ! S))) T / F / T / T / T / F / T / T / T / F / T / T / T / F / T / T / ¬S F T F T F T F T F T F T F T F T [Q p.15] [Contents] 97 Answers 3.5.1 1. No. None of our connectives has a truth table which matches the outputs of this truth function in all cases. [Q p.15] 2. input output of function f 42 output of function f 52 (T,T) (T,F) (F,T) (F,F) T F F F T F T T [Q p.15] 3. T [Q p.15] 4. (iv) You do not need to know any truth values. Whether ( A ! B) is T or F, ?( A ! B) is T, because ? sends both possible inputs (T and F) to T. [Q p.15] 5. !. The outputs of g are as follows: input (T,T) (T,F) (F,T) (F,F) output of g T F T T These outputs match the truth table for ! in every case. NB To get the output of g where the input is ( x, y), we first take x as input to f 21 , and then take the output of this, and y—in that order—as the inputs to f 32 . The output of that is the output of g for input ( x, y): x y T T T F F T F F f 21 ( x ) F F T T f 32 ( f 21 ( x ), y) T F T T The rightmost column of this table gives us the outputs of g for all possible values of x and y, since g( x, y) = f 32 ( f 21 ( x ), y). [Q p.15] [Contents] 98 Chapter 4 Uses of Truth Tables Answers 4.1.2 1. Valid A T T T T F F F F [Q p.16] B T T F F T T F F C T F T F T F T F A _ T T T T T T F F B A ! C T F T F T T T T (B ! C) ! C T / T F / T T / T T / F T / T F / T T / T T / F 2. Invalid. Counterexample: A false, B false, C false (row 8). [Q p.16] A T T T T F F F ⇤F B T T F F T T F F C T F T F T F T F ¬A F F F F T T T T ¬ (( A ! F / T / T / T / T / F / T / F / F / T / T / T / F / T / F / T / 99 B) ^ ( B ! C )) _ C T / T / T F / F / T F / T / T F / T / T T / T / T F / F / T T / T / T T / T / F 3. Invalid. Counterexample: A true, B true, C false (row 2). A T ⇤T T T F F F F B T T F F T T F F C T F T F T F T F (A ^ F / F / T / T / F / F / F / F / ¬ B) ! C / F T / F T T / T T / F / F T / F T T / T T / T ¬C F T F T F T F T ¬A F F F F T T T T 4. Valid. A T T T T F F F F B T T F F T T F F [Q p.16] C T F T F T F T F (A ^ T / T / F / F / F / F / F / F / B) $ C T F F T F T F T C ! T T F T T T F T B 5. Valid. A T T T T F F F F B T T F F T T F F [Q p.16] [Q p.16] C T F T F T F T F (¬ A ^ ¬ B) $ ¬C F / F / F / T F / F / F / F / F T / F / F / T / T F / F / F / T / F T / T / F / F / T F / T / F / F / F T / T / T / T / F F / T / T / T / T T / 100 ¬( A F F F F F F T T _ T / T / T / T / T / T / F / F / B) C! F T F T F T F T ¬C F / T / F / T / F / T / F / T / 6. Valid A T T T T F F F F [Q p.16] B T T F F T T F F C T F T F T F T F A_B T T T T T T F F ¬A F / F / F / F / T / T / T / T / _ C T F T F T T T T B!C T F T T T F T T 7. Invalid. Counterexample: A false, B false, C false (row 8). [Q p.17] A T T T T F F F ⇤F B T T F F T T F F C T F T F T F T F ¬( A _ B) $ ¬C F / T / T / F F / T / F T / F / T / T F / F / T / F T / F / T / T F / F / T / F T / T / F / F F / T / F / T T / ¬ A ^ ¬B F / F F / F / F F / F / F T / F / F T / T / F F / T / F F / T / T T / T / T T / C ^ F F F F F F F F ¬C F / T / F / T / F / T / F / T / 8. Invalid. Counterexample: A true, B false, C true (row 3). A T T ⇤T T F F F F B T T F F T T F F C T F T F T F T F ¬( A ^ B) F / T / F / T / T / F / T / F / T / F / T / F / T / F / T / F / ! T T T T T F T F (C _ A) T / T / T / T / T / F / T / F / 101 ¬A F / F / F / F / T / T / T / T / _ F F T T T T T T ¬B F / F / T / T / F / F / T / T / ¬ (C _ F T / F T / F T / F T / F T / F T / F T / F T / [Q p.17] ¬C ) F / T / / F T / / F T / / F T / 9. Valid. A T T T T F F F F B T T F F T T F F [Q p.17] C T F T F T F T F A ! T F F F T T T T (B ^ C) T / / F / F / F T / / F / F / F B $ F T T F F T T F ¬C F / T / F / T / F / T / F / T / 10. Valid. A T T T T F F F F B T T F F T T F F ¬A F F F F T T T T [Q p.17] C T F T F T F T F A!B T T F F T T T T B!C T F T T T F T T ¬C F T F T F T F T ¬A F F F F T T T T [Contents] Answers 4.2.1 1. Neither P T T F F Q T F T F [Q p.17] (( P _ Q) ! T / T T / T T / F F / T P) 102 2. Neither P T T T T F F F F Q T T F F T T F F [Q p.17] R T F T F T F T F (¬ P ^ ( Q F / F F / F F / F F / F T / T T / T T / T T / F _ T / T / T / F / T / T / T / F / R)) 3. Contradiction P T T F F Q T F T F ((¬ P F / F / T / T / [Q p.17] _ Q) $ (P ^ T / F F / F / F T / T / F F / T / F F / ¬ Q)) / F T / F / T / 4. Tautology P T T T T F F F F Q T T F F T T F F R T F T F T F T F [Q p.17] (P ! (Q ! (R ! T T / T / T T / T / T T / T / T T / T / T F / F / T T / T / T T / F / T T / T / P))) 5. Tautology P T T F F Q T F T F [Q p.17] ( P ! (( P ! Q) ! Q)) T T / T / T F / T / T T / T / T T / F / 103 6. Neither P T T F F Q T F T F [Q p.17] ( P ! (( Q ! P) T T / F T / T F / T T / ! Q)) T / F / T / / F 7. Tautology P T T F F Q T F T F (( P ! Q) T / F / T / T / [Q p.17] _ T T T T ¬( Q ^ ¬ Q)) T / F / F / T / / T F / T / F / F / T / F / T / 8. Tautology P T T F F Q T F T F (( P ! Q) T / F / T / T / [Q p.17] _ T T T T ¬( Q T / T / F / T / ^ F / / F T / F / ¬ P)) F / F / T / T / 9. Neither P T T F F Q T F T F [Q p.17] (( P ^ Q) T / F / F / F / $ T T T F ( Q $ P)) T / F / F / T / 10. Contradiction P T T F F Q T F T F ¬(( P ^ Q) F T / F / F F / F F / F [Q p.17] ! T / T / T / T / ( Q $ P)) T / /... Purchase answer to see full attachment

  
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