CS计算机代考程序代写 flex ant LOGIC

LOGIC

The Drill

Nicholas J.J. Smith and John Cusbert

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 ap-
pear in Logic: The Laws of Truth by Nicholas J.J. Smith (Princeton Univer-
sity 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:

The latest version of this work can be found at:

http://www.personal.usyd.edu.au/~njjsmith/lawsoftruth/

Any significant revisions (e.g. corrections or additions to the exercises or
answers) will be documented on the copyright page.

iii

mailto:
http://www.personal.usyd.edu.au/~njjsmith/lawsoftruth/

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

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

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

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

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

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

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

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]

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 ev-
eryone 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. [A p.79]

2. All dogs are mammals.
All dogs are animals.

All mammals are animals. [A p.79]

3. All dogs are mammals.
No fish are mammals.

No fish are dogs. [A p.79]

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]

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 propo-
sitions?

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]

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 re-
mains interested in art, her work will one day be of the highest qual-
ity. [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]

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]

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]

(iii) ¬(R→ ¬W) [A p.85]
(iv) ((R ∨W)→ (R ∧ ¬W)) [A p.85]
(v) ((W ∧W) ∨ (R ∧ ¬B)) [A p.85]

(vi) (G ∨ (W → R)) [A p.85]
(vii) ((Y ↔ Y) ∧ (¬Y ↔ ¬Y)) [A p.85]

(viii) ((B→W)→ (¬W → ¬B)) [A p.85]
(ix) (((R ∧W) ∧ B)→ (Y ∨ G)) [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]

(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]

(iii) (A→ (A→ A)) [A p.87]
(iv) A→ ((A→ A)) [A p.87]
(v) ((A ∧ B)∧)A [A p.87]

(vi) (A ∨ (A ∨ (A ∨ (A ∨ (A ∨ (A ∨ (A ∨ 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]

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]
(ii) ¬ ∧ ∨ PQR [A p.90]

(iii) ∧¬ ∨ 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]

(ii) ([P→ (Q ∨ R)]→ S) [A p.91]
(iii) [(P→ Q) ∨ (R→ S)] [A p.91]
(iv) (P→ [(Q ∨ R)→ S]) [A p.91]
(v) [(¬P ∧ ¬¬P)→ (P ∧ ¬P)] [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]

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 proposi-
tions.

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]

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 24 and f
2
5 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 23 ( f
1
2 (x), y)

[A p.98]

[Contents]

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 [A p.99]

3. (A ∧ ¬B)→ C
¬C
B
∴ ¬A [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]

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]

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]

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]

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 replace-
ments 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]

2. State whether the given propositions are instances of the given form.
If so, show what replacements of variables by propositions are re-
quired to obtain the proposition from the form.

(i) Form: ¬(α→ β)
Propositions:

(a) ¬(P→ Q) [A p.113]
(b) ¬(R→ Q) [A p.113]
(c) ¬(R→ (R→ Q)) [A p.113]

(ii) Form: ¬(α→ (α→ β))
Propositions:

(a) ¬(P→ (P→ Q)) [A p.113]
(b) ¬(P→ (P→ P)) [A p.113]
(c) ¬(P→ (Q→ P)) [A p.113]

(iii) Form: (α ∨ β)→ (α ∧ β)
Propositions:

(a) (¬P ∨Q)→ (¬P ∧Q) [A p.113]
(b) (P ∨ ¬P)→ (P ∧ ¬P) [A p.113]
(c) ¬(R ∨ S)→ ¬(R ∧ S) [A p.113]

(iv) Form: α ∨ (¬β ∨ α)
Propositions:

(a) (P ∨Q) ∨ (Q ∨ (P ∨Q)) [A p.113]
(b) Q ∨ (¬Q ∨ (Q ∧Q)) [A p.113]
(c) ¬P ∨ (¬¬P ∨ ¬P) [A p.113]

[Contents]

Exercises 5.3.1

For each of the following arguments, give four correct answers to the ques-
tion “what is the form of this argument?” For each form, show what re-
placements of variables by propositions are required to obtain the argu-
ment from the form.

1. ¬(R→ (R→ Q))
∴ R ∨ (R→ Q) [A p.114]

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:

α
α→ β
∴ β

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]

Exercises 5.5.1

1. (i) Show by producing a truth table for the following argument
form that it is invalid:

α
∴ β [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 argu-
ment form is an invalid argument, there are some invalid argument
forms whose instances are always invalid arguments. Give an exam-
ple of such an argument form. [A p.117]

[Contents]

Chapter 6

Connectives: Translation and
Adequacy

Exercises 6.5.1

Translate the following arguments into PL and then assess them for valid-
ity (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 light-
ning. [A p.121]

4. The car started only if you turned the key and pressed the accelera-
tor. 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 ac-
celerator 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]

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. Ei-
ther 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]

(iii) {15} (The connective 15 is often symbolized by ↓;
another common symbol for this connective is NOR.) [A p.130]

(iv) {→,∧} [A p.130]
(v) {¬, 12} [A p.131]

(vi) {∨, 4} [A p.131]

2. Give the truth table for each of the following propositions.

(i) B 14 A [A p.131]

(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:

α β γ ](α, β, γ) \(α, β, γ)
T T T T F
T T F F F
T F T T T
T F F T T
F T T T T
F T F F T
F F T T F
F F F T 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]

(iii) ¬A ∨ ¬B [A p.132]
(iv) ¬(¬A ∨ B) [A p.132]
(v) A↔ B [A p.132]

(vi) (A→ B) ∨ (B→ A) [A p.132]

5. (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]

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]

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) [A p.137]

2. (A ∨ B)
∴ B [A p.137]

3. (A ∨ B)
(A→ C)
(B→ D)
∴ (C ∨ D) [A p.138]

4. ((A ∨ ¬B)→ C)
(B→ ¬D)
D
∴ C [A p.138]

5. B
(A→ B)
∴ A [A p.138]

6. A
(A→ B)
∴ B [A p.138]

7. (A ∨ (B ∧ C))
(A→ B)
(B↔ D)
∴ (B ∧ D) [A p.139]

8. ¬(¬A→ B)
¬(C ↔ A)
(A ∨ C)
¬(C → B)
∴ ¬(A→ B) [A p.139]

9. (A↔ B)
(B→ C)
(¬B→ ¬C)
(A ∨ (B ∧ ¬B))
∴ C [A p.140]

10. (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 contradic-
tions. 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]

(iii) (A→ B) ∧ ¬(A ∨ B) [A p.141]
(iv) (A→ ¬(A ∨ B)) ∧ ¬(¬(A ∨ B) ∨ B) [A p.141]
(v) ¬((¬B ∨ C)↔ (B→ C)) [A p.142]

(vi) (A↔ ¬A) ∨ (A→ ¬(B ∨ C)) [A p.142]

2. Using trees, test whether the following sets of propositions are sat-
isfiable. 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, contradic-
tories, 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]

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]

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]

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]

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.

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. ∀x(Fx → Gx) [A p.161]

2. ∀x¬Gx [A p.161]

3. ¬∃x(Fx ∧ Gx) [A p.161]

4. (Fa ∧ ¬∃x¬Fx) [A p.161]

5. ∀x(Fx ∧ ∃y(Gx → Gy)) [A p.162]

6. (∀x(Fx → Gx) ∧ Fa) [A p.162]

7. ((¬Fa ∧ ¬Fb)→ ∀x¬Fx) [A p.162]

8. ∀x∀y((Fx ∧ Fy)→ Gx) [A p.163]

9. ∀x(Fx → ∀yFy) [A p.163]

10. (∀xFx → ∀yFy) [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. ∃xTx ∧ ∃xFx [A p.163]

4. ∃xTx ∧ ∀yFx [A p.163]

5. ∃xTx ∧ Fx [A p.164]

6. ∃x(Tx ∧ Fx) [A p.164]

7. ∀y∃xTy [A p.164]

8. ∃x(∀xTx → ∃yFx) [A p.164]

9. ∃y∀xTx → ∃yFx [A p.164]

10. ∀x(∃xTx ∧ Fx) [A p.164]

11. ∀x∃xTx ∧ Fx [A p.164]

12. ∃xTy [A p.164]

13. ∀xTx → ∃xFx [A p.164]

14. ∃x∀y(Tx ∨ Fy) [A p.164]

15. ∀xFx ∧ Gx [A p.164]

16. ∀x∀yFx → Gy [A p.164]

17. ∀x∀y(Fx → ∀xGy) [A p.164]

18. ∃yGb ∧ Gc [A p.164]

19. ∃yGy ∧ ∀x(Fx → Gy) [A p.164]

20. ∀x((Fx → ∃xGx) ∧ Gx) [A p.164]

[Contents]

Chapter 9

Semantics of Monadic Predicate
Logic

Exercises 9.1.1

For each of the propositions:

(i) Pa (ii) ∃xPx (iii) ∀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]

2That is, the set of positive integers.
3That is, the set of odd numbers.
4That is, the set of even numbers.

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↔ ∃xPx)

(iv) (∃xPx ∨ ¬Pa)

(v) ¬(∀xPx ∧ ¬∃xPx) [Answers p.165]

[Contents]

Exercises 9.3.1

1. If α(x) is (Fx ∧ Ga), what is

(i) α(a/x) [A p.166]

(ii) α(b/x) [A p.166]

2. If α(x) is ∀y(Fx → Gy), what is

(i) α(a/x) [A p.166]

(ii) α(b/x) [A p.166]

3. If α(x) is ∀x(Fx → Gx) ∧ Fx, what is

(i) α(a/x) [A p.166]

(ii) α(b/x) [A p.166]

4. If α(x) is ∀x(Fx ∧ Ga), what is

(i) α(a/x) [A p.166]

(ii) α(b/x) [A p.166]

5. If α(y) is ∃x(Gx → Gy), what is

(i) α(a/y) [A p.166]

(ii) α(b/y) [A p.166]

6. If α(x) is ∃y(∀x(Fx → Fy) ∨ Fx), what is

(i) α(a/x) [A p.166]

(ii) α(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) ∀xEx [A p.167]
(ii) ∀x(Ex ∨Ox) [A p.167]

(iii) ∃xEx [A p.167]
(iv) ∃x(Ex ∧Ox) [A p.167]
(v) ∀x(¬Ex → Ox) [A p.167]

(vi) ∀xEx ∨ ∃x¬Ex [A p.167]

2. State whether the given proposition is true or false in the given mod-
els.

(i) ∀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} [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) ∃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) ∃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)→ ∃x(Mx ∧ Fx) [A p.167]
(ii) ∀x∀y(Mx → My) [A p.167]

(iii) (¬Ma ∨ ¬Jr)→ ∃x∃y(Mx ∧ Fy) [A p.167]
(iv) ∀xMx → ∀xJx [A p.167]
(v) ∃x∃y(Mx ∧ Fy ∧ Sr) [A p.167]

(vi) ∃x(Fx ∧ Sx)→ ∀x(Fx → Sx) [A p.167]

5That is, the set of prime numbers.

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) ∀x(Fx → Gx) [A p.167]
(ii) ∀xFx ∧ ¬Fa [A p.167]

(iii) ∃xFx ∧ ¬Fa [A p.168]
(iv) ∃x(Fx ∧ Gx) [A p.168]
(v) ∀x(Fx → Fx) [A p.168]

(vi) ∃xFx ∧ ∃xGx [A p.168]
(vii) ∀xFx → ∃xFx [A p.168]

(viii) ∃x(Fx ∧ ¬Fx) [A p.168]
(ix) ∃xFx ∧ ∃x¬Fx [A p.168]
(x) ∃x(Fx → Fx) [A p.169]

(xi) ∃xFx → ∃xGx [A p.169]
(xii) ∃xFx → ∀xGx [A p.169]

(xiii) ∀xFx → Fa [A p.169]
(xiv) ∀x(Fx → Fa) [A p.169]
(xv) Fa→ Fb [A p.169]

(xvi) ∀x(Fx ∨ Gx) [A p.170]
(xvii) ∃x(Fx ∨ Gx) [A p.170]

(xviii) ∀x(Fx ∧ ¬Fx) [A p.170]
(xix) ∀x∃y(Fx → Gy) [A p.170]
(xx) ∀x(Fx → ∃yGy) [A p.170]

5. (i) Is ∀x(Fx → Gx) true or false in a model in which the extension
of F is the empty set? [A p.170]

(ii) Is ∃x(Fx ∧ Gx) true in every model in which ∀x(Fx → Gx) is
true? [A p.170]

[Contents]

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. ∃xFx ∧ ∃xGx
∴ ∃x(Fx ∧ Gx) [A p.171]

2. ∃x(Fx ∧ Gx)
∴ ∃xFx ∧ ∃xGx [A p.171]

3. ∀x(Fx ∨ Gx)
¬∀xFx
∴ ∀xGx [A p.171]

4. ∀x(Fx → Gx)
∀x(Gx → Hx)
∴ ∀x(Fx → Hx) [A p.171]

5. ∀x(Fx → Gx)
∀x(Gx → Hx)
∴ ∀x(Hx → Fx) [A p.171]

[Contents]

Chapter 10

Trees for Monadic Predicate Logic

Exercises 10.2.2

1. Using trees, determine whether the following propositions are log-
ical truths. For any proposition that is not a logical truth, read off
from your tree a model in which it is false.

(i) Fa→ ∃xFx [A p.172]
(ii) ∃xFx → ¬∀x¬Fx [A p.172]

(iii) ∀x((Fx ∧ ¬Gx)→ ∃xGx) [A p.173]
(iv) ∀xFx → ∃xFx [A p.173]
(v) (Fa ∧ (Fb ∧ Fc))→ ∀xFx [A p.173]

(vi) ∃xFx ∧ ∃x¬Fx [A p.174]
(vii) ∃x(Fx → ∀yFy) [A p.174]

(viii) ∀x(Fx → Gx)→ (Fa→ Ga) [A p.174]
(ix) ¬∀x(Fx ∧ Gx)↔ ∃x¬(Fx ∧ Gx) [A p.175]
(x) ¬∃x(Fx ∧ Gx)↔ ∀x(¬Fx ∧ ¬Gx) [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) ∃xFx ∧ ∃xGx
∴ ∃x(Fx ∧ Gx) [A p.176]

(ii) ∃x∀y(Fx → Gy)
∴ ∀y∃x(Fx → Gy) [A p.176]

(iii) Fa→ ∀xGx
∴ ∀x(Fa→ Gx) [A p.177]

(iv) Fa→ ∀xGx
∴ ∃x(Fa→ Gx) [A p.177]

(v) ∀x(Fx ∨ Gx)
¬∀xFx
∴ ∀xGx [A p.177]

(vi) ∃x(Fx ∧ Gx)
∴ ∃xFx ∧ ∃xGx [A p.178]

(vii) ∀x(Fx → Gx)
Fa
∴ Ga [A p.178]

(viii) ¬∀x(Fx ∨ Gx)
∴ ∃x(¬Fx ∧ ¬Gx) [A p.178]

(ix) ∀x(Fx → Gx)
∀x(Gx → Hx)
∴ ¬∃x(¬Fx ∧ Hx) [A p.179]

(x) ∀x(Fx ∨ Gx)
∴ ¬∃x(Fx ∧ Gx) [A p.179]

[Contents]

Exercises 10.3.8

Translate the following arguments into MPL, and then test for validity us-
ing 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]

5. Someone here is not smoking. Therefore, not everyone here is smok-
ing. [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 peo-
ple 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 intellectu-
al, or he isn’t tall. [A p.189]

[Contents]

Chapter 11

Models, Propositions, and Ways
the World Could Be

There are no exercises for chapter 11. [Contents]

Chapter 12

General Predicate Logic

Exercises 12.1.3.1

State whether each of the following is a wff of GPL.

1. ∀xF1y [A p.191]

2. ∀x∃yF1y [A p.191]

3. ∀xR2xy [A p.191]

4. ∀x∃xR2yy [A p.191]

5. R2x [A p.191]

6. ∀xR2x [A p.191]

7. ∀x(F1x → R2x) [A p.191]

8. ∀x∃y(F1x → R2xy) [A p.191]

9. ∀x∃y(F1xy→ R2y) [A p.191]

10. ∀x∃y∀x∃yR2xy [A p.191]

[Contents]

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]

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 big-
ger 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]

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]
6Read “himself” here as gender-neutral—that is, the claim is that no one self-admires.
7Read “himself” here as gender-neutral—that is, the claim is that Dave admires any-

one who doesn’t self-admire.

Exercises 12.2.2

1. Here is a model:
Domain: {1, 2, 3, . . .}
Referents: a: 1 b: 2 c: 3
Extensions: E: {2, 4, 6, . . .}

P: {2, 3, 5, 7, 11, . . .}8
L: {〈1, 2〉, 〈1, 3〉, 〈1, 4〉, . . . , 〈2, 3〉, 〈2, 4〉, . . . , 〈3, 4〉, . . .} 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) ∃xLxb [A p.196]
(v) ∃xLxa [A p.196]

(vi) ∃xLxx [A p.197]
(vii) ∀x∃yLxy [A p.197]

(viii) ∀x∃yLyx [A p.197]
(ix) ∃x(Px ∧ Lxb) [A p.197]
(x) ∃x(Px ∧ Lcx) [A p.197]

(xi) ∀x∃y(Ey ∧ Lxy) [A p.197]
(xii) ∀x∃y(Py ∧ Lxy) [A p.197]

(xiii) ∀x(Lcx → Ex) [A p.197]
(xiv) ∀x((Lax ∧ Lxc)→ Ex) [A p.197]
(xv) ∀x∀y(Lxy ∨ Lyx) [A p.197]

(xvi) ∃x∃y∃z(Ex ∧ Py ∧ Ez ∧ Pz ∧ Lxy ∧ Lyz) [A p.197]
(xvii) ∃x∃y∃z(Lxy ∧ Lyz ∧ Lzx) [A p.197]

(xviii) ∀x∀y∀z((Lxy ∧ Lyz)→ Lxz) [A p.197]
8That is, the set of prime numbers.
9That is, the set of all pairs 〈x, y〉 such that x is less than y. A more compact way of

writing this set is {〈x, y〉 : x < y}. See §16.1 for an explanation of this kind of notation for sets. 52 2. Here is a model: Domain: {1, 2, 3} Referents: a: 1 b: 2 c: 3 Extensions: F: {1, 2} G: {2, 3} R: {〈1, 2〉, 〈2, 1〉, 〈2, 3〉} S: {〈1, 2, 3〉} State whether each of the following propositions is true or false in this model. (i) ∀x∀y(Rxy→ Ryx) [A p.197] (ii) ∀x∀y(Ryx → Rxy) [A p.197] (iii) ∀x∃y(Gy ∧ Rxy) [A p.197] (iv) ∀x(Fx → ∃y(Gy ∧ Rxy)) [A p.197] (v) ∃x∃y∃zSxyz [A p.197] (vi) ∃x∃ySxay [A p.197] (vii) ∃x∃ySxby [A p.197] (viii) ∃xSxxx [A p.197] (ix) ∃x∃y(Fx ∧ Fy ∧ Sxby) [A p.197] (x) ∃x∃y(Fx ∧ Gy ∧ Sxby) [A p.197] 3. Here is a model: Domain: {Alice, Bob, Carol, Dave, Edwina, Frank} Referents: a:Alice b: Bob c: Carol d: Dave e: Edwina f : Frank Extensions: M: {Bob, Dave, Frank} F: {Alice, Carol, Edwina} L: {〈Alice, Carol〉, 〈Alice, Dave〉, 〈Alice, Alice〉, 〈Dave, Carol〉, 〈Edwina, Dave〉, 〈Frank, Bob〉} S: {〈Alice, Bob〉, 〈Alice, Dave〉, 〈Bob, Alice〉, 〈Bob, Dave〉〈Dave, Bob〉, 〈Dave, Alice〉} State whether each of the following propositions is true or false in this model. (i) ∀x∀y(Lxy→ Lyx) [A p.197] (ii) ∃xLxx [A p.198] (iii) ¬∃xSxx [A p.198] 53 (iv) ∀x∀y(Sxy→ Syx) [A p.198] (v) ∀x∀y∀z((Sxy ∧ Syz)→ Sxz) [A p.198] (vi) ∀x(Mx → ∃yLyx) [A p.198] (vii) ∀x(Fx → ∃yLyx) [A p.198] (viii) ∀x(Fx → ∃yLxy) [A p.198] (ix) ∃x∃y(Lax ∧ Lyb) [A p.198] (x) ∀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) ∀xFxx [A p.198] (ii) ∀x∀y(Fxy→ Fyx) [A p.198] (iii) ∀x∀y(Fxy↔ Fyx) [A p.198] (iv) ∃x∀yFxy [A p.198] (v) ∀x∃yFxy [A p.198] (vi) ∃x∃yFxy [A p.198] (vii) ∀x∀yFxy [A p.199] (viii) ∃x∃yFxy ∧ ¬Faa [A p.199] (ix) ∀x∀yFxy ∧ ¬Faa [A p.199] (x) ∀x∀y(Fxy↔ Fyx) ∧ Fab ∧ ¬Fba [A p.199] [Contents] Exercises 12.3.1 1. Using trees, determine whether the following propositions are log- ical truths. For any proposition that is not a logical truth, read off from your tree a model in which it is false. (i) ∀x(Rxx → ∃yRxy) [A p.200] (ii) ∀x(∃yRxy→ ∃zRzx) [A p.200] (iii) ∀xRax → ∀x∃yRyx [A p.200] (iv) ∀x∃y∃zRyxz→ ∃x∃yRxay [A p.201] 54 (v) ¬∀x∃yRxy [A p.201] (vi) ∀x∀y∀z((Rxy ∧ Ryz)→ Rxz) [A p.202] (vii) ∃x∀yRxy→ ∀x∃yRxy [A p.202] (viii) ∃y∀xRxy→ ∀x∃yRxy [A p.203] (ix) ∃x∀yRxy→ ∃x∃yRxy [A p.203] (x) ∀x∀y∃zRxyz ∨ ∀x∀y∀z¬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) ∀x∀y∀z((Rxy ∧ Ryz)→ Rxz) Rab Rba ∴ ∃xRxx [A p.204] (ii) ∀xFxa→ ∃xFax ∃xFxa ∴ ∃xFax [A p.205] (iii) ∃x∃y∃z(Rxy ∧ Rzy) ∴ ∃xRxx [A p.205] (iv) ∀x∀y(Rxy→ Ryx) ∃xRxa ∴ ∃xRax [A p.206] (v) ∀x∀y(¬Rxy→ Ryx) ∴ ∀x∃yRyx [A p.206] (vi) ∀x∀y(Rxy→ (Fx ∧ Gy)) ∴ ¬∃xRxx [A p.206] (vii) ∀x(Fx → (∀yRxy ∨ ¬∃yRxy)) Fa ¬Rab ∴ ¬Raa [A p.207] (viii) ∀x∀y(∃z(Rzx ∧ Rzy)→ Rxy) ∀xRax ∴ ∀x∀yRxy [A p.207] (ix) ∀x∃yRxy ∴ ∃xRxb [A p.208] 55 (x) ∃x∀y(Fy→ Rxy) ∃x∀y¬Ryx ∴ ∃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 some- body. [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 Can- berra. 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 success- ful 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 Budden- brooks. [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. (∀xPx ∨ ∀xQx) [A p.220] 2. (∃xPx ∨ ∃xQx) [A p.220] 3. (∀xPx → ∀xPx) [A p.220] 4. (∀xPx ↔ ∀xPx) [A p.220] 5. ¬∀x(Sx ∧ (∃yTy→ ∃zUxz)) [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 televi- sion 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: {Clark, Bruce, Peter} Referents: a: Clark b: Clark e: Peter f : Peter Extensions: F: {Bruce, Peter} R: {〈Clark, Bruce〉, 〈Clark, Peter〉, 〈Bruce, Bruce〉, 〈Peter, Peter〉} State whether each of the following propositions is true or false in this model. (i) ∀x(¬Fx → x = a) [A p.223] (ii) ∀x(x = a→ ∀yRxy) [A p.223] (iii) ∃x(x 6= f ∧ F f ∧ Rx f ) [A p.223] (iv) ∀x(x 6= b→ Rax) [A p.223] (v) ∃x(x 6= a ∧ ∀y(Fy→ Rxy)) [A p.223] (vi) ∃x(x 6= e ∧ Rxx) [A p.223] 2. Here is a model: Domain: {1, 2, 3, . . .} Referents: a: 1 b: 1 c: 2 e: 4 Extensions: F: {1, 2, 3} G: {1, 3, 5, . . .} R: {〈1, 2〉, 〈2, 3〉, 〈3, 4〉, 〈4, 5〉, . . .} State whether each of the following propositions is true or false in this model. (i) ∃x(Rax ∧ ¬Rbx) [A p.223] (ii) ∀x((Fx ∧ ¬Gx)→ x = c) [A p.223] (iii) ∀x(x 6= a→ ∃yRyx) [A p.223] (iv) ∀x(Gx → ∃y∃z(Rxy ∧ Ryz ∧ Gz)) [A p.223] (v) ∀x((x = a ∨ x = b)→ x 6= c) [A p.223] (vi) ∃x(¬Fx ∧ x 6= e ∧ ∃y(Fy ∧ Ryx)) [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) ∀x(Fx → x = a) [A p.223] (ii) ∃x(x = a ∧ x = b) [A p.223] (iii) ∃x∀y(x 6= y→ Rxy) [A p.223] (iv) ∀x∀y(Rxy→ x = y) [A p.224] (v) ∀x∀y(x 6= y→ ∃zRxyz) [A p.224] (vi) ∃x(x = a ∧ a 6= x) [A p.224] (vii) ∀x∀y((Fx ∧ Fy)→ x = y) [A p.224] (viii) ∃x(Fx ∧ ∀y(Gy→ x = y)) [A p.224] (ix) ∀x(Fx → ∃y(x 6= y ∧ Rxy)) [A p.224] (x) ∀x((Fx ∧ Rax)→ x 6= a) [A p.224] (xi) ∃x∃y∃z(x 6= y ∧ y 6= z ∧ x 6= z ∧ Rxyz) [A p.224] (xii) ∀x∀y∀z(Rxyz→ (x 6= y ∧ y 6= z ∧ x 6= z)) [A p.224] (xiii) ∀x∀y(x 6= y→ (Fx ∨ Fy)) [A p.225] (xiv) ∃x(Fx ∧ ∀y((Fy ∧ x 6= y)→ Rxy)) [A p.225] (xv) ∀x∀y∀z(Rxyz→ (Rxxx ∧ Ryyy ∧ Rzzz)) [A p.225] (xvi) ∀x(Rxx → ∀y(x = y→ Rxy)) [A p.225] (xvii) (Fa ∧ Fb) ∧ ∀x∀y((Fx ∧ Fy)→ x = y) [A p.225] (xviii) ∃x∃y(Fx ∧ Fy ∧ ∀z[Fz→ (x = z ∨ y = z)]) [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] (ii) {Rab, ¬Rbc, a = b} [A p.226] (iii) {∀x(Fx → x = a), Fa, a 6= b} [A p.227] (iv) {∀x(Fx → Gx), ∃xFx, ¬Ga, a = b} [A p.227] (v) {∀x(x 6= a→ Rax), ∀x¬Rxb, a 6= b} [A p.228] (vi) {∃x∀y(Fy→ x = y), Fa, Fb} [A p.228] 61 (vii) {∀x∀y(Rxy→ x = y), Rab, a 6= b} [A p.229] (viii) {∀x((Fx ∧ Rxa)→ x 6= a), Fb ∧ Rba, a = b} [A p.229] (ix) {∃x∃y∃zRxyz, ∀x(x = x → x = a)} [A p.230] (x) {∀x¬Rxx, ∀x∀yx = y, ∃xRax} [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) ∃xFx ∃yGy ∀x∀yx = y ∴ ∃x(Fx ∧ Gx) [A p.231] (ii) ∃x∃y(Fx ∧ Gy ∧ ∀z(z = x ∨ z = y)) ∴ ∃x(Fx ∧ Gx) [A p.232] (iii) Rab ∴ ∀x∀y∀z(((Rxy ∧ Ryz) ∧ x = z)→ Ryy) [A p.233] (iv) ∀x∀y(Rxy→ Ryx) ∃x(Rax ∧ x 6= b) ∴ ∃x(Rxa ∧ x 6= b) [A p.233] (v) ∀x∀yx = y ∴ ∀x∀y(Rxy→ Ryx) [A p.234] (vi) ∀x∀y∀z((Rxy ∧ Rxz)→ y = z) Rab ∧ Rcd b 6= d ∴ a 6= c [A p.234] (vii) ∃x∃y(Rxy ∧ x = y) ∴ ¬∀xRxx [A p.235] (viii) ∀x(x = a ∨ x = b) ∴ ∀xx = a [A p.235] (ix) ∀xRax ¬∀x∀yx = y ∴ ∃x∃y∃z(Rxy ∧ Rxz ∧ y 6= z) [A p.236] (x) ∀xx = a ∴ ∀xx = b [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 valid- ity 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 de- scriptions. 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 Con- rad. [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 defi- nite 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 assump- tions 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: {Alison, Bruce, Calvin, Delilah} Referents: a: Alison b: Bruce c: Calvin d: Delilah Extensions: F: {Alison, Delilah} M: {Bruce, Calvin} S: {〈Alison, Bruce〉, 〈Alison, Calvin〉, 〈Alison,Delilah〉,〈Bruce, Calvin〉, 〈Bruce, Delilah〉,〈Calvin, Delilah〉} Values of function symbols: f : {〈Alison, Bruce〉,〈Bruce, Calvin〉, 〈Calvin, Bruce〉, 〈Delilah, Calvin〉} m: {〈Alison, Delilah〉, 〈Bruce, Alison〉, {〈Calvin, Delilah〉, 〈Delilah, Alison〉} s : {〈Alison, Alison, Bruce〉, 〈Alison, Bruce, Calvin〉, 〈Alison, Calvin, Delilah〉, 〈Alison, Delilah, Alison〉, 〈Bruce, Alison, Calvin〉, 〈Bruce, Bruce, Calvin〉, 〈Bruce, Calvin, Delilah〉, 〈Bruce, Delilah, Alison〉, 〈Calvin, Alison, Delilah〉, 〈Calvin, Bruce,Delilah〉, 〈Calvin, Calvin, Delilah〉, 〈Calvin, Delilah, Alison〉, 〈Delilah, Alison, Alison〉, 〈Delilah, Bruce, Alison〉, 〈Delilah, Calvin, Alison〉, 〈Delilah, Delilah, Alison〉} State whether each of the following propositions is true or false in this model. (i) ∀xM f (x) [A p.262] (ii) ∃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) ∀x∀y∃z∀w(s(x, y) = w↔ w = z) [A p.262] (vii) ∃x∃y∃z∃w(s(x, y) = z ∧ s(x, y) = w ∧ z 6= w) [A p.262] 67 (viii) s(s(b, a), s(d, a)) = s(b, c) [A p.262] (ix) ∃x∃ys(x, y) = m(y) [A p.262] (x) ∀x∃ys(y, x) = x [A p.262] 3. Here is a model: Domain: {1, 2, 3, . . .} Referents: a1: 1 a2: 2 a3: 3 . . . Extensions: E: {2, 4, 6, . . .} O: {1, 2, 3, . . .} L: {〈x, y〉 : x < y} 10 Values of function symbols: q: {〈x, y〉 : y = x2} 11 s {〈x, y, z〉 : z = x + y} 12 p {〈x, y, z〉 : 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) ∀x∀yq(s(x, y)) = p(s(x, y), s(x, y)) [A p.262] (vi) ∀x∀yq(s(x, y)) = s(s(q(x), p(a2, p(x, y))), q(y)) [A p.262] (vii) ∀xEp(a2, x) [A p.262] (viii) ∀x((Ox → Op(a3, x)) ∧ (Ex → Ep(a3, x))) [A p.262] (ix) ∃xLp(a5, x)p(a5, x) [A p.262] (x) ∀x∀y(Lyx → Lp(a3, x)p(a4, y)) [A p.262] 10That is, {〈1, 2〉, 〈1, 3〉, 〈2, 3〉, 〈1, 4〉, 〈2, 4〉, 〈3, 4〉, 〈1, 5〉, 〈2, 5〉, 〈3, 5〉, 〈4, 5〉, . . .}. 11That is, {〈1, 1〉, 〈2, 4〉, 〈3, 9〉, 〈4, 16〉, . . .}. 12That is, {〈1, 1, 2〉, 〈2, 1, 3〉, 〈2, 2, 4〉, 〈1, 2, 3〉, 〈3, 1, 4〉, 〈3, 2, 5〉, 〈3, 3, 6〉, 〈2, 3, 5〉, 〈1, 3, 4〉, 〈4, 1, 5〉, . . .}. 13That is, {〈1, 1, 1〉, 〈2, 1, 2〉, 〈2, 2, 4〉, 〈1, 2, 2〉, 〈3, 1, 3〉, 〈3, 2, 6〉, 〈3, 3, 9〉, 〈2, 3, 6〉, 〈1, 3, 3〉, 〈4, 1, 4〉, . . .}. 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] (iii) f (a) 6= f (a) [A p.263] (iv) ∀x∃y f (x) = y [A p.263] (v) ∃x∀y f (x) = y [A p.263] (vi) ∀x∀ys(x, y) = s(y, x) [A p.263] (vii) ∀x∀y f (s(x, y)) = s( f (x), f (y)) [A p.263] (viii) ∃x∃ys(x, y) = f (x)→ ∃x∃ys(x, y) = f (y) [A p.264] (ix) ∃x∃ys(x, y) = f (x)→ ∃x∃y f (s(x, y)) = f (x) [A p.264] (x) ∀x∀y∃z∀w(s(x, y) = w↔ w = z) [A p.264] [Contents] 69 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 → ∀x(Fx → Gx)) [A p.265] 3. ∀xx = x [A p.265] 4. ∀x∃y¬Rxy [A p.265] 5. ¬∀xa 6= x [A p.265] 6. ∀x(Fx → ∃yRxy) [A p.265] 7. (∀xa = x ∧ ¬∃xa 6= x) [A p.265] 8. (Fa ∧ (Fa ∧ (Fa ∧ (Fa ∧ (Fa ∧ Fa))))) [A p.265] 9. ∀x(Fx → ∀x(Fx → ∀x(Fx → ∀x(Fx → ∀x(Fx → Fx))))) [A p.265] 10. (((¬∃x(¬Fx ∨ Gx) ∧ a 6= b)→ ¬Fa) ∨ (¬∃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) dis- junction 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. γ is of the form ¬α, and α’s main operator is conjunction. [A p.267] 2. γ is of the form ¬α, and α’s main operator is the conditional. [A p.267] 3. γ is of the form ¬α, and α’s main operator is the biconditional. [A p.268] 4. γ is of the form ¬α, and α’s main operator is the existential quantifier. [A p.268] 5. γ’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] (iii) ¬Q ` (¬P→ Q)→ P [A p.269] (iv) ` P→ P [A p.269] (v) ¬(P→ ¬Q) ` Q [A p.270] (vi) P,¬P ` 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] (iii) ` ((P→ Q)→ (P→ R))→ (P→ (Q→ R)) [A p.271] (iv) ` (P→ Q)→ (¬Q→ ¬P) [A p.271] (v) P→ Q, P→ ¬Q ` ¬P [A p.271] (vi) P→ Q,¬Q→ P ` Q [A p.272] (vii) ` (P→ (Q→ R))→ (Q→ (P→ R)) [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] (iii) P→ Q ` ¬Q→ ¬P [A p.273] (iv) ` ¬Q→ (Q→ P) [A p.273] (v) P ∧Q ` P→ Q [A p.273] (vi) ¬Q ` (P ∨Q)→ P [A p.274] (vii) ¬P ∧ ¬Q ` ¬(P ∨Q) [A p.274] (viii) ¬(P ∨Q) ` ¬P ∧ ¬Q [A p.275] (ix) ` ¬(P ∧ ¬P) [A p.275] (x) ` (P ∧ ¬P)→ Q [A p.276] (xi) ` P↔ P [A p.276] (xii) ` P→ (¬P→ Q) [A p.277] 4. Show the following in A∀=1 by producing formal or informal proofs. (i) ∀x(Fx → Gx), Fa ` Ga [A p.277] (ii) ∀xFx ` ∀x(Gx ∨ Fx) [A p.277] (iii) ∀x∀y(Rxy→ Ryx), Rab ` Rba [A p.277] (iv) ∃xFx → ¬Ga ` Ga→ ∀x¬Fx [A p.278] (v) ` Fa→ ∃xFx [A p.279] (vi) Fa, a = b ` Fb [A p.280] (vii) ∀x∀yx = y ` a = b [A p.280] (viii) a = b, a = c ` c = b [A p.280] (ix) ` a = b→ b = a [A p.281] (x) Fa,¬Fb ` ¬a = b [A p.281] (xi) ¬b = a, ∀x(¬Fx → x = a) ` Fb [A p.283] (xii) ` ∀xFx → ∀yFy [A p.283] 5. Explain why the original unrestricted deduction theorem does not hold in A∀=1 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] (iii) ` ¬¬P→ P [A p.284] (iv) ¬(A ∨ B) ` ¬A ∧ ¬B [A p.285] (v) A,¬A ` B [A p.285] (vi) A→ B, B→ C ` A→ C [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 [A p.287] (x) P→ Q ` ¬(P ∧ ¬Q) [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] (iii) ` ¬¬A→ A [A p.289] (iv) ` ¬(A ∧ ¬A) [A p.291] 3. Show the following in N∀∃=1 . (i) ` ∀x(Fx → Fx) [A p.291] (ii) ∃x(Fx ∧ Gx) ` ∃xFx ∧ ∃xGx [A p.291] (iii) ∀x(Fx → Gx),¬∃xGx ` ¬∃xFx [A p.292] (iv) ∀x(Fx → x = a) ` Fb→ a = b [A p.292] (v) ∀x∀yx = y, Raa ` ∀x∀yRxy [A p.293] (vi) ` ∀xRxx → ∀x∃yRxy [A p.293] (vii) ` ∃xFx → ¬∀x¬Fx [A p.294] (viii) ¬∃xFx ` ∀x¬Fx [A p.294] (ix) ∀xx = a ` b = c [A p.294] (x) ` ∀x∀y((Fx ∧ ¬Fy)→ ¬x = y) [A p.295] 74 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 α is: (a) a contradiction [A p.303] (b) satisfiable [A p.303] (ii) Propositions α and β are: (a) jointly satisfiable [A p.303] (b) equivalent [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 us- ing the sequent calculus S∀∃=1 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. [Contents] 76 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. [Q p.3] 78 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. [Q p.3] 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 deal- ings. [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. [Q p.5] 80 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 tomor- row’. The antecedent is a conjunction with conjuncts ‘It will be sunny tomorrow’ and ‘It will be windy tomorrow’. The consequent is a dis- junction 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 ne- gand ‘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 con- juncts ‘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 conjunc- tion with conjuncts ‘The unemployment rate will increase in the next quarter’ and ‘The unemployment rate will decrease in the next quar- ter’. [Q p.6] 6. This is a conditional with antecedent ‘You don’t stop swimming dur- ing 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 be- come 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 conjunc- tion 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 dis- junction, 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] (iii) ¬(R→ ¬W) [Q p.9] (iv) ((E ∧ R)→ (R ∨ ¬E)) [Q p.9] (v) ((J ↔ M) ∧ (M→ ¬N)) [Q p.9] (vi) (J → (N ∨M)) [Q p.9] (vii) (J → (M ∨ ¬N)) [Q p.9] (viii) ((W ∧M) ∨ (W ∧ ¬M)) [Q p.9] (ix) ((J ∧ ¬J)→ (B ∧ ¬B)) [Q p.9] (x) ((M ∧ B)→ (J ∧ ¬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] (iii) ((K →W) ∧ (L→W)) [Q p.9] (iv) ((L ∨ K) ∨ G) [Q p.9] (v) (W ↔ L) [Q p.9] (vi) (G → (W ∨ S)) [Q p.9] (vii) (G → (W ∧ S)) [Q p.9] (viii) (U → (W → K)) [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 in- teger’. 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− 2. (c) Nothing else is an odd number. [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 in- teger’. 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− 5. (c) Nothing else is divisible by five. [Q p.10] (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. [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. P / (2i) 2. Q / (2i) 3. R / (2i) 4. ¬P 1 / (2ii¬) 5. (Q ∧ R) 2, 3 / (2ii∧) 6. (¬P ∨ (Q ∧ R)) 4, 5 / (2ii∨) Main connective is ∨. [Q p.11] 2. 1. P / (2i) 2. Q / (2i) 3. R / (2i) 4. (Q ∨ R) 2, 3 / (2ii∨) 5. (P ∧ (Q ∨ R)) 1, 4 / (2ii∧) 6. ¬(P ∧ (Q ∨ R)) 5 / (2ii¬) Main connective is ¬. [Q p.11] 3. 1. P / (2i) 2. Q / (2i) 3. R / (2i) 4. ¬P 1 / (2ii¬) 5. ¬Q 2 / (2ii¬) 6. ¬R 3 / (2ii¬) 7. (¬P ∧ ¬Q) 4, 5 / (2ii∧) 8. ((¬P ∧ ¬Q) ∨ ¬R) 6, 7 / (2ii∨) Main connective is ∨. [Q p.11] 4. 1. P / (2i) 2. Q / (2i) 3. R / (2i) 4. S / (2i) 5. (P→ Q) 1, 2 / (2ii→) 6. (R→ S) 3, 4 / (2ii→) 7. ((P→ Q) ∨ (R→ S)) 5, 6 / (2ii∨) Main connective is ∨. [Q p.11] 89 5. 1. P / (2i) 2. Q / (2i) 3. R / (2i) 4. S / (2i) 5. (P↔ Q) 1, 2 / (2ii↔) 6. ((P↔ Q)↔ R) 3, 5 / (2ii↔) 7. (((P↔ Q)↔ R)↔ S) 4, 6 / (2ii↔) Main connective is↔. [Q p.11] 6. 1. P / (2i) 2. ¬P 1 / (2ii¬) 3. ¬¬P 2 / (2ii¬) 4. (¬P ∧ ¬¬P) 2, 3 / (2ii∧) 5. (P ∧ ¬P) 1, 2 / (2ii∧) 6. ((¬P ∧ ¬¬P)→ (P ∧ ¬P)) 4, 5 / (2ii→) Main connective is→. [Q p.11] [Contents] Answers 2.5.4.1 1. ordering disambiguation 2 2 3 3 4 4 5 2 6 5 [Q p.11] [Contents] Answers 2.5.5.1 1. (i) (¬P ∨ (Q ∧ R)) [Q p.11] (ii) ¬((P ∨Q) ∧ R) [Q p.11] (iii) (¬(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] (iii) ∨ → PQ→ RS [Q p.12] (iv) → P→ ∨QRS [Q p.12] (v) → ∧¬P¬¬P ∧ P¬P [Q p.12] [Contents] 91 Chapter 3 Semantics of Propositional Logic Answers 3.2.1 1. (¬ P ∧ (Q ∨ R)) phase 0: T T F phase 1: F T phase 2: F [Q p.13] 2. ¬ (P ∨ (Q → R)) phase 0: T T F phase 1: F phase 2: T phase 3: F [Q p.13] 3. (¬¬ P ∧ (Q → (R ∨ P))) phase 0: F T T F phase 1: T T phase 2: F T phase 3: F [Q p.13] 4. (¬¬ P ∧ (Q → (R ∨ P))) phase 0: T F F T phase 1: F T phase 2: T T phase 3: T [Q p.13] 5. ((P ∨ Q) → (P ∨ P)) phase 0: F T F F phase 1: T F phase 2: F [Q p.13] 92 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] 8. (P → (Q → (R → S))) phase 0: F T F T phase 1: T phase 2: T phase 3: T [Q p.13] 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 Q ((P ∧ Q) ∨ P) T T T/ T T F F/ T F T F/ F F F F/ F [Q p.14] 93 2. P (P ∧ (P ∨ P)) T T T/ F F F/ [Q p.14] 3. P Q ¬(¬P ∧ ¬Q) T T T F/ F/ F/ T F T F/ F/ T/ F T T T/ F/ F/ F F F T/ T/ T/ [Q p.14] 4. Q (Q → (Q ∧ ¬Q)) T F F/ F/ F T F/ T/ [Q p.14] 5. P Q R (P → (Q → R)) T T T T T/ T T F F F/ T F T T T/ T F F T T/ F T T T T/ F T F T F/ F F T T T/ F F F T T/ [Q p.14] 6. P Q ((P ∨ Q) ↔ (P ∧ Q)) T T T/ T T/ T F T/ F F/ F T T/ F F/ F F F/ T F/ [Q p.14] 7. P Q ¬((P ∧ Q) ↔ Q) T T F T/ T/ T F F F/ T/ F T T F/ F/ F F F F/ T/ [Q p.14] 8. P (((P → ¬P) → ¬P) → ¬P) T F/ F/ T/ F/ F F/ F T/ T/ T/ T/ T T/ [Q p.14] 94 9. P Q R ¬(P ∧ (Q ∧ R)) T T T F T/ T/ T T F T F/ F/ T F T T F/ F/ T F F T F/ F/ F T T T F/ T/ F T F T F/ F/ F F T T F/ F/ F F F T F/ F/ [Q p.14] 10. R S T ((¬R ∨ S) ∧ (S ∨ ¬T)) T T T F/ T/ T T/ F/ T T F F/ T/ T T/ T/ T F T F/ F/ F F/ F/ T F F F/ F/ F T/ T/ F T T T/ T/ T T/ F/ F T F T/ T/ T T/ T/ F F T T/ T/ F F/ F/ F F F T/ T/ T T/ T/ [Q p.14] [Contents] Answers 3.4.1 1. P Q (P→ Q) (Q→ P) T T T T T F F T F T T F F F T T [Q p.14] 2. P Q ¬ (P↔ Q) ((P ∨Q) ∧ ¬ (P ∧Q)) T T F T/ T/ F F/ T/ T F T F/ T/ T T/ F/ F T T F/ T/ T T/ F/ F F F T/ F/ F T/ F/ [Q p.14] 3. P Q ¬(P ∧ ¬Q) ¬Q T T T F/ F/ F T F F T/ T/ T F T T F/ F/ F F F T F/ T/ T [Q p.14] 95 4. P Q R ((P → Q) ∧ R)) (P ∨ (Q ∨ R)) T T T T/ T T T/ T T F T/ F T T/ T F T F/ F T T/ T F F F/ F T F/ F T T T/ T T T/ F T F T/ F T T/ F F T T/ T T T/ F F F T/ F F F/ [Q p.14] 5. P Q R S ((P ∧ Q) ∧ (¬R ∧ ¬S)) ((P ∨ (R → Q)) ∧ S) T T T T T/ F F/ F/ F/ T/ T/ T T T T F T/ F F/ F/ T/ T/ T/ F T T F T T/ F T/ F/ F/ T/ T/ T T T F F T/ T T/ T/ T/ T/ T/ F T F T T F/ F F/ F/ F/ T/ F/ T T F T F F/ F F/ F/ T/ T/ F/ F T F F T F/ F T/ F/ F/ T/ T/ T T F F F F/ F T/ T/ T/ T/ T/ F F T T T F/ F F/ F/ F/ T/ T/ T F T T F F/ F F/ F/ T/ T/ T/ F F T F T F/ F T/ F/ F/ T/ T/ T F T F F F/ F T/ T/ T/ T/ T/ F F F T T F/ F F/ F/ F/ F/ F/ F F F T F F/ F F/ F/ T/ F/ F/ F F F F T F/ F T/ F/ F/ T/ T/ T F F F F F/ F T/ T/ T/ T/ T/ F [Q p.14] 6. P Q (P ∧ ¬P) (Q ∧ ¬Q) T T F F/ F F/ T F F F/ F T/ F T F T/ F F/ F F F T/ F T/ [Q p.15] 7. P Q R (P ∨ (Q↔ R)) ((Q→ P) ∧ Q) T T T T T/ T/ T T T F T F/ T/ T T F T T F/ T/ F T F F T T/ T/ F F T T T T/ F/ F F T F F F/ F/ F F F T F F/ T/ F F F F T T/ T/ F [Q p.15] 96 8. P Q R ¬ ((P ∧Q) ∧ R) ((P→ Q) ↔ (P→ R)) T T T F T/ T/ T/ T T/ T T F T T/ F/ T/ F F/ T F T T F/ F/ F/ F T/ T F F T F/ F/ F/ T F/ F T T T F/ F/ T/ T T/ F T F T F/ F/ T/ T T/ F F T T F/ F/ T/ T T/ F F F T F/ F/ T/ T T/ [Q p.15] 9. P Q (P ∨Q) ¬P (Q ∨Q) T T T F T T F T F F F T T T T F F F T F [Q p.15] 10. P Q R S (P → (Q → (R→ S))) ¬S T T T T T T/ T/ F T T T F F F/ F/ T T T F T T T/ T/ F T T F F T T/ T/ T T F T T T T/ T/ F T F T F T T/ F/ T T F F T T T/ T/ F T F F F T T/ T/ T F T T T T T/ T/ F F T T F T F/ F/ T F T F T T T/ T/ F F T F F T T/ T/ T F F T T T T/ T/ F F F T F T T/ F/ T F F F T T T/ T/ F F F F F T T/ T/ 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 24 output of function f 25 (T,T) T T (T,F) F F (F,T) F T (F,F) F 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 output of g (T,T) T (T,F) F (F,T) T (F,F) 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 12 , and then take the output of this, and y—in that order—as the inputs to f 23 . The output of that is the output of g for input (x, y): x y f 12 (x) f 2 3 ( f 1 2 (x), y) T T F T T F F F F T T T F 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 23 ( f 1 2 (x), y). [Q p.15] [Contents] 98 Chapter 4 Uses of Truth Tables Answers 4.1.2 1. Valid [Q p.16] A B C A ∨ B A → C (B → C) → C T T T T T T/ T T T F T F F/ T T F T T T T/ T T F F T F T/ F F T T T T T/ T F T F T T F/ T F F T F T T/ T F F F F T T/ F 2. Invalid. Counterexample: A false, B false, C false (row 8). [Q p.16] A B C ¬A ¬ ((A → B) ∧ (B → C)) ∨ C T T T F F/ T/ T/ T/ T T T F F T/ T/ F/ F/ T T F T F T/ F/ F/ T/ T T F F F T/ F/ F/ T/ T F T T T F/ T/ T/ T/ T F T F T T/ T/ F/ F/ T F F T T F/ T/ T/ T/ T ∗ F F F T F/ T/ T/ T/ F 99 3. Invalid. Counterexample: A true, B true, C false (row 2). [Q p.16] A B C (A ∧ ¬B) → C ¬C ¬A T T T F/ F/ T F F ∗ T T F F/ F/ T T F T F T T/ T/ T F F T F F T/ T/ F T F F T T F/ F/ T F T F T F F/ F/ T T T F F T F/ T/ T F T F F F F/ T/ T T T 4. Valid. [Q p.16] A B C (A ∧ B) ↔ C C → B T T T T/ T T T T F T/ F T T F T F/ F F T F F F/ T T F T T F/ F T F T F F/ T T F F T F/ F F F F F F/ T T 5. Valid. [Q p.16] A B C (¬A ∧ ¬B) ↔ ¬C ¬(A ∨ B) C → ¬C T T T F/ F/ F/ T F/ F T/ F F/ T T F F/ F/ F/ F T/ F T/ T T/ T F T F/ F/ T/ T F/ F T/ F F/ T F F F/ F/ T/ F T/ F T/ T T/ F T T T/ F/ F/ T F/ F T/ F F/ F T F T/ F/ F/ F T/ F T/ T T/ F F T T/ T/ T/ F F/ T F/ F F/ F F F T/ T/ T/ T T/ T F/ T T/ 100 6. Valid [Q p.16] A B C A ∨ B ¬A ∨ C B→ C T T T T F/ T T T T F T F/ F F T F T T F/ T T T F F T F/ F T F T T T T/ T T F T F T T/ T F F F T F T/ T T F F F F T/ T T 7. Invalid. Counterexample: A false, B false, C false (row 8). [Q p.17] A B C ¬(A ∨ B) ↔ ¬C ¬A ∧ ¬B C ∧ ¬C T T T F/ T/ T F/ F/ F F/ F F/ T T F F/ T/ F T/ F/ F F/ F T/ T F T F/ T/ T F/ F/ F T/ F F/ T F F F/ T/ F T/ F/ F T/ F T/ F T T F/ T/ T F/ T/ F F/ F F/ F T F F/ T/ F T/ T/ F F/ F T/ F F T T/ F/ F F/ T/ T T/ F F/ ∗ F F F T/ F/ T T/ T/ T T/ F T/ 8. Invalid. Counterexample: A true, B false, C true (row 3). [Q p.17] A B C ¬(A ∧ B) → (C ∨ A) ¬A ∨ ¬B ¬ (C ∨ ¬C) T T T F/ T/ T T/ F/ F F/ F T/ F/ T T F F/ T/ T T/ F/ F F/ F T/ T/ ∗ T F T T/ F/ T T/ F/ T T/ F T/ F/ T F F T/ F/ T T/ F/ T T/ F T/ T/ F T T T/ F/ T T/ T/ T F/ F T/ F/ F T F T/ F/ F F/ T/ T F/ F T/ T/ F F T T/ F/ T T/ T/ T T/ F T/ F/ F F F T/ F/ F F/ T/ T T/ F T/ T/ 101 9. Valid. [Q p.17] A B C A → (B ∧ C) B ↔ ¬C ¬A T T T T T/ F F/ F T T F F F/ T T/ F T F T F F/ T F/ F T F F F F/ F T/ F F T T T T/ F F/ T F T F T F/ T T/ T F F T T F/ T F/ T F F F T F/ F T/ T 10. Valid. [Q p.17] A B C A→ B B→ C ¬C ¬A T T T T T F F T T F T F T F T F T F T F F T F F F T T F F T T T T F T F T F T F T T F F T T T F T F F F T T T T [Contents] Answers 4.2.1 1. Neither [Q p.17] P Q ((P ∨ Q) → P) T T T/ T T F T/ T F T T/ F F F F/ T 102 2. Neither [Q p.17] P Q R (¬P ∧ (Q ∨ R)) T T T F/ F T/ T T F F/ F T/ T F T F/ F T/ T F F F/ F F/ F T T T/ T T/ F T F T/ T T/ F F T T/ T T/ F F F T/ F F/ 3. Contradiction [Q p.17] P Q ((¬P ∨ Q) ↔ (P ∧ ¬Q)) T T F/ T/ F F/ F/ T F F/ F/ F T/ T/ F T T/ T/ F F/ F/ F F T/ T/ F F/ T/ 4. Tautology [Q p.17] P Q R (P → (Q → (R → P))) T T T T T/ T/ T T F T T/ T/ T F T T T/ T/ T F F T T/ T/ F T T T F/ F/ F T F T T/ T/ F F T T T/ F/ F F F T T/ T/ 5. Tautology [Q p.17] P Q (P → ((P → Q) → Q)) T T T T/ T/ T F T F/ T/ F T T T/ T/ F F T T/ F/ 103 6. Neither [Q p.17] P Q (P→ ((Q→ P) → Q)) T T T T/ T/ T F F T/ F/ F T T F/ T/ F F T T/ F/ 7. Tautology [Q p.17] P Q ((P→ Q) ∨ ¬(Q ∧ ¬Q)) T T T/ T T/ F/ F/ T F F/ T T/ F/ T/ F T T/ T T/ F/ F/ F F T/ T T/ F/ T/ 8. Tautology [Q p.17] P Q ((P→ Q) ∨ ¬(Q ∧ ¬P)) T T T/ T T/ F/ F/ T F F/ T T/ F/ F/ F T T/ T F/ T/ T/ F F T/ T T/ F/ T/ 9. Neither [Q p.17] P Q ((P ∧Q) ↔ (Q↔ P)) T T T/ T T/ T F F/ T F/ F T F/ T F/ F F F/ F T/ 10. Contradiction [Q p.17] P Q ¬((P ∧Q) → (Q↔ P)) T T F T/ T/ T/ T F F F/ T/ F/ F T F F/ T/ F/ F F F F/ T/ T/ [Contents] 104 Answers 4.3.1 1. P Q (P → Q) ¬(P ∧ ¬Q) ∗ T T T T F/ F/ T F F F T/ T/ F T T T F/ F/ F F T T F/ T/ (a) jointly satisfiable, because both true on e.g. ∗’ed row. (b) equivalent, because same truth value on every row (c) not contradictory, because jointly satisfiable (d) not contrary, because jointly satisfiable [Q p.18] 2. P Q (P ∧ Q) (P ∧ ¬Q) ∗ T T T F F/ T F F T T/ † F T F F F/ F F F F T/ (a) jointly unsatisfiable, because no row on which both true (b) not equivalent, because different truth values on e.g. ∗’ed row. (c) not contradictory, because both false on e.g. †’ed row. (d) contrary because jointly unsatisfiable and not contradictory [Q p.18] 3. P Q ¬(P ↔ Q) ¬(P → Q) ∨ ¬(P ∨ ¬Q) T T F T/ F/ T/ F F/ T/ F/ ∗ T F T F/ T/ F/ T F/ T/ T/ F T T F/ F/ T/ T T/ F/ F/ F F F T/ F/ T/ F F/ T/ T/ (a) jointly satisfiable, because both true on e.g. ∗’ed row. (b) equivalent because same truth value on every row (c) not contradictory, because jointly satisfiable (d) not contrary, because jointly satisfiable [Q p.18] 105 4. P Q R (P → (Q → R)) ((P → Q) → R) ∗ T T T T T/ T/ T T T F F F/ T/ F T F T T T/ F/ T T F F T T/ F/ T F T T T T/ T/ T † F T F T F/ T/ F F F T T T/ T/ T F F F T T/ T/ F (a) jointly satisfiable because both true on e.g. ∗’ed row. (b) not equivalent because different truth values on e.g. †’ed row. (c) not contradictory, because jointly satisfiable (d) not contrary, because jointly satisfiable [Q p.18] 5. P Q R (P ∧ (Q ∧ ¬Q)) ¬(Q → ¬(R ∧ ¬Q)) ∗ T T T F F/ F/ F T/ T/ F/ F/ T T F F F/ F/ F T/ T/ F/ F/ T F T F F/ T/ F T/ F/ T/ T/ T F F F F/ T/ F T/ T/ F/ T/ F T T F F/ F/ F T/ T/ F/ F/ F T F F F/ F/ F T/ T/ F/ F/ F F T F F/ T/ F T/ F/ T/ T/ F F F F F/ T/ F T/ T/ F/ T/ (a) jointly unsatisfiable, because no row on which both true (b) equivalent because same truth value on every row (c) not contradictory, because both false on e.g. ∗’ed row. (d) contrary, because jointly unsatisfiable and not contradictory [Q p.18] 106 6. P R (P ∧ ¬P) (R ∨ ¬R) ∗ T T F F/ T F/ T F F F/ T T/ F T F T/ T F/ F F F T/ T T/ (a) jointly unsatisfiable, because no row on which both true (b) not equivalent because different truth values on e.g. ∗’ed row. (c) contradictory, because jointly unsatisfiable and no row on which both false (d) not contrary, because no row on which both false [Q p.18] 7. P Q (P ∧ ¬P) ¬( Q → Q) ∗ T T F F/ F T/ T F F F/ F T/ F T F T/ F T/ F F F T/ F T/ (a) jointly unsatisfiable, because no row on which both true (b) equivalent because same truth value on every row (c) not contradictory, because both false on e.g. ∗’ed row. (d) contrary, because jointly unsatisfiable and not contradictory [Q p.18] 107 8. P Q R ((P→ Q) → R) ¬(P ∨ ¬ ( Q ∧ ¬R)) ∗ T T T T/ T F T/ T/ F/ F/ † T T F T/ F F T/ F/ T/ T/ T F T F/ T F T/ T/ F/ F/ T F F F/ T F T/ T/ F/ T/ F T T T/ T F T/ T/ F/ F/ F T F T/ F T F/ F/ T/ T/ F F T T/ T F T/ T/ F/ F/ F F F T/ F F T/ T/ F/ T/ (a) jointly unsatisfiable, because no row on which both true (b) not equivalent because different truth values on e.g. ∗’ed row. (c) not contradictory, because both false on e.g. †’ed row. (d) contrary, because jointly unsatisfiable and not contradictory [Q p.18] 9. P Q (P↔ Q) ((P ∧Q) ∨ (¬P ∧ ¬Q)) ∗ T T T T/ T F/ F/ F/ T F F F/ F F/ F/ T/ F T F F/ F T/ F/ F/ F F T F/ T T/ T/ T/ (a) jointly satisfiable, because both true on e.g. row 1 (b) equivalent because same truth value on every row (c) not contradictory, because jointly satisfiable (d) not contrary, because jointly satisfiable [Q p.18] 108 10. P Q (P↔ Q) ((P ∧Q) ∨ (¬P ∧ ¬Q)) ∗ T T T T/ T F/ F/ F/ T F F F/ F F/ F/ T/ F T F F/ F T/ F/ F/ F F T F/ T T/ T/ T/ (a) jointly satisfiable, because both true on e.g. row 1 (b) equivalent because same truth value on every row (c) not contradictory, because jointly satisfiable (d) not contrary, because jointly satisfiable [Q p.18] [Contents] Answers 4.4.1 1. Satisfiable (∗’ed row) [Q p.18] P Q (P ∨ Q) ¬ (P ∧ Q) T T T F T/ ∗ T F T T F/ F T T T F/ F F F T F/ 2. Unsatisfiable [Q p.18] P Q ¬ (P → Q) (P↔ Q) (¬P ∨ Q) T T F T/ T F/ T T F T F/ F F/ F F T F T/ F T/ T F F F T/ T T/ T 3. Unsatisfiable [Q p.18] P (P → ¬P) (P ∨ ¬P) (¬P → P) T F F/ T F/ F/ T F T T/ T T/ T/ F 109 4. Unsatisfiable [Q p.18] P Q R (P ∨Q) ∨ R (¬P → ¬Q) (¬Q → ¬R) ¬P T T T T/ T F/ T F/ F/ T F/ F T T F T/ T F/ T F/ F/ T T/ F T F T T/ T F/ T T/ T/ F F/ F T F F T/ T F/ T T/ T/ T T/ F F T T T/ T T/ F F/ F/ T F/ T F T F T/ T T/ F F/ F/ T T/ T F F T F/ T T/ T T/ T/ F F/ T F F F F/ F T/ T T/ T/ T T/ T 5. Satisfiable (∗’ed row) [Q p.18] P Q R (P↔ Q) (Q ∨ R) (R→ P) ∗ T T T T T T T T F T T T T F T F T T T F F F F T F T T F T F F T F F T T F F T T T F F F F T F T 6. Satisfiable (∗’ed row) [Q p.18] P Q (¬P → ¬Q) (P↔ Q) ∗ T T F/ T F/ T T F F/ T T/ F F T T/ F F/ F F F T/ T T/ T 7. Unsatisfiable [Q p.18] P ¬P (P → (P→ P)) (¬P ↔ P) T F T T/ F F F T T T/ T F 110 8. Unsatisfiable [Q p.19] P Q R (P ∨ ¬Q) (P→ R) ¬R (¬R → Q) T T T T F/ T F F/ T T T F T F/ F T T/ T T F T T T/ T F F/ T T F F T T/ F T T/ F F T T F F/ T F F/ T F T F F F/ T T T/ T F F T T T/ T F F/ T F F F T T/ T T T/ F 9. Satisfiable (∗’ed row) [Q p.19] P Q R ¬R ¬P ((Q → ¬Q) → R) T T T F F F/ F/ T T T F T F F/ F/ T T F T F F T/ T/ T T F F T F T/ T/ F F T T F T F/ F/ T ∗ F T F T T F/ F/ T F F T F T T/ T/ T F F F T T T/ T/ F 10. Unsatisfiable [Q p.19] P Q (¬P ∨ ¬Q) ¬ (P ∧ ¬Q) (P ∨ ¬Q) ¬ (¬P ∧ ¬Q) T T F/ F F/ T F/ F/ T F/ T F/ F/ F/ T F F/ T T/ F T/ T/ T T/ T F/ F/ T/ F T T/ T F/ T F/ F/ F F/ T T/ F/ F/ F F T/ T T/ T F/ T/ T T/ F T/ T/ T/ [Contents] 111 Chapter 5 Logical Form Answers 5.1.1 Note: There are also other correct answers to questions 1–4. 1. (i) ¬(α→ β) (ii) ¬(α→ (β→ γ)) (iii) ¬(α→ (α→ β)) [Q p.20] 2. (i) (α→ β) (ii) (α→ α) (iii) ((α ∨ β)→ (α ∨ β)) [Q p.20] 3. (i) α (ii) α ∧ β (iii) α ∧ (β→ γ) [Q p.20] 4. (i) α (ii) α↔ β (iii) (α ∧ β)↔ γ [Q p.20] [Contents] 112 Answers 5.2.1 1. First form: ¬¬α (i) α : C (ii) α : (A ∧ B) (iii) α : (C ∧ ¬D) Second form: ¬α (i) α : ¬C (ii) α : ¬(A ∧ B) (iii) α : ¬(C ∧ ¬D) Third form: α (i) α : ¬¬C (ii) α : ¬¬(A ∧ B) (iii) α : ¬¬(C ∧ ¬D) [Q p.20] 2. (i) (a) Yes: α : P ; β : Q. [Q p.21] (b) Yes: α : R ; β : Q. [Q p.21] (c) Yes: α : R ; β : (R→ Q). [Q p.21] (ii) (a) Yes: α : P ; β : Q. [Q p.21] (b) Yes: α : P ; β : P. [Q p.21] (c) No. [Q p.21] (iii) (a) Yes: α : ¬P ; β : Q. [Q p.21] (b) Yes: α : P ; β : ¬P. [Q p.21] (c) No. [Q p.21] (iv) (a) No. [Q p.21] (b) No. [Q p.21] (c) Yes: α : ¬P ; β : ¬P [Q p.21] [Contents] 113 Answers 5.3.1 Note: There are also other correct answers to questions 1–4. 1. (i) ¬(α→ (α→ β)) ∴ (α ∨ (α→ β)) replacements: α : R ; β : Q (ii) ¬(α→ β) ∴ (α ∨ β) replacements: α : R ; β : (R→ Q) (iii) ¬(α→ (β→ γ)) ∴ (α ∨ (β→ γ)) replacements: α : R ; β : R ; γ : Q (iv) α ∴ β replacements: α : ¬(R→ (R→ Q)) ; β : R ∨ (R→ Q) [Q p.21] 2. (i) (α ∧ β)→ β ¬β ∴ ¬(α ∧ β) replacements: α : P ; β : Q (ii) α→ β ¬β ∴ ¬α replacements: α : (P ∧Q) ; β : Q (iii) α→ β γ ∴ ¬α replacements: α : (P ∧Q) ; β : Q ; γ : ¬Q (iv) α β ∴ γ replacements: α : (P ∧Q)→ Q ; β : ¬Q ; γ : ¬(P ∧Q) [Q p.22] 114 3. (i) ¬α→ (β→ γ) ¬α ∴ β→ γ replacements: α : Q ; β : R ; γ : S (ii) ¬α→ β ¬α ∴ β replacements: α : Q ; β : (R→ S) (iii) α→ β α ∴ β replacements: α : ¬Q ; β : (R→ S) (iv) α β ∴ γ replacements: α : ¬Q→ (R→ S) ; β : ¬Q ; γ : (R→ S) [Q p.22] 4. (i) (α→ ¬β) ∨ (¬β→ α) ¬(¬β→ α) ∴ α→ ¬β replacements: α : P ; β : Q (ii) (α→ β) ∨ (β→ α) ¬(β→ α) ∴ α→ β replacements: α : P ; β : ¬Q (iii) α ∨ β ¬β ∴ α replacements: α : (P→ ¬Q) ; β : (¬Q→ P) (iv) α β ∴ γ replacements: α : (P→ ¬Q) ∨ (¬Q→ P) ; β : ¬(¬Q→ P) ; γ : (P→ ¬Q) [Q p.22] [Contents] 115 Answers 5.4.1 1. (i) α : P ; β : Q (ii) P Q P→ Q T T T T F F F T T F F T [Q p.22] 2. (i) α : (A ∧ B) ; β : (B ∨ C) (ii) A B C A ∧ B (A ∧ B)→ (B ∨ C) B ∨ C T T T T T T T T F T T T T F T F T T T F F F T F F T T F T T F T F F T T F F T F T T F F F F T F [Q p.22] 3. (i) α : (A ∨ ¬A) ; β : (A ∧ ¬A) (ii) P (A ∨ ¬A) (A ∨ ¬A)→ (A ∧ ¬A) (A ∧ ¬A) T T F F F T F F [Q p.22] 4. (i) α : (P→ ¬P) ; β : (P→ (Q ∧ ¬R)) (ii) P Q R (P→ ¬P) (P→ ¬P) → (P→ (Q ∧ ¬R)) (P → (Q ∧ ¬R)) T T T F T F T T F F T T T F T F T F T F F F T F F T T T T T F T F T T T F F T T T T F F F T T T [Q p.22] [Contents] 116 Answers 5.5.1 1. (i) α β T T ∗ T F F T F F Invalid: in ∗’ed row, the premise is T and the conclusion is F. [Q p.23] (ii) Instance: P ∴ P Truth table: P T F Valid: there is no row in which the premise (P) is true and the conclusion (P) is false. [Q p.23] 2. α ∨ ¬α ∴ α ∧ ¬α [Q p.23] [Contents] 117 Chapter 6 Connectives: Translation and Adequacy Answers 6.5.1 1. Glossary: B: Bob is happy. R: It is raining. S: The sun is shining. Translation: (B↔ R) (R ∨ S) ∴ (B→ ¬S) Truth Table: B R S (B↔ R) (R ∨ S) (B→¬S) ∗ T T T T T F T T F T T T T F T F × T F F F × F T T F × F T F F × F F T T T T F F F T F × [Q p.24] Invalid. Counterexample (∗’ed row), where B is T, R is T and S is T. 118 2. Glossary: M: I have money. C: I have a card. W: I shall walk. T: I shall get tired. R: I shall have a rest. Translation: (¬M ∧ ¬C)→W W → (T ∨ R) ∴ (R→ M) 119 Truth Table: M C W T R (¬M ∧ ¬C) → W W → (T ∨ R) R→ M T T T T T F/ T T T/ T T T T T F F/ T T T/ T T T T F T F/ T T T/ T T T T F F F/ T F F/ T T T F T T F/ T T T/ T T T F T F F/ T T T/ T T T F F T F/ T T T/ T T T F F F F/ T T F/ T T F T T T F/ T T T/ T T F T T F F/ T T T/ T T F T F T F/ T T T/ T T F T F F F/ T F F/ T T F F T T F/ T T T/ T T F F T F F/ T T T/ T T F F F T F/ T T T/ T T F F F F F/ T T F/ T F T T T T F/ T T T/ F F T T T F F/ T T T/ T F T T F T F/ T T T/ F F T T F F F/ T F F/ T F T F T T F/ T T T/ F F T F T F F/ T T T/ T F T F F T F/ T T T/ F F T F F F F/ T T F/ T ∗ F F T T T T/ T T T/ F F F T T F T/ T T T/ T F F T F T T/ T T T/ F F F T F F T/ T F F/ T F F F T T T/ F T T/ F F F F T F T/ F T T/ T F F F F T T/ F T T/ F F F F F F T/ F T F/ T Invalid. Counterexample (∗’ed row), where M is F, C is F, W is T, T is T and R is T. [Q p.24] 120 3. Glossary: M: Maisy is upset. T: There is thunder. L: There is lightning. Translation: M→ T T → L ∴ ¬M ∨ L Truth Table: M T L M→ T T → L ¬M ∨ L T T T T T F/ T T T F T F F/ F T F T F T F/ T T F F F T F/ F F T T T T T/ T F T F T F T/ T F F T T T T/ T F F F T T T/ T Valid. [Q p.24] 121 4. Glossary: C: The car started. K: You turned the key. A: You pressed the accelerator. Translation: C → (K ∧ A) (K ∧ ¬A)→ ¬C ¬C ∴ (A ∧ ¬K) ∨ (¬K ∧ ¬A) Truth Table: C K A C → (K ∧ A) (K ∧ ¬A) → ¬C ¬C (A ∧ ¬K) ∨ (¬K ∧ ¬A) T T T T T F× T T F F× T F T F× T F F F× ∗ F T T T T T F F T F T T T F F F T T T T T F F F T T T T Invalid. Counterexample (∗’ed row), where C is F, K is T and A is T. [Q p.24] 122 5. Glossary: B: Maisy is barking. R: There is a robber outside. A: Maisy is asleep. D: Maisy is depressed. Translation: (¬B ∨ R) (R ∧ ¬B)→ (A ∨ D) ¬A ∧ ¬D ∴ (B↔ R) Truth Table: B R A D ¬B ∨ R (R ∧ ¬B) → (A ∨ D) ¬A ∧ ¬D B↔ R T T T T T T F T T T T F T T F T T T F T T T F T T T F F T T T T T F T T F T F F T F T F F T F F T F F T F T F F T F F F F T T F F T T T T T F F F T T F T T F F F T F T T T F F F T F F T F T F F F T T T T F T F F T F T T F T F F F T T T F T F F F F T T T T Valid. [Q p.24] 123 6. Glossary: S: It is sunny. W: It is too windy. L: We are sailing. F: We are having fun. Translation: ¬S→ (W ∨ L) L→ F ¬S ∧ ¬W ∴ F Truth Table: S W L F ¬S → (W ∨ L) (L→ F) (¬S ∧ ¬W) T T T T F/ T T/ T F T T T F F/ T T/ F× T T F T F/ T T/ T F T T F F F/ T T/ T F T F T T F/ T T/ T F T F T F F/ T T/ F× T F F T F/ T F/ T F T F F F F/ T F/ T F F T T T T/ T T/ T F F T T F T/ T T/ F F T F T T/ T T/ T F F T F F T/ T T/ T F F F T T T/ T T/ T T F F T F T/ T T/ F× F F F T T/ F× F/ F F F F T/ F× F/ Valid. [Q p.25] 124 7. Glossary: S: You came through Singleton. M: You came through Maitland. N: You came through Newcastle. C: You came through Cessnock. Translation: (S ∧M) ∨ N ¬(S ∨M) ∧ C ∴ (N ∧ C) Truth Table: S M N C (S ∧M) ∨ N ¬ (S ∨M) ∧ C (N ∧ C) T T T T T/ T F/ T/ F× T T T F T/ T F/ T/ F× T T F T T/ T F/ T/ F× T T F F T/ T F/ T/ F× T F T T F/ T F/ T/ F× T F T F F/ T F/ T/ F× T F F T F/ F× T F F F F/ F× F T T T F/ T F/ T/ F× F T T F F/ T F/ T/ F× F T F T F/ F× F T F F F/ F× F F T T F/ T T/ F/ T T F F T F F/ T T/ F/ F× F F F T F/ F× F F F F F/ F× Valid. [Q p.25] 125 8. Glossary: S: The shop will be open. L: We shall have lobster for lunch. T: It is Sunday. R: We shall go to a restaurant. Translation: S→ L S ∨ T T → (R ∧ L) ∴ L Truth Table: S L T R S→ L S ∨ T T → (R ∧ L) T T T T T T T T/ T T T F T T F F/ T T F T T T T T/ T T F F T T T F/ T F T T F× T F T F F× T F F T F× T F F F F× F T T T T T T T/ F T T F T T F F/ F T F T T F× F T F F T F× F F T T T T F F/ F F T F T T F F/ F F F T T F× F F F F T F× Valid. [Q p.25] 126 9. Glossary: C: You will catch Billy a fish. D: You will feed Billy for a day. T: You will teach Billy to fish. L: You will feed Billy for life. Translation: C → D T → L ∴ ¬L ∨ T Truth Table: C D T L C → D T → L ¬L ∨ T T T T T T T F/ T T T T F T F× ∗ T T F T T T F/ F T T F F T T T/ T T F T T F× T F T F F× T F F T F× T F F F F× F T T T T T F/ T F T T F T F× F T F T T T F/ F F T F F T T T/ T F F T T T T F/ T F F T F T F× F F F T T T F/ F F F F F T T T/ T Invalid. Counterexample (∗’ed row), where C is T, D is T and T is F and L is T. [Q p.25] 127 10. Glossary: H: I shall be happy. W: The Tigers will win. D: It will be a draw. Translation: W → H W ∨ ¬W ¬W → D ∴ (¬D ∧ ¬W)→ H Truth Table: W H D W → H W ∨ ¬W ¬W → D (¬D ∧ ¬W) → H T T T T T F/ F/ T F/ F/ F/ T T T F T T F/ F/ T T/ F/ F/ T T F T F× T F F F× F T T T T T/ T/ T F/ F/ T/ T F T F T T T/ T/ F× F F T T T T/ T/ T F/ F/ T/ T F F F T T T/ T/ F× Valid. [Q p.25] [Contents] Answers 6.6.3 1. (i) Functionally complete: α β ¬(α→ ¬β) (¬α→ β) T T T T T F F T F T F T F F F F The second last column is the same as the truth table for α ∧ β, and the last column is the same as the truth table for α∨ β, so we have defined ∧ and ∨ in terms of ¬ and →. We already know 128 that {∧,∨,¬} is functionally complete, so this establishes that {¬,→} is functionally complete. (Where do the columns come from? That is, it is easy to see, once the formula¬(α→ ¬β) is given, that this formula is equiv- alent to (α ∧ β)—and similarly for (¬α → β) and (α ∨ β)—but where do these formulas come from in the first place? The an- swer is: they come by trial and error, guided by knowledge of the relevant truth tables. We know what truth table we want to end up with—say, the truth table for (α ∧ β)—and we know what connectives we are allowed to use—in this case ¬ and →—and what their truth tables are; we then play around with formulas involving different combinations of the allowed con- nectives until we find one that has the desired truth table.) [A p.25] (ii) Not functionally complete: we cannot define any connective which has an odd number of T’s in its truth table; e.g. → (one T), ∨ (three T’s) or ∧ (one T). Consider the truth table for α ↔ β. (We make no assumptions about how complex α and β are—i.e. about how many connec- tives and basic propositions they contain—hence no assump- tions about how many rows there are in this truth table.) α↔ β is T iff α and β have the same truth value (both T or both F); α ↔ β is F iff α and β have opposite truth values (one T and the other F). Let us call a row in which α ↔ β is true a ‘T row’ and a row in which it is false an ‘F row’, and let us say that a T is worth 1 point and an F is worth 0 points (this has no deep significance: it simply allows the following discussion to be pre- sented in a simple way). If we sum the number of points in the α and β columns (combined), each F row contributes 1 point, and each T row contributes either 2 points or 0 points. Now suppose there is an odd number of T rows (and hence an odd number of F rows, as there is an even number of rows in total in any truth table). Then the total number of points in the α and β columns (combined) is an odd number (the number of F rows) plus some 2’s and 0’s—i.e. an odd number. That means that either α is true in an odd number of rows and β is true in an even number of rows, or vice versa. Either way, it follows that α ↔ β is true in an odd number of rows iff one of α and β is true in an odd number of rows. Consider the truth table for α Y β. α Y β is T iff α and β have op- 129 posite truth values (one T and the other F); α Y β is F iff α and β have the same truth value (both T or both F). If we sum the num- ber of points in the α and β columns (combined), each T row contributes 1 point, and each F row contributes either 2 points or 0 points. Now suppose there is an odd number of T rows (and hence an odd number of F rows). Then the total number of points in the α and β columns (combined) is an odd number (the number of T rows) plus some 2’s and 0’s—i.e. an odd num- ber. That means that either α is true in an odd number of rows and β is true in an even number of rows, or vice versa. Either way, it follows that α Y β is true in an odd number of rows iff one of α and β is true in an odd number of rows. Now think about formulas that we can define using only ↔, Y and basic propositions. Each basic proposition is true in an even number of rows (half the rows in the table: recall how the matrix is laid out); and as we have just seen, any proposition built from ↔ or Y and propositions which are true in an even number of rows, is itself true in an even number of rows. So every proposition that we can define using only↔, Y and basic propositions has an even number of T’s in its truth table. [Q p.25] (iii) Functionally complete: α β α ↓ α (α ↓ α) ↓ (β ↓ β) (α ↓ β) ↓ (α ↓ β) T T F T T T F F F T F T T F T F F T F F The third last column is the same as the truth table for ¬α, the second last column is the same as the truth table for α ∧ β, and the last column is the same as the truth table for α ∨ β, so we have defined ¬, ∧ and ∨ in terms of ↓. We already know that {∧,∨,¬} is functionally complete, so this establishes that {↓} is functionally complete. [Q p.25] (iv) Not functionally complete: we cannot define any connective which has an F in the top row. When α and β are both T, (α→ β) is T and (α∧ β) is T—hence so is any formula, however complex, built up from α’s, β’s, →’s and ∧’s. Hence no such formula is equivalent to ¬α, which is F when α is T. [Q p.25] 130 (v) Functionally complete: α β ¬(¬α12β) α12¬β T T T T T F T F F T T F F F F F The second last column is the same as the truth table for α ∨ β, and the last column is the same as the truth table for α ∧ β, so we have defined ∧ and ∨ in terms of ¬ and 12. We already know that {∧,∨,¬} is functionally complete, so this establishes that {¬, 12} is functionally complete. [Q p.25] (vi) Not functionally complete: (α4β) and (α4α) are both equiv- alent to α, and (β4β) and (β4α) are both equivalent to β, so we cannot express anything more with α’s, β’s, ∨’s and 4’s than we can with just α’s, β’s and ∨’s. But {∨} is not a func- tionally complete set of connectives (why not?), hence neither is {∨, 4}. [Q p.25] 2. (i) A B (B14A) T T F T F T F T F F F F [Q p.25] (ii) A B ((A11B) 15 B) T T F/ F T F T/ F F T F/ F F F T/ F [Q p.26] (iii) A B ¬ (A ∨ (A6B)) T T F T/ T/ T F F T/ F/ F T F T/ T/ F F T F/ F/ [Q p.26] (iv) A B A ↔ (A 3 ¬B) T T T T/ F/ T F T T/ T/ F T F T/ F/ F F T F/ T/ [Q p.26] 131 (v) A B (A 12 B) Y (B 12 A) T T F/ F F/ T F T/ T F/ F T F/ T T/ F F F/ F F/ [Q p.26] (vi) A B (A 12 B) Y (B 16 A) T T F/ F F/ T F T/ T F/ F T F/ F F/ F F F/ F F/ [Q p.26] 3. Here are two ways to answer each part. First, we can form a disjunction of row descriptions, in the way ex- plained in §6.6.2, pp.129–131. This gives: (i) (α ∧ β ∧ γ) ∨ (α ∧ ¬β ∧ γ) ∨ (α ∧ ¬β ∧ ¬γ) ∨ (¬α ∧ β ∧ γ) ∨ (¬α ∧ ¬β ∧ γ) ∨ (¬α ∧ ¬β ∧ ¬γ) [Q p.26] (ii) (α ∧ ¬β ∧ γ) ∨ (α ∧ ¬β ∧ ¬γ) ∨ (¬α ∧ β ∧ γ) ∨ (¬α ∧ β ∧ ¬γ) [Q p.26] Alternatively, we can use the following hints, to get: (i) (¬β ∨ γ) or ¬(β ∧ ¬γ) [Q p.26] (Hint: Look at the truth table for ](α, β, γ), and note that it is equivalent to (β→ γ).) (ii) (α ∨ β) ∧ ¬(α ∧ β) [Q p.26] (Hint: Look at the truth table for \(α, β, γ), and note that it is equivalent to (α Y β).) There are also other correct answers to (i) and (ii). 4. (i) (A ∧ ¬B) [Q p.26] (ii) (¬A ∧ ¬B) [Q p.26] (iii) ¬(A ∧ B) [Q p.26] (iv) A ∧ ¬B [Q p.26] (v) ¬(A ∧ ¬B) ∧ ¬(B ∧ ¬A) [Q p.26] (vi) ¬(A ∧ ¬A) [Q p.26] 5. (i) ¬4 [Q p.26] 132 (ii) 14 [Q p.26] (iii) ¬2 and ¬ [Q p.26] (iv) 4, 6, 11 and 13 [Q p.26] [Contents] 133 Chapter 7 Trees for Propositional Logic Answers 7.2.1.1 1. (¬A ∨ ¬B) X ¬A ¬B [Q p.27] 2. (¬A→ B) X ¬¬A B [Q p.27] 3. (A→ B) ∧ B X (A→ B) B [Q p.27] 4. ((A↔ B)↔ B) X (A↔ B) B ¬(A↔ B) ¬B [Q p.27] 5. ¬(A↔ ¬¬A) X A ¬¬¬A ¬A ¬¬A [Q p.27] 134 6. ¬(¬A ∨ B) X ¬¬A ¬B [Q p.27] [Contents] Answers 7.2.2.1 1. (A→ B)→ B X ¬(A→ B) X A ¬B B [Q p.27] 2. (A→ B) ∨ (B→ A) X (A→ B) X ¬A B (B→ A) X ¬B A [Q p.27] 3. ¬(¬A→ (A ∨ B)) X ¬A ¬(A ∨ B) X ¬A ¬B [Q p.27] 4. ¬¬((A ∧ B) ∨ (A ∧ ¬B)) X ((A ∧ B) ∨ (A ∧ ¬B)) X (A ∧ B) X A B (A ∧ ¬B) X A ¬B [Q p.27] [Contents] 135 Answers 7.2.3.1 1. ¬(A→ (B→ A)) X A ¬(B→ A) X B ¬A × [Q p.28] 2. (A→ B) ∨ (¬A ∨ B) X (A→ B) X ¬A B (¬A ∨ B) X ¬A B [Q p.28] 3. ¬((A→ B) ∨ (¬A ∨ B)) X ¬(A→ B) X ¬(¬A ∨ B) X A ¬B ¬¬A X ¬B A [Q p.28] 4. ¬¬¬(A ∨ B) X ¬(A ∨ B) X ¬A ¬B [Q p.28] 5. ¬(A ∧ ¬A) X ¬A ¬¬A X A [Q p.28] 136 6. ¬(¬(A ∧ B)↔ (¬A ∨ ¬B)) X ¬(A ∧ B) X ¬(¬A ∨ ¬B) X ¬¬A X ¬¬B X A B ¬A × ¬B × ¬¬(A ∧ B) X (¬A ∨ ¬B) X (A ∧ B) X A B ¬A × ¬B × [Q p.28] [Contents] Answers 7.3.1.1 1. Valid. [Q p.28] A ¬(A ∨ B) X ¬A ¬B × 2. Invalid. Counterexample: A is T, B is F. [Q p.28] (A ∨ B) X ¬B A ↑ B × 137 3. Valid. [Q p.28] (A ∨ B) X (A→ C) X (B→ D) X ¬(C ∨ D) X ¬C ¬D ¬A A × B ¬B × D × C × 4. Valid. [Q p.28] ((A ∨ ¬B)→ C) X (B→ ¬D) X D ¬C ¬(A ∨ ¬B) X ¬A ¬¬B X B ¬B × ¬D × C × 5. Invalid. Counterexample: A is F, B is T. [Q p.28] B (A→ B) X ¬A ¬A B ↑ 6. Valid. [Q p.29] A (A→ B) X ¬B ¬A × B × 138 7. Valid. [Q p.29] (A ∨ (B ∧ C)) X (A→ B) X (B↔ D) X ¬(B ∧ D) X B D ¬B × ¬D × ¬B ¬D ¬B ¬A A × B ∧ C X B C × B × ¬D ¬A A × B ∧ C X B C × B × 8. Invalid. Counterexample: A is F, B is F, C is T. [Q p.29] ¬(¬A→ B) X ¬(C ↔ A) X (A ∨ C) X ¬(C → B) X ¬¬(A→ B) X A→ B X ¬A ¬B C ¬B A × C C ¬A ¬A ↑ B × ¬C A × 139 9. Valid. [Q p.29] (A↔ B) X (B→ C) X (¬B→ ¬C) (A ∨ (B ∧ ¬B)) X ¬C A A B ¬B × C × ¬A ¬B × B ∧ ¬B X B ¬B × 10. Valid. [Q p.29] (A→ B) X (B→ C) X (C → D) X (D → E) X ¬¬(A ∧ ¬E) X (A ∧ ¬E) X A ¬E ¬A × B ¬B × C ¬C × D ¬D × E × [Contents] 140 Answers 7.3.2.1 1. (i) Contradiction. [Q p.29] (A ∧ ¬A) X A ¬A × (ii) Contradiction. [Q p.29] (A ∨ B) ∧ ¬(A ∨ B) X (A ∨ B) X ¬(A ∨ B) X ¬A ¬B A × B × (iii) Satisfiable. True when A is F and B is F. [Q p.29] (A→ B) ∧ ¬(A ∨ B) X (A→ B) X ¬(A ∨ B) X ¬A ¬B ¬A ↑ B × (iv) Contradiction. [Q p.29] (A→ ¬(A ∨ B)) ∧ ¬(¬(A ∨ B) ∨ B) X (A→ ¬(A ∨ B)) X ¬(¬(A ∨ B) ∨ B) X ¬¬(A ∨ B) X ¬B (A ∨ B) X ¬A A × B × ¬(A ∨ B) × 141 (v) Contradiction. [Q p.29] ¬((¬B ∨ C)↔ (B→ C)) X ¬B ∨ C X ¬(B→ C) X B ¬C ¬B × C × ¬(¬B ∨ C) X B→ C X ¬¬B X ¬C B ¬B × C × (vi) Satisfiable. True when A is F, B is F and C is F. [Q p.29] (A↔ ¬A) ∨ (A→ ¬(B ∨ C)) X (A↔ ¬A) X A ¬A × ¬A ¬¬A X A × A→ ¬(B ∨ C) X ¬A ¬(B ∨ C) X ¬B ¬C ↑ 2. (i) Unsatisfiable. [Q p.30] (A ∨ B) X ¬B (A→ B) X A ¬A × B × B × (ii) Satisfiable. All true when A is F, B is T and C is F. [Q p.30] (A ∨ B) X (B ∨ C) X ¬(A ∨ C) X ¬A ¬C A × B B ↑ C × 142 (iii) Satisfiable. All true when A is F, B is F and C is T. [Q p.30] ¬(¬A→ B) X ¬(C ↔ A) X (A ∨ C) X ¬(C → B) X (A→ B) X ¬A ¬B C ¬B A × C ¬A C ¬A ↑ ¬C A × B × (iv) Satisfiable. All true when A is T, B is T and C is F. [Q p.30] A↔ B X ¬(A→ C) X (C → A) X (A ∧ B) ∨ (A ∧ C) X A ¬C A B ¬C (A ∧ B) X A B ↑ (A ∧ C) X A C × A (A ∧ B) X A B (A ∧ C) X A C × ¬A ¬B × [Contents] 143 Answers 7.3.3.1 1. Can both be true, e.g. when A is T and B is F: (¬A→ B) X (B→ A) X ¬¬A X A ¬B ↑ A B ¬B × A Therefore, jointly satisfiable. [Q p.30] 2. Cannot both be true: (A→ B) ¬(A→ (A→ B)) X A ¬(A→ B) × Cannot both be false: ¬(A→ B) X ¬¬(A→ (A→ B)) X (A→ (A→ B)) X A ¬B ¬A × (A→ B) X ¬A × B × Therefore, contradictories. [Q p.30] 144 3. Cannot both be true: ¬(A↔ ¬B) X ¬(A ∨ ¬B) X ¬A ¬¬B X B A ¬¬B × ¬A ¬B × Can both be false, e.g. when A is T and B is F: ¬¬(A↔ ¬B) X ¬¬(A ∨ ¬B) X (A↔ ¬B) X (A ∨ ¬B) X A ¬B A ↑ ¬B ¬A ¬¬B X B A × ¬B × Therefore, contraries. [Q p.30] 145 4. Cannot both be true: ¬(A ∨ ¬B) X (¬A→ ¬B) X ¬A ¬¬B X B ¬¬A X A × ¬B × Cannot both be false: ¬¬(A ∨ ¬B) X ¬(¬A→ ¬B) X (A ∨ ¬B) X ¬A ¬¬B X B A × ¬B × Therefore, contradictories. [Q p.30] 146 5. Cannot both be true: (¬A ∧ (A→ B)) X ¬(¬A→ (A→ B)) X ¬A (A→ B) X ¬A ¬(A→ B) X A ¬B × Can both be false, e.g. when A is T and B is T: ¬(¬A ∧ (A→ B)) X ¬¬(¬A→ (A→ B)) X (¬A→ (A→ B)) X ¬¬A X A ¬¬A X A (A→ B) X ¬A × B ↑ ¬(A→ B) X A ¬B ¬¬A X A (A→ B) X ¬A × B × Therefore, contraries. [Q p.30] 6. Can both be true, e.g. when A is T and B is F: (A→ B)↔ B X ¬(A→ B) X A ¬B (A→ B) X B × ¬(A→ B) X ¬B A ¬B ↑ Therefore, jointly satisfiable. [Q p.30] [Contents] 147 Answers 7.3.4.1 1. Tautology: [Q p.30] ¬(A→ (B→ A)) X A ¬(B→ A) X B ¬A × 2. Not a tautology. False when A is T and B is F: [Q p.30] ¬(A→ (A→ B)) X A ¬(A→ B) X A ¬B ↑ 3. Tautology: [Q p.31] ¬(((A ∧ B) ∨ ¬(A→ B))→ (C → A)) X ((A ∧ B) ∨ ¬(A→ B)) X ¬(C → A) X C ¬A (A ∧ B) X A B × ¬(A→ B) X A ¬B × 148 4. Tautology: [Q p.31] ¬((A ∧ (B ∨ C))↔ ((A ∧ B) ∨ (A ∧ C))) X A ∧ (B ∨ C) X ¬((A ∧ B) ∨ (A ∧ C)) X ¬(A ∧ B) X ¬(A ∧ C) X A (B ∨ C) X B ¬A × ¬B × C ¬A × ¬B ¬A × ¬C × ¬(A ∧ (B ∨ C)) X (A ∧ B) ∨ (A ∧ C) X (A ∧ B) X A B ¬A × ¬(B ∨ C) X ¬B ¬C × (A ∧ C) X A C ¬A × ¬(B ∨ C) X ¬B ¬C × 5. Not a tautology. False when A is T and B is T: [Q p.31] ¬(¬A ∨ ¬(A ∧ B)) X ¬¬A X ¬¬(A ∧ B) X A (A ∧ B) X A B ↑ 6. Not a tautology. False when A is F and B is T: [Q p.31] ¬(A ∨ (¬A ∧ ¬B)) X ¬A ¬(¬A ∧ ¬B) X ¬¬A X A × ¬¬B X B ↑ 149 7. Tautology. [Q p.31] ¬((A→ B) ∨ (A ∧ ¬B)) X ¬(A→ B) X ¬(A ∧ ¬B) X A ¬B ¬A × ¬¬B X B × 8. Not a tautology. False when A is F and B is T: [Q p.31] ¬((B ∧ ¬A)↔ (A↔ B)) X (B ∧ ¬A) X ¬(A↔ B) X B ¬A A ¬B × ¬A B ↑ ¬(B ∧ ¬A) X (A↔ B) X A B ¬B × ¬¬A X A ¬A ¬B ¬B ¬¬A X A × 9. Tautology. [Q p.31] ¬((A ∨ (B ∨ C))↔ ((A ∨ B) ∨ C)) X (A ∨ (B ∨ C)) X ¬((A ∨ B) ∨ C) X ¬(A ∨ B) X ¬C ¬A ¬B A × (B ∨ C) X B × C × ¬(A ∨ (B ∨ C)) X ((A ∨ B) ∨ C) X ¬A ¬(B ∨ C) X ¬B ¬C (A ∨ B) X A × B × C × 150 10. Not a tautology. False when, e.g., A is T, B is T and C is F: [Q p.31] ¬((A ∧ (B ∨ C))↔ ((A ∨ B) ∧ C)) X (A ∧ (B ∨ C)) X ¬((A ∨ B) ∧ C) X A (B ∨ C) X ¬(A ∨ B) X ¬A ¬B × ¬C B ↑ C × ¬(A ∧ (B ∨ C)) X ((A ∨ B) ∧ C) X (A ∨ B) X C ¬A A × B ¬(B ∨ C) X ¬B ¬C × [Contents] Answers 7.3.5.1 1. Equivalent: [Q p.31] ¬(P↔ (P ∧ P)) X ¬P (P ∧ P) X P P × P ¬(P ∧ P) X ¬P × ¬P × 2. Equivalent: [Q p.31] ¬((P→ (Q ∨ ¬Q))↔ (R→ R)) X (P→ (Q ∨ ¬Q)) ¬(R→ R) X R ¬R × ¬(P→ (Q ∨ ¬Q)) X (R→ R) P ¬(Q ∨ ¬Q) X ¬Q ¬¬Q X Q × 151 3. Equivalent: [Q p.31] ¬(¬(A ∨ B)↔ (¬A ∧ ¬B)) X ¬(A ∨ B) X ¬(¬A ∧ ¬B) X ¬A ¬B ¬¬A X A × ¬¬B X B × ¬¬(A ∨ B) X (¬A ∧ ¬B) X (A ∨ B) X ¬A ¬B A × B × 4. Not equivalent. Different truth values when, e.g., A is F and B is T: [Q p.31] ¬(¬(A ∨ B)↔ (¬A ∨ ¬B)) X ¬(A ∨ B) X ¬(¬A ∨ ¬B) X ¬A ¬B ¬¬A X ¬¬B A × ¬¬(A ∨ B) X (¬A ∨ ¬B) X (A ∨ B) X ¬A A × B ↑ ¬B A B × 5. Not equivalent. Different truth values when, e.g., A is F and B is T: [Q p.31] ¬(¬(A ∧ B)↔ (¬A ∧ ¬B)) X ¬(A ∧ B) X ¬(¬A ∧ ¬B) X ¬A ¬¬A X A × ¬¬B X B ↑ ¬B ¬¬A X A ¬¬B X B × ¬¬(A ∧ B) X (¬A ∧ ¬B) X (A ∧ B) X A B ¬A ¬B × 152 6. Equivalent. [Q p.31] ¬(¬(A ∧ B)↔ (¬A ∨ ¬B)) X ¬(A ∧ B) X ¬(¬A ∨ ¬B) X ¬¬A X ¬¬B X A B ¬A × ¬B × ¬¬(A ∧ B) X (¬A ∨ ¬B) X (A ∧ B) X A B ¬A × ¬B × 7. Equivalent. [Q p.31] ¬(A↔ ((A ∧ B) ∨ (A ∧ ¬B))) X A ¬((A ∧ B) ∨ (A ∧ ¬B)) X ¬(A ∧ B) X ¬(A ∧ ¬B) X ¬A × ¬B ¬A × ¬¬B X B × ¬A ((A ∧ B) ∨ (A ∧ ¬B)) X (A ∧ B) X A B × (A ∧ ¬B) X A ¬B × 153 8. Equivalent. [Q p.31] ¬(¬(P↔ Q)↔ ((P ∧ ¬Q) ∨ (¬P ∧Q))) X ¬(P↔ Q) X ¬((P ∧ ¬Q) ∨ (¬P ∧Q)) X ¬(P ∧ ¬Q) X ¬(¬P ∧Q) X ¬P P ¬Q × ¬P Q ¬¬P X P × ¬Q × ¬¬Q X Q P ¬Q × ¬P Q ¬¬P X P × ¬Q × ¬¬(P↔ Q) X ((P ∧ ¬Q) ∨ (¬P ∧Q)) X (P↔ Q) X (P ∧ ¬Q) X P ¬Q P Q × ¬P ¬Q × (¬P ∧Q) X ¬P Q P Q × ¬P ¬Q × 9. Equivalent. [Q p.31] ¬(((P ∧Q)→ R)↔ (P→ (¬Q ∨ R))) X ((P ∧Q)→ R) X ¬(P→ (¬Q ∨ R)) X P ¬(¬Q ∨ R) X ¬¬Q X ¬R Q ¬(P ∧Q) X ¬P × ¬Q × R × ¬((P ∧Q)→ R) X (P→ (¬Q ∨ R)) X (P ∧Q) X ¬R P Q ¬P × (¬Q ∨ R) X ¬Q × R × 154 10. Not equivalent. Different truth values when P is T and Q is F: [Q p.31] ¬(¬(P↔ Q)↔ (Q ∧ ¬P)) X ¬(P↔ Q) X ¬(Q ∧ ¬P) X P ¬Q ¬Q ↑ ¬¬P X P ¬P Q ¬Q × ¬¬P X P × ¬¬(P↔ Q) X (Q ∧ ¬P) X (P↔ Q) X Q ¬P P Q × ¬P ¬Q × [Contents] 155 Chapter 8 The Language of Monadic Predicate Logic Answers 8.2.1 Glossary: b: Bill B: is beautiful c: Canberra C: is a capital city d: Ben E: is even e: Jenny G: is a gardener j: John H: is happy l: Pluto K: is kite flying m: Mary N: is nice n: New York P: is a prime number p: The Pacific Ocean R: is grumpy q: seven (7) S: is small r: Rover T: is a planet s: Steve V: is heavily populated t: two (2) W: is winning A: is sailing Y: is tiny Translations: 1. Bp [Q p.32] 2. Vn [Q p.32] 3. Nm [Q p.32] 156 4. Rj [Q p.32] 5. Pq [Q p.32] 6. Tl [Q p.32] 7. Gb ∧ Gd [Q p.32] 8. (Am ∨ Ke)→ (Rb ∧ Rd) [Q p.32] 9. ¬(Am ∨ Km) [Q p.32] 10. Ke→ Am [Q p.32] 11. (Aj ∨ Kj) ∧ ¬(Aj ∧ Kj) [Q p.32] 12. ¬Am→ (¬Kj→ Aj) [Q p.32] 13. Ae→ (Am ∧ Aj) [Q p.33] 14. (Aj ∨ Am)→ Ae [Q p.33] 15. Am↔ Ke [Q p.33] 16. Ws→ ¬Hm [Q p.33] 17. Pt ∧ Et [Q p.33] 18. Sc ∧ ¬Yc ∧ Cc [Q p.33] 19. Kr → ¬Pt [Q p.33] 20. Hm↔ ¬He [Q p.33] [Contents] 157 Answers 8.3.2 Glossary: Cx: x is certain Rx: x is red Ex: x is expensive Wx: x is worthwhile Fx: x is fun i: Independence Hall Gx: x is green k: Kermit Hx: x is heavy p: Oscar’s piano Px: x is probable s: Spondulix Translations: 1. Ri→ ∃xRx [Q p.33] 2. ∀xRx → Ri [Q p.33] 3. ¬∃x(Gx ∧ Rx) [Q p.33] 4. ¬¬∃x(Gx ∧ Rx) [Q p.33] 5. This can be interpreted in two ways. The most natural interpretation is (i) no red thing is green; but we could also take it to mean (ii) not every red thing is green (i.e. some red things are not green). (i) ∀x(Rx → ¬Gx) (“Pick anything at all: if it is red, then it is not green.”) or equivalently: ¬∃x(Rx ∧ Gx) (“There does not exist anything which is both red and green.”) (ii) ¬∀x(Rx → Gx) or equivalently: ∃x(Rx ∧ ¬Gx) [Q p.33] 6. ∀x(Rx → (Hx ∨ Ex)) [Q p.33] 7. ∀x((Rx ∧ ¬Hx)→ Ex) [Q p.33] (“Pick anything at all: if it is red and not heavy, then it is expensive.”) 8. ∀x(Rx → Hx) ∧ ∃x(Gx ∧ ¬Hx) [Q p.33] 9. ∀x(Rx → Hx) ∧ ¬∀x(Hx → Rx) [Q p.33] 10. ∃x(Rx ∧ Hx) ∧ ∃x(Gx ∧ Hx) [Q p.33] 11. ∃x(Rx ∧ ¬Hx) ∧ ∃x(Hx ∧ ¬Rx) [Q p.33] 158 12. (Gk ∧ Rk)→ ¬¬∃x(Gx ∧ Rx) [Q p.34] 13. Hp ∧ ¬(Rp ∨ Ep) [Q p.34] 14. (Hs ∧ Es ∧ ∀x(Ex → Rx) ∧ ∀x(Hx → Gx))→ (Rs ∧ Gs) [Q p.34] 15. Hk→ ∃x(Gx ∧ Hx) [Q p.34] 16. ∀xFx → ¬∃xWx [Q p.34] 17. ∃xFx ∧ ∃xWx ∧ ¬∃x(Fx ∧Wx) [Q p.34] 18. ¬∃xCx → ¬∃xPx [Q p.34] 19. ∃xPx ∧ ∃x¬Px ∧ ¬∃xCx [Q p.34] 20. ∀x(Cx → Px) [Q p.34] [Contents] Answers 8.3.5 Glossary: Ax: x can stay Rx: x is telling the truth Bx: x is brown Sx: x is sad Cx: x works at this company Tx: x is in trouble Fx: x is a leaf Ux: x is laughing Gx: x is grey Yx: x is lying Hx: x is happy g: Gary Lx: x is laughing s: the sky Ox: x is in this room t: Stephanie Px: x is a person Translations: 1. ∀x(Px → Hx) [Q p.34] 2. ∃x(Px ∧ Sx) [Q p.34] 3. ¬∃x(Px ∧ Hx ∧ Sx) [Q p.34] 4. ∃x(Px ∧ Sx)→ ¬∀x(Px → Hx) [Q p.34] 159 5. ∀x(Px → (¬Hx → ¬Lx)) or ∀x((Px ∧ Lx)→ Hx) or ¬∃x(Px ∧ Lx ∧ ¬Hx) [Q p.34] 6. Lg→ ∃x(Px ∧ Hx) [Q p.34] 7. ∀x((Px ∧ Lx)→ Hx) [Q p.34] 8. Lg→ ∀x(Px → Lx) [Q p.35] 9. ∃x(Px ∧ Sx) ∧ ¬∀x(Px → Sx) ∧ ¬Sg [Q p.35] 10. ¬∀x(Px → Sx)→ ¬Hg [Q p.35] 11. ∀x(Fx → Bx) ∧ Gs [Q p.35] 12. ∃x(Fx ∧ Bx) ∧ ¬∀x(Fx → Bx) [Q p.35] 13. ∀x(Bx → Fx) [Q p.35] 14. This could be saying either of two things: (i) that the only leaves that can stay are the brown ones (but maybe non-leaves can stay too): ∀x((Ax ∧ Fx)→ Bx) or ∀x(Fx → (Ax → Bx)) (ii) that the only things that can stay are brown leaves: ∀x(Ax → (Fx ∧ Bx)) [Q p.35] 15. ¬Hg→ ∀x(Px → Tx) [Q p.35] 16. ¬Hg→ ∀x((Px ∧ Cx)→ Tx) [Q p.35] 17. Rt→ ∃x(Px ∧Yx) [Q p.35] 18. ¬∃x(Px ∧Yx)→ Rt [Q p.35] 19. Yt ∨ (¬∃x(Px ∧ Rx) ∧ ∀x(Px → Tx)) [Q p.35] 20. Yg→ ¬∀x((Px ∧Ox)→ Rx) [Q p.35] [Contents] 160 Answers 8.4.3.1 1. Main operator: ∀ [Q p.35] Step Wff constructed at this step from steps/by clause: 1. Fx / (3i) 2. Gx / (3i) 3. (Fx → Gx) 1, 2 / (3ii) line 5 4. ∀x(Fx → Gx) 3 / (3ii) line 7 2. Main operator: ∀ [Q p.35] Step Wff constructed at this step from steps/by clause: 1. Gx / (3i) 2. ¬Gx 1 / (3ii) line 1 3. ∀x¬Gx 2 / (3ii) line 7 3. Main operator: ¬ [Q p.35] Step Wff constructed at this step from steps/by clause: 1. Fx / (3i) 2. Gx / (3i) 3. (Fx ∧ Gx) 1, 2 / (3ii) line 2 4. ∃x(Fx ∧ Gx) 3 / (3ii) line 8 5. ¬∃x(Fx ∧ Gx) 4 / (3ii) line 1 4. Main operator: ∧ [Q p.35] Step Wff constructed at this step from steps/by clause: 1. Fa / (3i) 2. Fx / (3i) 3. ¬Fx 2/ (3ii) line 1 4. ∃x¬Fx 3/ (3ii) line 8 5. ¬∃x¬Fx 4/ (3ii) line 1 6. (Fa ∧ ¬∃x¬Fx) 1,5/ (3ii) line 2 161 5. Main operator: ∀ [Q p.35] Step Wff constructed at this step from steps/by clause: 1. Fx / (3i) 2. Gx / (3i) 3. Gy / (3i) 4. (Gx → Gy) 2, 3 / (3ii) line 5 5. ∃y(Gx → Gy) 4 / (3ii) line 8 6. Fx ∧ ∃y(Gx → Gy) 1, 5 / (3ii) line 2 7. ∀x(Fx ∧ ∃y(Gx → Gy)) 6 / (3ii) line 7 6. Main operator: ∧ [Q p.36] Step Wff constructed at this step from steps/by clause: 1. Fx / (3i) 2. Gx / (3i) 3. Fx → Gx 1, 2 / (3ii) line 5 4. ∀x(Fx → Gx) 3 / (3ii) line 7 5. Fa / (3i) 6. (∀x(Fx → Gx) ∧ Fa) 4, 5 / (3ii) line 2 7. Main operator: → [Q p.36] Step Wff constructed at this step from steps/by clause: 1. Fa / (3i) 2. Fb / (3i) 3. ¬Fa 1 / (3ii) line 1 4. ¬Fb 2 / (3ii) line 1 5. (¬Fa ∧ ¬Fb) 3, 4 / (3ii) line 2 6. Fx / (3i) 7. ¬Fx 6 / (3ii) line 1 8. ∀x¬Fx 7 / (3ii) line 7 9. ((¬Fa ∧ ¬Fb)→ ∀x¬Fx) 5, 8 / (3ii) line 5 162 8. Main operator: ∀ [Q p.36] Step Wff constructed at this step from steps/by clause: 1. Fx / (3i) 2. Fy / (3i) 3. (Fx ∧ Fy) 1, 2 / (3ii) line 2 4. Gx / (3i) 5. ((Fx ∧ Fy)→ Gx) 3, 4 / (3ii) line 5 6. ∀y((Fx ∧ Fy)→ Gx) 5 / (3ii) line 7 7. ∀x∀y((Fx ∧ Fy)→ Gx) 6 / (3ii) line 7 9. Main operator: ∀ [Q p.36] Step Wff constructed at this step from steps/by clause: 1. Fx / (3i) 2. Fy / (3i) 3. ∀yFy 2 / (3ii) line 7 4. (Fx → ∀yFy) 1, 3 / (3ii) line 5 5. ∀x(Fx → ∀yFy) 4 / (3ii) line 7 10. Main operator: → [Q p.36] Step Wff constructed at this step from steps/by clause: 1. Fx / (3i) 2. ∀xFx 1 / (3ii) line 7 3. Fy / (3i) 4. ∀yFy 3 / (3ii) line 7 5. (∀xFx → ∀yFy) 2, 4 / (3ii) line 5 [Contents] Answers 8.4.5.1 Free variables are underlined: 1. Tx ∧ Fx Open [Q p.36] 2. Tx ∧ Ty Open [Q p.36] 3. ∃xTx ∧ ∃xFx Closed [Q p.36] 4. ∃xTx ∧ ∀yFx Open [Q p.36] 163 5. ∃xTx ∧ Fx Open [Q p.36] 6. ∃x(Tx ∧ Fx) Closed [Q p.36] 7. ∀y∃xTy Closed [Q p.36] 8. ∃x(∀xTx → ∃yFx) Closed [Q p.36] 9. ∃y∀xTx → ∃yFx Open [Q p.36] 10. ∀x(∃xTx ∧ Fx) Closed [Q p.36] 11. ∀x∃xTx ∧ Fx Open [Q p.36] 12. ∃xTy Open [Q p.36] 13. ∀xTx → ∃xFx Closed [Q p.36] 14. ∃x∀y(Tx ∨ Fy) Closed [Q p.36] 15. ∀xFx ∧ Gx Open [Q p.36] 16. ∀x∀yFx → Gy Open [Q p.37] 17. ∀x∀y(Fx → ∀xGy) Closed [Q p.37] 18. ∃yGb ∧ Gc Closed [Q p.37] 19. ∃yGy ∧ ∀x(Fx → Gy) Open [Q p.37] 20. ∀x((Fx → ∃xGx) ∧ Gx) Closed [Q p.37] [Contents] 164 Chapter 9 Semantics of Monadic Predicate Logic Answers 9.1.1 1. (i) True (ii) True (iii) False [Q p.38] 2. (i) False (ii) True (iii) False [Q p.38] 3. (i) False (ii) True (iii) False [Q p.38] 4. (i) True (ii) True (iii) False [Q p.38] 5. (i) True (ii) True (iii) True [Q p.39] 6. (i) False (ii) False (iii) False [Q p.39] [Contents] Answers 9.2.1 Model 1: (i) False (ii) True (iii) True (iv) True (v) True Model 2: (i) True (ii) False (iii) False (iv) True (v) True [Questions p.39] 165 Model 3: (i) True (ii) False (iii) False (iv) True (v) True Model 4: (i) False (ii) True (iii) True (iv) True (v) True Model 5: (i) False (ii) True (iii) True (iv) True (v) True Model 6: (i) True (ii) False (iii) True (iv) True [Questions p.39] (v) True [Contents] Answers 9.3.1 1. (i) (Fa ∧ Ga) [Q p.39] (ii) (Fb ∧ Ga) [Q p.39] 2. (i) ∀y(Fa→ Gy) [Q p.39] (ii) ∀y(Fb→ Gy) [Q p.39] 3. (i) ∀x(Fx → Gx) ∧ Fa [Q p.39] (ii) ∀x(Fx → Gx) ∧ Fb [Q p.40] 4. (i) ∀x(Fx ∧ Ga) [Q p.40] (ii) ∀x(Fx ∧ Ga) [Q p.40] NB α(x), i.e. ∀x(Fx ∧ Ga), contains no free occurrences of x, so for any name a, α(a/x) is just α(x). For we replace all free occurrences of x with a; if there are no free occurrences, nothing gets replaced. 5. (i) ∃x(Gx → Ga) [Q p.40] (ii) ∃x(Gx → Gb) [Q p.40] 6. (i) ∃y(∀x(Fx → Fy) ∨ Fa) [Q p.40] (ii) ∃y(∀x(Fx → Fy) ∨ Fb) [Q p.40] 166 [Contents] Answers 9.4.3 1. (i) False [Q p.40] (ii) True [Q p.40] (iii) True [Q p.40] (iv) False [Q p.40] (v) True [Q p.40] (vi) True [Q p.40] 2. (i) (a) False [Q p.40] (b) True [Q p.41] (ii) (a) True [Q p.41] (b) False [Q p.41] (iii) (a) True [Q p.41] (b) False [Q p.41] 3. (i) True [Q p.41] (ii) False [Q p.41] (iii) True [Q p.41] (iv) True [Q p.41] (v) True [Q p.41] (vi) False [Q p.41] 4. (i) (a) Domain: {1, 2, 3, . . . } Extensions: F : {1, 2} G : {1, 2, 3} (b) Domain: {1, 2, 3, . . . } Extensions: F : {1, 2} G : {1} [Q p.42] (ii) (a) No such model. For the formula to be true on a model, it would have to be the case that all members of the (non- empty) domain were in the extension of F (so that the first conjunct were true) and also that a certain member of the domain were not in the extension of F (so that the second conjunct were true). 167 (b) Domain: {1, 2, 3, . . . } Referent of a: 2 Extension of F : {2, 4, 6, . . . } [Q p.42] (iii) (a) Domain: {1, 2, 3, . . . } Referent of a : 1 Extension of F : {2, 4, 6, . . . } (b) Domain: {1, 2, 3, . . . } Referent of a : 1 Extension of F : {1, 3, 5, . . . } [Q p.42] (iv) (a) Domain: {1, 2, 3, . . . } Extensions: F : {1, 2, 3, . . . } G : {2, 4, 6, . . . } (b) Domain: {1, 2, 3, . . . } Extensions: F : {1, 3, 5, . . . } G : {2, 4, 6, . . . } [Q p.42] (v) (a) Domain: {1, 2, 3, . . . } Extension of F : {1, 3, 5, . . . } (b) No such model. For the formula to be false on a model, it would have to be the case that, in that model, some member of the domain were both in the extension of F and not in the extension of F [Q p.42] (vi) (a) Domain: {1, 2, 3, . . . } Extensions: F : {1, 3, 5, . . . } G : {2, 4, 6, . . . } (b) Domain: {1, 2, 3, . . . } Extensions: F : {1, 3, 5, . . . } G : ∅ [Q p.42] (vii) (a) Domain: {1, 2, 3, . . . } Extension of F : {1, 3, 5, . . . } (b) No such model. For the formula to be false on a model, it would have to be the case that all members of the (non- empty) domain were in the extension of F (to make the an- tecedent true) and that no members of the domain were in the extension of F (to make the consequent false). [Q p.42] (viii) (a) No such model. For the formula to be true on a model, it would have to be the case that, in that model, a single member of the domain were both in the extension of F and not in the extension of F. (b) Domain: {1, 2, 3, . . . } Extension of F : {1, 3, 5, . . . } [Q p.42] (ix) (a) Domain: {1, 2, 3, . . . } Extension of F : {1, 3, 5, . . . } 168 (b) Domain: {1, 2, 3, . . . } Extension of F : ∅ [Q p.42] (x) (a) Domain: {1, 2, 3, . . . } Extension of F : {1, 3, 5, . . . } (b) No such model. For the formula to be false on a model, it would have to be the case that, in that model, a single member of the domain were both in the extension of F and not in the extension of F. [Q p.42] (xi) (a) Domain: {1, 2, 3, . . . } Extensions: F : {1} G : {2} (b) Domain: {1, 2, 3, . . . } Extensions: F : {1} G : ∅ [Q p.42] (xii) (a) Domain: {1, 2, 3, . . . } Extensions: F : {1} G : {1, 2, 3, . . . } (b) Domain: {1, 2, 3, . . . } Extensions: F : {1} G : {1} [Q p.42] (xiii) (a) Domain: {1, 2, 3, . . . } Referents: a : 1 Extensions: F : {1} (b) No such model. For the formula to be false on a model, it would have to be that all members of the domain were in the extension of F, but the referent of a was not. [Q p.42] (xiv) (a) Domain: {1, 2, 3, . . . } Referents: a : 1 Extensions: F : {1, 2, 3, . . . } (b) Domain: {1, 2, 3, . . . } Referents: a : 2 Extensions: F : {1} [Q p.42] (xv) (a) Domain: {1, 2} Referents: a : 1 b : 2 Extensions: F : {1, 2} (b) Domain: {1, 2} Referents: a : 1 b : 2 Extensions: F : {1} [Q p.42] 169 (xvi) (a) Domain: {1, 2, 3, . . . } Extensions: F : {1, 3, 5, . . . } G : {2, 4, 6, . . . } (b) Domain: {1, 2, 3, . . . } Extensions: F : {1} G : {2} [Q p.42] (xvii) (a) Domain: {1, 2, 3, . . . } Extensions: F : {1} G : ∅ (b) Domain: {1, 2, 3, . . . } Extensions: F : ∅ G : ∅ [Q p.42] (xviii) (a) No such model. For the formula to be true on a model, it would have to be that each member of the domain was both in the extension of F and also not in the extension of F. (b) Domain: {1, 2, 3, . . . } Extensions: F : {1} [Q p.42] (xix) (a) Domain: {1, 2, 3, . . . } Extensions: F : ∅ G : {1} (b) Domain: {1, 2, 3, . . . } Extensions: F : {1, 2, 3, . . . } G : ∅ [Q p.42] (xx) (a) Domain: {1, 2, 3, . . . } Extensions: F : ∅ G : {1} (b) Domain: {1, 2, 3, . . . } Extensions: F : {1, 2, 3, . . . } G : ∅ [Q p.42] 5. (i) True. Suppose there are no F’s. Then, whatever in the domain d (a new name) refers to, Fd is false—so Fd→ Gd is true (because its antecedent is false). So ∀x(Fx → Gx) is true when there are no F’s. [Q p.42] (ii) No. If there are no F’s, ∀x(Fx → Gx) is true (previous question), but ∃x(Fx ∧ Gx) is false. [Q p.42] [Contents] 170 Answers 9.5.1 1. Countermodel: Domain: {1, 2} Extensions: F : {1} G : {2} [Q p.43] 2. No countermodel. For the premise to be true, there must be at least one object in the domain which is in the extension of F and is also in the extension of G. If this is so, then any such object is a fortiori in the extension of F—making the left conjunct of the conclusion true—and in the extension of G—making the right conjunct of the conclusion true: hence the conclusion is true. [Q p.43] 3. Countermodel: Domain: {1, 2} Extensions: F : {2} G : {1} [Q p.43] 4. No countermodel. For the first premise to be true, the extension of F must be a subset of the extension of G. For the second premise to be true, the extension of G must be a subset of the extension of H. It follows that the extension of F is a subset of the extension of H—and this makes the conclusion true. [Q p.43] 5. Countermodel: Domain: {1, 2, 3} Extensions: F : {1} G : {1, 2} H : {1, 2, 3} [Q p.43] [Contents] 171 Chapter 10 Trees for Monadic Predicate Logic Answers 10.2.2 1. (i) ¬(Fa→ ∃xFx) X Fa ¬∃xFx X ∀x¬Fx \a ¬Fa × Logical truth. [Q p.44] (ii) ¬(∃xFx → ¬∀x¬Fx) X ∃xFxXa ¬¬∀x¬Fx X ∀x¬Fx \a Fa ¬Fa × Logical truth. [Q p.44] 172 (iii) ¬∀x((Fx ∧ ¬Gx)→ ∃xGx) X ∃x¬((Fx ∧ ¬Gx)→ ∃xGx) Xa ¬((Fa ∧ ¬Ga)→ ∃xGx) X Fa ∧ ¬Ga X ¬∃xGx X Fa ¬Ga ∀x¬Gx \a X ¬Ga ↑ Not a logical truth. Countermodel: Domain: {1} Extensions: F : {1} G : ∅ [Q p.44] (iv) ¬(∀xFx → ∃xFx) X ∀xFx \a ¬∃xFx X ∀x¬Fx \a Fa ¬Fa × Logical truth. [Q p.44] (v) ¬((Fa ∧ (Fb ∧ Fc))→ ∀xFx) X Fa ∧ (Fb ∧ Fc) X ¬∀xFx X Fa (Fb ∧ Fc) X Fb Fc ∃x¬Fx Xd ¬Fd X ↑ Not a logical truth. Countermodel: Domain: {1, 2, 3, 4} Extension of F : {1, 2, 3} [Q p.44] 173 (vi) ¬(∃xFx ∧ ∃x¬Fx) X ¬∃xFx X ∀x¬Fx \a ¬Fa ↑ ¬∃x¬Fx X ∀x¬¬Fx \a ¬¬Fa X Fa Not a logical truth. Countermodel: Domain: {1} Extension of F : ∅ [Q p.44] (vii) ¬∃x(Fx → ∀yFy) X ∀x¬(Fx → ∀yFy) \ab ¬(Fa→ ∀yFy) X Fa ¬∀yFy X ∃y¬Fy X b ¬Fb ¬(Fb→ ∀yFy) X Fb ¬∀yFy × Logical truth. [Q p.44] (viii) ¬(∀x(Fx → Gx)→ (Fa→ Ga)) X ∀x(Fx → Gx) \ a ¬(Fa→ Ga) X Fa ¬Ga Fa→ Ga X ¬Fa × Ga × Logical truth. [Q p.44] 174 (ix) ¬(¬∀x(Fx ∧ Gx)↔ ∃x¬(Fx ∧ Gx)) X ¬∀x(Fx ∧ Gx) X ¬∃x¬(Fx ∧ Gx) X ∃x¬(Fx ∧ Gx) Xa ∀x¬¬(Fx ∧ Gx) \a ¬(Fa ∧ Ga) X ¬¬(Fa ∧ Ga) X Fa ∧ Ga X Fa Ga ¬Fa × ¬Ga × ¬¬∀x(Fx ∧ Gx) X ∃x¬(Fx ∧ Gx) Xa ∀x(Fx ∧ Gx) \a ¬(Fa ∧ Ga) X Fa ∧ Ga X Fa Ga ¬Fa × ¬Ga × Logical truth. [Q p.44] (x) ¬(¬∃x(Fx ∧ Gx)↔ ∀x(¬Fx ∧ ¬Gx)) X ¬∃x(Fx ∧ Gx) X ¬∀x(¬Fx ∧ ¬Gx) X ∀x¬(Fx ∧ Gx) \a ∃x¬(¬Fx ∧ ¬Gx) Xa ¬(¬Fa ∧ ¬Ga) X ¬(Fa ∧ Ga) X ¬¬Fa X Fa ¬Fa × ¬Ga ↑ ¬¬Ga X Ga ¬Fa ¬Ga × ¬¬∃x(Fx ∧ Gx) X ∀x(¬Fx ∧ ¬Gx) \a ∃x(Fx ∧ Gx) Xa Fa ∧ Ga X Fa Ga ¬Fa ∧ ¬Ga X ¬Fa ¬Ga × Not a logical truth. Countermodel: Domain: {1} Extension of F : {1} G : ∅ [Q p.44] 175 2. (i) ∃xFx ∧ ∃xGx X ¬∃x(Fx ∧ Gx) X ∃xFx Xa ∃xGx Xb ∀x¬(Fx ∧ Gx) \ab Fa Gb ¬(Fa ∧ Ga) ¬Fa × ¬Ga ¬(Fb ∧ Gb) X ¬Fb ↑ ¬Gb × Invalid. Countermodel: Domain: {1, 2} Extensions: F : {1} G : {2} [Q p.44] (ii) ∃x∀y(Fx → Gy) Xa ¬∀y∃x(Fx → Gy) X ∃y¬∃x(Fx → Gy) Xb ∀y(Fa→ Gy) \b ¬∃x(Fx → Gb) X ∀x¬(Fx → Gb) \a Fa→ Gb X ¬(Fa→ Gb) X × Valid. [Q p.44] 176 (iii) Fa→ ∀xGx X ¬∀x(Fa→ Gx) X ¬Fa ∃x¬(Fa→ Gx) Xb ¬(Fa→ Gb) X Fa ¬Gb × ∀xGx \b ∃x¬(Fa→ Gx) Xb ¬(Fa→ Gb) X Fa ¬Gb Gb × Valid. [Q p.45] (iv) Fa→ ∀xGx X ¬∃x(Fa→ Gx) X ∀x¬(Fa→ Gx) \a ¬(Fa→ Ga) X Fa ¬Ga ¬Fa × ∀xGx \a Ga × Valid. [Q p.45] (v) ∀x(Fx ∨ Gx) \ab ¬∀xFx X ¬∀xGx X ∃x¬Fx Xa ∃x¬Gx Xb ¬Fa ¬Gb Fa ∨ Ga Fa × Ga Fb ∨ Gb Fb ↑ Gb × Invalid. Countermodel: Domain: {1, 2} Extensions: F : {2} G : {1} [Q p.45] 177 (vi) ∃x(Fx ∧ Gx) Xa ¬(∃xFx ∧ ∃xGx) X Fa ∧ Ga X Fa Ga ¬∃xFx X ∀x¬Fx \a ¬Fa × ¬∃xGx X ∀x¬Gx \a ¬Ga × Valid. [Q p.45] (vii) ∀x(Fx → Gx) \a Fa ¬Ga Fa→ Ga X ¬Fa × Ga × Valid. [Q p.45] (viii) ¬∀x(Fx ∨ Gx) X ¬∃x(¬Fx ∧ ¬Gx) X ∃x¬(Fx ∨ Gx) Xa ∀x¬(¬Fx ∧ ¬Gx) \a ¬(Fa ∨ Ga) X ¬Fa ¬Ga ¬(¬Fa ∧ ¬Ga) X ¬¬Fa X Fa × ¬¬Ga X Ga × Valid. [Q p.45] 178 (ix) ∀x(Fx → Gx) \a ∀x(Gx → Hx) \a ¬¬∃x(¬Fx ∧ Hx) X ∃x(¬Fx ∧ Hx) Xa ¬Fa ∧ Ha X ¬Fa Ha Fa→ Ga X ¬Fa Ga→ Ha X ¬Ga ↑ Ha Ga Ga→ Ha X ¬Ga × Ha Invalid. Countermodel: Domain: {1} Extensions: F : ∅ G : ∅ H : {1} [Q p.45] (x) ∀x(Fx ∨ Gx) \a ¬¬∃x(Fx ∧ Gx) X ∃x(Fx ∧ Gx) Xa Fa ∧ Ga X Fa Ga Fa ∨ Ga X Fa ↑ Ga Invalid. Countermodel: Domain: {1} Extensions: F : {1} G : {1} [Q p.45] [Contents] 179 Answers 10.3.8 1. Dx: x is a dog Mx: x is a mammal Ax: x is an animal ∀x(Dx → Mx) ∀x(Mx → Ax) ∴ ∀x(Dx → Ax) ∀x(Dx → Mx) \a ∀x(Mx → Ax) \a ¬∀x(Dx → Ax) X ∃x¬(Dx → Ax) Xa ¬(Da→ Aa) X Da ¬Aa Da→ Ma X ¬Da × Ma Ma→ Aa X ¬Ma × Aa × Valid. [Q p.45] 180 2. Fx: x is frozen Cx: x is cold ∀xFx → ∀xCx ∴ ∀x(Fx → Cx) ∀xFx → ∀xCx X ¬∀x(Fx → Cx) X ∃x¬(Fx → Cx) Xa ¬∀xFx X ¬(Fa→ Ca) X Fa ¬Ca ∃x¬Fx Xb ¬Fb ↑ ∀xCx \a ¬(Fa→ Ca) X Fa ¬Ca Ca × Invalid. Countermodel: Domain: {1, 2} Extensions: F : {1} C : ∅ [Q p.45] 181 3. Cx: x is conscious Gx: x is a divine being Sx: x has a sonic screwdriver ∀x(Cx → (∃yGy ∨ Sx)) ¬∃xSx ∴ ¬∀xCx ∀x(Cx → (∃yGy ∨ Sx)) \ab ¬∃xSx X ¬¬∀xCx X ∀xCx \abc ∀x¬Sx \abc Ca ¬Sa Ca→ (∃yGy ∨ Sa) X ¬Ca × ∃yGy ∨ Sa X ∃yGy Xb Gb Cb ¬Sb Cb→ (∃yGy ∨ Sb) X ¬Cb × ∃yGy ∨ Sb X ∃yGy Xc Gc Cc ¬Sc ... ↑ Sb × Sa × Invalid. Countermodel: Domain: {1, 2, 3, . . . } Extensions: C : {1, 2, 3, . . . } G : {2, 3, 4, . . . } S : ∅ [Q p.45] 182 4. Cx: x is a cow Sx: x is a scientist Fx: x can fly ∀x(Cx → Sx) ¬∃x(Sx ∧ Fx) ∴ ¬∃x(Cx ∧ Fx) ∀x(Cx → Sx) \a ¬∃x(Sx ∧ Fx) X ¬¬∃x(Cx ∧ Fx) X ∃x(Cx ∧ Fx) Xa ∀x¬(Sx ∧ Fx) \a Ca ∧ Fa X Ca Fa Ca→ Sa X ¬Ca × Sa ¬(Sa ∧ Fa) X ¬Sa × ¬Fa × Valid. [Q p.45] 183 5. Px: x is a person Hx: x is here Sx: x is smoking ∃x((Px ∧ Hx) ∧ ¬Sx) ∴ ¬∀x((Px ∧ Hx)→ Sx) ∃x((Px ∧ Hx) ∧ ¬Sx) Xa ¬¬∀x((Px ∧ Hx)→ Sx) X ∀x((Px ∧ Hx)→ Sx) \a (Pa ∧ Ha) ∧ ¬Sa X (Pa ∧ Ha) X ¬Sa Pa Ha (Pa ∧ Ha)→ Sa X ¬(Pa ∧ Ha) X ¬Pa × ¬Ha × Sa × Valid. [Q p.46] 184 6. Cx: x is a coward Rx: x rocks up Sx: x will shake c: Catwoman s: Superman Rs→ ∀x(Cx → Sx) ¬Cc ∴ ¬Sc Rs→ ∀x(Cx → Sx) X ¬Cc ¬¬Sc X Sc ¬Rs ∀x(Cx → Sx) \cs Cc→ Sc X Cs→ Ss X ¬Cc ¬Cs ↑ Ss Sc ¬Cs Ss Invalid. Countermodel: Domain: {1, 2} Referents: c : 1 s : 2 Extensions: C : ∅ R : ∅ S : {1} [Q p.46] 185 7. Bx: x is blue Cx: x a car Dx: x is defective Rx: x is red ∀x(Cx → (Rx ∨ Bx)) ∀x((Cx ∧ Rx)→ Dx) ∃x(Bx ∧ Cx ∧ ¬Dx) ∴ ∃x(Cx ∧ Dx) ∧ ∃x(Cx ∧ ¬Dx) ∀x(Cx → (Rx ∨ Bx)) \a ∀x((Cx ∧ Rx)→ Dx) \a ∃x(Bx ∧ Cx ∧ ¬Dx) Xa ¬(∃x(Cx ∧ Dx) ∧ ∃x(Cx ∧ ¬Dx)) X Ba ∧ Ca ∧ ¬Da X Ba Ca ¬Da Ca→ (Ra ∨ Ba) X ¬Ca × Ra ∨ Ba X (Ca ∧ Ra)→ Da X ¬(Ca ∧ Ra) X ¬Ca × ¬Ra Ra × Ba ¬∃x(Cx ∧ Dx) X ∀x¬(Cx ∧ Dx) \a ¬(Ca ∧ Da) X ¬Ca × ¬Da ↑ ¬∃x(Cx ∧ ¬Dx) X ∀x¬(Cx ∧ ¬Dx) \a ¬(Ca ∧ ¬Da) X ¬Ca × ¬¬Da X Da × Da × 186 Invalid. Countermodel: Domain: {1} Extensions: B : {1} C : {1} D : ∅ R : ∅ [Q p.46] 8. Fx: x is a fish Sx: x swims ∀x(Sx → ∃yFy) ∴ ∃x¬Sx ∀x(Sx → ∃yFy) \abc ¬∃x¬Sx ∀x¬¬Sx \abc ¬¬Sa X Sa Sa→ ∃yFy X ¬Sa × ∃yFy Xb Fb ¬¬Sb X Sb Sb→ ∃yFy X ¬Sb × ∃yFy Xc Fc ¬¬Sc X Sc Sc→ ∃yFy X ¬Sc × ∃yFy Xd Fd ... Invalid. Countermodel: Domain: {1, 2, 3, . . . } Extensions: F : {2, 3, 4, . . . } S : {1, 2, 3, . . . } [Q p.46] 187 9. Bx: x was built before 1970 Kx: x runs on kerosene Rx: x is a robot a: Autovac 23E ∀x((Rx ∧ Bx)→ Kx) Ba ∧ ¬Ka ∴ ¬Ra ∀x((Rx ∧ Bx)→ Kx) \a Ba ∧ ¬Ka X ¬¬Ra X Ra Ba ¬Ka (Ra ∧ Ba)→ Ka X ¬(Ra ∧ Ba) X ¬Ra × ¬Ba × Ka × Valid. [Q p.46] 188 10. Ax: x is an athlete Ix: x is an intellectual Px: x is a person Tx: x is tall g: Graham ∀x((Tx ∧ Px)→ (Ax ∨ Ix)) ∃x(Px ∧ Ax ∧ Ix) ∀x((Px ∧ Ax ∧ Ix)→ ¬Tx) Pg ∴ Ag→ (¬Ig ∨ ¬Tg) ∀x((Tx ∧ Px)→ (Ax ∨ Ix)) ∃x(Px ∧ Ax ∧ Ix) ∀x((Px ∧ Ax ∧ Ix)→ ¬Tx) \g Pg ¬(Ag→ (¬Ig ∨ ¬Tg)) X Ag ¬(¬Ig ∨ ¬Tg) X ¬¬Ig X ¬¬Tg X Ig Tg (Pg ∧ Ag ∧ Ig)→ ¬Tg X ¬(Pg ∧ Ag ∧ Ig) X ¬Pg × ¬Ag × ¬Ig × ¬Tg × Valid. [Q p.46] [Contents] 189 Chapter 11 Models, Propositions, and Ways the World Could Be There are no exercises for chapter 11. [Contents] 190 Chapter 12 General Predicate Logic Answers 12.1.3.1 1. Yes [Q p.48] 2. Yes [Q p.48] 3. Yes [Q p.48] 4. Yes [Q p.48] 5. No [Q p.48] 6. No [Q p.48] 7. No [Q p.48] 8. Yes [Q p.48] 9. No [Q p.48] 10. Yes [Q p.48] [Contents] 191 Answers 12.1.6 Glossary: a: Alice b: Bill c: Clare d: Dave e: Edward f : Fiesta j: The Bell Jar m: Mary t: the Eiffel tower Tx: x is tall Hxy: x heard y Lxy: x likes y Rxy: x has read y Txy: x is taller than y Pxyz: x prefers y to z Translations: 1. Hba [Q p.49] 2. ¬Hba [Q p.49] 3. Hba ∧ ¬Hab [Q p.49] 4. Hba→ Hab [Q p.49] 5. Hba↔ Haa [Q p.49] 6. Hba ∨ Hab [Q p.49] 7. Tcd ∧ ¬Tce [Q p.49] 8. Pmac [Q p.49] 9. ¬Pmdc ∧ ¬Pmcd [Q p.49] 10. Tec ∧ ¬Te [Q p.49] 11. Ttc ∧ Ttd [Q p.49] 12. Tdt→ Td [Q p.49] 192 13. Ttd ∧ Pcdt [Q p.49] 14. Tad→ Pdda [Q p.49] 15. Pdec→ Tet [Q p.49] 16. Pdec→ ¬Tc [Q p.49] 17. Rm f ∧ Lm f [Q p.49] 18. ¬Ld f ∧ ¬Rd f [Q p.49] 19. ¬Ldj→ ¬Rdj [Q p.49] 20. Pdj f ∧ ¬Rdj ∧ ¬Rd f [Q p.49] [Contents] Answers 12.1.9 1. Glossary: a: Alice b: Bill Cx: x is a chair Bx: x is broken Rx: x is a room Bxy: x is bigger than y Cxy: x contains y Translations: (i) ∃x∀yBxy [Q p.50] (ii) ∃x∀yByx [Q p.50] (iii) Bab→ ∃xBxb [Q p.50] (iv) ∀xBxb→ Bab [Q p.50] (v) ∃x∀yBxy→ ∃xBxx [Q p.50] (vi) (Bab ∧ Bba)→ ∀xBxx [Q p.50] (vii) ∃x∀y(Bay→ Bxy) [Q p.50] (viii) ∀x(Bxa→ ∀y(Bay→ Bxy)) [Q p.50] (ix) ∀x(Rx → ∃y(Cy ∧ Cxy)) [Q p.50] 193 (x) ∃x(Rx ∧ ∃y(Cy ∧ Cxy ∧ By)) ∧ ∃x(Rx ∧ ∀y((Cy ∧ Cxy)→ By)) ∧ ¬∃x(Rx ∧ ∀y((Cy ∧ Cxy)→ ¬By)) [Q p.50] 2. Glossary: Bx: x is a beagle Cx: x is a chihuahua Dx: x is a dog Px: x is a person Bxy: x is bigger than y Hxy: x is happier than y Oxy: x owns y Translations: (i) ∀x(Px → ∃y(Dy ∧Oxy)) [Q p.50] (ii) ∀x(Dx → ∃y(Py ∧Oyx)) [Q p.50] (iii) ∃x∃y(Bx ∧ Cy ∧Oxy) [Q p.50] (iv) ¬∃x(Bx ∧Oxx) [Q p.50] (v) ¬∃x∃y(Bx ∧ Cy ∧ Byx) [Q p.50] (vi) ∃x∃y(Cx ∧ By ∧ Bxy) [Q p.50] (vii) ∃x(Dx ∧ ∀y(Py→ Hxy)) [Q p.50] (viii) ∀x∀y((Px ∧ Py ∧ ∃z(Dz ∧Oxz) ∧ ¬∃w(Dw ∧Oyw))→ Hxy) [Q p.50] (ix) ∀x∀y((Dx ∧ Dy)→ (Bxy→ Hxy)) [Q p.50] (x) ∃x(Bx ∧ ∀y(Cy→ Bxy) ∧ ∀z(Pz→ Bzx)) [Q p.50] 194 3. Glossary: a: Alice b: Bill o: Woolworths Cx: x is a cat Dx: x is a dog Gx: x is grey Tx: x is timid Bxy: x is bigger than y Gxy: x growls at y Wxyz: x wants to buy y from z Translations: (i) Ta ∧ Da ∧ ∃x(Cx ∧ Bxa) [Q p.51] (ii) ∀x((Dx ∧ Bxa)→ Bxb) [Q p.51] (iii) Tb ∧ Cb ∧ ∀x(Dx → Bxb) [Q p.51] (iv) ∀x((Tx ∧ Dx)→ ∃y(Gy ∧ Cy ∧ Gxy)) [Q p.51] (v) ∀x(Dx → ∀y((Ty ∧ Cy)→ Gxy)) [Q p.51] (vi) ∃x(Tx ∧ Dx ∧ ∀y((Gy ∧ Cy)→ Gxy)) [Q p.51] (vii) ¬∃x∃y(Tx ∧ Dx ∧ Gy ∧ Cy ∧ Gxy) [Q p.51] (viii) ∃xWaxo ∧ ¬∃xWbxo [Q p.51] (ix) ∃x(Waxo ∧ ¬Wbxo) [Q p.51] (x) ∀x(Waxo → Gbx) [Q p.51] 195 4. Glossary: d: Dave e: Elvis f : Frank r: the Rolling Stones Px: x is a person Sx: x is a song Tx: x was in the top twenty Axy: x is admires y Rxy: x recorded y Pxyz: x prefers y to z Translations: (i) ∀x(Px → Adx) [Q p.51] (ii) ¬∃x(Px ∧ Axd) [Q p.51] (iii) ¬Add [Q p.51] (iv) ¬∃x(Px ∧ Axx) [Q p.51] (v) ∀x((Px ∧ ¬Axx)→ Adx) [Q p.51] (vi) ∀x((Px ∧ Axx)→ Axd) [Q p.51] (vii) A f e ∧ P f re [Q p.51] (viii) ∀x∀y((Sx ∧ Rrx ∧ Sy ∧ Rey)→ P f xy) [Q p.51] (ix) ∃x(Sx ∧ Tx ∧ Rrx) ∧ ¬∃x(Sx ∧ Tx ∧ Rex) [Q p.51] (x) ∀x∀y((Sx ∧ Tx ∧ Rrx ∧ Sy ∧ Rey)→ Pexy) [Q p.51] [Contents] Answers 12.2.2 1. (i) False [Q p.52] (ii) True [Q p.52] (iii) False [Q p.52] (iv) True (Ldb is true if we let the new name d refer to 1.) [Q p.52] (v) False (The extension of L contains no ordered pair whose sec- ond member is 1, which is the referent of a; so no matter what the new name d refers to, Lda is false.) [Q p.52] 196 (vi) False (The extension of L contains no ordered pair which has the same object in both first and second place, so there is no possible choice of referent for the new name d which makes Ldd true.) [Q p.52] (vii) True (No matter what we pick as the referent of d, we can then pick a referent for e such that Lde is true.) [Q p.52] (viii) False (If we pick 1 as the referent of d, then we cannot pick a referent for e such that Led is true.) [Q p.52] (ix) False (There is no object in the domain which is both in the ex- tension of P, and in the first place of an ordered pair in the ex- tension of L which has 2, which is the referent of b, in second place; so there is no possible choice of referent for the new name d which makes both Pd and Ldb true.) [Q p.52] (x) True [Q p.52] (xi) True [Q p.52] (xii) True [Q p.52] (xiii) False [Q p.52] (xiv) True [Q p.52] (xv) False [Q p.52] (xvi) False [Q p.52] (xvii) False [Q p.52] (xviii) True [Q p.52] 2. (i) False [Q p.53] (ii) False [Q p.53] (iii) False [Q p.53] (iv) True [Q p.53] (v) True [Q p.53] (vi) False [Q p.53] (vii) True [Q p.53] (viii) False [Q p.53] (ix) False [Q p.53] (x) True [Q p.53] 3. (i) False [Q p.53] 197 (ii) True [Q p.53] (iii) True [Q p.53] (iv) True [Q p.54] (v) False (e.g. 〈Alice, Bob〉 and 〈Bob, Alice〉 are in the extension of S but 〈Alice, Alice〉 isn’t. ) [Q p.54] (vi) False [Q p.54] (vii) False [Q p.54] (viii) False [Q p.54] (ix) True [Q p.54] (x) True [Q p.54] 4. (i) (a) Domain: {1, 2, 3} Extension of F : {〈1, 1〉, 〈2, 2〉, 〈3, 3〉} (b) Domain: {1, 2, 3} Extension of F : {〈1, 1〉, 〈2, 2〉} [Q p.54] (ii) (a) Domain: {1, 2, 3} Extension of F : {〈1, 2〉, 〈2, 1〉} (b) Domain: {1, 2, 3} Extension of F : {〈1, 2〉} [Q p.54] (iii) (a) Domain: {1, 2, 3} Extension of F : {〈1, 1〉, 〈2, 3〉, 〈3, 2〉} (b) Domain: {1, 2, 3} Extension of F : {〈1, 1〉, 〈2, 3〉} [Q p.54] (iv) (a) Domain: {1, 2, 3} Extension of F : {〈1, 1〉, 〈1, 2〉, 〈1, 3〉} (b) Domain: {1, 2, 3} Extension of F : {〈1, 2〉, 〈1, 3〉} [Q p.54] (v) (a) Domain: {1, 2, 3} Extension of F : {〈1, 2〉, 〈2, 3〉, 〈3, 1〉} (b) Domain: {1, 2, 3} Extension of F : {〈1, 1〉, 〈1, 2〉, 〈1, 3〉} [Q p.54] (vi) (a) Domain: {1, 2, 3} Extension of F : {〈1, 1〉} (b) Domain: {1, 2, 3} Extension of F : ∅ [Q p.54] 198 (vii) (a) Domain: {1} Extension of F : {〈1, 1〉} (b) Domain: {1} Extension of F : {∅} [Q p.54] (viii) (a) Domain: {1, 2, 3} Referent of a: 1 Extension of F : {〈1, 2〉} (b) Domain: {1, 2, 3} Referent of a: 1 Extension of F : {〈1, 1〉} [Q p.54] (ix) (a) No such model. For the proposition to be true, the ordered pair consisting of the referent of a, followed by the referent of a, would have to both be in the extension of F and not in the extension of F (b) Domain: {1, 2, 3} Referent of a: 1 Extension of F : {〈1, 1〉} [Q p.54] (x) (a) No such model. For the second and third conjuncts to be true, the extension of F must contain the ordered pair of the referent of a followed by the referent of b, but must not contain the ordered pair obtained by switching the order of these two referents. This is incompatible with the first conjunct. (b) Domain: {1, 2, 3} Referents: a: 1 b: 2 Extension of F : {〈1, 1〉} [Q p.54] [Contents] 199 Answers 12.3.1 1. (i) ¬∀x(Rxx → ∃yRxy) X ∃x¬(Rxx → ∃yRxy) Xa ¬(Raa→ ∃yRay) X Raa ¬∃yRay X ∀y¬Ray \a ¬Raa × Logical truth. [Q p.54] (ii) ¬∀x(∃yRxy→ ∃zRzx) X ∃x¬(∃yRxy→ ∃zRzx) Xa ¬(∃yRay→ ∃zRza) X ∃yRay Xb ¬∃zRza X ∀z¬Rza \ab Rab ¬Raa ¬Rba ↑ Not a logical truth. Countermodel: Domain: {1, 2} Extension of R : {〈1, 2〉} [Q p.54] (iii) ¬(∀xRax → ∀x∃yRyx) X ∀xRax \b ¬∀x∃yRyx X ∃x¬∃yRyx Xb ¬∃yRyb X ∀y¬Ryb \a Rab ¬Rab × Logical truth. [Q p.54] 200 (iv) ¬(∀x∃y∃zRyxz→ ∃x∃yRxay) ∀x∃y∃zRyxz \a ¬∃x∃yRxay X ∀x¬∃yRxay \b ∃y∃zRyaz Xb ∃zRbaz Xc Rbac ¬∃yRbay X ∀y¬Rbay \c ¬Rbac × Logical truth. [Q p.54] (v) ¬¬∀x∃yRxy X ∀x∃yRxy \abc ∃yRay Xb Rab ∃yRby Xc Rbc ∃yRcy Xd Rcd ... ↑ Not a logical truth. Countermodel: Domain: {1, 2, 3, . . . } Extension of R : {〈1, 2〉, 〈2, 3〉, 〈3, 4〉, . . . } [Q p.55] 201 (vi) ¬∀x∀y∀z((Rxy ∧ Ryz)→ Rxz) X ∃x¬∀y∀z((Rxy ∧ Ryz)→ Rxz) Xa ¬∀y∀z((Ray ∧ Ryz)→ Raz) X ∃y¬∀z((Ray ∧ Ryz)→ Raz) Xb ¬∀z((Rab ∧ Rbz)→ Raz) X ∃z¬((Rab ∧ Rbz)→ Raz) Xc ¬((Rab ∧ Rbc)→ Rac) X Rab ∧ Rbc X ¬Rac Rab Rbc ↑ Not a logical truth. Countermodel: Domain: {1, 2, 3} Extension of R : {〈1, 2〉, 〈2, 3〉} [Q p.55] (vii) ¬(∃x∀yRxy→ ∀x∃yRxy) X ∃x∀yRxy Xa ¬∀x∃yRxy X ∃x¬∃yRxy Xb ∀yRay \ab ¬∃yRby X ∀y¬Rby \ab Raa Rab ¬Rba ¬Rbb ↑ Not a logical truth. Countermodel: Domain: {1, 2} Extension of R : {〈1, 1〉, 〈1, 2〉} [Q p.55] 202 (viii) ¬(∃y∀xRxy→ ∀x∃yRxy) X ∃y∀xRxy Xa ¬∀x∃yRxy X ∃x¬∃yRxy Xb ∀xRxa \b ¬∃yRby X ∀y¬Rby \a Rba ¬Rba × Logical truth. [Q p.55] (ix) ¬(∃x∀yRxy→ ∃x∃yRxy) X ∃x∀yRxy Xa ¬∃x∃yRxy X ∀x¬∃yRxy \a ∀yRay \a ¬∃yRay X ∀y¬Ray \a Raa ¬Raa × Logical truth. [Q p.55] 203 (x) ¬(∀x∀y∃zRxyz ∨ ∀x∀y∀z¬Rxyz) X ¬∀x∀y∃zRxyz X ¬∀x∀y∀z¬Rxyz X ∃x¬∀y∃zRxyz Xa ∃x¬∀y∀z¬Rxyz Xb ¬∀y∃zRayz X ∃y¬∃zRayz Xc ¬∀y∀z¬Rbyz X ∃y¬∀z¬Rbyz Xd ¬∃zRacz X ∀z¬Racz \abcde ¬∀z¬Rbdz X ∃z¬¬Rbdz Xe ¬¬Rbde X Rbde ¬Raca ¬Racb ¬Racc ¬Racd ¬Race ↑ Not a logical truth. Countermodel: Domain: {1, 2, 3, 4, 5} Extension of R : {〈2, 4, 5〉} [Q p.55] 2. (i) ∀x∀y∀z((Rxy ∧ Ryz)→ Rxz) \a Rab Rba ¬∃xRxx X ∀x¬Rxx \a ∀y∀z((Ray ∧ Ryz)→ Raz) \b ∀z((Rab ∧ Rbz)→ Raz) \a (Rab ∧ Rba)→ Raa X ¬(Rab ∧ Rba) ¬Rab × ¬Rba × Raa ¬Raa × Valid. [Q p.55] 204 (ii) ∀xFxa→ ∃xFax X ∃xFxa Xb ¬∃xFax X ∀x¬Fax \abc ¬∀xFxa X ∃x¬Fxa Xc Fba ¬Fca ¬Faa ¬Fab ¬Fac ↑ ∃xFax Xc Fac ¬Fac × Invalid. Countermodel: Domain: {1, 2, 3} Referent of a : 1 Extension of F : {〈2, 1〉} [Q p.55] (iii) ∃x∃y∃z(Rxy ∧ Rzy) Xa ¬∃xRxx X ∀x¬Rxx \abc ∃y∃z(Ray ∧ Rzy) Xb ∃z(Rab ∧ Rzb) Xc Rab ∧ Rcb X Rab Rcb ¬Raa ¬Rbb ¬Rcc ↑ Invalid. Countermodel: Domain: {1, 2, 3} Extension of F : {〈1, 2〉, 〈3, 2〉} [Q p.55] 205 (iv) ∀x∀y(Rxy→ Ryx) \b ∃xRxa Xb ¬∃xRax X ∀x¬Rax \b Rba ∀y(Rby→ Ryb) \a Rba→ Rab X ¬Rba × Rab ¬Rab × Valid. [Q p.55] (v) ∀x∀y(¬Rxy→ Ryx) \a ¬∀x∃yRyx X ∃x¬∃yRyx Xa ¬∃yRya X ∀y¬Rya \a ¬Raa ∀y(¬Ray→ Rya) \a ¬Raa→ Raa X ¬¬Raa Raa × Raa × Valid. [Q p.55] (vi) ∀x∀y(Rxy→ (Fx ∧ Gy)) \a ¬¬∃xRxx X ∃xRxx Xa Raa ∀y(Ray→ (Fa ∧ Gy)) \a Raa→ (Fa ∧ Ga) X ¬Raa Fa ∧ Ga X Fa Ga ↑ Invalid. Countermodel: Domain: {1} Extensions: F : {1} G : {1} R : {〈1, 1〉} [Q p.55] 206 (vii) ∀x(Fx → (∀yRxy ∨ ¬∃yRxy)) \a Fa ¬Rab ¬¬Raa X Raa Fa→ (∀yRay ∨ ¬∃yRay) X ¬Fa × ∀yRay ∨ ¬∃yRay X ∀yRay \b Rab × ¬∃yRay X ∀y¬Ray \a ¬Raa × Valid. [Q p.55] (viii) ∀x∀y(∃z(Rzx ∧ Rzy)→ Rxy) \b ∀xRax \bc ¬∀x∀yRxy X ∃x¬∀yRxy Xb ¬∀yRby X ∃y¬Rby Xc ¬Rbc ∀y(∃z(Rzb ∧ Rzy)→ Rby) \c ∃z(Rzb ∧ Rzc)→ Rbc X ¬∃z(Rzb ∧ Rzc) X ∀z¬(Rzb ∧ Rzc) \a ¬(Rab ∧ Rac) X Rab Rac ¬Rab × ¬Rac × Rbc × Valid. [Q p.55] 207 (ix) ∀x∃yRxy \bacde ¬∃xRxb X ∀x¬Rxb \b ¬Rbb ∃yRby Xa Rba ∃yRay Xc Rac ∃yRcy Xd Rcd ∃yRdy Xe Rde ∃yRey X f Re f ... ↑ Invalid. Countermodel: Domain: {1, 2, 3, 4, 5, . . . } Referent of b: 2 Extension of R : {〈2, 1〉, 〈1, 3〉, 〈3, 4〉, 〈4, 5〉, 〈5, 6〉, . . . } [Q p.55] (x) ∃x∀y(Fy→ Rxy) Xa ∃x∀y¬Ryx Xb ¬∃x¬Fx X ∀x¬¬Fx \b ∀y(Fy→ Ray) \b ∀y¬Ryb \a ¬Rab ¬¬Fb X Fb Fb→ Rab X ¬Fb × Rab × Valid. [Q p.56] 208 3. (i) a: Alice b: Bill c: Carol Oxy: x is older than y Oab Obc ∴ Oac Oab Obc ¬Oac ↑ Invalid. Countermodel: Domain: {1, 2, 3} Referents: a : 1 b : 2 c : 3 Extension of O : {〈1, 2〉, 〈2, 3〉} [Q p.56] (ii) a: Alice b: Bill c: Carol Oxy: x is older than y Oab Obc ∀x∀y∀z((Oxy ∧Oyz)→ Oxz)) ∴ Oac Oab Obc ∀x∀y∀z((Oxy ∧Oyz)→ Oxz)) \a ¬Oac ∀y∀z((Oay ∧Oyz)→ Oaz)) \b ∀z((Oab ∧Obz)→ Oaz)) \c ((Oab ∧Obc)→ Oac)) X ¬(Oab ∧Obc) X ¬Oab × ¬Obc × Oac × Valid. [Q p.56] 209 (iii) a: me b: you d: Dave Bx: x is a banker Txy: x trusts y ∀x(Tbx → Tax) ∀x(Bx → Tbx) Bd ∴ Tad ∀x(Tbx → Tax) \d ∀x(Bx → Tbx) \d Bd ¬Tad Bd→ Tbd X ¬Bd × Tbd Tbd→ Tad X ¬Tbd × Tad × Valid. [Q p.56] 210 (iv) Px: x is a person Lxy: x loves y ∀x(Px → ∃y(Py ∧ Lxy)) ∴ ∀x(Px → ∃y(Py ∧ Lyx)) ∀x(Px → ∃y(Py ∧ Lxy)) \ab ¬∀x(Px → ∃y(Py ∧ Lyx)) X ∃x¬(Px → ∃y(Py ∧ Lyx)) Xa ¬(Pa→ ∃y(Py ∧ Lya)) X Pa ¬∃y(Py ∧ Lya) X ∀y¬(Py ∧ Lya) \ab ¬(Pa ∧ Laa) X ¬Pa × ¬Laa Pa→ ∃y(Py ∧ Lay) X ¬Pa × ∃y(Py ∧ Lay) Xb Pb ∧ Lab X Pb Lab ¬(Pb ∧ Lba) X ¬Pb × ¬Lba Pb→ ∃y(Py ∧ Lby) X ¬Pb × ∃y(Py ∧ Lby) Xc Pc ∧ Lbc X Pc Lbc ... ↑ Invalid. Countermodel: Domain: {1, 2, 3 . . . } Extensions: L : {〈1, 2〉, 〈2, 3〉, 〈3, 4〉 . . . } P : {1, 2, 3, . . . } [Q p.56] 211 (v) n: Nancy Rx: x is a restaurateur Axy: x can afford to feed y Wx: x is very wealthy Rn ∀x(Rx → (Anx ↔ ¬Axx)) ∴ Wn Rn ∀x(Rx → (Anx ↔ ¬Axx)) \n ¬Wn Rn→ (Ann↔ ¬Ann) X ¬Rn × Ann↔ ¬Ann X Ann ¬Ann × ¬Ann ¬¬Ann X Ann × Valid. [Q p.56] 212 (vi) e: the Eiffel tower l: Lake Burley Griffin Cx: x is in Canberra Px: x is in Paris Bxy: x is more beautiful than y ∀x∀y((Px ∧ Cy)→ Bxy) Pe ∧ Cl ∴ Bel ∀x∀y((Px ∧ Cy)→ Bxy) \e Pe ∧ Cl X ¬Bel Pe Cl ∀y((Pe ∧ Cy)→ Bey) \l ((Pe ∧ Cl)→ Bel) X ¬(Pe ∧ Cl) X ¬Pe × ¬Cl × Bel × Valid. [Q p.56] 213 (vii) Jx: x is a journalist Px: x is a politician Txy: x talks to y ∀x∀y((Px ∧ Txy)→ Py) ¬∃x(Px ∧ Jx) ∴ ¬∃x∃y(Px ∧ Jy ∧ Txy) ∀x∀y((Px ∧ Txy)→ Py) \a ¬∃x(Px ∧ Jx) X ¬¬∃x∃y(Px ∧ Jy ∧ Txy) X ∃x∃y(Px ∧ Jy ∧ Txy) Xa ∃y(Pa ∧ Jy ∧ Tay) Xb Pa ∧ Jb ∧ Tab X Pa Jb Tab ∀x¬(Px ∧ Jx) \ab ¬(Pa ∧ Ja) X ¬Pa × ¬Ja ¬(Pb ∧ Jb) X ¬Pb ∀y((Pa ∧ Tay)→ Py) \b (Pa ∧ Tab)→ Pb X ¬(Pa ∧ Tab) X ¬Pa × ¬Tab × Pb × ¬Jb × Valid. [Q p.56] 214 (viii) Sxy: x is smaller than y ¬∃x∀ySxy ∴ ¬∃x∀ySyx ¬∃x∀ySxy X ¬¬∃x∀ySyx X ∃x∀ySyx Xa ∀x¬∀ySxy Xabcd ∀ySya \abcd Saa ¬∀ySay X ∃y¬Say Xb ¬Sab Sba ¬∀ySby X ∃y¬Sby Xc ¬Sbc Sca ¬∀yScy X ∃y¬Scy Xd ¬Scd Sda ¬∀ySdy X ∃y¬Sdy ... ↑ Invalid. Countermodel: Domain: {1, 2, 3, 4, . . . } Extension of S : {〈1, 1〉, 〈2, 1〉, 〈3, 1〉, 〈4, 1〉 . . . } [Q p.56] 215 (ix) d: David m: Margaret Fx: x is French Mx: x is a movie Sx: x is commercially successful Lxy: x likes y ∀x(Mx → (¬Sx ∨ (Lmx ∧ Ldx))) ¬∃x(Fx ∧Mx ∧ Lmx ∧ Ldx) ∴ ¬∃x(Fx ∧Mx ∧ Sx) ∀x(Mx → (¬Sx ∨ (Lmx ∧ Ldx))) \a ¬∃x(Fx ∧Mx ∧ Lmx ∧ Ldx) X ¬¬∃x(Fx ∧Mx ∧ Sx) X ∃x(Fx ∧Mx ∧ Sx) Xa ∀x¬(Fx ∧Mx ∧ Lmx ∧ Ldx) \a Fa ∧Ma ∧ Sa X Fa Ma Sa Ma→ (¬Sa ∨ (Lma ∧ Lda)) X ¬Ma × ¬Sa ∨ (Lma ∧ Lda) X ¬Sa × Lma ∧ Lda X Lma Lda ¬(Fa ∧Ma ∧ Lma ∧ Lda) X ¬Fa × ¬Ma × ¬Lma × ¬Lda × Valid. [Q p.56] 216 (x) Cxy: x causes y ∃x∀yCxy ∴ ¬∃x∀yCyx ∃x∀yCxy Xa ¬¬∃x∀yCyx X ∃x∀yCyx Xb ∀yCay \ab ∀yCyb \ab Caa Cab Cab Cbb ↑ Invalid. Countermodel: Domain: {1, 2} Extension of C : {〈1, 1〉, 〈1, 2〉, 〈2, 2〉} [Q p.56] [Contents] Answers 12.4.1 The trees are not given in these answers. 1. Glossary: e: that egg r: Roger Fx: x is a food Exy: x will eat y Translation: ∀x(Fx → Erx) ∴ Ere Postulate: Fe [Q p.57] 217 2. Glossary: a: 180 b: 170 l: Bill n: Ben Gxy: x is greater than y Wxy: x weighs y pounds Vxy: x is heavier than y Translation: Wla Wnb ∴ Vln Postulates: Gab ∀x∀y∀z∀w((Wxy ∧Wzw ∧ Gyw)→ Vxz) [Q p.57] 3. Glossary: a: 5 b: 10 j: John n: Nancy Fxy: x ran further than y Gxy: x is greater than y Rxy: x ran y miles Translation: Rja Rnb ∴ Fnj Postulates: Gba ∀x∀y∀z∀w((Rxy ∧ Rzw ∧ Gyw)→ Fxz) [Q p.57] 218 4. Glossary: b: Buddenbrooks s: Sophie t: Thomas Mann Nx: x is a novel Axy: x is the author of y Exy: x enjoys y Translation: ∀x((Nx ∧ Atx)→ Esx) ∴ Esb Postulates: Nb Atb [Q p.57] 5. Glossary: b: Borges c: Chris Nx: x is a novel Axy: x is the author of y Exy: x enjoys y Translation: ∀x(Ecx ↔ Nx) ∴ ¬∃x(Abx ∧ Ecx) Postulate: ¬∃x(Nx ∧ Abx) [Q p.57] [Contents] 219 Answers 12.5.4 Note that a formula may have more than one prenex equivalent (i.e. there may be other correct answers). 1. ∀x∀y(Px ∨Qy) [Q p.57] 2. ∃x∃y(Px ∨Qy) [Q p.57] 3. ∃x∀y(Px → Py) [Q p.57] 4. ∃x∀y∃z∀w((Pz→ Pw) ∧ (Px → Py)) [Q p.57] 5. ∃x∃y∀z¬(Sx ∧ (Ty→ Uxz)) [Q p.57] [Contents] 220 Chapter 13 Identity Answers 13.2.2 Glossary a: Adam Hx: x is happy b: Ben Mx: x is a man c: Chris Ox: x is a town d: Diane Px: x is a person e: Edward Tx: x is a television show f : Family Guy Wx: x is a woman g: you Cxy: x is colder than y h: Sydney Kxy: x knows y i: Jindabyne Lxy: x is larger than y j: Jonathon Oxy: x owns y k: Melbourne Sxy: x is by y’s side l: Canberra Txy: x is taller than y m: Mary Vxy: x watches y n: I/me Wxy: x wants y s: Seinfeld Bxyz: y is between x and z Cx: x is a chihuahua Pxyz: x prefers y to z Dx: x is a dog Translations: 1. ∀x(x 6= c→ Lcx) [Q p.58] 2. ∀x((x 6= c ∧ Dx)→ Bx) ∧ Cc [Q p.58] 3. ∃x(Dx ∧ Sxb ∧ x 6= c)→ Hb [Q p.58] 4. ∀x((Px ∧ Scx ∧ x 6= j)→ Hc) [Q p.58] 221 5. ∀x(Dx → Ljx) [Q p.58] 6. ∀x(Wmx → ∃y(Py ∧ y 6= m ∧Oyx)) [Q p.58] 7. ∃x∃y(Px ∧ x 6= m ∧Wxy ∧Omy) [Q p.58] 8. ∃x(Omx ∧ ¬Wmx) [Q p.58] 9. ∃x(Bx ∧Omx)→ ∀y∀z((Py ∧ Bz ∧ y 6= m)→ ¬Oyz) [Q p.58] 10. ∀x∀y((Px ∧ x 6= m ∧Wmy)→ ¬Oxy) [Q p.58] 11. ∀x(Px → Pxs f ) [Q p.58] 12. ∀x((Tx ∧ x 6= s)→ Pasx) [Q p.58] 13. ∀x((Tx ∧ x 6= f )→ Pax f ) [Q p.58] 14. Vj f ∧ ∀x((Tx ∧ x 6= f )→ ¬Vjx) [Q p.59] 15. Vj f ∧ ∀x((Px ∧ x 6= j)→ ¬Vx f ) [Q p.59] 16. Wd ∧ ∀x((Wx ∧ x 6= d)→ Tdx) [Q p.59] 17. ∀x((Mx ∧ Txd)↔ x = e) [Q p.59] 18. Ted ∧ ∃x(Wx ∧ x 6= d ∧ Tex) [Q p.59] 19. ¬∃x(Px ∧ Tdx ∧ Txe) [Q p.59] 20. ∃x(Px ∧ x 6= e ∧ x 6= d) [Q p.59] 21. ∀x((Px ∧ Kxb)↔ x = g) [Q p.59] 22. ∃x(Px ∧ Knx ∧ x 6= b) [Q p.59] 23. ∀x((Kbx ∧ Px ∧ x 6= c ∧ x 6= n)→ Hx) [Q p.59] 24. ∀x((Px ∧ Hx ∧ Knx)↔ x = b) [Q p.59] 25. Pb ∧ Hb ∧ Knb ∧ ∀x((Px ∧ Hx ∧ Knx ∧ x 6= b)→ Tbx) [Q p.59] 26. Oi ∧ Bhik ∧ ∀x((Ox ∧ Bhxk ∧ x 6= i)→ Cix) [Q p.59] 27. ∃x(Ox ∧ Bhxk ∧ Cxl) [Q p.59] 28. ∀x((Ox ∧ x 6= i)→ ∃y(Oy ∧ Cyx)) [Q p.59] 29. ¬∃x(Ox ∧ Bhxk ∧ (Lxl ∨ Cxi)) [Q p.59] 222 30. Oi ∧ Bhik ∧ ∀x((Ox ∧ Bhxk ∧ x 6= i)→ Pnix) [Q p.59] [Contents] Answers 13.3.1 1. (i) True [Q p.60] (ii) False [Q p.60] (iii) True [Q p.60] (iv) True [Q p.60] (v) False [Q p.60] (vi) True [Q p.60] 2. (i) False [Q p.60] (ii) True [Q p.60] (iii) True [Q p.60] (iv) True [Q p.60] (v) True [Q p.60] (vi) False [Q p.60] 3. (i) (a) Domain: {1, 2, 3} Referent of a : 1 Extension of F : {1} (b) Domain: {1, 2, 3} Referent of a : 1 Extension of F : {1, 2} [Q p.61] (ii) (a) Domain: {1, 2} Referents: a : 1 b : 1 (b) Domain: {1, 2} Referents: a : 1 b : 2 [Q p.61] (iii) (a) Domain: {1, 2, 3} Extension of R : {〈1, 2〉, 〈1, 3〉} (b) Domain: {1, 2, 3} Extension of R : {〈1, 2〉} [Q p.61] 223 (iv) (a) Domain: {1, 2, 3} Extension of R : {〈1, 1〉, 〈2, 2〉} (b) Domain: {1, 2, 3} Extension of R : {〈1, 1〉, 〈1, 2〉} [Q p.61] (v) (a) Domain: {1, 2, 3} Extension of R : {〈1, 2, 1〉, 〈1, 3, 1〉, 〈2, 1, 2〉, 〈2, 3, 2〉, 〈3, 1, 3〉, 〈3, 2, 3〉} (b) Domain: {1, 2, 3} Extension of R : {〈1, 2, 1〉, 〈1, 3, 1〉} [Q p.61] (vi) (a) No such model: symmetry of identity. (b) Domain: {1, 2, 3} Referent of a : 1 [Q p.61] (vii) (a) Domain: {1, 2, 3} Extension of F : {1} (b) Domain: {1, 2, 3} Extension of F : {1, 2} [Q p.61] (viii) (a) Domain: {1, 2, 3} Extensions: F : {1, 2} G : {1} (b) Domain: {1, 2, 3} Extensions: F : {1} G : {1, 2} [Q p.61] (ix) (a) Domain: {1, 2, 3} Extensions: F : {1, 2} R : {〈1, 2〉, 〈2, 1〉} (b) Domain: {1, 2, 3} Extensions: F : {1, 2} R : {〈1, 2〉, 〈2, 2〉} [Q p.61] (x) (a) Domain: {1, 2, 3} Referents: a : 1 Extensions: F : {1} R : ∅ (b) Domain: {1, 2, 3} Referents: a : 1 Extensions: F : {1} R : {〈1, 1〉} [Q p.61] (xi) (a) Domain: {1, 2, 3} Extensions: R : {〈1, 2, 3〉} (b) Domain: {1, 2, 3} Extensions: R : {〈1, 2, 1〉} [Q p.61] (xii) (a) Domain: {1, 2, 3} Extensions: R : {〈1, 2, 3〉} (b) Domain: {1, 2, 3} Extensions: R : {〈1, 2, 1〉} [Q p.61] 224 (xiii) (a) Domain: {1, 2, 3} Extensions: F : {2, 3} (b) Domain: {1, 2, 3} Extensions: F : {2} [Q p.61] (xiv) (a) Domain: {1, 2, 3} Extensions: F : {1, 2} R : {〈1, 2〉} (b) Domain: {1, 2, 3} Extensions: F : {1, 2} R : {〈1, 1〉} [Q p.61] (xv) (a) Domain: {1, 2, 3} Extensions: R : {〈1, 2, 3〉, 〈1, 1, 1〉, 〈2, 2, 2〉, 〈3, 3, 3〉} (b) Domain: {1, 2, 3} Extensions: R : {〈1, 2, 3〉, 〈1, 1, 1〉, 〈2, 2, 2〉} [Q p.61] (xvi) (a) Domain: {1, 2, 3} Extensions: R : {〈1, 1〉} (b) No such model: the only y identical to x is x itself. [Q p.61] (xvii) (a) Domain: {1, 2, 3} Referents: a : 1 b : 1 Extensions: F : {1} (b) Domain: {1, 2, 3} Referents: a : 1 b : 2 Extensions: F : {1, 2} [Q p.61] (xviii) (a) Domain: {1, 2, 3} Extensions: F : {1, 2} (b) Domain: {1, 2, 3} Extensions: F : {1, 2, 3} [Q p.61] [Contents] 225 Answers 13.4.3 1. (i) Rab→ ¬Rba X Rab a = b ¬Rab × ¬Rba Raa Rba × Unsatisfiable. [Q p.61] (ii) Rab ¬Rbc a = b Raa Rba Rbb ¬Rac ↑ Satisfiable. Domain: {1, 2} Referents: a : 1 b : 1 c : 2 Extension of R : {〈1, 1〉} [Q p.61] 226 (iii) ∀x(Fx → x = a) \ab Fa a 6= b Fa→ a = a X ¬Fa × a = a Fb→ b = a X ¬Fb ↑ b = a a 6= a × Satisfiable. Domain: {1, 2} Referents: a : 1 b : 2 Extension of F : {1} [Q p.61] (iv) ∀x(Fx → Gx) \abc ∃xFx Xc ¬Ga a = b Fc Fa→ Ga X ¬Fa ¬Fb Fb→ Gb X ¬Fb Fc→ Gc X ¬Fc × Gc ¬Gb ↑ Gb ¬Gb × Ga × Satisfiable. Domain: {1, 2} Referents: a : 1 b : 1 c : 2 Extensions: F : {2} G : {2} [Q p.61] 227 (v) ∀x(x 6= a→ Rax) \b ∀x¬Rxb \ab a 6= b ¬Rab ¬Rbb b 6= a→ Rab X ¬(b 6= a) X b = a a 6= a × Rab × Unsatisfiable. [Q p.61] (vi) ∃x∀y(Fy→ x = y) Xc Fa Fb ∀y(Fy→ c = y) \abc Fa→ c = a X ¬Fa × c = a Fb→ c = b X ¬Fb × c = b Fc→ c = c ¬Fc ¬Fa × c = c ↑ Satisfiable. Domain: {1} Referents: a : 1 b : 1 c : 1 Extensions: F : {1} [Q p.61] 228 (vii) ∀x∀y(Rxy→ x = y) \a Rab a 6= b ∀y(Ray→ a = y) \b Rab→ a = b X ¬Rab × a = b a 6= a × Unsatisfiable. [Q p.62] (viii) ∀x((Fx ∧ Rxa)→ x 6= a) \b Fb ∧ Rba X a = b Fb Rba (Fb ∧ Rba)→ b 6= a X ¬(Fb ∧ Rba) X ¬Fb × ¬Rba × b 6= a a 6= a × Unsatisfiable. [Q p.62] 229 (ix) ∃x∃y∃zRxyz Xa ∀x(x = x → x = a) \abc ∃y∃zRayz Xb ∃zRabz Xc Rabc a = a→ a = a X ¬a = a × a = a b = b→ b = a X ¬b = b × b = a c = c→ c = a X ¬c = c × c = a Raba Raaa Raab Raac Rabb Raca Racb Racc Rbaa Rbab Rbac Rbba Rbbb Rbbc Rbca Rbcb Rbcc Rcaa Rcab Rcac Rcba Rcbb Rcbc Rcca Rccb Rccc ↑ Satisfiable. (See next page for model.) 230 Domain: {1} Referents: a : 1 b : 1 c : 1 Extensions: R : {〈1, 1, 1〉} [Q p.62] (x) ∀x¬Rxx \a ∀x∀yx = y \a ∃xRax Xb Rab ∀ya = y \b a = b Raa ¬Raa × Unsatisfiable. [Q p.62] 2. (i) ∃xFx Xa ∃yGy Xb ∀x∀yx = y \a ¬∃x(Fx ∧ Gx) X ∀x¬(Fx ∧ Gx) \a Fa Gb ∀ya = y \b a = b Ga ¬(Fa ∧ Ga) X ¬Fa × ¬Ga × Valid. [Q p.62] 231 (ii) ∃x∃y(Fx ∧ Gy ∧ ∀z(z = x ∨ z = y)) Xa ¬∃x(Fx ∧ Gx) X ∀x¬(Fx ∧ Gx) \ab ∃y(Fa ∧ Gy ∧ ∀z(z = a ∨ z = y)) Xb Fa ∧ Gb ∧ ∀z(z = a ∨ z = b) X Fa Gb ∀z(z = a ∨ z = b) \ab ¬(Fa ∧ Ga) X ¬Fa × ¬Ga ¬(Fb ∧ Gb) X ¬Fb a = a ∨ a = b X a = a b = a ∨ b = b X b = a ¬Fa × b = b ↑ a = b ¬Fa × ¬Gb × Invalid. Countermodel: Domain: {1, 2} Referents: a : 1 b : 2 Extensions: F : {1} G : {2} [Q p.62] 232 (iii) Rab ¬∀x∀y∀z(((Rxy ∧ Ryz) ∧ x = z)→ Ryy) X ∃x¬∀y∀z(((Rxy ∧ Ryz) ∧ x = z)→ Ryy) Xc ¬∀y∀z(((Rcy ∧ Ryz) ∧ c = z)→ Ryy) X ∃y¬∀z(((Rcy ∧ Ryz) ∧ c = z)→ Ryy) Xd ¬∀z(((Rcd ∧ Rdz) ∧ c = z)→ Rdd) X ∃z¬(((Rcd ∧ Rdz) ∧ c = z)→ Rdd) Xe ¬(((Rcd ∧ Rde) ∧ c = e)→ Rdd) X (Rcd ∧ Rde) ∧ c = e) X ¬Rdd Rcd ∧ Rde X c = e Rcd Rde Red Rdc ↑ Invalid. Countermodel: Domain: {1, 2, 3, 4} Referents: a : 1 b : 2 c : 3 d : 4 e : 3 Extension of R : {〈1, 2〉, 〈3, 4〉, 〈4, 3〉} [Q p.62] (iv) ∀x∀y(Rxy→ Ryx) \a ∃x(Rax ∧ x 6= b) Xc ¬∃x(Rxa ∧ x 6= b) X ∀x¬(Rxa ∧ x 6= b) \c Rac ∧ c 6= b X Rac c 6= b ∀y(Ray→ Rya) \c Rac→ Rca X ¬Rac × Rca ¬(Rca ∧ c 6= b) X ¬Rca × ¬(c 6= b) X c = b × Valid. [Q p.62] 233 (v) ∀x∀yx = y \a ¬∀x∀y(Rxy→ Ryx) X ∃x¬∀y(Rxy→ Ryx) Xa ¬∀y(Ray→ Rya) X ∃y¬(Ray→ Rya) Xb ¬(Rab→ Rba) X Rab ¬Rba ∀ya = y \b a = b Raa Rba × Valid. [Q p.62] (vi) ∀x∀y∀z((Rxy ∧ Rxz)→ y = z) \a Rab ∧ Rcd X b 6= d ¬a 6= c X a = c Rab Rcd ∀y∀z((Ray ∧ Raz)→ y = z) \b ∀z((Rab ∧ Raz)→ b = z) \d (Rab ∧ Rad)→ b = d X ¬(Rab ∧ Rad) X ¬Rab × ¬Rad ¬Rcd × b = d b 6= b × Valid. [Q p.62] 234 (vii) ∃x∃y(Rxy ∧ x = y) Xa ¬¬∀xRxx X ∀xRxx \ab ∃y(Ray ∧ a = y) Xb Rab ∧ a = b X Rab a = b Raa Rbb Rba ↑ Invalid. Countermodel: Domain: {1} Referents: a : 1 b : 1 Extension of R : {〈1, 1〉} [Q p.62] (viii) ∀x(x = a ∨ x = b) \cab ¬∀xx = a X ∃x¬x = a Xc ¬c = a c = a ∨ c = b X c = a ¬c = c × c = b a = a ∨ a = b X a = a b = a ∨ b = b X b = a c = a ¬c = c × b = b ¬b = a ↑ a = b a = c ¬c = c × Invalid. Countermodel: Domain: {1, 2} Referents: a : 1 b : 2 c : 2 [Q p.62] 235 (ix) ∀xRax \abc ¬∀x∀yx = y X ¬∃x∃y∃z(Rxy ∧ Rxz ∧ y 6= z) X ∃x¬∀yx = y Xb ¬∀yb = y X ∃y¬b = y Xc ¬b = c Raa Rab Rac ∀x¬∃y∃z(Rxy ∧ Rxz ∧ y 6= z) \a ¬∃y∃z(Ray ∧ Raz ∧ y 6= z) X ∀y¬∃z(Ray ∧ Raz ∧ y 6= z) \b ¬∃z(Rab ∧ Raz ∧ b 6= z) X ∀z¬(Rab ∧ Raz ∧ b 6= z) \c ¬(Rab ∧ Rac ∧ b 6= c) X ¬Rab × ¬Rac × ¬b 6= c X b = c ¬b = b × Valid. [Q p.62] (x) ∀xx = a \bc ¬∀xx = b X ∃x¬x = b Xc ¬c = b b = a c = a b = c ¬b = b × Valid. [Q p.62] 236 3. (i) s: Stan Fx: x is a firefighter (Fs ∧ ∀x(Fx → x = s))→ ¬∃x(Fx ∧ x 6= s) ¬((Fs ∧ ∀x(Fx → x = s))→ ¬∃x(Fx ∧ x 6= s)) Fs ∧ ∀x(Fx → x = s) X ¬¬∃x(Fx ∧ x 6= s) X ∃x(Fx ∧ x 6= s) Xa Fs ∀x(Fs→ x = s) \a Fa ∧ a 6= s X Fa a 6= s Fs→ a = s X ¬Fs × a = s × Logical truth. [Q p.63] (ii) c: Lewis Carroll j: Julius Caesar Lx: x is left-handed (Lj ∧ ¬Lc)→ c 6= j ¬((Lj ∧ ¬Lc)→ c 6= j) X Lj ∧ ¬Lc X ¬c 6= j X c = j Lj ¬Lc ¬Lj × Logical truth. [Q p.63] 237 (iii) a: Apollo s: the sun Wxy: x is warming y ∀x(s 6= x ↔Wsx)→Wsa ¬(∀x(s 6= x ↔Wsx)→Wsa) ∀x(s 6= x ↔Wsx) \as ¬Wsa s 6= a↔Wsa X s 6= a Wsa × ¬s 6= a X ¬Wsa s = a s 6= s↔Wss X s 6= s Wss × ¬s 6= s ¬Wss s = s ¬Was ¬Waa ↑ Not a logical truth. Countermodel: Domain: {1} Referents: a : 1 s : 1 Extension of W : ∅ [Q p.63] (iv) k: Kevin Bacon m: Michael J. Fox k 6= k→ k = m ¬(k 6= k→ k = m) X k 6= k k 6= m × Logical truth. (k 6= k is logically false; any conditional with a logically false antecedent is logically true.) [Q p.63] 238 (v) c: Clemens t: Twain Ax: x is an author Wx: x is witty (∀x((Ax ∧Wx)→ x = t) ∧ Ac)→ ¬Wc ¬((∀x((Ax ∧Wx)→ x = t) ∧ Ac)→ ¬Wc) X ∀x((Ax ∧Wx)→ x = t) ∧ Ac X ¬ ¬Wc X Wc ∀x((Ax ∧Wx)→ x = t) \ct Ac (Ac ∧Wc)→ c = t X ¬(Ac ∧Wc) X ¬Ac × ¬Wc × c = t (At ∧Wt)→ t = t X ¬(At ∧Wt) X ¬At ¬Ac × ¬Wt ¬Wc × t = t Wt At ↑ Not a logical truth. Countermodel: Domain: {1} Referents: c : 1 t : 1 Extensions: A : {1} W : {1} [Q p.63] 239 (vi) Sx: x is a spy Txy: x trusts y ∀x∀y((Sx ∧ Sy ∧ x 6= y)→ ¬Txy) ¬∀x∀y((Sx ∧ Sy ∧ x 6= y)→ ¬Txy) X ∃x¬∀y((Sx ∧ Sy ∧ x 6= y)→ ¬Txy) Xa ¬∀y((Sa ∧ Sy ∧ a 6= y)→ ¬Tay) X ∃y¬((Sa ∧ Sy ∧ a 6= y)→ ¬Tay) Xb ¬((Sa ∧ Sb ∧ a 6= b)→ ¬Tab) X Sa ∧ Sb ∧ a 6= b X ¬¬Tab X Tab Sa Sb a 6= b ↑ Not a logical truth. Countermodel: Domain: {1, 2} Referents: a : 1 b : 2 Extensions: S : {1, 2} T : {〈1, 2〉} [Q p.63] (vii) a: this ant ∀xx = a ∨ ¬∃xx = a ¬(∀xx = a ∨ ¬∃xx = a) X ¬∀xx = a X ¬¬∃xx = a X ∃xx = a Xb ∃x¬x = a Xc b = a ¬c = a ¬c = b ↑ Not a logical truth. Countermodel: Domain: {1, 2} Referents: a : 1 b : 1 c : 2 [Q p.63] 240 (viii) d: Doug s: Santa Claus Axy: x is afraid of y ∀x(x 6= s→ Adx)→ (Add ∨ d = s) ¬(∀x(x 6= s→ Adx)→ (Add ∨ d = s)) X ∀x(x 6= s→ Adx) \d ¬(Add ∨ d = s) X ¬Add ¬d = s d 6= s→ Add X ¬d 6= s X d = s ¬d = d × Add × Logical truth. [Q p.63] 241 (ix) m: Mark s: Samuel Rxy: x respects y (Rms ∧ ∀x(Rmx → x = s))→ ¬Rmm ¬((Rms ∧ ∀x(Rmx → x = s))→ ¬Rmm) X Rms ∧ ∀x(Rmx → x = s) X ¬¬Rmm X Rmm Rms ∀x(Rmx → x = s) \ms Rmm→ m = s X ¬Rmm × m = s Rms→ s = s X ¬Rms × s = s Rss Rsm ↑ Not a logical truth. Countermodel: Domain: {1} Referents: m : 1 s : 1 Extension: R : {〈1, 1〉} [Q p.63] 242 (x) m: I/me Px: x is a physical body Pm ∨ ∃x(x = m ∧ ¬Px) ¬(Pm ∨ ∃x(x = m ∧ ¬Px)) X ¬Pm ¬∃x(x = m ∧ ¬Px) X ∀x¬(x = m ∧ ¬Px) \m ¬(m = m ∧ ¬Pm) X ¬m = m × ¬¬Pm X Pm × Logical truth. [Q p.63] [Contents] 243 Answers 13.5.1 1. (i) Gx: x is a gremlin ∀x∀y∀z((Gx ∧ Gy ∧ Gz)→ (x = y ∨ x = z ∨ y = z)) ¬∀x∀y∀z((Gx ∧ Gy ∧ Gz)→ (x = y ∨ x = z ∨ y = z)) X ∃x¬∀y∀z((Gx ∧ Gy ∧ Gz)→ (x = y ∨ x = z ∨ y = z)) Xa ¬∀y∀z((Ga ∧ Gy ∧ Gz)→ (a = y ∨ a = z ∨ y = z)) X ∃y¬∀z((Ga ∧ Gy ∧ Gz)→ (a = y ∨ a = z ∨ y = z)) Xb ¬∀z((Ga ∧ Gb ∧ Gz)→ (a = b ∨ a = z ∨ b = z)) X ∃z¬((Ga ∧ Gb ∧ Gz)→ (a = b ∨ a = z ∨ b = z)) Xc ¬((Ga ∧ Gb ∧ Gc)→ (a = b ∨ a = c ∨ b = c)) X Ga ∧ Gb ∧ Gc X ¬(a = b ∨ a = c ∨ b = c) X Ga Gb Gc ¬a = b ¬a = c ¬b = c ↑ Not a logical truth. Countermodel: Domain: {1, 2, 3} Referents: a : 1 b : 2 c : 3 Extension of G : {1, 2, 3} [Q p.63] 244 (ii) Bx: x is a Beatle ∃x∃y∃z(Bx ∧ By ∧ Bz ∧ (x 6= y ∧ x 6= z ∧ y 6= z)) ¬∃x∃y∃z(Bx ∧ By ∧ Bz ∧ (x 6= y ∧ x 6= z ∧ y 6= z)) X ∀x¬∃y∃z(Bx ∧ By ∧ Bz ∧ (x 6= y ∧ x 6= z ∧ y 6= z)) \a ¬∃y∃z(Ba ∧ By ∧ Bz ∧ (a 6= y ∧ a 6= z ∧ y 6= z)) X ∀y¬∃z(Ba ∧ By ∧ Bz ∧ (a 6= y ∧ a 6= z ∧ y 6= z)) \a ¬∃z(Ba ∧ Ba ∧ Bz ∧ (a 6= a ∧ a 6= z ∧ a 6= z)) X ∀z¬(Ba ∧ Ba ∧ Bz ∧ (a 6= a ∧ a 6= z ∧ a 6= z)) \a ¬(Ba ∧ Ba ∧ Ba ∧ (a 6= a ∧ a 6= a ∧ a 6= a)) X ¬Ba ↑ ¬Ba ↑ ¬Ba ↑ ¬(a 6= a ∧ a 6= a ∧ a 6= a) X ¬(a 6= a) X a = a ↑ ¬(a 6= a) X a = a ↑ ¬(a 6= a) X a = a ↑ Not a logical truth. Countermodel: Domain: {1} Referent of a : 1 Extension of B : ∅ [Q p.63] 245 (iii) k: Kevin Bacon ∃xx = k ∧ ∀x∀y((x = k ∧ y = k)→ x = y) ¬(∃xx = k ∧ ∀x∀y((x = k ∧ y = k)→ x = y)) X ¬∃xx = k X ∀x¬x = k \k ¬k = k × ¬∀x∀y((x = k ∧ y = k)→ x = y) X ∃x¬∀y((x = k ∧ y = k)→ x = y) Xa ¬∀y((a = k ∧ y = k)→ a = y) X ∃y¬((a = k ∧ y = k)→ a = y) Xb ¬((a = k ∧ b = k)→ a = b) X a = k ∧ b = k X ¬a = b a = k b = k ¬k = b ¬k = k × Logical truth. (This may seem odd at first sight—but note what it means: on every model of the fragment of GPLI used to state the wff in question—the fragment which contains the name k— the wff comes out true. Of course!—that’s not strange at all: every such model assigns exactly one referent to k.) [Q p.63] 246 (iv) Ox: x is an ocean ∃x∃y(Ox ∧Oy ∧ x 6= y)→ ∃xOx ¬(∃x∃y(Ox ∧Oy ∧ x 6= y)→ ∃xOx) ∃x∃y(Ox ∧Oy ∧ x 6= y) Xa ¬∃xOx X ∀x¬Ox \a ∃y(Oa ∧Oy ∧ a 6= y) Xb Oa ∧Ob ∧ a 6= b X Oa Ob a 6= b ¬Oa × Logical truth. [Q p.63] (v) Dx: x is a dog Lxy: x is larger than y ∀x∀y((Dx ∧ Dy ∧ x 6= y ∧ Lxy)→ ¬Lyx) ¬∀x∀y((Dx ∧ Dy ∧ x 6= y ∧ Lxy)→ ¬Lyx) X ∃x¬∀y((Dx ∧ Dy ∧ x 6= y ∧ Lxy)→ Lyx) Xa ¬∀y((Da ∧ Dy ∧ a 6= y ∧ Lay)→ ¬Lya) X ∃y¬((Da ∧ Dy ∧ a 6= y ∧ Lay)→ ¬Lya) Xb ¬((Da ∧ Db ∧ a 6= b ∧ Lab)→ ¬Lba) X Da ∧ Db ∧ a 6= b ∧ Lab X ¬¬Lba X Lba Da Db a 6= b Lab ↑ Not a logical truth. Countermodel: Domain: {1, 2} Referents: a : 1 b : 2 Extensions: D : {1, 2} L : {〈1, 2〉, 〈2, 1〉} [Q p.64] 247 (vi) Ax: x is an apple (∃xAx ∧ ∀x∀y((Ax ∧ Ay)→ x = y))→ ∃xAx ¬((∃xAx ∧ ∀x∀y((Ax ∧ Ay)→ x = y))→ ∃xAx) X ∃xAx ∧ ∀x∀y((Ax ∧ Ay)→ x = y) X ¬∃xAx X ∀x¬Ax \a ∃xAx Xa ∀x∀y((Ax ∧ Ay)→ x = y) Aa ¬Aa × Logical truth. [Q p.64] (vii) Ax: x is an apple ¬(∃x∃y(Ax ∧ Ay ∧ x 6= y) ∧ ∀x∀y((Ax ∧ Ay)→ x = y)) ¬¬(∃x∃y(Ax ∧ Ay ∧ x 6= y) ∧ ∀x∀y((Ax ∧ Ay)→ x = y)) X ∃x∃y(Ax ∧ Ay ∧ x 6= y) ∧ ∀x∀y((Ax ∧ Ay)→ x = y) X ∃x∃y(Ax ∧ Ay ∧ x 6= y) Xa ∀x∀y((Ax ∧ Ay)→ x = y) \a ∃y(Aa ∧ Ay ∧ a 6= y) Xb Aa ∧ Ab ∧ a 6= b X Aa Ab a 6= b ∀y((Aa ∧ Ay)→ a = y) \b (Aa ∧ Ab)→ a = b X ¬(Aa ∧ Ab) X ¬Aa × ¬Ab × a = b a 6= a × Logical truth. [Q p.64] 248 (viii) Sx: x is a snake ¬∃xSx ∨ ∃x∃y(Sx ∧ Sy ∧ x 6= y) ¬(¬∃xSx ∨ ∃x∃y(Sx ∧ Sy ∧ x 6= y)) X ¬¬∃xSx X ¬∃x∃y(Sx ∧ Sy ∧ x 6= y) X ∃xSx X Sa ∀x¬∃y(Sx ∧ Sy ∧ x 6= y) \a ¬∃y(Sa ∧ Sy ∧ a 6= y) X ∀y¬(Sa ∧ Sy ∧ a 6= y) \a ¬(Sa ∧ Sa ∧ a 6= a) X ¬Sa × ¬Sa × ¬a 6= a X a = a ↑ Not a logical truth. Countermodel: Domain: {1} Referent of a : 1 Extension of S : {1} [Q p.64] 249 2. (i) Rx: x is in the room ∃x∃y∃z(Rx ∧ Ry ∧ Rz ∧ x 6= y ∧ x 6= z ∧ y 6= z) ∴ ∃x∃y(Rx ∧ Ry ∧ x 6= y) ∃x∃y∃z(Rx ∧ Ry ∧ Rz ∧ x 6= y ∧ x 6= z ∧ y 6= z) Xa ¬∃x∃y(Rx ∧ Ry ∧ x 6= y) X ∃y∃z(Ra ∧ Ry ∧ Rz ∧ a 6= y ∧ a 6= z ∧ y 6= z) Xb ∃z(Ra ∧ Rb ∧ Rz ∧ a 6= b ∧ a 6= z ∧ b 6= z) Xc Ra ∧ Rb ∧ Rc ∧ a 6= b ∧ a 6= c ∧ b 6= c X Ra Rb Rc a 6= b a 6= c b 6= c ∀x¬∃y(Rx ∧ Ry ∧ x 6= y) \a ¬∃y(Ra ∧ Ry ∧ a 6= y) X ∀y¬(Ra ∧ Ry ∧ a 6= y) \b ¬(Ra ∧ Rb ∧ a 6= b) X ¬Ra × ¬Rb × ¬a 6= b × Valid. [Q p.64] 250 (ii) c: Canada Bx: x is a bear Nxy: x is in y ∃x∃y(Bx ∧ By ∧ Nxc ∧ Nyc ∧ x 6= y) ∴ ∀x∀y∀z((Bx ∧ By ∧ Bz ∧ Nxc ∧ Nyc ∧ Nzc)→ (x = y ∨ x = z ∨ y = z)) ∃x∃y(Bx ∧ By ∧ Nxc ∧ Nyc ∧ x 6= y) Xa ¬∀x∀y∀z((Bx ∧ By ∧ Bz ∧ Nxc ∧ Nyc ∧ Nzc)→ (x = y ∨ x = z ∨ y = z)) X ∃y(Ba ∧ By ∧ Nac ∧ Nyc ∧ a 6= y) Xb Ba ∧ Bb ∧ Nac ∧ Nbc ∧ a 6= b X Ba Bb Nac Nbc a 6= b ∃x¬∀y∀z((Bx ∧ By ∧ Bz ∧ Nxc ∧ Nyc ∧ Nzc)→ (x = y ∨ x = z ∨ y = z)) Xd ¬∀y∀z((Bd ∧ By ∧ Bz ∧ Ndc ∧ Nyc ∧ Nzc)→ (d = y ∨ d = z ∨ y = z)) X ∃y¬∀z((Bd ∧ By ∧ Bz ∧ Ndc ∧ Nyc ∧ Nzc)→ (d = y ∨ d = z ∨ y = z)) Xe ¬∀z((Bd ∧ Be ∧ Bz ∧ Ndc ∧ Nec ∧ Nzc)→ (d = e ∨ d = z ∨ e = z)) X ∃z¬((Bd ∧ Be ∧ Bz ∧ Ndc ∧ Nec ∧ Nzc)→ (d = e ∨ d = z ∨ e = z)) X f ¬((Bd ∧ Be ∧ B f ∧ Ndc ∧ Nec ∧ N f c)→ (d = e ∨ d = f ∨ e = f )) X Bd ∧ Be ∧ B f ∧ Ndc ∧ Nec ∧ N f c ¬(d = e ∨ d = f ∨ e = f ) Bd Be B f Ndc Nec N f c ¬d = e ¬d = f ¬e = f ↑ Invalid. Countermodel: Domain: {1, 2, 3, 4, 5, 6} Referents: a : 1 b : 2 c : 3 d : 4 e : 5 f : 6 Extensions: B : {1, 2, 4, 5, 6} N : {〈1, 3〉, 〈2, 3〉, 〈4, 3〉, 〈5, 3〉, 〈6, 3〉} [Q p.64] 251 (iii) Bx: x is a barber Hxy: x cuts y’s hair ∀x∀y((Bx ∧ By)→ x = y) ∴ ∀x(Bx → Hxx) ∨ ∀x∀y((Bx ∧ By)→ ¬Hxy) ∀x∀y((Bx ∧ By)→ x = y) \a ¬(∀x(Bx → Hxx) ∨ ∀x∀y((Bx ∧ By)→ ¬Hxy)) X ¬∀x(Bx → Hxx) X ¬∀x∀y((Bx ∧ By)→ ¬Hxy) X ∃x¬(Bx → Hxx) Xa ∃x¬∀y((Bx ∧ By)→ ¬Hxy) Xb ¬(Ba→ Haa) X Ba ¬Haa ¬∀y((Bb ∧ By)→ ¬Hby) X ∃y¬((Bb ∧ By)→ ¬Hby) Xc ¬((Bb ∧ Bc)→ ¬Hbc) X Bb ∧ Bc X ¬¬Hbc X Bb Bc Hbc ∀y((Ba ∧ By)→ a = y) \bc (Ba ∧ Bb)→ a = b X ¬(Ba ∧ Bb) X ¬Ba × ¬Bb × a = b Hac (Ba ∧ Bc)→ a = c X ¬(Ba ∧ Bc) X ¬Ba × ¬Bc × a = c Haa × Valid. [Q p.64] 252 (iv) Hxy: x is heavier than y ∀x∀y∀z(x = y ∨ x = z ∨ y = z) ∀x∀y(Hxy ∨ Hyx) ∴ ∀x(∀y(x 6= y→ Hxy) ∨ ∀y(x 6= y→ Hyx)) ∀x∀y∀z(x = y ∨ x = z ∨ y = z) \a ∀x∀y(Hxy ∨ Hyx) \a ¬∀x(∀y(x 6= y→ Hxy) ∨ ∀y(x 6= y→ Hyx)) X ∃x¬(∀y(x 6= y→ Hxy) ∨ ∀y(x 6= y→ Hyx)) Xa ¬(∀y(a 6= y→ Hay) ∨ ∀y(a 6= y→ Hya)) X ¬∀y(a 6= y→ Hay) X ¬∀y(a 6= y→ Hya) X ∃y¬(a 6= y→ Hay) Xb ¬(a 6= b→ Hab) X a 6= b ¬Hab ∃y¬(a 6= y→ Hya) Xc ¬(a 6= c→ Hca) X a 6= c ¬Hca ∀y∀z(a = y ∨ a = z ∨ y = z) \b ∀z(a = b ∨ a = z ∨ b = z) \c a = b ∨ a = c ∨ b = c X a = b × a = c × b = c ¬Hac ∀y(Hay ∨ Hya) \c (Hac ∨ Hca) X Hac × Hca × Valid. [Q p.64] 253 (v) Ax: x is an athlete Fx: x is a footballer Gx: x is a golfer ∃x(Fx ∧ Ax) ∃x(Gx ∧ Ax) ∴ ∃x∃y(Ax ∧ Ay ∧ x 6= y) ∃x(Fx ∧ Ax) Xa ∃x(Gx ∧ Ax) Xb ¬∃x∃y(Ax ∧ Ay ∧ x 6= y) X ∀x¬∃y(Ax ∧ Ay ∧ x 6= y) \ab Fa ∧ Aa X Fa Aa Gb ∧ Ab X Gb Ab ¬∃y(Aa ∧ Ay ∧ a 6= y) X ∀y¬(Aa ∧ Ay ∧ a 6= y) \ab ¬(Aa ∧ Aa ∧ a 6= a) X ¬Aa × ¬Aa × ¬a 6= a X a = a ¬(Aa ∧ Ab ∧ a 6= b) X ¬Aa × ¬Ab × ¬a 6= b X a = b ¬∃y(Ab ∧ Ay ∧ b 6= y) X ∀y¬(Ab ∧ Ay ∧ b 6= y) \ab ¬(Ab ∧ Aa ∧ b 6= a) X ¬Ab × ¬Aa × ¬b 6= a X b = a ¬(Ab ∧ Ab ∧ b 6= b) X ¬Ab × ¬Ab × ¬b 6= b X b = b Fb Ga ↑ Invalid. Countermodel: Domain: {1} Referents: a : 1 b : 1 Extensions: F : {1} G : {1} A : {1} [Q p.64] 254 (vi) Pxy: x is a part of y ∀xPxx ∴ ∀x∃y∃z(Pyx ∧ Pzx ∧ y 6= z) ∀xPxx \a ¬∀x∃y∃z(Pyx ∧ Pzx ∧ y 6= z) X ∃x¬∃y∃z(Pyx ∧ Pzx ∧ y 6= z) Xa ¬∃y∃z(Pya ∧ Pza ∧ y 6= z) X ∀y¬∃z(Pya ∧ Pza ∧ y 6= z) \a ¬∃z(Paa ∧ Pza ∧ a 6= z) X ∀z¬(Paa ∧ Pza ∧ a 6= z) \a ¬(Paa ∧ Paa ∧ a 6= a) X ¬Paa Paa × ¬Paa Paa × ¬a 6= a X a = a Paa ↑ Invalid. Countermodel: Domain: {1} Referent of a : 1 Extension of P : {〈1, 1〉} [Q p.64] 255 (vii) e: the Eiffel tower ∃x∃y(x = e ∧ y = e ∧ x 6= y) ∴ ¬∃xx = e ∃x∃y(x = e ∧ y = e ∧ x 6= y) Xa ¬¬∃xx = e ∃y(a = e ∧ y = e ∧ a 6= y) Xb a = e ∧ b = e ∧ a 6= b X a = e b = e a 6= b e 6= b e 6= e × Valid. (The premise is logically false, hence any conclusion at all follows logically from it.) [Q p.64] 256 (viii) c: the Chief of Police j: Jemima m: I/me Axy: x is afraid of y Amj ∧ Amc ∴ j = c ∨ ∃x∃y(Amx ∧ Amy ∧ x 6= y) Amj ∧ Amc X ¬(j = c ∨ ∃x∃y(Amx ∧ Amy ∧ x 6= y)) X Amj Amc j 6= c ¬∃x∃y(Amx ∧ Amy ∧ x 6= y) X ∀x¬∃y(Amx ∧ Amy ∧ x 6= y) \j ¬∃y(Amj ∧ Amy ∧ j 6= y) X ∀y¬(Amj ∧ Amy ∧ j 6= y) \c ¬(Amj ∧ Amc ∧ j 6= c) X ¬Amj × ¬Amc × ¬j 6= c × Valid. [Q p.64] [Contents] 257 Answers 13.6.1.1 a: Vance c: Joseph Conrad i: The Inheritors l: Lord Jim s: The Shadow Line Axy: x authored y Fxy: x is father of y Rxy: x reads y Txy: x is taller than y 1. ∃x(∀y(Ays↔ y = x) ∧ c = x) [Q p.65] 2. ∃x(∀y(Ays↔ y = x) ∧ Axl) [Q p.65] 3. ∃x(∀y(Ays↔ y = x) ∧ ∃z(∀y(Ayl ↔ y = z) ∧ x = z)) [Q p.65] 4. ∃x(∀y(Ayl ↔ y = x) ∧ ∀y(Axy→ Ray)) [Q p.65] 5. Aci ∧ ¬∃x(∀y(Ayi↔ y = x) ∧ c = x) Another possible translation: Aci ∧ ∃x(∀y(Ayi↔ y = x) ∧ c 6= x) The first translation is preferable if we assume that the person mak- ing the claim being translated knows that The Inheritors was authored by two persons, one of whom was Joseph Conrad. [Q p.65] 6. ∃x(∀y(Ays↔ y = x) ∧ ∀y(Ayl → Txy)) [Q p.65] 7. ∃x(∀y(Ays↔ y = x) ∧ ∃yTyx) [Q p.65] 8. ∃x(∀y(Ays↔ y = x) ∧ Txc) ∧ ∃x(∀y(Ayl ↔ y = x) ∧ Tcx) [Q p.65] 9. ∃x(∀y(Ays↔ y = x) ∧ ∃z(∀y(Fyx ↔ y = z) ∧ Tzc)) [Q p.65] 10. ∃x(∀y(Ays↔ y = x) ∧ ∃z(∀y(Fyx ↔ y = z) ∧ Tzx)) [Q p.65] [Contents] 258 Answers 13.6.2.1 Glossary: a: Vance c: Joseph Conrad i: The Inheritors l: Lord Jim s: The Shadow Line Axy: x authored y Fxy: x is father of y Rxy: x reads y Txy: x is taller than y 1. c = ιxAxs [Q p.65] 2. A ιxAxsl [Q p.65] 3. ιxAxs = ιxAxl [Q p.65] 4. ∀y(A ιxAxly→ Ray)) [Q p.65] 5. Aci ∧ c 6= ιxAxi Note that this corresponds to the second translation given in An- swers 13.6.1.1 Question 5 (see p.258). [Q p.65] 6. ∀y(Ayl → T ιxAxsy) [Q p.65] 7. ∃yTy ιxAxs [Q p.65] 8. T ιxAxsc ∧ Tc ιxAxl [Q p.65] 9. T ιyFy ιxAxsc [Q p.65] 10. T ιyFy ιxAxs ιxAxs [Q p.65] [Contents] 259 Answers 13.6.3.1 Glossary: a: Vance c: Joseph Conrad i: The Inheritors l: Lord Jim s: The Shadow Line f : the father of the author of The Shadow Line i2: the author of The Inheritors l2: the author of Lord Jim s2: the author of The Shadow Line Axy: x authored y Fxy: x is father of y Rxy: x reads y Txy: x is taller than y 1. translation: c = s2 uniqueness postulate for s2 (the author of The Shadow Line): ∀x(Axs↔ x = s2) [Q p.66] 2. translation: As2l uniqueness postulate for s2: as above [Q p.66] 3. translation: s2 = l2 uniqueness postulate for s2: as above uniqueness postulate for l2 (the author of Lord Jim): ∀x(Axl ↔ x = l2) [Q p.66] 4. translation: ∀x(Al2x → Rax) uniqueness postulate for l2: as above [Q p.66] 5. translation: Aci ∧ c 6= i2 uniqueness postulate for i2 (the author of The Inheritors): ∀x(Axi↔ x = i2) Note that this corresponds to the second translation given in An- swers 13.6.1.1 Question 5 (see p.258). [Q p.66] 260 6. translation: ∀x(Axl → Ts2x) uniqueness postulate for s2: as above [Q p.66] 7. translation: ∃xTxs2 uniqueness postulate for s2: as above [Q p.66] 8. translation: Ts2c ∧ Tcl2 uniqueness postulates for s2 and l2: as above [Q p.66] 9. translation: T f c uniqueness postulate for f (the father of the author of The Shadow Line): ∀x(Fxs2 ↔ x = f ) uniqueness postulate for s2 (which features in the uniqueness postu- late for f ): as above [Q p.66] 10. translation: T f s2 uniqueness postulates for f and s2: as above [Q p.66] [Contents] Answers 13.7.4 1. Glossary: an: n s(x, y): x + y p(x, y): x× y q(x): x squared Ex: x is even Ox: x is odd Lxy: x < y (i) s(a2, a2) = a4 [Q p.66] (ii) p(a2, a2) = a4 [Q p.66] (iii) s(a2, a2) = p(a2, a2) [Q p.66] (iv) q(a2) = p(a2, a2) [Q p.66] (v) ∀x∀yq(s(x, y)) = p(s(x, y), s(x, y)) [Q p.66] 261 (vi) ∀x∀yq(s(x, y)) = s(s(q(x), p(a2, p(x, y))), q(y)) Note that we have represented 2xy using the two-place function symbol p. This means that we have to represent it as p(a2, p(x, y))— i.e. 2× (x× y)—or as p(p(a2, x), y)—i.e. (2× x)× y. Both choices are equally good. Similar comments apply to representing x2 + 2xy + y2 using the two-place function symbol s. [Q p.66] (vii) ∀x((Ex ∨Ox)→ Ep(a2, x)) [Q p.66] (viii) ∀x((Ox → Op(a3, x)) ∧ (Ex → Ep(a3, x))) [Q p.66] (ix) ∀xLp(a5, x)p(a6, x) [Q p.66] (x) ∀x∀y(Lxy→ Lp(a3, x)p(a4, y)) [Q p.66] 2. (i) True [Q p.67] (ii) False [Q p.67] (iii) True [Q p.67] (iv) True [Q p.67] (v) True [Q p.67] (vi) True [Q p.67] (vii) False [Q p.67] (viii) True [Q p.68] (ix) True [Q p.68] (x) False [Q p.68] 3. (i) False [Q p.68] (ii) False [Q p.68] (iii) True [Q p.68] (iv) False [Q p.68] (v) True [Q p.68] (vi) True [Q p.68] (vii) True [Q p.68] (viii) True [Q p.68] (ix) False [Q p.68] (x) False [Q p.68] 262 4. (i) (a) Domain: {1, 2} Referents: a: 1 b: 2 Value of f : {〈1, 1〉, 〈2, 1〉} (b) Domain: {1, 2} Referents: a: 1 b: 2 Value of f : {〈1, 1〉, 〈2, 2〉} [Q p.69] (ii) (a) Domain: {1, 2} Referents: a: 1 b: 2 Value of f : {〈1, 1〉, 〈2, 2〉} (b) Domain: {1, 2} Referents: a: 1 b: 2 Value of f : {〈1, 1〉, 〈2, 1〉} [Q p.69] (iii) (a) No model: reflexivity of identity. (b) Domain: {1, 2} Referents: a: 1 b: 2 Value of f : {〈1, 1〉, 〈2, 1〉} [Q p.69] (iv) (a) Domain: {1, 2} Value of f : {〈1, 1〉, 〈2, 1〉} (b) No model: functions are totally defined. [Q p.69] (v) (a) Domain: {1} Value of f : {〈1, 1〉} (b) Domain: {1, 2} Value of f : {〈1, 1〉, 〈2, 1〉} [Q p.69] (vi) (a) Domain: {1, 2} Value of s: {〈1, 1, 1〉, 〈1, 2, 2〉, 〈2, 1, 2〉, 〈2, 2, 1〉} (b) Domain: {1, 2} Value of s: {〈1, 1, 1〉, 〈1, 2, 1〉, 〈2, 1, 2〉, 〈2, 2, 1〉} [Q p.69] (vii) (a) Domain: {1, 2} Values: f : {〈1, 1〉, 〈2, 2〉} s: {〈1, 1, 1〉, 〈1, 2, 2〉, 〈2, 1, 2〉, 〈2, 2, 1〉} (b) Domain: {1, 2} Values: f : {〈1, 2〉, 〈2, 2〉} s: {〈1, 1, 1〉, 〈1, 2, 2〉, 〈2, 1, 2〉, 〈2, 2, 1〉} [Q p.69] 263 (viii) (a) Domain: {1, 2} Values: f : {〈1, 2〉, 〈2, 1〉} s: {〈1, 1, 1〉, 〈1, 2, 2〉, 〈2, 1, 2〉, 〈2, 2, 1〉} (b) Domain: {1, 2} Values: f : {〈1, 2〉, 〈2, 1〉} s: {〈1, 1, 1〉, 〈1, 2, 2〉, 〈2, 1, 1〉, 〈2, 2, 2〉} [Q p.69] (ix) (a) Domain: {1, 2} Values: f : {〈1, 2〉, 〈2, 1〉} s: {〈1, 1, 1〉, 〈1, 2, 1〉, 〈2, 1, 2〉, 〈2, 2, 2〉} (b) Domain: {1, 2} Values: f : {〈1, 2〉, 〈2, 1〉} s: {〈1, 1, 2〉, 〈1, 2, 2〉, 〈2, 1, 1〉, 〈2, 2, 1〉} [Q p.69] (x) (a) Domain: {1, 2} Value of s: {〈1, 1, 1〉, 〈1, 2, 1〉, 〈2, 1, 2〉, 〈2, 2, 1〉} (b) No model: the proposition says that for every x and y in the domain, there is exactly one object z in the domain such that s(x, y) = z; it is built into the semantics of GPLIF that on every model, s (a two-place function symbol) is assigned a value of a sort that makes this true. [Q p.69] [Contents] 264 Chapter 14 Metatheory Answers 14.1.1.1 1. 0 [Q p.70] 2. 3 [Q p.70] 3. 1 [Q p.70] 4. 3 [Q p.70] 5. 3 [Q p.70] 6. 3 [Q p.70] 7. 5 [Q p.70] 8. 5 [Q p.70] 9. 10 [Q p.70] 10. 15 [Q p.70] [Contents] Answers 14.1.2.1 1. Let p be a path featuring (α ∧ β). Suppose there is a model M on which every proposition on p is true. Let p′ be the path obtained from p by adding α and β. By clause 3 of §9.4.2, we know that α must be true on M, since (α∧ β) is. By clause 3 of §9.4.2, we know that β must 265 be true on M, since (α∧ β) is. Since α and β are the only propositions that were added to p to get p′, we know that every proposition on p′ is true on M. So there is a model on which all propositions on p′ are true. Therefore the tree rule for unnegated conjunction is truth- preserving. [Q p.71] 2. Let p be a path featuring ¬(α ∧ β). Suppose there is a model M on which every proposition on p is true. Let q be the path obtained from p by adding ¬α, and let r be the path obtained from p by adding ¬β. Since ¬(α ∧ β) is true on M, we know that (α ∧ β) is false on M, by clause 2 of §9.4.2. Thus, either α is false on M or β is false on M, by clause 3 of §9.4.2. Thus, either ¬α is true on M or ¬β is true on M, by clause 2 of §9.4.2. Thus, either all propositions on q are true on M, or all propositions on r are true on M. So either there is a model on which every proposition on q is true, or there is a model on which every proposition on r is true. Therefore the tree rule for negated conjunction is truth-preserving. [Q p.71] 3. Let p be a path featuring (α → β). Suppose there is a model M on which every proposition on p is true. Let q be the path obtained from p by adding ¬α, and let r be the path obtained from p by adding β. Since (α → β) is true on M, we know that either α is false on M or β is true on M, by clause 6 of §9.4.2. Thus, either ¬α is true on M or β is true on M, by clause 2 of §9.4.2. Thus, either all propositions on q are true on M, or all propositions on r are true on M. So either there is a model on which every proposition on q is true, or there is a model on which every proposition on r is true. Therefore the tree rule for negated conjunction is truth-preserving. [Q p.71] 4. Let p be a path featuring ¬(α → β). Suppose there is a model M on which every proposition on p is true. Let p′ be the path obtained from p by adding α and ¬β. Since ¬(α → β) is true on M, we know that α → β is false on M, by clause 2 of §9.4.2. So α is true on M and β is false on M, by clause 6 of §9.4.2. So α is true on M and ¬β is true on M, by clause 2 of §9.4.2. So all propositions on p′ are true on M. So there is a model on which all propositions on p′ are true. Therefore the tree rule for negated conditional is truth-preserving. [Q p.71] 5. Let p be a path featuring (α ↔ β). Suppose there is a model M on which every proposition on p is true. Let q be the path obtained from p by adding α and β, and let r be the path obtained from p by adding ¬α and ¬β. Either α and β are both true on M, or α and β are both 266 false on M, by clause 7 of §9.4.2. Thus either α and β are both true on M, or ¬α and ¬β are both true on M, by clause 2 of §9.4.2. .So either there is a model on which every proposition on q is true, or there is a model on which every proposition on r is true. Therefore the tree rule for unnegated biconditional is truth-preserving. [Q p.71] 6. Let p be a path featuring ¬(α ↔ β). Suppose there is a model M on which every proposition on p is true. Let q be the path obtained from p by adding α and ¬β, and let r be the path obtained from p by adding ¬α and β. We know that (α ↔ β) is false on M, by clause 2 of §9.4.2. So either α is true on M and β is false on M, or α is false on M and β is true on M, by clause 7 of §9.4.2. So either α and ¬β are true on M, or ¬α and β are true on M, by clause 2 of §9.4.2. So either every proposition on q is true on M, or every proposition on r is true on M. So either there is a model on which every proposition on q is true, or there is a model on which every proposition on r is true. Therefore the tree rule for negated biconditional is truth-preserving. [Q p.71] 7. Let p be a path featuring ¬¬α. Suppose there is a model M on which every proposition on p is true. Let p′ be the path obtained from p by adding α. We know that ¬α is false on M, by clause 2 of §9.4.2. And so we know that α is true on M, again by clause 2 of §9.4.2. So there is a model on which all propositions on p′ are true. Therefore the tree rule for negated conditional is truth-preserving. [Q p.71] [Contents] Answers 14.1.3.1 1. α is of the form (β ∧ δ); so the formula ¬α which we are considering is of the form ¬(β ∧ δ). Then ¬β or ¬δ also occurs on p. The com- plexities of these wffs are less than the complexity of ¬(β ∧ δ), so by the induction hypothesis, whichever of them is on p is true on M. So ¬(β ∧ δ) is also true on M. [Q p.71] 2. α is of the form (β→ δ); so the formula ¬α which we are considering is of the form ¬(β → δ). Then β and ¬δ also occur on p. The com- plexities of these wffs are less than the complexity of ¬(β → δ), so by the induction hypothesis, β and ¬δ are true on M. So ¬(β → δ) is also true on M. [Q p.71] 267 3. α is of the form (β ↔ δ); so the formula ¬α which we are consid- ering is of the form ¬(β ↔ δ). Then either β and ¬δ, or ¬β and δ, also occur on p. The complexities of all these wffs are less than the complexity of ¬(β ↔ δ), so by the induction hypothesis, whichever pair of them is on p, both formulas in the pair are true on M. Either way, ¬(β↔ δ) is also true on M. [Q p.71] 4. α is of the form ∃xβ; so the formula ¬α which we are considering is of the form ¬∃xβ. Then ∀x¬β also occurs on p. By the clause earlier in step (III) covering the case of wffs on p whose main operator is the universal quantifier, ∀x¬β is true on M. So by the reasoning in §10.1.1 (which establishes that ¬∃xβ and ∀x¬β are true and false in exactly the same models), ¬∃xβ is also true on M. [Q p.71] 5. γ is of the form (α ↔ β). Then either α and β, or ¬α and ¬β, also occur on p. (i) Suppose it is the former pair. The complexities of α and β are less than the complexity of (α ↔ β), so by the induction hypothesis α and β are true on M. So by the rule governing the truth of biconditionals in models, (α ↔ β) is also true on M. (ii) Suppose it is the latter pair. The complexities of ¬α and ¬β are not necessarily less than the complexity of (α ↔ β): if β is an atomic wff (i.e. it has complexity 0), then ¬α and (α ↔ β) have the same complexity; if α is an atomic wff, then ¬β and (α ↔ β) have the same complexity. If neither α nor β is atomic, then the complexities of ¬α and ¬β are less than the complexity of (α ↔ β); so by the induction hypothesis ¬α and ¬β are true on M; and so by the rule governing the truth of biconditionals in models, (α ↔ β) is also true on M. If one or both of α or β is atomic, we need to reason in a way similar to that used in the clause earlier in step (III) covering the case of wffs on p whose main operator is the conditional. [Q p.71] [Contents] 268 Chapter 15 Other Methods of Proof Answers 15.1.5 1. (i) 1. ¬P→ Q A 2. ¬P A 3. Q 1, 2 (MP) [Q p.72] (ii) 1. P A 2. P→ (¬Q→ P) (A1) 3. ¬Q→ P 1, 2 (MP) [Q p.72] (iii) 1. ¬Q A 2. ¬Q→ (¬P→ ¬Q) (A1) 3. (¬P→ ¬Q)→ ((¬P→ Q)→ P) (A3) 4. (¬P→ ¬Q) 1, 2 (MP) 5. (¬P→ Q)→ P 3, 4 (MP) [Q p.72] (iv) 1. P→ (P→ P) (A1) 2. P→ ((P→ P)→ P) (A1) 3. (P→ ((P→ P)→ P))→ ((P→ (P→ P))→ (P→ P)) (A2) 4. (P→ (P→ P))→ (P→ P) 2, 3 (MP) 5. P→ P 1, 4 (MP) [Q p.72] 269 (v) 1. ¬(P→ ¬Q) A 2. (¬Q→ ¬(P→ ¬Q))→ ((¬Q→ (P→ ¬Q))→ Q) (A3) 3. ¬(P→ ¬Q)→ (¬Q→ ¬(P→ ¬Q)) (A1) 4. ¬Q→ ¬(P→ ¬Q) 1, 3 (MP) 5. (¬Q→ (P→ ¬Q))→ Q 2, 4 (MP) 6. ¬Q→ (P→ ¬Q) (A1) 7. Q 5, 6 (MP) [Q p.72] (vi) 1. P A 2. ¬P A 3. (¬Q→ ¬P)→ ((¬Q→ P)→ Q) (A3) 4. ¬P→ (¬Q→ ¬P) (A1) 5. ¬Q→ ¬P 2, 4 (MP) 6. (¬Q→ P)→ Q 3, 5 (MP) 7. P→ (¬Q→ P) (A1) 8. ¬Q→ P 1, 7 (MP) 9. Q 6, 8 (MP) [Q p.72] (vii) P ∧Q := ¬(P→ ¬Q) 1. ¬(P→ ¬Q) A 2. ¬(P→ ¬Q)→ ((P→ ¬Q)→ ¬(P→ ¬Q)) (A1) 3. (P→ ¬Q)→ ¬(P→ ¬Q) 1, 2 (MP) [Q p.72] 2. (i) 1. ¬(P→ ¬Q) ` Q ∗ 2. ` ¬(P→ ¬Q)→ Q 1, DT ∗ Lines 1–7 of proof in Answer 1v. [Q p.72] (ii) (P ∨Q) := (¬P→ Q) 1. P,¬P ` Q ∗ 2. P ` (¬P→ Q) 1, DT 3. ` P→ (¬P→ Q) 2, DT ∗ Lines 1–9 of proof in Answer 1vi. [Q p.72] 270 (iii) 1. ((P→ Q)→ (P→ R)) A 2. P A 3. Q A 4. (Q→ (P→ Q)) (A1) 5. (P→ Q) 3, 4 (MP) 6. (P→ R) 1, 5 (MP) 7. R 2, 6 (MP) 8. ((P→ Q)→ (P→ R)), P, Q ` R 1–7 9. ((P→ Q)→ (P→ R)), P ` (Q→ R) 8, DT 10. ((P→ Q)→ (P→ R)) ` (P→ (Q→ R)) 9, DT 11. ` ((P→ Q)→ (P→ R))→ (P→ (Q→ R)) 10, DT [Q p.72] (iv) 1. (P→ Q) A 2. ¬Q A 3. (¬Q→ (¬¬P→ ¬Q)) (A1) 4. ((¬¬P→ ¬Q)→ ((¬¬P→ Q)→ ¬P)) (A3) 5. (¬¬P→ ¬Q) 2, 3 (MP) 6. ((¬¬P→ Q)→ ¬P) 4, 5 (MP) 7. (¬¬P→ Q) ∗ 8. ¬P 6, 7 (MP) 9. (P→ Q),¬Q ` ¬P 1–8 10. (P→ Q) ` (¬Q→ ¬P) 9, DT 11. ` (P→ Q)→ (¬Q→ ¬P) 10, DT ∗ 1, line 7 of proof in Fig. 15.8 (p.396). [Q p.72] (v) 1. P→ Q A 2. P→ ¬Q A 3. (¬¬P→ ¬Q)→ ((¬¬P→ Q)→ ¬P) (A3) 4. ¬¬P→ Q ∗ 5. ¬¬P→ ¬Q † 6. (¬¬P→ Q)→ ¬P 3, 5 (MP) 7. ¬P 4, 6 (MP) ∗ 1, line 7 of proof in Fig. 15.8 (p.396). † 1, line 7 of a proof which is just like that in Fig. 15.8 (p.396) except that it has ¬Q in place of Q throughout. [Q p.72] 271 (vi) 1. P→ Q A 2. ¬Q→ P A 3. (P→ Q)→ (¬Q→ ¬P) ∗ 4. ¬Q→ ¬P 1, 3 (MP) 5. (¬Q→ ¬P)→ ((¬Q→ P)→ Q) (A3) 6. (¬Q→ P)→ Q 4, 5 (MP) 7. Q 2, 6 (MP) ∗ Line 11 of proof in Answer 2iv. [Q p.72] (vii) 1. P→ (Q→ R) A 2. Q A 3. P A 4. Q→ R 1, 3 (MP) 5. R 2, 4 (MP) 6. P→ (Q→ R), Q, P ` R 1–5 7. P→ (Q→ R), Q ` P→ R 6, DT 8. P→ (Q→ R) ` Q→ (P→ R) 7, DT 9. ` (P→ (Q→ R))→ (Q→ (P→ R)) 8, DT [Q p.72] 3. (i) 1. P A 2. (¬P→ P)→ ((¬P→ ¬P)→ ¬¬P) (A9′) 3. P→ (¬P→ P) (A1) 4. ¬P→ P 1, 3 (MP) 5. (¬P→ ¬P)→ ¬¬P 2, 4 (MP) 6. ¬P→ (¬P→ ¬P) (A1) 7. (¬P→ (¬P→ ¬P))→ ((¬P→ ((¬P→ ¬P)→ ¬P))→ (¬P→ ¬P)) (A2′) 8. (¬P→ ((¬P→ ¬P)→ ¬P))→ (¬P→ ¬P) 6, 7 (MP) 9. ¬P→ ((¬P→ ¬P)→ ¬P) (A1) 10. ¬P→ ¬P 8, 9 (MP) 11. ¬¬P 5, 10 (MP) 12. P ` ¬¬P 1–11 13. ` P→ ¬¬P 12, DT [Q p.72] 272 (ii) 1. P→ ¬P A 2. (P→ ¬P)→ ((P→ ¬¬P)→ ¬P) (A9′) 3. (P→ ¬¬P)→ ¬P 1, 2 (MP) 4. P→ ¬¬P ∗ 5. ¬P 3, 4 (MP) ∗ Line 13 of proof in Answer 3i [Q p.73] (iii) 1. P→ Q A 2. ¬Q A 3. (P→ Q)→ ((P→ ¬Q)→ ¬P) (A9′) 4. (P→ ¬Q)→ ¬P 1, 3 (MP) 5. ¬Q→ (P→ ¬Q) (A1) 6. P→ ¬Q 2, 5 (MP) 7. ¬P 4, 6 (MP) 8. P→ Q,¬Q ` ¬P 1–7 9. P→ Q ` ¬Q→ ¬P 8, DT [Q p.73] (iv) 1. ¬Q A 2. Q A 3. Q→ (¬P→ Q) (A1) 4. ¬Q→ (¬P→ ¬Q) (A1) 5. ¬P→ Q 2, 3 (MP) 6. ¬P→ ¬Q 1, 4 (MP) 7. (¬P→ Q)→ ((¬P→ ¬Q)→ ¬¬P) (A9′) 8. (¬P→ ¬Q)→ ¬¬P 5, 7 (MP) 9. ¬¬P 6, 8 (MP) 10. ¬¬P→ P (A10′) 11. P 9, 10 (MP) 12. ¬Q, Q ` P 1–11 13. ¬Q ` Q→ P 12, DT 14. ` ¬Q→ (Q→ P) 13, DT [Q p.73] (v) 1. P ∧Q A 2. (P ∧Q)→ Q (A5′) 3. Q 1, 2 (MP) 4. Q→ (P→ Q) (A1) 5. P→ Q 3, 4 (MP) [Q p.73] 273 (vi) 1. ¬Q A 2. (P→ P)→ ((Q→ P)→ ((P ∨Q)→ P)) (A8′) 3. P→ P ∗ 4. (Q→ P)→ ((P ∨Q)→ P) 2, 3 (MP) 5. ¬Q→ (Q→ P) † 6. Q→ P 1, 5 (MP) 7. (P ∨Q)→ P 4, 6 (MP) ∗ Mimic lines 6–10 of the proof in Answer 3i, with P in place of ¬P throughout. Alternatively, line 3 of the following proof: 1. P A 2. P ` P 1 (i.e. 1–1) 3. ` P→ P 2, DT † line 14 of proof in Answer 3iv. [Q p.73] (vii) 1. ¬P ∧ ¬Q A 2. (¬P ∧ ¬Q)→ ¬P (A4′) 3. (¬P ∧ ¬Q)→ ¬Q (A5′) 4. ¬P 1, 2 (MP) 5. ¬Q 1, 3 (MP) 6. ¬P→ ((P ∨Q)→ ¬P) (A1) 7. (P ∨Q)→ ¬P 4, 6 (MP) 8. (P ∨Q)→ P ∗ 9. ((P ∨Q)→ P)→ (((P ∨Q)→ ¬P)→ ¬(P ∨Q)) (A9′) 10. ((P ∨Q)→ ¬P)→ ¬(P ∨Q) 8, 9 (MP) 11. ¬(P ∨Q) 7, 10 (MP) ∗ 5, line 7 of proof in Answer 3vi. [Q p.73] 274 (viii) 1. ¬(P ∨Q) A 2. P→ (P ∨Q) (A6′) 3. ¬(P ∨Q)→ (P→ ¬(P ∨Q)) (A1) 4. P→ ¬(P ∨Q) 1, 3 (MP) 5. (P→ (P ∨Q))→ ((P→ ¬(P ∨Q))→ ¬P) (A9′) 6. (P→ ¬(P ∨Q))→ ¬P 2, 5 (MP) 7. ¬P 4, 6 (MP) 8. Q→ (P ∨Q) (A7′) 9. ¬(P ∨Q)→ (Q→ ¬(P ∨Q)) (A1) 10. Q→ ¬(P ∨Q) 1, 9 (MP) 11. (Q→ (P ∨Q))→ ((Q→ ¬(P ∨Q))→ ¬Q) (A9′) 12. (Q→ ¬(P ∨Q))→ ¬Q 8, 11 (MP) 13. ¬Q 10, 12 (MP) 14. ¬P→ (¬Q→ (¬P ∧ ¬Q)) (A3′) 15. ¬Q→ (¬P ∧ ¬Q) 7, 14 (MP) 16. ¬P ∧ ¬Q 13, 15 (MP) [Q p.73] (ix) 1. (P ∧ ¬P)→ P (A4′) 2. (P ∧ ¬P)→ ¬P (A5′) 3. ((P ∧ ¬P)→ P)→ (((P ∧ ¬P)→ ¬P)→ ¬(P ∧ ¬P)) (A9′) 4. ((P ∧ ¬P)→ ¬P)→ ¬(P ∧ ¬P) 1, 3 (MP) 5. ¬(P ∧ ¬P) 2, 4 (MP) [Q p.73] 275 (x) 1. P ∧ ¬P A 2. (P ∧ ¬P)→ P (A4′) 3. (P ∧ ¬P)→ ¬P (A5′) 4. P 1, 2 (MP) 5. ¬P 1, 3 (MP) 6. P→ (¬Q→ P) (A1) 7. ¬P→ (¬Q→ ¬P) (A1) 8. ¬Q→ P 4, 6 (MP) 9. ¬Q→ ¬P 5, 7 (MP) 10. (¬Q→ P)→ ((¬Q→ ¬P)→ ¬¬Q) (A9′) 11. (¬Q→ ¬P)→ ¬¬Q 8, 10 (MP) 12. ¬¬Q 9, 11 (MP) 13. ¬¬Q→ Q (A10′) 14. Q 12, 13 (MP) 15. P ∧ ¬P ` Q 1–14 16. ` (P ∧ ¬P)→ Q 15, DT [Q p.73] (xi) P↔ P := ¬((P→ P)→ ¬(P→ P)) 1. P→ P ∗ 2. (P→ P)→ (((P→ P)→ ¬(P→ P))→ (P→ P)) (A1) 3. ((P→ P)→ ¬(P→ P))→ (P→ P) 1, 2 (MP) 4. (((P→ P)→ ¬(P→ P))→ (P→ P))→ ((((P→ P)→ ¬(P→ P))→ ¬(P→ P))→ ¬((P→ P)→ ¬(P→ P))) (A9) 5. (((P→ P)→ ¬(P→ P))→ ¬(P→ P))→ ¬((P→ P)→ ¬(P→ P)) 3, 4 (MP) 6. ((P→ P)→ ¬(P→ P))→ ¬(P→ P) † 7. ¬((P→ P)→ ¬(P→ P)) 5, 6 (MP) ∗ See line 3 of the proof in Answer 3vi. † The proof in Answer 3ii plus DT gives ` (P → ¬P) → ¬P. Mimic this proof, with (P → P) in place of P throughout, and the resulting proof plus DT gives: ` ((P→ P)→ ¬(P→ P))→ ¬(P→ P). [Q p.73] 276 (xii) 1. P A 2. ¬P A 3. P→ (¬Q→ P) (A1) 4. ¬Q→ P 1, 3 (MP) 5. (¬Q→ P)→ ((¬Q→ ¬P)→ ¬¬Q) (A9′) 6. (¬Q→ ¬P)→ ¬¬Q 4, 5 (MP) 7. ¬P→ (¬Q→ ¬P) (A1) 8. ¬Q→ ¬P 2, 7 (MP) 9. ¬¬Q 6, 8 (MP) 10. ¬¬Q→ Q (A10′) 11. Q 9, 10 (MP) 12. P,¬P ` Q 1–11 13. P ` ¬P→ Q 12, DT 14. ` P→ (¬P→ Q) 13, DT [Q p.73] 4. (i) 1. ∀x(Fx → Gx) A 2. Fa A 3. ∀x(Fx → Gx)→ (Fa→ Ga) (A4) 4. Fa→ Ga 1, 3 (MP) 5. Ga 2, 4 (MP) [Q p.73] (ii) ∀x(Gx ∨ Fx) := ∀x(¬Gx → Fx) 1. ∀xFx A 2. ∀xFx → Fx (A4) 3. Fx 1, 2 (MP) 4. Fx → (¬Gx → Fx) (A1) 5. (¬Gx → Fx) 3, 4 (MP) 6. ∀x(¬Gx → Fx) 4 (Gen) [Q p.73] (iii) 1. ∀x∀y(Rxy→ Ryx) A 2. Rab A 3. ∀x∀y(Rxy→ Ryx)→ ∀y(Ray→ Rya) (A4) 4. ∀y(Ray→ Rya) 1, 3 (MP) 5. ∀y(Ray→ Rya)→ (Rab→ Rba) (A4) 6. Rab→ Rba 4, 5 (MP) 7. Rba 2, 6 (MP) [Q p.73] 277 (iv) ∃xFx := ¬∀x¬Fx 1. ¬∀x¬Fx → ¬Ga A 2. Ga A 3. Ga→ (¬∀x¬Fx → Ga) (A1) 4. ¬∀x¬Fx → Ga 2, 3 (MP) 5. (¬∀x¬Fx → ¬Ga)→ ((¬∀x¬Fx → Ga)→ ∀x¬Fx) (A3) 6. (¬∀x¬Fx → Ga)→ ∀x¬Fx 1, 5 (MP) 7. ∀x¬Fx 4, 6 (MP) 8. ¬∀x¬Fx → ¬Ga, Ga ` ∀x¬Fx 1–7 9. ¬∀x¬Fx → ¬Ga ` Ga→ ∀x¬Fx 8, DT [Q p.73] 278 (v) Auxiliary proof: 1. ¬¬∀x¬Fx A 2. ¬¬∀x¬Fx → (¬∀x¬Fx → ¬¬∀x¬Fx) (A1) 3. (¬∀x¬Fx → ¬¬∀x¬Fx)→ ((¬∀x¬Fx → ¬∀x¬Fx)→ ∀x¬Fx) (A3) 4. ¬∀x¬Fx → ¬¬∀x¬Fx 1, 2 (MP) 5. (¬∀x¬Fx → ¬∀x¬Fx)→ ∀x¬Fx 4, 3 (MP) 6. ¬∀x¬Fx → ¬∀x¬Fx † 7. ∀x¬Fx 5, 6 (MP) 8. ∀x¬Fx → ¬Fa (A4) 9. ¬Fa 7, 8 (MP) 10. ¬¬∀x¬Fx ` ¬Fa 1–9 11. ` ¬¬∀x¬Fx → ¬Fa 10, DT † Look at the proof that `A2 P→ P in the commentary on line 3 of the proof in Answer 3vi (marked ∗). Mimic this proof, with any wff α in place of P throughout, and the resulting proof establishes `A∀=1 α→ α. (Note that the proof does not use (Gen), so the restriction on DT in A∀=1 is automatically satisfied.) Main proof: ∃xFx := ¬∀x¬Fx 1. Fa A 2. (¬¬∀x¬Fx → ¬Fa)→ ((¬¬∀x¬Fx → Fa)→ ¬∀x¬Fx) (A3) 3. ¬¬∀x¬Fx → ¬Fa ∗ 4. (¬¬∀x¬Fx → Fa)→ ¬∀x¬Fx 2, 3 (MP) 5. Fa→ (¬¬∀x¬Fx → Fa) (A1) 6. ¬¬∀x¬Fx → Fa 1, 5 (MP) 7. ¬∀x¬Fx 4, 6 (MP) 8. Fa ` ¬∀x¬Fx 1–7 9. ` Fa→ ¬∀x¬Fx 8, DT ∗ Line 11 of Auxiliary (above). [Q p.73] 279 (vi) 1. Fa A 2. a = b A 3. x = y→ (Fx → Fy) (A7) 4. ∀y(x = y→ (Fx → Fy)) 3 (Gen) 5. ∀x∀y(x = y→ (Fx → Fy)) 4 (Gen) 6. ∀x∀y(x = y→ (Fx → Fy))→ ∀y(a = y→ (Fa→ Fy)) (A4) 7. ∀y(a = y→ (Fa→ Fy)) 5, 6 (MP) 8. ∀y(a = y→ (Fa→ Fy))→ (a = b→ (Fa→ Fb)) (A4) 9. a = b→ (Fa→ Fb) 7, 8 (MP) 10. Fa→ Fb 2, 9 (MP) 11. Fb 1, 10 (MP) [Q p.73] (vii) 1. ∀x∀yx = y A 2. ∀x∀yx = y→ ∀ya = y (A4) 3. ∀ya = y 1, 2 (MP) 4. ∀ya = y→ a = b (A4) 5. a = b 3, 4 (MP) [Q p.73] (viii) 1. a = b A 2. a = c A 3. (x = y→ (x = b→ y = b)) (A7) 4. ∀x(x = y→ (x = b→ y = b)) 3 (Gen) 5. ∀x(x = y→ (x = b→ y = b))→ (a = y→ (a = b→ y = b)) (A4) 6. (a = y→ (a = b→ y = b)) 4, 5 (MP) 7. ∀y(a = y→ (a = b→ y = b)) 6 (Gen) 8. ∀y(a = y→ (a = b→ y = b))→ (a = c→ (a = b→ c = b)) (A4) 9. (a = c→ (a = b→ c = b)) 7, 8 (MP) 10. (a = b→ c = b) 2, 9 (MP) 11. c = b 1, 10 (MP) [Q p.73] 280 (ix) 1. a = b A 2. (x = y→ (x = a→ y = a)) (A7) 3. ∀x(x = y→ (x = a→ y = a)) 2 (Gen) 4. ∀x(x = y→ (x = a→ y = a))→ (a = y→ (a = a→ y = a)) (A4) 5. (a = y→ (a = a→ y = a)) 3, 4 (MP) 6. ∀y(a = y→ (a = a→ y = a)) 5 (Gen) 7. ∀y(a = y→ (a = a→ y = a))→ (a = b→ (a = a→ b = a)) (A4) 8. a = b→ (a = a→ b = a) 6, 7 (MP) 9. a = a→ b = a 1, 8 (MP) 10. ∀xx = x (A6) 11. ∀xx = x → a = a (A4) 12. a = a 10, 11 (MP) 13. b = a 9, 12 (MP) 14. a = b ` b = a 1–13 15. ` a = b→ b = a 14, DT [Q p.73] (x) Auxiliary proof A: 1. ¬¬a = b A 2. ¬¬a = b→ (¬a = b→ ¬¬a = b) (A1) 3. (¬a = b→ ¬¬a = b)→ ((¬a = b→ ¬a = b)→ a = b) (A3) 4. ¬a = b→ ¬¬a = b 1, 2 (MP) 5. (¬a = b→ ¬a = b)→ a = b 4, 3 (MP) 6. ¬a = b→ ¬a = b ∗ 7. a = b 6, 5 (MP) 8. ¬¬a = b ` a = b 1–7 9. ` ¬¬a = b→ a = b 8, DT ∗ See comment on line 6 of proof in Answer 4v (marked †). 281 Auxiliary proof B: 1. ¬Fb A 2. ¬¬a = b A 3. ¬¬a = b→ a = b ∗ 4. a = b 2, 3 (MP) 5. a = b→ b = a † 6. b = a 4, 5 (MP) 7. b = a→ (¬Fb→ ¬Fa) ‡ 8. ¬Fb→ ¬Fa 6, 7 (MP) 9. ¬Fa 1, 8 (MP) 10. ¬Fb,¬¬a = b ` ¬Fa 1–9 11. ¬Fb ` ¬¬a = b→ ¬Fa 10, DT ∗ Line 9 of Auxiliary proof A (above) † Line 15 of proof in Answer 4ix. ‡ To get this line, use the same strategy as in lines 3–9 of the proof in Answer 4vi. Main proof: 1. Fa A 2. ¬Fb A 3. (¬¬a = b→ ¬Fa)→ ((¬¬a = b→ Fa)→ ¬a = b) (A3) 4. ¬¬a = b→ ¬Fa ∗ 5. (¬¬a = b→ Fa)→ ¬a = b 3, 4 (MP) 6. Fa→ (¬¬a = b→ Fa) (A1) 7. ¬¬a = b→ Fa 1, 6 (MP) 8. ¬a = b 5, 7 (MP) ∗ 2, line 11 of Auxiliary proof B (above). [Q p.73] 282 (xi) 1. ¬b = a A 2. ∀x(¬Fx → x = a) A 3. ∀x(¬Fx → x = a)→ (¬Fb→ b = a) (A4) 4. ¬Fb→ b = a 2, 3 (MP) 5. ¬b = a→ (¬Fb→ ¬b = a) (A1) 6. ¬Fb→ ¬b = a 1, 5 (MP) 7. (¬Fb→ ¬b = a)→ ((¬Fb→ b = a)→ Fb) (A3) 8. (¬Fb→ b = a)→ Fb 6, 7 (MP) 9. Fb 4, 8 (MP) [Q p.73] (xii) 1. ∀xFx A 2. ∀xFx → Fy (A4) 3. Fy 1, 2 (MP) 4. ∀yFy 3 (Gen) 5. ∀xFx ` ∀yFy 1–4 6. ` ∀xFx → ∀yFy 5, DT [Q p.73] 5. Sketch of answer: Adding new axioms does not affect the proof of DT in §15.1.1.1 (provided we retain (MP) and the existing axioms which feature in the proof). Adding new rules does affect the proof: each rule requires separate treatment in the induction step. For the rule (MP), the treatment of it in the induction step employs axiom (A2) (see pp.394–5). For the rule (Gen), the treatment of it in the induction step must employ axiom (A5). But (A5) includes a restriction (i.e. “α contains no free x”). This means that only uses of (Gen) with a corresponding restriction (i.e. the rule is not applied “using a variable which is free in β”) can be handled. [Q p.73] [Contents] 283 Answers 15.2.3 1. (i) 1 ¬P→ P 2 ¬P 3 P 1, 2 (→ E) 4 ¬P 2 (RI) 5 P 2–4 (¬E) 6 (¬P→ P)→ P 1–5 (→ I) [Q p.74] (ii) 1 (A→ C) 2 (B→ C) 3 (A ∨ B) 4 A 5 C 1, 4 (→ E) 6 B 7 C 2, 6 (→ E) 8 C 3, 4–5, 6–7 (∨E) [Q p.74] (iii) 1 ¬¬P 2 ¬P 3 ¬P 2 (RI) 4 ¬¬P 1 (RI) 5 P 2–4 (¬E) 6 ¬¬P→ P 1–5 (→ I) [Q p.74] 284 (iv) 1 ¬(A ∨ B) 2 A 3 A ∨ B 2 (∨I) 4 ¬(A ∨ B) 1 (RI) 5 ¬A 2–4 (¬I) 6 B 7 A ∨ B 6 (∨I) 8 ¬(A ∨ B) 1 (RI) 9 ¬B 6–8 (¬I) 10 ¬A ∧ ¬B 5, 9 (∧I) [Q p.74] (v) 1 A 2 ¬A 3 ¬B 4 A 1 (RI) 5 ¬A 2 (RI) 6 B 3–5 (¬E) [Q p.74] (vi) 1 A→ B 2 B→ C 3 A 4 B 1, 3 (→ E) 5 C 2, 4 (→ E) 6 A→ C 3–5 (→ I) [Q p.74] 285 (vii) 1 P→ Q 2 ¬Q 3 P 4 Q 1, 3 (→ E) 5 ¬Q 2 (RI) 6 ¬P 3–5 (¬I) 7 ¬Q→ ¬P 2–6 (→ I) [Q p.74] (viii) 1 A ∨ B 2 ¬A 3 A ∨ B 1 (RI) 4 A 5 ¬B 6 A 4 (RI) 7 ¬A 2 (RI) 8 B 5–7 (¬E) 9 B 10 B 8 (RI) 11 B 3, 4–8, 9–10 (∨E) [Q p.74] 286 (ix) 1 P→ R 2 Q→ R 3 P ∨Q 4 P 5 R 1, 4 (→ E) 6 Q 7 R 2, 6 (→ E) 8 R 3, 4–5, 6–7 (∨E) [Q p.74] (x) 1 P→ Q 2 P ∧ ¬Q 3 P 2 (∧E) 4 Q 1, 3 (→ E) 5 ¬Q 2 (∧E) 6 ¬(P ∧ ¬Q) 2–5 (¬I) [Q p.74] 287 2. (i) `N2 A ∨ ¬A 1 ¬(A ∨ ¬A) 2 A 3 A ∨ ¬A 2 (∨I) 4 (A ∨ ¬A) ∧ ¬(A ∨ ¬A) 1, 3 (∧I) 5 ¬A 2–4 (¬I′) 6 ¬A 7 A ∨ ¬A 6 (∨I) 8 (A ∨ ¬A) ∧ ¬(A ∨ ¬A) 1, 7 (∧I) 9 ¬¬A 6–8 (¬I′) 10 A 9 (¬¬E) 11 A ∧ ¬A 5, 10 (∧I) 12 ¬¬(A ∨ ¬A) 1–11 (¬I′) 13 A ∨ ¬A 12 (¬¬E) `N3 A ∨ (A→ ⊥) 1 A ∨ (A→ ⊥) (TND) `N4 A ∨ (A→ ⊥) 1 A 2 A ∨ (A→ ⊥) 1 (∨I) 3 A→ ⊥ 4 A ∨ (A→ ⊥) 3 (∨I) 5 A ∨ (A→ ⊥) 1–2, 3–4 (NCD) 288 `N5 A ∨ (A→ ⊥) 1 ((A ∨ (A→ ⊥))→ ⊥ 2 A→ ⊥ 3 A ∨ (A→ ⊥) 2 (∨I) 4 ⊥ 1, 3 (→ E) 5 A 2–4 (RAA) 6 A ∨ (A→ ⊥) 5 (∨I) 7 ⊥ 1, 6 (→ E) 8 A ∨ (A→ ⊥) 1–7 (RAA) [Q p.74] (ii) (A ∧ ¬A) `N2 B 1 (A ∧ ¬A) 2 B 1 (¬E′) A ∧ (A→ ⊥) `N3/N4/N5 B 1 A ∧ (A→ ⊥) 2 A 1 (∧E) 3 A→ ⊥ 1 (∧E) 4 ⊥ 2, 3 (→ E) 5 B 4 (⊥E) [Q p.74] (iii) `N2 (¬¬A→ A) 1 ¬¬A 2 A 1 (¬¬E) 3 ¬¬A→ A 1–2 (→ I) 289 `N3 ((A→ ⊥)→ ⊥)→ A 1 (A→ ⊥)→ ⊥ 2 A ∨ (A→ ⊥) (TND) 3 A 4 A 3 (RI) 5 A→ ⊥ 6 ⊥ 1, 5 (→ E) 7 A 6 (⊥E) 8 A 2, 3–4, 5–7 (∨E) 9 ((A→ ⊥)→ ⊥)→ A 1–8 (→I) `N4 ((A→ ⊥)→ ⊥)→ A 1 (A→ ⊥)→ ⊥ 2 A 3 A 2 (RI) 4 A→ ⊥ 5 ⊥ 1, 4 (→E) 6 A 5 (⊥E) 7 A 2–3, 4–6 (NCD) 8 ((A→ ⊥)→ ⊥)→ A 1–7 (→ I) `N5 ((A→ ⊥)→ ⊥)→ A 1 (A→ ⊥)→ ⊥ 2 (A→ ⊥) 3 ⊥ 1, 2 (→ E) 4 A 2–3 (RAA) 5 ((A→ ⊥)→ ⊥)→ A 1–4 (→ I) [Q p.74] 290 (iv) `N2 ¬(A ∧ ¬A) 1 A ∧ ¬A 2 A ∧ ¬A 1 (RI) 3 ¬(A ∧ ¬A) 1–2 (¬I′) `N3/N4/N5 (A ∧ (A→ ⊥))→ ⊥ 1 A ∧ (A→ ⊥) 2 A 1 (∧E) 3 A→ ⊥ 1 (∧E) 4 ⊥ 2, 3 (→E) 5 (A ∧ (A→ ⊥))→ ⊥ 1–4 (→ I) [Q p.74] 3. (i) 1 Fa 2 Fa 1 (RI) 3 Fa→ Fa 1–2 (→ I) 4 ∀x(Fx → Fx) 3 (∀I) [Q p.74] (ii) 1 ∃x(Fx ∧ Gx) 2 Fa ∧ Ga 3 Fa 2 (∧E) 4 ∃xFx 3 (∃I) 5 Ga 2 (∧E) 6 ∃xGx 5 (∃I) 7 ∃xFx ∧ ∃xGx 4, 6 (∧I) 8 ∃xFx ∧ ∃xGx 1, 2–7 (∃E) [Q p.74] 291 (iii) 1 ∀x(Fx → Gx) 2 ¬∃xGx 3 ∃xFx 4 Fa 5 Fa→ Ga 1 (∀E) 6 Ga 4, 5 (→ E) 7 ∃xGx 6 (∃I) 8 ∃xGx 3, 4–7 (∃E) 9 ¬∃xGx 2 (RI) 10 ¬∃xFx 3–9 (¬I) [Q p.74] (iv) 1 ∀x(Fx → x = a) 2 Fb 3 Fb→ b = a 1 (∀E) 4 b = a 2, 3 (→ E) 5 a = a (= I) 6 a = b 4, 5 (= E) 7 Fb→ a = b 2–6 (→ I) [Q p.74] 292 (v) 1 ∀x∀yx = y 2 Raa 3 ¬Rbc 4 ∀yb = y 1 (∀E) 5 b = a 4 (∀E) 6 ∀y(a = y) 1 (∀E) 7 a = c 6 (∀E) 8 ¬Rac 3, 5 (= E) 9 Raa 2 (RI) 10 ¬Raa 7, 8 (= E) 11 Rbc 3–10 (¬E) 12 ∀yRby 11 (∀I) 13 ∀x∀yRxy 12 (∀I) [Q p.74] (vi) 1 ∀xRxx 2 Raa 1 (∀E) 3 ∃yRay 2 (∃I) 4 ∀x∃yRxy 3 (∀I) 5 ∀xRxx → ∀x∃yRxy 1–4 (→I) [Q p.74] 293 (vii) 1 ∃xFx 2 Fa 3 ∀x¬Fx 4 Fa 2 (RI) 5 ¬Fa 3 (∀E) 6 ¬∀x¬Fx 3–5 (¬I) 7 ¬∀x¬Fx 1, 2–6 (∃E) 8 ∃xFx → ¬∀x¬Fx 1–7 (→ I) [Q p.74] (viii) 1 ¬∃xFx 2 Fa 3 ∃xFx 2 (∃I) 4 ¬∃xFx 1 (RI) 5 ¬Fa 2–4 (¬I) 6 ∀x¬Fx 5 (∀I) [Q p.74] (ix) 1 ∀xx = a 2 ¬b = c 3 b = a 1 (∀E) 4 c = a 1 (∀E) 5 ¬a = c 2, 3 (= E) 6 a = a (= I) 7 ¬a = a 4, 5 (= E) 8 b = c 2–7 (¬E) [Q p.74] 294 (x) 1 Fa ∧ ¬Fb 2 a = b 3 Fa ∧ ¬Fa 1, 2 (= E) 4 Fa 3 (∧E) 5 ¬Fa 3 (∧E) 6 ¬a = b 2–5 (¬I) 7 (Fa ∧ ¬Fb)→ ¬a = b 1–6 (→ I) 8 ∀y((Fa ∧ ¬Fy)→ ¬a = y) 7 (∀I) 9 ∀x∀y((Fx ∧ ¬Fy)→ ¬x = y) 8 (∀I) [Q p.74] 4. (i) • Rules of N1 reformulated in list style: → introduction: {m} m. α A ∆ ∪ {m} n. β ∆ k. α→ β m, n (→ I) → elimination: Γ m. α ∆ n. α→ β Γ ∪ ∆ k. β m, n (→ E) ∧ introduction: Γ m. α ∆ n. β Γ ∪ ∆ k. α ∧ β m, n (∧I) ∧ elimination: Γ m. α ∧ β Γ k. α (or β) m (∧E) ¬ introduction: {m} m. α A ∆ or ∆ ∪ {m} n. β Γ or Γ ∪ {m} o. ¬β ∆ ∪ Γ k. ¬α m, n, o (¬I) 295 ¬ elimination: {m} m. ¬α A ∆ or ∆ ∪ {m} n. β Γ or Γ ∪ {m} o. ¬β ∆ ∪ Γ k. α m, n, o (¬E) ∨ introduction: Γ m. α Γ k. α ∨ β (or β ∨ α) m (∨I) ∨ elimination: Γ m. α ∨ β {n} n. α A ∆ ∪ {n} o. γ {p} p. β A Λ ∪ {p} q. γ Γ ∪ ∆ ∪Λ k. γ m, n, o, p, q (∨E) • Answers to Question 1, reformulated in list style: 1.(i) {1} 1. ¬P→ P A {2} 2. ¬P A {1, 2} 3. P 1, 2 (→ E) {1} 4. P 2, 3 (¬E) ∅ 5. (¬P→ P)→ P 1, 4 (→ I) [Q p.74] 1.(ii) {1} 1. A→ C A {2} 2. B→ C A {3} 3. A ∨ B A {4} 4. A A {1, 4} 5. C 1, 4 (→ E) {6} 6. B A {2, 6} 7. C 2, 6 (→ E) {1, 2, 3} 8. C 3, 4, 5, 6, 7 (∨E) [Q p.74] 1.(iii) {1} 1. ¬¬P A {2} 2. ¬P A {1} 3. P 1, 2 (¬E) ∅ 4. ¬¬P→ P 1, 3 (→ I) [Q p.74] 296 1.(iv) {1} 1. ¬(A ∨ B) A {2} 2. A A {2} 3. A ∨ B 2 (∨I) {1} 4. ¬A 1, 2, 3 (¬I) {6} 5. B A {6} 6. A ∨ B 5 (∨I) {1} 7. ¬B 1, 5, 6 (¬I) {1} 8. ¬A ∧ ¬B 4, 7 (∧I) [Q p.74] 1.(v) {1} 1. A A {2} 2. ¬A A {3} 3. ¬B A {1, 2} 4. B 1, 2, 3 (¬E) [Q p.74] 1.(vi) {1} 1. A→ B A {2} 2. B→ C A {3} 3. A A {1, 3} 4. B 1, 3 (→ E) {1, 2, 3} 5. C 2, 4 (→ E) {1, 2} 6. A→ C 3, 5 (→ I) [Q p.74] 1(vii) {1} 1. P→ Q A {2} 2. ¬Q A {3} 3. P A {1, 3} 4. Q 1, 3 (→ E) {1, 2} 5. ¬P 2, 3, 4 (¬I) {1} 6. ¬Q→ ¬P 2, 5 (→ I) [Q p.74] 1.(viii) {1} 1. A ∨ B A {2} 2. ¬A A {3} 3. A A {4} 4. ¬B A {2, 3} 5. B 2, 3, 4 (¬E) {6} 6. B A {1, 2} 7. B 1, 3, 5, 6 (∨I) [Q p.74] 297 1.(ix) {1} 1. P→ R A {2} 2. Q→ R A {3} 3. P ∨Q A {4} 4. P A {1, 4} 5. R 1, 4 (→ E) {6} 6. Q A {2, 6} 7. R 2, 6 (→ E) {1, 2, 3} 8. R 3, 4, 5, 6, 7 (∨E) [Q p.74] 1.(x) {1} 1. P→ Q A {2} 2. P ∧ ¬Q A {2} 3. P 2 (∧E) {1, 2} 4. Q 1, 3 (→ E) {2} 5. ¬Q 2 (∧E) {1} 6. ¬(P ∧ ¬Q) 2, 4, 5 (¬I) [Q p.74] 298 (ii) • Rules of N1 reformulated in stack style: → introduction: [α]n ... β (→ I)n α→ β → elimination: α α→ β (→ E) β ∧ introduction: α β (∧I) α ∧ β ∧ elimination: α ∧ β (∧E) α (or β) ¬ introduction: [α]n ... β ¬β (¬I)n¬α ¬ elimination: [¬α]n ... β ¬β (¬E)nα ∨ introduction: α (∨I) α ∨ β (or β ∨ α) 299 ∨ elimination: α ∨ β [α]n ... γ [β]n ... γ (∨E)nγ repetition (R): α β (R)α (This is the analogue of repetition inward (RI). We need this rule to facilitate certain applications of (¬I) and (¬E); see how it is used in the answers below. Not every stack-style natural deduction system requires this rule. For example, it is not used in the system in Dirk van Dalen, Logic and Structure [Springer, Berlin, fourth edition, 2004], which is a stack form of something similar to system N5 in the text pp.412–3.) • Answers to Question 1, reformulated in stack style: 1.(i) [¬P→ P]2 [¬P]1 (→ E) P (¬E)1P (→ I)2 (¬P→ P)→ P [Q p.74] 1.(ii) A ∨ B A→ C [A]1 (→ E) C B→ C [B]1 (→ E) C (∨E)1C [Q p.74] 1.(iii) [¬¬P]2 [¬P]1 (R)¬¬P (¬E)1P (→ I)2¬¬P→ P [Q p.74] 300 1.(iv) ¬(A ∨ B) [A]1 (∨I) A ∨ B (R) ¬(A ∨ B) (¬I)1¬A ¬(A ∨ B) [B]2 (∨I) A ∨ B (R) ¬(A ∨ B) (¬I)2¬B (∧I)¬A ∧ ¬B [Q p.74] 1.(v) A [¬B]1 (R) A ¬A (R)¬A (¬E)1B [Q p.74] 1.(vi) A→ B [A]1 (→ E) B B→ C (→ E) C (→ I)1A→ C [Q p.74] 1.(vii) P→ Q [P]1 (→ E) Q [¬Q]2 (R)¬Q (¬I)1¬P (→ I)2¬Q→ ¬P [Q p.74] 1.(viii) A ∨ B [A]2 [¬B]1 (R) A ¬A (R)¬A (¬E)1B [B]2 (∨E)2B [Q p.74] 1.(ix) P ∨Q P→ R [P]1 (→ E) R Q→ R [Q]1 (→ E) R (∨E)1R [Q p.74] 301 1.(x) P→ Q [P ∧ ¬Q]1 (∧E) P (→ E) Q [P ∧ ¬Q]2 (∧E)¬Q (R)¬Q (¬I)1¬(P ∧ ¬Q) (¬I)2¬(P ∧ ¬Q) [Q p.74] 5. Introduction: α ... β β ... α I α↔ β Elimination: α↔ β α (or β) I β (or α, if β above) [Q p.75] [Contents] 302 Answers 15.3.3 1. (i) (a) {α} ⇒ ∅ holds logically [Q p.75] (b) {α} ⇒ ∅ does not hold logically [Q p.75] (ii) (a) {α, β} ⇒ ∅ does not hold logically [Q p.75] (b) {α} ⇒ {β} and {β} ⇒ {α} both hold logically [Q p.75] 2. No answers supplied. [Q p.75] 3. No answers supplied. [Q p.75] 4. {α, β} ∪ Γ⇒∆ Γ⇒∆ ∪ {α, β} (↔⇒) {α↔ β} ∪ Γ⇒∆ {α} ∪ Γ⇒∆ ∪ {β} {β} ∪ Γ⇒∆ ∪ {α} (⇒↔) Γ⇒∆ ∪ {α↔ β} [Q p.75] 5. β α ¬α [Q p.75] [Contents] 303 Chapter 16 Set Theory There are no exercises for chapter 16. [Contents] 304 Preface Questions*5mm 1. Propositions and Arguments Exercises 1.2.1 Exercises 1.3.1 Exercises 1.4.1 Exercises 1.5.1 Exercises 1.6.1.1 Exercises 1.6.2.1 Exercises 1.6.4.1 Exercises 1.6.6*5mm 2. The Language of Propositional Logic Exercises 2.3.3 Exercises 2.3.5 Exercises 2.3.8 Exercises 2.5.1 Exercises 2.5.3.1 Exercises 2.5.4.1 Exercises 2.5.5.1*5mm 3. Semantics of Propositional Logic Exercises 3.2.1 Exercises 3.3.1 Exercises 3.4.1 Exercises 3.5.1*5mm 4. Uses of Truth Tables Exercises 4.1.2 Exercises 4.2.1 Exercises 4.3.1 Exercises 4.4.1*5mm 5. Logical Form Exercises 5.1.1 Exercises 5.2.1 Exercises 5.3.1 Exercises 5.4.1 Exercises 5.5.1*5mm 6. Connectives: Translation and Adequacy Exercises 6.5.1 Exercises 6.6.3*5mm 7. Trees for Propositional Logic Exercises 7.2.1.1 Exercises 7.2.2.1 Exercises 7.2.3.1 Exercises 7.3.1.1 Exercises 7.3.2.1 Exercises 7.3.3.1 Exercises 7.3.4.1 Exercises 7.3.5.1*5mm 8. The Language of Monadic Predicate Logic Exercises 8.2.1 Exercises 8.3.2 Exercises 8.3.5 Exercises 8.4.3.1 Exercises 8.4.5.1*5mm 9. Semantics of Monadic Predicate Logic Exercises 9.1.1 Exercises 9.2.1 Exercises 9.3.1 Exercises 9.4.3 Exercises 9.5.1*5mm 10. Trees for Monadic Predicate Logic Exercises 10.2.2 Exercises 10.3.8*5mm 11. Models, Propositions, and Ways the World Could Be*5mm 12. General Predicate Logic Exercises 12.1.3.1 Exercises 12.1.6 Exercises 12.1.9 Exercises 12.2.2 Exercises 12.3.1 Exercises 12.4.1 Exercises 12.5.4*5mm 13. Identity Exercises 13.2.2 Exercises 13.3.1 Exercises 13.4.3 Exercises 13.5.1 Exercises 13.6.1.1 Exercises 13.6.2.1 Exercises 13.6.3.1 Exercises 13.7.4*5mm 14. Metatheory Exercises 14.1.1.1 Exercises 14.1.2.1 Exercises 14.1.3.1*5mm 15. Other Methods of Proof Exercises 15.1.5 Exercises 15.2.3 Exercises 15.3.3*5mm 16. Set Theory*5mm Answers*5mm 1. Propositions and Arguments Answers 1.2.1 Answers 1.3.1 Answers 1.4.1 Answers 1.5.1 Answers 1.6.1.1 Answers 1.6.2.1 Answers 1.6.4.1 Answers 1.6.6*5mm 2. The Language of Propositional Logic Answers 2.3.3 Answers 2.3.5 Answers 2.3.8 Answers 2.5.1 Answers 2.5.3.1 Answers 2.5.4.1 Answers 2.5.5.1*5mm 3. Semantics of Propositional Logic Answers 3.2.1 Answers 3.3.1 Answers 3.4.1 Answers 3.5.1*5mm 4. Uses of Truth Tables Answers 4.1.2 Answers 4.2.1 Answers 4.3.1 Answers 4.4.1*5mm 5. Logical Form Answers 5.1.1 Answers 5.2.1 Answers 5.3.1 Answers 5.4.1 Answers 5.5.1*5mm 6. Connectives: Translation and Adequacy Answers 6.5.1 Answers 6.6.3*5mm 7. Trees for Propositional Logic Answers 7.2.1.1 Answers 7.2.2.1 Answers 7.2.3.1 Answers 7.3.1.1 Answers 7.3.2.1 Answers 7.3.3.1 Answers 7.3.4.1 Answers 7.3.5.1*5mm 8. The Language of Monadic Predicate Logic Answers 8.2.1 Answers 8.3.2 Answers 8.3.5 Answers 8.4.3.1 Answers 8.4.5.1*5mm 9. Semantics of Monadic Predicate Logic Answers 9.1.1 Answers 9.2.1 Answers 9.3.1 Answers 9.4.3 Answers 9.5.1*5mm 10. Trees for Monadic Predicate Logic Answers 10.2.2 Answers 10.3.8*5mm 11. Models, Propositions, and Ways the World Could Be*5mm 12. General Predicate Logic Answers 12.1.3.1 Answers 12.1.6 Answers 12.1.9 Answers 12.2.2 Answers 12.3.1 Answers 12.4.1 Answers 12.5.4*5mm 13. Identity Answers 13.2.2 Answers 13.3.1 Answers 13.4.3 Answers 13.5.1 Answers 13.6.1.1 Answers 13.6.2.1 Answers 13.6.3.1 Answers 13.7.4*5mm 14. Metatheory Answers 14.1.1.1 Answers 14.1.2.1 Answers 14.1.3.1*5mm 15. Other Methods of Proof Answers 15.1.5 Answers 15.2.3 Answers 15.3.3*5mm 16. Set Theory

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