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Logic Primer

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Logic_Primer_-_2nd_Edition/list.txtLogic Primer - 2nd Edition\preface.pdf Logic Primer - 2nd Edition\chap1.pdf Logic Primer - 2nd Edition\chap2.pdf Logic Primer - 2nd Edition\chap3.pdf Logic Primer - 2nd Edition\chap4.pdf Logic Primer - 2nd Edition\acknowledgements.pdf Logic Primer - 2nd Edition\answers.pdf

Logic_Primer_-_2nd_Edition/Logic Primer - 2nd Edition/acknowledgements.pdf This excerpt is provided, in screen-viewable form, for personal use only by members of MIT CogNet. Unauthorized use or dissemination of this information is expressly forbidden. If you have any questions about this material, please contact [email protected]

Acknowledgements

We were unfortunately remiss in the first edition in failing to thankHarry Stanton for his encouragement to write this text . We gratefullyacknowledge the comments we have received from colleagues whohave taught from the first edition , particularly Jon K vanvig and ChrisMenzel . A number of typo graphical errors were identified by anextremely meticulous self-study reader from the Midwest whoseidentity has become lost to us and whom we encourage to contact usagain (and to accept our apologies). Chris Menzel also receives creditfor his extensive (and voluntary ) work on the web software. Finally ,Amy Kind deserves special thanks for her help with the website andfor her most useful comments on the manuscript for the second edition .

This excerpt is provided, in screen-viewable form, for personal use only by members of MIT CogNet. Unauthorized use or dissemination of this information is expressly forbidden. If you have any questions about this material, please contact [email protected]

acknowledgements.pdf/var/tmp/PDF25889.pdfThis except is provided, in screen-viewable form, for personal use only by members of MIT CogNet

Logic_Primer_-_2nd_Edition/Logic Primer - 2nd Edition/answers.pdf This excerpt is provided, in screen-viewable form, for personal use only by members of MIT CogNet. Unauthorized use or dissemination of this information is expressly forbidden. If you have any questions about this material, please contact [email protected]

Answers to Selected Exercises

Conditional: Antecedent A; Consequent BNot wffConditional: Antecedent A; Consequent ( 8 - + C)Conditional: Antecedent (P & Q); Consequent RDisjunction: Left disjunct (A & B); Right disjunct (C - + ( 0 H 0 NegationNot wff ; requires outer parentheses to be a disjunctionNot wffNot wffConjunction: Left conjunct - (P & F); Right conjunct (P H (Q V - Q Biconditional: Left side -

8 v P) & C); Right side 0 v - 0 ) - + H)Not wff

Note : In almost all cases the answers given are not the only con-ect answers possible .

Chapter 1

Exerdse 1.1

FalseFalseFalseTrueFalseTrueFalseTrueTrueFalse

Exerdse 1.2.1

Atomic SentenceNot wffNot wff

iiiiiiivvviviiviiiixx

iiiiiiivv

viviiviiiixxxixiixiiixivxv

Ambiguous

Exercise 1.3

114 Answers to Chapter 1 Exercises

Exerdse 1.2.2

ivviviiviiixiiixiv

A -+BA -+ (8 -+ C)P& Q-+R(A & B) v (C -+ (D +-+ 0 - (P& P) v (p +-+ Q v -Q)-

8 v P) & C) +-+ D v - 0 -+ HExerdse 1.2.3

iiiillivvviviiviiiixx

P& -Q- P-+-TP-+TT -+P-P v T(or -T -+ -F)(T -+ P) -+ - U(Q & - g) -+ R- (P v R) -+-T-T v (P v R) (or the same as 8)(P& R) -+TT & - (P v R)R -+ (Q -+ P)T -+U-T -+ - (P v R)

Unambiguous: P & Q) H (-R v S AmbiguousUnambiguous: P -+ (Q & -R H -S v T) -+ U

Unambiguous: (P H (-Q v R Unambiguous: P v Q) -+ (R & S Unambiguous: (P v Q) -+ R) H S)AmbiguousAmbiguousAmbiguous

1234567891011121314

Answers to Chapter 1 Exercises ~ll

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Chapter

Exerdse 1.4.2

AAAl &El &E5&E4,6 &12,7-+E3,8 v E

116 Answers to . I Exercises

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1,31,2,3

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5312 (2)3 (3)3 (4)1,3 (5)1,3 (6)1,2,3 (7)

-Q-QvR-5

S4I2

P & (Q & R), P & R -+ -5, 5 v T J- T(1) P& (Q& R)(2) P & R -+ -5

SvTPQ& RRP& R- ST

3 (3)1 (4)1 (5)1 (6)1 (7)1,2 (8)1,2,3 (9)

-Q v R -+ -5P& TP

Sl123

1,3 v E2,4 -+E

AAA3 &E1,3 - +E5 v I2,6- +E

Answers to Chapter Exercises 117

8712

-P, R v -P H P v Q ~ Q(I) -P(2) R v -P H P v Q(3) Rv -P(4) Rv -PHPV Q(5) P v Q(6) Q

AA1 v I2HE3,4 -+E1,5 v E

I21,21,2

59 -P -+ Q & R, -P v 5 -+ - TU & -P J- (U & R) & - T1 (1) -P-+ Q& R2 (2) -Pv5 -+-T3 (3) U & -P3 (4) -P1,3 (5) Q& R1,3 (6) R3 (7) U1,3 (8) U & R3 (9) -P v 52,3 (10) -T1,2,3 (11) (U& R) & -TExerdse 1.5.1

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AAA3&E1,4 -+E5&E3&E6,7 &I4vl2,9 -+E8,10 &1

118 Answers to Chapter I Exercises

812I231,31,2

AAA1,3 -+E2,4 RAA (3)

514123

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1,31,2

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Answers to Chapter I Exercises 611

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Answers to Chapter I Exercises

-(P v Q) -it- -p & -Q-(P v Q) t- -p & -Q(1) -(P v Q)(2) P

215511

PvQ- PQPvQ-Q- P & -Q

(b)11141,41

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Al &El &EA2,4 v E3,5 RAA (4)

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21

(3)(4)(5)(6)(7)(8) 4,7 &I

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Chapter 1 ExercisesAnswers to

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Answers to Chapter

AI &EI &E3&E3&E2.4 &15.6 &I

1 Exercises122

II

(3)(4)

0' 0 '

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Al &El &E2&E2&E3,5 &I4,6 &1

P v (Q v R) -il - (P v Q) v RP v (Q v R) I- (P v Q) v R(1) P v (Q v R)(2) - P v Q) v R)(3) -P(4) Q v R(5) -Q(6) R(7) (P v Q) v R(8) P(9) P v Q(10) (P v Q) v R(11) Q(12) P v Q(13) (P v Q) v R(14) (P v Q) v R

AAA

1,3,51,3,51,2,51,2,51,2,51,21,21,21

I HE2,3 HI

P & (Q & R) -i~ (P & Q) & RP& (Q& R) ~ (P& Q) & R(1) P & (Q & R)(2) P(3) Q & R(4) Q(5) R(6) P& Q(7) (P& Q) & R

1,3 v EA4,5 v E6vI2,7 RAA (3)8 v I9vI2,10 RAA (5)11 v I12 v I2,13 RAA (2)

ExercisesAnswers to Chapter I 123

(b)I233266

(P v Q) v R r P v (Q v R)(I) (p v Q) v R(2) - (P v (Q v R (3) P(4) P v (Q v R)(5) -P(6) R(7) QvR(8) P v (Q v R)(9) -R(10) P v Q(II ) Q(12) Q v R(13) P v (Q v R)(14) P v (Q v R)

621,21,21,21,21

542(a)121151.51.5

1.21.21.21.2

1

AA [for RAA]l &El &EA [for RAA]3,5 &I6vI2,7 RAA (5)

2,11 RAA (2)

-e

- Nff

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(p & Q) v (p & R) J- P & (Q v R)(1) (P& Q) v (P& R)(2) -P(3) P&Q(4) P

AA [for RAA]A [for RAA]3&E

4,8 v B3,9 &110 v I

AAA3 v I2,4 RAA (3)A6vI7vI2,8 RAA (6)1,9 v E5,10 v E11 v

I12 v I2,13 RAA (2)

P & (Q v R) -il - (P & Q) v (P & R)P & (Q v R) I- (P & Q) v (P & R)(I) P & (Q v R)(2) -

P & Q) v (p & R (3) P(4) Q v R(5) Q(6) P& Q(7) (p & Q) v (p & R)(8) -Q(9) R(10) P& R(II ) (P& Q) v (P& R)(12) (p & Q) v (p & R)

Chapter

Exerdse 1.5.2553 PH Q~"- (p & Q) v (-P & -Q)(a) PH Q"- (p & Q) v (-P & -Q)I (I) PHQI (2) P -+ QI (3) Q -+ P4 (4) - P & Q) v (-P & -Q 5 (5) -P6 (6) Q1,6 (7) P1,5 (8) -Q1,5 (9) -P & -Q1,5 (10) (p & Q) v (-P & -Q)1,4 (II ) P1,4 (12) Q1,4 (13) P & Q1,4 (14) (p & Q) v (-P & -Q)I (IS) (p & Q) v (-P & -Q)

Answers to , I Exercises

(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)

A1 HE1 HE

(P & Q) v (-P & -Q) .- P H Q(1) (P & Q) v (-P & -Q)(2) P

(b)12

AA

2,4 RAA (3)1,5 v E6&E2,7 RAA (2)A [for RAA]A3&E10,11 RAA (3)1,12vE13&E14 vi9,15 RAA (10)16 vi9,17 RAA (9)8,18 &1

124

21,21,219103101,101,101,101,91,911

-(P& Q)P& RPP-(Q v R)-QQ-(P& Q)P& RRQvRQQvRQvRP & (Q v R)

AAA3,6 -+E5,7 RAA (6)5,8 &19 vi4,10 RAA (5)2,11 -+E11,12 &113 vi4,14 RAA (4)

~

51

.- 51

0 ' 0 ' I

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Answers to Chapter I 125

34431,31,31,2111121211I,llI,ll11

(3)(4)(5)(6)(7)(8)(9)(10)(II )(12)(13)(14)(IS)(16)(17)(18)

sss (-P v Q) & R, Q -+ SIP -+ (R -+ S)(I) (-PvQ)&R-

M ~ ~

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I1,31,2,31,2,31,2

l &E3,5 v E2,6 -+E7 -+1 (4)8 -+1 (3)

QsR-+SP -+ (R -+ S)

5561

Q & R, Q -+ P v S, - (S & R) ~ P(I) Q& R(2) Q -+ P v S

AAAl &El &E2,4 -+EA [for RAA]6,7 v E5,8 &13,9 RAA (7)

23111,271,2,71,2,71,2,3

PvS-PSS& RP

(2)(3)(4)(5)(6)(7)(8)(9)

(3)(4)(5)(6)(7)(8)(9)(10)

-(5 & R)QR

AA4&E3,5 RAA (4)1,6 v E7&E2,8 RAA (3)9 -+1 (2)A [for -+1]A [for RAA]12&E11,13 RAA (12)1,14 v E15&E16 -+1 (II )10,17 HI

Q-+SpR-PvQ

Exerdse 1.5.3

Substitution(P/R; Q/S)(p/-P; Q/Q v R; RlS)(pIP & Q; Q/R)(pIP v Q; Q/-R; RI-S)(P/R v S)(pIP v R; Q/S)(pIP; Q/Q v R)(p/- (P & Q) ; Q/R)(pIP & Q; Q/R & S)(pIP; Q/R v S; RlQ & R)

Answers to Chapter 1 Exercises126

Exerdse l .5A

S661231,31,3171,71,7

PHQr(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)

M

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AtHEtHEA4v - +

11441.4

PvQ -+P& QPHQPVQPQP& QP-+P& QQPP& QQ-+P& QP& QPVQ-+P& Q

2,5 Spec Dll

FormTransHSv CommDist &IvDNNeg~v~TCDMSim Oil

iiiiiivvviviiviiiixx

P +-+ Q -i .- - (P v Q) v - ( -P v -Q)P +-+ Q'- - (P v Q) v - (-P v -Q)(1) P +-+ Q(2) P-+Q(3) Q -+ P(4) PvQ(5) -P -+ Q(6) Q

AA [for -+1)A1,3 BP3,4 &I5 -+1 (3)A1,7 BP7,8 &I9 -+1 (7)2,6,10 Si I D Oil11 -+1 (2)

ChapterAnswers to . I Exercises 127

41,41,41,411

(7)(8)(9)(10)(11)(12)

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==

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Q

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0 - ~ > ( r ) Q ~ ~ > O \ ON - > a \ . , : ,

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~ ~ O \ - - N ~ ~ ~ ~ < ~ ~ - - - ~ N ( r ) ~ r - . ~ ~ ~ = ~

(b)I

- (p v Q) v -( -p v - Q) J- P H Q(1) -(P v Q) v -(-P v -Q)(2) P2

21,21,21,21881,81,81,811

S73

-

NN

N

677

6

I

(3) PvQ(4) - (-P v -Q)(5) P& Q(6) Q(7) P -+ Q(8) Q(9) PvQ(10) - (-P v -Q)(11) P& Q(12) P(13) Q -+ P(14) P +-+ Q

(P & Q) v (R v S) I- p & Q) v R) v S(1) (p & Q) v (R v S)(2) P& Q(3) (p & Q) v R(4) p & Q) v R) v S(5) P & Q -+ p & Q) v R) v S(6) R v S(7) R(8) (p & Q) v R(9) R -+ (p & Q) v R(10) S -+ S(11) P & Q) v R) v S(12) R v S -+

P & Q) v R) v S(13) P & Q) v R) v S

Answers to Chapter 1 Exercises8' l1

I9L

S

vv

v

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( II )

d

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(' 8 ~ l - ( L )

.

1 - ( 9 )

S 1 \ d - ( S )

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.

1 + - ' 8 ~ l ) ( )

(S 1 \ d - ) ~ ( ' 8 + - l ( z )

l) 1 \ d ( I )

S 1 \. 1 - f . L + - ' 8 ~ l ) ' ( S 1 \ d - ) ~ ( ' 8 + - l ' l ) 1 \ d

b + - . L ~ S ( 91 ) ' Z ' I

b

( ~ I )

d- ( tl )

.L H

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1 )

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S

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+- UH

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l

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H

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6

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t

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t

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H

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Answers to Chapter 1 Exercises 129

879I23

(P H -Q) -+ -R, (-P & S) v (Q & T) , S v T -+ R ~ Q -+ P(1) (P H -Q) -+- R(2) (- P & S) v (Q & T)(3) S v T -+ R

AAAA [for -+1]A [for RAA]5FA4FA6,7 HI1,8 -+E3,9 MTr10DMII &E12vl13DMA15&E15&E17DN16,18 &119 -+1 (IS)14,20 MTr2,21 v E22&EII &E23,24 RAA (5)25 -+1 (4)

880I233

AA

666

Q- p

A [for -+1]3FA4 -+1 (3)A6&E7TC8 -+1 (6)

P-+-Q-Q-+PPH - Q- R- (5 v T)-5 & - T- 5Pv -5- ( -P & - - 5)-P& 5- P5- - 5-P & - - 5-P& 5 -+ -P& - - 5- (-P & 5)Q& TT-TPQ-+P

- s v (S & R), (S -+ R) -+ P .- P(1) - S v (S & R)(2) (S -+ R) -+ P(3) - S(4) S -+ R(5) - S -+ (S -+ R)(6) S & R(7) R(8) S -+ R(9) S & R -+ (S -+ R)

4 (4 )5 ( 5 )5 ( 6 )4 ( 7 )4 ,5 ( 8 )1,4 ,5 ( 9 )1,3 ,4 ,5 ( 10 )1,3 ,4 ,5 ( 11 )1,3 ,4 ,5 ( 12 )1,3 ,4 ,5 ( 13 )1,3 ,4 ,5 ( 14 )15 ( 15 )15 ( 16 )15 ( 17 )15 ( 18 )15 ( 19 )

( 20 )1,3 ,4 ,5 ( 21 )1,2 ,3 ,4 ,5 ( 22 )1,2 ,3 ,4 ,5 ( 23 )1,3 ,4 ,5 ( 24 )1,2 ,3 ,4 ( 25 )1,2 ,3 ( 26 )

ChapterAnswers to . 1 Exercises

d

( 11 )

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+ - S ( 01 )

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rn

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Ad.

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.

L + - O

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'8 + - S ( 81 )

'8 ( LI )

b+ - d ( 91 )

b

( ~ I )

b- - ( tl )

(' 8 + - S ) + - S - ( 1 )

'8 + - S ( ' ll )

S- ( II )

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(b - ~ S ) - ( 6 )

.

L + - b - ~ S ( 8 )

.

L - ( L )

(' 8 + - S ) + - ' 8 ( 9 )

'8 + - S ( ~ )

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(J , + - b - ) + - S ( V

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'l8

S

vv

tt

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IIL

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Answers to Chapter 1 Exercises 131

1,2,3 (22) 8 -+ R1,2 (23) R v -T -+ (8 -+ R)

3,6,21 Sim Oil22 -+1 (3)

AA [for -+1]2 Neg-+A [for -+1]3&E4,5 &I1,6 -+E3&E7,8 v E9 -+1 (4)10 -+1 (2)11 v-+

S84 (p -+ Q) & (R -+ P), (p v R) & - (Q & R) .- (p & Q) & -RI (I) (p -+ Q) & (R -+ P) A2 t2) (p v R) & -(Q & R) AI (3) P -+ Q I &EI (4) R -+ P I &E2 (5) PvR 2&E2 (6) - (Q & R) 2 &E2 (7) -Qv -R 6DM2 (8) -P -+ R 5 v-+I (9) -P -+ -R 4 Trans1,2 (10) - -P 8.9 IA1,2 (II ) P 10 DNI (12) R -+ Q 3,4 HSI (13) -Q -+ -R 12 Trans2 (14) Q -+ -R 7 v-+1,2 (IS) -R 13,14 Spec Dll1,2 (16) Q 3,11 -+E1,2 (17) P & Q 11,16 &I1,2 (18) (p & Q) & -R 15,17 &I

S83 P & Q -+ R v S J- (P -+ R) v (Q -+ S)1 (1) P& Q-+RvS2 (2) - (P -+ R)2 (3) P & -R4 (4) Q2 (5) P2,4 (6) P& Q1,2,4 (7) R v S2 (8) -R1,2,4 (9) S1,2 (10) Q -+ S1 (11) -(P -+ R) -+ (Q -+ S)1 (12) (P -+ R) v (Q -+ S)

-4.5 &I1.6 -+E7&E7&E9 Neg-+A5,11 &12,12 -+E13 -+1 (II )10,14 IA8,15 v E3,16 -+E4,17 v E5TC19 Neg-+18,20 v E21 &E15,22 RAA (5)22 -+1 (4)

( I )(2)

3 ( 3 )4 (4 )5 ( 5 )4 ,5 ( 6 )1,4 ,5 ( 7 )1,4 ,5 ( 8 )1,4 ,5 ( 9 )1,4 ,5 ( 10 )II ( II )5 , 11 ( 12 )2 ,5 , 11 ( 13 )2 ,5 ( 14 )1,2 ,4 ,5 ( 15 )1,2 ,4 ,5 ( 16 )1,2 ,3 ,4 ,5 ( 17 )1,2 ,3 ,4 ,5 ( 18 )5 ( 19 )5 ( 20 )1,2 ,3 ,4 ,5 ( 21 )1,2 ,3 ,4 ,5 ( 22 )1,2 ,3 ,4 (23 )1,2 ,3 ( 24 )

132 Answers to Chapter 1 Exercises

S85 P& Q-+ (RvS) & - (R& S), R & Q-+ S,

pQP& Q(R v S) &RvS-(R & S)R -+ -SRR& QSR-+S-RS

A [for -+1]A [for RAA]

R & Q) v (- R & - Q v -P(R& Q) v (- R& -Q)-R -+ Q- (- R& -Q)R& QR-QP -+ -Q

Exerdse 1.6.1

Tl1

I- P-+P(I)(2)

T2 I- P v -P(i) primitive roles only1 (I) - (P v - f )2 (2) P2 (3) P v -P

A [for RAA]A [for RAA]2vI

-(R & S)

p -+p

s - . R & Q) v (-R& -Q v -Pr p -. -Q

P& Q- . (RvS) & - (R& S) AR& Q-. S AS -.

R & Q) v (-R& -Q v -P A

A1 - +1 (I )

A [for RAA]A [for RAA]2vl1,3 RAA (2)A [for RAA]A [for -+1]A [for RAA]5,6 RAA (7)8 -+1 (6)4,9 RAA (5)A [for -+1]10 -+1 (II )12vl1,13 RAA (I)

Answers to Chapter I Exercises 133

- pPv - PPv - P

1,3 RAA (2)4vI1,5 RAA (1)

(4)(5)(6)

(ii ) derived rules allowedI A

1 - +1 (1)2v - +

Q. .

I Q . .

t I

Q. . Q . . >

I I Q . .

-

- - -

-

N ( f ' ) " ! t '

-

- - -

A [for -+1]A [for -+1]1 -+1 (2)3 -+1 (1)

(ii) derived mles allowed1 (1) P1 (2) Q -+ P

(3) P -+ (Q -+ P)

AITC2 -+1 (I)

-

- -

-

N ( f ' )-

- -

Q-+PP -+ (Q -+ P)

II

T4 t- P -+ (Q -+ P)(i) primitive rules onlyI P2 QI

T5 I- (P -+ Q) v (Q -+ P)(i) primitive roles onlyI (I) - P -+ Q) v (Q -+ P 2 (2) P -+ Q2 (3) (P -+ Q) v (Q -+ P)I (4) - (P -+ Q)5 (5) -P6 (6) P7 (7) -Q5,6 (8) Q5 (9) P -+ QI (10) PII (II ) QI (12) Q -+ PI (13) (P -+ Q) v (Q -+ P)

(14) (P -+ Q) v (Q -+ P)

Answers to Chapter Exerci~134

(ii) derived rules allowed1 (I) -

P -+ Q) v (Q -+ P I (2) - (p -+ Q) & -(Q -+ P)1 (3) - (p -+ Q)1 (4) -(Q -+ P)1 (5) P& -Q1 (6) Q & -P1 (7) P1 (8) -P

(9) (P -+ Q) v (Q -+ P)

--11

-

~ ~

~

~ ~ ll ~ ~ ~ ~ ~

-

NN ( f " ) " f1 ' In \ oro : t ~ to ' t

5I] ~ O ' ~ O ' ~ I ~ O ' I ~ ~ O ' ~ ~ e : - g ~ I ~ O ' ~ O ' ~

tu~

e

- - - - - - - - - - - - - - -

0 - - - - - - _ - - O - N ~ ~ ~ ~ ~ ~ ~ O

N

~ ~

IU- N ~ ~ ~ ~ ~ ~ ~ - - - - - - - - - - ~ N

NN

N

~

> - - - - - - - - - - - - - - - - - - - - - - - -

:. c :

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~

i ~ 0 NN

~

-

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~

~

~ ~ ~ O ~ ~ ~ ~

~

~ - N ~ ~ NN

~ - - - - ~ ~ - -

- - - ~ - - N -3.11 v E12 -+1 (10)9.13 RAA (3)14 -+1 (2)IS -+1 (1)A [for -+1]A [for -+1]A [for RAA]A18.20 -+E19.21 RAA (20)A17.23 -+E

AtOy I

144

41,41,41991,91,911

ChapterAnswers to I Exercises 145

17,18,19 (25) - (Q ~ P)26 (26) Q26 (27) Q v P19,26 (28) P19 (29) Q ~ P17,18 (30) Q17 (31) (P ~ Q) ~ Q

(32) Q ~ P) ~ P) ~

P ~ Q) ~ Q)(33) (P~ Q) ~ QH (Q~ P) ~ P

22.24 RAA (23)

A1 Neg -+2&E2&E4 Neg -+5&E5&E7FA3,8 -+E6,9 -+E7,10 RAA (1)A12 Neg -+13&E13&E15 Neg -+16&E16&E18FA14,19 -+E17,20 -+E18,21 RAA (12)11,22 HI

A26 v I19,27 v E28 -+1 (26)25,29 RAA (19)30 -+1 (18)31 -+1 (17)16,32 HI

Chapter

20,28 RAA (22)28 v I20,30 RAA (20)27 v I21,32 RAA (22)33 v I21,34 RAA (21)30,35 &I

(i) primitive rules onlyI (I) (p & Q) v (R & S)2 (2) - (P v R)3 (3) - (P v S)4 (4) -P5 (5) P & Q5 (6) P4 (7) - (P& Q)1,4 (8) R & S1,4 (9) R1,4 (10) S1,4 (11) P v R1,2 (12) P1,2 (13) P v RI (14) P v R1,4 (15) P v S1,3 (16) P1,3 (17) P v SI (18) PvSI (19) (P v R) & (P v S)20 (20) -(Q v R)21 (21) - (Q v S)22 (22) -Q5 (23) Q22 (24) -(P & Q)1,22 (25) R & S1,22 (26) R1,22 (27) S1,22 (28) Q v R1,20 (29) Q1,20 (30) Q v RI (31) Q v R1,22 (32) Q v S1,21 (33) Q1,21 (34) Q v SI (35) Q v SI (36) (Q v R) & (Q v S)

1 Exercises

~

0-

~

Answers to

J- (P & 0) v (R & S) H (P v R) & (P v S & Q v R) & (Q v S A [for -+1]A [for RAA]A [for RAA]AA5&E4,6 RAA (5)

1,7 v E8&E8&E

25&E26 v I

9 v I2,11 RAA (4)12 v I2,13 RAA (2)10 v I3,15 RAA (4)16 v I3,17 RAA (3)14,18 &1A [for RAA]A [for RAA]A5&E22,23 RAA (5)1,24 v E25&E

Chapter

mleg

Ot

UaM

SUY

I Exercises Ltl

(8E )

(LE )

s Ab

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(SA

d) ~ ( ' HA

d)

S A

b ) ~ ( ' H A b ~ SAd

) ~ ( ' HAd

S A

b ) ~ ( ' H A b ~ SAd

) ~ ( ' HAd

+- ( S ~ ' H > A m ~ d )

S A

b ) ~ ( ' H A b ~ SAd

) ~ ( ' HAd

(I ) I + - L

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9 ' 61

S- : W ' H

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HS

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II

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d

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d-

(S ~ ' H ) A ( 0 ~ d )

(s ~

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b ) ~ ( SA

d)

(' H

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)

(Z9 )

(19 ) 6

(09 ) 9t ' 6

(6 ~ ) 9t ' 6

(8 ~ ) 9t ' 6

(L ~ ) 9t ' 6

(9 ~ ) 9t ' 6

(~ ~ ) Lt ' 9t ' 6

(t ~ ) Lt ' 9t ' 6

( ~ ) Lt ' 9t ' 6

(Z ~ ) Lt ' 9t ' 6

(I ~ ) Lt ' 9t

(O ~ ) 8t ' Lt

(6t ) 8t ' Lt

(8t ) 8t

(Lt ) Lt

(9t ) 9t

(~ t ) 6

(tt ) 6

( t ) 6

(Zt ) 6

(It ) 6

(Qt ) 6

(6 ) 6

(6 ) I + - 19

(9t ) VY

H

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6 ~

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3A

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H

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Exercises148 Answers to Chapter I

-

- - - - - -

5&E6&E7,8 &15&E6&E10,11 &19,12 &113 -+1 (1)

15151515151515151515151515

(IS)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)(28)

A15&E15&E16&E16&E17&E17&E18,20 &119,21 &122 Dist23 Dist24,25 &1

(P v R) & (P v S)(Q v R) & (Q v S)PvRPvSQvRQvS(P v R) & (Q v R)(P v S) & (Q v S)(P& Q) v R(P& Q) v S

(29) (P & Q) v (R & S) H

T31 J- (P v Q) & (R v S) H P & R) v (P & S v Q & R) v (Q & S (i) primitive rules only1 (I) (P v Q) & (R v S)2 (2) -

(P & R) v (P & S v Q & R) v (Q & S )3 (3) P & R3 (4) (P & R) v (P & S)3 (5)

P & R) v (P & S v Q & R) v (Q & S 2 (6) - (P & R)7 (7) P & S

A [for -+1]A [for RAA]A3 v I4vI2,5 RAA (3)A

(P & Q) v (R & S) P v R) & (P v S &

Q v R) & (Q v S

26 Dist27 -+1 (15)

14,28 HI

(7) PvR(8) P v S(9) (P v R) & (P v S)(10) Q v R(11) Q v S(12) (Q v R) & (Q v S)(13) P v R) & (P v S &

Q v R) & (Q v S (14) (P & Q) v (R & S) -+

P v R) & (P v S & Q v R) & (Q v S P v R) & (P v S & Q v R) & (Q v S

P & Q) v R) & P & Q) v S)(P & Q) v (R & S) P v R) & (P v S &

Q v R) & (Q v S -+

A21,22 &16,23 RAA (22)20,24 v E21,25 &110,26 RAA (21)19,27 v E22,28 &114,29 RAA (22)20,30 v E28,31 &118,32 RAA (2)33 -+1 (1)

(8)(9)(10)(II )(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)(28)(29)(30)(31)(32)(33)(34)

M

- - M

M- C " ! . C " ! . C " ! .

-

M " ' : C " ! . C " ! . C " ! . C " ! . C " ! . C " ! . C " ! . C " ! . C " ! .

-

- MM

MM

- - - - - - - - -

r- . r - . r - .

tf' ) tf ' ) tf ' )

'ot $ ' ot $ ' ot $ \ O

tf' ) tf ' ) tf ' ) tf ' )

149Answers to Chapter 1 Exercises

7 v I8 v I2,9 RAA (7)A

772IIIIII21515152

-

-

-

In-

-

'- ' ' - '

-

- ~

- - ~

>

> ~

>

> r - -

-

N ~ . . . . . In \ O ~-

- N . . . . - - N(35)(36)(37)(38)(39)(40)(41)(42)(43)(44)(45)

A for - +135363738383836

-

-

~

r - .~

~

~

-

-

~

tIJ _ ~ ~ tIJ _ ~

cS

citd

> ~ ~ citd

> ~

~

~

~

~

.. . . . . . . . . . . . . . ~ O \ \ O

r- . N ~ \ O

.. . . . . . . . . . . ~ ~ ~ ~ ~ ~ ~

I &El &EA

(P & R) v (P & S) P & R) v (P & S v Q & R) v (Q & S - (P & S)Q& R(Q & R) v (Q & S) P & R) v (P & S v Q & R) v (Q & S - (Q & R)Q& S(Q & R) v (Q & S) P & R) v (P & S v Q & R) v (Q & S -(Q & S)PvQRvSPRP& R-RSP& S-PQQ& R-RSQ& S P & R) v (P & S v Q & R) v (Q & S (P v Q) & (R v S) -+ P & R) v (P & S v Q & R) v (Q & S P & R) v (P & S v Q & R) v (Q & S -(pvQ)(P & R) v (P & S)P& RPPvQ-(P & R)P& SPPvQ-

P & R) v (P & S

Answers to Chapter Exercises

35,36 (46)47 (47)47 (48)47 (49)36 (50)35,3635,3635,3635 36,53 RAA (36)

A [for RAA]AA57&E58 v I55,58 RAA (57)56,00 v E61&E62 v I55,63 RAA (56)35,64 v EA66&E67 v I55,68 RAA (66)

55565757575555,5655,5655,565535,556666665535,5535,5535,553535

150

Q & R) v (Q & S Q& RQPvQ-(Q & R)Q& SQPvQPvQ-(R v S)(Q & R) v (Q & S)Q& RR

-~

~

~ ~

~

~ > ~ ~ ~ >

.

n < r - - ~ ~ ~ - N

f" ) ~ ~ f " ) ~ V ) V )

-

- - - - - - - - - - - - - - - - - - - - - - - -

-

-

N ~ ~ ~ ~ ~ ~ ~ O - N ~ ~ ~ ~ ~ ~ ~ ON

~ ~ ~

~

~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

~

-

- - - - - - - - - - - - - - - - - - - - - - - -

-

--r

. f )~

v ;~

~

>

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-

2 >

-

~

-

~

r. f ) ~ r . f ) r . f ) ~ ~ ~ r . f ) ~ r . f )

>O ' ~ > ~ ~ ~ >~

~

~

' T ' 0 ' r . f ) ~ ' T ' ~ ~ ~ ~ 1 ~ r . f )

RvSRvS(P v Q) & (R v S) P & R) v (P & S

v Q & R) v (Q & S -+

(P v Q) & (R v S)(P v Q) & (R v S) H P & R) v (P & S v Q & R) v (Q & S

71 v I55,72 RAA (55)54,73 &174 --+1 (35)

34,7S HI

(ii) derived rules allowed1 (1) (P v Q) & (R v S)1 (2) (P & (R v S v (Q & (R v S 3 (3) P & (R v S)

A1 DistA

65,69 v E70&E

PvQ 21 v I(P v Q) & (R v S) 22.23 &I(Q & R) v (Q & S) -+ (P v Q) & (R v S) 24 -+1 (19)(P v Q) & (R v S) 11.18.25 SimDil P & R) v (P & S v Q & R) v (Q & S -+ 26 -+1 (11)

primitive rules only(1) (p -+ Q) & (R -+ S) A [for -+1](2) - ( -P & -R) v(-P & S v Q & -R) v(Q & S ) A [for RAA](3) P -+ Q 1 &E(4) R -+ S 1 &E(5) (-P & -R) v (-P & S) A

Answers to Chapter 1 Exercises 151

(4)(5)(6)(7)(8)(9)(10)

3 Dist4 -+1 (3)A6 Dist7 -+1 (6)

\0 \ 0

11121212121212

-

- - - - - - - - - - - - - - - -

-

N ~ ~ ~ ~ ~ ~ ~ O - N ~ ~ ~ ~ ~

-

- - - - - - - - NN

NN

NN

NN

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T32 .- (P -+ Q) & (R -+ S) H -P & -R) v (-P & S v Q & -R) v (Q & S (i)12115

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(P v Q) & (R v S) P & R) v (P & S v Q & R) v (Q & S

(P & R) v (P & S)P & (R v S) -+ (P & R) v (P & S)Q & (R v S)(Q& R) v (Q& S)Q & (R v S) -+ (Q & R) v (Q & S) P & R) v (P & S v Q & R) v (Q & S (P v Q) & (R v S) -+ P & R) v (P & S v Q & R) v (Q & S P & R) v (P & S v Q & R) v (Q & S A

A12 Dist13&E13&E14vl15.16 &I17 -+1 (12)A19 Dist20&E20&E

1919

11

Answers to Chapter I Exercises152

-

- - - - - - - - - - - - - - - - - - - - - - - - - -

- - - - - - -

-

- - - O - N ~ " ' f ' ~ ~ ~ ~ ~ O - N ~ ~ ~ ~ ~ ~ ~ ON

~ " ' f ' ~ ~ ~ ~ ~ O - ~ ~

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NN

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- - - - - - -

-P & -R) v (-P & S

v Q & -R) v (Q & S 5 v I

-

-P & -R) v (-P & S

2,6 RAA (5)(Q & -R) v (Q & S) A

-P & -R) v (-P & S

v Q & -R) v (Q & S 8 v I

-

Q & -R) v (Q & S 2,9 RAA (8)(-P & -R) A(-P & -R) v (-P & S) 11 v I- (-P & -R) 7,12 RAA (11)-P & S A(-P & -R) v (-P & S) 14 v I-(-P & S) 7,15 RAA (14)Q & -R A(Q & -R) v (Q & S) 17 v I-(Q & -R) 10,18 RAA (17)Q& S A(Q & -R) v (Q & S) 20 v I- (Q & S) 10,21 RAA (20)-P ARAP

& -R 23,24 &1R 13,25 RAA (24)S 4,26 -+E-P & S 23,27 &1P 16,28 RAA (23)Q 3,29 -+EQ & -R 24,30 &1R 19,31 RAA (24)S 4,32 -+EQ & S 30,33 &1

~

ff ' ) ~ ~ ~

r- . 00

ff ' ) ~ ~ ~ ~ ~ ~ ~ ~ N . ~ N .

-

NN

NN

NM

NN

- - - - - - - -

-P & -R) v (-P & S

v Q & -R) v (Q & S 22,34 RAA (2)

(P -+ Q) & (R -+ S) -+ 35 -+1 (1)

-P & -R) v (-P & S v Q & -R) v (Q & S

-P & -R) v (-P & S

v Q & -R) v (Q & S A [for -+1]

P A [for -+1]37383940403943

-QQ& -RQ-(Q & -R)Q& S

A [for RAA]A4O&E39,41 RAA (40)A

52882111121414217

Answers to Chapter 1 Exercises

A42,46 v B

(ii) derived rules allowed1 (1) (p -+ Q) & (R -+ S)1 (2) P -+ Q

153

43 &E

45,47 RAA (46)37,48 v EA50&E38,51 RAA (50)49,52 v E53&E38,54 RAA (39)55 ~ I (38)A [for ~ I]

-P & -R) v (-P & S v Q & -R) v (Q & S -+ 74 -+1 (37)(P -+ Q) & (R -+ S)

(76) (P -+ Q) & (R -+ S) H 36,7S HI

-P & -R) V (-P & S V Q & -R) V (Q & S

Al &E

39,44 RAA (43)

(-P & -R) v (-P & S)- P & -R-R- ( -P & -R)- P& SSSR-+S(P -+ Q) & (R -+ S)

37,65 v EA67&E57,68 RAA (67)66,69 v E70&E58,71 RAA (58)72 -+1 (57)56,73 &1

43 (44)39 (45)46 (46)39,46 (47)39 (48)37,39 (49)50 (50)50 (51)38 (52)37,38,39 (53)37,38,39 (54)37,38 (55)37 (56)57 (57)58 (58)59 (59)59 (60)58 (61)62 (62)58,62 (63)58,62 (64)57,58 (65)37,57,58 (66)67 (67)67 (68)57 (69)37,57,58 (70)37,57,58 (71)37,57 (72)37 (73)37 (74)

(75)

A [for RAA]A59&E58,60 RAA (59)A61,62 v E63&E57,64 RAA (62)

Answers to Chapter Exercises154

1111188

(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)

-

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161717

(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)(28)(29)(30)

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Answers to Chapter 2 Exercises 155

ii- {P& Q) v PTTFT TTFTFTFTTFTFFTFT

O' ~ tI . o ~ tI . o

~

~ ~ tI . otl

. oPv (-PvQ ) PQ

TF TTF FTT TTT T

iiiPQTTTFFTFF

iv(pv Q) v (-P& Q)

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viRH -Pv (R& Q)

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(P H - Q) H (-P H - Q)FF F F TFTT FF FTTF FT FFFT FT TT

QRTTTFFTFFTTTFFTFF

Chapter 2

Exerdse 2.1

156 2 ExercisesAnswers to Chapter

ixPQRTTTTTFTFTTFFFTTFTFFFTFFF

A [for RAA]l &E2,3 BPl &E4,5 RAA (2)

(p+-+Q) +-+(pv R-+ (- Q -+ R T T T T F TT T T T F TF F T T T TF T T FTFF F T TFTF F F T F TT T T T T TT T F T T F

xP Q R S (P & Q) v (R & S) -+ (P & R) v (Q & S)TT T T T T T T T T TTT T F T T F T T T FTTFT T T F T F T TTTFF T T F F F F FTF T T F T T T T T FTF T F F F F T T T FTF F T F F F T F F FTF F F F F F T F F FFT T T F T T T F T TFT T F F F F T F F FFTFT F F F T F T TFTFF F F F T F F FFF T T F T T F F F FFF T F F F F T F F FFF F T F F F T F F FFF F F F F F T F F F

Exerdse 2.2

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Chapter

INVALIDP:F Q:F

158 Answers to 2 Exercises

-

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3 -+1 (2)A [for -+1]SPA6 -+1 (5)1,4,7 Corn Oil

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VALID(1)(2)(3)(4)(5)(6)(7)(8)(9)

xiv VALID1 (1) P -+ (Q & R -+ S)2 (2) P3 (3) -S4 (4) Q& R1,2 (5) Q & R -+ S1,2,4 (6) S1,2,3 (7) - (Q & R)

AAAA [for RAA]1,2 -+E4,5 -+E2,6 RAA (4)

xvR:T

R& -PRQQvR-(Q v R)- (Q H R)

55

1

P -+ (-P -+ R)Q-Q-+RQ -+ (-Q -+ R)(-P -+ R) v (-Q -+ R)

Answers to Chapter 2 Exercises 159

,-

'>

' >

~

~

INV A LffiP:T Q:F R:F S:F T:FVALID

1234

(1)(2)(3)(4)(5)(6)(7)(8)

AAAA [for -+1]4 Neg-+5 Comm3.6 BP

A [for RAA]8 v I2.9 M1T1.10 -+E7.11 -+E

12&E10

.13 RAA (8)14 -+1 (4)15 v-+

0-

I~

o - ~~

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Q:F R:F S:F T:F U:F V:T W:FExerdse 2.4.2

P:FPiTP:TP:FPiTP:FPiTP:FPiTP:TPiTP:FVALID(1)

Q:TQ:FQ:TQ:FQ:FQ:FQ:FQ:FQ:TQ:FQ:FQ:T

R:FRiTR:TR:FRiTR:TR:FRiTR:FR:F

f- . ~

f - . ~ ~ f - .

ri5ri5

ri5

ri5ri5

ri5

T:T

T:F

T:TTiT

P - + (- Q - + - R & - 8)

Q-+(P-+R& -Q)-Q-+- (TvV )U & SHP- (S -+ -V)S& UU & SPTTvVQP-+R& -Q

iiiiiiivvviviiviiiixxxixiixiii1

443,488 (9)2,8 (10)1,2,8 (II )1,2,3,4,8 (12)1,2,3,4,8 (13)1,2,3,4 (14)1,2,3 (15) - (5 -+ -U) -+-T1,2,3 (16) (5 -+ -U) v -Tx viii INV ALID

VALWAA2TC2FA3,4 &11,5 -+E2,6 MTr7DM8 Neg-+9 -+1 (2)

Answers to Chapter 2 Exercises160

2341,41,3,41,3,41,3,41,3,41,3,41,3,41,2,3

-

- -

-

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-CI)

:t

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AAA

- Q - + - R & - S- R & - S- R- SR - + SS - + RRHS- p

1,4 -+E3,5 -+E6 &E6&E7FA8FA9,10 HI

P:F Q:T RiT S:FPiT Q:F RiT S:FPiT Q:F RiT S:FVALID(1) - (p -+ -Q & R)(2) -R H -P(3) P & - (-Q & R)(4) P(5) - (-Q & R)(6) - -Q v -R(7) R(8) - -Q(9) Q(10) P& Q

xivxvxvixviiI

T:FT:F U:F

AA2

111

1 Neg-+3&E3&E50M2,4 BT6,7 v ESON4,9 &1

(p -+ Q) & (-Q -+ P & R) -+ (5 v T -+ -Q)QP-+Q-Q-+P& R(p -+ Q) & (-Q -+ P & R)5vT -+-Q

-

NN

NN

(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)

1,21,21,21,21

2,11 RAA (4)

- (S v T)- S & - T- ( -S -+ T)Q -+ - (-S -+ T)

11,21,21,21,2

xviii

A [for -+1]A [for RAA]2vl2,3 -+E4&E2,5 RAA(2)6 -+1(1)

Exerdse 3.1.1

given the

Answers to Chapter 3 Exercises 161

xix V A LillI (I)2 (2)2 (3)1,2 (4)1,2 (5)I (6)

(7)8 (8)9 (9)8,9 (10)8,9 (II )8 (12)

(13)(14)

P v - Q - + - P & - QPPv - Q- P & - Q- P- P

-

~

~

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tt _ 0 : : - 00 ' -

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.. . . . . ~ ~ ' t = : : ' t = : : : : I : ~ ~

c2c2

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V A Lillxx121,21111

(I)(2)(3)(4)(5)(6)(7)

QH -QQ-Q-QQQV (PH -P)PH -P 4,6 v B

.-

.-

. -

.-

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Chapter 3

Not a wffNot a wffUniversalExistentialNot a wffNot a wffUniversalNot a wff ( but acceptable biconditional abbreviationdropping convention )

P v - Q) -+ (-P & -Q -+- P- PPv - Q-Q- P & -Q(P v -Q) -+ (- P & -Q)-P -+

P v -Q) -+ (- P & -Q

P v - Q) -+ (-P & -Q +-+- P

Chapter

Biconditional

Exercise 3.1.2Open FonnulaFzNoneGcax

3 Exercises162 Answers to

xiixiiixivxvxvixviixviiixixxxxxixxiixxiiixxivxxx Not a wff

Example WFF3zFzi

iiilliv

3y- VxGxy- Vy- HyVxAx3xFxx

v

vi

vii

GxyGyx(Gxy & Gyx)'v'y(Gxy & Gyx)GxyHyAxFxxFxyHxyzJz(Hxyz & Jz)'v'z(Hxyz & Jz)(Fxy -+ 'v'z(Hxyz & Jz 'v'y(Fxy -+ 'v'z(Hxyz & Jz

Not a wffNot a wffNot a wffUniversalNot a wffNot a wffNegation

NegationNot a wff (but acceptable conditional abbreviation given the parenthesis-dropping convention)Not a wff (but acceptable conditional abbreviation given the parenthesis-dropping convention)NegationNot a wffExistentialNot a wffAtomic sentence

ixx

Xl

Chapter

Translation scheme is provided only where it is not obvious.alt: indicates an alternative, logically equivalent ttanslation.amb: indicates non

-equivalent rendering of an ambiguous sentence.inc ! indicates a common, but incorrect answer.

all:inc!

163Answers to , 3 Exercises

viii FxxFxy

'v'xFx

Exercise 3.1.

Vx(Dx -+ Mx)Vx(Dx & Mx)3x(Sx & Ox)3x(Sx -+ Ox)Vx(Fx -+ -Ex)-3x(Fx & Ex)-Vx(Fx -+ Ex)- Vx(Fx -+ Px)3x(Fx & - Px)

alt: 'v'x( Rx v Ax - + - Ex)inc! 'v'x( Rx & Ax - + - Ex)

'v'x( Rx - + - Ex) & 'v'x(Ax - + - Ex)

'v'yFxyHzIx(Hz v Ix)3z(Hz v Ix)- 3z(Hz v Ix)Fxx(Ha v Fxx)- (Ha v Fxx)NoneFxFxFyyy(Fyyy & P)FzxHxyz(Fzy H Hxyz)

xixiixiiixiv

xv

Iinc!

2inc!

3

regional

Translation scheme I using single-place predicates only:Ta : a is a trickWa : a is a whaleSa: Shamu can do aCa: a can do a tricks: ShamuNote: Strictly we should use a letter from a-d for a name, but the use of s forShamu is more perspicuous.

164 Answers to Chapter 3 Exercises

3x(Px & Ax) & 3x(Rx & Ax)3x

Px & Rx) & Ax)3x

Px v Rx) & Ax)'v'x(Ox -+ Lx)'v'x(Lx -+ Ox)'v'x(Sx -+ (Px -+ Tx v Bx 'v'x(Sx & Px -+ Tx v Bx)'v'x(Mx H Px)'v'x

Mx -+ Px) & (Px -+ Mx 'v'x(Fx -+ -Wx vEx )'v'x(Fx & Wx -+ Ex)

inc!inc!

dialects of English]

(3x(Ox & Cx) & 3x(Ox & Mx & - 3x(Cx & Mx)'V'x(lx - + Px)'V'x( - Px - + - Ix)'v'x(Px - + Ix)'v'x(Ax - + (- Wx - + Nx

'v'x(Ax - + (Nx - + - Wx

'v'x(Mx & Wx -+ Bx)3x

Mx & Wx) & Fx)

7inc!

8alt:

9alt:

10alt:

1112alt:inc!

131415alt:

161718alt:

19arnb:

202122

23-29

3x(Sx & (Px & Fx & -Vx(Px & Fx -+ Sx)3x(Sx & (Px & Fx & 3x Px & Fx) & -Sx)Wa & 'v'x( Wx -+ Mx) -+ Ma'v'x(Sx -+ (- Nx v Mx 'v'x(Ox & Ex -+ -Px)-3x

Ox & Ex) & Px)'v'x(Px -+ - Hx)- 'v'x(Px -+ Hx) [possible reading in some- 'v'x(Px -+ Cx)

23 'v'x(Tx -+ Sx)24 'v'x(Tx -+ Sx)25 - 'v'x(Tx -+ Sx)26 'v'x(Tx -+ -Sx)alternative translation that is not logically equivalent: -Cs

27 3x(Wx & Cx) -+ Cs'v'x( Wx & Cx -+ Cs) Note difference in scope.

amb i: 'v'x( Wx -+ Cx) -+ Cs Less natural reading.amb ii : If any whale can do a trick, Shamu can do that same trick.This reading is not expressible using single-place predicates only.

28 'v'x( Wx -+ Cx) -+ Cs Note scope again.29 3x( Wx & Cx) -+ 'v'x( Wx -+ Cx)amb 'v'x( Wx -+ Cx) -+ 'v'x( Wx -+ Cx) This reading is less natural.

23-29 Translation Scheme nTa : a is a hickCap: a can do pWa: a is a whales: Shamu

lI~ing predicates

165Answers to Chapter 3 Exercises

many -place

232425

'v'x(Tx -+ Csx)'v'x(Tx -+ Csx)- 'v'x(Tx -+ Csx)

alt: 3x(Tx & - Csx)26 'v'x(Tx -+ - Csx)27 3xy Wx & Ty) & Cxy) -+ 3z(Tz & Csz)alt: 'v'xy ( Wx & Ty) & Cxy) -+ 3z(Tz & Csz Scope!amb-i: 'v'x3y(Wx -+ Ty & Cxy) -+ 3z(Tz & Csz)amb-ii : 'v'xy ( Wx &Ty) & Cxy) -+ Csy)

This is the ambiguous reading not expressible with the previous b' anslation scheme.28 'v'x3y( Wx -+ (Ty & Cxy -+ 3z(Tz & Csz)alt: 'v'x( Wx -+ 3y(Ty & Cxy -+ 3z(Tz & Csz)

29 3xy Wx & Ty) & Cxy) -+ 'v'x( Wx -+ 3y(Ty & Cxy amb: 'v'x3y( Wx -+ (Ty & Cxy -+ 'v'x( Wx -+ 3y(Ty & Cxy

3031

Agb3xAxb

Chapter166

32 3xAgx33 VxAbx34 VxAxb35 3xyAxy36 3x Vy Axyamb: Vy3x Axy

37 Vx3y Axyamb: 3y Vx Axy

38 VxyAxy39 VxAxx40 3xAxx41 Vx-Axxalt: - 3xAxx

42 3xVy-Axyalt: 3x-3yAxy

Translation Scheme for 43-46.Sa Jiy; a said Ii to 'Yfa : a is a person

43 Vx(Px -+ 3yVz(Pz -+ Sxyz amb: Vxy(Px & Py -+ 3zSxzy

44 Vx(Px -+ 3yz(Pz & Sxyz 45 Vx(Px -+ 3y(Py & -3zSxzy 46 Vxy(Px & Py -+ -3zSxzy)47 3xyz Rx & (Cy & Sxy & (Dz & Lxz 48 3x(Fx & Vy(Hy -+ Sxy alt: 3xVy(Fx & (Hy -+ Sxy amb: 3x(Fx & 3y( Hy & Sxy

49 3x(Fx & Vy( My -+ Sxy 50 3x( Wx & Vy(Fy & Exy -+ My amb: 3x( Wx & Vy(Exy -+ Fy & My

51 3x( Wx & Vy(Fy & My -+ -Exy 52 3xy My & Fy) & Exy) -+ Vx(Sx -+ 3y My & Fy) & Exy amb: 3xy My & Fy) & Exy) -+ 3x(Sx & 3y My & Fy) & Exy

53 Vwxyz Jw & Txw) & (Oy & Tzy) -+ Lxz) [Tali: a is Ii's tail]

Answers to 3 Exercises

'v'wxyz (Jw & Tx) & Bxw) & Oy & Tz) & Bzy) -+ Lxz) (Bali: a belongsto Ii]'v'x(3y(Cy & Sxy) -+ Ax)'v'xy(Cy & Sxy -+ Ax)'v'uvxy(Bu & Pvu - + (Hx & Pyx - + Mvy &'v'uvwx(Ou & Evu - + Mw v Bw) & Exw - + Avx & - Mvx Ambiguous.i. The amount eaten by some whales is more than the amount eaten by anyfish.Translation scheme:Au : a is an amount (of food)Fa: a is a fishEafi : a eats fi amount (of food)Gafi : a is greater than fi

167Answers to Chapter 3 Exercises

-- alt :

(Using identity)58 3x(Cx & 'v'y(Cy -+ y=x 59 3x x=p

alt : 3x(x=p & 'v'y(y=p -+ y=x 60 3xy (Tx & Ty) & x * y) & (Ebx & Eby 61 'v'x(x~ -+ Exb) & -Ebb62 'v'x(Dx -+ 3y Ty & Byx) & 'v'z(Tz & Bzx -+ y=z

3x( Wx & 3y Ay & Exy) & 'v'zw(Fz & Aw & Ezw - + Gyw )

ii . The amount eaten by some whales is more than the amount eaten by allthe fishes combined.Addition to translation scheme:a: the amount eaten by all the fishes combined

3xy Wx & Ay) & (Exy & Gya 3x( Mx & 'v'y( My -+ (Gxy H -Gyy )

54alt:

55

56

Exerdse 3.3.1

i is an instance of vii is an instance of viii is an instance ix

Chapter

S88122

168 Answers to 3 Exercises

S87 3x(Gx & -Fx), 'v'x(Gx -+ Hx) ~ 3x(Hx & -Fx)1 (1) 3x(Gx & -Fx)2 (2) 'v'x(Gx -+ Hx)2 (3) Ga -+ Ha4 (4) Ga & -Fa4 (5) Ga2,4 (6) Ha4 (7) -Fa2,4 (8) Ha & -Fa2,4 (9) 3x(Hx & -Fx)1,2 (10) 3x(Hx & -Fx)

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Chapter172 Answers to . 3 Exercises

A531

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AA [for -+1]A [for RAA]A4313,5 RAA (4)6VI2,7 RAA (3)A [for 3E on 8]I ' VE9,10 BT8,11 3E (9)A [for RAA]A [for 3E on 13]10,14 BP13,153E(14)12,16 RAA (13)A [for RAA]183117,19 RAA (18)20 VI

'v'xFx I- -3xOx +-+ - (3x(Fx & Ox) & 'v'y(Oy -+ Fy 'v'xFx-3xOx3x(Fx & Ox) & 'v'y(Oy -+ Fy)3x(Fx & Ox)Fa & OaOa3xOx3xOx- (3x(Fx & Ox) & 'v'y(Oy -+ Fy -3xOx -+ - (3x(Fx & Ox) & 'v'y(Oy -+ Fy 3xOxOaFaOa -+ Fa'v'x(Ox -+ Fx)Fa & Oa3x(Fx & Ox)3x(Fx & Ox) & 'v'x(Ox -+ Fx)3x(Fx & Ox) & 'v'x(Ox -+ Fx)3xOx -+ 3x(Fx & Ox) & 'v'x(Ox -+ Fx)- (3x(Fx & Ox) & 'v'x(Ox -+ Fx -+ - 3xOx-3xOx +-+ - (3x(Fx & Ox) & 'v'y(Oy -+ Fy

12 'v'x(3yFyx -+ 'v'zFxz) ~ 'v'yx(Fyx(1) 'v'x(3yFyx -+ 'v'zFxz)(2) Fab(3) 3yFyb

173Answers to Chapter 3 Exercises

313

(3)(4)(5)(6)(7)(8)(9)(10)

'v'y(Fy - + Gy)Da - + FaFa - + GaDa - + GaGa

1,31,2,31,211

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S130*123416222,642,4,62,4,62,4,62,4,61,2,41,2,31,2

175

S127*I233363

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' v' x3y Gyx , ' v' xy ( Gxy - + - Gyx ) r - 3y ' v' x ( x : #y - + Gyx )( I ) ' v' x3y Gyx( 2 ) ' v' xy ( Gxy - + - Gyx )( 3 ) 3y ' v' x ( x : #y - + Gyx )(4 ) ' v' x ( x ~ - + Gax )( 5 ) 3yGya( 6 ) Gba( 7 ) ' v' y ( Gay - + - Gya )( 8 ) Gab - + - Gba

( 9 ) - Gab( 10 ) ~ - + Gab( II ) b = a( 12 ) Gaa( 13 ) - Gaa( 14 ) - 3y ' v' x ( x : #y - + Gyx )( IS ) - 3y ' v' x ( x : #y - + Gyx )( 16 ) - 3y ' v' x ( x : #y - + Gyx )( 17 ) - 3y ' v' x ( x : #y - + Gyx )

A [for RAA]A [for 3E]I ' VEA [for 3E]2 ' VE7 ' VE6,8 ~ E4 ' VE9,10MTr6,11=E9,11=E12,13 RAA (3)5,14 3E (6)3,15 3E (4)3,16 RAA (3)

Exercise 3.4.1

.VxPx -1.- 3x- Px

'v'x(Px -+ Q) -it- 3xPx -+ Q'v'x(Px -+ Q) t- 3xPx -+ Q

'v'x(Px -+ Q)3xPxPa-+QPaQQ3xPx -+ Q

176 Answers to Chapter 3 Exercises

5150(a)123322

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2I41,41,21

3,4 -+E2,5 3E (4)6 -+1 (2)

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(b)122

AA231

Answers to Chapter 3 Exercises

3xQx3xPx & 3xQx3xPx & 3xQx3xPx & 3xQx

177

1.3 -+E4 -+1 (2)5 VI

1,211

(4)(5)(6)

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515612222

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2VE3 v I4 VI

777

10 -+1 (7)1,6,11 Sim Dll

5157

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(a)1233333321

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- 3x(P -+ Qx)P -+ Qa3x(P -+ Qx)- (p -+ Qa)P& -Qa

AAA2,3 -+E431S -+1 (2)1,6 3E (3)

Exercise 3.4.2

Answers to Chapter 3 Exercises

QbPa& Qb3y(pa & Qy)3xy(Px & Qy)3xy(Px & Qy)3xy(Px & Qy)

(5)(6)(7)(8)(9)(10)

A4,5 &16317313,8 3E (5)2,9 3E (4)

P -+ 3xQx ~J- 3x(P -+ Qx)P -+ 3xQx J- 3x(P -+ Qx)(1) P -+ 3xQx(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)

(a)1 A

A2332221,2210101,21

1,7 -+E6&EA9,10 RAA (2)8,11 3E (10)2,12 RAA (2)

3x(P -+ Qx) ~ P -+ 3xQx3x(P -+ Qx)PP -+ Qa

(b)1232,32,331

-

- - - - - -

-

N ~ ~ V ) \ Or - .

-

- - - - - -

Qa3xQxp ~ 3xQxp ~ 3xQx

T4012

.- 'v'x(Fx - + Gx) - + ('v'xFx - + 'v'xGx)(1)(2)

A [for - +1]A [for - +1]

P3xQx-QaQa

-

- -

666

t t t

~

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mm

m

'v'x(Fx -+ Ox)VxFx

178

54,54,54,51,41

SI60

A3312,4 RAA (3)5 Neg-+6&E

3xFx v 3xGx3xFxFaFa v Ga3x(Fx v Gx)3x(Fx v Gx)3xFx -+ 3x(Fx v Gx)

(22)(23)(24)(25)(26)(27)(28)

22,25 3E (23)26 -+1 (22)15,21,27Sirn Dll

Answers to Chapter 3 Exercises 179

(3)(4)(5)(6)(7)(8)

Fa - + GaFaGaVxGx

i 've2VE3,4 -+ES'v16 -+1 (2)7 -+1 (I)

I21,21,21

T42I A [for -+1]

A [for RAA]

-

- - - - - - - - - - -

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20M3 &E3 &E4QE5QE6VE7VEA [for 3E on I ]8,10 v E9,11 RAA (2)1,123E(10)13 -+1 (I )A [for -+1]AA [for 3E on 16]17 v I183116,19 3E (17)20 -+1 (16)AA [for 3E on 22]

102,10101

151617171716

3xGxGaFa v Ga3x(Fx v Gx)3x(Fx v Gx)3xGx -+ 3x(Fx v Gx)3x(Fx v Gx)

'V'xFx - + 'V'xGx'v'x(Fx - + Gx) - + ('v'xFx - + 'v'xGx)

23 v I2431

r 3x(Fx v Ox) H 3xFx v 3xOx3x(Fx v Ox)- (3xFx v 3xOx)-3xFx & -3xOx-3xFx-3xOx'v'x-Fx'v'x-Ox-Fa-OaFa v OaOa3xFx v 3xOx3xFx v 3xOx3x(Fx v Ox) -+ (3xFx v 3xOx)

2223232322

15

J- (3xFx -+ 3xGx) -+ 3x(Fx -+ Gx)(1) 3xFx -+ 3xGx(2) -3x(Fx -+ Gx)(3) 'v'x- (Fx -+ Gx)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)

180 Answers to Chapter 3 Exercises

(29) 3xFx v 3xOx -+ 3x(Fx v Ox)(30) 3x(Fx v Ox) H 3xFx v 3xOx

T461222

A [for -+1]A [for RAA]

2221,29291,21

1,7 -+EA [for 3E on 8]5&E9,10 RAA (2)8,11 3E (9)2,12 RAA (2)13 -+1 (I)

T59122251,51,5191,91,21,21,21,21,21,21

(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)

AA2QE3 'v'EA1,5 SP6 'v'E7 -+1 (5)A8,9 HI4,10 RAA (9)II Neg-+12&E12&E13 VI1,15 BP14,16 RAA (2)

28 -+1 (15)14,29 HI

-3x(Fx H P)'v'X- (Fx H P)- (Fa H P)P'v'xFxFaP -+ FaFa -+ PFaHP- (Fa -+ P)Fa & - PFaP

'v'xFxP3x(Fx H P)

2QE3'v'E4 Neg~5&E631

Rxerci~ ~

(xz

o

- + - x , { : I ) - zA , { EX

A

(, { H + - x : I - ) , { EX

A

(n , { o + - ex : I ) , { xA

(zr

~

Z , { H + - , { : I ) zA , { E

( Z ' X ' } J + - Xd ) z : x A

(qO

~

q . . J ~ 80

~ 8d ) -

(80

~

8d ) -

(: > H - ~ : > d ) 1 \ ( QH - ~ Q : J ) 1 \ ( 8H - ~ 8d ) H : > d ~ q . . J ~ 8d

(QH - ~ Q : J ) 1 \ ( 8H - ~ 8d ) H q . . J ~ 8d

Answers to Chapter 4 181

(d H

x : I ) xE

+ - ( d H x : fX

A ) ( 81 )

(I ) I + - LI

r; - tO as

p. Ia

x3

~t

q! flA

8! flA

:> HA

qHA

8HA

:> ! A

q! A

8! A

:> A

qA8A

:> A !

qA!

8A!

:> m

qm8m

:> H

qH8H

:> !

q!

8!

~

~ 8 : : 1

8: : 1

qO

1 \ 80

+ - q . . J ~ 8d

80

+ - 8d

d ~

( . ? d 1 \ q . . J 1 \ 8tI )

d ~

( q . . J 1 \ 8tI )d

~ 8d

.? d ~ q . . J ~ 8d

UH-

~ 8d

+ - +

8d

~

H A ~ d A

~ A

~ A 8H

A 8d

lele

t

asp

. I3x3

t . . 8I

d8

& 1 : >

~

"

~

"

eH

"

ed

eH

"

ed

:> 0 " qO

"

eo

"

eH

qO

" eo

" 88

eo

" 88

(~ H

~

~

~ 88 ) " d H : > 0 ) ~ ( d H qO ) ~ ( d H eo

(~

~ 88 ) " d H qo ) ~ ( d H eo

eH

" ( d H eo )

:> 0 " qo

"

eo

+ - ~ d ~ ~ ~ ed

Answers to Chapter 4 Exercises

Gb )

NU

~

: a

~

, C U ~ , C U

-

' >

, ~

, ~ , ~

~

~

~

- (Fa & Ga & fb & Gb & Fc & Gc)- (Fa & -Ga)-

Fa & - (Ga & Gb & (Fb & -(Ga &

Exerdse 4.1.2

TFFTFTTTFF

iblibiiibivbvbvibviibviiibixbxb

tl. 4E - - E - - E - - E - - E - - tI . 4E - - E - - E - -

uu

u

u

.u

:~

u

u

u

. -

> u . . . u

.- : =

: =

. - >

> ' ~

. ~

~

~

~ ~ ~ ~ ~ ~ ~ ~ ~

~

-

~

~

-

~

.~

~ : =

~

-

~

. ~ . > . ~ . . . -

.- : =

: =

. - >

>

>

. es

~

Exerdse 4.2

U:{a,b} F:{a} G:{ }

-

Fa &- (Oa&Ob&Gc &(Fb&--(Oa& Ob &Gc &(Fc &- (Oa&Gb &Gc )- (Oa H .Ha & -Fa)- (Oa & Ob H (Ha & -Fa) v (Hb & - fOb - (Oa & Ob & Gc H (Ha & -Fa) v (Hb & - fOb) v (Hc & -Fc

Fa & fb -+ Ga & Gb.- (Fa -+ Ga) & (fb -+ Gb)U:{a,b} F:{a} G:{b}Fa v fb -+ Ga v Gb.- (Fa -+ Ga) & (fb -+ Gb)Same model as ii(Fa v fb ) & (Ga v Gb) .- (Fa & Ga) v (fb & Gb)Same model as i(Fa v Ga) v (fb v Gb) .- (Fa & fb ) v (Ga & Gb)

Answers to Chapter 4 Exercises

v Same model as i

Fa v fOb H Oa & Ob, - Fa - + Ha) & ( Fb - + Hb ~ Ha v Hb - +- Oa v - Gb

183

(Fa -+ Ga) v (Fb -+ Gb) r (Fa v fb) -+ (Ga v Gb)U:{a,b} F:{a,b} G:{a}

xiii

U:{a} F:{a} G:{a}Ga v - Ha, Ga & Fa .- - Ha

H:(a}

(Fa -+ Ga) v (fb -+ Gb) I- (Fa & fb ) -+ (Ga & Gb)Same model as iFa& FbH Ga& Gbl- (FaH Ga) & ( fbH Gb)Same model as iiFa v fb H Ga v Gb I- (Fa H Ga) & (fb H Gb)U:{a,b} F:{a} P is FALSE(Fa & fb ) H PI - (FaH P) & ( fbH P)U:{a,b} F:{a} Pis T RUE(Fa v fb ) H PI- (FaH P) & (fbH P)Same model as ix(FaH P) v (fbH P) I- (Fa v fb ) H P

Same model as x(FaH P) v (fbH P) I- (Fa & fb ) H P

U:{a} F:{ } G:{ } H:{a}Fa -+ Ga, Ga -+ Ha I- Ha -+ Fa

U:{a} F:{ } G:{ } H:{ }Fa -+ -Ha, Ha -+ -Ga I- Fa & Ga

U:{a,b} F:{a} G:{a,b} H:{b}

Chapter

(q ~ ' W

ew

A ( qed

' Wee

t J ) - . . ( q ~ A ew

' W ( qed

A eet

J )

t81

01

~ MSU

V

S~ ! : > J ~ XH

t

(~

+ - qO

A~ ~

( UM

+ - UO

AUtO

- i ~ AUM

' qO

A U O ' ~ A Ud

{e } : d { q ' e } : n

qO

1 \ 80

+ - ~

1 \ 8d

- i ( qO

1 \ 80

+ - ~

1 \ ( qO

1 \ 80

+ - 8 : J )

! S8

I ~ W

~ W

8S

~

-

: w

8d - - i ( ~ : w 8 : J ) -

l

r . - t asp

. I3x3

{ ( 8 ' 8 ) } : d

{q ' e } : n

!! su

I ~ PO

W

~ wes

{( e ' e ) } : O { } : d { e } : n

qq ~ ' W

eqW

' W ( qqeA

' Weq

eA

A qe ~ ' W

eeW

' W ( que

A ' W

un

A - i

qq ~ ' Weq

W

A ( qe ~ ' W

eeW

) ' W

qqeA

' Weq

eA

> A ( que

A ' W

un

A

{q } : H { e } : o { e } : d { q ' e } : n ! ! ! AX

eo

- i eH

' W

ed

' eM

+ - eo

' W

ed

{e } : H { } : O { e } : d { e } : n ! ! AX

q~

A 88 : : 1 - i ( q ~ A qu : : l ) ~ ( u ~ A 88 : : 1 )

{ ( U ' q ) ' ( q ' e ) } : : : 1 { q ' e } : n

q~

~ qe : : l ~ U ~ ~ 88 : : 1 - i q ~ A 88 : : 1

{ ( 8 ' q ' q ) ' ( q ' 8 ' 8 ) ' ( q ' q ' q ) ' ( 8 ' 8 ' 8 ) } : A

{q ' 8 } : n

no

- A 8 : : 1 - f no

+ - 8 : : 1

A! S ' B

I ~ PO

W

~ wes

no

-

- f n : : l - + - no

' 88 : : 1 -

- (Taa & Tab) & - (Tba & Thb) ..- - (Taa v Tab) & -(Tha v Thb)

G:{(a,b)}G: { (a,b), (b,a) }

Answers to Chapter 4 Exercises

vii U:{a,b} T:{(a,b)}

185

viii

ix

x

U:{a.b,c}U: {a.b,c}U:{a.b} F: { (a,a) }

Exerdse 4.4

i U: {m, n} b: m d: na: m c: nm = m , n = n ~ m = n

U: {a, b, c} F: {a, b}[ Fa & Fa) & ~ ) v Fa & Fb) & a$b) v Fa & Fc) & a#C)]v [ Fb & Fa) & ~ ) v Fb & Fb) & ~ ) v Fb & Fc) & ~ )]v [ Fc & Fa) & ~ ) v Fc & Fb) & ~ ) v Fc & Fc) & t * )]I- Fa & Fb & Fc

F: {(a.b), (h,c),(a.a),(c,a)}F: {(a.b), (h,c), (c,a)}

U: {a, b,c} F: {(a, b), (a. c) } Faa H ~ ) & (Fab H ~ ) & (Fac H ~ v

Fha H b:I:a) & (Fbb H b#b) (Fbc H b#c))v

(Fca H c:#:a) &: (Fcb H ~ ) (Fcc H t * ))t- (Faa & Faa - + a=a) & (Faa & Fab - + a=b) & (Faa & Fac - + a=c)& (Fab & Faa - + b=a) & (Fab & Fab - + b=b) & (Fab & Fac - + b=c)& (Fac & Faa - + c=a) & (Fac & Fab - + c=b) & (Fac & Fac - + c=c)& (Fha & Fha - + a=a) & (Fha & Fbb - + a=b) & (Fha &Fbc - + a=c)& (Fbb & Fha - + b=a) & (Fbb & Fbb - + b=b) & (Fbb & Fbc - + b=c)& (Fbc & Fha - + c=a) & (Fbc & Fbb - + c=b) & (Fbc & Fbc - + c=c)& (Fca & Fca - + a=a) & (Fca & Fcb - + a=b) & (Fca &Fcc - + a=c)& (Fcb & Fca - + b=a) & (Fcb & Fcb - + b=b) & (Fbb & Fbc - + b=c)& (Fcc & Fca - + c=a) & (Fcc & Fcb - + c=b) & (Fbc & Fbc - + c=c)

Exercises

I st premise: T2nd premise: TConclusion: F

Premise: T.('Every number is less than some othernumber, and if this other number is

Conclusion: F.('Some number

I st premise: T('For each number there is an even number that is greater.')2nd premise: T('If Y is an even number greater than x, and z is an even number greater thany, then z is an even number greater than x.')3rd premise: T('No number is an even number greater than itself.')

numberConclusion: F.('If x is even and y is odd, then either x is an evenis an even number greater than x.')

Answers to Chapter 4186

Exerdse 4.5.2

'v'xyz(Fxy & Fyz -+ Fxz), 'v'x3y Fxy r 3xFxxU: N.F: {(m,o) : mO &; (n=2k(m+l ) - 1 or n=2k(m+l ) }

'v'xyz(Oxy &; Oyz - + Oxz), 'v'xy(Oxy - + - Gyx),'v'x3y Gyx, 'v'x(x~ - + Ou ) J- 3y'v'x(x* y - + Oyx)

U: N

IJ~ N-

G: {{m, n) : m > n }a: zero

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ChapterlSententialLogic

Basic logicalnotion.

Definition. An ARGUMENT is a pair of things:. a set of sentences, the PREMISES. a sentence, the CONCLUSION .

Comment. All arguments have conclusions, but not allarguments have premises: the set of premises can bethe empty set! Later we shall examine this idea insome detail.

argument ,premises,conclusion

validity

Definition . An argument is V A Lm if and only if it isnecessary that if all its premises are true, its conclusionis true.

1.1

Comment. If the sentences involved belong to English (or any othernatural language), we need to specify that the premises and theconclusion are sentences that can be true or false. That is, thepremises and the conclusion must all be declarative (or indicative)sentences such as 'The cat is on the mat' or 'I am here' , and notsentences such as 'Is the cat on the mat?' (interrogative) or 'Comehere!' (imperative). We are going to construct some formallanguages in which every sentence is either true or false. Thus thisqualification is not present in the definition above.

Comment. The intuitive idea captured by this definition is this: If it is possible for the conclusion of an

argument to be false when its premises are all b1Je,then the argument is not reliable (that is, it is invalid).

Chapter 1

entaUment

Every premise of a valid argument is hUe.Every invalid argument has a false conclusion .Every valid argument has exactly two premises.Some valid arguments have false conclusions.Some valid arguments have a false conclusion despitehaving premises that are all hUe.

If true premises guarantee a true conclusion then theargument is valid.

Alternate formulation of the definition. An argument isV A Lm if and only if it is impossible for all thepremises to be true while the conclusion is false.

Definition. When an argument is valid we say that itspremises EN T All.., its conclusion.

soundness I if and only if it is

Comment. It follows that all sound arguments havehue conclusions.

than soundness.

Exercise 1.1 Indicate whether each of the following sentences isTrue or False.

i*ii *iii *iv*v*

Definition . An argument is SOUNDvalid and all its premises are b"ue.

Comment. An argument may be unsound in either oftwo ways: it is invalid , or it has one or more falsepremises.

Comment. The rest of this book is concerned with validity rather

Chapterl

A sound argument cannot have a false conclusion .Some sound arguments are invalid .Some unsound arguments have true premises.Premises of sound arguments entail their conclusions.If an argument has true premises and a true conclusionthen it is sound.

1.2 A Formal Language for Sentential Logic

fonnal Comment. To represent similarities among argumentsof a natural language, logicians introduce formallanguages. The first formal language we will introduceis the language of sentential logic (also known aspropositional logic). In chapter 3 we introduce a moresophisticated language: that of predicate logic.

vocabulary -LOGIC consists of. SENTENCE LETTERS,. CONNECTIVES, and. PARENTHESES.

Definition. A SENTENCE LE If E R is any symbolletter from the following list:

sentencevariable

vi *vii *

...Vlll *ix *x*

language

Definition. The VOCABULARY OF SENTENTIAL

sentence

A, ... ,Z, Ao' ... ,Zo' A I' ... ,Zit ....

Comment. By the use of subscripts we make availablean infinite number of sentence letters. These sentenceletters are also sometimes called SENTENCE VARI -ABLES, because we use them to stand for sentencesof natural languages.

Chapter 1

connectives Definition. The SENTENTIAL CONNECTIVES(often just called CONNECTIVES ) are the membersof the following list: - , & , v , - +, H .

Comment. The sentential connectives correspond tovarious words in natural languages that serve toconnect declarative sentences.

tilde - The TILDE corresponds to the English 'It is not thecase that' . (In this case the use of the term 'connective'is odd, since only one declarative sentence is negatedat a time.)

ampersand & The AMPERSAND corresponds to the English 'Both

wedge v The WEDGE corresponds. . .

' in its inclusive sense.

~ The ARROW corresponds to the English then

... and ...' .

to the English 'Either ... or

an - ow,

. . . .

double -

arrow

The DOUBLE - ARROW corresponds to the English'if and only if ' .

Chapterl

Definition. An EXPRESSION ofany sequence of sentence letters,rives, or left and right parentheses.

sententialsentential

metavariable

representlogic, but it may be used tosentential logic.

) and (

.

expression logic is

: an expression of

Comment. Natural languages typically provide more than one wayto express a given connection between sentences. For instance, thesentence 'John is dancing but Mary is sitting down' express es thesame logical relationship as 'John is dancing and Mary is sittingdown' . The issue of translation from English to the formallanguage is taken up in section 1.3.

The right and left parentheses are used as punctuationmarks for the language .

Examples.(P ~ Q) is an expression of sentential logic.)PQ~ - is also an expression of sentential logic.(3 ~ 4) is not an expression of sentential logic.

Definition. Greek letters such as , and 'I' are used asMETA VARIABLES . They are not themselves parts ofthe language of sentential logic, but they stand forexpressions of the language.

Comment. (, ~ '1') is not an expression of sentential

Chapter I

weD-formed Definition. A WELL - FORMED FORMULA ( WFF)of sentential logic is any expression that accords withthe following seven rules:

f Or Dlu1a

(1) A sentence letter standing alone is a wff .

sentence language

(2) If , is a wff , then the expression denoted by - , is

negation

conjunction

(4) If cj) and 'If are both wffs, then the expression

disjunction

(5) If (j) and 'I' are both wffs, then the expressiondenoted by (j) - + '1') is a wff .

atomic [Definition. The sentence letters are the ATOMIC: of sentential logic.]

also awIf .

SENTENCES of the

[Definition. A wff of this form is known as a CON-JUNCTION. ct> and 'If are known as the left and rightCONJUNCTS, respectively.]

denoted by (~ v '1/) is a wff .

[Definition. A wff of this form is known as a DIS-JUNCTION . ~ and '1/ are the left and rightDISJUNCTS, respectively.]

[Definition. A wff of this form is known as a NEGA -TION , and - ~ is known as the NEGATION OF ~.]

(3) If ~ and 'I' are both wffs, then the expressiondenoted by (~ & '1') is a wff .

conditional ,antecedent ,consequent

denoted by (cj) H '1') is a wff .

Definition. & , v , ~ , and H are BINARY CONNEC-TIVES , since they connect two wffs together. - is aUNARY CONNECTIVE , since it attaches to a singlewff .

Definition. A SENTENCE of the fonnallanguage is awff that is not part of a larger wff .

sentence

denial Definition . The DENIAL of a wff (j) that is not anegation is - (j). A negation, - (j), has two DENIALS : (j)and - - (j).

Chapter 1

binaryand unaryconnectives

[Definition. A wffofthis Conn is known as a O: NX-nO N A L. The wff , is known as the ANTECEDENTof the conditional . The wff 'If is known as theCONSEQUENT of the conditional.]

(6) If , and 'If are both wiTs, then the expressi


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