ADDITIONAL SOLUTIONS for exercises in Introductory Symbolic Logic Without Formal Proofs

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1 1 ADDITIONAL SOLUTIONS for exercises in Introductory Symbolic Logic Without Formal Proofs P P v ~J 3. ~P ~J 4. ~P (~A ~J) 5. ~(J v A) 6. [J v (~A ~J)] 7. J A 8. J (A P) 9. J (A v P) 11. P v (A J) 12. P v ~(J A); this could also be interpreted as P v (~J ~A) 13. ~(A J) 14. ~(J v A) or (~J ~A)16. A J 18. J P P. 42 (6 & 7 are logical equivalences; 2, and 5 are logical implications.) Pp I. 1. AB A B B 2. AC AvC A ~C TT T T TT T T F TF F F TF T T T FT F T FT T F F FF F F FF F F T 3. AB A B ~B ~A 4. AC A v C ~A C 5. DE D E E D TT T F F TT T F T TT T T T TF F T F TF T F F TF F F T FT T F T FT T T T FT F T F FF T T T FF F T F FF T F F II. The solutions below utilize the shorter truth table. 1. ST Sv~T T S 2. SR ~SvR R S TT T T T TT T T T TF T F TF F FT F FT T T F FF T F FF T F 3. PC P C ~C ~P 4. EP ~(E P) P ~E 5. PC ~(P C) C P TT T F TT F T TT F T TF F TF T F F TF T F F FT T F FT T F T T FT F T FF T T T FF T F F FF F T 7. JOH ~J ~O ~O H JvH 8. CTP C (TvP) ~T C P TTT F T F T T TTT T T T F TTF F T F T T TTF T T T F TFT F T T TT T TFT T T T T T TFF F T T FF TFF T F F FTT T FF FTT F T F FTF T FF FTF F T F FFT T TT T TT T FFT F T T T FFF T TT T FF FFF F T T T

2 2 9. APN A (PvN) ~N P 10. QD Q ~D ~Q D TTT T T T F TT FF TTF T T T T T TF TT F TFT T T T F FT TF T T TFF T F F FF FT FTT F T F FTF F T T T FFT F T F FFF F T T F P. 58 In what follows, (a) gives the truth table; (b), the argument form; (c), the argument proposition; (d), the statement form. 1. (a) AB A B ~B ~A (b) p q TT T F F T ~q ~p TF F FT T F T (c) (A B) (~B ~A) FF T T T T (d) (p q) (~q ~p) 2. (a) ABC A (B C) B C (b) [p (p r)] (p q) TTT T T T T TTF T F F (c) [A (B C)] (B C) TFT T T T T TFF T T T T (d) [p (q r)] (p r) FTT F F T FTF F F FFT F F FFF F F 3. (a) A B (A v B) (A B) (A v B) (A v B) A B T T T T T T T T F T F F F T T F F F F F T F (b) (p v q) (p q) (c) {[(A v B) (A B)] (A v B)} (A B) (p v q) (d) [(p v q) (p q)] (p v q)} (p q) (p q) 4. (a) P C ~(P C) C P (b) ~(p q) T T F T q p T F T F F F T F T (c) ~(P C) (C P) F F (d) ~(p q) (q p) 5. (a) E A G ~E v A A G ~(G ~E) (b) ~p v q T T T F T T T T T F F q r T T F F T T F ~(r ~p) T F T F F F T F F F F F (c) [(~E v A) (A G)] ~(G ~E) F T T T T T T T T F T T T F T F (d) [(~p v q) (q r)] ~(r ~p) F F T F F F

3 3 6. (a) H M H M M H (b) p q (c) (H M) (M H) T F T T q p T F F F T T F (d) (p q) (q p) PP I. When first stating what is in each path in (c) below, I begin with the first path on the left and move to the right. 1. (a) 6 branches; (b) 4 paths; (c) BCA; BC~D; BDA; BD~D 2. (a) 2 " (b) 2 " (c) BDA~E; BD~A. 3. (a) 6 " (b) 4 " (c) BDAE ~E; BDAEF; BD~C~E; BD~C F. 4. (a) O " (b) 1 " (c) CE~F. 5. (a) 4 " (b) 3 " (c) CFG; CFHA; CFHB or 4 " (b) 3 " (c) AG; AE; AF AGET; AG~BC~E F~E; FG; FH E~BG; E~BH; E~BI;F. II. For each of the problems, I indicate first whether the tree is open or closed and, if it is open, I indicate which paths are open, counting from the left. 1. open tree; left & right-hand paths are open 5. closed tree 2. closed tree 6. closed tree 3. open tree; left path is open 7. open tree; right path is open 4. open tree; left path is open 8. closed tree P. 76 In these exercises, I simply put statements on the tree in the order in which they appear in the argument, rather than in the economical way we do this later, to keep things simple. Note that #4 is an example where the student should add all of the information in premise 3 before the path is closed off. That is, as soon as the student adds "~C," the second and fourth paths close off. Nevertheless, for the sake of clarity, I require the students to add all of premise 3 "~C D," and then check for contradictions. 1. /\ 2. B 3. B A B / \ C / \ ~B A / \ ~B C ~B C x / \ A ~B ~C ~C x ~C ~B C / \ x ~A x x x ~C ~C H x x ~A x 4. /\ 5. / \ 6. B 7. B A B B H G ~A / \ / \ ~G ~G ~G C ~B A ~A C ~A C A A A / \ X ~C x ~C ~C ~C / \ x ~H G ~C D D D G C G C / \ /\ invalid: x ~C x x ~C x ~C ~B H ~B H B,A,~C invalid: x x x x x ~A B,~A,~C,D invalid: B,~A,C,G,H

4 4 8. / \ 9. / \ 10. ~D A C X ~X / \ B \ T ~T A B / ~B ~C / / \ / \ / \ ~B ~C ~B x A B A B D ~A D ~A x ~B X ~T X ~T x ~D x ~D x / / / \ \ x / \ ~A ~B ~A ~B ~A ~B ~A ~B G I x ~X ~X x x ~X ~X x H ~I invalid: x x x ~I x C,~B invalid invalid: ~X.~T,B,~A ~D,B,~A,G,H,~I P (1) p q ~q (2)~q p (3) ~p / \ (4) ~~p N.C.,3 ~p q (5) p D.N.,4 x x (6) ~p v q Impl.,1 2. (1) p q ~q (2) ~q p (3) ~p / \ (4) ~~p N.C.,3 p ~p (5) p D.N.,4 q ~q (6) (p q) v (~p ~q) Equiv.,1 x x 3. (1) p p (2) ~q ~q (3) p v q ~p (4) ~(p v q) N.C.,3 ~q (5) ~p ~q DeM., 4 x ~r 4. (1) ~r ~p ~p (2) p v q / \ (3) q v r p q (4) p q x / \ (5) ~(p q) N.C., 4 q r (6) ~pv~q DeM., 5 / \ x ~p ~q x r 5. (1) r ~q (2) p q / \ (3) r p ~p q (4) q / \ x (5) ~q N.C., 4 r ~r (6) ~p v q Impl.,2 p ~p (7) (r p) v (~r ~p) Equiv.,3 x x P. 88 I. 1. 1) A (B C) 2. 1) (C D) (S D) 2) ~A v (B C) Impl.,1 2) ~(C D) v (S D) Impl.,2 3) ~A v (~B v C) Impl.,2 3) (~C v ~D) v (S D) DeM.,2 4) (~C v ~D) v (~S v D) Impl, ) (A v B) C 2) ~(A v B) v C Impl.,1 4. 1) ~(E F) D 3) (~A ~B) v C DeM., 2 2) ~~(E F) v D Impl.,1 3) (E F) v D D.N.,2

5 5. 1) ~(A B) 6. 1) G (B D) 2) A ~B N.I.,1 2) ~G v (B D) Impl., ) A ~(C v E) 8. 1) D ~E 2) ~A v ~(C v E) Impl.,1 2) (D ~E) v (~D ~~E) Equiv.,1 3) ~A v (~C ~E) DeM.,3 3) (D ~E) v (~D E) D.N.,3 9. 1) ~[(A B) ~D] 10. 1) ~A (~B C) 2) (A B) ~D N.I.,1 2) ~~A v (~B C) Impl.,1 3) A v (~B C) D.N..,2 4) A v (~~B v C) Impl.,3 5) A v (B v C) D.N.,4 II. 1. 1) S D S 2. 1) (~D v ~P) C ~C 2) D F ~F 2)..C / \ 3).. S F / \ 3) ~C NC,2 D C 4) ~(S F) NC,3 ~S D 4) ~(~D v ~P) v C Impl,1 P x 5) S ~F NI,4 x / \ 5) (~~D ~~P) v C DeM,4 6) ~S v D Impl,1 ~D F 6) (D P) v C (2) DN,5* invalid: ~C,D,P 7) ~D v F Impl,2 x x ~A P. 95 I. 1. 5) ~A NC,4 / \ ~A C 6) ~A v C Impl,1 / \ / \ 7) ~C v ~D Impl,2 ~C ~D ~C ~D Invalid: X ~A,~C,D ~E 2. 3) B A B 5) ~E NC,4 A 6) ~A v B Impl,1 / \ 7) C (~B v E) Impl,2 ~A B x C / \ ~B E x x 3. 5) ~C ~C 6) ~B v C Impl,1 / \ 7) ~E v (~B v F) (2) Impl,2* ~B C 8) (B E) v (~B ~E) Equiv,3 / \ X B ~B E ~E INVALID: ~C,~B,~E X / \ ~E /\ B F *NOTE: I have applied the same rule twice in a single step, indicating in parentheses how many times I applied the rule. B ~F 4. 5) ~(B F) NC,4 / \ 6) B ~F NI,5 ~F E 7) ~F v E Impl,2 / \ / \ A ~B A ~B 8) ~(B A) v C Impl,3 C x C X invalid: 9) (~B v ~A) v C DeM,8 / \ \ / \ \ B,~F,A,C ~B ~A C ~B ~A C x x x x

6 6 5. 5) ~[(C A) B] NC,4 C 6) (C A) ~B NI,5 A 7) ~(F v C) v A Impl,1 ~B 8) (~F ~C) v A DeM,7 / \ 9) ~C v B Impl,2 ~C B 10) ~C v (~A v D) (2)Impl,3 X X C ~H II. 2. 1) C (V L) / \ 2) T (H v A)* ~C V 3) L T X L 4) C H / \ 5) ~(C H) NC,4 ~L T 6) C ~H NI,5 X /\ 7) ~C v (V L) Impl,1 ~T / \ 8) ~T v (H v A) Impl,2 invalid: X H A 9) ~L v T Impl,3 C,~H,V,L,T,A X *The consequent here is really an exclusive "or." It would be translated as T [(HvA) ~(H A)]. This will add another pair of branches to the open path, but one will remain open anyway, with the same information in the open path as this tree. T 3. 1) (P T) (I v D) ~I 2) P D / \ 3) T I / / \ 4) ~(T I) NC,3 ~P ~T I D 5) T ~I NI,4 / \ X X / \ 6) ~(P T) v (I v D) Impl,1 P ~P P ~P 7) (~P v ~T) v (I v D) DeM,6 D ~D D ~D invalid: 8) (P D) v (~P ~D) Equiv,2 X X T,~I,~P,~D or T,~I,D,P 4. 1) J (S C) S 2) S (J ~C) / \ 3)..~S ~S J 4) ~~S NC,3 X ~C 5) S DN,4 / \ 6) ~J v (~S v C) (2) Impl,1 ~J /\ 7) ~S v (J ~C) Impl,2 X ~S C X X P ) P J ~H 2) J H / \ 3) ~H ~J H 4) ~P ~J / \ x 5) ~(~P ~J) NC,4 ~P J 6) ~~P v ~~J DeM,5 / \ x 7) P v J (2)DN,6 P J 8) ~P v J Impl,1 x x 9) ~J v H Impl,2 6. 1) S v G ~R 2) (S v ~T) R / \ 3) G (R v S) ~S R 4)..R T X 5) ~R NC,4 / \ 6) ~(S v ~T) v R Impl,2 ~G \ 7) (~S ~~T) v R DeM,6 / R S 8) (~S T) v R DN,7 S G X X 9) ~G v (R v S) Impl,3 X X

7 7. 1) S (C I) S 2) C (~D ~H) ~D 3) ~H ~S / \ 4)..S D ~S C 5) ~(S D) NC,4 X I 6) S ~D NI,5 / \ 7) ~S v (C I) Impl,1 ~C / \ 8) ~C v (~~D v ~H) (2) Impl,2 X D ~H 9) ~C v (D v ~H) DN,8 X / \ 10) ~~H v ~S Impl,3 H ~S 11) H v ~S DN,10 X X ) I (C ~L) 10) L v ~D DN,9 I 2) ~L ~D 11) ~D v R Impl,3 R 3) D R / \ 4)..I ~R ~I C 5) ~(I ~R) NC,5 X ~L 6) I ~~R NI,5 / \ invalid: 7) I R DN,6 L ~D I,R,C,~L.~D 8) ~I v (C ~L) Impl,1 X / \ 9) ~~L v ~D Impl,2 ~D R 9. 1) I (~C ~T) I 2) ~(T v C) F ~R 3) F R / \ 4)..I R ~I ~C 5) ~(I R) NC,4 X ~T 6) I ~R NI, 5 / \ 7) ~I v (~C ~T) Impl,1 / F 8) ~~(T v C) v F Impl,2 T C / \ 9) (T v C) v F DN,8 X X ~F R 10) ~F v R Impl,3 X X P. 97 III. A. 5) ~B NC,4 ~B C. 5) ~B NC,4 / \ ~F B ~B / \ X G ~D F / \ ~E X B ~A A X / \ \ / \ H~F C A B D ~D B E ~G ~F C X E X / \ x x X X A C F D D D ~C~C~C x x x B. 5) ~~C NC,4 C C DN,5 / \ ~C ~D X / \ D ~G ~F X / \ \ F A F A / / \ \ \ invalid: C,~D,~F,A,G ~F D~F D~FD~F D G E G E GE G E x x x x xx x

8 IV. A. 5) ~(G E) NC,4 B. 5) ~(~C ~P) NC,4 6) ~G v ~E DeM,5 6) ~C ~~P) NI,5 7) (A E) v (~A ~E) Equiv,1 7) ~C P DN,6 8) ~A v C Impl,2 8) G [(H ~I)v(~H ~~I)] Equiv,1 9) ~~A v (~E v G) Impl,3 9) G [(H ~I)v(~HvI) DN,8 10) A v (~E v G) DN,9 10) ~(HvI)v(~HvC) (2)Impl,2 11) (~H ~I)v(~HvC) DeM,10 12) ~(~HvF)vP Impl,3 13) (~~H ~F)vP DeM,12 14) (H ~F)vP DN,13 P. 98 C. 5) ~(~CvAvI) NC,4 D. 5) ~[C (BvD)] NC,4 6) ~~C ~A ~I Dem,5 6) C ~(BvD) NI,5 7) C ~A ~I DN,6 7) C (~B ~D) DeM,6 8) (~I B)v(~~I ~B) Equiv,1 8) ~(~D H)v(L A) Impl,2 9) (~I B)v(I ~B) DN,8 9) (~~Dv~H)v(L A) DeM,8 10) ~(GvB)vI Impl,2 10) (Dv~H)v(L A) DN,9 11) (~G ~B)vI DeM,9 11) [(A B) C]v[~(A B) ~C] Equiv,1 12) [(AvB) C]v[~(AvB) ~C] Equiv,3 12) [(A B) C]v[(A ~B) ~C] NI,11 13) [(AvB) C]v[(~A ~B) ~C] DeM,12 13) [(~A v B) C]v[(A ~B) ~C] Impl, 13 14) ~~Hv(~AvD) (2)Impl,3 15) Hv(~AvD) DN,14 8 E. 5) ~[(~Cv~D)v(H ~I)] NC,4 V. C=there is a contracted pelvis 6) ~(~Cv~D) ~(H ~I) DeM,5 T=fetus is in the tranverse position 7) (~~C.~~D) (~Hv~~I) (2) DeM,6 P=Caesarian section should be performed 8) (C D) (~HvI) (3)DN,7 9) ~~Hv(~CvB) Impl,1 First question: (C T) P ~C 10) Hv(~CvB) DN,9 Second question: T P T 11) ~~Iv(BvH) Impl,2 Inference: C P 12) Iv(BvH) DN,11 / \ 13) ~HvI Impl,3 invalid ~C ~T P X P. l08 1.The argument is of course invalid. (The U.S. Supreme Court subsequently overturned the decision on free speech grounds.) 1) (TvO) V 2) V L 3) R (W V) 4) M 5) V O P. 113 I. 1. Only the left path closes off. 2. Only the right path closes off. 3. None of the paths closes off. 4. Both paths close off. P Only the right path remains open. 6. None of the paths closes off. 7. Only the middle path closes off. 8. Both paths close off. 9. Both paths close off. 10. All paths close off Hj Rj 2. Rs Ms 3. (Hs Ms) Rs 4. (HjvMj) Rj 5. (Hs Rs) Ms P (x)(ox Tx) 2. (x)(px Yx) 3. (x)(hx Rx) 4. (x)[hx (Rx v Mx] 5. (x)[(ox Ex) Vx] 6. (x)[(ox Bx) Dx] 7. (x)[(ox Bx) (Dx v Ex)]

9 9 P ) (x)[tx (Sx v Kx)] 2) (x)(kx Gx) Tj 3) Tj ~Kj 4) Kj / \ 5) ~Kj NC,4 ~Tj / \ 6) Tj (Sj v Kj) UI,1 x Sj Kj 7) Kj Gj UI,2 / \ x 8) ~Tj v (Sj v Kj) Impl,6 ~Kj Gj 9) ~Kj v Gj Impl,7 invalid: Tj,~Kj,Sj 2. 1) (x)[(gx v Kx) Tx] ~Tj 2) ~Tj Kj 3) ~Kj / \ 4) ~~Kj NC,3 ~Gj Tj 5) Kj DN,4 ~Kj x 6) (Gj v Kj) Tj UI,1 x 7) ~(Gj v Kj) v Tj Impl,6 8) (~Gj ~Kj) v Tj DeM,7 Df 3. 1) (x)(dx Fx) 7) ~Df v Ff Impl,6 ~Pf 2) (x)(fx Px) 8) Ff Pf UI,2 / \ 3) Df 9) ~Ff v Pf Impl,8 ~Df Ff 4) Pf x / \ 5) ~Pf NC,4 ~Ff Pf 6) Df Ff UI,1 x x 4. 1) (x)[(dx Tx) (Cx Tx)] ~Tr 2) ~Tr Cr 3) ~Cr / \ 4) ~~Cr NC,3 5) Cr DN,4 ~Dr Tr 6) (Dr Tr) (Cr Tr) UI,1 / \ 7) (~Dr v Cr) (~Cr v Tr) (2)Impl,6 ~Cr Tr ~Cr Tr x x x x (The first premise does not refer to all x's that are both dogs and cats, so it is not (x)[(dx Cx) Tx]. One could also read the "and" as an "or" yielding (x) (Dx v Cx) Tx.) 5. 1) (x)[(dx (Ix v Vx)] Dl 2) (x)[(ix Dx) Tx] ~Vl 3) Dl ~Vl ~Il 4) Il / \ 5) ~Il NC,4 ~Dl \ 6) Dl (Il v Vl) UI,1 x Il Vl 7) ~Dl v (Il v Vl) Impl,6 x x 8) (Ia Da) Ta UI,2 9) ~(Ia Da) v Ta Impl,8 10) (~Ia v ~Da) v Ta) DeM,9

10 10 P. 125 Dr 2. 1) (x)(dx ~Vx) ~Pr 2) (x)(vx ~Px) / \ 3) Dr ~Dr ~Vr 4) Pr x / \ 5) ~Pr NC,4 ~Vr ~Pr 6) Dr ~Vr UI,1 invalid: Dr,~Pr,~Vr 7) ~Dr v ~Vr Impl,6 8) Vr ~Pr UI,2 Dr Vr Pr Dr ~Vr Vr ~Pr Dr Pr 9) ~Vr v ~Pr Impl,8 T F F T T T F T T F 3. 1) (x)[dx ~(Ix.Gx)] Dr 2) Dr ~Ir 3) ~Ir ~Gr 4) Gr / \ 5) ~Gr NC,4 ~Dr / \ 6) Dr ~(Ir.Gr) UI,1 x ~Ir ~Gr 7) ~Dr v ~(Ir.Gr) Impl,1 8) ~Dr v (~Ir v ~Gr) Dem,7 invalid: Dr, ~Ir,~Gr 4. 1) (x)[(bx Sx) ~Nx] 2) (x)(tx Bx) Tr 3) Tr Nr 4) ~Nr / \ 5) ~~Nr NC,4 / \ ~Nr 6) Nr DN,5 ~Br ~Sr x 7) (Br Sr) ~Nr UI,1 / \ / \ 8) ~(Br Sr) v ~Nr Impl,7 ~Tr Br ~Tr Br 9) (~Br v ~Sr) v ~Nr DeM,8 x x x 10) Tr Br UI,2 11) ~Tr v Br Impl,10 invalid: Tr,Nr,~Sr,Br Br Sr Nr Tr (Br.Sr) ~Nr Tr. Br Tr ~Nr T F T T T F F T T T T F P ( x)(dx Vx) 2. ( x)(dx ~Gx) 3. ( x)[dx (Bx Gx)] 4. ( x)[px (Tx ~Ex)] 5. ( x)[px ~(Ix Hx)] P. 141, I. 1. ( x)~cx 2. ( x)(cx) 3. ( x)~(ax Bx) 4. ( x)(bx Cx) 5. (x)~bx 6. (x)~(ax Bx) 7. (x)(ax Bx) 8. (x)~(ax Cx Dx) P. 142 II. 1. 4) ~( x)ax NC,3 ~Aa 5) (x)~ax QE,4 Ba 6) ~Aa UI,5 / \ invalid 7) Aa Ba UI,1 ~Aa Ba 8) ~Aa v Ba Impl,7 9) Ba UI,2

11 11 / \ 2. 4) ~( x)(~bx Ax) NC,3 Ba ~Aa 5) (x)~(~bx Ax) QE,4 / \ \ 6) ~(~Ba Aa) UI,5 ~Ba Ca ~Ba Ca 7) ~~Ba v ~Aa DeM,6 x / \ \ 8) Ba v ~Aa DN,7 ~Aa Ba ~Aa Ba ~Aa Ba 9) Aa Ba UI,1 x 10) ~Aa v Ba Impl,9 invalid: Ba,Ca,~Aa or Ba, Ca or 11) Ba Ca UI,2 ~Aa,~Ba or ~Aa,Ca or ~Aa, 12) ~Ba v Ca Impl,11 Ca,Ba 3. 4) ~( x)(bx v Dx) NC,3 ~Ba 5) (x)~(bx v Dx) QE,4 ~Da 6) ~(Ba v Da) UI,5 / \ invalid 7) ~Ba ~Da DeM,6 ~Aa Ca 8)(Aa v Ba) Ca UI,1 ~Ba \ 9) ~(AavBa) v Ca Impl,8 / \ ~Ba Da 10) (~Aa ~Ba) v Ca DeM,9 ~Ba Da x 11) Ba Da UI,2 x 12) ~Ba v Da Impl, ) ~(~Aa.~Ba) NC,4 ~Ca 6) ~~Aa v ~~Ba DeM,5 / \ 7) Aa v Ba (2)DN,5 Aa Ba 8) Aa Ba UI,1 / \ / \ 9) ~Aa v Ba Impl,8 ~Aa Ba ~Aa Ba 10) Ba Ca UI,2 x / \ \ 11) ~Ba v Ca Impl,10 ~Ba Ca ~Ba Ca ~Ba Ca x x x x x x 5. 1) (x)(px Bx) 2) (x) (Mx ~Bx) / \ 3) ( x)(px ~Mx) ~Pa Ma 4) ~( x)(px ~Mx) NC,3 / \ / \ 5) x)~(px ~Mx) QE,4 ~Ma ~Ba ~Ma ~Ba 6)~(Pa ~Ma) UI,5 /\ / \ x / \ 7) ~Pa v ~~Ma DeM,6 ~Pa Ba ~Pa Ba ~Pa Ba 8) ~Pa v Ma DN,7 x x 9) Pa Ba UI, 1 10) ~Pa v Ba Impl,9 invalid 11) Ma ~Ba UI,2 12) ~Ma v ~Ba Impl, ) (x)[px (Cx v Ix)] / \ 2) (x)(gx ~Ix) / \ 3) ( x)(gx Cx) / / \ 4) ~( x)(gx Cx) NC,3 ~Pa Ca Ia 5) (x) ~(Gx Cx) QE,4 / \ / \ / \ 6) ~(Ga Ca) UI,5 ~Ga ~Ia ~Ga ~Ia ~Ga ~Ia 7) ~Ga v ~Ca DeM,6 / \ / \ / \ / \ / \ x 8) Pa (Ca v Ia) UI,1 ~Ga~Ca~Ga~Ca~Ga~Ca~Ga~Ca~Ga~Ca 9) ~Pa v (Ca v Ia) Impl,8 x x 10) Ga ~Ia UI,2 11) ~Ga v ~Ia Impl,10

12 12 P ) ~( x)cx NC,3 Aa 5) (x)~cx QE,4 ~Ca 6) Aa EI,2 / \ 7) ~Ca UI,5 ~Aa Ca 8) (Aa v Ba) Ca UI,1 ~Ba x 9) ~(Aa v Ba) v Ca Impl,8 x 10) (~Aa ~Ba) v Ca DeM,9 P ) ~( x)dx NC,4 Aa 6) (x)~dx QE,5 Ba 7) Aa Ba EI,1 ~Da 8) ~Da UI,6 / \ 9) (Ba Ca) Da UI,2 ~Aa Da 10) ~(Ba Ca) v Da Impl,9 x x (second premise is 11) (~Ba v ~Ca) v Da DeM,10 not really needed) 12) Aa Da UI,3 13) ~Aa v Da Impl, ) ~(x)(ex Ax) NC,4 14) Aa Ba UI,1 Ea 6) ( x)~(ex ~Ax) QE,5 15) ~Aa v Ba Impl,14 Aa 7) (x)(ex ~Dx) QE,3 16) Ba Da UI,2 / \ 8) ~(Ea ~Aa) EI,6 17) ~Ba v Da Impl,16 ~Ea ~Da 9) Ea ~~Aa NI,8 x / \ 10) Ea Aa DN,9 ~Ba Da 11) (x)(x ~Dx) QE,3 / \ x 12) Ea ~Da UI,11 ~Aa Ba 13) ~Ea v ~Da Impl,12 x x 5. 5) ~( x)~ax NC,4 12) ~(Ca v ~Da) v ~Ba Impl,11 Ca 6) (x)ax QE,5 13) (~Ca ~Da) v ~Ba DeM, 12 Aa 7) Ca EI,3 / \ 8) Aa UI,6 ~Ca ~Ba 9) Aa Ba UI,1 ~Da / \ 10) ~Aa v Ba Impl,9 x ~Aa Ba 11) (Ca v Da) ~Ba UI,2 x x 6. 1) (x)(cx Lx) 7) Ca ~Wa NI,6 Ca 2) (x)(lx Wx) 8) Ca La UI,1 ~Wa 3) (x)(cx Wx) 9) ~Ca v La Impl,8 / \ 4) ~(x)(cx Wx) NC,3 10) La Wa UI,2 ~Ca La 5) ( x)~(cx Wx) QE,4 11) ~La v Wa Impl,10 x / \ 6) ~(Ca Wa) EI,5 ~La Wa x x 7. Ca 1) (x)(fx Sx) 6) Ca Fa EI,2 Fa 2) ( x)(cx Fx) 7) ~(Ca Sa) UI,5 / \ 3) ( x)(cx Sx) 8) ~Ca v ~Sa DeM,7 ~Ca ~Sa 4) ~( x)(cx Sx) NC,3 9) Fa Sa UI,1 x / \ 5) (x)~(cx Sx) QE,4 10) ~Fa v Sa Impl,9 ~Fa Sa x x

13 ) ( x)(lx.~jx) 8) ~La v ~~Sa DeM,7 La 2) (x)[(lx.~jx) ~S] 9) ~La v Sa DN,8 ~Ja 3) ( x)(lx ~Sx) 10) (La ~Ja) ~Sa UI,2 / \ 4) ~( x)(lx ~Sx) NC3 11) ~(La ~Ja) v ~Sa Impl,10 ~La Sa 5) (x)~(lx ~Sx) QE,4 12) (~La v ~~Ja) v ~Sa DeM,11 x / \ 6) (La ~Ja) EI,1 13) (~La v Ja) v ~Sa DN,12 / \ ~Sa 7) ~(La ~Sa) UI,5 ~La Ja x x x 9. 1) (x)[px (Cx Ix)] 9) Pa (Ca Ia) UI,1 Pa 2) ( x)(px Lx) 10) ~Pa v (Ca Ia) Impl,9 La 3) ( x)(lx Cx) / \ 4) ~( x)(lx Cx) NC,3 ~La ~Ca 5) (x)~(lx Cx) QE,4 x / \ 6) (Pa La) EI,2 ~Pa Ca 7) ~(La Ca) UI,5 x Ia 8) ~La v ~Ca DeM,7 x 10. Ca 1) (x)[ca (Ix v Vx)] 9) Ca (~~Ra ~Fa) DeM,8 Ra 2) (x)(ix ~Rx) 10) Ca (Ra ~Fa) DN,9 ~Fa 3) (x)(vx Fx) 11) Ca (Ia v Va) UI,1 / \ 4) (x)[cx (~Rx v Fx)] 12) ~Ca v (Ia v Va) Impl,11 ~Ia ~Ra 5) ~(x)[cx (~Rx v Fx)] NC,4 13) Ia ~Ra UI,2 / \ x 6) ( x)~[cx (~Rx v Fx)] QE,5 14) ~Ia v ~Ra Impl,13 ~Ca / \ 7) ~[Ca (~Ra v Fa)] EI,6 15) Va Fa UI,3 x Ia Va 8) Ca ~(~Ra v Fa) NI,7 16) ~Va v Fa Impl,15 x / ~Va Fa x x P ) (x) (Cx Ex) 8) Ca ~Ma NI,7 Ca 2) (x)[ex (Mx v Ix)] 9) Ca Ea UI,1 ~Ma 3) (x)(cx ~Ix) 10) ~Ca v Ea Impl,9 / \ 4) (x)(cx Mx) 11) Ea (Ma v Ia) UI,2 ~Ca Ea 5) ~(x)(cx Mx) NC,4 12) ~Ea v (Ma v Ia) Impl,11 x / \ 6) ( x)~[cx Mx) QE,5 13) Ca ~Ia UI,3 ~Ea Ma Ia 7) ~(Ca Ma) EI,6 14) ~Ca v ~Ia Impl,13 x x / \ 2. La 1) ( x)(px.cx) 7) ~(La Ca) EI, 6 ~Ca 2) ~(x)(lx Cx) 8) La ~Ca NI,7 Pb 3) ( x)(lx ~Px) 9) Pb Cb EI,1 Cb 4) ~( x)(lx ~Px) NC,3 10) ~(La ~Pa) UI,5 / \ 5) (x)~(lx ~Px) QE,4 11) ~La v ~~Pa DeM,10 ~La Pa 6) ( x)~(lx Cx) QE,2 12) ~La v Pa DN,11 x 3. 1) (x) (Bx Fx) 10) Ba Fa UI,1 Ca 2) (x)(rx Bx) 11) ~Ba v Fa Impl,10 Ra 3) ( x)(cx Rx) 12) Ra Ba UI,2 / \ 4) ( x)(cx Fx) 13) ~Ra v Ba Impl,12 ~Ca ~Fa 5) ~( x)(cx Fx) NC,4 x / \ 6) (x)~(cx Fx) QE,5 ~Ba Fa 7) Ca Ra EI,3 / \ x 8) ~(Ca Fa) UI,6 ~Ra Ba 9) ~Ca v ~Fa DeM,8 x x ~Ca ~Ia x x

14 ) (x)[(px v Cx) Fx] Aa 2) ( x)(ax Px) Pa 3) ( x)(ax Cx) / \ 4) ~( x)(ax Cx) NC,3 ~Aa ~Ca 5) (x)~(ax Cx) QE,4 x / \ 6) Aa Pa EI,3 ~Pa Fa 7) ~(Aa Ca) UI,5 ~Ca 8) ~Aa v ~Ca DeM,7 x 9) (Pa v Ca) Fa UI,1 10) ~(Pa v Ca) v Fa Impl,9 11) (~Pa ~Ca) v Fa DeM, ) (x)[(px v Cx) Rx] 9) Pa Sa DN,8 Pa 2) (x)(ax ~Sx) 10) (Pa v Ca) Ra UI,1 Sa 3) (x)(cx ~Ax) 11) ~(Pa v Ca) v Ra Impl,10 / \ 4) (x)(px ~Sx) 12) (~Pa.~Ca) v Ra DeM,11 ~Pa Ra 5) ~(x)(px ~Sx) NC,4 13) Aa ~Sa UI,2 ~Ca / \ 6) ( x)~(px ~Sx) QE,5 14) ~Aa v ~Sa Impl,13 x ~Aa ~Sa 7) ~(Pa ~Sa) EI,6 15) Ca ~Aa UI,3 / \ x 8) Pa ~~Sa NI,7 16) ~Ca v ~Aa Impl,15 ~Ca ~Aa 6. 1) (x)(hx ~Vx) 7) La ~~Va) NI,6 La 2) (x)(lx Hx) 8) La Va DN,7 Va 3) (x)(lx ~Vx) 9) Ha ~Va UI,1 / \ 4) ~(x)(lx ~Vx) NC,3 10) ~Ha v ~Va) Impl,9 ~Ha ~Va 5) ( x)~(lx ~Vx) QE,4 11) La Ha UI,2 / \ x 6) ~(La ~Va) EI,5 12) ~La v Ha Impl,11 ~La Ha x x 7. 1) (x)(bx Fx) Oa 2) (x)(ox ~Tx) Ba 3) (x)[(fx ~Tx) Px] ~Pa 4) (x)[(ox Bx) Px] / \ 5) ~(x)[(ox Bx) Px] NC,4 ~Oa ~Ta 6) ( x)~[(ox Bx) Px] QE,5 x / \ 7) ~[(Oa Ba) Pa] EI,6 / \ Pa 8) (Oa Ba) ~Pa NI,7 ~Fa Ta x 9) Ba Fa UI,1 / \ x 10) ~Ba v Fa Impl,9 ~Ba Fa 11) Oa ~Ta UI,2 x x 12) ~Oa v ~Ta Impl,11 13) (Fa ~Ta) Pa UI,3 14) ~(Fa ~Ta) v Pa Impl,13 15) (~Fa ~~Ta) v Pa DeM,14 16) (~Fa v Ta) v Pa DN,15

15 ) (x)(bx Fx) Oa 2) (x)[(fx Rx) ~Px] Ba 3) ( x)[(ox Bx) (Fx Rx)] Fa 4) ( x)(bx ~Px) Ra 5) ~( x)(bx ~Px) NC,4 / \ 6) (x)~(bx ~Px) QE,5 / \ ~Pa 7) Oa Ba (Fa Ra) EI,3 ~Fa ~Ra \ 8) Ba Fa UI,1 x x ~Ba Pa 9) ~Ba v Fa Impl,8 x x 10) (Fa Ra) ~Pa UI,2 11) ~(Fa Ra) v ~Pa Impl,10 first premise is not needed 12) (~Fa v ~Ra) v ~Pa DeM,11 13) ~(Ba ~Pa) UI,6 14) ~Ba v ~~Pa DeM,13 15) ~Ba v Pa DN,14 Aa 9. 1) (x)(ax Fx) 8) ~(Fa ~Ea) UI,6 / \ 2) ( x)(ax (Rx v ~Hx) 9) ~Fa v ~~Ea DeM,8 Ra ~Ha 3) (x)(rx ~Ex) 10) ~Fa v Ea DN,9 / \ / \ 4) ( x)(fx ~Ex) 11) Aa Fa UI,1 ~Aa Fa ~Aa Fa 5) ~( x)(fx ~Ex) NC,4 12) ~Aa v Fa Impl,11 x / \ x / \ 6) (x)~(fx ~Ex) QE,5 13) Ra ~Ea UI,3 ~Ra ~Ea ~Ra ~Ea 7) Aa (Ra v ~Ha) EI,2 14) ~Ra v ~Ea Impl,13 x / \ / / \ (diagram 7,12,14,10) ~Fa Ea~Fa Ea ~Fa Ea x x x x x 10. 1) ( x)(bx ~Dx) 2) (x)(bx ~Sx) 3) (x)[(dx Bx) Hx] 4) ( x)(hx ~Sx 5) ~( x)(hx ~Sx) NC,4 Ba 6) (x)~(hx ~Sx) QE,5 Da 7) Ba ~Da EI,1 / \ 8) ~(Ha ~Sa) UI,6 ~Ba ~Sa 9) ~Ha v ~~Sa DeM,8 x / \ 10) ~Ha v Sa DN,9 / \ \ 11) Ba ~Sa UI,2 ~Da ~Ba Ha 12) ~Ba v ~Sa Impl,11 x x / \ 13) (Da Ba) Ha UI,3 ~Ha Sa 14) ~(Da Ba) v Ha Impl,13 x x 15) (~Da v ~Ba) v Ha DeM,14 (diagramming 7,12,15,10) P. 153 I. 3. [(x)(~bx)] [( x)(ex)] 1. [( x)(bx)] [(x)(gx)] 4. [(x)(bx Gx)] v [( x)[~px)] 2. [(x)~bx] [( x)ex] 5. [(x)(cx)] v (( x)(ex)] P. 154, II, 2. Ca 1) (x)[cx (Dx v ~Lx)] 9) Ca Da DN,8 Da 2) (x)~(lx ~Dx) 10) Cb ~Lb EI,3 Cb 3) ( x)(cx ~Lx) 11) Ca (Da v ~La) UI,1 ~Lb 4) (x)(cx ~Dx) 12) ~Ca v (Da v ~La) Impl,11 / \ 5) ~(x)(cx ~Dx) NC,4 13) ~(La ~Da) UI,2 / / \ 6) ( x)~(cx ~Dx) QE,5 14) ~La v ~~Da DeM,13 ~Ca Da ~La 7) ~(Ca ~Da) EI,6 15) ~La v Da DN,14 x / \ / \ 8) Ca ~~Da NI,7 ~La Da ~La Da

16 3. Ea 1) (x)(ax v Sx) 7) ( x)~[(ex ~Sx) Dx] QE,3 ~Da 2) ( x)ex 8) Ea EI,2 Eb 3) ~(x)[(ex ~Sx) Dx] 9) ~[(Eb ~Sb) Db] EI,7 ~Sb 4) ( x)dx 10) (Eb ~Sb) ~Db NI,9 ~Db 5) ~( x)dx NC,4 11) ~Da UI,6 / \ 6) (x)~dx QE,5 12) Aa v Sa UI,1 Aa Sa ) (x)(ex ~Sx) 2) ( x)(ex) 8) ~Da UI,6 Ea 3) (x)(ex Dx) 9) Ea ~Sa UI,1 ~Da 4) ( x)dx 10) ~Ea v ~Sa UI,3 / \ 5) ~( x)dx NC,4 11) Ea Da UI,3 ~Ea Da 6) (x)~dx QE,5 12) ~Ea v Da Impl,11 x x 7) Ea EI,2 (diagramming 7,8,12 only) 5. 1) (x)[(vx Dx) Mx] 2) (x) (Rx Vx) 3) ( x)(rx Dx) Ra 4) ( x)(rx Mx) Da 5) ~( x)(rx Mx) NC,4 / \ 6) (x)~(rx Mx) QE,5 ~Ra ~Ma 7) Ra Da EI,3 x / \ 8)~(Ra Ma) UI,6 / \ \ 9) ~Ra v ~Ma DeM,8 ~Va ~Da Ma 10) (Va Da) Ma UI,1 / \ x x 11) ~(Va Da) v Ma Impl,10 ~Ra Va 12) (~Va v ~Da) v Ma DeM,11 x x 13) (Ra Va) UI,2 14) ~Ra v Va Impl, ) (x)(hx Dx) 2) (x)(dx Ix) or (x) (~Ix ~Dx) Ia 3) (x)(ix Hx) ~Ha 4) ~(x)(ix Hx) NC,3 / \ 5) ( x)~(ix Hx) QE,4 ~Ha Da 6) ~(Ia Ha) EI,5 / \ / \ 7) Ia ~Ha NI,6 ~Da Ia ~Da Ia 8) Ha Da UI,1 x 9) ~Ha v Da Impl,8 10) Da Ia UI,2 11) ~Da v Ia Impl,10 Ia 7. ~Wa 1) (x)(ax Wx) 8) ~(Ia ~Da) UI,6 / \ 2) ( x)(ix ~Wx) 9) ~Ia v ~~Da DeM,8 ~Ia Da 3) (x)(dx Ax) 10) ~Ia v Da DN,9 x / \ 4) ( x)(ix ~Dx) 11) Aa Wa UI,1 ~Aa Wa 5) ~( x)(ix ~Dx) NC,4 12) ~Aa v Wa Impl,11 / \ x 6) (x)~(ix ~Dx) QE,5 13) Da Aa UI,3 ~Da Aa 7) Ia ~Wa EI,2 14) ~Da v Aa Impl,13 x x

17 ) (x)(cx Ex) 9) Sa Ta DN,8 Sa 2) (x)(sx ~Cx) 10) Cb Tb EI,3 Ta 3) ( x)(cx Tx) 11) Ca Ea UI,1 Cb 4) (x)(sx ~Tx) 12) ~Ca v Ea Impl,11 Tb 5) ~(x)(sx ~Tx) NC,4 13) Sa ~Ca UI,2 / \ 6) ( x)~(sx ~Tx) QE,5 14) ~Sa v ~Ca Impl,13 ~Sa ~Ca 7) ~(Sa ~Ta) EI,6 x / \ 8) Sa ~~Ta NI,7 ~Ca Ea P ( y) Py (x) (Px Sxy) 3. ( y) Py (x) (Px Syx) 4. (x)(y) {[(Px Py) Ixy] Sxy} 5. ~(x)(y) {[(Px Py) Sxy] Ixy} P. 198 I. 1. Abusive ad hominem; 2. circumstantial ad hominem; 3. appeal to force; 4. pointing to the other wrong; 5. appeal to self-righteousness; 6. appeal to tradition; 7. arguing beside the point; 8. appeal to authority. P. 199 II. 1. ad hominem (more circumstantial than abusive); 2. appeal to force; 3. equivocation on "war" (formally declared war as opposed to the commonsense understanding of war); 4. circumstantial ad hominem; 5. begging the question; 6. abusive ad hominem. P quantitative, classical; 3. quantitative; 4. quantitative, relative frequency; 5. quantitative (since this appears to be more precise than just a ballpark guess), relative frequency. Pp a..7; b..2 x.5 =.1; c =.6 2. a. 1/6 (of the 36 possible combinations totaling 7: 6+1, 1+6, 3+4, 4+3, 5+2, 2+5) b.2/9 (in addition to the above six combinations, there are two more: 5+6 and 6+5, yielding a total of 8 out of 36) 3.a. 4/52 = 7.7%; b. 4/51 = 7.9%; c. 4/53 x 4/51 = 16/2652 =.6%; d. 1-4/52 = 48/52 = 9.2% 5.1/365. The first person has a 100% chance of having a birthday. The odds that the second person has the same birthdate are 1/365. (1 x 1/365 1/365). (Note that these people are chosen at random. This is not the same question as the one at issue when recruits in the military are suckered into betting that two recruits in a room of, say, fifty people, share a birthday. In that situation, one keeps looking until one finds a match. The odds increase dramatically because one does not simply stop with the first two people picked at random.) 6. a. 1/2 x 1/90 = 1/180; b. 1/4 x 1/180 = 1/720 Solution to #8, p. 255 On the one hand, it would seem that it would not matter, because one of the two remaining doors contains the prize anyway, so your chances of winning seem to be the same whether you switch or not. This would certainly be true if the host opened door number two before you announced your choice to the host. But does it make a difference that he opened the door after you made your choice? Another way of analyzing the situation is this: Ask what the chances are of being wrong before receiving any help from the host. The answer is that the

18 odds are two out of three. And then ask whether those odds could possibly be improved by a policy of switching. That is, given all the possible combinations (of doors with goats and a Ferrari behind them and of initial choices one might make before the host opens a door with a goat), would a policy of switching be more likely to open the door with the Ferrari than with the remaining goat? 18 P The principle does not hold for every assignment, but it does hold when p = 1 or when P = 0. However, the principle that we discussed in Chapters III and V held only for those assignments anyway, because we introduced contradiction in crisp logic. 2. Where S =.25 and J =.8, the value of J S is max 1 - J, S, or max.2,.25, or Two reasons for believing crisp logic is a special case of fuzzy logic are the results of the principle of contradiction and of the application of disjunction and conjunction. Since the law of contradiction holds for all of crisp logic but only for special cases in fuzzy logic, it would seem that crisp logic is a special case of fuzzy logic. Similarly, the rules for fuzzy disjunction and conjunction yield the correct result regardless of the fuzzy assignment, including the extreme cases that fall into crisp logic, but the crisp logic rules hold only for crisp logic. 4. It means: the value of fuzzy p and fuzzy q is the lesser of fuzzy p and fuzzy q. This is false. 5. The translation is P Q = min P, Q. This is correct.