Exercise Set 11 (Predicate Logic) Keith Burgess-Jackson 2 November 2017

Transcription

1 Exercise Set 11 (Predicate Logic) Keith Burgess-Jackson 2 November 2017 Exercises I. Translate each of the following English sentences into the logical notation of propositional functions and quantifiers, in each case using the abbreviations suggested, and having each formula begin with a quantifier, not with a negation symbol. 1. Bats are mammals. (Bx: x is a bat; Mx: x is a mammal.) 2. Sparrows are not mammals. (Sx: x is a sparrow; Mx: x is a mammal.) 3. Reporters are present. (Rx: x is a reporter; Px: x is present.) 4. Nurses are always considerate. (Nx: x is a nurse; Cx: x is considerate.) 5. Diplomats are not always rich. (Dx: x is a diplomat; Rx: x is rich.) 6. Ambassadors are always dignified. (Ax: x is an ambassador; Dx: x is dignified.) 7. No boy scout ever cheats. (Bx: x is a boy scout; Cx: x cheats.) 8. Only licensed physicians can charge for medical treatment. (Lx: x is a licensed physician; Cx: x can charge for medical treatment.) 9. Snake bites are sometimes fatal. (Sx: x is a snake bite; Fx: x is fatal.) 10. The common cold is never fatal. (Cx: x is a common cold; Fx: x is fatal.) 11. A child pointed his finger at the emperor. (Cx: x is a child; Px: x pointed his finger at the emperor.) 12. Not all children pointed their fingers at the emperor. (Cx: x is a child; Px: x pointed his finger at the emperor.) 1

2 13. All that glitters is not gold. (Gx: x glitters; Ax: x is gold.) 14. None but the brave deserve the fair. (Bx: x is brave; Dx: x deserves the fair.) 15. Only citizens of the United States can vote in U.S. elections. (Cx: x is a citizen of the United States; Vx: x can vote in U.S. elections.) 16. Citizens of the United States can vote only in U.S. elections. (Ex: x is an election in which citizens of the United States can vote; Ux: x is a U.S. election.) 17. There are honest politicians. (Hx: x is honest; Px: x is a politician.) 18. Not every applicant was hired. (Ax: x is an applicant; Hx: x was hired.) 19. Not any applicant was hired. (Ax: x is an applicant; Hx: x was hired.) 20. Nothing of importance was said. (Ix: x is of importance; Sx: x was said.) II. Translate each of the following English sentences into the logical notation of propositional functions and quantifiers. 1. Some zebras are not mammals. 2. Something is orange. 3. All boxes are sturdy. 4. Everything is blue. 5. Some tigers are lions. 6. Nothing is red. 7. Something is not purple. 8. No sores are gangrenous. III. Translate the following symbolized expressions into English. Choose appropriate names and properties. 2

3 1. Tk 2. Dx 3. (x)cx 4. (Ǝx)(Wx Kx) 5. Φx 6. (Ǝx)Lx 7. (x)(ex כ Fx) 8. Mn 9. (x)(gx כ ~Hx) 10. (x)~zx 11. (Ǝx)Ox 12. Φr 13. (Ǝx)~Sx 14. (Ǝx)(Tx ~Bx) Which of your translated propositions are true and which false? Which are neither true nor false? IV. Translate the following symbolized expressions into English. Choose appropriate names and properties. 1. (x)ax É (x)bx 2. (x)cx É (Ǝx)Dx 3. (x)ex É Fm 4. (Ǝx)Gx É (x)hx 5. (Ǝx)Ix É (Ǝx)Jx 6. (Ǝx)Kx É Lt 3

4 7. Me É (x)nx 8. Oe É (Ǝx)Px 9. Qe É Rf 10. [(x)sx (Ǝx)Tx] É Uk 11. (Ǝx)Vx v (Ǝx)Wx 12. (x)xx º ~(Ǝx)~Yx 13. (Zc v Ad) É (Za v Ag) 14. (x)(bx É Cx) 15. (x)(dx É Ex) v Fn 16. (Ǝx)(Gx Hx) É (x)(ix É ~Jx) 17. (x)kx É (x)(kx É Lx) 18. (Ǝx)Mx v (Ǝx)(Nx ~Ox) 19. [Pa v (x)qx] É [(Ǝx)(Rx Sx) º Ts] 20. Us ~(x)vx 21. Wr É (Xo v Yi) 22. (x)zx É (Ǝx)Zx V. For each of the following, find a normal-form formula (NFF) that is logically equivalent to the given one. 1. ~(x)(ax כ Bx) 2. ~(x)(cx כ ~Dx) 3. ~(Ǝx)(Ex Fx) 4. ~(Ǝx)(Gx ~Hx) 4

5 5. ~(x)(~ix Ú Jx) 6. ~(x)(~kx Ú ~Lx) 7. ~(Ǝx)[~(Mx Ú Nx)] 8. ~(Ǝx)[~(Ox Ú ~Px)] 9. ~(Ǝx)[~(~Qx Ú Rx)] 10. ~(x)[~(sx ~Tx)] 11. ~(x)[~(~ux ~Vx)] 12. ~(Ǝx)[~(~Wx Ú Xx)] 13. ~(x)(y)(z)[(rxy Ryz) כ Rxz] The 13th exercise differs from the previous 12 in that R is a dyadic (binary, two-place) predicate rather than a monadic (one-place) predicate. The technique, however, is the same. VI. Construct a formal proof of validity for each of the following arguments (x)(ax כ ~Bx) 2. (Ǝx)(Cx Ax) / (Ǝx)(Cx ~Bx) (x)(dx כ ~Ex) 2. (x)(fx כ Ex) / (x)(fx כ ~ Dx) (x)(gx כ Hx) 2. (x)(ix כ ~Hx) / (x)(ix כ ~Gx) (Ǝx)(Jx Kx) 2. (x)(jx כ Lx) / (Ǝx)(Lx Kx) (x)(mx כ Nx) 2. (Ǝx)(Mx Ox) / (Ǝx)(Ox Nx) (Ǝx)(Px ~Qx) 2. (x)(px כ Rx) / (Ǝx)(Rx ~Qx) (x)(sx כ ~Tx) 5

6 2. (Ǝx)(Sx Ux) / (Ǝx)(Ux ~Tx) (x)(vx כ Wx) 2. (x)(wx כ ~Xx) / (x)(xx כ ~Vx) (Ǝx)(Yx Zx) 2. (x)(zx כ Ax) / (Ǝx)(Ax Yx) (x)(bx כ ~Cx) 2. (Ǝx)(Cx Dx) / (Ǝx)(Dx ~Bx) (x)(fx כ Gx) 2. (Ǝx)(Fx ~Gx) / (Ǝx)(Gx ~Fx) VII. Construct a formal proof of validity for each of the following arguments, in each case using the suggested notations. 1. No athletes are bookworms. Carol is a bookworm. Therefore, Carol is not an athlete. (Ax, Bx, c) 2. All dancers are exuberant. Some fencers are not exuberant. Therefore, some fencers are not dancers. (Dx, Ex, Fx) 3. No gamblers are happy. Some idealists are happy. Therefore, some idealists are not gamblers. (Gx, Hx, Ix) 4. All jesters are knaves. No knaves are lucky. Therefore, no jesters are lucky. (Jx, Kx, Lx) 5. All mountaineers are neighborly. Some outlaws are mountaineers. Therefore, some outlaws are neighborly. (Mx, Nx, Ox) 6. Only pacifists are Quakers. There are religious Quakers. Therefore, pacifists are sometimes religious. (Px, Qx, Rx) 7. To be a swindler is to be a thief. None but the underprivileged are thieves. Therefore, swindlers are always underprivileged. (Sx, Tx, Ux) 8. No violinists are not wealthy. There are no wealthy xylophonists. Therefore, violinists are never xylophonists. (Vx, Wx, Xx) 9. None but the brave deserve the fair. Only soldiers are brave. Therefore, the fair are deserved only by soldiers. 6

7 (Dx: x deserves the fair; Bx: x is brave; Sx: x is a soldier) 10. Everyone that asketh receiveth. Simon receiveth not. Therefore, Simon asketh not. (Ax, Rx, s) 11. All men are mortal. Socrates is a man. Therefore, Socrates is mortal. (Ex, Ox, s) VIII. Complete the justifications of the following proofs (Ǝx)Gx כ (x)hx 2. Ga / Ha 3. (Ǝx)Gx 4. (x)hx 5. Ha If something is G, then everything is H; a is G; therefore, a is H Fa ~Ga 2. (x)[fx כ (Gx Ú Hx)] / Ha 3. Fa כ (Ga Ú Ha) 4. Fa 5. Ga Ú Ha 6. ~Ga Fa 7. ~Ga 8. Ha a is F and a is not G; all F is either G or H; therefore, a is H (x)[(fx Ú Gx) כ Hx] 2. ~Ha / (Ǝx)~Gx 3. (Fa Ú Ga) כ Ha 4. ~(Fa Ú Ga) 5. ~Fa ~Ga 6. ~Ga ~Fa 7. ~Ga 8. (Ǝx)~Gx Anything that is F or G is H; a is not H; therefore, something is not G ~(Ǝx)Dx / Da כ Ga 2. (x)~dx 7

8 Solutions 3. ~Da 4. ~Da Ú Ga 5. Da כ Ga Nothing is D; therefore, if a is D, then a is G (x)~gx 2. (x)fx כ (Ǝx)Gx / (Ǝx)~Fx 3. ~(Ǝx)Gx 4. ~(x)fx 5. (Ǝx)~Fx Nothing is G; if everything is F, then something is G; therefore, something is not F ~(x)fx 2. Ga º Hb 3. (Ǝx)~Fx כ ~(Ǝx)Gx / ~Hb 4. (Ǝx)~Fx 5. ~(Ǝx)Gx 6. (x)~gx 7. ~Ga 8. (Ga כ Hb) (Hb כ Ga) 9. (Hb כ Ga) (Ga כ Hb) 10. Hb כ Ga 11. ~Hb Not everything is F; a is G iff b is H; if something is not F, then nothing is G; therefore, b is not H. I. Translate each of the following English sentences into the logical notation of propositional functions and quantifiers, in each case using the abbreviations suggested, and having each formula begin with a quantifier, not with a negation symbol. 1. Bats are mammals. (Bx: x is a bat; Mx: x is a mammal.) (x)(bx כ Mx) 2. Sparrows are not mammals. (Sx: x is a sparrow; Mx: x is a mammal.) (x)(sx כ ~Mx) 3. Reporters are present. (Rx: x is a reporter; Px: x is 8

9 present.) (Ǝx)(Rx Px) :: (Ǝx)(Px Rx) 4. Nurses are always considerate. (Nx: x is a nurse; Cx: x is considerate.) (x)(nx כ Cx) 5. Diplomats are not always rich. (Dx: x is a diplomat; Rx: x is rich.) (Ǝx)(Dx ~Rx) 6. Ambassadors are always dignified. (Ax: x is an ambassador; Dx: x is dignified.) (x)(ax כ Dx) 7. No boy scout ever cheats. (Bx: x is a boy scout; Cx: x cheats.) (x)(bx כ ~Cx) 8. Only licensed physicians can charge for medical treatment. (Lx: x is a licensed physician; Cx: x can charge for medical treatment.) (x)(cx כ Lx) :: (x)(~lx כ ~Cx) 9. Snake bites are sometimes fatal. (Sx: x is a snake bite; Fx: x is fatal.) (Ǝx)(Sx Fx) 10. The common cold is never fatal. (Cx: x is a common cold; Fx: x is fatal.) (x)(cx כ ~Fx) 11. A child pointed his finger at the emperor. (Cx: x is a child; Px: x pointed his finger at the emperor.) (Ǝx)(Cx Px) 12. Not all children pointed their fingers at the emperor. (Cx: x is a child; Px: x pointed his finger at the emperor.) (Ǝx)(Cx ~Px) 13. All that glitters is not gold. (Gx: x glitters; Ax: x is gold.) (Ǝx)(Gx ~Ax) :: (Ǝx)(~Ax Gx) 14. None but the brave deserve the fair. (Bx: x is brave; Dx: x deserves the fair.) 9

10 (x)(dx כ Bx) :: (x)(~bx כ ~Dx) 15. Only citizens of the United States can vote in U.S. elections. (Cx: x is a citizen of the United States; Vx: x can vote in U.S. elections.) (x)(vx כ Cx) :: (x)(~cx כ ~Vx) 16. Citizens of the United States can vote only in U.S. elections. (Ex: x is an election in which citizens of the United States can vote; Ux: x is a U.S. election.) (x)(ex כ Ux) :: (x)(~ux כ ~Ex) 17. There are honest politicians. (Hx: x is honest; Px: x is a politician.) (Ǝx)(Px Hx) :: (Ǝx)(Hx Px) 18. Not every applicant was hired. (Ax: x is an applicant; Hx: x was hired.) (Ǝx)(Ax ~Hx) 19. Not any applicant was hired. (Ax: x is an applicant; Hx: x was hired.) (x)(ax כ ~Hx) :: (x)(hx כ ~Ax) 20. Nothing of importance was said. (Ix: x is of importance; Sx: x was said.) (x)(sx כ ~Ix) :: (x)(ix כ ~Sx) II. Translate each of the following English sentences into the logical notation of propositional functions and quantifiers. 1. Some zebras are not mammals. (Ǝx)(Zx ~Mx) 2. Something is orange. (Ǝx)Ox 3. All boxes are sturdy. (x)(bx כ Sx) 4. Everything is blue. (x)bx 5. Some tigers are lions. (Ǝx)(Tx Lx) 10

11 6. Nothing is red. (x)~rx 7. Something is not purple. (Ǝx)~Px 8. No sores are gangrenous. (x)(sx כ ~Gx) III. Translate the following symbolized expressions into English. Choose appropriate names and properties. 1. Tk Keith is tall. 2. Dx x is a dog. 3. (x)cx Everything is cool. 4. (Ǝx)(Wx Kx) Some Wisconsinites are killers. 5. Φx x is Φ. 6. (Ǝx)Lx Something is laconic. 7. (x)(ex כ Fx) All elephants are fabulous. 8. Mn Nick is a Moonie. 9. (x)(gx כ ~Hx) No giraffes are hilarious. 10. (x)~zx Nothing is a zebra. 11. (Ǝx)Ox Something is outrageous. 12. Φr Ralph is Φ. 11

12 13. (Ǝx)~Sx Something is not sad. 14. (Ǝx)(Tx ~Bx) Some tyrants are not bitter. IV. Translate the following symbolized expressions into English. Choose appropriate names and properties. 1. (x)ax É (x)bx If everything is an apple, then everything is a banana. 2. (x)cx É (Ǝx)Dx If everything is a cat, then something is a dog. 3. (x)ex É Fm If everything is an elephant, then Mark is fat. 4. (Ǝx)Gx É (x)hx If something is a giraffe, then everything is a horse. 5. (Ǝx)Ix É (Ǝx)Jx If something is intransigent, then something is juvenile. 6. (Ǝx)Kx É Lt If something is a Kansan, then Tim is a lieutenant. 7. Me É (x)nx If Edward is married, then everything is naked. 8. Oe É (Ǝx)Px If Ernie is obese, then something is poor. 9. Qe É Rf If Edgar is a quartermaster, then Frances is religious. 10. [(x)sx (Ǝx)Tx] É Uk If everything is small and something is tall, then Karen is unpopular. 11. (Ǝx)Vx v (Ǝx)Wx Either something is virtuous or something is wicked. 12. (x)xx º ~(Ǝx)~Yx 12

13 Everything is a xylophone if and only if it s not the case that something is not yellow. 13. (Zc v Ad) É (Za v Ag) If either Carl is a zebra or David is an accountant, then either Andre is a zebra or Georgia is an accountant. 14. (x)(bx É Cx) All bazookas are contraband. 15. (x)(dx É Ex) v Fn Either all dogs are elephants or Nick is a freak. 16. (Ǝx)(Gx Hx) É (x)(ix É ~Jx) If some goofballs are happy, then no insane people are jealous. 17. (x)kx É (x)(kx É Lx) If everything is a Kentuckian, then all Kentuckians are lazy. 18. (Ǝx)Mx v (Ǝx)(Nx ~Ox) Either something is married or some neighbors are not orthodox. 19. [Pa v (x)qx] É [(Ǝx)(Rx Sx) º Ts] If either Annabel is poor or everything is quiet, then some rabbits are snakes if and only if Susie is a tarantula. 20. Us ~(x)vx Stillmon is unmarried and it s not the case that everything is valuable. 21. Wr É (Xo v Yi) If Rachel is wrong, then either Oliver is an x-ray or Ivan is a Yalie. 22. (x)zx É (Ǝx)Zx If everything is a Zanzibarian, then something is a Zanzibarian. V. For each of the following, find a normal-form formula (NFF) that is logically equivalent to the given one. 1. ~(x)(ax כ Bx) (Ǝx)~(Ax כ Bx) (Ǝx)~(~Ax Ú Bx) (Ǝx)(~~Ax ~Bx) MI 13

14 (Ǝx)(Ax ~Bx) DN 2. ~(x)(cx כ ~Dx) (Ǝx)~(Cx כ ~Dx) (Ǝx)~(~Cx Ú ~Dx) MI -or- (Ǝx)~~(Cx Dx) (Ǝx)(~~Cx ~~Dx) (Ǝx)(Cx Dx) DN (Ǝx)(Cx ~~Dx) DN (Ǝx)(Cx Dx) DN 3. ~(Ǝx)(Ex Fx) (x)~(ex Fx) (x)(~ex Ú ~Fx) -or- (x)(ex כ ~Fx) 4. ~(Ǝx)(Gx ~Hx) (x)~(gx ~Hx) (x)(~gx Ú ~~Hx) (x)(~gx Ú Hx) -or- (x)(gx כ Hx) 5. ~(x)(~ix Ú Jx) (Ǝx)~(~Ix Ú Jx) (Ǝx)(~~Ix ~Jx) (Ǝx)(Ix ~Jx) 6. ~(x)(~kx Ú ~Lx) (Ǝx)~(~Kx Ú ~Lx) (Ǝx)~~(Kx Lx) (Ǝx)(Kx Lx) 7. ~(Ǝx)[~(Mx Ú Nx)] (x)~[~(mx Ú Nx)] (x)~(~mx ~Nx) (x)~~(mx Ú Nx) (x)(mx Ú Nx) 8. ~(Ǝx)[~(Ox Ú ~Px)] (x)~[~(ox Ú ~Px)] (x)~(~ox ~~Px) (x)~~(ox Ú ~Px) MI DN MI DN DN DN 14

15 (x)(ox Ú ~Px) -or- (x)(~px Ú Ox) -or- (x)(px כ Ox) 9. ~(Ǝx)[~(~Qx Ú Rx)] (x)~[~(~qx Ú Rx)] (x)~(~~qx ~Rx) (x)~~(~qx Ú Rx) (x)(~qx Ú Rx) -or- (x)(qx כ Rx) 10. ~(x)[~(sx ~Tx)] (Ǝx)~[~(Sx ~Tx)] (Ǝx)~(~Sx Ú ~~Tx) (Ǝx)~~(Sx ~Tx) (Ǝx)(Sx ~Tx) 11. ~(x)[~(~ux ~Vx)] (Ǝx)~[~(~Ux ~Vx)] (Ǝx)~[~~(Ux Ú Vx)] (Ǝx)~(Ux Ú Vx) (Ǝx)(~Ux ~Vx) 12. ~(Ǝx)[~(~Wx Ú Xx)] (x)~[~(~wx Ú Xx)] (x)~(~~wx ~Xx) (x)~~(wx Ú Xx) (x)(wx Ú Xx) DN Com MI DN MI DN DN DN 13. ~(x)(y)(z)[(rxy Ryz) כ Rxz] (Ǝx)~(y)(z)[(Rxy Ryz) כ Rxz] (Ǝx)(Ǝy)~(z)[(Rxy Ryz) כ Rxz] (Ǝx)(Ǝy)(Ǝz)~[(Rxy Ryz) כ Rxz] (Ǝx)(Ǝy)(Ǝz)~[~(Rxy Ryz) Ú Rxz] (Ǝx)(Ǝy)(Ǝz)[~~(Rxy Ryz) ~Rxz] (Ǝx)(Ǝy)(Ǝz)[(Rxy Ryz) ~Rxz] MI DN VI. Construct a formal proof of validity for each of the following arguments. 15

16 1. 1. (x)(ax כ ~Bx) 2. (Ǝx)(Cx Ax) / (Ǝx)(Cx ~Bx) 3. Ca Aa 2, EI 4. Aa כ ~Ba 1, UI 5. Ca 3, Simp 6. Aa Ca 3, Com 7. Aa 6, Simp 8. ~Ba 4, 7, MP 9. Ca ~Ba 5, 8, Conj 10. (Ǝx)(Cx ~Bx) 9, EG Note: This is EIO-1, which is unconditionally valid (x)(dx כ ~Ex) 2. (x)(fx כ Ex) / (x)(fx כ ~Dx) 3. Dy כ ~Ey 1, UI 4. Fy כ Ey 2, UI 5. ~~Ey כ ~Dy 3, Trans 6. Ey כ ~Dy 5, DN 7. Fy כ ~Dy 4, 6, HS 8. (x)(fx כ ~Dx) 7, UG Note: This is EAE-2,which is unconditionally valid (x)(gx כ Hx) 2. (x)(ix כ ~Hx) / (x)(ix כ ~Gx) 3. Gy כ Hy 1, UI 4. Iy כ ~Hy 2, UI 5. ~Hy כ ~Gy 3, Trans 6. Iy כ ~Gy 4, 5, HS 7. (x)(ix כ ~Gx) 6, UG Note: This is AEE-2, which is unconditionally valid (Ǝx)(Jx Kx) 2. (x)(jx כ Lx) / (Ǝx)(Lx Kx) 3. Ja Ka 1, EI 4. Ja כ La 2, UI 5. Ja 3, Simp 6. La 4, 5, MP 7. Ka Ja 3, Com 8. Ka 7, Simp 9. La Ka 6, 8, Conj 10. (Ǝx)(Lx Kx) 9, EG 16

17 Note: This is IAI-3, which is unconditionally valid (x)(mx כ Nx) 2. (Ǝx)(Mx Ox) / (Ǝx)(Ox Nx) 3. Ma Oa 2, EI 4. Ma כ Na 1, UI 5. Oa Ma 3, Com 6. Oa 5, Simp 7. Ma 3, Simp 8. Na 4, 7, MP 9. Oa Na 6, 8, Conj 10. (Ǝx)(Ox Nx) 9, EG Note: This is AII-3, which is unconditionally valid (Ǝx)(Px ~Qx) 2. (x)(px כ Rx) / (Ǝx)(Rx ~Qx) 3. Pa ~Qa 1, EI 4. Pa כ Ra 2, UI 5. Pa 3, Simp 6. Ra 4, 5, MP 7. ~Qa Pa 3, Com 8. ~Qa 7, Simp 9. Ra ~Qa 6, 8, Conj 10. (Ǝx)(Rx ~Qx) 9, EG Note: This is OAO-3, which is unconditionally valid (x)(sx כ ~Tx) 2. (Ǝx)(Sx Ux) / (Ǝx)(Ux ~Tx) 3. Sa Ua 2, EI 4. Sa כ ~Ta 1, UI 5. Ua Sa 3, Com 6. Ua 5, Simp 7. Sa 3, Simp 8. ~Ta 4, 7, MP 9. Ua ~Ta 6, 8, Conj 10. (Ǝx)(Ux ~Tx) 9, EG Note: This is EIO-3, which is unconditionally valid (x)(vx כ Wx) 2. (x)(wx כ ~Xx) / (x)(xx כ ~Vx) 3. Vy כ Wy 1, UI 17

18 4. Wy כ ~Xy 2, UI 5. Vy כ ~Xy 3, 4, HS 6. ~~Xy כ ~Vy 5, Trans 7. Xy כ ~Vy 6, DN 8. (x)(xx כ ~Vx) 7, UG Note: This is AEE-4, which is unconditionally valid (Ǝx)(Yx Zx) 2. (x)(zx כ Ax) / (Ǝx)(Ax Yx) 3. Ya Za 1, EI 4. Za כ Aa 2, UI 5. Ya 3, Simp 6. Za Ya 3, Com 7. Za 6, Simp 8. Aa 4, 7, MP 9. Aa Ya 8, 5, Conj 10. (Ǝx)(Ax Yx) 9, EG Note: This is IAI-4, which is unconditionally valid (x)(bx כ ~Cx) 2. (Ǝx)(Cx Dx) / (Ǝx)(Dx ~Bx) 3. Ca Da 2, EI 4. Ba כ ~Ca 1, UI 5. Da Ca 3, Com 6. Da 5, Simp 7. Ca 3, Simp 8. ~~Ca 7, DN 9. ~Ba 4, 8, MT 10. Da ~Ba 6, 9, Conj 11. (Ǝx)(Dx ~Bx) 10, EG Note: This is EIO-4, which is unconditionally valid (x)(fx כ Gx) 2. (Ǝx)(Fx ~Gx) / (Ǝx)(Gx ~Fx) 3. Fa ~Ga 2, EI 4. Fa כ Ga 1, UI 5. Fa 3, Simp 6. Ga 4, 5, MP 7. ~Ga Fa 3, Com 8. ~Ga 7, Simp 9. ~Fa 4, 8, MT 18

19 10. Ga ~Fa 6, 9, Conj 11. (Ǝx)(Gx ~Fx) 10, EG or 9. Ga Ú (Ǝx)(Gx ~Fx) 6, Add 10. (Ǝx)(Gx ~Fx) 9, 8, DS Note: This is AOO-2, which is unconditionally valid. VII. Construct a formal proof of validity for each of the following arguments, in each case using the suggested notations. 1. No athletes are bookworms. Carol is a bookworm. Therefore, Carol is not an athlete. (Ax, Bx, c) 1. (x)(ax כ ~Bx) 2. Bc / ~Ac 3. Ac כ ~Bc 1, UI 4. ~~Bc 2, DN 5. ~Ac 3, 4, MT 2. All dancers are exuberant. Some fencers are not exuberant. Therefore, some fencers are not dancers. (Dx, Ex, Fx) 1. (x)(dx כ Ex) 2. (Ǝx)(Fx ~Ex) / (Ǝx)(Fx ~Dx) 3. Fa ~Ea 2, EI 4. Da כ Ea 1, UI 5. Fa 3, Simp 6. ~Ea Fa 3, Com 7. ~Ea 6, Simp 8. ~Da 4, 7, MT 9. Fa ~Da 5, 8, Conj 10. (Ǝx)(Fx ~Dx) 9, EG Note: This is AOO-2, which is unconditionally valid. 3. No gamblers are happy. Some idealists are happy. Therefore, some idealists are not gamblers. (Gx, Hx, Ix) 1. (x)(gx כ ~Hx) 2. (Ǝx)(Ix Hx) / (Ǝx)(Ix ~Gx) 3. Ia Ha 2, EI 19

20 4. Ga כ ~Ha 1, UI 5. Ia 3, Simp 6. Ha Ia 3, Com 7. Ha 6, Simp 8. ~~Ha 7, DN 9. ~Ga 4, 8, MT 10. Ia ~Ga 5, 9, Conj 11. (Ǝx)(Ix ~Gx) 10, EG Note: This is EIO-2, which is unconditionally valid. 4. All jesters are knaves. No knaves are lucky. Therefore, no jesters are lucky. (Jx, Kx, Lx) 1. (x)(jx כ Kx) 2. (x)(kx כ ~Lx) / (x)(jx כ ~Lx) 3. Jy כ Ky 1, UI 4. Ky כ ~Ly 2, UI 5. Jy כ ~Ly 3, 4, HS 6. (x)(jx כ ~Lx) 5, UG Note: This is EAE-1, which is unconditionally valid. 5. All mountaineers are neighborly. Some outlaws are mountaineers. Therefore, some outlaws are neighborly. (Mx, Nx, Ox) 1. (x)(mx כ Nx) 2. (Ǝx)(Ox Mx) / (Ǝx)(Ox Nx) 3. Oa Ma 2, EI 4. Ma כ Na 1, UI 5. Oa 3, Simp 6. Ma Oa 3, Com 7. Ma 6, Simp 8. Na 4, 7, MP 9. Oa Na 5, 8, Conj 10. (Ǝx)(Ox Nx) 9, EG Note: This is AII-1, which is unconditionally valid. 6. Only pacifists are Quakers. There are religious Quakers. Therefore, pacifists are sometimes religious. (Px, Qx, Rx) 1. (x)(qx כ Px) 2. (Ǝx)(Rx Qx) / (Ǝx)(Px Rx) 3. Ra Qa 2, EI 20

21 4. Qa כ Pa 1, UI 5. Ra 3, Simp 6. Qa Ra 3, Com 7. Qa 6, Simp 8. Pa 4, 7, MP 9. Pa Ra 8, 5, Conj 10. (Ǝx)(Px Rx) 9, EG Note: This is IAI-4, which is unconditionally valid. 7. To be a swindler is to be a thief. None but the underprivileged are thieves. Therefore, swindlers are always underprivileged. (Sx, Tx, Ux) 1. (x)(sx כ Tx) 2. (x)(tx כ Ux) / (x)(sx כ Ux) 3. Sy כ Ty 1, UI 4. Ty כ Uy 2, UI 5. Sy כ Uy 3, 4, HS 6. (x)(sx כ Ux) 5, UG Note: This is AAA-1, which is unconditionally valid. 8. No violinists are not wealthy. There are no wealthy xylophonists. Therefore, violinists are never xylophonists. (Vx, Wx, Xx) 1. (x)(vx כ Wx) 2. (x)(wx כ ~Xx) / (x)(vx כ ~Xx) 3. Vy כ Wy 1, UI 4. Wy כ ~Xy 2, UI 5. Vy כ ~Xy 3, 4, HS 6. (x)(vx כ ~Xx) 5, UG Note: This is EAE-1, which is unconditionally valid. 9. None but the brave deserve the fair. Only soldiers are brave. Therefore, the fair are deserved only by soldiers. (Dx: x deserves the fair; Bx: x is brave; Sx: x is a soldier) 1. (x)(dx כ Bx) 2. (x)(bx כ Sx) / (x)(dx כ Sx) 3. Dy כ By 1, UI 4. By כ Sy 2, UI 5. Dy כ Sy 3, 4, HS 6. (x)(dx כ Sx) 5, UG 21

22 Note: This is AAA-1, which is unconditionally valid. 10. Everyone that asketh receiveth. Simon receiveth not. Therefore, Simon asketh not. (Ax, Rx, s) 1. (x)(ax כ Rx) 2. ~Rs / ~As 3. As כ Rs 1, UI 4. ~As 3, 2, MT 11. All men are mortal. Socrates is a man. Therefore, Socrates is mortal. (Ex, Ox, s) 1. (x)(ex כ Ox) 2. Es / Os 3. Es כ Os 1, UI 4. Os 3, 2, MP VIII. Complete the justifications of the following proofs (Ǝx)Gx כ (x)hx 2. Ga / Ha 3. (Ǝx)Gx 2, EG 4. (x)hx 1, 3, MP 5. Ha 4, UI If something is G, then everything is H; a is G; therefore, a is H Fa ~Ga 2. (x)[fx כ (Gx Ú Hx)] / Ha 3. Fa כ (Ga Ú Ha) 2, UI 4. Fa 1, Simp 5. Ga Ú Ha 3, 4, MP 6. ~Ga Fa 1, Com 7. ~Ga 6, Simp 8. Ha 5, 7, DS a is F and a is not G; all F is either G or H; therefore, a is H (x)[(fx Ú Gx) כ Hx] 2. ~Ha / (Ǝx)~Gx 3. (Fa Ú Ga) כ Ha 1, UI 4. ~(Fa Ú Ga) 3, 2, MT 22

23 5. ~Fa ~Ga 4, 6. ~Ga ~Fa 5, Com 7. ~Ga 6, Simp 8. (Ǝx)~Gx 7, EG Anything that is F or G is H; a is not H; therefore, something is not G ~(Ǝx)Dx / Da כ Ga 2. (x)~dx 1, 3. ~Da 2, UI 4. ~Da Ú Ga 3, Add 5. Da כ Ga 4, MI Nothing is D; therefore, if a is D, then a is G (x)~gx 2. (x)fx כ (Ǝx)Gx / (Ǝx)~Fx 3. ~(Ǝx)Gx 1, 4. ~(x)fx 2, 3, MT 5. (Ǝx)~Fx 4, Nothing is G; if everything is F, then something is G; therefore, something is not F ~(x)fx 2. Ga º Hb 3. (Ǝx)~Fx כ ~(Ǝx)Gx / ~Hb 4. (Ǝx)~Fx 1, 5. ~(Ǝx)Gx 3, 4, MP 6. (x)~gx 5, 7. ~Ga 6, UI 8. (Ga כ Hb) (Hb כ Ga) 2, ME 9. (Hb כ Ga) (Ga כ Hb) 8, Com 10. Hb כ Ga 9, Simp 11. ~Hb 10, 7, MT Not everything is F; a is G iff b is H; if something is not F, then nothing is G; therefore, b is not H. 23