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Shantila's Inside Logic #27

Answers to Exercises in #24-26

24.1 The identification of fallacious reasoning requires interpretation within a certain context, and sometimes it is not totally clear cut when reasoning is fallacious. But assuming that each of the following is asserted or thought in a context in which it is fallacious, which fallacy (of those discussed in this chapter) is most clearly exhibited by each of these?

1. Noted psychologist Dr. Laura believes investing in gold is how to get rich fast, so that is what you should do. Appeal to authority

2. You may argue that God doesn't exist, but you are just following a fad like you always do. Ad hominem

3. I got sideswiped by this Italian guy name Calvino, which means all Italians are lousy drivers. Hasty generalization

4. Father Dave's arguments against abortion are no good. He's a priest, so he has to say that abortion is wrong, and besides lots of priests have molested children. Ad hominem

5. She asserts that we need more military spending, but that is false, since she is only saying it because she works for a defense contractor. Ad hominem

6. Since I'm not lying, it follows that I'm telling the truth. Begging the question

7. The sign says "Fine for parking here." The sign said it was fine, so you shouldn't be writing me that ticket, sir. Equivocation

8. We know that God exists, since the Bible says God exists. What the Bible says must be true, since God inspired it and God never lies. Begging the question

9. All of my friends are in favor of the federal tax increase, so the tax increase is very popular. Hasty generalization

10. The great baseball player Rafael Palmeiro endorsed Viagra on tv, so you should get some right away. Appeal to authority

11. The belief in God is universal. After all, everyone believes in God. Begging the question

12. All men are all sexist pigs, which I know all too well since every one of my last four boyfriends turned out to be a sexist pig. Hasty generalization

13. If such actions were not illegal, then they would not be prohibited by the law. Begging the question

**

25.2 Symbolize each of the following sentences.

(a) Joe is the same as Ben.

j=b

(b) Madonna is not the same as Kid Rock.

~m=k

(c) There is only one turtle.

(Ex)(Tx & (y)(Ty > y=x))

(d) There are at least two turtles.

(Ex)(Tx & (Ey)(Ty & ~y=x))

(e) There is at most one turtle.

(x)(Tx > (y)(Ty > y=x))

(f) The only turtle is lonely.

(Ex)(Tx & ((y)(Ty > y=x) & Lx))

(g) Madonna is the only pop star in town.

(Pm & Tm) & (x)((Px & Tx) > x=m)

(h) At most one rapper got sued by his mother.

(x)((Rx & Sx) > (y)((Ry & Sy) > y=x))

(i) At least two people are friendly.

(Ex)((Px & Fx) & (Ey)((Py & Fy) & ~y=x))

25.3 Construct derivations to show that each of the following sequents is valid, using our Rules A, MP, DN, MT, CP, &I, &E, vI, DS, RA, UI, EG, and =E.

137 Ka, Fc, a=c } (Ex)(Kx & Fx)

  1 1. Ka A        
  2 2. Fc A
  3 3. a=c A        
  2,3 4. Fa 2,3 =E        
  1,2,3 5. Ka & Fa 1,4 &I        
  1,2,3 6. (Ex)(Kx & Fx) 5 EG        
                 

note: 137 also can be done by writing Kc on line 4 (based on 1,3 =E); then Kc & Fc on line 5 (based on 2,5 &I).

138 Fa, ~Fc } ~a=c

  1 1. Fa A        
  2 2. ~Fc A
  3 3. a=c A (for RA)        
  1,3 4. Fc 1,3 =E        
  1,2,3 5. Fc & ~Fc 2,4 &I        
  1,2 6. ~a=c 3,5 RA  
                 

139 Fa & (y)(Fy > y=a), Fc } c=a

  1 1. Fa & (y)(Fy > y=a) A        
  2 2. Fc A
  1 3. (y)(Fy > y=a) 1, &E        
  1 4. Fc > c=a 3 UI        
  1,2 5. c=a 2,4 MP                  

140 a=b, Fa } ~(Fc & ~Fb)

  1 1. a=b A        
  2 2. Fa A
  3 3. Fc & ~Fb A (for RA)        
  3 4. ~Fb 3 &E        
  1,3 5. ~Fa 1,4 =E        
  1,2,3 6. Fa & ~Fa 2,5 &I        
  1,2 7. ~(Fc & ~Fb) 3,6 RA        
                 

141 Fa > a=c, Fc > Hc } Fa > Hc

  1 1. Fa > a=c A        
  2 2. Fc > Hc A
  3 3. Fa A (for CP)        
  1,3 4. a=c 1,3 MP        
  1,3 5. Fc 3,4 =E        
  1,2,3 6. Hc 2,5 MP        
  1,2 7. Fa > Hc 3,6 CP        
                 
 

25.4 Symbolize each of the following and construct derivations to show that the reasoning is valid in each case.

(a) Jim is cold. At most one person is cold. Bob is cold. So Bob is the same as Jim. (Cx: x is cold; j: Jim; b: Bob)

  1 1. Cj A        
  2 2. (x)(Cx > (y)(Cy > y=x)) A
  3 3. Cb A        
  2 4. Cj > (y)(Cy > y=j) 2 UI        
  1,2 5. (y)(Cy > y=j) 1,4 MP        
  1,2 6. Cb > b=j 5 UI        
  1,2,3 7. b=j 3,6 MP        
 

(b) Jim is cold. My fridge is cold. Jim is not the same my fridge. So at least two things are cold. (f: my fridge)

  1 1. Cj A        
  2 2. Cf A
  3 3. ~j=f A        
  1,2 4. Cj & Cf 1,2 &I        
  1,2,3 5. (Cj & Cb) & ~j=f 3,4 &I        
  1,2,3 6. (Ey)((Cj & Cy) & ~j=y) 5 EG        
  1,2,3 7. (Ex)(Ey)((Cx & Cy) & ~x=y) 6 EG        

(c) Madonna is a friendly person. Eminem is a friendly person. Madonna is not Eminem. So there are at least two friendly people. (Fx: x is friendly; Px: x is a person; m: Madonna; e: Eminem).

This one is similar to (b).

(d) All criminals definitely should be in jail. All thiefs are criminals. Professor X is a thief. Professor X is identical with Mister Blazer. So Mister Blazer definitely should be in jail. (Cx: x is a criminal; Jx: x definitely should be in jail; Tx: x is a thief; c: Professor X; b: Mister Blazer)

  1 1. (x)(Cx > Jx) A        
  2 2. (x)(Tx > Cx) A
  3 3. Tc A        
  4 4. c=b A        
  2 5. Tc > Cc 2 UI        
  2,3 6. Cc 3,5 MP        
  1 7. Cc > Jc 1 UI      
  1,2,3 8. Jc 6,7 MP        
  1,2,3,4 9. Jb 4,8 =I        

25.5 Extra credit

(a) Joe is the only guy who married Sally. Joe is friendly. So the guy who married Sally is friendly. (Let Mx: x is a guy who married Sally; Fx: x is friendly; j: Joe)

  1 1. Mj & (x)(Mx > x=j) A        
  2 2. Fj A
  1,2 3. (Mj & (x)(Mx > x=j)) & Fj 1,2 &I        
  1,2 4. (Ey)((My & (x)(Mx > x=y)) & Fy) 3 EG        

(b) Eminem was born in Detroit. Eminem is only the rapper who got sued by his mother. So the rapper who got sued by his mother was born in Detroit. (Dx: x was born in Detroit; Rx: x is a rapper; Sx: x got sued by his mother; e: Eminem)

  1 1. De A      
  2 2. (Re & Se) & (x)((Rx & Sx) > x=e) A
  1,2 3. ((Re & Se) & (x)((Rx & Sx) > x=e)) & De 1,2 &I      
  1,2 4. (Ey)(((Ry & Sy) & (x)((Rx & Sx)> x=y)) & Dy) 3 EG      

(c) Bob is the only guy who is nice. Bob just left. So the guy who is nice just left. (Nx: x is nice; Lx: x just left)

  1 1. Nb & (x)(Nx > x=b) A        
  2 2. Lb A
  1,2 3. (Nb & (x)(Nx > x=b)) & Lb 1,2 &I        
  1,2 4. (Ey)((Ny & (x)(Nx > x=y)) & Ly) 3 EG        
   
25. 6 Extra Credit

142 a=b } Ka v ~Kb

  1 1. a=b A        
  2 2. ~(Ka v ~Kb) A (for RA)
  3 3. Ka A (for RA)        
  3 4. Ka v ~Kb 3 vI        
  2,3 5. (Ka v ~Kb) & ~(Ka v ~Kb) 2,4 &I        
  2 6. ~Ka 3,5 RA        
  1,2 7. ~Kb 1,6 =E        
  1,2 8. Ka v ~Kb 7, vI        
  1,2 9. (Ka v ~Kb) & ~(Ka v ~Kb) 2,8 &I        
  1 10. ~~(Ka v ~Kb) 2,9 RA        
  1 11. Ka v ~Kb 10 DN        
 
**

26.1 The identification of fallacious reasoning requires interpretation within a certain context, and it is not always clear cut when reasoning is fallacious. But assuming that each of the following is asserted or thought in a context in which it is fallacious, which fallacy (of the 6 discussed in this chapter) is most clearly exhibited by each of these?

1. The petition will be passed around now, and you should sign it because if you don't you will lose your membership in this organization. Ad baculum

2. You should pay your taxes. People don't want to pay because they want to keep their money. But there are more important things in life than money. Straw man (also might be red herring)

3. You should vote for Sally for President. Our budget is tight and we need to cut expenses. You don't want our club to go into debt, do you? Red herring

4. Of course, God exists. For one thing, if you don't believe in God you are going to end up in hell. Ad baculum

5. Senator X's arguments against our going to war with Iraq are incredible to me. I cannot believe that an honorable man like Senator X would support a vicious dictator like Saddam Hussein. Straw man

6. You should give me the job because if I don't get it my mother is going to be heartbroken. Ad misericordiam

7. Have you stopped cheating on your income taxes? Complex question

8. What do you mean the foul is on me? C'mon, they already are ahead by 15 points! Ad misericordiam (or Red herring)

9. Bowling Green has a very, very boring logic professor there. I know because I saw it online. Internet fallacy

10. Tom suggests we should send some money to help with the AIDS crisis in Africa. But why should we be spending all of our money in Africa? Straw man

11. My father is the chief of police. He's the one who's been firing lots of cops. So you shouldn't be writing me that ticket, sir. Ad baculum

12. She asserts that we need more military spending, but are we seriously going to accept the entire federal budget being devoted to the military? Straw man