Exercises: Two sets

This page contains  two sets of exercises to start with. In the simpler one the task is to figure out what the justification of each line is. Once you can do these, then move on to the next set, which calls for you to come up with the intermediate lines (as well as the reasons or justifications).

Keep a printed set of the rules in front of you as you work on these.  Remember that until you have the 8 rules of inference thoroughly memorized, so that you know what each of them does (what kind of statement it generates, and what kind of statements it works on) you will be struggling to see anything at all. So work on memorizing and learning them. It is also true that the Rules of Equivalence will not suggest themselves to you for use until you know them: what they change and what they result in. But I will not be as insistent that you memorize them, and I will let you have a list of them on the exam to refer to so you can be sure you are conceiving them accurately. Those of you who work like A students will, of course, be memorizing most of them as well.

Using MP, MT, HS, DS, CD, SM, CN, AD, CM, AS, and DN, see if you can draw inferences allowed by the rules, given these premises.

 

a.

1.  (D v M) > T

2.  ~T

3. ______      ______

 

b.

1. (P ≡  (U v I)) v (B . N)

2.  ~ (P = (U v I)

3. _______    ______

 

c.

1. M > (P . L)

2. (G > I) > M

3. ________     _____

 

d.

1. E > (Y > U)

2. (E > (Y > U)) > (C v (G . M)

3. _________     ______

 

e.

1. (B v Y)

2. (V > O)

3. ________    _____

 

f.

1. P . (I > L)

2. T > C

3. _______      ______

 

g.

1. P . (I > L)

2. T > C

3. _______    _____

4. I > L    3  SM

 

h.

1. L v (C . G)

2. (L > S) . ((C . G) > (T ≡  R))

3. _____________    _______

 

i.

1. (C . G) > (T ≡  R)

2. ~ (T ≡  R)

3. _______  _____

 

j.

1. ((N v (J v L)) > T

2. ~ T

3. __________     ____

4. ~((N v J) v L)     3,  AS

 

k.

1. F > (P v C)

2. (B . C) > F

3. ~(C v P)

4. ________   ___

5._________  ___

6. ~(B . C)    ________

 

l.

1.  T > (I . O)

2.  W v (B v T)

3. ~(W v B)

4. ________     ______

5. ________     ______

6. ________     ______

7. I                   _______

 

 

 

 

Proofs: first rules

 

You should be able to do these with just the first four rules, but you need not limit yourself to those if you can see applications of others.  

 More often than not, there is more than one way to do a proof, so you might not have chosen all the same moves every time in the same order as you will see below. For instance, in #1, it is a matter of indifference whether you first combine 1 with 3 (MP) or 2 with 4 (MP). What is not a matter of indifference is that you cannot do HS to 1 and 2; their consequents must be “extracted” onto lines in which their operators are the main operators of the lines ( 5 and 6) before HS can be applied to them.

 

1)

1. P > (G > T)

2. Q > (T > E)

3. P

4.  Q         /  G > E

 

2)

1.~W > [~W > (X > W)]

2. ~W      /  ~ X

 

3)

1.  J > (K > L)

2.  L v J

3.  ~L         / ~K

 

4)

1.  (L ≡ K) > C

2.  (L ≡ K) v (P > ~E)

3.  ~E > C

4.  ~C         / ~P

 

5)

1.  ~J > [~A > (D > A)]

2.  J  v ~A

3.  ~J        / ~D

 

6)

1. (B >  ~M) > (T > ~S)

2.  B > K

3.  K > ~M

4.  ~S > N      /  T > N

 

7)

1. H v (Q v F)

2. R v (Q > R)

3. R v ~H

4. ~R     /  F

 

8. If a tenth planet exists, its orbit is perpendicular to that of the other planets. Either a tenth planet is responsible for the death of the dinosaurs or its orbit is not perpendicular to the others. It is not responsible for the death of the dinosaurs, so no such planet exists.

 

9. If terrorists take more hostages, then their demands will be met if and only if the media give them full coverage. Either the media voluntarily limits the flow of information or if the media recognize that they are being exploited by terrorists, they will voluntarily limit the flow of information. Either they will recognize they are being exploited by terrorists or terrorists will take more hostages. They will not voluntarily limit the flow of information, so terrorist demands will be met if and only if the media give them full coverage.

 

10. Either we take recycling seriously or we will be buried in garbage. If we incinerate our garbage only if our health is jeopardized, then we do not take it seriously. If our landfills are becoming exhausted, then if we incinerate our garbage, then toxic ash will be produced. If that happens, then our health is jeopardized. Our landfills are becoming exhausted, so we will be buried in garbage.

 

 

 

Proofs: first rules        SOLUTIONS

 

You should be able to do these with just the first four rules, but you need not limit yourself to those if you can see applications of others.  

 More often than not, there is more than one way to do a proof, so you might not have chosen all the same moves every time in the same order as you will see below. For instance, in #1, it is a matter of indifference whether you first combine 1 with 3 (MP) or 2 with 4 (MP). What is not a matter of indifference is that you cannot do HS to 1 and 2; their consequents must be “extracted” onto lines in which their operators are the main operators of the lines ( 5 and 6) before HS can be applied to them.

 

1)

1. P > (G > T)

2. Q > (T > E)

3. P

4.  Q         /  G > E

5. G > T  MP  1,3

6. T>  E  MP 2 ,4

7. G > E  HS  5,6

 

 

2)

1.~W > [~W > (X > W)]

2. ~W      /  ~ X

3. ~W >( X > W)  mp 1,2

4. X > W  mp 2,3

5. ~X    mt  2,4

 

 

 

3)

1.  J > (K > L)

2.  L v J

3.  ~L         / ~K

4. J  2,3 ds

5. K > L  4,1 mp

6. ~K   mt  4,1

 

4)

1.  (L ≡  K) > C

2.  (L ≡  K) v (P > ~E)

3.  ~E > C

4.  ~C         / ~P

5.  ~(L ≡  K)   1,4  mt

6.  P > ~E  2,5  ds

7.  P > C    6, 3  hs

8.  ~P     4, 7  mt

 

 

 

 

5)

1.  ~J > [~A > (D > A)]

2.  J  v ~A

3.  ~J        / ~D

4. ~A       2,3 ds

5. ~A > (D > A)      3,1   mp

6.  D > A           4,5  mp

7.  ~D        6,4  mt

 

 

6)

1. (B >  ~M) > (T > ~S)

2.  B > K

3.  K > ~M

4.  ~S > N      /  T > N

5.  B > ~M      hs 2,3

6.  T > ~S      1,5  mp

7.  T > N       6, 4  hs

 

 

 

7)

1. H v (Q v F)

2. R v (Q > R)

3. R v ~H

4. ~R     /  F

5. ~H      3,4 ds

6.  Q > R   2,4  ds

7.  Q v F    1,5  ds

8.  ~Q         4,6  mt

9.  F          7,8  ds

 

 

8. If a tenth planet exists, its orbit is perpendicular to that of the other planets. Either a tenth planet is responsible for the death of the dinosaurs or its orbit is not perpendicular to the others. It is not responsible for the death of the dinosaurs, so no such planet exists.

1.  T > P

2.  D v ~P

3.  ~D    /  ~T

4.   ~D   2,3  ds

5.  ~T   4,1      mt

 

9. If terrorists take more hostages, then their demands will be met if and only if the media give them full coverage. Either the media voluntarily limits the flow of information or if the media recognize that they are being exploited by terrorists, they will voluntarily limit the flow of information. Either they will recognize they are being exploited by terrorists or terrorists will take more hostages. They will not voluntarily limit the flow of information, so terrorist demands will be met if and only if the media give them full coverage.

1.  M > (D ≡  F)

2.  V v (R > V)

3.  R v M

4.  ~V     / D ≡  F

5. R > V  2,4  ds

6.  ~R   4,5  mt

7.  M       3,6  ds

8.   D ≡  F     1,7  mp

 

 

 

 

 

10. Either we take recycling seriously or we will be buried in garbage. If we incinerate our garbage only if our health is jeopardized, then we do not take it seriously. If our landfills are becoming exhausted, then if we incinerate our garbage, then toxic ash will be produced. If that happens, then our health is jeopardized. Our landfills are becoming exhausted, so we will be buried in garbage.

 

1.  T v B

2.   (I > J) >  ~ T

3.  E > (I > W)

4.  W > J

5.  E     /  B

6.  I > W      3,5  mp

7.  I > J       6,4  hs

8.  ~T         2,7  mp

9.  B         1,8   ds

 

 

 

 

 

 

 

 

 

 

 

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