Chapter 9 Exercises: Proofs (Symbolic Logic)

proof

series of inferences, each one justified by a valid inference rule, that leads from the premises to the conclusion of the argument

1. Premise

2. Premise

3. Premise

4. [Proof Step] /Conclusion

5. [Proof Step] [Justification]

…

n. Conclusion [Justification]

How is a proof constructed?

True

True or False: Only valid arguments have proofs.

False

• Just because you can’t find a proof doesn’t mean that there isn’t one. You might just not have found it. 

True or False: If you’ve tried and failed to find a proof for an argument, then it follows that the argument is invalid. 

True

True or False: There is more than one correct proof for any valid argument.

1. Modus Pollens

2. Modus Tollens

3. Disjunctive Syllogism

4. Hypothetical Syllogism

5. Simplification

6. Conjunction

7. Addition

8. Constructive Dilemma

Which inference rules are introduced in this chapter?

True

True or False: The step on line 4, below, is a correct application of Modus Ponens. 

1. A

2. C ● D

3. A ⊃ B

4. B              1, 3 MP

False

• This tells us that B implies C IF A is the case, but we don’t actually know that A is true.

True or False: The step on line 3, below, is a correct application of modus ponens. 

1. A ⊃ (B ⊃ C)

2. B

3. C                  1, 2 MP

True

True or False: The step on line 3 is a correct application of disjunctive syllogism.

1. K ∨ (M ● N)

2. ∼K

3. M ● N            1, 2 DS

True

True or False: The step on line 3 is a correct application of hypothetical syllogism.

1. R ⊃ (S ∨ T)

2. (S ∨ T) ⊃ ∼O

3. R ⊃ ∼O 1, 2 HS

1. A

2. B ⊃ C

3. ∼C

4. ∼B           2, 3 MT

Fill in the missing justification on line 4:

1. A

2. B ⊃ C

3. ∼C

4. ∼B [?]

1. R ∨ (∼S ⊃ T)

2. S ⊃ O

3. ∼O

4. ∼S ⊃ R         /T

5. ∼S               2, 3 MT

6. ∼R               4, 5 MP

7. ∼S ⊃ T         1, 6 DS

8. T              5, 7 MP

The proof below requires only one more step to complete it. Fill in the final step and its justification:

1. R ∨ (∼S ⊃ T)

2. S ⊃ O

3. ∼O

4. ∼S ⊃ ∼R         /T

5. ∼S               2, 3 MT

6. ∼R               4, 5 MP

7. ∼S ⊃ T         1, 6 DS

8. [?]              [?]

False

• Addition cannot occur with a conjunction (dot). Addition can only occur with a wedge.

True or False: The step below is a valid application of the addition rule (Add).

1. R

2. R ● S          1, Add

True

True or False: The step below is a valid application of the simplification rule (Simp.)

1. R ● (T ⊃ ∼Q)

2. T ⊃ ∼Q           1, Simp

False

• You can only use simplification on a conjunction (dot), not a disjunction (wedge).

True or False: The step below is a valid application of the simplification rule (Simp)

1. M ∨ (K ● T)

2. K               1, Simp

True

True or False: The proof below is correct.

1. A

2. A ⊃ B

3. C ⊃ D                        /B ∨ D

4. (A ⊃ B) ● (C ⊃ D)      2, 3 Conj

5. A ∨ C                       1, Add

6. B ∨ D                       4, 5 CD

1. J ∨ S

2. ∼S

3. M ● N             /J ● N

4. J                      [1, 2 DS]

5. N                     [3, Simp]

6. J ● N               [4, 5 Conj]

Fill in the missing justifications in the proof below:

1. J ∨ S

2. ∼S

3. M ● N             /J ● N

4. J                      [?]

5. N                     [?]

6. J ● N               [?]

1. A ⊃ B

2. ∼B

3. ∼A ⊃ C                   /C

4. ∼A                          1, 2 MT

5. C                            3, 4 MP

Construct a proof for the argument below:

1. A ⊃ B

2. ∼B

3. ∼A ⊃ C                   /C

1. A ∨ (B ∨ C)

2. ∼B

3. ∼B ⊃ ∼A              /C

4. ∼A                   2, 3 MP

5. B ∨ C              1, 4 DS

6. C                     2, 5 DS

Construct a proof for the argument below:

1. A ∨ (B ∨ C)

2. ∼B

3. ∼B ⊃ ∼A              /C

1. A ⊃ (B ⊃ C)

2. A

3. ∼C

4. B ∨ D               /D

5. B ⊃ C               1, 2 MP

6. ∼B 3, 5 MT

7. D 4, 6 DS

Construct a proof for the argument below:

1. A ⊃ (B ⊃ C)

2. A

3. ∼C

4. B ∨ D               /D

1. T ∨ (H ⊃ L)

2. T ⊃ S

3. ∼S

4. L ⊃ O              /H ⊃ O

5. ∼T                  2, 3 MT

6. H ⊃ L              1, 5 DS

7. H ⊃ O             4, 6 HS

Construct a proof for the argument below:

1. T ∨ (H ⊃ L)

2. T ⊃ S

3. ∼S

4. L ⊃ O /H ⊃ O

1. D ⊃ (C ≡ F)

2. (C ≡ F) ⊃ G

3. G ⊃ H

4. ∼H                   /∼D

5. D ⊃ G              1, 2 HS

6. D ⊃ H               3, 5 HS

7. ∼D                    4, 6 MT

Alternate: 

6. ∼G                    3, 4 MT

7. ∼D 5, 6 MT

Construct a proof for the argument below:

1. D ⊃ (C ≡ F)

2. (C ≡ F) ⊃ G

3. G ⊃ H

4. ∼H                  /∼D

1. F ⊃ ∼G

2. ∼∼G

3. F ∨ ∼H

4. I ⊃ H            /∼F ● ∼I

5. ∼F                1, 2 MT

6. ∼H                3, 5 DS

7. ∼I                  4, 6 MT

8. ∼F ● ∼I         5, 7 Conj 

Construct a proof for the argument below:

1. F ⊃ ∼G

2. ∼∼G

3. F ∨ ∼H

4. I ⊃ H            /∼F ● ∼I

1. A ⊃ (E ⊃ ∼F)

2. H ∨ (∼F ⊃ M)

3. A ● ∼H                      /E ⊃ M

4. A                              3, Simp

5. ∼H                           3, Simp

6. E ⊃ ∼F                     1, 4 MP

7. ∼F ⊃ M                    2, 5 DS

8. E ⊃ M                      6, 7 HS

Construct a proof for the argument below:

1. A ⊃ (E ⊃ ∼F)

2. H ∨ (∼F ⊃ M)

3. A ● ∼H /E ⊃ M

1. (R ⊃ S) ● (P ⊃ Q)

2. M ● R                 /S ∨ Q

3. R                        2, Simp

4. R ∨ P                 3, Add

5. S ∨ Q                 1, 4 CD

Construct a proof for the argument below:

1. (R ⊃ S) ● (P ⊃ Q)

2. M ● R /S ∨ Q

1. L ⊃ M

2. L ∨ (M ⊃ ∼S)

3. ∼M                    /L ⊃ ∼S

4. ∼L                     1, 3 MT

5. M ⊃ ∼S             2, 4 DS

6. L ⊃ ∼S              1, 5 HS

Construct a proof for the argument below:

1. L ⊃ M

2. L ∨ (M ⊃ ∼S)

3. ∼M /L ⊃ ∼S

1. D ● E

2. E ⊃ (A ∨ B)

3. ∼A                         /B ∨ C

4. E                            1, Simp

5. A ∨ B                      2, 4 MP

6. B                             3, 5 DS

7. B ∨ C                      6, Add

Construct a proof for the argument below:

1. D ● E

2. E ⊃ (A ∨ B)

3. ∼A                         /B ∨ C

1. D ● E

2. E ⊃ (A ∨ B)

3. ∼A                   /B ● D

4. D                      1, Simp

5. E                       1, Simp

6. A ∨ B                2, 5 MP

7. B                        3, 6 DS

8. B ● D                 4, 7 Conj

Construct a proof for the argument below:

1. D ● E

2. E ⊃ (A ∨ B)

3. ∼A                   /B ● D

1. (∼S ● R) ⊃ P

2. S ⊃ T

3. R ● ∼T                  /P

4. R                           3, Simp

5. ∼T                         3, Simp

6. ∼S                         2, 5 MT

7. ∼S ● R                   4, 6 Conj

8. P                            1, 7 MP

Construct a proof for the argument below:

1. (∼S ● R) ⊃ P

2. S ⊃ T

3. R ● ∼T                  /P

1. A ⊃ B

2. C ⊃∼D

3. E ● (A ∨ C)                /B ∨ ∼D

4. (A ⊃ B) ● (C ⊃∼D)     1, 2 Conj

5. A ∨ C                          3, Simp

6. B ∨ ∼D                        4, 5 CD

Construct a proof for the argument below:

1. A ⊃ B

2. C ⊃∼D

3. E ● (A ∨ C) /B ∨ ∼D

1. A ⊃ C

2. H ● ∼D

3. H ⊃ (A ∨ B)

4. B ⊃ D                   /C ● H

5. H                          2, Simp

6. A ∨ B                 3, 5 MP

7. ∼D                     2, Simp

8. ∼B                     4, 7 MT

9. A                        6, 8 DS

10. C                        1, 9 MP

11. C ● H                 5, 10 Conj

Construct a proof for the argument below:

1. A ⊃ C

2. H ● ∼D

3. H ⊃ (A ∨ B)

4. B ⊃ D /C ● H

1. (∼Q ∨ T) ⊃ S

2. (H ● M) ⊃ ∼Q

3. H

4. M ● N                  /S

5. M                         4, Simp

6. H ● M                   3, 5 Conj

7. ∼Q                       2, 6 MP

8. ∼Q ∨ T                7, Add

9. S                          1, 8 MP

Construct a proof for the argument below:

1. (∼Q ∨ T) ⊃ S

2. (H ● M) ⊃ ∼Q

3. H

4. M ● N                  /S

1. (D ⊃ M) ⊃ (R ● A)

2. D ⊃ J

3. R ⊃ (S ≡ ∼T)

4. J ⊃ M                      /S ≡ ∼T

5. D ⊃ M                     2, 4 HS

6. R ● A                      1, 5 MP

7. R                             6, Simp

8. S ≡ ∼T                    3, 7 MP

Construct a proof for the argument below:

1. (D ⊃ M) ⊃ (R ● A)

2. D ⊃ J

3. R ⊃ (S ≡ ∼T)

4. J ⊃ M                 /S ≡ ∼T

1. (F ⊃ L) ● (R ⊃ Q)

2. D ● F

3. (L ∨ O) ⊃ T            /T ● D

4. D                            2, Simp

5. F                            2, Simp

6. F ∨ R                    5, Add

7. L ∨ O                    1, 6 CD

8. T                            3, 7 MP

9. T ● D                     4, 8 Conj

Construct a proof for the argument below:

1. (F ⊃ L) ● (R ⊃ Q)

2. D ● F

3. (L ∨ O) ⊃ T /T ● D

1. A ⊃ B

2. C ⊃ (D ● ∼B)

3. A ● C /G

4. A                     3, Simp

5. B                    1, 4 MP

6. C                     3, Simp

7. D ● ∼B            2, 6 MP

8. ∼B                   7, Simp

9. B ∨ G              5, Add

10. G                    9, DS

Construct a proof for the argument below:

1. A ⊃ B

2. C ⊃ (D ● ∼B)

3. A ● C /G

1. G ⊃ ∼S

2. ∼S ⊃ ∼L

3. ∼L ⊃ ∼C

4. R ∨ G

5. ∼R                        /∼C

6. G                          4, 5 DS

7. ∼S                        1, 6 MP

8. ∼L                         2, 7 MP

9. ∼C                        3, 8 MP

Alternate:

6. G ⊃ ∼L                    1, 2 HS

7. G ⊃ ∼C                    3, 6 HS

8. G                          4, 5 DS

9. ∼C                        7, 8 MP

For this last exercise, we’ll go through all the familiar steps in the logic process. Identify the premises and the conclusion in the argument below, symbolize it using the indicated letters for the simple statements, and then test it by constructing a proof for it:

If Jupiter is a gaseous planet, then it doesn’t have a solid surface. If Jupiter has no solid surface, then you can’t land on it. If you can’t land on Jupiter, then it can’t be colonized. So, you can’t colonize Jupiter. For either the planet is rocky, or it’s gaseous. And Jupiter is not rocky.

Let G = Jupiter is a gaseous planet.
Let R = Jupiter is rocky.
Let S = Jupiter has a solid surface.
Let L = You can land on Jupiter.
Let C = Jupiter can be colonized.