Solutions to Chapter 3 Other logical notions

A. For each of the following: Is it necessarily true, necessarily false, or contingent?

1. 1.
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   Caesar crossed the Rubicon. Contingent

2. 2.
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   Someone once crossed the Rubicon. Contingent

3. 3.
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   No one has ever crossed the Rubicon. Contingent

4. 4.
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   If Caesar crossed the Rubicon, then someone has. Necessarily true

5. 5.
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   Even though Caesar crossed the Rubicon, no one has ever crossed the Rubicon. Necessarily false

6. 6.
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   If anyone has ever crossed the Rubicon, it was Caesar. Contingent

B. For each of the following: Is it a necessary truth, a necessary falsehood, or contingent?

1. 1.
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   Elephants dissolve in water.

2. 2.
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   Wood is a light, durable substance useful for building things.

3. 3.
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   If wood were a good building material, it would be useful for building things.

4. 4.
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   I live in a three story building that is two stories tall.

5. 5.
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   If gerbils were mammals they would nurse their young.

C. Which of the following pairs of sentences are necessarily equivalent?

D. Which of the following pairs of sentences are necessarily equivalent?

E. Consider the following sentences:

1. G1.
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   There are at least four giraffes at the wild animal park.

2. G2.
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   There are exactly seven gorillas at the wild animal park.

3. G3.
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   There are not more than two Martians at the wild animal park.

4. G4.
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   Every giraffe at the wild animal park is a Martian.

F. Consider the following sentences.

1. M1.
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   All people are mortal.

2. M2.
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   Socrates is a person.

3. M3.
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   Socrates will never die.

4. M4.
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   Socrates is mortal.

Which combinations of sentences are jointly possible? Mark each “possible” or “impossible.”

1. 1.
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   Sentences M1, M2, and M3

2. 2.
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   Sentences M2, M3, and M4

3. 3.
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   Sentences M2 and M3

4. 4.
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   Sentences M1 and M4

5. 5.
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   Sentences M1, M2, M3, and M4

G. Which of the following is possible? If it is possible, give an example. If it is not possible, explain why.

1. 1.
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   A valid argument that has one false premise and one true premise

2. Yes: ‘All whales are mammals (true). All mammals are plants (false). So all whales are plants.’

3. 2.
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   A valid argument that has a false conclusion

4. Yes. (See example from previous exercise.)

5. 3.
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   A valid argument, the conclusion of which is a necessary falsehood

6. Yes: ‘$1+1=3$. So $1+2=4$.’

7. 4.
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   An invalid argument, the conclusion of which is a necessary truth

8. No. If the conclusion is necessarily true, then there is no way to make it false, and hence no way to make it false whilst making all the premises true.

9. 5.
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   A necessary truth that is contingent

10. No. If a sentence is a necessary truth, it cannot possibly be false, but a contingent sentence can be false.

11. 6.
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    Two necessarily equivalent sentences, both of which are necessary truths

12. Yes: ‘4 is even’, ‘4 is divisible by 2’.

13. 7.
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    Two necessarily equivalent sentences, one of which is a necessary truth and one of which is contingent

14. No. A necessary truth cannot possibly be false, while a contingent sentence can be false. So in any situation in which the contingent sentence is false, it will have a different truth value from the necessary truth. Thus they will not necessarily have the same truth value, and so will not be equivalent.

15. 8.
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    Two necessarily equivalent sentences that together are jointly impossible

16. Yes: ‘$1+1=4$’ and ‘$1+1=3$’.

17. 9.
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    A jointly possible collection of sentences that contains a necessary falsehood

18. No. If a sentence is necessarily false, there is no way to make it true, let alone it along with all the other sentences.

19. 10.
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    A jointly impossible set of sentences that contains a necessary truth

20. Yes: ‘$1+1=4$’ and ‘$1+1=2$’.

H. Which of the following is possible? If it is possible, give an example. If it is not possible, explain why.