Chapter 3 Other logical notions

Consider these two sentences:

1. B1.
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   Jane’s only brother is shorter than her.

2. B2.
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   Jane’s only brother is taller than her.

Logic alone cannot tell us which, if either, of these sentences is true. Yet we can say that if the first sentence (B1) is true, then the second sentence (B2) must be false. Similarly, if B2 is true, then B1 must be false. There is no possible scenario where both sentences are true together. These sentences are incompatible with each other, they cannot all be true at the same time. This motivates the following definition:

Sentences are jointly possible if and only if there is a case where they are all true together.

B1 and B2 are jointly impossible, while, say, the following two sentences are jointly possible:

1. B1.
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   Jane’s only brother is shorter than her.

2. B2.
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   Jane’s only brother is younger than her.

We can ask about the joint possibility of any number of sentences. For example, consider the following four 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.

G1 and G4 together entail that there are at least four Martian giraffes at the park. This conflicts with G3, which implies that there are no more than two Martian giraffes there. So the sentences G1–G4 are jointly impossible. They cannot all be true together. (Note that the sentences G1, G3 and G4 are jointly impossible. But if sentences are already jointly impossible, adding an extra sentence to the mix cannot make them jointly possible!)

There is one thing worth pointing out: You might think that an argument only “makes sense” if its premises are jointly possible. But neither our definition of what an argument is, nor of when it is valid, requires this. In fact, according to our definition, any argument with jointly impossible premises is automatically valid! (Exercise: convince yourself that this is true.)

In assessing arguments for validity, we care about what would be true if the premises were true, but some sentences just must be true. Consider these sentences:

1. 1.
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   It is raining.

2. 2.
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   Either it is raining here, or it is not.

3. 3.
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   It is both raining here and not raining here.

In order to know if sentence 1 is true, you would need to look outside or check the weather channel. It might be true; it might be false. A sentence which is capable of being true and capable of being false (in different circumstances, of course) is called contingent .

Sentence 2 is different. You do not need to look outside to know that it is true. Regardless of what the weather is like, it is either raining or it is not. That is a necessary truth .

Equally, you do not need to check the weather to determine whether or not sentence 3 is true. It must be false, simply as a matter of logic. It might be raining here and not raining across town; it might be raining now but stop raining even as you finish this sentence; but it is impossible for it to be both raining and not raining in the same place and at the same time. So, whatever the world is like, it is not both raining here and not raining here. It is a necessary falsehood .

Something might always be true and still be contingent. For instance, if there never were a time when the universe contained fewer than seven things, then the sentence ‘At least seven things exist’ would always be true. Yet the sentence is contingent: the world could have been much, much smaller than it is, and then the sentence would have been false.

We can also ask about the logical relations between two sentences. For example:

1. ​

   John went to the store after he washed the dishes.

2. ​

   John washed the dishes before he went to the store.

These two sentences are both contingent, since John might not have gone to the store or washed dishes at all. Yet they must have the same truth-value. If either of the sentences is true, then they both are; if either of the sentences is false, then they both are. When two sentences have the same truth value in every case, we say that they are necessarily equivalent .

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  An argument is valid if there is no case where the premises are all true and the conclusion is not; it is invalid otherwise.

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  A necessary truth is a sentence that is true in every case.

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  A necessary falsehood is a sentence that is false in every case.

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  A contingent sentence is neither a necessary truth nor a necessary falsehood; a sentence that is true in some case and false in some other case.

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  Two sentences are necessarily equivalent if, in every case, they are both true or both false.

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  A collection of sentences is jointly possible if there is a case where they are all true together; it is jointly impossible otherwise.

1. 1.
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   Elephants dissolve in water.
   If you put an elephant in water, it will disintegrate.

2. 2.
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   All mammals dissolve in water.
   If you put an elephant in water, it will disintegrate.

3. 3.
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   George Bush was the 43rd president.
   Barack Obama is the 44th president.

4. 4.
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   Barack Obama is the 44th president.
   Barack Obama was president immediately after the 43rd president.

5. 5.
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   Elephants dissolve in water.
   All mammals dissolve in water.

1. 1.
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   Thelonious Monk played piano.
   John Coltrane played tenor sax.

2. 2.
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   Thelonious Monk played gigs with John Coltrane.
   John Coltrane played gigs with Thelonious Monk.

3. 3.
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   All professional piano players have big hands.
   Piano player Bud Powell had big hands.

4. 4.
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   Bud Powell suffered from severe mental illness.
   All piano players suffer from severe mental illness.

5. 5.
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   John Coltrane was deeply religious.
   John Coltrane viewed music as an expression of spirituality.

Now consider each of the following collections of sentences. Which are jointly possible? Which are jointly impossible?

1. 1.
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   Sentences G2, G3, and G4

2. 2.
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   Sentences G1, G3, and G4

3. 3.
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   Sentences G1, G2, and G4

4. 4.
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   Sentences G1, G2, and G3

1. 1.
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   A valid argument, whose premises are all necessary truths, and whose conclusion is contingent

2. 2.
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   A valid argument with true premises and a false conclusion

3. 3.
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   A jointly possible collection of sentences that contains two sentences that are not necessarily equivalent

4. 4.
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   A jointly possible collection of sentences, all of which are contingent

5. 5.
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   A false necessary truth

6. 6.
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   A valid argument with false premises

7. 7.
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   A necessarily equivalent pair of sentences that are not jointly possible

8. 8.
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   A necessary truth that is also a necessary falsehood

9. 9.
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   A jointly possible collection of sentences that are all necessary falsehoods