Section 5: Other logical notions
In addition to deductive validity, we will be interested in some other logical concepts.
Truth-values
True or false is said to be the truth-value of a sentence. We defined sentences as things that could be true or false; we could have said instead that sentences are things that can have truth-values.
Logical truth
In considering arguments formally, we care about what would be true if the premises were true. Generally, we are not concerned with the actual truth value of any particular sentences— whether they are actually true or false. Yet there are some sentences that must be true, just as a matter of logic.
Consider these sentences:
1. It is raining.
2. Either it is raining, or it is not.
3. It is both raining and not raining.
In order to know if sentence 1 is true, you would need to look outside or check the weather channel. Logically speaking, it might be either true or false. Sentences like this are called contingent sentences.
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 not. This sentence is logically true ; it is true merely as a matter of logic, regardless of what the world is actually like. A logically true sentence is called a tautology .
You do not need to check the weather to know about sentence 3, either. 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 read this, but it is impossible for it to be both raining and not raining here at this moment. The third sentence is logically false ; it is false regardless of what the world is like. A logically false sentence is called a contradiction .
To be precise, we can define a contingent sentence as a sentence that is neither a tautology nor a contradiction.
A sentence 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; its truth is not a matter of logic. There is no contradiction in considering a possible world in which there are fewer than seven things. The important question is whether the sentence must be true, just on account of logic.
Logical equivalence
We can also ask about the logical relations between two sentences. For example:
John went to the store after he washed the dishes.
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 necessarily have the same truth value, we say that they are logically equivalent .
Consistency
Consider these two sentences:
B1
My only brother is taller than I am.
B2
My only brother is shorter than I am.
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. And if B2 is true, then B1 must be false. It cannot be the case that both of these sentences are true.
If a set of sentences could not all be true at the same time, like B1–B2, they are said to be inconsistent . Otherwise, they are consistent .
We can ask about the consistency of any number of sentences. For example, consider the following list of sentences:
G1
There are at least four giraffes at the wild animal park.
G2
There are exactly seven gorillas at the wild animal park.
G3
There are not more than two martians at the wild animal park.
G4
Every giraffe at the wild animal park is a martian.
G1 and G4 together imply 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 set of sentences G1–G4 is inconsistent. Notice that the inconsistency has nothing at all to do with G2. G2 just happens to be part of an inconsistent set.
Sometimes, people will say that an inconsistent set of sentences ‘contains a contradiction.’ By this, they mean that it would be logically impossible for all of the sentences to be true at once. A set might be inconsistent even if each of the sentences in it is either contingent or tautologous. When a single sentence is a contradiction, then that sentence alone cannot be true.