Section Progress:

2.2  Second Weird Case of Validity

Next, evaluate this argument:

1. Quinn is telling the truth.
2. Quinn is not telling the truth.
Thus:
3. The moon is made of green cheese.

If you thought it couldn't get weirder than circular reasoning, here you go: reasoning from a contradiction.

A contradiction occurs when two sentences say the opposite of each other, like the premises above.

A contradiction can also be a single sentence that contradicts itself, like this complex sentence:

"Quinn is telling the truth and is not telling the truth."

The concept of a contradiction will be important to the rest of this book, and it is sometimes used in different ways in logic, so let's be explicit about how we'll use it.

Contradiction: a logical falsehood; a sentence that is necessarily false.

By a contradiction, or contradictory sentence, we will mean a sentence that can't possibly be true. In other words, a contradiction is a logically false sentence, a sentence that is necessarily false because of logic.

Here are some other examples of contradictory sentences:

  • "Some dogs are not dogs."
  • "Quinn isn't Quinn."
  • "Nobody knew the answer, but somebody knew the answer."

We'll also talk about a contradictory set of sentences, which is a group of sentences that can't possibly be true simultaneously. Like the set of premises in the previous argument:

{Quinn is telling the truth, Quinn is not telling the truth}

Contradictory set: a group of sentences that can't be true at once.

Neither of those sentences is a contradiction itself, but together they make a contradictory set.

Here's another way to make a contradictory set: take a sentence that is itself a contradiction and put it in the set. Since that sentence can never be true, the whole set becomes contradictory. A contradictory set is one whose members can't be all true simultaneously, and if that sentence can't be true ever, then it follows that you can't make all the members true simultaneously.

Let's practice using these notions. Place all the contradictions into the bin.

Now that you understand contradictions, we can understand the second weird case of validity.

This is the argument we were considering:

1. Quinn is telling the truth.
2. Quinn is not telling the truth.
Thus:
3. The moon is made of green cheese.

That argument is odd for several reasons. First, the conclusion is ridiculous and has nothing to do with the premises. Second, the premises form a contradictory set.

Weird case #2: An argument with contradictory premises is valid.

But it's that last fact that actually makes the argument valid: any argument with contradictory premises is valid.

Why? Let's consider our definition of validity.

Validity says that whenever ALL the premises are true, the conclusion must be true.

Here's the point that is hard to grasp: when we have contradictory premises, they satisfy that definition trivially, because the premises can never all be true together.

The conclusion is true "whenever all the premises are true", just because the premises can never all be true.

The only way to show an argument is invalid is to find a way to make the premises true and the conclusion false. If an argument has contradictory premises, you can't do that.

Think about it like this: the only way to violate that definition is if it's possible to have the argument's  premises true and conclusion false. But an argument with contradictory premises can never do that, because it can't have true premises.

We said in Chapter 1 that there are several ways of expressing the idea validity. Here's one of them: an argument is valid if it meets this condition: it is impossible for the premises to be true and the conclusion false.

That way of thinking about it makes weird case #2 easier to understand. All arguments with contradictory premises meet that condition, because it is impossible for their premises to be true.

If it is impossible for their premises to be true, then it is impossible for the premises to be true and the conclusion false.

Now let's apply the idea.

2.2 Second Weird Case of Validity