# 2.3 Third Weird Case of Validity

So far we’ve seen two weird cases of validity. Your next job is to assess this argument:

1. If all police dogs are smart, then Rufus is smart.
2. Rufus is smart.
Thus:
3. All police dogs are smart.

That argument is tempting but flawed–it is another type of fallacy which we’ll discuss later in the book.

We just mixed it in so you didn’t answer “valid” without thinking about it.

Now that we know you’re paying attention, here’s the argument we really want you to think about:

[no premises]
Thus:
1. All police dogs are dogs.

The set of premises can be empty.

We said that an argument is just a collection of sentences: a set of premises and a conclusion. Here’s a complication we didn’t tell you about: the set of premises might be empty.

As odd as that sounds, we can still use the definition of validity to assess the argument.

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

Now, considering that the set of premises might be empty, the definition means this: “whenever the premises are true (should there be any), the conclusion must be true.”

That means that if there are no premises, the conclusion must always be true. So we can check the validity of an argument with no premises by seeing if the conclusion is necessarily true.

Talk about necessary truth might make you recall the notion of a contradiction from last section. That’s great: the ideas are related; one is the polar opposite of the other.

Logical truth: a necessarily truth due to logical laws.

A logical truth is a sentence that is necessarily true because of the laws of logic. A logical truth is not possibly false. The opposite of a logical truth is impossible, a contradiction.

Logical falsity = a contradiction: a necessary falsehood due to logical laws.

A logical falsity is a sentence that is necessarily false because of the laws of logic. That’s just another name for the notion of a contradiction, which you’ve already learned. A logical falsity is not possibly true, and the opposite of a logical falsity must be true. So the opposite of a contradiction is a logical truth.

Let’s try some examples.

[no premises]
Thus:
1. All police dogs are dogs.

Now that you know its conclusion is a logical truth, you know why it is valid: since its conclusion is necessarily true, it meets the condition that the conclusion must be true “whenever the premises are true (should there be any).”

Weird case #3: an argument with a logically true conclusion is always valid.

This point generalizes. It’s not just when an argument has no premises that an argument with a logically true conclusion is valid. No matter what, an argument with a logically true conclusion is valid. That is weird case #3.

Think about it like this. One way of thinking about validity is that it is impossible for the premises to be true and the conclusion false.

An argument with a logically true conclusion always satisfies that condition. When the conclusion is a logical truth, then it is impossible for the conclusion to be false.

That means it is impossible for the premises to be true and conclusion to be false.

Now let’s apply the idea. Drag just ONE sentence in order to make a valid argument.