Section Progress:

3.4  Bivalence

You already know that we are concerned with sentences that make claims which can be true or false.

Truth values: true and false.

Truth and falsity are called truth values. In the logical systems we study in this book, those are the only possible truth values.

Bivalence: two possible truth values.

Logics like that are called bivalent: "bi" for two and "valent" for values.

That means every sentence must have exactly one truth value, true or false. If a sentence were true and  false, that would be like having another truth value of "both", and if a sentence neither true nor false, that would be like having another value of "none of the above".

In either of those cases, there wouldn't be just two truth values after all.

It follows that in bivalent systems, every sentence has exactly one truth value, true or false.

Like the other features of our logical systems, bivalence is a useful feature to build into our models. It makes them easy to learn and very powerful.

But you can also build logical systems for other purposes that aren't bivalent: logics can have three values, four values, even an infinite number of values.

Now that you know about bivalence, we can finish our detective story.

Because of bivalence, you can reason this way:

1. Raquel's testimony must be either true or false.
2. She is speaking falsely only if she has something to gain.
3. She doesn't have anything to gain from speaking falsely.

If Raquel is speaking truly, then one of the original pieces of evidence is false.

Then it dawns on you. Only two people had access to the file before it landed on your desk: the Sergeant and the other detective, Murphy.

One of them must have planted the false evidence. So you hatch a plan: whomever you can catch in a lie is the culprit.

First, you ask Murphy if the Sergeant is trustworthy. Murphy says, "He never lies."

Then you ask the Sergeant if Murphy is trustworthy, and the Sergeant says, "He never tells the truth."

You smile to yourself--you've got all the information you need. Here are the premises:

1. Whoever is speaking falsely planted the false evidence.
2. Murphy said, "The Sergeant never lies."
3. The Sergeant said, "Murphy isn't telling the truth."
4. Because of bivalence, each of them must be speaking truly or falsely.

Finding this solution is difficult, but we never said being a detective was going to be easy.

Because of bivalence, you know that Murphy is speaking truly or falsely.

First consider what follows if he's speaking truly. He said that the Sergeant never lies, so that means what the Sergeant said must be true too. But the Sergeant said that Murphy never tells the truth, which contradicts the assumption that Murphy's speaking truly.

If one case leads to a contradiction and the other doesn't, we can know the first case is impossible and the second must be true.

The contradiction means it is impossible for Murphy to be speaking truly. So it follows from the first premise that Murphy planted the false evidence.

We'll leave it to you to confirm that the Sergeant is telling the truth.

If that method of reasoning still seems confusing, don't worry. Later in the book we will learn how to carefully reconstruct how it works.

Once you know Murphy is the culprit, you bring Raquel in to testify to the Sergeant. He launches an investigation and releases Quinn, now proven innocent.

As you watch Quinn regain his freedom, that sick feeling in your stomach finally starts to fade.

The Sergeant sees the look of relief on your face.

"You did good kid," he says with a smile. "Don't let it go to your head."

3.4 Bivalence