Skip to main content

Subsection 3.3.3 Double Negation

Double Negation : p ≡ ¬(¬ p ).

In other words, in logic, two negatives cancel each other out.

To prove this claim, we build a truth table that has one column for the left hand side of the equivalence and one column for the right hand side. Then we build a final column that corresponds to the claim that those two are the same. We have a proof if that final column contains all T ’s. (Notice that there may be additional working columns as well. We don’t care what their values are as long as they lead to the two critical columns being the same.) We’ve already seen the first three columns of the truth table that we need. We showed them when we pointed out that applying not twice gets us back where we started. Here’s the whole table that we need to build to prove the double negation identity:

p ¬ p ¬(¬ p ) p ≡ ¬(¬ p ).
T F T T
F T F T

We’ve highlighted the two columns that correspond to the two sides of the equivalence. Notice that they are identical. So the final column contains all T ’s.

English Aside

Double negation shows up much more often in English than you might imagine. One reason it happens is that many words other than “not” actually mean not.

Let P be the statement: “I will go to the game.”

Then: “No way will I miss that game,” is \(\neg \neg \) P

Since: “miss” means \(\neg\) P.

So you know P: “I will go to the game.”

We should remind you though that, in some dialects, double negation means the same thing as single negation. For example:

“Don’t nobody love me.”

Nifty Aside

The Fork in the Road

A weary traveler approaches a fork in the road. Not knowing which way to go, he decides he should ask the one native he sees. But he knows that there’s a problem. This is the land of the liars and the truth-tellers. Everyone is either a liar (who lies absolutely all the time) or a truth-teller (who tells the truth absolutely all the time. Unfortunately, these folks don’t wear affinity tee shirts. There’s no way to tell, when you’re talking to someone, which camp he’s in. And there’s one more thing: It’s well known that strangers are allowed to ask only a single question before they must move on.

Fortunately, our traveler is a logician. He asks a single question and, without knowing whether he’s talking to a liar or a truth-teller, gets the answer that he needs. What question does he ask?

This puzzle is fun. See if you can work it out.

Suppose that we take as a premise:

It’s not impossible that Riley will win.

Is it possible that Riley will win?

Answer.
Yes.
Solution.
Explanation: If it is not not possible that Riley will win, then, by double negation, it is possible that Riley will win.