Negation

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Section 1.7 Negation

The negation of a statement \(\phi\text{,}\) which we shall denote \(\quine{\neg \phi}\text{,}\) is the statement “\(\phi\) is false.” For example, the negation of the statement “\(57\) is prime” is “\(57\) is not prime.” The negation of the statement “\(57\) is not prime” is “\(57\) is prime.” Depending on the statement, its negation can be written differently from just inserting a “not”:

If \(S\) is a set, then the statement “\(s \notin S\)” is the negation of the statement “\(s \in S\text{.}\)”
If \(\psi\) is the statement “\(x = y\text{,}\)” then the negation \(\quine{\neg \psi}\) is the sentence “\(x \neq y\text{.}\)”

The negation of a statement has the opposite truth value: if \(\phi\) is true, then \(\neg\phi\) is false; if \(\phi\) is false, then \(\neg\phi\) is true.

\(\phi\)	\(\quine{\neg\phi}\)

T	F
F	T

Example 1.7.1. Every statement is either true or false.

Let's consider the statement \(\quine{\phi \vee \neg\phi}\text{.}\) That's the statement that either \(\phi\) is true or \(\phi\) is not true. In other words, it's the assertion that either \(\phi\) is true or false. Here's the truth table:

\(\phi\)	\(\quine{\neg\phi}\)	\(\quine{\phi \vee \neg\phi}\)

T	F	T
F	T	T

Note that we only have T's in the last column. That means that the sentence \(\quine{\phi \vee \neg\phi}\) is always true, irrespective of what the truth-value of \(\phi\) is. This statement is sometimes called the Law of Excluded Middle.

Example 1.7.2. No statement is both true and false.

Now let's contemplate the statement \(\quine{\phi \wedge \neg\phi}\text{.}\) Here's the truth table:

\(\phi\)	\(\quine{\neg\phi}\)	\(\quine{\phi \vee \neg\phi}\)

T	F	F
F	T	F

Note that we only have F's in the last column. That means that \(\quine{\phi \wedge \neg\phi}\) is always false, no matter what the truth-value of \(\phi\) is. This statement is sometimes just called a Contradiction.