Definition 2.2.1.
Let \(P\) be a statement. The negation of \(P\) is denoted \(\neg P\text{.}\) When the original statement \(P\) is true, the negation \(\neg P\) is false. And when the original statement is false, the negation is true.
Section 2.2 Negation
Given a statement, \(P\text{,}\) we can form a new statement which is the negation of the original, which we denote \(\neg P\text{;}\) this little squiggle is called a tilde.
You will also see the negation written as \(!P\) or \(\lnot P\text{.}\) Since all three are quite commonly use, you should recognise all three. To not unduly confuse your reader, you should pick one and stick with it. You should recognise all three notations, as all three are in common use; we’ll use the tilde notation in this text 18 .
For our general statement \(P\) we can summarise its truth values and the corresponding truth values of its negation in a table:
| \(P\) | \(\neg P\) | \(\neg(\neg P)\) |
| T | F | T |
| F | T | F |
This table is called a truth table and we’ll use them quite a bit. They can be a bit dull and mechanical to use, but they make the truth values very clear and precise and can help us reduce the problem of understanding the truth value of some complicated combination of statements to a simple procedure of filling in entries of a table.
We have included a column for the double-negation of a statement, \(\neg(\neg P)\text{.}\) Notice that the truth values of the double-negation are the same as those of the original statement. It is related to the law of the excluded middle — a statement is true or its negation is true — there is no third (middle) option. Thus the mean of negations in mathematics is quite different from what can happen in written and spoken English 21 . Also notice that if we only have the negation to play with then we cannot really do very much at all. We need some ways of combining statements. To do this we start with the logical “conjunction” and “disjunction” — “and” and “or”.
This notation for the negation of a statement goes back at least as far as an Giuseppe Peano (1897) and Bertrand Russell (1908). The use of \(\lnot\) is due to Arend Heyting (1930) — many thanks to this website. The authors could not track down the earliest use of ! to denote the negation, but we do note that it is very commonly used in programming languages.
The statement “I can write with my right hand” is not the negation of “I can write with my left hand”. Just because someone cannot write with the left-hand does not mean that they can write with their right. For most of human history people could not write with either hand.
In this case, because 4 is an integer we know that if it is not even then it must be odd. However, this is not that case for non-integers. For example, the negation of “\(\pi\) is even” is “\(\pi\) is not even” rather than “\(\pi\) is odd”. We’ll come back to even and odd in Chapter 3.
In written and spoken English a double-negation can sometimes a negation, “We don’t need no education.”; sometimes it is ambiguous: “I do not disagree.”; and sometimes positive: “The time you have is not unlimited.”. In many languages a double-negation serves as a means of emphasising the negation. “Yeah, right” is a good example of a double-positive being a (sarcastic) negative.
