Definitions

# Negation

[ni-gey-shuhn]

In logic and mathematics, negation or not is an operation on logical values, for example, the logical value of a proposition, that sends true to false and false to true. Intuitively, the negation of a proposition holds exactly when that proposition does not hold. In grammar, nor is an adverb which acts as a coordinating conjunction.

## Definition

Logical negation is an operation on one logical value, typically the value of a proposition, that produces a value of true when its operand is false and a value of false when its operand is true. So, if statement A is true, then ¬A (pronounced "not A") would therefore be false; and conversely, if ¬A is true, then A would be false.

The truth table of ¬p (also written as ~p or NOT p) is as follows:

p ~p
T F
F T

The logical negation of a proposition p is notated in different ways in various contexts of discussion and fields of application. Among these variants are the following:

Variant Notations
Notation Vocalization
$bar\left\{p\right\}$ bar p,

p bar

$p\text{'}!$ p prime,

p complement

$!p!$ bang p

No matter how it is notated or symbolized, the logical negation ¬p is read as "it is not the case that p", or usually more simply as "not p".

• Within a system of classical logic, double negation, that is, the negation of the negation of a proposition p, is logically equivalent to the initial proposition p. Expressed in symbolic terms, ¬(¬p) ⇔ p.
• Within a system of intuitionistic logic, however, ¬¬p is a weaker statement than p. On the other hand, the logical equivalence ¬¬¬p ⇔ ¬p remains valid.

Logical negation can be defined in terms of other logical operations. For example, ¬p can be defined as pF, where "→" is logical implication and F is absolute falsehood. Conversely, one can define F as p & ¬p for any proposition p, where "&" is logical conjunction. The idea here is that any contradiction is false. While these ideas work in both classical and intuitionistic logic, they don't work in Brazilian logic, where contradictions are not necessarily false. But in classical logic, we get a further identity: pq can be defined as ¬pq, where "∨" is logical disjunction.

Algebraically, logical negation corresponds to the complement in a Boolean algebra (for classical logic) or a Heyting algebra (for intuitionistic logic).

## Properties

• distributivity: ~$\left(a equiv b\right) equiv \left($~$a equiv b\right)$
• linear: In Boolean algebra, a linear function is one such that:

If there exists a0, a1, ... , an $in$ {0,1} such that f(b1, ... , bn) = a0 ⊕ (a1 $land$ b1) ⊕ ... ⊕ (an $land$ bn), for all b1, ... , bn $in$ {0,1}.

Another way to express this is that each variable always makes a difference in the truth-value of the operation or it never makes a difference. Negation is a linear logical operator.

If f(a1, ... , an) = ~f(~a1, ... , ~an) for all a1, ... , an $in$ {0,1}. Negation is a self dual logical operator.

## Rules of Inference

Like the other logical connectors, negation has two major rules of inference associated with it. The first could be called negation introduction, though it is more commonly known as reductio ad absurdum, or proof by contradiction. In propositional logic, this allows you to claim the negation of a proposition, if it would lead to a contradiction if it were true. This is also a common technique in mathematical and rhetorical arguments.

The other rule is known as double negation elimination. This is a feature of standard first order logic, that allows you to remove double negatives in front of propositions. Under the standard framework, double negatives can also introduced in front of propositions anywhere in a sentence without affecting the sentence's truth value. However, this is not true in intuitionist logic.

## Computer science

As in mathematics, negation is used in computer science to construct logical statements.

`   if (!(r == t))`
`   {`
`       ...statements executed when r does NOT equal t...`
}

The "!" signifies logical NOT in B, C, and languages with a C-inspired syntax such as C++, Java, JavaScript, Perl, and PHP. "NOT" is the operator used in ALGOL 60, BASIC, and languages with an ALGOL-inspired syntax such as Pascal, Ada, and Eiffel. Some languages (C++, Perl, etc.) provide more than one operator for negation. Few languages, like Ratfor, use ¬ for negation. Some modern computers and operating systems will display ¬ as ! on files encoded ASCII.

In computer science there is also bitwise negation. This takes the value given and switches all the binary 1s to 0s and 0s to 1s. See bitwise operation. This is often used to create ones' complement or "~" in C or C++ and two's complement (just simplified to "-" or the negative sign since this is equivalent to taking the arithmetic negative value of the number) as it basically creates the opposite (negative value equivalent) or mathematical complement of the value (where both values are added together they create a whole).

Take the following for example:

Say we wanted to get the absolute (positive equivalent) value of a given integer to following would work as the "-" changes it from negative to positive (we know it is negative because it is true that "x < 0")

``` ```

``   unsigned int abs(int x)``
``` { if (x < 0) return -x; else return x; } ```

To demonstrate logical negation:

``` ```

``   unsigned int abs(int x)``
``` { if (!(x < 0)) return x; else return -x; } ```

Inverting the condition and reversing the outcomes produces code that is logically equivalent to the original code, i.e. will have identical results for any input. (Note that depending on the compiler used, the actual instructions performed by the computer may differ.)

Similarly, the following is equivalent on two's complement machines:

``` ```

``   unsigned int abs(int x)``
``` { if (x<0) return ((~x) + 1); else return x; } ```

However, since this relies on the binary representation of integers, it will not work on machines that do not use a two's-complement representation for negative numbers.

## References

• Gabbay, Dov, and Wansing, Heinrich, eds., 1999. What is Negation? Kluwer.
• Horn, L., 2001. A Natural History of Negation. Univ. of Chicago Press.
• G. H. von Wright, 1953-59, "On the Logic of Negation", Commentationes Physico-Mathematicae 22.
• Wansing, Heinrich, 2001, "Negation," in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic. Blackwell.