# User Contributed Dictionary

### Verb

negated- past of negate

# Extensive Definition

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:

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:

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 p
→ F, 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: p → q can be
defined as ¬p ∨ q, 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: ~(a \equiv b) \equiv (~a \equiv b)

- linear: In Boolean algebra, a linear function is one such that:

If there exists a0, a1, ... , an \in such that
f(b1, ... , bn) = a0 ⊕ (a1 \land b1) ⊕ ...
⊕ (an \land bn), for all b1, ... , bn \in .

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.

- self dual: In Boolean algebra a self dual function is one such that:

If f(a1, ... , an) = ~f(~a1, ... , ~an) for all
a1, ... , an \in . Negation is a self dual logical operator.

## Computer science

As in mathematics, negation is used in computer science to construct logical statements.if (!(r == 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 unsigned int abs(int x)