negation

### Noun

1 a negative statement; a statement that is a
refusal or denial of some other statement

2 the speech act of negating

3 (logic) a proposition that is true if and only
if another proposition is false

### Noun

negation (

countable and

uncountable; plural
negations)

- the act of negating
something
- a denial or contradiction
- In the context of "logic|countable": a proposition that is true if and only if another one is
false; a proposition
resulting from the application to another proposition of the
logical operation "not"

#### Translations

act of negating something

denial or contradiction

negation in logic

- Czech: negace
- German: Negation

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

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.

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)

abnegation,

abolishment,

abolition,

absence,

annihilation,

annulment,

antagonism,

argumentation,

bucking,

challenge,

choking, choking off,

clashing,

conflict,

contention,

contradiction,

contraposition,

contravention,

contraversion,

controversy,

counteraction,

counterworking,

crosscurrent,

crossing,

declension,

declination,

declinature,

declining,

denial,

deprivation,

deracination,

difference,

disaccord,

disaccordance,

disagreement,

disallowance,

disclaimer,

disclamation,

disconfirmation,

discord,

discordance,

discordancy,

discrepancy,

disharmony,

disobedience,

disparity,

disproof,

disproval,

disproving,

dissension,

dissent,

dissidence,

dissonance,

disunion,

disunity,

divergence,

diversity,

elimination,

emptiness,

eradication,

explosion,

expose,

exposure,

extermination,

extinction,

extinguishment,

extirpation,

faction,

gainsaying, head wind,
holding back,

impugnation,

impugnment,

inaccordance,

inequality,

invalidation,

jarring,

liquidation,

nay,

negative, negative answer,

negativeness,

negativity,

nihility,

nix,

no,

nonacceptance,

nonbeing,

noncompliance,

nonconsent,

nonentity,

nonexistence,

nonobservance,

nonoccurrence,

nonreality,

nonsubsistence,
not-being,

nothingness,

nullification,

nullity,

opposing,

opposition,

opposure,

oppugnancy,

oppugnation,

purge,

rebutment,

rebuttal,

recantation,

redargution, reductio ad
absurdum,

refusal,

rejection,

repudiation,

repugnance,

resistance,

retention, rooting out,

silencing, snuffing
out, standing against,

stifling,

strangulation,

suffocation,

suppression, thumbs-down,

traversal,

turndown,

unactuality,

undercurrent,

unharmoniousness,

unreality,

unwillingness,

uprooting,

vacancy,

vacuity,

vacuum,

variance,

void,

voiding,

withholding