Negation

For other uses, see Negation (disambiguation).

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. Logic is the study of the principles of valid demonstration and Inference. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In its simplest meaning in Mathematics and Logic, an operation is an action or procedure which produces a new value from one or more input values In Logic and Mathematics, a logical value, also called a truth value, is a value indicating the extent to which a Proposition is true In Logic and Philosophy, proposition refers to either (a the content or Meaning of a meaningful Declarative sentence See False for other meanings of false FALSE is an esoteric programming language designed by Wouter van Oortmerssen in See False for other meanings of false FALSE is an esoteric programming language designed by Wouter van Oortmerssen in 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. Grammar is the field of Linguistics that covers the Rules governing the use of any given natural language.

1 Definition
2 Properties
3 Computer science
4 References
5 See also
6 External links

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. Table of logic symbolsIn Logic, two sentences (either in a formal language or a natural language may be joined by means of a logical connective to form a compound sentence In Logic and Mathematics, a logical value, also called a truth value, is a value indicating the extent to which a Proposition is true In Logic and Philosophy, proposition refers to either (a the content or Meaning of a meaningful Declarative sentence 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. A truth table is a Mathematical table used in Logic — specifically in connection with Boolean algebra, Boolean functions and Propositional Among these variants are the following:

**Variant Notations**
Notation	Vocalization
$\bar{p}$	bar p, p bar
$p'\!$	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. Classical logic identifies a class of Formal logics that have been most intensively studied and most widely used In Logic, statements p and q are logically equivalent if they have the same logical content Expressed in symbolic terms, ¬(¬p) ⇔ p.

Within a system of intuitionistic logic, however, ¬¬p is a weaker statement than p. Intuitionistic logic, or constructivist logic, is the Symbolic logic system originally developed by Arend Heyting to provide a formal basis for Brouwer 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. In Logic and Mathematics, logical implication is a logical relation that holds between a set T of formulae and a formula B when every Conversely, one can define F as p & ¬p for any proposition p, where "&" is logical conjunction. In Logic and/or Mathematics, logical conjunction or and is a two-place Logical operation that results in a value of true if both of The idea here is that any contradiction is false. In Classical logic, a contradiction consists of a logical incompatibility between two or more Propositions It occurs when the propositions taken together yield While these ideas work in both classical and intuitionistic logic, they don't work in Brazilian logic, where contradictions are not necessarily false. A paraconsistent logic is a Logical system that attempts to deal with Contradictions in a discriminating way 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). In Abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. In Mathematics, Heyting algebras are special Partially ordered sets that constitute a generalization of Boolean algebras named after Arend Heyting

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, . In Mathematics, and in particular in Abstract algebra, distributivity is a property of Binary operations that generalises the distributive law The word linear comes from the Latin word linearis, which means created by lines. Boolean algebra (or Boolean logic) is a logical calculus of truth values, developed by George Boole in the late 1830s . . , a_n $\in$ {0,1} such that f(b₁, . . . , b_n) = a₀ ⊕ (a₁ $\land$ b₁) ⊕ . . . ⊕ (a_n $\land$ b_n), for all b₁, . . . , b_n $\in$ {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.

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

If f(a₁, . In Mathematics, duality has numerous meanings Generally speaking duality is a metamathematical involution. Boolean algebra (or Boolean logic) is a logical calculus of truth values, developed by George Boole in the late 1830s . . , a_n) = ~f(~a₁, . . . , ~a_n) for all a₁, . . . , a_n $\in$ {0,1}. Negation is a self dual logical operator.

Computer science

As in mathematics, negation is used in computer science to construct logical statements. Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their

   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. B is a programming language that was developed at Bell Labs. It is almost extinct as it was replaced by the C language. tags please moot on the talk page first! --> In Computing, C is a general-purpose cross-platform block structured C++ (" C Plus Plus " ˌsiːˌplʌsˈplʌs is a general-purpose Programming language. JavaScript is a Scripting language most often used for Client-side web development NOTES FOR EDITORS "Perl" is not an acronym (read the "Name" section below PHP is a computer Scripting language. Originally designed for producing Dynamic web pages it has evolved to include a Command line interface capability "NOT" is the operator used in ALGOL 60, BASIC, and languages with an ALGOL-inspired syntax such as Pascal, Ada, and Eiffel. Algol (β Per / Beta Persei known colloquially as the Demon Star, is a bright Star in the Constellation Perseus. In Computer programming, BASIC (an Acronym for Beginner's All-purpose Symbolic Instruction Code) is a family of High-level programming languages Pascal is an influential imperative and procedural Programming language, designed in 1968/9 and published in 1970 by Niklaus Wirth as a small Ada is a structured, Statically typed, imperative, and object-oriented high-level computer Programming language Eiffel is an ISO -standardized Object-oriented Programming language designed to enable programmers to efficiently develop extensible reusable reliable Some languages (C++, Perl, etc. ) provide more than one operator for negation. Few languages, like Ratfor, use ¬ for negation. Ratfor (short for RAT ional FOR tran is a Programming language implemented as a Preprocessor for Fortran 66 Some modern computers and operating systems will display ¬ as ! on files encoded ASCII. An operating system (commonly abbreviated OS and O/S) is the software component of a Computer system that is responsible for the management and coordination American Standard Code for Information Interchange ( 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. The binary numeral system, or base-2 number system, is a Numeral system that represents numeric values using two symbols usually 0 and 1. See bitwise operation. In Computer programming, a bitwise operation operates on one or two Bit patterns or binary numerals at the level of their individual Bits On most 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). In Mathematics, negative numbers in any base are represented in the usual way by prefixing them with a "&minus" sign The two's complement of a Binary number is defined as the value obtained by subtracting the number from a large power of two (specifically from 2 N for

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. Georg Henrik von Wright (pronounced roughly fon vrikt, IPA hɛnrik fɔn-vrikt ( June 14, 1916 &ndash June 16, 2003) was
Wansing, Heinrich, 2001, "Negation," in Goble, Lou, ed. , The Blackwell Guide to Philosophical Logic. Blackwell.

External links

NOT, on MathWorld

In Boolean logic, logical nor or joint denial is a truth-functional operator which produces a result that is the inverse of logical or. In Computer programming, a bitwise operation operates on one or two Bit patterns or binary numerals at the level of their individual Bits On most In Propositional logic, the inference rules double negative elimination (also called double negation elimination, double negative introduction, double Grammatical polarity is the distinction of affirmative and negative which indicates the truth or falsehood of a statement respectively In Logic and Mathematics, the minimal negation operator \nu\! is a Multigrade operator (\nu_{k}_{k \in \mathbb{N}} where each Negation as failure ( NAF, for short is a non-monotonic inference rule in Logic programming, used to derive not~p from failure to derive p MathWorld is an online Mathematics reference work created and largely written by Eric W

Dictionary

-noun

(uncountable) The act of negating something.
(countable) A denial or contradiction.
(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".

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