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Encyclopedia > Negation

Negation (i.e. negative, negatory) represents the change of the truth value of a statement to its opposite, used in logic (philosophy of arguments), grammar (linguistics), and mathematics (symbolic logic). Negative has meaning in several contexts: Look up negative in Wiktionary, the free dictionary Negative and non-negative numbers Negative (photography) In optics, diverging lenses are also called negative lenses. ... In logic, a truth value, or truth-value, is a value indicating to what extent a statement is true. ... Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of arguments, although the exact definition of logic is a matter of controversy among philosophers. ... Look up argument in Wiktionary, the free dictionary. ... Grammar is the study of rules governing the use of language. ... Linguistics is the scientific study of human language, and someone who engages in this study is called a linguist or linguistician. ... Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ...

Contents


Definition and symbol

In logic, logical negation is a unary logical operator that reverses the truth value of its operand. Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of arguments, although the exact definition of logic is a matter of controversy among philosophers. ... In mathematics, a unary operation is an operation with only one operand. ... In logical calculus, logical operators or logical connectors serve to connect statements into more complicated compound statements. ... In logic, a truth value, or truth-value, is a value indicating to what extent a statement is true. ... In mathematics, an operand is one of the inputs of an operator. ...


The negation of the statement p is written in various ways:

  • p (which is p with a bar over it)
  • ~p
  • ¬p
  • NOT p
  •  !p

It is read as "It is not the case that p", or simply "not p".

Truth Table
A not A
F T
T F

~p is true if and only if p is false. For instance, if p denotes the statement "today is Saturday", then its negation ~p is the statement "today is not Saturday". Truth tables are a type of mathematical table used in logic to determine whether an expression is true or whether an argument is valid. ... When someone sincerely agrees with an assertion, they might claim that it is the truth. ... ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P... FALSE is an esoteric programming language designed by Wouter van Oortmerssen in 1993, named after his favourite boolean value. ...


In classical logic, double negation means affirmation; i.e., the statements p and ~(~p) are logically equivalent. In intuitionistic logic, however, ~~p is a weaker statement than p. Nevertheless, ~~~p and ~p are logically equivalent. 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. ... Intuitionistic logic, or constructivist logic, is the logic used in mathematical intuitionism and other forms of mathematical constructivism. ...


Logical negation can be defined in terms of other logical operations. For example, ~p can be defined as pF, where → is material 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. In logical calculus of mathematics, the logical conditional (also known as the material implication, sometimes material conditional) is a binary logical operator connecting two statements, if p then q where p is a hypothesis (or antecedent) and q is a conclusion (or consequent). ... AND Logic Gate In mathematics, logical conjunction (usual symbol and) is a logical operator that results in false if either of the operands is false. ... Broadly speaking, a contradiction is an incompatibility between two or more statements, ideas, or actions. ... In logic, Brazilian logic is a name given by Chris Mortensen, in his book Inconsistent Mathematics, to a system R# of relevance logic. ... OR logic gate In mathematics, logical disjunction (usual symbol or) is a logical operator that results in true if either of the operands is true. ...


Algebraically, logical negation corresponds to the complement in a Boolean algebra (for classical logic) or a Heyting algebra (for intuitionistic logic). Wikibooks has more about Boolean logic, under the somewhat misleading title Boolean Algebra For a basic intro to sets, Boolean operations, Venn diagrams, truth tables, and Boolean applications, see Boolean logic. ... In mathematics, Heyting algebras are special partially ordered sets that constitute a generalization of Boolean algebras. ...


Grammar

In grammar, negation is the process that turns an affirmative statement (I am the walrus) into its opposite denial (I am not the walrus). Nouns as well as verbs can be grammatically negated, by the use of a negative adjective (There is no walrus), a negative pronoun (Nobody is the walrus), or a negative adverb (I never was the walrus). Grammar is the study of rules governing the use of language. ... I Am the Walrus is a 1967 Beatles song, written by John Lennon (though credited to both John Lennon and Paul McCartney). ... A noun, or noun substantive, is a part of speech (a word or phrase) which can co-occur with (in)definite articles and attributive adjectives, and function as the head of a noun phrase. ... A verb is a part of speech that usually denotes action (bring, read), occurrence (decompose, glitter), or a state of being (exist, stand). Depending on the language, a verb may vary in form according to many factors, possibly including its tense, aspect, mood and voice. ... An adjective is a part of speech which modifies a noun, usually describing it or making its meaning more specific. ... In linguistics and grammar, a pronoun is a pro-form that substitutes for a noun phrase. ... An adverb as a part of speech. ...


In English, negation for most verbs other than be and have, or verb phrases in which be, have or do already occur, requires the recasting of the sentence using the dummy auxiliary verb do, which adds little to the meaning of the negative phrase, but serves as a place to attach the negative particles not, or its contracted form -n't, to: The English language is a West Germanic language that originates in England. ... The word copula originates from the Latin noun for a link or tie that connects two different things. ... A verb phrase (VP) is a phrase whose head is a verb. ... In linguistics, an auxiliary or helping verb is a verb whose function it is to give further semantic information about the main or full verb which follows it. ...

  • I have a walrus.
  • I haven't a walrus. (rare, but it is still possible to negate have without the auxiliary do.)
  • I don't have a walrus. (the most common way in contemporary English.)

In Middle English, the particle not could be attached to any verb: Middle English is the name given by historical linguistics to the diverse forms of the English language spoken between the Norman invasion in 1066 and the mid-to-late 15th century, when the Chancery Standard, a form of London-based English, began to become widespread, a process aided by the...

  • I see not the walrus.

In Modern English, these forms fell out of use, and the use of an auxiliary such as do or be is obligatory in most cases: For the 80s pop band, see Modern English (band). ...

  • I do not see the walrus.
  • I am not seeing the walrus.
  • I have not seen the walrus.

The verb do also follows this rule, and therefore requires a second instance of itself in order to be marked for negation:

  • The walrus doesn't do tricks
not
  • The walrus doesn't tricks.

In English, as in most other Germanic languages, the use of double negatives as grammatical intensifiers was formerly in frequent use: The Germanic languages form one of the branches of the Indo-European (IE) language family. ... A double negative occurs when two or more ways to express negation are used in the same sentence. ...

  • We don't have no walruses here.

Usage prescriptivists consider this use of double negatives to be a solecism, and condemn it. It makes the rhetorical figure of litotes ambiguous. It remains common in colloquial English. In Ancient Greek, a simple negative (οὐ or μὴ) following another simple or compound negative (e.g., οὐδείς, no one) results in an affirmation, whereas a compound negative following a simple or compound negative strengthens the negation. In linguistics, prescription is the laying down or prescribing of normative rules for a language. ... In linguistic prescriptivism, a solecism is a grammatical or other mistake or absurdity. ... Rhetoric (from Greek ρήτωρ, rhêtôr, orator) is one of the three original liberal arts or trivium (the other members are dialectic and grammar) in Western culture. ... In rhetoric, litotes is a figure of speech in which the speaker emphasizes the magnitude of a statement by denying its opposite. ... Note: This article contains special characters. ...

  • οὐδείς οὐκ ἔπασχε τι, everyone was suffering, literally no one was not suffering something.
  • μὴ θορυβήσῃ μηδείς, let no one raise an uproar, literally do not let no one raise an uproar.

Other languages have simpler forms of negation; in Latin, simple negation is a matter of adding the negative particles non or ne to the verb. In French, the most basic form of verb negation involves adding the circumflexion ne ... pas to the main verb or its auxiliary; je veux un morse ("I want a walrus"); je ne veux pas un morse ("I do not want a walrus.") Latin is an ancient Indo-European language originally spoken in the region around Rome called Latium. ... A circumfix or circumflection is an affix, a morpheme which is placed around another morpheme. ...


Philologically, from the Latin non: no, not indeed, a categoric negative root concept found in languages, even if in different forms. "Not that I know of", expressive of categoric negative assertion, egotistic, defensive, cognitive. Also a negative prefix to concepts, especially as expressed in L. nihil, Eng. emphatic no, definitively not. L. nemo is person oriented, and opposite to L. nihil and means no man, nobody. ne hemo (old form) = no man (homo). Nihil, no+thing, nothing is thing oriented, opposite to nemo. L. nullus means no, not, none (of all those or anything involved). ne ullus = not any one, where unulus is the diminutive of unus, one. Both person and thing oriented, where emphasis is on insignificance. None has ever been so - emphatic, person oriented expression, emphasis being here also denoted by ever (L. aevum, Gr. aion}which here really means: No (one + ever) has been.


Computer science

As in mathematics, negation is used in computer science to prove the logic of a statement. Computer science is the study of information and computation. ...

 if (!(r == t)) { //the statements that happen when r does NOT equal t } 

The ! signifies logical NOT in C, C++, Java, JavaScript, Perl and PHP. "NOT" is the operation used in ALGOL 60, BASIC, Pascal, Ada, Eiffel and Seed7. The ! operator has proven so popular over the years that the operation "!=" is used for not equal to. However, "!(r == t)" and "r != t" are often equivalent after compilation (but not under the X86 processor, as it has a separate "not equal" assembly). This is called logical not which changes a truth to a false or vice versa. The C Programming Language, Brian Kernighan and Dennis Ritchie, the original edition that served for many years as an informal specification of the language The C programming language is a standardized imperative computer programming language developed in the early 1970s by Dennis Ritchie for use on the Unix operating system. ... C++ (pronounced see plus plus, IPA: ) is a general-purpose computer programming language. ... Java is an object-oriented programming language developed by James Gosling and colleagues at Sun Microsystems in the early 1990s. ... JavaScript is an object-based scripting programming language based on the concept of prototypes. ... Perl, also Practical Extraction and Report Language (a backronym, see below) is an interpreted procedural programming language designed by Larry Wall. ... PHP, short for PHP: Hypertext Preprocessor, is an open-source, reflective programming language used mainly for developing server-side applications and dynamic web content, and more recently, other software. ... ALGOL (short for ALGOrithmic Language) is a programming language originally developed in the mid 1950s which became the de facto standard way to report algorithms in print for almost the next 30 years. ... BASIC (Beginners All-purpose Symbolic Instruction Code) is a family of high-level programming languages. ... Pascal is an imperative computer programming language, developed in 1970 by Niklaus Wirth as a language particularly suitable for structured programming. ... Ada is a structured, statically typed imperative computer programming language designed by a team lead by Jean Ichbiah of CII Honeywell Bull during 1977–1983. ... Eiffel is an object-oriented programming language which emphasizes the production of robust software. ... Seed7 is a general-purpose programming language designed by Thomas Mertes. ... x86 or 80x86 is the generic name of a microprocessor architecture first developed and manufactured by Intel. ... This page includes English translations of several Latin phrases and abbreviations such as . ...


In computer science there is also bitwise negation. This takes the value given and switches all the binary 1's to 0's and 0's to 1's. 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) 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). Look up binary in Wiktionary, the free dictionary. ... In computer programming, a bitwise operation operates on one or two bit patterns or binary numerals at the level of their individual bits. ... In mathematics, negative numbers in any base are represented in the usual way, by prefixing them with a - sign. ... Twos complement is the most popular method of signifying negative integers in computer science. ...


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 prove logical negation the following should work

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

We have applied NOT to the if statement boolean variable to create the opposite effect. This still works as we have also swapped the outcome to ensure the same thing happens. However this is less efficient as the if now takes an extra operation and is larger when converted into binary. To make similarities with mathematics ~~q is equivalent to q but is less strong.


Similarly the following would do the same on two's complement machines.

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

However this is machine-dependent and will not work on machines that use sign bits or one's complement.


See also

Apophasis (Late Latin, from Greek apophanai, to say no [1]) refers, in general, to mentioning by not mentioning. Apophasis has specific meanings when used a figure of speech or as a logical device. ...

References

  • Horn, L., A Natural History of Negation, Stanford 2001
  • von Wright, G.H., On the Logic of Negation, Commentationes physico-mathematicae, vol. XXII, 1953-59

  Results from FactBites:
 
Negation - Wikipedia, the free encyclopedia (1142 words)
In linguistics, logic, and mathematics, negation is an operation on logical values, for example, the logical value of a proposition, that sends true to false and false to true.
In logic, logical negation is a unary logical operator that reverses the truth value of its operand.
As in mathematics, negation is used in computer science to prove the logic of a statement.
Untitled Document (5478 words)
The appearance of negation as refusal to comply with a request or command is missing is many mammals because there is a deficit in their natural social environment that makes it unnecessary for them to grasp it.
Negation is often used for the concept of zero, or non-existence, as when we say 'there is no way to get there from here' or an infant notes an unexpected absence by saying 'no car'.
Negation is not only used to indicate the total absence of a quality, but can also be used to indicate a quantity less or greater than another to which it is compared.
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