Negation
This article includes a list of general references, but it lacks sufficient corresponding inline citations. (March 2013) |
NOT | |
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Definition | |
Truth table | |
Logic gate | |
Normal forms | |
Disjunctive | |
Conjunctive | |
Zhegalkin polynomial | |
Post's lattices | |
0-preserving | no |
1-preserving | no |
Monotone | no |
Affine | yes |
Logical connectives | ||||||||||||||||||||||
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Category | ||||||||||||||||||||||
In
An operand of a negation is a negand,[3] or negatum.[3]
Definition
Classical negation is an
The truth table of is as follows:
True False False True
Negation can be defined in terms of other logical operations. For example, can be defined as (where is logical consequence and is absolute falsehood). Conversely, one can define as for any proposition Q (where is logical conjunction). The idea here is that any contradiction is false, and while these ideas work in both classical and intuitionistic logic, they do not work in paraconsistent logic, where contradictions are not necessarily false. In classical logic, we also get a further identity, can be defined as , where is logical disjunction.
Algebraically, classical negation corresponds to
Notation
The negation of a proposition p is notated in different ways, in various contexts of discussion and fields of application. The following table documents some of these variants:
Notation | Plain text | Vocalization |
---|---|---|
¬p , 7p[4] | Not p | |
~p | Not p | |
-p | Not p | |
En p | ||
p' |
| |
̅p |
| |
!p |
|
The notation is Polish notation.
In set theory, is also used to indicate 'not in the set of': is the set of all members of U that are not members of A.
Regardless how it is notated or symbolized, the negation can be read as "it is not the case that P", "not that P", or usually more simply as "not P".
Precedence
As a way of reducing the number of necessary parentheses, one may introduce
Here is a table that shows a commonly used precedence of logical operators.[5]
Operator | Precedence |
---|---|
1 | |
2 | |
3 | |
4 | |
5 |
Properties
Double negation
Within a system of classical logic, double negation, that is, the negation of the negation of a proposition , is
However, in intuitionistic logic, the weaker equivalence does hold. This is because in intuitionistic logic, is just a shorthand for , and we also have . Composing that last implication with triple negation implies that .
As a result, in the propositional case, a sentence is classically provable if its double negation is intuitionistically provable. This result is known as Glivenko's theorem.
Distributivity
De Morgan's laws provide a way of distributing negation over disjunction and conjunction:
- , and
- .
Linearity
Let denote the logical
If there exists , , for all .
Another way to express this is that each variable always makes a difference in the
Self dual
In
for all . Negation is a self dual logical operator.
Negations of quantifiers
In first-order logic, there are two quantifiers, one is the universal quantifier (means "for all") and the other is the existential quantifier (means "there exists"). The negation of one quantifier is the other quantifier ( and ). For example, with the predicate P as "x is mortal" and the domain of x as the collection of all humans, means "a person x in all humans is mortal" or "all humans are mortal". The negation of it is , meaning "there exists a person x in all humans who is not mortal", or "there exists someone who lives forever".
Rules of inference
There are a number of equivalent ways to formulate rules for negation. One usual way to formulate classical negation in a natural deduction setting is to take as primitive rules of inference negation introduction (from a derivation of to both and , infer ; this rule also being called reductio ad absurdum), negation elimination (from and infer ; this rule also being called ex falso quodlibet), and double negation elimination (from infer ). One obtains the rules for intuitionistic negation the same way but by excluding double negation elimination.
Negation introduction states that if an absurdity can be drawn as conclusion from then must not be the case (i.e. is false (classically) or refutable (intuitionistically) or etc.). Negation elimination states that anything follows from an absurdity. Sometimes negation elimination is formulated using a primitive absurdity sign . In this case the rule says that from and follows an absurdity. Together with double negation elimination one may infer our originally formulated rule, namely that anything follows from an absurdity.
Typically the intuitionistic negation of is defined as . Then negation introduction and elimination are just special cases of implication introduction (conditional proof) and elimination (modus ponens). In this case one must also add as a primitive rule ex falso quodlibet.
Programming language and ordinary language
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
¬
for negation. Most modern languages allow the above statement to be shortened from if (!(r == t))
to if (r != t)
, which allows sometimes, when the compiler/interpreter is not able to optimize it, faster programs.
In computer science there is also bitwise negation. This takes the value given and switches all the
~
" in C or C++ and two's complement-
" 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).
To get the absolute (positive equivalent) value of a given integer the following would work as the "-
" changes it from negative to positive (it is negative because "x < 0
" yields true)
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 (depending on the compiler used, the actual instructions performed by the computer may differ).
This convention occasionally surfaces in ordinary written speech, as computer-related slang for not. For example, the phrase !voting
means "not voting". Another example is the phrase !clue
which is used as a synonym for "no-clue" or "clueless".[6][7]
Kripke semantics
In
See also
- Affirmation and negation (grammatical polarity)
- Ampheck
- Apophasis
- Binary opposition
- Bitwise NOT
- Contraposition
- Cyclic negation
- Negation as failure
- NOT gate
- Plato's beard
- Square of opposition
References
- ^ Weisstein, Eric W. "Negation". mathworld.wolfram.com. Retrieved 2 September 2020.
- ^ "Logic and Mathematical Statements - Worked Examples". www.math.toronto.edu. Retrieved 2 September 2020.
- ^ ISBN 978-0-203-85155-5.
- .
- ISBN 9781846285981.
- ^ Raymond, Eric and Steele, Guy. The New Hacker's Dictionary, p. 18 (MIT Press 1996).
- ^ Munat, Judith. Lexical Creativity, Texts and Context, p. 148 (John Benjamins Publishing, 2007).
Further reading
- Gabbay, Dov, and Wansing, Heinrich, eds., 1999. What is Negation?, Kluwer.
- Horn, L., 2001. A Natural History of Negation, University 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.
- Tettamanti, Marco; Manenti, Rosa; Della Rosa, Pasquale A.; Falini, Andrea; Perani, Daniela; Cappa, Stefano F.; Moro, Andrea (2008). "Negation in the brain: Modulating action representation". NeuroImage. 43 (2): 358–367. S2CID 17658822.
External links
- Horn, Laurence R.; Wansing, Heinrich. "Negation". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.
- "Negation", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- NOT, on MathWorld
- Tables of Truth of composite clauses
- "Table of truth for a NOT clause applied to an END sentence". Archived from the original on 1 March 2000.
- "NOT clause of an END sentence". Archived from the original on 1 March 2000.
- "NOT clause of an OR sentence". Archived from the original on 17 January 2000.
- "NOT clause of an IF...THEN period". Archived from the original on 1 March 2000.