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{{Refimprove|date=February 2013}}
Hello. Let me introduce the writer. Her title is Refugia Shryock. Bookkeeping is what I do. To collect badges is what her family members and her appreciate. California is where I've always been living and I adore every day residing right here.<br><br>Feel free to surf to my web site - [http://www.ubi-cation.com/ubication/node/6056 www.ubi-cation.com]
{{ SpecialChars
| special    = logic symbols
| fix        = Help:Special_characters
| characters = logic symbols
}}
{{See also|Logical connective}}
__NOTOC__
In [[logic]], a set of symbols is commonly used to express logical representation. As logicians are familiar with these symbols, they are not explained each time they are used. So, for students of logic, the following table lists many common symbols together with their name, pronunciation, and the related field of mathematics. Additionally, the third column contains an informal definition, and the fourth column gives a short example.
 
Be aware that, outside of logic, different symbols have the same meaning, and the same symbol has, depending on the context, different meanings.
 
==Basic logic symbols==
{| class="wikitable"
|- bgcolor=#a0e0a0
! rowspan="3" align=center|<div style="font-
size:150%;">Symbol</div>
!align=left|Name
! rowspan="3" |Explanation
! rowspan="3" |Examples
! rowspan="3" |Unicode<br/>Value
! rowspan="3" |HTML<br/>Entity
! rowspan="3" |[[LaTeX]]<br/>symbol
|- bgcolor=#a0e0a0
!align=center|Should be read as
|- bgcolor=#a0e0a0
!align=right|Category
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|<div style="font-size:200%;">⇒<br/><br/>→<br/><br/>⊃</div>
||[[material conditional|material implication]]
| rowspan=3|''A'' ⇒ ''B'' is true just in the case that either ''A'' is false or ''B'' is true, or both.<br/><br/>→ may mean the same as ⇒ (the symbol may also indicate the domain and codomain of a [[function (mathematics)|function]]; see [[table of mathematical symbols]]).<br/><br/>⊃ may mean the same as ⇒ (the symbol may also mean [[superset]]).
| rowspan=3|''x'' = 2&nbsp;&nbsp;⇒&nbsp; ''x''<sup>2</sup> = 4 is true, but ''x''<sup>2</sup> = 4 &nbsp;&nbsp;⇒&nbsp; ''x'' = 2 is in general false (since ''x'' could be −2).
! rowspan="3" |U+21D2<br/><br/>U+2192<br/><br/>U+2283
! rowspan="3" |&amp;rArr;<br/><br/>&amp;rarr;<br/><br/>&amp;sup;
! rowspan="3" | <div><math>\Rightarrow</math>\Rightarrow<br/><math>\to</math>\to<br/><math>\supset</math>\supset<br/><math>\implies</math>\implies</div>
|-
|align=center|implies; if .. then
|-
|align=right|[[propositional logic]], [[Heyting algebra]]
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|<div style="font-size:200%;">⇔<br/><br/>≡<br/><br/>↔</div>
||[[material equivalence]]
| rowspan=3|''A''&nbsp;⇔ ''B'' is true just in case either both ''A'' and ''B'' are false, or both ''A'' and ''B'' are true.
| rowspan=3|''x''&nbsp;+&nbsp;5&nbsp;=&nbsp;''y''&nbsp;+&nbsp;2&nbsp;&nbsp;⇔&nbsp;&nbsp;''x''&nbsp;+&nbsp;3&nbsp;= ''y''
! rowspan="3" |U+21D4<br/><br/>U+2261<br/><br/>U+2194
! rowspan="3" |&amp;hArr;<br/><br/>&amp;equiv;<br/><br/>&amp;harr;
! rowspan="3" | <div> <math>\Leftrightarrow</math>\Leftrightarrow<br/><math>\equiv</math>\equiv<br/><math>\leftrightarrow</math>\leftrightarrow<br/><math>\iff</math>\iff</div>
|-
|align=center|if and only if; iff; means the same as
|-
|align=right|[[propositional logic]]
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|<div style="font-size:200%;">¬<br/><br/>˜<br/><br/>!</div>
||[[negation]]
| rowspan=3|The statement ¬''A'' is true if and only if ''A'' is false.<br/><br/>A slash placed through another operator is the same as "¬" placed in front.
| rowspan=3|¬(¬''A'')&nbsp;⇔ ''A'' <br/> ''x''&nbsp;≠&nbsp;''y''&nbsp;&nbsp;⇔&nbsp; ¬(''x''&nbsp;=&nbsp;''y'')
! rowspan="3" |U+00AC<br/><br/>U+02DC
! rowspan="3" |&amp;not;<br/><br/>&amp;tilde; ~
! rowspan="3" | <div><math>\neg</math>\lnot or \neg<br/><math>\sim</math>\sim</div>
|-
|align=center|not
|-
|align=right|[[propositional logic]]
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|<div style="font-size:200%;">∧ <br/><br/>•<br/><br/>&</div>
||[[logical conjunction]]
| rowspan=3|The statement ''A'' ∧ ''B'' is true if ''A'' and ''B'' are both true; else it is false.
| rowspan=3|''n''&nbsp;< 4&nbsp;&nbsp;∧&nbsp; ''n''&nbsp;>2&nbsp;&nbsp;⇔&nbsp; ''n''&nbsp;= 3 when ''n'' is a [[natural number]].
! rowspan="3" |U+2227<br/><br/>U+0026
! rowspan="3" |&amp;and;<br/><br/>&amp;amp;
! rowspan="3" | <math>\wedge</math>\wedge or \land<br/>\&<ref>Although this character is available in LaTeX, the [[MediaWiki]] TeX system doesn't support this character.</ref>
|-
|align=center|and
|-
|align=right|[[propositional logic]], [[Boolean algebra (logic)|Boolean algebra]]
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|<div style="font-size:200%;">∨<br/><br/>+<br/><br/>ǀǀ</div>
||[[logical disjunction]]
| rowspan=3|The statement ''A'' ∨ ''B'' is true if ''A'' or ''B'' (or both) are true; if both are false, the statement is false.
| rowspan=3|''n''&nbsp;≥ 4&nbsp;&nbsp;∨&nbsp; ''n''&nbsp;≤ 2&nbsp;&nbsp;⇔ ''n''&nbsp;≠ 3 when ''n'' is a [[natural number]].
! rowspan="3" |U+2228
! rowspan="3" |&amp;or;
! rowspan="3" | <math>\lor</math>\lor or \vee
|-
|align=center|or
|-
|align=right|[[propositional logic]], [[Boolean algebra (logic)|Boolean algebra]]
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|<br/><div style="font-size:200%;">⊕<br/><br/>{{Unicode|⊻}}</div> ||[[exclusive or|exclusive disjunction]]
| rowspan=3| The statement ''A'' ⊕ ''B'' is true when either A or B, but not both, are true. ''A'' {{Unicode|⊻}} ''B'' means the same.
| rowspan=3| (¬''A'')  ⊕ ''A'' is always true, ''A'' ⊕ ''A'' is always false.
! rowspan="3" |U+2295<br/><br/>U+22BB
! rowspan="3" |&amp;oplus;
! rowspan="3" | <math>\oplus</math>\oplus<br/><math>\veebar</math>\veebar
|-
|align=center|xor
|-
|align=right|[[propositional logic]], [[Boolean algebra (logic)|Boolean algebra]]
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|<br/><div style="font-size:200%;">⊤<br/><br/>T<br/><br/>1</div> ||[[Tautology (logic)|Tautology]]
| rowspan=3| The statement ⊤ is unconditionally true.
| rowspan=3| ''A'' ⇒ ⊤ is always true.
! rowspan="3" |U+22A4
! rowspan="3" |T
! rowspan="3" | <math>\top</math>\top
|-
|align=center|top, verum
|-
|align=right|[[propositional logic]], [[Boolean algebra (logic)|Boolean algebra]]
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|<br/><div style="font-size:200%;">⊥<br/><br/>F<br/><br/>0</div> ||[[Contradiction]]
| rowspan=3| The statement ⊥ is unconditionally false.
| rowspan=3| ⊥ ⇒ ''A'' is always true.
! rowspan="3" |U+22A5
! rowspan="3" |&amp;perp; F
! rowspan="3" |<math>\bot</math>\bot
|-
|align=center|bottom, falsum
|-
|align=right|[[propositional logic]], [[Boolean algebra (logic)|Boolean algebra]]
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|<div style="font-size:200%;">∀<br/><br/>()</div>
||[[universal quantification]]
| rowspan=3|∀&nbsp;''x'':&nbsp;''P''(''x'') or (''x'')&nbsp;''P''(''x'') means ''P''(''x'') is true for all ''x''.
| rowspan=3|∀&nbsp;''n''&nbsp;∈ {{Unicode|&#x2115;}}: ''n''<sup>2</sup>&nbsp;≥ ''n''.
! rowspan="3" |U+2200
! rowspan="3" |&amp;forall;
! rowspan="3" | <math>\forall</math>\forall
|-
|align=center|for all; for any; for each
|-
|align=right|[[first-order logic]]
|-
| rowspan=3  bgcolor=#d0f0d0 align=center|<div style="font-size:200%;">∃</div>
||[[existential quantification]]
| rowspan=3|∃&nbsp;''x'': ''P''(''x'') means there is at least one ''x'' such that ''P''(''x'') is true.
| rowspan=3|∃&nbsp;''n''&nbsp;∈ {{Unicode|&#x2115;}}: ''n'' is even.
! rowspan="3" |[[Turned E|U+2203]]
! rowspan="3" |&amp;exist;
! rowspan="3" | <math>\exists</math>\exists
|-
|align=center|there exists
|-
|align=right|[[first-order logic]]
|-
| rowspan=3  bgcolor=#d0f0d0 align=center|<div style="font-size:200%;">∃!</div>
||[[uniqueness quantification]]
| rowspan=3|∃!&nbsp;''x'': ''P''(''x'') means there is exactly one ''x'' such that ''P''(''x'') is true.
| rowspan=3|∃!&nbsp;''n''&nbsp;∈ {{Unicode|&#x2115;}}: ''n''&nbsp;+ 5&nbsp;= 2''n''.
! rowspan="3" |U+2203&nbsp;U+0021
! rowspan="3" |&amp;exist; !
! rowspan="3" |<math>\exists !</math>\exists !
|-
|align=center|there exists exactly one
|-
|align=right|[[first-order logic]]
|-
| rowspan=3 bgcolor=#d0f0d0 align=center|<div style="font-size:200%;">:=<br/><br/>≡<br/><br/>:⇔</div>
||[[definition]]
| rowspan=3|''x''&nbsp;:= ''y'' or ''x''&nbsp;≡ ''y'' means ''x'' is defined to be another name for ''y'' (but note that ≡ can also mean other things, such as [[congruence relation|congruence]]).<br/><br/>''P''&nbsp;:⇔ ''Q'' means ''P'' is defined to be [[Logical equivalence|logically equivalent]] to ''Q''.
| rowspan=3|cosh&nbsp;''x''&nbsp;:= (1/2)(exp&nbsp;''x''&nbsp;+  exp&nbsp;(−''x''))<br/><br/>''A''&nbsp;XOR&nbsp;''B'' :⇔ (''A''&nbsp;∨&nbsp;''B'')&nbsp;∧&nbsp;¬(''A''&nbsp;∧&nbsp;''B'')
! rowspan="3" |U+2254 (U+003A&nbsp;U+003D)<br/><br/>U+2261<br/><br/>U+003A&nbsp;U+229C
! rowspan="3" |:=<br/>:<br/><br/>&amp;equiv;<br/><br/>&amp;hArr;
! rowspan="3" | <div><math>:=</math>:=<br/><math>\equiv</math>\equiv<br/><math>\Leftrightarrow</math>\Leftrightarrow</div>
|-
|align=center|is defined as
|-
|align=right|everywhere
|-
| rowspan="3"  bgcolor=#d0f0d0 align=center|<div style="font-size:200%;">( )</div>
| precedence grouping
| rowspan="3" | Perform the operations inside the parentheses first.
| rowspan="3" |(8 ÷ 4) ÷ 2&nbsp;= 2 ÷ 2&nbsp;= 1, but 8 ÷ (4 ÷ 2)&nbsp;= 8 ÷ 2&nbsp;= 4.
! rowspan="3" | U+0028&nbsp;U+0029
! rowspan="3" |( )
! rowspan="3" | <math>(~)</math> ( )
|-
|align=center|parentheses, brackets
|-
|align=right|everywhere
|-
| rowspan=3 bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">{{Unicode|⊢}}</div>
||[[Turnstile (symbol)|Turnstile]]
| rowspan=3|''x'' {{Unicode|⊢}} ''y'' means ''y'' is provable from ''x'' (in some specified formal system).
| rowspan=3| ''A'' → ''B'' {{Unicode|⊢}} ¬''B'' → ¬''A''
! rowspan="3" |U+22A2
! rowspan="3" |&amp;#8866;
! rowspan="3" | <math>\vdash</math>\vdash
|-
|align=center|provable
|-
|align=right|[[propositional logic]], [[first-order logic]]
|-
| rowspan=3 bgcolor=#d0f0d0 align=center| <div style="font-size:200%;">⊨</div>
||[[double turnstile]]
| rowspan=3|''x'' ⊨ ''y'' means ''x'' semantically entails ''y''
| rowspan=3| ''A'' → ''B'' ⊨ ¬''B'' → ¬''A''
! rowspan="3" |U+22A8
! rowspan="3" |&amp;#8872;
! rowspan="3" | <math>\models</math>\models
|-
|align=center|entails
|-
|align=right|[[propositional logic]], [[first-order logic]]
|}
 
==Advanced and rarely used logical symbols==
<!--sum1 plz make a table 4 this, and also some copy editing i am horrible at writing anything that does not look like P&Q...-->
These symbols are sorted by their Unicode value:
 
* {{unichar|00B7|MIDDLE DOT}}, an outdated way for denoting AND{{Citation needed|date=November 2010}}, still in use in electronics; for example "A·B" is the same as "A&B"
*<span style="text-decoration: overline">·</span>: Center dot with a line above it. Outdated way for denoting NAND, for example "A<span style="text-decoration: overline">·</span>B" is the same as "A NAND B" or "A|B" or "¬(A & B)". See also Unicode {{unichar|22C5|dot operator}}.
 
*{{unichar|0305|COMBINING OVERLINE|nlink=overline|cwith=&nbsp;}}, used as abbreviation for standard numerals. For example, using HTML style "{{unicode|4&#x0305;}}" is a shorthand for the standard numeral "SSSS0".
* Overline, is also a rarely used format for denoting [[Gödel numbering|Gödel numbers]], for example "<span style="text-decoration: overline">AVB</span>" says the Gödel number of "(AVB)"
* Overline is also an outdated way for denoting negation, still in use in electronics; for example "<span style="text-decoration: overline">AVB</span>" is the same as "¬(AVB)"
 
* {{unichar|2191|UPWARDS ARROW}} or {{unichar|007C|VERTICAL LINE}}: [[Sheffer stroke]], the sign for the NAND operator.
* {{unichar|2201|Complement|nlink=Complement (set theory)}}
* {{unichar|2204|THERE DOES NOT EXIST}}: strike out existential quantifier same as "¬∃"
* {{unichar|2234|Therefore|nlink=Therefore sign}}
* {{unichar|2235|Because|nlink=Therefore sign#Related_signs}}
* {{unichar|22A7|Models}}: is a [[Model theory|model]] of
* {{unichar|22A8|True}}: is true of
* {{unichar|22AC|DOES NOT PROVE}}: negated ⊢, the sign for "does not prove", for example ''T'' ⊬ ''P'' says "''P'' is not a theorem of ''T''"
* {{unichar|22AD|Not true}}: is not true of
* {{unichar|22BC|NAND}}: another NAND operator, can also be rendered as <span style="text-decoration: overline">∧</span>
* {{unichar|22BD|Nor}}: another NOR operator, can also be rendered as <span style="text-decoration: overline">V</span>
* {{unichar|22C4|DIAMOND OPERATOR}}: modal operator for "it is possible that", "it is not necessarily not" or rarely "it is not provable not" (in most modal logics it is defined as "¬◻¬")
* {{unichar|22C6|STAR OPERATOR}}: usually used for ad-hoc operators
* {{unichar|22A5|UP TACK}} or {{unichar|2193|DOWNWARDS ARROW}}: Webb-operator or Peirce arrow, the sign for [[Logical NOR|NOR]]. Confusingly, "⊥" is also the sign for contradiction or absurdity.
 
* {{unichar|2310|REVERSED NOT SIGN}}
 
* {{unichar|231C|TOP LEFT CORNER}} and {{unichar|231D|TOP RIGHT CORNER}}: corner quotes, also called "Quine quotes"; for quasi-quotation, i.e. quoting specific context of unspecified ("variable") expressions;<ref>[[Willard Van Orman Quine|Quine, W.V.]] (1981): ''Mathematical Logic'', §6</ref> also the standard symbol{{Citation needed|date=September 2013}} used for denoting [[Gödel number]]; for example "⌜G⌝" denotes the Gödel number of G.  (Typographical note: although the quotes appears as  a "pair" in unicode (231C and 231D), they are not symmetrical in some fonts. And in some fonts (for example Arial) they are only symmetrical in certain sizes.  Alternatively the quotes can be rendered as ⌈ and ⌉ (U+2308 and U+2309) or by using a negation symbol and a reversed negation symbol ⌐ ¬ in superscript mode. )
 
* {{unichar|25FB|WHITE MEDIUM SQUARE}} or  {{unichar|25A1|WHITE SQUARE}}: modal operator for "it is necessary that" (in [[modal logic]]), or "it is provable that" (in [[provability logic]]), or "it is obligatory that" (in [[deontic logic]]), or "it is believed that" (in [[doxastic logic]]).
 
Note that the following operators are rarely supported by natively installed fonts.  If you wish to use these in a web page, you should always embed the necessary fonts so the page viewer can see the web page without having the necessary fonts installed in their computer.
* {{unichar|27E1|WHITE CONCAVE-SIDED DIAMOND}}
* {{unichar|27E2|WHITE CONCAVE-SIDED DIAMOND WITH LEFTWARDS TICK}}: modal operator for was never
* {{unichar|27E3|WHITE CONCAVE-SIDED DIAMOND WITH RIGHTWARDS TICK}}: modal operator for will never be
* {{unichar|27E4|WHITE SQUARE WITH LEFTWARDS TICK}}: modal operator for was always
* {{unichar|27E5|WHITE SQUARE WITH RIGHTWARDS TICK}}: modal operator for will always be
* {{unichar|297D|RIGHT FISH TAIL}}: sometimes used for "relation", also used for denoting various ad hoc relations (for example, for denoting "witnessing" in the context of [[Rosser's trick]]) The fish hook is also used as strict implication by C.I.Lewis <math> p </math> {{unicode|&#x297D;}} <math> q \equiv \Box(p\rightarrow q)</math>, the corresponding LaTeX macro is \strictif.  [http://www.fileformat.info/info/unicode/char/297d/index.htm See here] for an image of glyph. Added to Unicode 3.2.0.
 
==See also==
{{Portal|Logic}}
* [[Logic Alphabet]], a suggested set of logical symbols
* [[Mathematical operators and symbols in Unicode]]
* [[Polish notation#Polish notation for logic|Polish notation]]
* [[List of mathematical symbols]]
 
==Notes==
<references/>
 
==External links==
*[http://www.w3.org/TR/WD-html40-970708/sgml/entities.html Named character entities] in [[HTML]] 4.0
 
{{Logic}}
 
[[Category:Mathematical notation|*]]
[[Category:Logic symbols| ]]

Latest revision as of 23:57, 13 November 2014

Hello. Let me introduce the writer. Her title is Refugia Shryock. Bookkeeping is what I do. To collect badges is what her family members and her appreciate. California is where I've always been living and I adore every day residing right here.

Feel free to surf to my web site - www.ubi-cation.com

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