View source for List of logic symbols

{{Short description|List of symbols used to express logical relations}}
{{ SpecialChars
| special    = logic symbols
| fix        = Help:Special_characters
| characters = logic symbols
}}

In [[Logic|logic]], a set of symbols is commonly used to express logical representation. The following table lists many common symbols, together with their name, how they should be read out loud, and the related field of [[Mathematics|mathematics]]. Additionally, the subsequent columns contains an informal explanation, a short example, the [[Unicode]] location, the name for use in [[HTML]] documents,<ref>{{cite web|title=Named character references|url=http://www.w3.org/html/wg/drafts/html/master/syntax.html#named-character-references|website=HTML 5.1 Nightly|publisher=W3C|access-date=9 September 2015}}</ref> and the [[Software:LaTeX|LaTeX]] symbol.

==Basic logic symbols==
{| class="wikitable"
|- bgcolor=#a0e0a0
! scope="col" |Symbol
! scope="col" |Unicode<br />value<br />(hexadecimal)
! scope="col" |HTML<br />value<br />(decimal)
! scope="col" |HTML<br />entity<br />(named)
! scope="col" |[[Software:LaTeX|LaTeX]]<br />symbol
!Logic Name 
!Read as
!Category
! scope="col" |Explanation
! scope="col" |Examples
|-
! scope"row" align="center" |<div style="font-size:200%;">⇒<br />→<br />⊃</div>
| style="text-align:left;font-family:monospace" |U+21D2<br /><br />U+2192<br /><br />U+2283
| style="text-align:left;font-family:monospace" |&amp;#8658;<br /><br />&amp;#8594;<br /><br />&amp;#8835;
| style="text-align:left;font-family:monospace" |&amp;rArr;<br /><br />&amp;rarr;<br /><br />&amp;sup;
| style="text-align:left;font-family:monospace" |<div><math>\Rightarrow</math>\Rightarrow<br /><math>\implies</math>\implies<br />
<math>\to</math>\to or \rightarrow<br /><math>\supset</math>\supset<br /></div>
||[[Material conditional|material conditional (material implication)]]
|implies,<br> if ... then ...,<br> it is not the case that ... and not ...
|propositional&nbsp;logic, Boolean&nbsp;algebra, [[Heyting algebra|Heyting&nbsp;algebra]]
|<math>A \Rightarrow B</math> is&nbsp;false when {{mvar|A}} is true and {{mvar|B}} is false but&nbsp;true&nbsp;otherwise.<br /><br /><math>\rightarrow</math> may mean the same as <math>\Rightarrow</math><br>(the symbol may also indicate the domain and codomain of a [[Function (mathematics)|function]]; see table of mathematical symbols).<br /><br /><math>\supset</math> may mean the same as <math>\Rightarrow</math> (the symbol may also mean superset).
| scope"row" align="center" |<div style="font-size:90%;"><math>x = 2 \Rightarrow x^2 = 4</math> is true, but <math>x^2 = 4 \Rightarrow x = 2</math> is in general&nbsp;false<br> (since {{mvar|x}} could be −2).</div>
|-
! scope"row" align="center" |<div style="font-size:200%;">⇔<br />↔<br />≡</div>
| style="text-align:left;font-family:monospace" |U+21D4<br /><br />U+2194<br /><br />U+2261
| style="text-align:left;font-family:monospace" |&amp;#8660;<br /><br />&amp;#8596;<br /><br />&amp;#8801;
| style="text-align:left;font-family:monospace" |&amp;hArr;<br /><br />&amp;LeftRightArrow;<br /><br />&amp;equiv;
| style="text-align:left;font-family:monospace" |<math>\Leftrightarrow</math>\Leftrightarrow<br /><math>\iff</math>\iff<br /><math>\leftrightarrow</math>\leftrightarrow<br /><math>\equiv</math>\equiv
||material biconditional (material equivalence)
|if and only if, iff, xnor
|propositional logic, Boolean&nbsp;algebra
|<math>A \Leftrightarrow B</math> is true only if both&nbsp;{{mvar|A}}&nbsp;and&nbsp;{{mvar|B}}&nbsp;are false, or both {{mvar|A}}&nbsp;and&nbsp;{{mvar|B}}&nbsp;are&nbsp;true. Whether a symbol means a material biconditional or a [[Logical equivalence|logical equivalence]], it depends on the author’s style.
| scope"row" align="center" |<div style="font-size:90%;"><math>x + 5 = y + 2 \Leftrightarrow x + 3 = y</math></div>
|-
! scope"row" align="center" |<div style="font-size:200%;">¬<br />~<br />!</div>
| style="text-align:left;font-family:monospace" |U+00AC<br /><br />U+007E<br /><br />U+0021
| style="text-align:left;font-family:monospace" |&amp;#172;<br /><br />&amp;#732;<br /><br />&amp;#33;
| style="text-align:left;font-family:monospace" |&amp;not;<br /><br />&amp;tilde;<br /><br />&amp;excl;
| style="text-align:left;font-family:monospace" |<math>\neg</math>\lnot or \neg<br /><br /><math>\sim</math>\sim<br /><br /><br />
||[[Negation|negation]]
|not
|propositional logic, Boolean&nbsp;algebra
|The statement <math>\lnot A</math> is true if&nbsp;and&nbsp;only&nbsp;if {{mvar|A}}&nbsp;is&nbsp;false.<br /><br />A slash placed through another operator is the same as <math>\neg</math> placed in front.
| scope"row" align="center" |<div style="font-size:90%;"><math>\neg (\neg A) \Leftrightarrow A</math><br /> <math>x \neq y \Leftrightarrow \neg (x = y)</math></div>
|-
! scope"row" align="center" |<div style="font-size:200%;">∧<br/>·<br/>&</div>
| style="text-align:left;font-family:monospace" |U+2227<br /><br />U+00B7<br /><br />U+0026
| style="text-align:left;font-family:monospace" |&amp;#8743;<br /><br />&amp;#183;<br /><br />&amp;#38;<br />
| style="text-align:left;font-family:monospace" |&amp;and;<br /><br />&amp;middot;<br /><br />&amp;amp;
| style="text-align:left;font-family:monospace" |<div><math>\wedge</math>\wedge or \land<br /><math>\cdot</math>\cdot
<br /><br /><math>\&</math>\&<ref>Although this character is available in LaTeX, the [[Software:MediaWiki|MediaWiki]] TeX system does not support it.</ref></div>
||[[Logical conjunction|logical conjunction]]
|and
|propositional logic, Boolean&nbsp;algebra
|The statement ''A''&nbsp;∧&nbsp;''B'' is true if ''A''&nbsp;and&nbsp;''B'' are both true; otherwise, it is false.
| scope"row" align="center" |<div style="font-size:90%;">{{math|''n''&nbsp;< 4&nbsp;&nbsp;∧&nbsp;}} {{math|''n''&nbsp;>2&nbsp;&nbsp;⇔&nbsp;}} {{math|1=''n''&nbsp;= 3}} when ''n'' is a [[Natural number|natural number]].</div>
|-
! scope"row" align="center" |<div style="font-size:200%;">∨<br />+<br />∥</div>
| style="text-align:left;font-family:monospace" |U+2228<br /><br />U+002B<br /><br />U+2225
| style="text-align:left;font-family:monospace" |&amp;#8744;<br /><br />&amp;#43;<br /><br />&amp;#8741;
| style="text-align:left;font-family:monospace" |&amp;or;<br /><br />&amp;plus;<br /><br />&amp;parallel;
| style="text-align:left;font-family:monospace" |<math>\lor</math>\lor or \vee<br /><br /><br /><br /><math>\parallel</math>\parallel
||[[Logical disjunction|logical (inclusive) disjunction]]
|or
|propositional logic, Boolean&nbsp;algebra
|The statement ''A''&nbsp;∨&nbsp;''B'' is true if ''A'' or ''B'' (or both) are true; if both are false, the statement is false.
| scope"row" align="center" |<div style="font-size:90%;">{{math|''n''&nbsp;≥ 4&nbsp;&nbsp;∨&nbsp; ''n''&nbsp;≤ 2&nbsp;&nbsp;⇔ ''n''&nbsp;≠ 3}} when ''n'' is a [[Natural number|natural number]].</div>
|-
! scope"row" align="center" |<br /><div style="font-size:200%;">⊻<br />⊕<br />↮<br />≢</div>
| style="text-align:left;font-family:monospace" |U+22BB<br /><br />U+2295<br /><br />U+21AE<br /><br />U+2262
| style="text-align:left;font-family:monospace" |&amp;#8891;<br /><br />&amp;#8853;<br /><br />&amp;#8622;<br /><br />&amp;#8802;
| style="text-align:left;font-family:monospace" |&amp;veebar;<br /><br />&amp;oplus;<br /><br /><br /><br />&amp;nequiv;
| style="text-align:left;font-family:monospace" |<math>\veebar</math>\veebar<br /><br /><math>\oplus</math>\oplus<br /><br /><br /><br /><math>\not\equiv</math>\not\equiv
| [[Exclusive or|exclusive disjunction]]
|xor,<br> either ... or ... (but not both)
|propositional logic, Boolean algebra
| The statement ''A''&nbsp;⊻&nbsp;''B'' is true when either A or B, but&nbsp;not both, are true. This is equivalent to<br> ¬(A ↔ B), hence the symbols <math>\nleftrightarrow</math> and <math>\not\equiv</math> .
| scope"row" align="center" |<div style="font-size:90%;"><math>\lnot A \veebar A</math> is always true and <math>A \veebar A</math> is always false (if [[Vacuous truth|vacuous truth]] is excluded).</div>
|-
! scope"row" align="center" |<br /><div style="font-size:200%;">⊤<br />T<br />1</div>
| style="text-align:left;font-family:monospace" |<br />U+22A4<br /><br /> <br /> <br /> <br /> <br />
| style="text-align:left;font-family:monospace" |&amp;#8868;<br /><br /> <br /> <br /> <br />
| style="text-align:left;font-family:monospace" |&amp;top;<br /><br /> <br /> <br /> <br />

| style="text-align:left;font-family:monospace" |<math>\top</math>\top<br /><br /> <br /> <br /> <br />
|[[Philosophy:Tautology (logic)|true (tautology)]]
|top, truth, tautology, verum, full clause
|propositional logic, Boolean algebra, [[First-order logic|first-order logic]]
|<math>\top</math> denotes a proposition that is always true.
| scope"row" align="center" |<div style="font-size:90%;">The proposition <math>\top \lor P</math> is always true since at least one of the two is unconditionally true.</div>
|-
! scope"row" align="center" |<br/><div style="font-size:200%;">⊥<br />F<br />0</div> 
| style="text-align:left;font-family:monospace" |U+22A5<br /><br /> <br /> <br /> <br />
| style="text-align:left;font-family:monospace" |&amp;#8869;<br /><br /> <br /> <br /> <br />
| style="text-align:left;font-family:monospace" |&amp;perp;<br /><br /> <br /> <br /> <br />
| style="text-align:left;font-family:monospace" |<math>\bot</math>\bot<br /><br /> <br /> <br /> <br />
|[[Philosophy:False (logic)|false (contradiction)]]
|bottom, falsity, contradiction, falsum, empty clause
|propositional logic, Boolean algebra, [[First-order logic|first-order logic]]
|<math>\bot</math> denotes a proposition that is always false. <br>The symbol ⊥ may also refer to [[Perpendicular|perpendicular]]&nbsp;lines.
| scope"row" align="center" |<div style="font-size:90%;">The proposition <math>\bot \wedge P</math> is always false since at least one of the two is unconditionally false.</div>
|-
! scope"row" align="center" |<div style="font-size:200%;">∀<br />()</div>
| style="text-align:left;font-family:monospace" |U+2200<br /><br /><br />
| style="text-align:left;font-family:monospace" |&amp;#8704;<br /><br /><br />
| style="text-align:left;font-family:monospace" |&amp;forall;<br /><br /><br />
| style="text-align:left;font-family:monospace" |<math>\forall</math>\forall<br /><br /><br />
||[[Universal quantification|universal quantification]]
|given any, for all, for every, for each, for any
|[[First-order logic|first-order logic]]
|<math>\forall x </math>&nbsp;<math>P(x) </math> or<br> <math>(x) </math>&nbsp;<math>P(x) </math> says “given any <math>x </math>, <math>x </math> has property <math>P</math>.”
| scope"row" align="center" |<div style="font-size:90%;"><math>\forall n \isin \mathbb{N}: n^2 \geq n.</math></div>
|-
! scope"row" align="center" |<div style="font-size:200%;">∃</div>
| style="text-align:left;font-family:monospace" |U+2203
| style="text-align:left;font-family:monospace" |&amp;#8707;
| style="text-align:left;font-family:monospace" |&amp;exist;
| style="text-align:left;font-family:monospace" |<math>\exists</math>\exists
||[[Existential quantification|existential quantification]]
|there exists
|[[First-order logic|first-order logic]]
|<math>\exists x </math>&nbsp;<math>P(x) </math> says “there exists an ''x'' (at least one) such that <math>x</math> has property <math>P</math>.”
| scope"row" align="center" |<div style="font-size:90%;"><math>\exists n \isin \mathbb{N}:</math> ''n'' is even.</div>
|-
! scope"row" align="center" |<div style="font-size:200%;">∃!</div>
| style="text-align:left;font-family:monospace" |U+2203&nbsp;U+0021
| style="text-align:left;font-family:monospace" |&amp;#8707; &amp;#33;
| style="text-align:left;font-family:monospace" |&amp;exist;!
| style="text-align:left;font-family:monospace" |<math>\exists !</math>\exists !
||[[Uniqueness quantification|uniqueness quantification]]
|there exists '''exactly&nbsp;one'''
|[[First-order logic|first-order logic]]
|<math>\exists! x</math> <math>P ( x )</math> says “there exists '''exactly one''' ''x'' such that ''x'' has property ''P''.” Only <math>\forall</math> and <math>\exists</math> are part of formal logic.<br> <math>\exists! x</math> <math>P ( x )</math> is a shorthand for <div style="font-size:86%;"><math>\exists x \forall y(P(y) \leftrightarrow y = x)</math></div>
| scope"row" align="center" |<div style="font-size:90%;"><math>\exists! n \isin \mathbb{N}: n+5=2n.</math></div>
|-
! scope"row" align="center" |<div style="font-size:200%;">( )</div>
| style="text-align:left;font-family:monospace" | U+0028&nbsp;U+0029
| style="text-align:left;font-family:monospace" |&amp;#40; &amp;#41;
| style="text-align:left;font-family:monospace" |&amp;lpar;<br> &amp;rpar;
| style="text-align:left;font-family:monospace" |<math>(~)</math> ( )
|precedence grouping
|parentheses; brackets
|everywhere
| Perform the operations inside the parentheses&nbsp;first.
| scope"row" align="center" |<div style="font-size:90%;">{{math|1=(8 ÷ 4) ÷ 2&nbsp;= 2 ÷ 2&nbsp;= 1}}, but {{math|1=8 ÷ (4 ÷ 2)&nbsp;= 8 ÷ 2&nbsp;= 4}}.</div>
|-
! scope"row" align="center" |<div style="font-size:200%;"><math>\mathbb{D}</math></div>
| style="text-align:left;font-family:monospace" |U+1D53B
| style="text-align:left;font-family:monospace" |&amp;#120123;
| style="text-align:left;font-family:monospace" |&amp;Dopf;
| style="text-align:left;font-family:monospace" |\mathbb{D}
||[[Philosophy:Domain of discourse|domain of discourse]]
|domain of discourse
|[[First-order logic#Semantics|first-order logic (semantics)]]
|
| scope"row" align="center" |<div style="font-size:90%;"><math>\mathbb D\mathbb :\mathbb R</math></div>
|-
! scope"row" align="center" |<div style="font-size:200%;">⊢</div>
| style="text-align:left;font-family:monospace" |U+22A2
| style="text-align:left;font-family:monospace" |&amp;#8866;
| style="text-align:left;font-family:monospace" |&amp;vdash;
| style="text-align:left;font-family:monospace" |<math>\vdash</math>\vdash
||[[Turnstile (symbol)|turnstile]]
|[[Logical consequence|syntactically entails (proves)]]
|[[Metalogic|metalogic]]
|<math>A \vdash B </math> says “<math>B</math> is<br> a theorem of <math>A</math>”.<br> In other words,<br> <math>A</math> proves <math>B</math> via a deductive system.
| scope"row" align="center" |<div style="font-size:90%;"><math>(A \rightarrow B) \vdash (\lnot B \rightarrow \lnot A) </math><br> (eg. by using [[Philosophy:Natural deduction|natural deduction]])</div>
|-
! scope"row" align="center" | <div style="font-size:200%;">⊨</div>
| style="text-align:left;font-family:monospace" |U+22A8
| style="text-align:left;font-family:monospace" |&amp;#8872;
| style="text-align:left;font-family:monospace" |&amp;vDash;
| style="text-align:left;font-family:monospace" |<math>\vDash</math>\vDash, \models
||[[Double turnstile|double turnstile]]
|[[Double turnstile|semantically entails]]
|[[Metalogic|metalogic]]
|<math>A \vDash B </math> says<br> “in every [[Structure (mathematical logic)|model]],<br> it is not the case that <math>A</math> is true and <math>B</math> is false”.
| scope"row" align="center" |<div style="font-size:90%;"><math>(A \rightarrow B) \vDash (\lnot B \rightarrow \lnot A) </math><br> (eg. by using truth tables)</div>
|-
!  scope"row" align="center" |<div style="font-size:200%;">≡<br />⟚<br />⇔</div>
| style="text-align:left;font-family:monospace" |U+2261<br /><br />U+27DA<br /><br />U+21D4
| style="text-align:left;font-family:monospace" |&amp;#8801;<br /><br /><br /><br />&amp;#8660;
| style="text-align:left;font-family:monospace" |&amp;equiv;<br /><br /><br /><br />&amp;hArr;
| style="text-align:left;font-family:monospace" |<math>:\equiv</math>\equiv<br /><br /><br /><math>\Leftrightarrow</math>\Leftrightarrow
|[[Logical equivalence|logical equivalence]]
|is logically equivalent to
|[[Metalogic|metalogic]]
|It’s when <math>A \vDash B</math> and <math>B \vDash A</math>. Whether a symbol means a material biconditional or a [[Logical equivalence|logical equivalence]], it depends on the author’s style.
| scope"row" align="center" |<div style="font-size:90%;"><math>(A \rightarrow B) \equiv (\lnot A \lor B)</math></div>
|-
!scope"row" align="center" |<div style="font-size:200%;">⊬</div>
|style="text-align:left;font-family:monospace" |U+22AC
|
|
|style="text-align:left;font-family:monospace" |⊬\nvdash
|
|does not syntactically entail (does not prove)
|[[Metalogic|metalogic]]
|<math>A \nvdash B </math> says “<math>B</math> is<br> '''not''' a theorem of <math>A</math>”.<br> In other words,<br> <math>B</math> is not derivable from <math>A</math> via a deductive system.
| scope"row" align="center" |<div style="font-size:90%;"><math>A \lor B \nvdash A \wedge B </math></div>
|-
!scope"row" align="center" |<div style="font-size:200%;">⊭</div>
|style="text-align:left;font-family:monospace" |U+22AD
|
|
|style="text-align:left;font-family:monospace" |⊭\nvDash
|
|does not semantically entail
|[[Metalogic|metalogic]]
|<math>A \nvDash B </math> says “<math>A</math> does not guarantee the truth of <math>B</math>&nbsp;”.<br> In other words,<br> <math>A</math> does not make <math>B</math> true.
| scope"row" align="center" |<div style="font-size:90%;"><math>A \lor B \nvDash A \wedge B </math></div>
|-
!scope"row" align="center" |<div style="font-size:200%;">□</div>
|style="text-align:left;font-family:monospace" |U+25A1
|
|
| style="text-align:left;font-family:monospace" |<math>\Box</math>\Box
|[[Modal logic#Modal logics in philosophy|logical necessity within a model]]
|box; it is necessary that
|[[Modal logic|modal logic]]
|modal operator for “it&nbsp;is&nbsp;necessary&nbsp;that”<br>in [[Modal logic|alethic logic]], “it&nbsp;is&nbsp;provable&nbsp;that”<br>in [[Philosophy:Provability logic|provability logic]], “it&nbsp;is&nbsp;obligatory&nbsp;that”<br>in [[Philosophy:Deontic logic|deontic logic]], “it&nbsp;is&nbsp;believed&nbsp;that”<br>in [[Philosophy:Doxastic logic|doxastic logic]].
| scope"row" align="center" |<div style="font-size:90%;"><math>\Box \forall x P(x)</math> says “it is necessary that everything has property P”</div>
|-
!scope"row" align="center" |<div style="font-size:200%;">◇</div>
|style="text-align:left;font-family:monospace" |U+25C7
|
|
| style="text-align:left;font-family:monospace" |<math>\Diamond</math>\Diamond
|[[Modal logic#Modal logics in philosophy|logical possibility within a model]]
|diamond; it is possible that
|[[Modal logic|modal logic]]
|modal operator for “it&nbsp;is&nbsp;possible&nbsp;that”, (in most modal logics it is defined as “¬□¬”, “it is not necessarily not”).
| scope"row" align="center" |<div style="font-size:90%;"><math>\Diamond \exists x P(x)</math> says “it is possible that something has property P”</div>
|-
!scope"row" align="center" |<div style="font-size:200%;">∴</div>
|style="text-align:left;font-family:monospace" |U+2234
|
|
|style="text-align:left;font-family:monospace" |∴\therefore
|[[Therefore sign|therefore]]
|[[Therefore sign|therefore]]
|[[Philosophy:Metalanguage|informal metalanguage]]
|shorthand for “therefore”.
|
|-
!scope"row" align="center" |<div style="font-size:200%;">∵</div>
|style="text-align:left;font-family:monospace" |U+2235
|
|
|style="text-align:left;font-family:monospace" |∵\because
|[[Therefore sign#Similar signs|because]]
|[[Therefore sign#Similar signs|because]]
|[[Philosophy:Metalanguage|informal metalanguage]]
|shorthand for “because”.
|
|-
!  scope"row" align="center" |<div style="font-size:200%;">≔<br />≡</div>
| style="text-align:left;font-family:monospace" |U+2254 (U+003A&nbsp;U+003D)<br /><br />U+2261
| style="text-align:left;font-family:monospace" |&amp;#8788; (&amp;#58; &amp;#61;)
<br />&amp;#8801;<br />
| style="text-align:left;font-family:monospace" |<br />&amp;coloneq;
<br /><br />&amp;equiv;
| style="text-align:left;font-family:monospace" |<br /><math>:=</math>:=
<br /><br /><math>\equiv</math>\equiv
|[[Philosophy:Definition|definition (between terms)]]
|is defined as
|[[Philosophy:Metalanguage|informal metalanguage]]
|<math>x:=y </math> (or <math>x \equiv y </math>) means <math>x</math> is defined to be another name for <math>y</math>. This notation seems to have its origin in [[Computer programming|coding]]. However, from the standpoint of formal logic, there is no difference between <math>=</math> and <math>:=</math> , since equality is a [[Symmetric relation|symmetric relation]].
| scope"row" align="center" |<div style="font-size:90%;"><math>\cosh x := \frac {e^x + e^{-x}} {2}</math></div>
|}

==Advanced or 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:
{| class="wikitable"
|- bgcolor="#a0e0a0"
! scope="col" |Symbol
!Unicode<br />value<br />(hexadecimal)
!HTML<br />value<br />(decimal)
! scope="col" |HTML<br />entity<br />(named)
! scope="col" |[[Software:LaTeX|LaTeX]]<br />symbol
! scope="col" |Logic Name
! scope="col" |Read as
! scope="col" |Category
! scope="col" |Explanation
!Examples 
|-
! scope"row" align="center" |<div style="font-size:200%;">̅</div>
|U+0305
|
|
|
|COMBINING OVERLINE
|
|
|used&nbsp;format&nbsp;for&nbsp;denoting [[Gödel numbering|Gödel&nbsp;numbers]].
denoting negation used primarily in electronics.
|using&nbsp;HTML&nbsp;style “4̅” is a shorthand for the standard&nbsp;numeral “SSSS0”.
“<span style="text-decoration: overline“>A ∨ B</span>” says the Gödel number of “(A ∨ B)”.
“<span style="text-decoration: overline“>A ∨ B</span>” is the same as “¬(A ∨ B)”.
|-
! scope"row" align="center" |<div style="font-size:200%;">↑<br><nowiki>|</nowiki></div>
|U+2191<br>U+007C
|
|
|
|UPWARDS ARROW<br>VERTICAL LINE
|
|
|[[Sheffer stroke]],<br>the sign for the NAND operator (negation of conjunction).
|
|-
!scope"row" align="center" |<div style="font-size:200%;">↓</div>
|U+2193
|
|
|
|DOWNWARDS ARROW
|
|
|Peirce Arrow, <br>the sign for the NOR operator (negation of disjunction).
|
|-
!scope"row" align="center" |<div style="font-size:200%;">⊙</div>
|U+2299
|
|
|<math>\odot</math>\odot
|CIRCLED DOT OPERATOR
|
|
|the sign for the XNOR&nbsp;operator (negation of exclusive disjunction).
|
|-
!scope"row" align="center" |<div style="font-size:200%;">∁</div>
|U+2201
|
|
|
|[[Complement (set theory)|COMPLEMENT]]
|
|
|
|
|-
!scope"row" align="center" |<div style="font-size:200%;">∄</div>
|U+2204
|
|
|∄\nexists
|THERE DOES NOT EXIST
|
|
|strike out existential&nbsp;quantifier, same as “¬∃”
|
|-
!scope"row" align="center" |<div style="font-size:200%;">⊧</div>
|U+22A7
|
|
|
|MODELS
|
|
|is a model of (or “is a [[Philosophy:Valuation (logic)|valuation]] satisfying”)
|
|-
!scope"row" align="center" |<div style="font-size:200%;">†</div>
|U+2020
|
|
|
|DAGGER
|it is true that ...
|
|Affirmation operator
|
|-
!scope"row" align="center" |<div style="font-size:200%;">⊼</div>
|U+22BC
|
|
|
|NAND
|
|
|NAND operator
|
|-
!scope"row" align="center" |<div style="font-size:200%;">⊽</div>
|U+22BD
|
|
|
|NOR
|
|
|NOR operator
|
|-
!scope"row" align="center" |<div style="font-size:200%;">⋆</div>
|U+22C6
|
|
|
|STAR OPERATOR
|
|
|usually used for ad-hoc operators
|
|-
!scope"row" align="center" |<div style="font-size:200%;">⊥<br>↓</div>
|U+22A5<br>U+2193
|
|
|
|[[Up tack|UP TACK]]<br>DOWNWARDS ARROW
|
|
|Webb-operator or Peirce arrow, the sign for [[Logical NOR|NOR]].<br> Confusingly, “⊥” is also the sign for contradiction or absurdity.
|
|-
!scope"row" align="center" |<div style="font-size:200%;">⌐</div>
|U+2310
|
|
|
|REVERSED NOT SIGN
|
|
|
|
|-
!scope"row" align="center" |<div style="font-size:200%;">⌜<br>⌝</div>
|U+231C<br>U+231D
|
|
|\ulcorner
\urcorner
|TOP LEFT CORNER<br>TOP RIGHT CORNER
|
|
|corner quotes, also called “Quine quotes”; for quasi-quotation, i.e. quoting specific context of unspecified (“variable”) expressions;<ref>[[Biography:Willard Van Orman Quine|Quine, W.V.]] (1981): ''Mathematical Logic'', §6</ref> also used for denoting Gödel number;<ref>{{citation|last=Hintikka|first=Jaakko|title=The Principles of Mathematics Revisited|url=https://books.google.com/books?id=JHBnE0EQ6VgC&pg=PA113|page=113|year=1998|publisher=Cambridge University Press|isbn=9780521624985}}.</ref> 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. 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. )
|
|-
!scope"row" align="center" |<div style="font-size:200%;">⟚</div>
|U+27DA
|
|
|
|LEFT AND RIGHT DOUBLE TURNSTILE
|
|semantic equivalent
|
|
|-
!scope"row" align="center" |<div style="font-size:200%;">⟛</div>
|U+27DB
|
|
|
|LEFT AND RIGHT TACK
|
|syntactic equivalent
|
|
|-
!scope"row" align="center" |<div style="font-size:200%;">⊩</div>
|U+22A9
|
|
|
|FORCES
|
|
|one of this symbol’s uses is to mean “models” in modal logic, as in  𝔐, 𝑤 ⊩ 𝑃 .
|
|-
!scope"row" align="center" |<div style="font-size:200%;">⟡</div>
|U+27E1
|
|
|
|WHITE CONCAVE-SIDED DIAMOND
|never
|modal operator
|
|
|-
!scope"row" align="center" |<div style="font-size:200%;">⟢</div>
|U+27E2
|
|
|
|WHITE CONCAVE-SIDED DIAMOND WITH LEFTWARDS TICK
|was never
|modal operator
|
|
|-
!scope"row" align="center" |<div style="font-size:200%;">⟣</div>
|U+27E3
|
|
|
|WHITE CONCAVE-SIDED DIAMOND WITH RIGHTWARDS TICK
|will never be
|modal operator
|
|
|-
!scope"row" align="center" |<div style="font-size:200%;">⟤</div>
|U+25A4
|
|
|
|WHITE SQUARE WITH LEFTWARDS TICK
|was always
|modal operator
|
|
|-
!scope"row" align="center" |<div style="font-size:200%;">⟥</div>
|U+25A5
|
|
|
|WHITE SQUARE WITH RIGHTWARDS TICK
|will always be
|modal operator
|
|
|-
!scope"row" align="center" |<div style="font-size:200%;">⥽</div>
|U+297D
|
|
|\strictif
|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> ⥽ <math> q \equiv \Box(p\rightarrow q)</math>.  [https://www.fileformat.info/info/unicode/char/297d/index.htm See here] for an image of glyph. Added to Unicode 3.2.0.
|
|-
!scope"row" align="center" |<div style="font-size:200%;">⨇</div>
|U+2A07
|
|
|
|TWO LOGICAL AND OPERATOR
|
|
|
|
|}

== Usage in various countries ==
=== Poland ===

=== Japan ===
The ⇒ symbol is often used in text to mean "result" or "conclusion", as in "We examined whether to sell the product ⇒ We will not sell it". Also, the → symbol is often used to denote "changed to", as in the sentence "The interest rate changed. March 20% → April 21%".

==See also==
* [[Biography:Józef Maria Bocheński|Józef Maria Bocheński]]
* List of notation used in Principia Mathematica
* [[List of mathematical symbols]]
* [[Logic alphabet]], a suggested set of logical symbols
* {{slink|Logic gate#Symbols}}
* [[Logical connective]]
* [[Mathematical operators and symbols in Unicode]]
* [[Non-logical symbol]]
* [[Polish notation#Polish notation for logic|Polish notation]]
* [[Truth function]]
* [[Truth table]]
* Wikipedia:WikiProject Logic/Standards for notation

==References==
<references/>

==Further reading==
* [[Biography:Józef Maria Bocheński|Józef Maria Bocheński]] (1959), ''A Précis of Mathematical Logic'', trans., Otto Bird, from the French and German editions, Dordrecht, South Holland:  D. Reidel.

==External links==
*[http://www.w3.org/TR/WD-html40-970708/sgml/entities.html Named character entities] in [[HTML]] 4.0

{{Common logical symbols}}
{{Logic}}

[[Category:Mathematical notation|*]]
[[Category:Logic symbols|*]]

{{Sourceattribution|List of logic symbols}}