# List of logic symbols

Short description: List of symbols used to express logical relations

In 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. Additionally, the subsequent columns contains an informal explanation, a short example, the Unicode location, the name for use in HTML documents,[1] and the LaTeX symbol.

## Basic logic symbols

Symbol Unicode
value
HTML
value
(decimal)
HTML
entity
(named)
LaTeX
symbol
Logic Name Read as Category Explanation Examples

U+21D2

U+2192

U+2283
&#8658;

&#8594;

&#8835;
&rArr;

&rarr;

&sup;
$\displaystyle{ \Rightarrow }$\Rightarrow
$\displaystyle{ \implies }$\implies
$\displaystyle{ \to }$\to or \rightarrow
$\displaystyle{ \supset }$\supset
material conditional (material implication) implies,
if ... then ...,
it is not the case that ... and not ...
propositional logic, Boolean algebra, Heyting algebra $\displaystyle{ A \Rightarrow B }$ is false when A is true and B is false but true otherwise.

$\displaystyle{ \rightarrow }$ may mean the same as $\displaystyle{ \Rightarrow }$
(the symbol may also indicate the domain and codomain of a function; see table of mathematical symbols).

$\displaystyle{ \supset }$ may mean the same as $\displaystyle{ \Rightarrow }$ (the symbol may also mean superset).
$\displaystyle{ x = 2 \Rightarrow x^2 = 4 }$ is true, but $\displaystyle{ x^2 = 4 \Rightarrow x = 2 }$ is in general false
(since x could be −2).

U+21D4

U+2194

U+2261
&#8660;

&#8596;

&#8801;
&hArr;

&LeftRightArrow;

&equiv;
$\displaystyle{ \Leftrightarrow }$\Leftrightarrow
$\displaystyle{ \iff }$\iff
$\displaystyle{ \leftrightarrow }$\leftrightarrow
$\displaystyle{ \equiv }$\equiv
material biconditional (material equivalence) if and only if, iff, xnor propositional logic, Boolean algebra $\displaystyle{ A \Leftrightarrow B }$ is true only if both A and B are false, or both A and B are true. Whether a symbol means a material biconditional or a logical equivalence, it depends on the author’s style.
$\displaystyle{ x + 5 = y + 2 \Leftrightarrow x + 3 = y }$
¬
~
!
U+00AC

U+007E

U+0021
&#172;

&#732;

&#33;
&not;

&tilde;

&excl;
$\displaystyle{ \neg }$\lnot or \neg

$\displaystyle{ \sim }$\sim

negation not propositional logic, Boolean algebra The statement $\displaystyle{ \lnot A }$ is true if and only if A is false.

A slash placed through another operator is the same as $\displaystyle{ \neg }$ placed in front.
$\displaystyle{ \neg (\neg A) \Leftrightarrow A }$
$\displaystyle{ x \neq y \Leftrightarrow \neg (x = y) }$

·
&
U+2227

U+00B7

U+0026
&#8743;

&#183;

&#38;
&and;

&middot;

&amp;
$\displaystyle{ \wedge }$\wedge or \land
$\displaystyle{ \cdot }$\cdot

$\displaystyle{ \& }$\&[2]
logical conjunction and propositional logic, Boolean algebra The statement A ∧ B is true if A and B are both true; otherwise, it is false.
n < 4  ∧  n >2  ⇔  n = 3 when n is a natural number.

+
U+2228

U+002B

U+2225
&#8744;

&#43;

&#8741;
&or;

&plus;

&parallel;
$\displaystyle{ \lor }$\lor or \vee

$\displaystyle{ \parallel }$\parallel
logical (inclusive) disjunction or propositional logic, Boolean algebra The statement A ∨ B is true if A or B (or both) are true; if both are false, the statement is false.
n ≥ 4  ∨  n ≤ 2  ⇔ n ≠ 3 when n is a natural number.

U+22BB

U+2295

U+21AE

U+2262
&#8891;

&#8853;

&#8622;

&#8802;
&veebar;

&oplus;

&nequiv;
$\displaystyle{ \veebar }$\veebar

$\displaystyle{ \oplus }$\oplus

$\displaystyle{ \not\equiv }$\not\equiv
exclusive disjunction xor,
either ... or ... (but not both)
propositional logic, Boolean algebra The statement A ⊻ B is true when either A or B, but not both, are true. This is equivalent to
¬(A ↔ B), hence the symbols $\displaystyle{ \nleftrightarrow }$ and $\displaystyle{ \not\equiv }$ .
$\displaystyle{ \lnot A \veebar A }$ is always true and $\displaystyle{ A \veebar A }$ is always false (if vacuous truth is excluded).

T
1

U+22A4

&#8868;

&top;

$\displaystyle{ \top }$\top

true (tautology) top, truth, tautology, verum, full clause propositional logic, Boolean algebra, first-order logic $\displaystyle{ \top }$ denotes a proposition that is always true.
The proposition $\displaystyle{ \top \lor P }$ is always true since at least one of the two is unconditionally true.

F
0
U+22A5

&#8869;

&perp;

$\displaystyle{ \bot }$\bot

false (contradiction) bottom, falsity, contradiction, falsum, empty clause propositional logic, Boolean algebra, first-order logic $\displaystyle{ \bot }$ denotes a proposition that is always false.
The symbol ⊥ may also refer to perpendicular lines.
The proposition $\displaystyle{ \bot \wedge P }$ is always false since at least one of the two is unconditionally false.

()
U+2200

&#8704;

&forall;

$\displaystyle{ \forall }$\forall

universal quantification given any, for all, for every, for each, for any first-order logic $\displaystyle{ \forall x }$ $\displaystyle{ P(x) }$ or
$\displaystyle{ (x) }$ $\displaystyle{ P(x) }$ says “given any $\displaystyle{ x }$, $\displaystyle{ x }$ has property $\displaystyle{ P }$.”
$\displaystyle{ \forall n \isin \mathbb{N}: n^2 \geq n. }$
U+2203 &#8707; &exist; $\displaystyle{ \exists }$\exists existential quantification there exists first-order logic $\displaystyle{ \exists x }$ $\displaystyle{ P(x) }$ says “there exists an x (at least one) such that $\displaystyle{ x }$ has property $\displaystyle{ P }$.”
$\displaystyle{ \exists n \isin \mathbb{N}: }$ n is even.
∃!
U+2203 U+0021 &#8707; &#33; &exist;! $\displaystyle{ \exists ! }$\exists ! uniqueness quantification there exists exactly one first-order logic $\displaystyle{ \exists! x }$ $\displaystyle{ P ( x ) }$ says “there exists exactly one x such that x has property P.” Only $\displaystyle{ \forall }$ and $\displaystyle{ \exists }$ are part of formal logic.
$\displaystyle{ \exists! x }$ $\displaystyle{ P ( x ) }$ is a shorthand for
$\displaystyle{ \exists x \forall y(P(y) \leftrightarrow y = x) }$
$\displaystyle{ \exists! n \isin \mathbb{N}: n+5=2n. }$
( )
U+0028 U+0029 &#40; &#41; &lpar;
&rpar;
$\displaystyle{ (~) }$ ( ) precedence grouping parentheses; brackets everywhere Perform the operations inside the parentheses first.
(8 ÷ 4) ÷ 2 = 2 ÷ 2 = 1, but 8 ÷ (4 ÷ 2) = 8 ÷ 2 = 4.
$\displaystyle{ \mathbb{D} }$
U+1D53B &#120123; &Dopf; \mathbb{D} domain of discourse domain of discourse first-order logic (semantics)
$\displaystyle{ \mathbb D\mathbb :\mathbb R }$
U+22A2 &#8866; &vdash; $\displaystyle{ \vdash }$\vdash turnstile syntactically entails (proves) metalogic $\displaystyle{ A \vdash B }$ says “$\displaystyle{ B }$ is
a theorem of $\displaystyle{ A }$”.
In other words,
$\displaystyle{ A }$ proves $\displaystyle{ B }$ via a deductive system.
$\displaystyle{ (A \rightarrow B) \vdash (\lnot B \rightarrow \lnot A) }$
(eg. by using natural deduction)
U+22A8 &#8872; &vDash; $\displaystyle{ \vDash }$\vDash, \models double turnstile semantically entails metalogic $\displaystyle{ A \vDash B }$ says
“in every model,
it is not the case that $\displaystyle{ A }$ is true and $\displaystyle{ B }$ is false”.
$\displaystyle{ (A \rightarrow B) \vDash (\lnot B \rightarrow \lnot A) }$
(eg. by using truth tables)

U+2261

U+27DA

U+21D4
&#8801;

&#8660;
&equiv;

&hArr;
$\displaystyle{ :\equiv }$\equiv

$\displaystyle{ \Leftrightarrow }$\Leftrightarrow
logical equivalence is logically equivalent to metalogic It’s when $\displaystyle{ A \vDash B }$ and $\displaystyle{ B \vDash A }$. Whether a symbol means a material biconditional or a logical equivalence, it depends on the author’s style.
$\displaystyle{ (A \rightarrow B) \equiv (\lnot A \lor B) }$
U+22AC ⊬\nvdash does not syntactically entail (does not prove) metalogic $\displaystyle{ A \nvdash B }$ says “$\displaystyle{ B }$ is
not a theorem of $\displaystyle{ A }$”.
In other words,
$\displaystyle{ B }$ is not derivable from $\displaystyle{ A }$ via a deductive system.
$\displaystyle{ A \lor B \nvdash A \wedge B }$
U+22AD ⊭\nvDash does not semantically entail metalogic $\displaystyle{ A \nvDash B }$ says “$\displaystyle{ A }$ does not guarantee the truth of $\displaystyle{ B }$ ”.
In other words,
$\displaystyle{ A }$ does not make $\displaystyle{ B }$ true.
$\displaystyle{ A \lor B \nvDash A \wedge B }$
U+25A1 $\displaystyle{ \Box }$\Box logical necessity within a model box; it is necessary that modal logic modal operator for “it is necessary that”
in alethic logic, “it is provable that”
in provability logic, “it is obligatory that”
in deontic logic, “it is believed that”
in doxastic logic.
$\displaystyle{ \Box \forall x P(x) }$ says “it is necessary that everything has property P”
U+25C7 $\displaystyle{ \Diamond }$\Diamond logical possibility within a model diamond; it is possible that modal logic modal operator for “it is possible that”, (in most modal logics it is defined as “¬□¬”, “it is not necessarily not”).
$\displaystyle{ \Diamond \exists x P(x) }$ says “it is possible that something has property P”
U+2234 ∴\therefore therefore therefore informal metalanguage shorthand for “therefore”.
U+2235 ∵\because because because informal metalanguage shorthand for “because”.

U+2254 (U+003A U+003D)

U+2261
&#8788; (&#58; &#61;)

&#8801;

&coloneq;

&equiv;

$\displaystyle{ := }$:=

$\displaystyle{ \equiv }$\equiv

definition (between terms) is defined as informal metalanguage $\displaystyle{ x:=y }$ (or $\displaystyle{ x \equiv y }$) means $\displaystyle{ x }$ is defined to be another name for $\displaystyle{ y }$. This notation seems to have its origin in coding. However, from the standpoint of formal logic, there is no difference between $\displaystyle{ = }$ and $\displaystyle{ := }$ , since equality is a symmetric relation.
$\displaystyle{ \cosh x := \frac {e^x + e^{-x}} {2} }$

## Advanced or rarely used logical symbols

These symbols are sorted by their Unicode value:

Symbol Unicode
value
HTML
value
(decimal)
HTML
entity
(named)
LaTeX
symbol
Logic Name Read as Category Explanation Examples
̅
U+0305 COMBINING OVERLINE used format for denoting Gödel numbers.

denoting negation used primarily in electronics.

using HTML style “4̅” is a shorthand for the standard numeral “SSSS0”.

A ∨ B” says the Gödel number of “(A ∨ B)”. “A ∨ B” is the same as “¬(A ∨ B)”.

|
U+2191
U+007C
UPWARDS ARROW
VERTICAL LINE
Sheffer stroke,
the sign for the NAND operator (negation of conjunction).
U+2193 DOWNWARDS ARROW Peirce Arrow,
the sign for the NOR operator (negation of disjunction).
U+2299 $\displaystyle{ \odot }$\odot CIRCLED DOT OPERATOR the sign for the XNOR operator (negation of exclusive disjunction).
U+2201 COMPLEMENT
U+2204 ∄\nexists THERE DOES NOT EXIST strike out existential quantifier, same as “¬∃”
U+22A7 MODELS is a model of (or “is a valuation satisfying”)
U+2020 DAGGER it is true that ... Affirmation operator
U+22BC NAND NAND operator
U+22BD NOR NOR operator
U+22C6 STAR OPERATOR usually used for ad-hoc operators

U+22A5
U+2193
UP TACK
DOWNWARDS ARROW
Webb-operator or Peirce arrow, the sign for NOR.
Confusingly, “⊥” is also the sign for contradiction or absurdity.
U+2310 REVERSED NOT SIGN

U+231C
U+231D
\ulcorner

\urcorner

TOP LEFT CORNER
TOP RIGHT CORNER
corner quotes, also called “Quine quotes”; for quasi-quotation, i.e. quoting specific context of unspecified (“variable”) expressions;[3] also used for denoting Gödel number;[4] 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. )
U+27DA LEFT AND RIGHT DOUBLE TURNSTILE semantic equivalent
U+27DB LEFT AND RIGHT TACK syntactic equivalent
U+22A9 FORCES one of this symbol’s uses is to mean “models” in modal logic, as in 𝔐, 𝑤 ⊩ 𝑃 .
U+27E1 WHITE CONCAVE-SIDED DIAMOND never modal operator
U+27E2 WHITE CONCAVE-SIDED DIAMOND WITH LEFTWARDS TICK was never modal operator
U+27E3 WHITE CONCAVE-SIDED DIAMOND WITH RIGHTWARDS TICK will never be modal operator
U+25A4 WHITE SQUARE WITH LEFTWARDS TICK was always modal operator
U+25A5 WHITE SQUARE WITH RIGHTWARDS TICK will always be modal operator
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 $\displaystyle{ p }$$\displaystyle{ q \equiv \Box(p\rightarrow q) }$. See here for an image of glyph. Added to Unicode 3.2.0.
U+2A07 TWO LOGICAL AND OPERATOR

## Usage in various countries

### 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%".