# List of logic symbols

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

material implication implies; if ... then propositional logic, Heyting algebra ${\displaystyle A\Rightarrow B}$  is false when A is true and B is false but true otherwise.[2][circular reference]

${\displaystyle \rightarrow }$  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+21D2

U+2192

U+2283
&#8658;

&#8594;

&#8835;
&rArr;

&rarr;

&sup;
${\displaystyle \Rightarrow }$ \Rightarrow
${\displaystyle \to }$ \to or \rightarrow
${\displaystyle \supset }$ \supset
${\displaystyle \implies }$ \implies

material equivalence if and only if; iff; means the same as propositional logic ${\displaystyle A\Leftrightarrow B}$  is true only if both A and B are false, or both A and B are true. ${\displaystyle x+5=y+2\Leftrightarrow x+3=y}$  U+21D4

U+2261

U+27F7
&#8660;

&#8801;

&#10231;
&hArr;

&equiv;

&#10231;
${\displaystyle \Leftrightarrow }$ \Leftrightarrow
${\displaystyle \equiv }$ \equiv
${\displaystyle \leftrightarrow }$ \leftrightarrow
${\displaystyle \iff }$ \iff
¬
˜
!
negation not propositional logic 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+00AC

U+02DC

U+0021
&#172;

&#732;

&#33;
&not;

&tilde;

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

${\displaystyle \sim }$ \sim

${\displaystyle \mathbb {D} }$
Domain of discourse Domain of predicate Predicate (mathematical logic) ${\displaystyle \mathbb {D} \mathbb {:} \mathbb {R} }$  U+1D53B &#120123; &Dopf; \mathbb{D}

·
&
logical conjunction and propositional logic, Boolean algebra The statement AB 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+2227

U+00B7

U+0026
&#8743;

&#183;

&#38;
&and;

&middot;

&amp;
${\displaystyle \wedge }$ \wedge or \land
${\displaystyle \cdot }$ \cdot ${\displaystyle \&}$ \&[3]

+
logical (inclusive) disjunction or propositional logic, Boolean algebra The statement AB 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+2228

U+002B

U+2225
&#8744;

&#43;

&#8741;
&or;

&plus;

&parallel;

${\displaystyle \lor }$ \lor or \vee

${\displaystyle \parallel }$ \parallel

${\displaystyle \nleftrightarrow }$

exclusive disjunction xor; either ... or propositional logic, Boolean algebra The statement A ${\displaystyle \nleftrightarrow }$  B is true when either A or B, but not both, are true. AB means the same. A) ${\displaystyle \nleftrightarrow }$  A is always true, and A ${\displaystyle \nleftrightarrow }$  A always false, if vacuous truth is excluded. U+2295

U+22BB

U+2262

&#8853;

&#8891;

&#8802;

&oplus;

&veebar;

&nequiv;

${\displaystyle \oplus }$ \oplus

${\displaystyle \veebar }$ \veebar

${\displaystyle \not \equiv }$ \not\equiv

T
1
Tautology top, truth, full clause propositional logic, Boolean algebra, first-order logic The statement is unconditionally true. ⊤(A) ⇒ A is always true. U+22A4

U+25A0

&#8868;

&top;

${\displaystyle \top }$ \top

F
0
Contradiction bottom, falsum, falsity, empty clause propositional logic, Boolean algebra, first-order logic The statement ⊥ is unconditionally false. (The symbol ⊥ may also refer to perpendicular lines.) ⊥(A) ⇒ A is always false. U+22A5

U+25A1

&#8869;

&perp;

${\displaystyle \bot }$ \bot

()
universal quantification for all; for any; for each first-order logic ∀ xP(x) or (xP(x) means P(x) is true for all x. ${\displaystyle \forall n\in \mathbb {N} :n^{2}\geq n.}$  U+2200

&#8704;

&forall;

${\displaystyle \forall }$ \forall
existential quantification there exists first-order logic ∃ x: P(x) means there is at least one x such that P(x) is true. ${\displaystyle \exists n\in \mathbb {N} :}$  n is even. U+2203 &#8707; &exist; ${\displaystyle \exists }$ \exists
∃!
uniqueness quantification there exists exactly one first-order logic ∃! x: P(x) means there is exactly one x such that P(x) is true. ${\displaystyle \exists !n\in \mathbb {N} :n+5=2n.}$  U+2203 U+0021 &#8707; &#33; &exist;! ${\displaystyle \exists !}$ \exists !

:⇔
definition is defined as everywhere x ≔ y or x ≡ y means x is defined to be another name for y (but note that ≡ can also mean other things, such as congruence).

P :⇔ Q means P is defined to be logically equivalent to Q.
${\displaystyle \cosh x:={\frac {e^{x}+e^{-x}}{2}}}$

A XOR B :⇔ (A ∨ B) ∧ ¬(A ∧ B)
U+2254 (U+003A U+003D)

U+2261

U+003A U+229C
&#8788; (&#58; &#61;)

&#8801;

&#8860;

&coloneq;

&equiv;

&hArr;

${\displaystyle :=}$ :=

${\displaystyle \equiv }$ \equiv

${\displaystyle :\Leftrightarrow }$ :\Leftrightarrow

( )
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. U+0028 U+0029 &#40; &#41; &lpar;

&rpar;

${\displaystyle (~)}$  ( )
turnstile proves propositional logic, first-order logic xy means x proves (syntactically entails) y (AB) ⊢ (¬B → ¬A) U+22A2 &#8866; &vdash; ${\displaystyle \vdash }$ \vdash
double turnstile models propositional logic, first-order logic xy means x models (semantically entails) y (AB) ⊨ (¬B → ¬A) U+22A8 &#8872; &vDash; ${\displaystyle \vDash }$ \vDash, \models

## Advanced and rarely used logical symbols

These symbols are sorted by their Unicode value:

Symbol Name Read as Category Explanation Examples Unicode
value
HTML
value
(decimal)
HTML
entity
(named)
LaTeX
symbol
̅
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+0305

|
UPWARDS ARROW
VERTICAL LINE
Sheffer stroke, the sign for the NAND operator (negation of conjunction). U+2191
U+007C
DOWNWARDS ARROW Peirce Arrow, the sign for the NOR operator (negation of disjunction). U+2193
CIRCLED DOT OPERATOR the sign for the XNOR operator (negation of exclusive disjunction). U+2299
COMPLEMENT U+2201
THERE DOES NOT EXIST strike out existential quantifier, same as "¬∃"[4] U+2204
THEREFORE Therefore[4] U+2234
BECAUSE because[4] U+2235
MODELS is a model of (or "is a valuation satisfying")[4] U+22A7
TRUE is true of U+22A8
DOES NOT PROVE negated ⊢, the sign for "does not prove" TP says "P is not a theorem of T"[4] U+22AC
NOT TRUE is not true of U+22AD
DAGGER it is true that ... Affirmation operator U+2020
NAND NAND operator U+22BC
NOR NOR operator U+22BD
WHITE DIAMOND modal operator for "it is possible that", "it is not necessarily not" or rarely "it is not probably not" (in most modal logics it is defined as "¬◻¬")[4] U+25C7
STAR OPERATOR usually used for ad-hoc operators U+22C6

UP TACK
DOWNWARDS ARROW
Webb-operator or Peirce arrow, the sign for NOR. Confusingly, "⊥" is also the sign for contradiction or absurdity.[4] U+22A5
U+2193
REVERSED NOT SIGN U+2310

TOP LEFT CORNER
TOP RIGHT CORNER
corner quotes, also called "Quine quotes"; for quasi-quotation, i.e. quoting specific context of unspecified ("variable") expressions;[5] also used for denoting Gödel number;[6] 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. ) U+231C
U+231D

WHITE MEDIUM SQUARE
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); also as empty clause (alternatives: ${\displaystyle \emptyset }$  and ⊥) U+25FB
U+25A1
LEFT AND RIGHT TACK semantic equivalent U+27DB
WHITE CONCAVE-SIDED DIAMOND never modal operator U+27E1
WHITE CONCAVE-SIDED DIAMOND WITH LEFTWARDS TICK was never modal operator U+27E2
WHITE CONCAVE-SIDED DIAMOND WITH RIGHTWARDS TICK will never be modal operator U+27E3
WHITE SQUARE always modal operator U+25A1
WHITE SQUARE WITH LEFTWARDS TICK was always modal operator U+25A4
WHITE SQUARE WITH RIGHTWARDS TIC will always be modal operator U+25A5
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 ${\displaystyle p}$ ${\displaystyle q\equiv \Box (p\rightarrow q)}$ , the corresponding LaTeX macro is \strictif. See here for an image of glyph. Added to Unicode 3.2.0. U+297D
TWO LOGICAL AND OPERATOR U+2A07

## Usage in various countries

### Poland and Germany

As of 2014 in Poland, the universal quantifier is sometimes written ${\displaystyle \wedge }$ , and the existential quantifier as ${\displaystyle \vee }$ .[7][8] The same applies for Germany.[9][10]

### 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%".

## References

1. ^ "Named character references". HTML 5.1 Nightly. W3C. Retrieved 9 September 2015.
2. ^
3. ^ Although this character is available in LaTeX, the MediaWiki TeX system does not support it.
4. "Comprehensive List of Logic Symbols". Math Vault. 2020-04-06. Retrieved 2020-08-20.
5. ^ Quine, W.V. (1981): Mathematical Logic, §6
6. ^ Hintikka, Jaakko (1998), The Principles of Mathematics Revisited, Cambridge University Press, p. 113, ISBN 9780521624985.
7. ^ "Kwantyfikator ogólny". 2 October 2017 – via Wikipedia.[circular reference]
8. ^ "Kwantyfikator egzystencjalny". 23 January 2016 – via Wikipedia.[circular reference]
9. ^ "Quantor". 21 January 2018 – via Wikipedia.[circular reference]
10. ^ Hermes, Hans. Einführung in die mathematische Logik: klassische Prädikatenlogik. Springer-Verlag, 2013.