List of logic symbols
Appearance
In
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 (hexadecimal) 
HTML codes 
LaTeX symbol 
Logic Name  Read as  Category  Explanation  Examples 

⇒
→ ⊃ 
U+21D2 U+2192 U+2283 
⇒ → ⊃ ⇒ 
\Rightarrow
\implies \to or \rightarrow \supset 
material conditional (material implication)  implies, if P then Q, it is not the case that P and not Q 
Boolean algebra, Heyting algebra

is false when A is true and B is false but true otherwise. may mean the same as (the symbol may also indicate the domain and codomain of a table of mathematical symbols ).may mean the same as (the symbol may also mean superset ).

is true, but is in general false
(since x could be −2). 
⇔
↔ ≡ 
U+21D4 U+2194 U+2261 
⇔ ↔ ≡ ⇔ 
\Leftrightarrow \iff \leftrightarrow \equiv 
material biconditional (material equivalence)

if and only if, iff, xnor  Boolean algebra

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 , depends on the author’s style.


¬
~ ! 
U+00AC U+007E U+0021 
¬ ˜ ! ¬ 
\lnot or \neg \sim 
negation  not  Boolean algebra

The statement is true if and only if A is false. A slash placed through another operator is the same as placed in front. 

∧
· & 
U+2227 U+00B7 U+0026 
∧ · & ∧ 
\wedge or \land
\cdot \&^{[2]} 
logical conjunction  and  Boolean algebra

The statement A ∧ B is true if A and B are both true; otherwise, it is false.  
∨
+ ∥ 
U+2228 U+002B U+2225 
∨ + ∥ ∨ 
\lor or \vee \parallel 
logical (inclusive) disjunction  or  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+2295 U+22BB U+21AE U+2262 
⊕ ⊻ ↮ ≢ ⊕ 
\oplus \veebar \not\equiv 
exclusive disjunction  xor, either ... or ... (but not both) 
Boolean algebra

The statement is true when either A or B, but not both, are true. This is equivalent to ¬(A ↔ B), hence the symbols and . 

⊤
T 1 
U+22A4 
⊤

\top 
true (tautology)  top, truth, tautology, verum, full clause  Boolean algebra, firstorder logic

denotes a proposition that is always true.  The proposition is always true since at least one of the two is unconditionally true.

⊥
F 0 
U+22A5 
⊥
⊥ 
\bot 
false (contradiction)  bottom, falsity, contradiction, falsum, empty clause  Boolean algebra, firstorder logic

denotes a proposition that is always false. The symbol ⊥ may also refer to perpendicular lines. 
The proposition is always false since at least one of the two is unconditionally false.

∀
() 
U+2200 
∀
∀ 
\forall 
universal quantification  given any, for all, for every, for each, for any  firstorder logic  or says “given any , has property .” 

∃

U+2203  ∃
∃ 
\exists  existential quantification  there exists, for some  firstorder logic  says “there exists an (at least one) such that has property .”  n is even.

∃!

U+2203 U+0021  ∃ !
∃! 
\exists !  uniqueness quantification  there exists exactly one  firstorder logic (abbreviation)  says “there exists exactly one such that has property .” Only and are part of formal logic. is an abbreviation for 

( )

U+0028 U+0029  ( )
( 
( )  precedence grouping

parentheses; brackets  almost all logic syntaxes, as well as metalanguage  Perform the operations inside the parentheses first.  (8 ÷ 4) ÷ 2 = 2 ÷ 2 = 1, but 8 ÷ (4 ÷ 2) = 8 ÷ 2 = 4.

U+1D53B  𝔻
𝔻 
\mathbb{D}  domain of discourse  domain of discourse  metalanguage (firstorder logic semantics)  
⊢

U+22A2  ⊢
⊢ 
\vdash  turnstile  syntactically entails (proves)  metalanguage (metalogic)  says “ is a theorem of ”. In other words, proves via a deductive system. 

⊨

U+22A8  ⊨
⊨ 
\vDash, \models  double turnstile  semantically entails  metalanguage (metalogic)  says “in every model, it is not the case that is true and is false”. 
(eg. by using truth tables ) 
≡
⟚ ⇔ 
U+2261 U+27DA U+21D4 
≡
— 
\equiv \Leftrightarrow 
logical equivalence  is logically equivalent to  metalanguage (metalogic)  It’s when and . Whether a symbol means a material biconditional or a logical equivalence , depends on the author’s style.


⊬

U+22AC  ⊬\nvdash  does not syntactically entail (does not prove)  metalanguage (metalogic)  says “ is not a theorem of ”. In other words, is not derivable from via a deductive system. 

⊭

U+22AD  ⊭\nvDash  does not semantically entail  metalanguage (metalogic)  says “ does not guarantee the truth of ”. In other words, does not make true. 

□

U+25A1  \Box  necessity (in 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. 
says “it is necessary that everything has property ”
 
◇

U+25C7  \Diamond  possibility (in 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”).  says “it is possible that something has property ”
 
∴

U+2234  ∴\therefore  therefore  therefore  metalanguage  abbreviation for “therefore”.  
∵

U+2235  ∵\because  because  because  metalanguage  abbreviation for “because”.  
≔
≜ ≝ 
U+2254 U+225C U+225D 
≔
≔ 
:=
\triangleq
\scriptscriptstyle \mathrm{def}}{=} 
definition  is defined as  metalanguage  means "from now on, is defined to be another name for ." This is a statement in the metalanguage, not the object language. The notation may occasionally be seen in physics, meaning the same as . 
Advanced or rarely used logical symbols
The following symbols are either advanced and contextsensitive or very rarely used:
Symbol  Unicode value (hexadecimal) 
HTML value (decimal) 
HTML entity (named) 
LaTeX symbol 
Logic Name  Read as  Category  Explanation 

⥽

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 ⥽ .  
̅

U+0305  combining overline  Used format for denoting Gödel numbers. Using HTML style “4̅” is an abbreviation for the standard numeral “SSSS0”.
It may also denote a negation (used primarily in electronics).  
⌜
⌝ 
U+231C U+231D 
\ulcorner
\urcorner 
top left corner top right corner 
Corner quotes, also called “Quine quotes”; for quasiquotation, i.e. quoting specific context of unspecified (“variable”) expressions;^{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+2204  \nexists  there does not exist  Strike out existential quantifier. “¬∃” is recommended instead. ^{[by whom?]}  
↑
 
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 , a sign for the NOR operator (negation of disjunction).  
⊼

U+22BC  NAND  A new symbol made specifically for the NAND operator.  
⊽

U+22BD  NOR  A new symbol made specifically for the NOR operator.  
⊙

U+2299  \odot  circled dot operator  A sign for the XNOR operator (material biconditional and XNOR are the same operation).  
⟛

U+27DB  left and right tack  “Proves and is proved by”.  
⊧

U+22A7  models  “Is a model of” or “is a valuation satisfying”.
 
⊩

U+22A9  forces  One of this symbol’s uses is to mean “truthmakes” in the truthmaker theory of truth. It is also used to mean “forces” in the set theory method of forcing.  
⟡

U+27E1  white concavesided diamond  never  modal operator  
⟢

U+27E2  white concavesided diamond with leftwards tick  was never  modal operator  
⟣

U+27E3  white concavesided 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+22C6  star operator  May sometimes be used for adhoc operators.  
⌐

U+2310  reversed not sign  
⨇

U+2A07  two logical AND operator 
See also
 Glossary of logic
 Józef Maria Bocheński
 List of notation used in Principia Mathematica
 List of mathematical symbols
 Logic alphabet, a suggested set of logical symbols
 Logic gate § Symbols
 Logical connective
 Mathematical operators and symbols in Unicode
 Nonlogical symbol
 Polish notation
 Truth function
 Truth table
 Wikipedia:WikiProject Logic/Standards for notation
References
 ^ "Named character references". HTML 5.1 Nightly. W3C. Retrieved 9 September 2015.
 ^ Although this character is available in LaTeX, the MediaWiki TeX system does not support it.
 ^ Quine, W.V. (1981): Mathematical Logic, §6
 ISBN 9780521624985.
Further reading
 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
 Named character entities in HTML 4.0