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

⇒
→ ⊃ 
U+21D2 U+2192 U+2283 
⇒ → ⊃ 
⇒ → ⊃ 
\Rightarrow
\to or \rightarrow \supset \implies 
material implication  implies; if ... then  propositional logic, Heyting algebra  is false when A is true and B is false but true otherwise.^{[2]}^{[circular reference]} may mean the same as (the symbol may also indicate the domain and codomain of a function; see 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+2261 U+2194 
⇔ ≡ ↔ 
⇔ ≡ ↔ 
\Leftrightarrow \equiv \leftrightarrow \iff 
material equivalence  if and only if; iff; means the same as  propositional logic  is true only if both A and B are false, or both A and B are true.  
¬
˜ ! 
U+00AC U+02DC U+0021 
¬ ˜ ! 
¬ ˜ ! 
\lnot or \neg

negation  not  propositional logic  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+1D53B  𝔻  𝔻  \mathbb{D}  Domain of discourse  Domain of predicate  Predicate (mathematical logic)  
∧
· & 
U+2227 U+00B7 U+0026 
∧ · & 
∧ · & 
\wedge or \land
\cdot \&^{[3]} 
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 
∨ + ∥ 
∨

\lor or \vee

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+21AE U+2295 U+22BB

↮ ⊕ ⊻

⊕

\oplus

exclusive disjunction  xor; either ... or  propositional logic, Boolean algebra  The statement A ↮ B is true when either A or B, but not both, are true. A ⊻ B means the same.  (¬A) ↮ A is always true, and A ↮ A always false, if vacuous truth is excluded. 
⊤
T 1 ■ 
U+22A4 U+25A0 
⊤ 
⊤  \top  Tautology  top, truth, full clause  propositional logic, Boolean algebra, firstorder logic  The statement ⊤ is unconditionally true.  ⊤(A) ⇒ A is always true. 
⊥
F 0 □ 
U+22A5 U+25A1 
⊥ 
⊥ 
\bot  Contradiction  bottom, falsum, falsity, empty clause  propositional logic, Boolean algebra, firstorder logic  The statement ⊥ is unconditionally false. (The symbol ⊥ may also refer to perpendicular lines.)  ⊥(A) ⇒ A is always false. 
∀
() 
U+2200 
∀ 
∀ 
\forall  universal quantification  for all; for any; for each  firstorder logic  ∀ x: P(x) or (x) P(x) means P(x) is true for all x.  
∃

U+2203  ∃  ∃  \exists  existential quantification  there exists  firstorder logic  ∃ x: P(x) means there is at least one x such that P(x) is true.  n is even. 
∃!

U+2203 U+0021  ∃ !  ∃!  \exists !  uniqueness quantification  there exists exactly one  firstorder logic  ∃! x: P(x) means there is exactly one x such that P(x) is true.  
≔
≡ :⇔ 
U+2254 (U+003A U+003D) U+2261 U+003A U+229C 
≔ (: =)

≔

:=
:\Leftrightarrow 
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. 
A XOR B :⇔ (A ∨ B) ∧ ¬(A ∧ B) 
( )

U+0028 U+0029  ( )  (
) 
( )  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+22A2  ⊢  ⊢  \vdash  turnstile  proves  propositional logic, firstorder logic  x ⊢ y means x proves (syntactically entails) y  (A → B) ⊢ (¬B → ¬A) 
⊨

U+22A8  ⊨  ⊨  \vDash, \models  double turnstile  models  propositional logic, firstorder logic  x ⊨ y means x models (semantically entails) y  (A → B) ⊨ (¬B → ¬A) 
Advanced and rarely used logical symbols
These symbols are sorted by their Unicode value:
Symbol  Unicode value (hexadecimal) 
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  \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+2234  ∴\therefore  THEREFORE  Therefore  
∵

U+2235  ∵\because  BECAUSE  because  
⊧

U+22A7  MODELS  is a model of (or "is a valuation satisfying")  
⊨

U+22A8  ⊨\vDash  TRUE  is true of  
⊬

U+22AC  ⊬\nvdash  DOES NOT PROVE  negated ⊢, the sign for "does not prove"  T ⊬ P says "P is not a theorem of T"  
⊭

U+22AD  ⊭\nvDash  NOT TRUE  is not true of  
†

U+2020  DAGGER  it is true that ...  Affirmation operator  
⊼

U+22BC  NAND  NAND operator  
⊽

U+22BD  NOR  NOR operator  
◇

U+25C7  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 "¬◻¬")  
⋆

U+22C6  STAR OPERATOR  usually used for adhoc operators  
⊥
↓ 
U+22A5 U+2193 
UP TACK DOWNWARDS ARROW 
Webboperator 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 quasiquotation, i.e. quoting specific context of unspecified ("variable") expressions;^{[4]} also used for denoting Gödel number;^{[5]} 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+25FB U+25A1 
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: and ⊥)  
⟛

U+27DB  LEFT AND RIGHT TACK  semantic equivalent  
⟡

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+25A1  WHITE SQUARE  always  modal operator  
⟤

U+25A4  WHITE SQUARE WITH LEFTWARDS TICK  was always  modal operator  
⟥

U+25A5  WHITE SQUARE WITH RIGHTWARDS TIC  will always be  modal operator  
⥽

U+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 ⥽ , the corresponding LaTeX macro is \strictif. See here for an image of glyph. Added to Unicode 3.2.0.  
⨇

U+2A07  TWO LOGICAL AND OPERATOR 
Usage in various countries
Poland and Germany
As of 2014^{[update]} in Poland, the universal quantifier is sometimes written , and the existential quantifier as .^{[6]}^{[7]} The same applies for Germany.^{[8]}^{[9]}
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
 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.
 ^ "Material conditional".
 ^ Although this character is available in LaTeX, the MediaWiki TeX system does not support it.
 ^ Quine, W.V. (1981): Mathematical Logic, §6
 ^ Hintikka, Jaakko (1998), The Principles of Mathematics Revisited, Cambridge University Press, p. 113, ISBN 9780521624985.
 ^ "Kwantyfikator ogólny". 2 October 2017 – via Wikipedia.^{[circular reference]}
 ^ "Kwantyfikator egzystencjalny". 23 January 2016 – via Wikipedia.^{[circular reference]}
 ^ "Quantor". 21 January 2018 – via Wikipedia.^{[circular reference]}
 ^ Hermes, Hans. Einführung in die mathematische Logik: klassische Prädikatenlogik. SpringerVerlag, 2013.
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