In logic, a set of symbols is commonly used to express logical representation. The following table lists many common symbols, together with their name, pronunciation, and the related field of mathematics. Additionally, the third column contains an informal definition, the fourth column gives a short example, the fifth and sixth give the Unicode location and name for use in HTML documents.^{[1]} The last column provides the LaTeX symbol.
Basic logic symbols
Symbol  Name  Read as  Category  Explanation  Examples  Unicode value (hexadecimal) 
HTML value (decimal) 
HTML entity (named) 
LaTeX symbol 

⇒
→ ⊃ 
material implication  implies; if ... then  propositional logic, Heyting algebra  is false when is true and 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 could be −2).  U+21D2 U+2192 U+2283 
⇒ → ⊃ 
⇒ → ⊃ 
\Rightarrow
\to or \rightarrow \supset \implies 
⇔
≡ ↔ 
material equivalence  if and only if; iff; means the same as  propositional logic  is true only if both and are false, or both and are true.  U+21D4 U+2261 U+2194 
⇔ ≡ ↔ 
⇔ ≡ ↔ 
\Leftrightarrow \equiv \leftrightarrow \iff  
¬
˜ ! 
negation  not  propositional logic  The statement is true if and only if is false. A slash placed through another operator is the same as placed in front. 
U+00AC U+02DC U+0021 
¬ ˜ ! 
¬ ˜ ! 
\lnot or \neg
 
Domain of discourse  Domain of predicate  Predicate (mathematical logic)  U+1D53B  𝔻  𝔻  \mathbb{D}  
∧
· & 
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+2227 U+00B7 U+0026 
∧ · & 
∧ · & 
\wedge or \land
\cdot \&^{[3]} 
∨
+ ∥ 
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+2228 U+002B U+2225 
∨ + ∥ 
∨

\lor or \vee

⊕ ⊻ ≢ 
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.  U+2295 U+22BB

⊕ ⊻

⊕

\oplus

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

\top 
⊥ F 0 
Contradiction  bottom, falsum, falsity  propositional logic, Boolean algebra  The statement ⊥ is unconditionally false. (The symbol ⊥ may also refer to perpendicular lines.)  ⊥(A) ⇒ A is always false.  U+22A5 
⊥ 
⊥ 
\bot 
∀
() 
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+2200 
∀ 
∀ 
\forall  
∃

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  ∃  ∃  \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+2203 U+0021  ∃ !  ∃!  \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. 
A XOR B :⇔ (A ∨ B) ∧ ¬(A ∧ B) 
U+2254 (U+003A U+003D) U+2261 U+003A U+229C 
≔ (: =)

≔

:=
:\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  ( )  (
) 
( ) 
⊢

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

double turnstile  models  propositional logic, firstorder logic  x ⊨ y means x models (semantically entails) y  (A → B) ⊨ (¬B → ¬A)  U+22A8  ⊨  ⊨  \vDash, \models 
Advanced and rarely used logical symbols
These symbols are sorted by their Unicode value:
 U+0305 ̅ COMBINING OVERLINE, used as abbreviation for standard numerals (Typographical Number Theory). For example, using HTML style "4̅" is a shorthand for the standard numeral "SSSS0".
 Overline is also a rarely used format for denoting Gödel numbers: for example, "A ∨ B" says the Gödel number of "(A ∨ B)".
 Overline is also an outdated^{[according to whom?]} way for denoting negation, still in use in electronics: for example, "A ∨ B" is the same as "¬(A ∨ B)".
 U+2191 ↑ UPWARDS ARROW or U+007C  VERTICAL LINE: Sheffer stroke, the sign for the NAND operator (negation of conjunction).^{[4]}
 U+2193 ↓ DOWNWARDS ARROW Peirce Arrow, the sign for the NOR operator (negation of disjunction).^{[4]}
 U+2299 ⊙ CIRCLED DOT OPERATOR the sign for the XNOR operator (negation of exclusive disjunction).
 U+2201 ∁ COMPLEMENT
 U+2204 ∄ THERE DOES NOT EXIST: strike out existential quantifier, same as "¬∃"^{[4]}
 U+2234 ∴ THEREFORE: Therefore^{[4]}
 U+2235 ∵ BECAUSE: because^{[4]}
 U+22A7 ⊧ MODELS: is a model of (or "is a valuation satisfying")^{[4]}
 U+22A8 ⊨ TRUE: is true of
 U+22AC ⊬ DOES NOT PROVE: negated ⊢, the sign for "does not prove", for example T ⊬ P says "P is not a theorem of T"^{[4]}
 U+22AD ⊭ NOT TRUE: is not true of
 U+2020 † DAGGER: Affirmation operator (read 'it is true that ...')
 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 provable not" (in most modal logics it is defined as "¬◻¬")^{[4]}
 U+22C6 ⋆ STAR OPERATOR: usually used for adhoc operators
 U+22A5 ⊥ UP TACK or U+2193 ↓ DOWNWARDS ARROW: Webboperator or Peirce arrow, the sign for NOR. Confusingly, "⊥" is also the sign for contradiction or absurdity.^{[4]}
 U+2310 ⌐ REVERSED NOT SIGN
 U+231C ⌜ TOP LEFT CORNER and U+231D ⌝ TOP RIGHT CORNER: corner quotes, also called "Quine quotes"; for quasiquotation, 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+25FB ◻ WHITE MEDIUM SQUARE or U+25A1 □ 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
The following operators are rarely supported by natively installed fonts.
 U+27E1 ⟡ WHITE CONCAVESIDED DIAMOND
 U+27E2 ⟢ WHITE CONCAVESIDED DIAMOND WITH LEFTWARDS TICK: modal operator for was never
 U+27E3 ⟣ WHITE CONCAVESIDED DIAMOND WITH RIGHTWARDS TICK: modal operator for will never be
 U+27E4 ⟤ WHITE SQUARE WITH LEFTWARDS TICK: modal operator for was always
 U+27E5 ⟥ WHITE SQUARE WITH RIGHTWARDS TICK: modal operator for will always be
 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 .^{[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%".
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.
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} ^{h} ^{i} "Comprehensive List of Logic Symbols". Math Vault. 20200406. Retrieved 20200820.
 ^ 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.
 ^ "Kwantyfikator egzystencjalny". 23 January 2016 – via Wikipedia.
 ^ "Quantor". 21 January 2018 – via Wikipedia.
 ^ 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