List of logic symbols
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In logic, a set of symbols is commonly used to express logical representation. As logicians are familiar with these symbols, they are not explained each time they are used. So, for students of logic, 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, and the fourth column gives a short example.
Be aware that, outside of logic, different symbols have the same meaning, and the same symbol has, depending on the context, different meanings.
Contents
Basic logic symbols
Symbol

Name  Explanation  Examples  Unicode Value 
HTML Entity 
LaTeX symbol 

Read as  
Category  
⇒
→ ⊃ 
material implication  A ⇒ B is true only in the case that either A is false or B is true, or both. → 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). 
x = 2 ⇒ x^{2} = 4 is true, but x^{2} = 4 ⇒ x = 2 is in general false (since x could be −2).  U+21D2 U+2192 U+2283 
⇒ → ⊃ 

implies; if .. then  
propositional logic, Heyting algebra  
⇔
≡ ↔ 
material equivalence  A ⇔ B is true only if both A and B are false, or both A and B are true.  x + 5 = y + 2 ⇔ x + 3 = y  U+21D4 U+2261 U+2194 
⇔ ≡ ↔ 

if and only if; iff; means the same as  
propositional logic  
¬
˜ ! 
negation  The statement ¬A is true if and only if A is false. A slash placed through another operator is the same as "¬" placed in front. 
¬(¬A) ⇔ A x ≠ y ⇔ ¬(x = y) 
U+00AC U+02DC 
¬ ˜ ~ 

not  
propositional logic  
∧
• & 
logical conjunction  The statement A ∧ B is true if A and B are both true; else it is false.  n < 4 ∧ n >2 ⇔ n = 3 when n is a natural number.  U+2227 U+0026 
∧ & 
\wedge or \land \&^{[1]} 
and  
propositional logic, Boolean algebra  
∨
+ ǀǀ 
logical (inclusive) disjunction  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  ∨  \lor or \vee 
or  
propositional logic, Boolean algebra  
⊕ ⊻ 
exclusive disjunction  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, A ⊕ A is always false.  U+2295 U+22BB 
⊕  \oplus \veebar 
xor  
propositional logic, Boolean algebra  
⊤ T 1 
Tautology  The statement ⊤ is unconditionally true.  A ⇒ ⊤ is always true.  U+22A4  T  \top 
top, verum  
propositional logic, Boolean algebra  
⊥ F 0 
Contradiction  The statement ⊥ is unconditionally false.  ⊥ ⇒ A is always true.  U+22A5  ⊥ F  \bot 
bottom, falsum  
propositional logic, Boolean algebra  
∀
() 
universal quantification  ∀ x: P(x) or (x) P(x) means P(x) is true for all x.  ∀ n ∈ ℕ: n^{2} ≥ n.  U+2200  ∀  \forall 
for all; for any; for each  
firstorder logic  
∃

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

uniqueness quantification  ∃! x: P(x) means there is exactly one x such that P(x) is true.  ∃! n ∈ ℕ: n + 5 = 2n.  U+2203 U+0021  ∃ !  \exists ! 
there exists exactly one  
firstorder logic  
:=
≡ :⇔ 
definition  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. 
cosh x := (1/2)(exp x + exp (−x)) A XOR B :⇔ (A ∨ B) ∧ ¬(A ∧ B) 
U+2254 (U+003A U+003D) U+2261 U+003A U+229C 
:= : ≡ ⇔ 

is defined as  
everywhere  
( )

precedence grouping  Perform the operations inside the parentheses first.  (8 ÷ 4) ÷ 2 = 2 ÷ 2 = 1, but 8 ÷ (4 ÷ 2) = 8 ÷ 2 = 4.  U+0028 U+0029  ( )  ( ) 
parentheses, brackets  
everywhere  
⊢

Turnstile  x ⊢ y means y is provable from x (in some specified formal system).  A → B ⊢ ¬B → ¬A  U+22A2  ⊢  \vdash 
provable  
propositional logic, firstorder logic  
⊨

double turnstile  x ⊨ y means x semantically entails y  A → B ⊨ ¬B → ¬A  U+22A8  ⊨  \models 
entails  
propositional logic, firstorder logic 
Advanced and rarely used logical symbols
These symbols are sorted by their Unicode value:
 Template:Unichar, an outdated way for denoting AND{{ safesubst:#invoke:Unsubstdate=__DATE__ $B=
{{#invoke:Category handlermain}}{{#invoke:Category handlermain}}^{[citation needed]} }}, still in use in electronics; for example "A·B" is the same as "A&B"
 ·: Center dot with a line above it. Outdated way for denoting NAND, for example "A·B" is the same as "A NAND B" or "AB" or "¬(A & B)". See also Unicode Template:Unichar.
 Template:Unichar, 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 "AVB" says the Gödel number of "(AVB)"
 Overline is also an outdated way for denoting negation, still in use in electronics; for example "AVB" is the same as "¬(AVB)"
 Template:Unichar or Template:Unichar: Sheffer stroke, the sign for the NAND operator.
 Template:Unichar
 Template:Unichar: strike out existential quantifier same as "¬∃"
 Template:Unichar
 Template:Unichar
 Template:Unichar: is a model of
 Template:Unichar: is true of
 Template:Unichar: negated ⊢, the sign for "does not prove", for example T ⊬ P says "P is not a theorem of T"
 Template:Unichar: is not true of
 Template:Unichar: another NAND operator, can also be rendered as ∧
 Template:Unichar: another NOR operator, can also be rendered as V
 Template:Unichar: 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 "¬◻¬")
 Template:Unichar: usually used for adhoc operators
 Template:Unichar or Template:Unichar: Webboperator or Peirce arrow, the sign for NOR. Confusingly, "⊥" is also the sign for contradiction or absurdity.
 Template:Unichar and Template:Unichar: corner quotes, also called "Quine quotes"; for quasiquotation, i.e. quoting specific context of unspecified ("variable") expressions;^{[2]} also the standard symbol{{ safesubst:#invoke:Unsubstdate=__DATE__ $B=
{{#invoke:Category handlermain}}{{#invoke:Category handlermain}}^{[citation needed]} }} used for denoting Gödel number; 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. )
 Template:Unichar or Template:Unichar: 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).
Note that the following operators are rarely supported by natively installed fonts. If you wish to use these in a web page, you should always embed the necessary fonts so the page viewer can see the web page without having the necessary fonts installed in their computer.
 Template:Unichar
 Template:Unichar: modal operator for was never
 Template:Unichar: modal operator for will never be
 Template:Unichar: modal operator for was always
 Template:Unichar: modal operator for will always be
 Template:Unichar: 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.
Poland
Template:As of in Poland, the universal quantifier is sometimes written and the existential quantifier as .
See also
{{#invoke:Portalportal}}
 List of notation used in Principia Mathematica
 List of mathematical symbols
 Logic alphabet, a suggested set of logical symbols
 Logical connective
 Mathematical operators and symbols in Unicode
 Polish notation
Notes
 ↑ Although this character is available in LaTeX, the MediaWiki TeX system doesn't support this character.
 ↑ Quine, W.V. (1981): Mathematical Logic, §6
External links
 Named character entities in HTML 4.0