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 
⇒ → ⊃ 
\Rightarrow
\to \supset \implies 
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 
⇔ ≡ ↔ 
\Leftrightarrow
\equiv \leftrightarrow \iff 
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 
¬ ˜ ~ 
\lnot or \neg
\sim 
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 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 
:= : ≡ ⇔ 
:=
\equiv \Leftrightarrow 
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:
 U+00B7 · middle dot, an outdated way for denoting AND^{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 U+22C5 ⋅ dot operator.
 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 "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)"
 U+2191 ↑ upwards arrow or U+007C  vertical line: Sheffer stroke, the sign for the NAND operator.
 U+2201 ∁ complement
 U+2204 ∄ there does not exist: strike out existential quantifier same as "¬∃"
 U+2234 ∴ therefore
 U+2235 ∵ because
 U+22A7 ⊧ models: is a model of
 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"
 U+22AD ⊭ not true: is not true of
 U+22BC ⊼ nand: another NAND operator, can also be rendered as ∧
 U+22BD ⊽ nor: another NOR operator, can also be rendered as V
 U+22C4 ⋄ diamond operator: 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 "¬◻¬")
 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.
 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;^{2} also the standard symbol^{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. )
 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).
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.
 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.
Poland
As of 2014^{[update]} in Poland, the universal quantifier is sometimes written and the existential quantifier as .
See also
 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


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