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.

Basic logic symbols

Symbol
Name Explanation Examples Unicode
Value
HTML
Entity
LaTeX
symbol
Read as
Category




material implication AB 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  ⇒  x2 = 4 is true, but x2 = 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 AB 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
&and;

&amp;
\wedge or \land
\&[1]
and
propositional logic, Boolean algebra


+

ǀǀ
logical (inclusive) disjunction The statement AB 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 &or; \lor or \vee
or
propositional logic, Boolean algebra



exclusive disjunction The statement AB is true when either A or B, but not both, are true. A B means the same. A) ⊕ A is always true, AA is always false. U+2295

U+22BB
&oplus; \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 &perp; F \bot
bottom, falsum
propositional logic, Boolean algebra


()
universal quantification ∀ xP(x) or (xP(x) means P(x) is true for all x. ∀ n ∈ : n2 ≥ n. U+2200 &forall; \forall
for all; for any; for each
first-order logic
existential quantification ∃ x: P(x) means there is at least one x such that P(x) is true. ∃ n ∈ : n is even. U+2203 &exist; \exists
there exists
first-order logic
∃!
uniqueness quantification ∃! x: P(x) means there is exactly one x such that P(x) is true. ∃! n ∈ : n + 5 = 2n. U+2203 U+0021 &exist; ! \exists !
there exists exactly one
first-order 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;

&hArr;
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). AB ¬B → ¬A U+22A2 &#8866; \vdash
provable
propositional logic, first-order logic
double turnstile xy means x semantically entails y AB ⊨ ¬B → ¬A U+22A8 &#8872; \models
entails
propositional logic, first-order 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:Unsubst||date=__DATE__ |$B=

{{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[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 "A|B" or "¬(A & B)". See also Unicode Template:Unichar.
  • Template:Unichar, used as abbreviation for standard numerals (Typographical Number Theory). For example, using HTML style "" 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 and Template:Unichar: corner quotes, also called "Quine quotes"; for quasi-quotation, i.e. quoting specific context of unspecified ("variable") expressions;[2] also the standard symbol{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B=

{{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[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. )

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.

Poland

Template:As of in Poland, the universal quantifier is sometimes written and the existential quantifier as .

See also

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Notes

  1. Although this character is available in LaTeX, the MediaWiki TeX system doesn't support this character.
  2. Quine, W.V. (1981): Mathematical Logic, §6

External links

Template:Common logical symbols Template:Logic