List of logic symbols

Jump to navigation Jump to search

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 Explanation Examples Unicode
Value
(hexadecimal)
HTML
Value
(decimal)
HTML
Entity
(named)
LaTeX
symbol
Read as
Category

material implication ${\displaystyle A\Rightarrow B}$ is true if and only if ${\displaystyle B}$ can be true and ${\displaystyle A}$ can be false but not vice versa .

${\displaystyle \rightarrow }$ may mean the same as ${\displaystyle \Rightarrow }$ (the symbol may also indicate the domain and codomain of a function; see table of mathematical symbols).

${\displaystyle \supset }$ may mean the same as ${\displaystyle \Rightarrow }$ (the symbol may also mean superset).
${\displaystyle x=2\Rightarrow x^{2}=4}$ is true, but ${\displaystyle x^{2}=4\Rightarrow x=2}$ is in general false (since ${\displaystyle x}$ could be −2). U+21D2

U+2192

U+2283
&#8658;

&#8594;

&#8835;
&rArr;

&rarr;

&sup;
${\displaystyle \Rightarrow }$\Rightarrow
${\displaystyle \to }$\to or \rightarrow
${\displaystyle \supset }$\supset
${\displaystyle \implies }$\implies
implies; if .. then
propositional logic, Heyting algebra

material equivalence ${\displaystyle A\Leftrightarrow B}$ is true only if both ${\displaystyle A}$ and ${\displaystyle B}$ are false, or both ${\displaystyle A}$ and ${\displaystyle B}$ are true. ${\displaystyle x+5=y+2\Leftrightarrow x+3=y}$ U+21D4

U+2261

U+2194
&#8660;

&#8801;

&#8596;
&hArr;

&equiv;

&harr;
${\displaystyle \Leftrightarrow }$\Leftrightarrow
${\displaystyle \equiv }$\equiv
${\displaystyle \leftrightarrow }$\leftrightarrow
${\displaystyle \iff }$\iff
if and only if; iff; means the same as
propositional logic
¬

˜

!
negation The statement ${\displaystyle \lnot A}$ is true if and only if ${\displaystyle A}$ is false.

A slash placed through another operator is the same as ${\displaystyle \neg }$ placed in front.
${\displaystyle \neg (\neg A)\Leftrightarrow A}$
${\displaystyle x\neq y\Leftrightarrow \neg (x=y)}$
U+00AC

U+02DC

U+0021
&#172;

&#732;

&#33;
&not;

&tilde;

&excl;
${\displaystyle \neg }$\lnot or \neg
${\displaystyle \sim }$\sim
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+00B7

U+0026
&#8743;

&#183;

&#38;
&and;

&middot;

&amp;
${\displaystyle \wedge }$\wedge or \land
${\displaystyle \&}$\&[2]
and
propositional logic, Boolean algebra

+

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

U+002B

U+2225
&#8744;

&#43;

&#8741;
&or;

${\displaystyle \lor }$\lor or \vee
${\displaystyle \parallel }$\parallel
or
propositional logic, Boolean algebra

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

U+22BB
&#8853;

&#8891;
&oplus;

${\displaystyle \oplus }$\oplus
${\displaystyle \veebar }$\veebar
xor
propositional logic, Boolean algebra

T

1
Tautology The statement ⊤ is unconditionally true. A ⇒ ⊤ is always true. U+22A4

&#8868;

${\displaystyle \top }$\top
top, verum
propositional logic, Boolean algebra

F

0
Contradiction The statement ⊥ is unconditionally false. (The symbol ⊥ may also refer to perpendicular lines.) ⊥ ⇒ A is always true. U+22A5

&#8869;

&perp;

${\displaystyle \bot }$\bot
bottom, falsum, falsity
propositional logic, Boolean algebra

()
universal quantification ∀ xP(x) or (xP(x) means P(x) is true for all x. ∀ n ∈ ℕ: n2 ≥ n. U+2200

&#8704;

&forall;

${\displaystyle \forall }$\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: n is even. U+2203 &#8707; &exist; ${\displaystyle \exists }$\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: n + 5 = 2n. U+2203 U+0021 &#8707; &#33; ${\displaystyle \exists !}$\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
&#8788; (&#58; &#61;)

&#8801;

&#8860;

&equiv;

&hArr;
${\displaystyle :=}$:=
${\displaystyle \equiv }$\equiv
${\displaystyle :\Leftrightarrow }$:\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 &#40; &#41; ${\displaystyle (~)}$ ( )
parentheses, brackets
everywhere
Turnstile xy means y is provable from x (in some specified formal system). AB ⊢ ¬B → ¬A U+22A2 &#8866; ${\displaystyle \vdash }$\vdash
provable
propositional logic, first-order logic
double turnstile xy means x semantically entails y AB ⊨ ¬B → ¬A U+22A8 &#8872; ${\displaystyle \vDash }$\vDash, \models
entails
propositional logic, first-order 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,[3] 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 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, "A V B" says the Gödel number of "(A V B)".
• Overline is also an outdated way for denoting negation, still in use in electronics: for example, "A V B" is the same as "¬(A V B)".
• U+2191 UPWARDS ARROW or U+007C | VERTICAL LINE: Sheffer stroke, the sign for the NAND operator.
• U+2193 DOWNWARDS ARROW Peirce Arrow, the sign for the NOR operator.
• U+2201 COMPLEMENT
• U+2204 THERE DOES NOT EXIST: strike out existential quantifier same as "¬∃"
• U+2234 THEREFORE: Therefore
• U+2235 BECAUSE: 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 TP says "P is not a theorem of T"
• U+22AD NOT TRUE: is not true of
• U+22BC NAND: NAND operator. In HTML, it can also be produced by <span style="text-decoration: overline">&and;</span>:
• U+22BD NOR: NOR operator. In HTML, it can also be produced by <span style="text-decoration: overline">&or;</span>:
• 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 "¬◻¬")
• U+22C6 STAR OPERATOR: usually used for ad-hoc operators
• U+22A5 UP TACK or U+2193 DOWNWARDS ARROW: Webb-operator 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 quasi-quotation, 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 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: ${\displaystyle \emptyset }$ and ⊥).

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 CONCAVE-SIDED DIAMOND
• U+27E2 WHITE CONCAVE-SIDED DIAMOND WITH LEFTWARDS TICK: modal operator for was never
• U+27E3 WHITE CONCAVE-SIDED 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 ${\displaystyle p}$${\displaystyle q\equiv \Box (p\rightarrow q)}$, 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 in Poland, the universal quantifier is sometimes written ${\displaystyle \wedge }$ and the existential quantifier as ${\displaystyle \vee }$.[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%".

References

1. ^ "Named character references". HTML 5.1 Nightly. W3C. Retrieved 9 September 2015.
2. ^ Although this character is available in LaTeX, the MediaWiki TeX system doesn't support this character.
3. ^ Brody, Baruch A. (1973), Logic: theoretical and applied, Prentice-Hall, p. 93, ISBN 9780135401460, We turn now to the second of our connective symbols, the centered dot, which is called the conjunction sign.
4. ^ Quine, W.V. (1981): Mathematical Logic, §6
5. ^ Hintikka, Jaakko (1998), The Principles of Mathematics Revisited, Cambridge University Press, p. 113, ISBN 9780521624985.
6. ^ "Kwantyfikator ogólny". 2 October 2017 – via Wikipedia.
7. ^ "Kwantyfikator egzystencjalny". 23 January 2016 – via Wikipedia.
8. ^ "Quantor". 21 January 2018 – via Wikipedia.
9. ^ Hermes, Hans. Einführung in die mathematische Logik: klassische Prädikatenlogik. Springer-Verlag, 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.
Retrieved from "https://en.wikipedia.org/w/index.php?title=List_of_logic_symbols&oldid=865675487"
This content was retrieved from Wikipedia : http://en.wikipedia.org/wiki/List_of_logic_symbols
This page is based on the copyrighted Wikipedia article "List of logic symbols"; it is used under the Creative Commons Attribution-ShareAlike 3.0 Unported License (CC-BY-SA). You may redistribute it, verbatim or modified, providing that you comply with the terms of the CC-BY-SA