List of logic symbols

This article contains logic symbols. Without proper rendering support, you may see question marks, boxes, or other symbols instead of logic symbols.

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.

Outside logic, different symbols have the same meaning, and the same symbol can have different meanings, depending on the context.

Basic logic symbols

Symbol	Name	Explanation	Examples	Unicode Value (hexdecimal)	HTML Value (decimal)	HTML Entity (named)	LaTeX symbol
	Read as
	Category
⇒ → ⊃	material implication	A ⇒ B is true only in the case that either A is false or B is true. → 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	⇒ → ⊃	⇒ → ⊃	$\Rightarrow$ \Rightarrow $\to$ \to $\supset$ \supset $\implies$ \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	⇔ ≡ ↔	⇔ &equiv; ↔	$\Leftrightarrow$ \Leftrightarrow $\equiv$ \equiv $\leftrightarrow$ \leftrightarrow $\iff$ \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 U+0021	¬ ˜ !	¬ &tilde; &excl;	$\neg$ \lnot or \neg $\sim$ \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+00B7 U+0026	∧ · &	&and; · &	$\wedge$ \wedge or \land $\&$ \&^[2]
	and
	propositional logic, Boolean algebra
∨ + ∥	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 U+002B U+2225	∨ + ∥	&or;	$\lor$ \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;	$\oplus$ \oplus $\veebar$ \veebar
	xor
	propositional logic, Boolean algebra
⊤ T 1	Tautology	The statement ⊤ is unconditionally true.	A ⇒ ⊤ is always true.	U+22A4	⊤		$\top$ \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	⊥	&perp;	$\bot$ \bot
	bottom, falsum, falsity
	propositional logic, Boolean algebra
∀ ()	universal quantification	∀ x: P(x) or (x) P(x) means P(x) is true for all x.	∀ n ∈ 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: n is even.	U+2203	∃	∃	$\exists$ \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	∃ !		$\exists !$ \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	≔ (: =) ≡ ⊜	&equiv; ⇔	$:=$ := $\equiv$ \equiv $:\Leftrightarrow$ :\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	( )		$(~)$ ( )
	parentheses, brackets
	everywhere
⊢	Turnstile	x ⊢ y means y is provable from x (in some specified formal system).	A → B ⊢ ¬B → ¬A	U+22A2	⊢		$\vdash$ \vdash
	provable
	propositional logic, first-order logic
⊨	double turnstile	x ⊨ y means x semantically entails y	A → B ⊨ ¬B → ¬A	U+22A8	⊨		$\vDash$ \vDash
	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+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 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+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).

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 $p$ ⥽ $q\equiv \Box (p\rightarrow q)$ , the corresponding LaTeX macro is \strictif. See here for an image of glyph. Added to Unicode 3.2.0.

Poland and Germany

As of 2014^[update] in Poland, the universal quantifier is sometimes written $\wedge$ and the existential quantifier as $\vee$ . The same applies for Germany.

References

^ "Named character references". HTML 5.1 Nightly. W3C. Retrieved 9 September 2015.
^ Although this character is available in LaTeX, the MediaWiki TeX system doesn't support this character.
^ 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.
^ Quine, W.V. (1981): Mathematical Logic, §6
^ Hintikka, Jaakko (1998), The Principles of Mathematics Revisited, Cambridge University Press, p. 113, ISBN 9780521624985.

External links

Named character entities in HTML 4.0

[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.

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Basic logic symbols

Advanced and rarely used logical symbols

Poland and Germany

See also

References

Further reading

External links