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, how they should be read out loud, and the related field of mathematics. Additionally, the subsequent columns contains an informal explanation, a short example, the Unicode location, the name for use in HTML documents,^[1] and the LaTeX symbol.

Basic logic symbols

Symbol	Name	Read as	Category	Explanation	Examples	Unicode value (hexadecimal)	HTML value (decimal)	HTML entity (named)	LaTeX symbol
⇒ → ⊃	material implication	implies; if ... then	propositional logic, Heyting algebra	$A\Rightarrow B$ is false when $A$ is true and $B$ is false but true otherwise.^[2]^{[circular reference]} $\rightarrow$ may mean the same as $\Rightarrow$ (the symbol may also indicate the domain and codomain of a function; see table of mathematical symbols). $\supset$ may mean the same as $\Rightarrow$ (the symbol may also mean superset).	$x=2\Rightarrow x^{2}=4$ is true, but $x^{2}=4\Rightarrow x=2$ is in general false (since $x$ could be −2).	U+21D2 U+2192 U+2283	⇒ → ⊃	⇒ → ⊃	$\Rightarrow$ \Rightarrow $\to$ \to or \rightarrow $\supset$ \supset $\implies$ \implies
⇔ ≡ ⟷	material equivalence	if and only if; iff; means the same as	propositional logic	$A\Leftrightarrow B$ is true only if both $A$ and $B$ are false, or both $A$ and $B$ are true.	$x+5=y+2\Leftrightarrow x+3=y$	U+21D4 U+2261 U+27F7	⇔ ≡ ⟷	⇔ &equiv; ⟷	$\Leftrightarrow$ \Leftrightarrow $\equiv$ \equiv $\leftrightarrow$ \leftrightarrow $\iff$ \iff
¬ ˜ !	negation	not	propositional logic	The statement $\lnot A$ is true if and only if $A$ is false. A slash placed through another operator is the same as $\neg$ placed in front.	$\neg (\neg A)\Leftrightarrow A$ $x\neq y\Leftrightarrow \neg (x=y)$	U+00AC U+02DC U+0021	¬ ˜ !	¬ &tilde; &excl;	$\neg$ \lnot or \neg $\sim$ \sim
$\mathbb {D}$	Domain of discourse	Domain of predicate	Predicate (mathematical logic)		$\mathbb {D} \mathbb {:} \mathbb {R}$	U+1D53B	𝔻	&Dopf;	\mathbb{D}
∧ · &	logical conjunction	and	propositional logic, Boolean algebra	The statement A ∧ B is true if A and B are both true; otherwise, it is false.	$n < 4 \land$ $n >2 \Leftrightarrow$ $n = 3$ when n is a natural number.	U+2227 U+00B7 U+0026	∧ · &	&and; · &	$\wedge$ \wedge or \land $\cdot$ \cdot $\&$ \&^[3]
∨ + ∥	logical (inclusive) disjunction	or	propositional logic, Boolean algebra	The statement A ∨ B is true if A or B (or both) are true; if both are false, the statement is false.	$n \geq 4 \lor n \leq 2 \Leftrightarrow n \neq 3$ when n is a natural number.	U+2228 U+002B U+2225	∨ + ∥	&or; + &parallel;	$\lor$ \lor or \vee $\parallel$ \parallel
↮ ⊕ ⊻ ≢	exclusive disjunction	xor; either ... or	propositional logic, Boolean algebra	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, and A ↮ A always false, if vacuous truth is excluded.	U+21AE U+2295 U+22BB U+2262	↮ ⊕ ⊻ ≢	&oplus; &veebar; &nequiv;	$\oplus$ \oplus $\veebar$ \veebar $\not \equiv$ \not\equiv
⊤ T 1 ■	Tautology	top, truth, full clause	propositional logic, Boolean algebra, first-order logic	The statement $⊤$ is unconditionally true.	⊤(A) ⇒ A is always true.	U+22A4 U+25A0	⊤	&top;	$\top$ \top
⊥ F 0 □	Contradiction	bottom, falsum, falsity, empty clause	propositional logic, Boolean algebra, first-order logic	The statement ⊥ is unconditionally false. (The symbol ⊥ may also refer to perpendicular lines.)	⊥(A) ⇒ A is always false.	U+22A5 U+25A1	⊥	&perp;	$\bot$ \bot
∀ ()	universal quantification	for all; for any; for each	first-order logic	$\forall x : P (x)$ or $(x) P (x)$ means P(x) is true for all x.	$\forall n\in \mathbb {N} :n^{2}\geq n.$	U+2200	∀	∀	$\forall$ \forall
∃	existential quantification	there exists	first-order logic	$\exists x : P (x)$ means there is at least one x such that P(x) is true.	$\exists n\in \mathbb {N} :$ n is even.	U+2203	∃	∃	$\exists$ \exists
$\exists!$	uniqueness quantification	there exists exactly one	first-order logic	$\exists! x : P (x)$ means there is exactly one x such that P(x) is true.	$\exists !n\in \mathbb {N} :n+5=2n.$	U+2203 U+0021	∃ !	∃!	$\exists !$ \exists !
≔ ≡ :⇔	definition	is defined as	everywhere	$x ≔ y$ or $x \equiv y$ means x is defined to be another name for y (but note that ≡ can also mean other things, such as congruence). $P :\Leftrightarrow Q$ means P is defined to be logically equivalent to Q.	$\cosh x:={\frac {e^{x}+e^{-x}}{2}}$ $A XOR B :\Leftrightarrow (A \lor B) \land \neg(A \land B)$	U+2254 (U+003A U+003D) U+2261 U+003A U+229C	≔ (: =) ≡ ⊜	&coloneq; &equiv; ⇔	$:=$ := $\equiv$ \equiv $:\Leftrightarrow$ :\Leftrightarrow
( )	precedence grouping	parentheses; brackets	everywhere	Perform the operations inside the parentheses first.	$(8 \div 4) \div 2 = 2 \div 2 = 1$ , but $8 \div (4 \div 2) = 8 \div 2 = 4$ .	U+0028 U+0029	( )	( )	$(~)$ ( )
⊢	turnstile	proves	propositional logic, first-order logic	x ⊢ y means x proves (syntactically entails) y	(A → B) ⊢ (¬B → ¬A)	U+22A2	⊢	&vdash;	$\vdash$ \vdash
⊨	double turnstile	models	propositional logic, first-order logic	x ⊨ y means x models (semantically entails) y	(A → B) ⊨ (¬B → ¬A)	U+22A8	⊨	&vDash;	$\vDash$ \vDash, \models

Advanced and rarely used logical symbols

These symbols are sorted by their Unicode value:

Symbol	Name	Read as	Category	Explanation	Examples	Unicode value (hexadecimal)	LaTeX symbol
̅	COMBINING OVERLINE			used format for denoting Gödel numbers. denoting negation used primarily in electronics.	using HTML style "4̅" is a shorthand for the standard numeral "SSSS0". "A ∨ B" says the Gödel number of "(A ∨ B)". "A ∨ B" is the same as "¬(A ∨ B)".	U+0305
↑ \|	UPWARDS ARROW VERTICAL LINE			Sheffer stroke, the sign for the NAND operator (negation of conjunction).		U+2191 U+007C
↓	DOWNWARDS ARROW			Peirce Arrow, the sign for the NOR operator (negation of disjunction).		U+2193
⊙	CIRCLED DOT OPERATOR			the sign for the XNOR operator (negation of exclusive disjunction).		U+2299	$\odot$ \odot
∁	COMPLEMENT					U+2201
∄	THERE DOES NOT EXIST			strike out existential quantifier, same as "¬∃"		U+2204
∴	THEREFORE	Therefore				U+2234	∴\therefore
∵	BECAUSE	because				U+2235
⊧	MODELS			is a model of (or "is a valuation satisfying")		U+22A7
⊨	TRUE	is true of				U+22A8
⊬	DOES NOT PROVE			negated ⊢, the sign for "does not prove"	T ⊬ P says "P is not a theorem of T"	U+22AC
⊭	NOT TRUE	is not true of				U+22AD
†	DAGGER	it is true that ...		Affirmation operator		U+2020
⊼	NAND			NAND operator		U+22BC
⊽	NOR			NOR operator		U+22BD
◇	WHITE DIAMOND			modal operator for "it is possible that", "it is not necessarily not" or rarely "it is not probably not" (in most modal logics it is defined as "¬◻¬")		U+25C7
⋆	STAR OPERATOR			usually used for ad-hoc operators		U+22C6
⊥ ↓	UP TACK DOWNWARDS ARROW			Webb-operator or Peirce arrow, the sign for NOR. Confusingly, "⊥" is also the sign for contradiction or absurdity.		U+22A5 U+2193
⌐	REVERSED NOT SIGN					U+2310
⌜ ⌝	TOP LEFT CORNER 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+231C U+231D	\ulcorner \urcorner
◻ □	WHITE MEDIUM SQUARE 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: $\emptyset$ and ⊥)		U+25FB U+25A1
⟛	LEFT AND RIGHT TACK		semantic equivalent			U+27DB
⟡	WHITE CONCAVE-SIDED DIAMOND	never	modal operator			U+27E1
⟢	WHITE CONCAVE-SIDED DIAMOND WITH LEFTWARDS TICK	was never	modal operator			U+27E2
⟣	WHITE CONCAVE-SIDED DIAMOND WITH RIGHTWARDS TICK	will never be	modal operator			U+27E3
□	WHITE SQUARE	always	modal operator			U+25A1
⟤	WHITE SQUARE WITH LEFTWARDS TICK	was always	modal operator			U+25A4
⟥	WHITE SQUARE WITH RIGHTWARDS TIC	will always be	modal operator			U+25A5
⥽	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.		U+297D
⨇	TWO LOGICAL AND OPERATOR					U+2A07

Usage in various countries

Poland and Germany

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

^ "Named character references". HTML 5.1 Nightly. W3C. Retrieved 9 September 2015.
^ "Material conditional".
^ Although this character is available in LaTeX, the MediaWiki TeX system does not support it.
^ Quine, W.V. (1981): Mathematical Logic, §6
^ Hintikka, Jaakko (1998), The Principles of Mathematics Revisited, Cambridge University Press, p. 113, ISBN 9780521624985.
^ "Kwantyfikator ogólny". 2 October 2017 – via Wikipedia.^{[circular reference]}
^ "Kwantyfikator egzystencjalny". 23 January 2016 – via Wikipedia.^{[circular reference]}
^ "Quantor". 21 January 2018 – via Wikipedia.^{[circular reference]}
^ Hermes, Hans. Einführung in die mathematische Logik: klassische Prädikatenlogik. Springer-Verlag, 2013.

External links

Named character entities in HTML 4.0

[1] "Named character references". HTML 5.1 Nightly. W3C. Retrieved 9 September 2015.

[2] "Material conditional".

[3] Although this character is available in LaTeX, the MediaWiki TeX system does not support it.

[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.^{[circular reference]}

[7] "Kwantyfikator egzystencjalny". 23 January 2016 – via Wikipedia.^{[circular reference]}

[8] "Quantor". 21 January 2018 – via Wikipedia.^{[circular reference]}

[9] Hermes, Hans. Einführung in die mathematische Logik: klassische Prädikatenlogik. Springer-Verlag, 2013.

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