List 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	Unicode value (hexadecimal)	HTML value (decimal)	HTML entity (named)	LaTeX symbol	Logic Name	Read as	Category	Explanation	Examples
⇒ → ⊃	U+21D2 U+2192 U+2283	⇒ → ⊃	⇒ → ⊃	$\Rightarrow$ \Rightarrow $\implies$ \implies $\to$ \to or \rightarrow $\supset$ \supset	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. $\rightarrow$ 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+21D4 U+2261 U+2194	⇔ ≡ ↔	⇔ &equiv; &LeftRightArrow;	$\Leftrightarrow$ \Leftrightarrow $\equiv$ \equiv $\leftrightarrow$ \leftrightarrow $\iff$ \iff	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+00AC U+02DC U+0021	¬ ˜ !	¬ &tilde; &excl;	$\neg$ \lnot or \neg $\sim$ \sim	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.	$\neg (\neg A)\Leftrightarrow A$ $x\neq y\Leftrightarrow \neg (x=y)$
$\mathbb {D}$	U+1D53B	𝔻	&Dopf;	\mathbb{D}	Domain of discourse	Domain of predicate	Predicate (mathematical logic)		$\mathbb {D} \mathbb {:} \mathbb {R}$
∧ · &	U+2227 U+00B7 U+0026	∧ · &	&and; · &	$\wedge$ \wedge or \land $\cdot$ \cdot $\&$ \&[2]	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+2228 U+002B U+2225	∨ + ∥	&or; + &parallel;	$\lor$ \lor or \vee $\parallel$ \parallel	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+21AE U+2295 U+22BB U+2262	↮ ⊕ ⊻ ≢	&oplus; &veebar; &nequiv;	$\oplus$ \oplus $\veebar$ \veebar $\not \equiv$ \not\equiv	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.
⊤ T 1 ■	U+22A4 U+25A0	⊤	&top;	$\top$ \top	Tautology	top, truth, full clause	propositional logic, Boolean algebra, first-order logic	$⊤$ is unconditionally true.	⊤(A) ⇒ A is always true.
⊥ F 0 □	U+22A5 U+25A1	⊥	&perp;	$\bot$ \bot	Contradiction	bottom, falsum, falsity, empty clause	propositional logic, Boolean algebra, first-order logic	⊥ is unconditionally false. (The symbol ⊥ may also refer to perpendicular lines.)	⊥(A) ⇒ A is always false.
∀ ()	U+2200	∀	∀	$\forall$ \forall	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+2203	∃	∃	$\exists$ \exists	existential quantification	there exists	first-order logic	$\exists x : P (x)$ means there exists at least one x such that P(x) is true.	$\exists n\in \mathbb {N$ n is even.
∃!	U+2203 U+0021	∃ !	∃!	${\displaystyle \exists$ \exists !	uniqueness quantification	there exists exactly one	first-order logic	$\exists! x : P (x)$ means there exists exactly one x such that P(x) is true.	$\exists !n\in \mathbb {N} :n+5=2n.$
≔ ≡ :⇔	U+2254 (U+003A U+003D) U+2261 U+003A U+21D4	≔ (: =) ≡ ⊜	&coloneq; &equiv; ⇔	${\displaystyle$ := $\equiv$ \equiv ${\displaystyle$ :\Leftrightarrow	definition	is defined as	everywhere	$x ≔ y$ or $x \equiv y$ means x is defined to be another name for y ( The symbol ≡ can also mean other things, such as congruence). $P :\Leftrightarrow Q$ means P is defined to be logically equivalent to Q.	$\cosh x:={\frac {e^{x}+e^{-x}}{2}}$ $A \oplus B :\Leftrightarrow (A \lor B) \land \neg(A \land B)$
( )	U+0028 U+0029	( )	( )	$(~)$ ( )	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+22A2	⊢	&vdash;	$\vdash$ \vdash	turnstile	proves	propositional logic, first-order logic	x ⊢ y means x proves (syntactically entails) y	(A → B) ⊢ (¬B → ¬A)
⊨	U+22A8	⊨	&vDash;	$\vDash$ \vDash, \models	double turnstile	models	propositional logic, first-order logic	x ⊨ y means x models (semantically entails) y	(A → B) ⊨ (¬B → ¬A)

Advanced and rarely used logical symbols

These symbols are sorted by their Unicode value:

Symbol	Unicode value (hexadecimal)	LaTeX symbol	Logic Name	Read as	Category	Explanation	Examples
̅	U+0305		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+2191 U+007C		UPWARDS ARROW VERTICAL LINE			Sheffer stroke, the sign for the NAND operator (negation of conjunction).
↓	U+2193		DOWNWARDS ARROW			Peirce Arrow, the sign for the NOR operator (negation of disjunction).
⊙	U+2299	$\odot$ \odot	CIRCLED DOT OPERATOR			the sign for the XNOR operator (negation of exclusive disjunction).
∁	U+2201		COMPLEMENT
∄	U+2204	∄\nexists	THERE DOES NOT EXIST			strike out existential quantifier, same as "¬∃"
∴	U+2234	∴\therefore	THEREFORE	Therefore
∵	U+2235	∵\because	BECAUSE	because
⊧	U+22A7		MODELS			is a model of (or "is a valuation satisfying")
⊨	U+22A8	⊨\vDash	TRUE	is true of
⊬	U+22AC	⊬\nvdash	DOES NOT PROVE			negated ⊢, the sign for "does not prove"	T ⊬ P says "P is not a theorem of T"
⊭	U+22AD	⊭\nvDash	NOT TRUE	is not true of
†	U+2020		DAGGER	it is true that ...		Affirmation operator
⊼	U+22BC		NAND			NAND operator
⊽	U+22BD		NOR			NOR operator
◇	U+25C7		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+22C6		STAR OPERATOR			usually used for ad-hoc operators
⊥ ↓	U+22A5 U+2193		UP TACK 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 U+231D	\ulcorner \urcorner	TOP LEFT CORNER TOP RIGHT CORNER			corner quotes, also called "Quine quotes"; for quasi-quotation, i.e. quoting specific context of unspecified ("variable") expressions;[3] also used for denoting Gödel number;[4] 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. 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 U+25A1		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+27DB		LEFT AND RIGHT TACK		semantic equivalent
⟡	U+27E1		WHITE CONCAVE-SIDED DIAMOND	never	modal operator
⟢	U+27E2		WHITE CONCAVE-SIDED DIAMOND WITH LEFTWARDS TICK	was never	modal operator
⟣	U+27E3		WHITE CONCAVE-SIDED DIAMOND WITH RIGHTWARDS TICK	will never be	modal operator
□	U+25A1		WHITE SQUARE	always	modal operator
⟤	U+25A4		WHITE SQUARE WITH LEFTWARDS TICK	was always	modal operator
⟥	U+25A5		WHITE SQUARE WITH RIGHTWARDS TIC	will always be	modal operator
⥽	U+297D	\strictif	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 $p$ ⥽ $q\equiv \Box (p\rightarrow q)$ . See here for an image of glyph. Added to Unicode 3.2.0.
⨇	U+2A07		TWO LOGICAL AND OPERATOR

Usage in various countries

Poland

As of 2014 in Poland, the universal quantifier is sometimes written ∧, and the existential quantifier as ∨. The same applies for Germany.

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.
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.

External links

Named character entities in HTML 4.0

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[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 does not support it.

[3] Quine, W.V. (1981): Mathematical Logic, §6

[4] Hintikka, Jaakko (1998), The Principles of Mathematics Revisited, Cambridge University Press, p. 113, ISBN 9780521624985.