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[edit]

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	⇒ → ⊃	⇒ → ⊃	${\displaystyle \Rightarrow }$\Rightarrow ${\displaystyle \implies }$\implies ${\displaystyle \to }$\to or \rightarrow ${\displaystyle \supset }$\supset	material conditional (material implication)	implies, if ... then ..., it is not the case that ... and not ...	propositional logic, Boolean algebra, Heyting algebra	${\displaystyle A\Rightarrow B}$ is false when A is true and B is false but true otherwise. ${\displaystyle \rightarrow }$ 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 x could be −2).
⇔ ↔ ≡	U+21D4 U+2194 U+2261	⇔ ↔ ≡	⇔ ↔ ≡	${\displaystyle \Leftrightarrow }$\Leftrightarrow ${\displaystyle \iff }$\iff ${\displaystyle \leftrightarrow }$\leftrightarrow ${\displaystyle \equiv }$\equiv	material biconditional (material equivalence)	if and only if, iff, xnor	propositional logic, Boolean algebra	${\displaystyle A\Leftrightarrow B}$ is true only if both A and B are false, or both A and B are true. Whether a symbol means a material biconditional or a logical equivalence, it depends on the author’s style.	${\displaystyle x+5=y+2\Leftrightarrow x+3=y}$
¬ ~ !	U+00AC U+007E U+0021	¬ ˜ !	¬ ˜ !	${\displaystyle \neg }$\lnot or \neg ${\displaystyle \sim }$\sim	negation	not	propositional logic, Boolean algebra	The statement ${\displaystyle \lnot A}$ is true if and only if 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+2227 U+00B7 U+0026	&#8743; &#183; &#38;	&and; &middot; &amp;	${\displaystyle \wedge }$\wedge or \land ${\displaystyle \cdot }$\cdot ${\displaystyle \&}$\&[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  ∧  n >2  ⇔  n = 3 when n is a natural number.
∨ + ∥	U+2228 U+002B U+2225	&#8744; &#43; &#8741;	&or; &plus; &parallel;	${\displaystyle \lor }$\lor or \vee ${\displaystyle \parallel }$\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 ≥ 4  ∨  n ≤ 2  ⇔ n ≠ 3 when n is a natural number.
⊕ ⊻ ↮ ≢	U+2295 U+22BB U+21AE U+2262	&#8853; &#8891; &#8622; &#8802;	&oplus; &veebar; &nequiv;	${\displaystyle \oplus }$\oplus ${\displaystyle \veebar }$\veebar ${\displaystyle \not \equiv }$\not\equiv	exclusive disjunction	xor, either ... or ... (but not both)	propositional logic, Boolean algebra	The statement ${\displaystyle A\oplus B}$ is true when either A or B, but not both, are true. This is equivalent to ¬(A ↔ B), hence the symbols ${\displaystyle \nleftrightarrow }$ and ${\displaystyle \not \equiv }$ .	${\displaystyle \lnot A\oplus A}$ is always true and ${\displaystyle A\oplus A}$ is always false (if vacuous truth is excluded).
⊤ T 1	U+22A4	&#8868;	&top;	${\displaystyle \top }$\top	true (tautology)	top, truth, tautology, verum, full clause	propositional logic, Boolean algebra, first-order logic	${\displaystyle \top }$ denotes a proposition that is always true.	The proposition ${\displaystyle \top \lor P}$ is always true since at least one of the two is unconditionally true.
⊥ F 0	U+22A5	&#8869;	&perp;	${\displaystyle \bot }$\bot	false (contradiction)	bottom, falsity, contradiction, falsum, empty clause	propositional logic, Boolean algebra, first-order logic	${\displaystyle \bot }$ denotes a proposition that is always false. The symbol ⊥ may also refer to perpendicular lines.	The proposition ${\displaystyle \bot \wedge P}$ is always false since at least one of the two is unconditionally false.
∀ ()	U+2200	&#8704;	&forall;	${\displaystyle \forall }$\forall	universal quantification	given any, for all, for every, for each, for any	first-order logic	${\displaystyle \forall x}$ ${\displaystyle P(x)}$ or ${\displaystyle (x)}$ ${\displaystyle P(x)}$ says “given any ${\displaystyle x}$, ${\displaystyle x}$ has property ${\displaystyle P}$.”	${\displaystyle \forall n\in \mathbb {N} :n^{2}\geq n.}$
∃	U+2203	&#8707;	&exist;	${\displaystyle \exists }$\exists	existential quantification	there exists, for some	first-order logic	${\displaystyle \exists x}$ ${\displaystyle P(x)}$ says “there exists an x (at least one) such that ${\displaystyle x}$ has property ${\displaystyle P}$.”	${\displaystyle \exists n\in \mathbb {N} :}$ n is even.
∃!	U+2203 U+0021	&#8707; &#33;	&exist;!	${\displaystyle \exists !}$\exists !	uniqueness quantification	there exists exactly one	first-order logic (abbreviation)	${\displaystyle \exists !x}$ ${\displaystyle P(x)}$ says “there exists exactly one x such that x has property P.” Only ${\displaystyle \forall }$ and ${\displaystyle \exists }$ are part of formal logic. ${\displaystyle \exists !x}$ ${\displaystyle P(x)}$ is a shorthand for ${\displaystyle \exists x\forall y(P(y)\leftrightarrow y=x)}$	${\displaystyle \exists !n\in \mathbb {N} :n+5=2n.}$
( )	U+0028 U+0029	&#40; &#41;	&lpar; &rpar;	${\displaystyle (~)}$ ( )	precedence grouping	parentheses; brackets	almost all logic syntaxes, as well as metalanguage	Perform the operations inside the parentheses first.	(8 ÷ 4) ÷ 2 = 2 ÷ 2 = 1, but 8 ÷ (4 ÷ 2) = 8 ÷ 2 = 4.
${\displaystyle \mathbb {D} }$	U+1D53B	&#120123;	&Dopf;	\mathbb{D}	domain of discourse	domain of discourse	metalanguage (first-order logic semantics)		${\displaystyle \mathbb {D} \mathbb {:} \mathbb {R} }$
⊢	U+22A2	&#8866;	&vdash;	${\displaystyle \vdash }$\vdash	turnstile	syntactically entails (proves)	metalanguage (metalogic)	${\displaystyle A\vdash B}$ says “${\displaystyle B}$ is a theorem of ${\displaystyle A}$”. In other words, ${\displaystyle A}$ proves ${\displaystyle B}$ via a deductive system.	${\displaystyle (A\rightarrow B)\vdash (\lnot B\rightarrow \lnot A)}$ (eg. by using natural deduction)
⊨	U+22A8	&#8872;	&vDash;	${\displaystyle \vDash }$\vDash, \models	double turnstile	semantically entails	metalanguage (metalogic)	${\displaystyle A\vDash B}$ says “in every model, it is not the case that ${\displaystyle A}$ is true and ${\displaystyle B}$ is false”.	${\displaystyle (A\rightarrow B)\vDash (\lnot B\rightarrow \lnot A)}$ (eg. by using truth tables)
≡ ⟚ ⇔	U+2261 U+27DA U+21D4	&#8801; &#8660;	&equiv; &hArr;	${\displaystyle :\equiv }$\equiv ${\displaystyle \Leftrightarrow }$\Leftrightarrow	logical equivalence	is logically equivalent to	metalanguage (metalogic)	It’s when ${\displaystyle A\vDash B}$ and ${\displaystyle B\vDash A}$. Whether a symbol means a material biconditional or a logical equivalence, it depends on the author’s style.	${\displaystyle (A\rightarrow B)\equiv (\lnot A\lor B)}$
⊬	U+22AC			⊬\nvdash		does not syntactically entail (does not prove)	metalanguage (metalogic)	${\displaystyle A\nvdash B}$ says “${\displaystyle B}$ is not a theorem of ${\displaystyle A}$”. In other words, ${\displaystyle B}$ is not derivable from ${\displaystyle A}$ via a deductive system.	${\displaystyle A\lor B\nvdash A\wedge B}$
⊭	U+22AD			⊭\nvDash		does not semantically entail	metalanguage (metalogic)	${\displaystyle A\nvDash B}$ says “${\displaystyle A}$ does not guarantee the truth of ${\displaystyle B}$ ”. In other words, ${\displaystyle A}$ does not make ${\displaystyle B}$ true.	${\displaystyle A\lor B\nvDash A\wedge B}$
□	U+25A1			${\displaystyle \Box }$\Box	necessity (in a model)	box; it is necessary that	modal logic	modal operator for “it is necessary that” in alethic logic, “it is provable that” in provability logic, “it is obligatory that” in deontic logic, “it is believed that” in doxastic logic.	${\displaystyle \Box \forall xP(x)}$ says “it is necessary that everything has property P”
◇	U+25C7			${\displaystyle \Diamond }$\Diamond	possibility (in a model)	diamond; it is possible that	modal logic	modal operator for “it is possible that”, (in most modal logics it is defined as “¬□¬”, “it is not necessarily not”).	${\displaystyle \Diamond \exists xP(x)}$ says “it is possible that something has property P”
∴	U+2234			∴\therefore	therefore	therefore	metalanguage	shorthand for “therefore”.
∵	U+2235			∵\because	because	because	metalanguage	shorthand for “because”.
≔ ≡	U+2254 (U+003A U+003D) U+2261	&#8788; (&#58; &#61;) &#8801;	&coloneq; &equiv;	${\displaystyle :=}$:= ${\displaystyle \equiv }$\equiv	definition (between terms)	is defined as	metalanguage	${\displaystyle a:=b}$ means "henceforth, ${\displaystyle a}$ is defined to be another name for ${\displaystyle b}$." This notation seems to have its origin in coding. This is a statement in the metalanguage. In the object language (for example, ZFC set theory), all mathematical objects exist simultaneously. The notation ${\displaystyle a\equiv b}$ may occasionally be seen in physics, meaning the same as ${\displaystyle a:=b}$.	${\displaystyle \cosh x:={\frac {e^{x}+e^{-x}}{2}}}$

Advanced or rarely used logical symbols[edit]

These symbols are sorted by their Unicode value:

Symbol	Unicode value (hexadecimal)	HTML value (decimal)	HTML entity (named)	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 an abbreviation 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			${\displaystyle \odot }$\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+22A7				MODELS			is a model of (or “is a valuation satisfying”)
†	U+2020				DAGGER	it is true that ...		Affirmation operator
⊼	U+22BC				NAND			NAND operator
⊽	U+22BD				NOR			NOR operator
⋆	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+27DA				LEFT AND RIGHT DOUBLE TURNSTILE		semantic equivalent
⟛	U+27DB				LEFT AND RIGHT TACK		syntactic equivalent
⊩	U+22A9				FORCES			one of this symbol’s uses is to mean “models” in modal logic, as in 𝔐, 𝑤 ⊩ 𝑃 .
⟡	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+25A4				WHITE SQUARE WITH LEFTWARDS TICK	was always	modal operator
⟥	U+25A5				WHITE SQUARE WITH RIGHTWARDS TICK	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 ${\displaystyle p}$ ⥽ ${\displaystyle q\equiv \Box (p\rightarrow q)}$. See here for an image of glyph. Added to Unicode 3.2.0.
⨇	U+2A07				TWO LOGICAL AND OPERATOR

Usage in various countries[edit]

Poland[edit]

As of 2014^[update] in Poland, the universal quantifier is sometimes written ${\displaystyle \bigwedge }$, and the existential quantifier as ${\displaystyle \bigvee }$^{[citation needed]}. The same applies for Germany^{[citation needed]}.

Japan[edit]

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%".

See also[edit]

• Józef Maria Bocheński
• List of notation used in Principia Mathematica
• List of mathematical symbols
• Logic alphabet, a suggested set of logical symbols
• Logic gate § Symbols
• Logical connective
• Mathematical operators and symbols in Unicode
• Non-logical symbol
• Polish notation
• Truth function
• Truth table
• Wikipedia:WikiProject Logic/Standards for notation

References[edit]

Further reading[edit]

• Józef Maria Bocheński (1959), A Précis of Mathematical Logic, trans., Otto Bird, from the French and German editions, Dordrecht, South Holland: D. Reidel.

External links[edit]

• Named character entities in HTML 4.0