// Workers AI · dad joke modeWhat did the list of logic symbols say? "I'm implying you'll understand

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. In addition, the subsequent columns contain an informal explanation, a short example, the Unicode location, the name for use in HTML documents,^[1] and the LaTeX symbol.

Propositional logic

Logical connectives
Logic Name	Symbol	Unicode value (hexadecimal)	HTML codes	LaTeX symbol	Read as	Explanation	Diagram
Negation	¬	U+00AC	¬ ¬	$\neg$ \lnot or \neg	NOT	$\neg (\neg A)\Leftrightarrow A$	Negation
	~	U+007E	˜ &tilde;	$\sim$ \sim
	!	U+0021	&excl;	$\neg$ \lnot or \neg
	′	U+2032	′	$'$ '
Conjunction	∧	U+2227	∧ &and;	$\wedge$ \wedge or \land	AND	$A\land B\Leftrightarrow \neg (\neg A\lor \neg B)$	Conjunction
	·	U+00B7	· ·	$\cdot$ \cdot
	&	U+0026	& &	$\&$ \&^[2]
Inclusive disjunction	∨	U+2228	∨ &or;	$\lor$ \lor or \vee	OR	$A\lor B\Leftrightarrow \neg (\neg A\land \neg B)$	Inclusive disjunction
	+	U+002B	+ +
	∥	U+2225	∥ &parallel;	$\parallel$ \parallel
Exclusive disjunction	⊕	U+2295	⊕ &oplus;	$\oplus$ \oplus	XOR, either ... or ...	$A\veebar B\Leftrightarrow (A\lor B)\land \neg (A\land B)$	Exclusive disjunction
	⊻	U+22BB	⊻ &veebar;	$\veebar$ \veebar
	↮	U+21AE	↮	$\not \leftrightarrow$ \nleftrightarrow
	≢	U+2262	≢ &nequiv;	$\not \equiv$ \not\equiv
Non-conjunction (Sheffer stroke)	↑	U+2191		$\uparrow$ \uparrow	NAND, neither ... nor ...	$A\uparrow B\Leftrightarrow \neg (A\land B)$	Non-conjunction
	\|	U+007C		$\mid$ \vert, \mid
	⊼	U+22BC		$\barwedge$ \barwedge
Non-disjunction	↓	U+2193		$\downarrow$ \downarrow	NOR	$A\downarrow B\Leftrightarrow \neg (A\lor B)$	Non-disjunction
Non-disjunction	⊽	U+22BD		${\overline {\vee }}$ \overline{\vee}	NOR	$A\downarrow B\Leftrightarrow \neg (A\lor B)$	Non-disjunction
Conditional	⇒	U+21D2	⇒ ⇒	$\Rightarrow$ \Rightarrow $\implies$ \implies	Implies, then	$(A\Rightarrow B)\Leftrightarrow \neg (A\land \neg B)$	Conditional
	→^[3]	U+2192	→ →	$\to$ \to or \rightarrow
	⊃^[4]	U+2283	⊃ ⊃	$\supset$ \supset
Biconditional	⇔	U+21D4	⇔ ⇔	$\Leftrightarrow$ \Leftrightarrow $\iff$ \iff	If and only if, iff, XNOR	$(A\Leftrightarrow B)\Leftrightarrow (\neg A\land \neg B)\lor (A\land B)$	Biconditional
	↔	U+2194	↔ ↔	$\leftrightarrow$ \leftrightarrow
	≡	U+2261	≡ &equiv;	$\equiv$ \equiv
	⊙	U+2299		$\odot$ \odot

Common propositional logic symbolsOther symbols
Logic Name	Symbol	Unicode value (hexadecimal)	HTML codes	LaTeX symbol	Read as	Category	Explanation	Examples
true (tautology)	⊤	U+22A4	⊤ &top;	$\top$ \top	top, truth, tautology, verum, full clause	propositional logic, Boolean algebra, first-order logic	$\top$ denotes a proposition that is always true.	The proposition $\top \lor P$ is always true since at least one of the two is unconditionally true.
	T
	1
false (contradiction)	⊥	U+22A5	⊥ &perp;	$\bot$ \bot	bottom, falsity, contradiction, falsum, empty clause	propositional logic, Boolean algebra, first-order logic	$\bot$ denotes a proposition that is always false. The symbol ⊥ may also refer to perpendicular lines.	The proposition $\bot \wedge P$ is always false since at least one of the two is unconditionally false.
	F
	0
precedence grouping	( )	U+0028 U+0029	( ) ( )	$(~)$ ( )	parentheses; brackets	almost all logic syntaxes, as well as metalanguage	Perform the operations inside the parentheses first.	$(8 \div 4) \div 2 = 2 \div 2 = 1$ , but $8 \div (4 \div 2) = 8 \div 2 = 4$ .

Advanced or rarely used

Advanced or rarely used propositional symbols
Symbol	Unicode value (hexadecimal)	Logic Name
⌐	U+2310	reversed not sign
⨇	U+2A07	two logical AND operator

First-order logic

Common first-order logic symbols
Symbol	Unicode value (hexadecimal)	HTML codes	LaTeX symbol	Logic Name	Read as	Category	Explanation	Examples
∀	U+2200	∀ ∀	$\forall$ \forall	universal quantification	given any, for all, for every, for each, for any	first-order logic	$\forall x$ $P(x)$ or $(x)$ $P(x)$ says "given any $x$ , $x$ has property $P$ ."	$\forall n\in \mathbb {N} :n^{2}\geq n.$
()				universal quantification	given any, for all, for every, for each, for any	first-order logic		$\forall n\in \mathbb {N} :n^{2}\geq n.$
∃	U+2203	∃ ∃	$\exists$ \exists	existential quantification	there exists, for some	first-order logic	$\exists x$ $P(x)$ says "there exists an $x$ (at least one) such that $x$ has property $P$ ."	$\exists n\in \mathbb {N$ n is even.
∃!	U+2203 U+0021	∃ ! ∃!	${\displaystyle \exists$ \exists !	uniqueness quantification	there exists exactly one	first-order logic (abbreviation)	$\exists !x$ $P(x)$ says "there exists exactly one $x$ such that $x$ has property $P$ ." Only $\forall$ and $\exists$ are part of formal logic. $\exists !x$ $P(x)$ is an abbreviation for $\exists x\forall y(P(y)\leftrightarrow y=x)$	$\exists !n\in \mathbb {N} :n+5=2n.$

Advanced or rarely used first-order symbols

Advanced or rarely used first-order logic symbols
Symbol	Unicode value (hexadecimal)	HTML value (decimal)	HTML entity (named)	LaTeX symbol	Logic Name	Read as	Category	Explanation
∄	U+2204			\nexists	there does not exist			Strike out existential quantifier. "¬∃" is sometimes used instead.

Metalanguage and metalogic

Common metalanguage and metalogic symbols
Symbol	Unicode value (hexadecimal)	HTML codes	LaTeX symbol	Logic Name	Read as	Category	Explanation	Examples
$\mathbb {D}$	U+1D53B	𝔻 &Dopf;	\mathbb{D	domain of discourse	domain of discourse	metalanguage (first-order logic semantics)		$\mathbb {D} \mathbb {:} \mathbb {R}$
⊢	U+22A2	⊢ &vdash;	$\vdash$ \vdash	syntactic consequence	proves, syntactically entails (single) turnstile	metalanguage (metalogic)	$A\vdash B$ says " $B$ is a theorem of $A$ ". In other words, $A$ proves $B$ via a deductive system.	$(A\rightarrow B)\vdash (\lnot B\rightarrow \lnot A)$ (eg. by using natural deduction)
⊨	U+22A8	⊨ &vDash;	$\vDash$ \vDash, \models	semantic consequence or satisfaction	(semantically) entails or satisfies, models double turnstile	metalanguage (metalogic)	$A\vDash B$ says "in every model, it is not the case that $A$ is true and $B$ is false". ${\mathcal {M}},\sigma \vDash B$ says a formula $B$ is true in a model ${\mathcal {M}}$ with variable assignment $\sigma$ .	$(A\rightarrow B)\vDash (\lnot B\rightarrow \lnot A)$ (eg. by using truth tables)
≡	U+2261	&equiv;	$\equiv$ \equiv	logical equivalence	is logically equivalent to	metalanguage (metalogic)	It's when $A\vDash B$ and $B\vDash A$ . Whether a symbol means a material biconditional or a logical equivalence, depends on the author's style.	$(A\rightarrow B)\equiv (\lnot A\lor B)$
⟚	U+27DA
⇔	U+21D4	⇔ ⇔	$\Leftrightarrow$ \Leftrightarrow
⊬	U+22AC		⊬ \nvdash		does not syntactically entail (does not prove)	metalanguage (metalogic)	$A\nvdash B$ says " $B$ is not a theorem of $A$ ". In other words, $B$ is not derivable from $A$ via a deductive system.	$A\lor B\nvdash A\wedge B$
⊭	U+22AD		⊭\nvDash		does not semantically entail	metalanguage (metalogic)	$A\nvDash B$ says " $A$ does not guarantee the truth of $B$ ". In other words, $A$ does not make $B$ true.	$A\lor B\nvDash A\wedge B$
∴	U+2234		∴\therefore	therefore	therefore	metalanguage	abbreviation for "therefore"
∵	U+2235		∵ \because	because	because	metalanguage	abbreviation for "because".
≔	U+2254	≔ &coloneq;	≔ \coloneqq ${\displaystyle$ :=	definition	is defined as	metalanguage	$a:=b$ means "from now on, $a$ is defined to be another name for $b$ ." This is a statement in the metalanguage, not the object language. The notation $a\equiv b$ may occasionally be seen in physics, meaning the same as $a:=b$ .	$\cosh x:={\frac {e^{x}+e^{-x}}{2}}$
≜	U+225C		$\triangleq$ \triangleq
≝	U+225D		${\stackrel {\scriptscriptstyle \mathrm {def} }{=}}$ \stackrel{ \scriptscriptstyle \mathrm{def}}{=}

Advanced or rarely used metalogical symbols

Advanced or rarely used metalogical symbols
Symbol	Unicode value (hexadecimal)	LaTeX symbol	Logic Name	Explanation
̅	U+0305	\overline{x}	combining overline	Used format for denoting Gödel numbers. Using HTML style "4̅" is an abbreviation for the standard numeral "SSSS0". It may also denote a negation (used primarily in electronics).
⌜ ⌝	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;^[5] also used for denoting Gödel number;^[6] 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+27DB		left and right tack	"Proves and is proved by".
⊩	U+22A9	\Vdash	forces	One of this symbol's uses is to mean "truthmakes" in the truthmaker theory of truth. It is also used to mean "forces" in the set theory method of forcing.
⋆	U+22C6	\star	star operator	May sometimes be used for ad-hoc operators.

Modal logic

Common modal logic symbols
Symbol	Unicode value (hexadecimal)	HTML codes	LaTeX symbol	Logic Name	Read as	Category	Explanation	Examples
□	U+25A1		$\Box$ \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, "it is known that" in autoepistemic logic.	$\Box \forall xP(x)$ says "it is necessary that everything has property $P$ "
◇	U+25C7		$\Diamond$ \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").	$\Diamond \exists xP(x)$ says "it is possible that something has property $P$ "

Advanced or rarely used modal symbols

Advanced or rarely used modal logic symbols
Symbol	Unicode value (hexadecimal)	LaTeX symbol	Logic Name	Read as	Category	Explanation
⥽	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 tail is also used as strict implication by C.I.Lewis $p$ ⥽ $q\equiv \Box (p\rightarrow q)$ .
⟡	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

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.
↑ In other mathematical contexts $\rightarrow$ may indicate the domain and codomain of a function
↑ In other mathematical contexts $\supset$ may mean superset.
↑ 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 does not support it.

[3] In other mathematical contexts $\rightarrow$ may indicate the domain and codomain of a function

[4] In other mathematical contexts $\supset$ may mean superset.

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

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

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v t e Common logical connectives
Tautology/True $\top$
Alternative denial (NAND gate) ${\overline {\wedge }}$ Converse implication $\Leftarrow$ Implication (IMPLY gate) $\Rightarrow$ Disjunction (OR gate) $\lor$
Negation (NOT gate) $\neg$ Exclusive or (XOR gate) $\oplus$ Biconditional (XNOR gate) $\odot$ Statement (Digital buffer)
Joint denial (NOR gate) ${\overline {\vee }}$ Nonimplication (NIMPLY gate) $\nRightarrow$ Converse nonimplication $\nLeftarrow$ Conjunction (AND gate) $\land$
Contradiction/False $\bot$
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