List of logic symbols

⇒→⊃

U+21D2U+2192U+2283

⇒→⊃ ⇒→⊃

⇒ {\displaystyle \Rightarrow } \Rightarrow ⟹ {\displaystyle \implies } \implies → {\displaystyle \to } \to or \rightarrow ⊃ {\displaystyle \supset } \supset

material conditional (material implication)

implies, if P then Q, it is not the case that P and not Q

propositional logic, Boolean algebra, Heyting algebra

A ⇒ B {\displaystyle A\Rightarrow B} is false when A is true and B is false but true otherwise.In other mathematical contexts, see glossary of mathematical symbols, → {\displaystyle \rightarrow } may indicate the domain and codomain of a function and ⊃ {\displaystyle \supset } may mean superset.

x = 2 ⇒ x 2 = 4 {\displaystyle x=2\Rightarrow x^{2}=4} is true, but x 2 = 4 ⇒ x = 2 {\displaystyle x^{2}=4\Rightarrow x=2} is in general false (since x could be −2).

⇔↔≡

U+21D4U+2194U+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

A ⇔ B {\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, depends on the author’s style.

x + 5 = y + 2 ⇔ x + 3 = y {\displaystyle x+5=y+2\Leftrightarrow x+3=y}

¬~! ′

U+00ACU+007EU+0021U+2032

¬˜!′ ¬˜!′

¬ {\displaystyle \neg } \lnot or \neg ∼ {\displaystyle \sim } \sim ′ {\displaystyle '} '

negation

not

propositional logic, Boolean algebra

The statement ¬ A {\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. The prime symbol is placed after the negated thing, e.g. p ′ {\displaystyle p'}

¬ ( ¬ A ) ⇔ A {\displaystyle \neg (\neg A)\Leftrightarrow A} x ≠ y ⇔ ¬ ( x = y ) {\displaystyle x\neq y\Leftrightarrow \neg (x=y)}

∧·&

U+2227U+00B7U+0026

∧·& ∧·&

∧ {\displaystyle \wedge } \wedge or \land ⋅ {\displaystyle \cdot } \cdot & {\displaystyle \&} \&

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+2228U+002BU+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+2295U+22BBU+21AEU+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 A ⊕ B {\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 } .

¬ A ⊕ A {\displaystyle \lnot A\oplus A} is always true and A ⊕ A {\displaystyle A\oplus A} is always false (if vacuous truth is excluded).

⊤T1

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 ⊤ ∨ P {\displaystyle \top \lor P} is always true since at least one of the two is unconditionally true.

⊥F0

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 ⊥ ∧ P {\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

∀ x {\displaystyle \forall x}   P ( x ) {\displaystyle P(x)} or ( x ) {\displaystyle (x)}   P ( x ) {\displaystyle P(x)} says “given any x {\displaystyle x} , x {\displaystyle x} has property P {\displaystyle P} .”

∀ n ∈ N : n 2 ≥ n . {\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

∃ x {\displaystyle \exists x}   P ( x ) {\displaystyle P(x)} says “there exists an x {\displaystyle x} (at least one) such that x {\displaystyle x} has property P {\displaystyle P} .”

∃ n ∈ N : {\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)

∃ ! x {\displaystyle \exists !x} P ( x ) {\displaystyle P(x)} says “there exists exactly one x {\displaystyle x} such that x {\displaystyle x} has property P {\displaystyle P} .” Only ∀ {\displaystyle \forall } and ∃ {\displaystyle \exists } are part of formal logic. ∃ ! x {\displaystyle \exists !x} P ( x ) {\displaystyle P(x)} is an abbreviation for ∃ x ∀ y ( P ( y ) ↔ y = x ) {\displaystyle \exists x\forall y(P(y)\leftrightarrow y=x)}

∃ ! n ∈ N : n + 5 = 2 n . {\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.

D {\displaystyle \mathbb {D} }

U+1D53B

&#120123; &Dopf;

\mathbb{D}

domain of discourse

domain of discourse

metalanguage (first-order logic semantics)

D : R {\displaystyle \mathbb {D} \mathbb {:} \mathbb {R} }

⊢

U+22A2

&#8866; &vdash;

⊢ {\displaystyle \vdash } \vdash

syntactic consequence

proves, syntactically entails (single) turnstile

metalanguage (metalogic)

A ⊢ B {\displaystyle A\vdash B} says “ B {\displaystyle B} is a theorem of A {\displaystyle A} ”. In other words, A {\displaystyle A} proves B {\displaystyle B} via a deductive system.

( A → B ) ⊢ ( ¬ B → ¬ A ) {\displaystyle (A\rightarrow B)\vdash (\lnot B\rightarrow \lnot A)} (eg. by using natural deduction)

⊨

U+22A8

&#8872; &vDash;

⊨ {\displaystyle \vDash } \vDash, \models

semantic consequence or satisfaction

(semantically) entails or satisfies, models double turnstile

metalanguage (metalogic)

A ⊨ B {\displaystyle A\vDash B} says “in every model, it is not the case that A {\displaystyle A} is true and B {\displaystyle B} is false”. M , σ ⊨ B {\displaystyle {\mathcal {M}},\sigma \vDash B} says a formula B {\displaystyle B} is true in a model M {\displaystyle {\mathcal {M}}} with variable assignment σ {\displaystyle \sigma } .

( A → B ) ⊨ ( ¬ B → ¬ A ) {\displaystyle (A\rightarrow B)\vDash (\lnot B\rightarrow \lnot A)} (eg. by using truth tables)

≡⟚⇔

U+2261U+27DAU+21D4

&#8801; —&#8660; &equiv; — &hArr;

≡ {\displaystyle \equiv } \equiv ⇔ {\displaystyle \Leftrightarrow } \Leftrightarrow

logical equivalence

is logically equivalent to

metalanguage (metalogic)

It’s when A ⊨ B {\displaystyle A\vDash B} and B ⊨ A {\displaystyle B\vDash A} . Whether a symbol means a material biconditional or a logical equivalence, depends on the author’s style.

( A → B ) ≡ ( ¬ A ∨ B ) {\displaystyle (A\rightarrow B)\equiv (\lnot A\lor 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.

◻ ∀ x P ( x ) {\displaystyle \Box \forall xP(x)} says “it is necessary that everything has property P {\displaystyle 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”).

◊ ∃ x P ( x ) {\displaystyle \Diamond \exists xP(x)} says “it is possible that something has property P {\displaystyle P} ”

∴

U+2234

∴\therefore

therefore

therefore

metalanguage

abbreviation for “therefore”.

∵

U+2235

∵\because

because

because

metalanguage

abbreviation for “because”.

≔≜≝

U+2254U+225CU+225D

&#8788; &coloneq;

≔ \coloneqq := {\displaystyle :=} := ≜ {\displaystyle \triangleq } \triangleq = d e f {\displaystyle {\stackrel {\scriptscriptstyle \mathrm {def} }{=}}} \stackrel{ \scriptscriptstyle \mathrm{def}}{=}

definition

is defined as

metalanguage

a := b {\displaystyle a:=b} means "from now on, a {\displaystyle a} is defined to be another name for b {\displaystyle b} ." This is a statement in the metalanguage, not the object language. The notation a ≡ b {\displaystyle a\equiv b} may occasionally be seen in physics, meaning the same as a := b {\displaystyle a:=b} .

cosh ⁡ x := e x + e − x 2 {\displaystyle \cosh x:={\frac {e^{x}+e^{-x}}{2}}}

↑| ⊼

U+2191U+007C U+22BC

↑ {\displaystyle \uparrow } \uparrow

Sheffer stroke, NAND

NAND, not both up arrow

Propositional logic

NAND is the negation of conjunction so A ↑ B {\displaystyle A\uparrow B} is true if and only if it's not the case that both A and B are true. See also NAND gate

↓ ⊽

U+2193 U+22BC

↓ {\displaystyle \downarrow } \downarrow

Peirce Arrow, NOR

nor down arrow

Propositional logic

NOR is the negation of conjunction so A ↓ B {\displaystyle A\downarrow B} is true if and only if both A and B are false. See also NOR gate