Logic symbols

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In logic, a basic set of logic symbols is used as a shorthand for logical constructions. As these symbols are often considered as familiar, they are not always explained. For convenience, the following table lists some common symbols together with their name, pronunciation and related field of mathematics. Additionally, the third column contains an informal definition, and the fourth column gives a short example.

Be aware that, outside of logic, different symbols have the same meaning, and the same symbol has, depending on the context, different meanings.

Note: This article contains logic symbols. Without proper rendering support, you may see question marks, boxes, or other symbols instead of logic symbols.

Symbol

| Name  | Explanation  | Examples  | Unicode
Value  | HTML
Entity  | LaTeX
symbol

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Should be read as
Category

| material implication | A ⇒ B means if A is true then B is also true; if A is false then nothing is said about B.

→ may mean the same as ⇒ (the symbol may also indicate the domain and codomain of a function; see table of mathematical symbols).

⊃ may mean the same as ⇒ (the symbol may also mean superset).  | x = 2 ⇒ x2 = 4 is true, but x2 = 4  ⇒ x = 2 is in general false (since x could be −2).  | U+21D2

U+2192

U+2283  | &rArr;
&rarr;
&sup;  |

⇒\Rightarrow
->\to
⊃\supset

implies; if .. then
propositional logic, Heyting algebra

| material equivalence | A ⇔ B means A is true if and only if B is true.  | x \+ 5 = y +2 ⇔ x \+ 3 = y | U+21D4

U+2261

U+2194  | &hArr;
&equiv;
&harr;  |

⇔\Leftrightarrow
≡\equiv
↔\leftrightarrow

if and only if; iff
propositional logic

| negation | The statement ¬A is true if and only if A is false.

A slash placed through another operator is the same as "¬" placed in front.  | ¬(¬A) ⇔ A
x ≠ y ⇔ ¬(x = y)  | U+00AC

U+02DC  | &not;
&tilde;
~  |

¬\lnot
∼\sim

not
propositional logic

| logical conjunction | The statement A ∧ B is true if A and B are both true; else it is false.  | n < 4 ∧ n >2 ⇔ n = 3 when n is a natural number.  | U+2227

U+0026  | &and;
&amp;  | ∧\wedge or \land
\&[1]
and
propositional logic

| logical disjunction | 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+2228  | &or;  | ∨\lor

propositional logic

| exclusive disjunction | 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, A ⊕ A is always false.  | U+2295

U+22BB  | &oplus;  | ⊕\oplus
⊻\veebar
xor
propositional logic, Boolean algebra

| Tautology | The statement ⊤ is unconditionally true.  | A ⇒ ⊤ is always true.  | U+22A4  | T  | ⊤\top
top
propositional logic, Boolean algebra

| Contradiction | The statement ⊥ is unconditionally false.  | ⊥ ⇒ A is always true.  | U+22A5  | &perp;
F  | ⊥\bot
bottom
propositional logic, Boolean algebra

| universal quantification | ∀ x: P(x) means P(x) is true for all x.  | ∀ n ∈ N: n2 ≥ n.  | U+2200  | &forall;  | ∀\forall
for all; for any; for each
predicate logic

| existential quantification | ∃ x: P(x) means there is at least one x such that P(x) is true.  | ∃ n ∈ N: n is even.  | U+2203  | &exist;  | ∃\exists
there exists
first-order logic

| uniqueness quantification | ∃! x: P(x) means there is exactly one x such that P(x) is true.  | ∃! n ∈ N: n \+ 5 = 2n.  | U+2203 U+0021  | &exist; !  | ∃!\exists !
there exists exactly one
first-order logic

| definition | x := y or x ≡ y means x is defined to be another name for y (but note that ≡ can also mean other things, such as congruence).

P :⇔ Q means P is defined to be logically equivalent to Q.  | cosh x := (1/2)(exp x \+ exp (−x))

A XOR B :⇔ (A ∨ B) ∧ ¬(A ∧ B)  | U+2254 (U+003A U+003D)

U+2261

U+003A U+229C  | :=
: &equiv;
&hArr;  |

:=:=
≡\equiv
⇔\Leftrightarrow

is defined as
everywhere

( )

| precedence grouping  | Perform the operations inside the parentheses first.  | (8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4.  | U+0028 U+0029  | ( )  | () ( )
everywhere

| inference | x ⊢ y means y is derived from x.  | A → B ⊢ ¬B → ¬A | U+22A2  |  | ⊢\vdash
infers or is derived from
propositional logic, first-order logic

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## Footnotes[edit]

1. ↑ Although this character is available in LaTeX, the Mediawiki TeX system doesn't support this character.