In logic, a set of symbols is commonly used to express logical representation. As logicians are familiar with these symbols, they are not explained each time they are used. So, for students of logic, the following table lists many 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.
Symbol
|
Name | Explanation | Examples | Unicode Value |
HTML Entity |
LaTeX symbol |
---|---|---|---|---|---|---|
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 |
⇒ → ⊃ |
\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 |
⇔ ≡ ↔ |
\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 |
¬ ˜ ~ |
\lnot or \neg
\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 |
∧ & |
\wedge or \land \&[1] |
and | ||||||
propositional logic | ||||||
∨
+ ǀǀ |
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 | ∨ | \lor or \vee |
or | ||||||
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 \veebar |
xor | ||||||
propositional logic, Boolean algebra | ||||||
⊤
T 1 |
Tautology | The statement ⊤ is unconditionally true. | A ⇒ ⊤ is always true. | U+22A4 | T | \top |
top | ||||||
propositional logic, Boolean algebra | ||||||
⊥
F 0 |
Contradiction | The statement ⊥ is unconditionally false. | ⊥ ⇒ A is always true. | U+22A5 | ⊥ 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 |
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 | ∃ | \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 | ∃ ! | \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 \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 | ||||||
⊢
|
turnstile | x ⊢ y means y is provable from x (in some specified formal system). | A → B ⊢ ¬B → ¬A | U+22A6 | &⊢ | \vdash |
provable | ||||||
propositional logic, first-order logic | ||||||
⊨
|
double turnstile | x ⊨ y means x semantically entails y | A → B ⊨ ¬B → ¬A | U+22A7 | ⊨ | \models |
entails | ||||||
propositional logic, first-order logic |
These symbols are sorted by their Unicode value:
Note that the following operators are rarely supported by natively installed fonts. If you wish to use these in a web page, you should always embed the necessary fonts so the page viewer can see the web page without having the necessary fonts installed in their computer.