Table of logic symbols

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In logic, a set of symbols is frequently used to express logical constructs. 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, in some cases, different symbols have the same meaning, and the same symbol has, depending on the context, different meanings.

Note: This article contains special characters.

[edit] Basic logic symbols

Symbol
 Name Explanation Examples
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).
implies; if .. then
propositional logic
⇔

↔
 material equivalence A ⇔ B means A is true if B is true and A is false if B is false. x + 5 = y +2  ⇔  x + 3 = y
if and only if; iff
propositional logic
¬

˜
 logical 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)
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.
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.
or
propositional logic

⊕


⊻
 exclusive or 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.
xor
propositional logic, Boolean algebra
∀
 universal quantification ∀ x: P(x) means P(x) is true for all x. ∀ n ∈ N: n2 ≥ n.
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.
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.
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)
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.
everywhere
⊢
 inference x ⊢ y means y is derived from x. A → B ⊢ ¬B → ¬A
infers or is derived from
propositional logic, first-order logic

[edit] See also

[edit] Special characters

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