Definition:Conditional/Notational Variants

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Various symbols are encountered that denote the concept of the conditional:

Symbol Origin Known as
$p \implies q$ Implies
$p \to q$ often used when space is limited
$p \supset q$ 1910: Alfred North Whitehead and Bertrand Russell: Principia Mathematica hook or horseshoe
$p \, \mathop {-\!\!\!<} q$ Charles Sanders Peirce sign of illation
$\operatorname C p q$ Łukasiewicz's Polish notation

In mathematics, as opposed to works concerned purely with logic, it is usual to use "$\implies$", as then it can be ensured that it is understood to mean exactly the same thing when we use it in the "mathematical" context. There are other uses in mathematics for the other symbols.

Sign of Illation

The sign of illation $-\!\!\!<$ is a notation invented by Charles Sanders Peirce to denote the conditional operator.

Peirce derives $-\!\!\!<$ as a variant of the sign $\le$ for less than or equal to, so as to denote that:

$A \mathop {-\!\!\!<} B$

represents the situation such that whenever a particular statement $A$ is true, then so is statement $B$.