Logical Implication

A logical implication is a relationship between two statements. If a statement $Q$ is always true when another statement $P$ is true, then we say that $P$ implies $Q$, which is denoted symbolically as $P\implies Q$. Note that if $P$ is false, $Q$ does not necessarily have to be false. For example, if $x>10$, then $x>0$, so we can say that "$x>10\implies x>0$". However, if x is less than 10, it doesn't necessarily mean that x isn't greater than 0. That is, $x>10\implies x>0$ does NOT mean that $x\leq 10\implies x\leq 0$. The truth table for logical implication is as follows:

$P$	$Q$	$P\Rightarrow Q$
T	T	T
T	F	F
F	T	T
F	F	T

Note that while the inverse of $P\implies Q$ (that is, $\neg P\implies \neg Q$) does not necessarily have the same truth value as $P\implies Q$, the contrapositive ($\neg Q\implies \neg P$) does. For example, $x>10\implies x>0$ and its contrapositive, $x\leq 0\implies x\leq 10$, are logically equivalent, and always have the same truth value for any number $x$.

Resources

• Truth Tables, Tautologies, and Logical Equivalences, Millersville University