Converse implication

Venn diagram of $A\leftarrow B$
(the white area shows where the statement is false)

Converse implication is the converse of implication, written ←. That is to say; that for any two propositions $P$ and $Q$, if $Q$ implies $P$, then $P$ is the converse implication of $Q$.

It is written $P\leftarrow Q$, but may also be notated $P\subset Q$, or "Bpq" (in Bocheński notation).

Definition

Truth table

The truth table of $P\leftarrow Q$

$P$	$Q$	$P\leftarrow Q$
T	T	T
T	F	T
F	T	F
F	F	T

Logical Equivalences

Converse implication is logically equivalent to the disjunction of $P$ and $\mathrm{\neg }Q$

$P\leftarrow Q$	$\iff$	$P$	$\vee$	$\mathrm{\neg }Q$
	$\iff$		$\vee$

Properties

truth-preserving: The interpretation under which all variables are assigned a truth value of 'true' produces a truth value of 'true' as a result of converse implication.

Symbol

←, ⇐ 

Natural language

"Not q without p."

"p if q."

See also

• Logical connective
• Material conditional