# Converse implication

Venn diagram of ${\displaystyle A\leftarrow B}$
(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 ${\displaystyle P}$ and ${\displaystyle Q}$, if ${\displaystyle Q}$ implies ${\displaystyle P}$, then ${\displaystyle P}$ is the converse implication of ${\displaystyle Q}$.

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

## Definition

### Truth table

The truth table of ${\displaystyle P\leftarrow Q}$

 ${\displaystyle P}$ ${\displaystyle Q}$ ${\displaystyle P\leftarrow Q}$ T T T T F T F T F F F T

### Logical Equivalences

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

 ${\displaystyle P\leftarrow Q}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle P}$ ${\displaystyle \lor }$ ${\displaystyle \neg Q}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle \lor }$

## 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.

←, ⇐

## Natural language

"Not q without p."

"p if q."