# 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 $\displaystyle{ \neg Q }$

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

## 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."