Converse implication

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 $\neg Q$

$P\leftarrow Q$	$\Leftrightarrow$	$P$	$\lor$	$\neg Q$
	$\Leftrightarrow$		$\lor$

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."

Logical connectives
Tautology/True $\top$
Alternative denial (NAND gate) $\uparrow$ Converse implication $\leftarrow$ Implication (IMPLY gate) $\rightarrow$ Disjunction (OR gate) $\lor$
Negation (NOT gate) $\neg$ Exclusive or (XOR gate) $\oplus$ Biconditional (XNOR gate) $\leftrightarrow$ Statement
Joint denial (NOR gate) $\downarrow$ Nonimplication $\nrightarrow$ Converse nonimplication $\nleftarrow$ Conjunction (AND gate) $\land$
Contradiction/False $\bot$