In logic, converse nonimplication is a logical connective which is the negation of converse implication (equivalently, the negation of the converse of implication).
Converse nonimplication is notated , or , and is logically equivalent to and .
The truth table of .
Converse nonimplication is notated , which is the left arrow from converse implication (), negated with a stroke ().
Alternatives include
falsehood-preserving: The interpretation under which all variables are assigned a truth value of 'false' produces a truth value of 'false' as a result of converse nonimplication
Example,
If it rains (P) then I get wet (Q), just because I am wet (Q) does not mean it is raining, in reality I went to a pool party with the co-ed staff, in my clothes (~P) and that is why I am facilitating this lecture in this state (Q).
Q does not imply P.
Not P, but Q.
<div id="Definition"> Converse nonimplication in a general Boolean algebra is defined as .
<div id="TwoElements"> Example of a 2-element Boolean algebra: the 2 elements {0,1} with 0 as zero and 1 as unity element, operators as complement operator, as join operator and as meet operator, build the Boolean algebra of propositional logic.
</div> <div id="DivisorsOfSix">
Example of a 4-element Boolean algebra: the 4 divisors {1,2,3,6} of 6 with 1 as zero and 6 as unity element, operators (co-divisor of 6) as complement operator, (least common multiple) as join operator and (greatest common divisor) as meet operator, build a Boolean algebra.
</div>
if and only if #s5 (In a two-element Boolean algebra the latter condition is reduced to or ). Hence in a nontrivial Boolean algebra converse nonimplication is nonassociative.
Clearly, it is associative if and only if .
</div>
<div id="NonCommutative">
</div> <div id="Dual">
</div>
An example for converse nonimplication in computer science can be found when performing a right outer join on a set of tables from a database, if records not matching the join-condition from the "left" table are being excluded.