Converse nonimplication

In logic, converse nonimplication[1] is a logical connective which is the negation of converse implication (equivalently, the negation of the converse of implication).

Converse nonimplication is notated

, and is logically equivalent to

[2] 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. Converse Nonimplication in a general Boolean algebra is defined as

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.

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.

#s5 (In a two-element Boolean algebra the latter condition is reduced to

Hence in a nontrivial Boolean algebra Converse Nonimplication is nonassociative.

(De Morgan's laws)

{\displaystyle {\begin{aligned}(r\nleftarrow q)\nleftarrow p&=r'q\nleftarrow p&{\text{(by definition)}}\\&=(r'q)'p&{\text{(by definition)}}\\&=(r+q')p&{\text{(De Morgan's laws)}}\\&=(r+r'q')p&{\text{(Absorption law)}}\\&=rp+r'q'p\\&=rp+r'(q\nleftarrow p)&{\text{(by definition)}}\\&=rp+r\nleftarrow (q\nleftarrow p)&{\text{(by definition)}}\\\end{aligned}}}

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.