In logic, converse nonimplication is a logical connective which is the negation of [converse implication](https://bliptext.com/articles/[converse](https://bliptext.com/articles/conversion-logic)-[implication](https://bliptext.com/articles/material-conditional)) (equivalently, the negation of the converse of implication).

Definition

Converse nonimplication is notated, or , and is logically equivalent to and.

Truth table

The truth table of.

Notation

Converse nonimplication is notated, which is the left arrow from converse implication (\leftarrow), negated with a stroke ( / ). Alternatives include / ).

Properties

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

Natural language

Grammatical

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).

Rhetorical

Q does not imply P.

Colloquial

Boolean algebra

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 \sim as complement operator, \vee as join operator and \wedge 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.

Properties

Non-associative

if and only if rp = 0 #s5 (In a two-element Boolean algebra the latter condition is reduced to r = 0 or p=0). Hence in a nontrivial Boolean algebra Converse Nonimplication is nonassociative. Clearly, it is associative if and only if rp=0.

Non-commutative

Neutral and absorbing elements

0 is a left neutral element and a right absorbing element.

Computer science

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.