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

Definition

.

Truth table

The truth table.

Notation

, which is the left arrow from converse implication), negated with a stroke ().

Alternatives include

• , which combines converse implication's, negated with a stroke ().

• , which combines converse implication's) with negation's).

• Mpq, in Bocheński notation

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

Not P, but Q.

Boolean algebra

Converse nonimplication in a general Boolean algebra.

as meet operator, build the Boolean algebra of propositional logic. (greatest common divisor) as meet operator, build a Boolean algebra.

Properties

Non-associative

#s5 (In a two-element Boolean algebra). Hence in a nontrivial Boolean algebra converse nonimplication is nonassociative

.

Non-commutative

• #s6. Hence converse nonimplication is noncommutative.

Neutral and absorbing elements

• is a left neutral element) and a right absorbing element).

• .

• #s7.

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.

References

External links