Converse nonimplication

Venn diagram of
(the red area is true)

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

Definition

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

Truth table

The truth table of .[2]

 T T F T F F F T T F F F

Notation

Converse nonimplication is notated , 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 left arrow( ) with Negation's tilde( ).
• 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

Classic passive aggressive: "yeah, no"

"not A but B"

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 as complement operator, as join operator and as meet operator, build the Boolean algebra of propositional logic.

 0 1 1 0 x
and
1 0 y 1 1 0 1 x
and
1 0 y 0 1 0 0 x
then means
1 0 y 0 0 0 1 x
(Negation) (Inclusive Or) (And) (Converse Nonimplication)

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 (codivisor of 6) as complement operator, (least common multiple) as join operator and (greatest common divisor) as meet operator, build a Boolean algebra.

 1 2 3 6 6 3 2 1 x
and
6 3 2 1 y 6 6 6 6 3 6 3 6 2 2 6 6 1 2 3 6 x
and
6 3 2 1 y 1 2 3 6 1 1 3 3 1 2 1 2 1 1 1 1 x
then means
6 3 2 1 y 1 1 1 1 1 2 1 2 1 1 3 3 1 2 3 6 x
(Codivisor 6) (Least Common Multiple) (Greatest Common Divisor) (x's greatest Divisor coprime with y)

Properties

Non-associative

iff #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 iff .

Non-commutative

• iff #s6. Hence Converse Nonimplication is noncommutative.

Neutral and absorbing elements

Converse Nonimplication is noncommutative
Step Make use of Resulting in
Definition
Definition
- expand Unit element
- evaluate expression
- regroup common factors
- join of complements equals unity
- evaluate expression

Implication is the dual of Converse Nonimplication
Step Make use of Resulting in
Definition
- .'s dual is +
- Involution complement
- De Morgan's laws applied once
- Commutative law

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.[3]

References

• Knuth, Donald E. (2011). The Art of Computer Programming, Volume 4A: Combinatorial Algorithms, Part 1 (1st ed.). Addison-Wesley Professional. ISBN 0-201-03804-8.
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