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).
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
falsehoodpreserving: 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"
Rhetorical
"not A but B"
Boolean algebra
Converse Nonimplication in a general Boolean algebra is defined as .
Example of a 2element 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.

and 

and 

then means 
 
(Negation)  (Inclusive Or)  (And)  (Converse Nonimplication) 
Example of a 4element 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.

and 

and 

then means 
 
(Codivisor 6)  (Least Common Multiple)  (Greatest Common Divisor)  (x's greatest Divisor coprime with y) 
Properties
Nonassociative
iff #s5 (In a twoelement Boolean algebra the latter condition is reduced to or ). Hence in a nontrivial Boolean algebra Converse Nonimplication is nonassociative.
Clearly, it is associative iff .
Noncommutative
 iff #s6. Hence Converse Nonimplication is noncommutative.
Neutral and absorbing elements
 0 is a left neutral element ( ) and a right absorbing element ( ).
 , , and .
 Implication is the dual of Converse Nonimplication #s7.
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 joincondition from the "left" table are being excluded.^{[3]}
References
 ↑ Lehtonen, Eero, and Poikonen, J.H.
 ↑ Knuth 2011, p. 49
 ↑ http://www.codinghorror.com/blog/2007/10/avisualexplanationofsqljoins.html
 Knuth, Donald E. (2011). The Art of Computer Programming, Volume 4A: Combinatorial Algorithms, Part 1 (1st ed.). AddisonWesley Professional. ISBN 0201038048.