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

Contents

DefinitionEdit

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

Truth tableEdit

The truth table of  .[2]

     
T T F
T F F
F T T
F F F

NotationEdit

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

PropertiesEdit

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 languageEdit

GrammaticalEdit

Classic passive aggressive: "yeah, no"

RhetoricalEdit

"not A but B"

ColloquialEdit

Boolean algebraEdit

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.

  1 0
x 0 1
and
y
1 1 1
0 0 1
  0 1 x
and
y
1 0 1
0 0 0
  0 1 x
then   means
y
1 0 0
0 0 1
  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.

  6 3 2 1
x 1 2 3 6
and
y
6 6 6 6 6
3 3 6 3 6
2 2 2 6 6
1 1 2 3 6
  1 2 3 6 x
and
y
6 1 2 3 6
3 1 1 3 3
2 1 2 1 2
1 1 1 1 1
  1 2 3 6 x
then   means
y
6 1 1 1 1
3 1 2 1 2
2 1 1 3 3
1 1 2 3 6
  1 2 3 6 x
(Codivisor 6) (Least common multiple) (Greatest common divisor) (x's greatest divisor coprime with y)

PropertiesEdit

Non-associativeEdit

  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-commutativeEdit

  •   iff   #s6. Hence Converse Nonimplication is noncommutative.

Neutral and absorbing elementsEdit

  • 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 scienceEdit

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]

ReferencesEdit

  • 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.