Venn diagram of 
  
    
      
        P
        ↚
        Q
      
    
    {\displaystyle P\nleftarrow Q}
  
(the red area is true)
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]

True True False
True False False
False True True
False False False

Notation

Converse nonimplication is notated , which is the left arrow from converse implication (), 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

This section is empty. You can help by adding to it. (February 2011)

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.

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)

Properties[edit]

Non-associative[edit]

if and only if #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 if and only if .

Non-commutative[edit]

Neutral and absorbing elements[edit]

Converse Nonimplication is noncommutative
Step Make use of Resulting in
s.1 Definition
s.2 Definition
s.3 s.1 s.2
s.4
s.5 s.4.right - expand Unit element
s.6 s.5.right - evaluate expression
s.7 s.4.left = s.6.right
s.8
s.9 s.8 - regroup common factors
s.10 s.9 - join of complements equals unity
s.11 s.10.right - evaluate expression
s.12 s.8 s.11
s.13
s.14 s.12 s.13
s.15 s.3 s.14

Implication is the dual of Converse Nonimplication
Step Make use of Resulting in
s.1 Definition
s.2 s.1.right - .'s dual is +
s.3 s.2.right - Involution complement
s.4 s.3.right - De Morgan's laws applied once
s.5 s.4.right - Commutative law
s.6 s.5.right
s.7 s.6.right
s.8 s.7.right
s.9 s.1.left = s.8.right

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

  1. ^ Lehtonen, Eero, and Poikonen, J.H.
  2. ^ Knuth 2011, p. 49
  3. ^ "A Visual Explanation of SQL Joins". 11 October 2007.