Converse nonimplication

Notation

Converse nonimplication is notated $p\nleftarrow q$, which is the left arrow from converse implication ($\leftarrow$), negated with a stroke (/).

Alternatives include

• $p\not\subset q$, which combines converse implication's ${\displaystyle \subset }$, negated with a stroke (/).
• $p\tilde{\leftarrow }q$, which combines converse implication's left arrow ($\leftarrow$) with negation's tilde ($\sim$).
• 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

Boolean algebra

Converse Nonimplication in a general Boolean algebra is defined as $q\nleftarrow p=q^{\prime }p$.

Example of a 2-element Boolean algebra: the 2 elements {0,1} with 0 as zero and 1 as unity element, operators $\sim$ as complement operator, $\vee$ as join operator and $\wedge$ as meet operator, build the Boolean algebra of propositional logic.

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 ${\displaystyle {}^c}$ (co-divisor of 6) as complement operator, ${\displaystyle {}_{\vee }}$ (least common multiple) as join operator and ${\displaystyle {}_{\wedge }}$ (greatest common divisor) as meet operator, build a Boolean algebra.

Properties

Non-associative

${\displaystyle r\nleftarrow (q\nleftarrow p)=(r\nleftarrow q)\nleftarrow p}$ if and only if ${\displaystyle rp=0}$ #s5 (In a two-element Boolean algebra the latter condition is reduced to ${\displaystyle r=0}$ or ${\displaystyle p=0}$). Hence in a nontrivial Boolean algebra Converse Nonimplication is nonassociative. $${\displaystyle \begin{array}{ccc}(r\nleftarrow q)\nleftarrow p& =r^{\prime }q\nleftarrow p& \text{(by definition)}\\ & =(r^{\prime }q{)}^{\prime }p& \text{(by definition)}\\ & =(r+q^{\prime })p& \text{(De Morgan's laws)}\\ & =(r+r^{\prime }{q}^{\prime })p& \text{(Absorption law)}\\ & =rp+r^{\prime }{q}^{\prime }p\\ & =rp+r^{\prime }(q\nleftarrow p)& \text{(by definition)}\\ & =rp+r\nleftarrow (q\nleftarrow p)& \text{(by definition)}\end{array}}$$

Clearly, it is associative if and only if ${\displaystyle rp=0}$.

Non-commutative

• ${\displaystyle q\nleftarrow p=p\nleftarrow q}$ if and only if ${\displaystyle q=p}$ #s6. Hence Converse Nonimplication is noncommutative.

Neutral and absorbing elements

• 0 is a left neutral element (${\displaystyle 0\nleftarrow p=p}$) and a right absorbing element (${\displaystyle p\nleftarrow 0=0}$).
• ${\displaystyle 1\nleftarrow p=0}$, ${\displaystyle p\nleftarrow 1=p^{\prime }}$, and ${\displaystyle p\nleftarrow p=0}$.
• Implication ${\displaystyle q\to p}$ is the dual of converse nonimplication ${\displaystyle q\nleftarrow p}$ #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.