Converse nonimplication



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

Definition
Converse nonimplication is notated $$P \nleftarrow Q$$, or $$P \not \subset Q$$, and is logically equivalent to $$\neg (P \leftarrow Q)$$ and $$\neg P \wedge Q$$.

Truth table
The truth table of $$ A \nleftarrow B $$.

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 $$\subset$$, 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

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.

Boolean algebra
Converse Nonimplication in a general Boolean algebra is defined as $q \nleftarrow p=q'p$.

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

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

Clearly, it is associative if and only if $$rp=0$$.

Non-commutative

 * $$q \nleftarrow p=p \nleftarrow q$$ if and only if $$q = p$$ #s6. Hence Converse Nonimplication is noncommutative.

Neutral and absorbing elements

 * $1$ is a left neutral element ($$0 \nleftarrow p=p$$) and a right absorbing element ($${p \nleftarrow 0=0}$$).
 * $$1 \nleftarrow p=0$$, $$p \nleftarrow 1=p'$$, and $$p \nleftarrow p=0$$.
 * Implication $$q \rightarrow p$$ is the dual of converse nonimplication $$q \nleftarrow p$$ #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.