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 ().
* $p \tilde q$, which combines converse implication's left arrow ($\leftarrow$) with negation's tilde ($\sim$).
* M''pq'', 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

Not P, but Q.

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 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 of 6 with 1 as zero and 6 as unity element, operators $\scriptstyle\!$ (co-divisor of 6) as complement operator, $\scriptstyle\!$ (least common multiple) as join operator and $\scriptstyle\!$ (greatest common divisor) as meet operator, build a Boolean algebra.

Properties

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.
$\begin (r \nleftarrow q) \nleftarrow p &= r'q \nleftarrow p & \text \\ &= (r'q)'p & \text \\ &= (r + q')p & \text \\ &= (r + r'q')p & \text \\ &= rp + r'q'p \\ &= rp + r'(q \nleftarrow p) & \text \\ &= rp + r \nleftarrow (q \nleftarrow p) & \text \\ \end$

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

* is a left neutral element ($0 \nleftarrow p=p$) and a right absorbing element ($$).
* $1 \nleftarrow p=0$, $p \nleftarrow 1=p'$, 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.

References

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External links

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{{DEFAULTSORT:Converse Nonimplication
Logical connectives