Converse nonimplication

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

Definition

$_{p\not\subset q}\!$ which is the same as $_{\sim(p\subset q)}\!$

Truth table

The truth table of $_{p\not\subset q}\!$.

p	q	$_{\not\subset}\!$
T	T	F
T	F	F
F	T	T
F	F	F

Venn diagram

The Venn Diagram of "It is not the case that B implies A" (the red area is true)

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

Symbol

Alternatives for $_{p\not\subset q}\!$ are

• $_{p\tilde{\leftarrow}q}\!$: $_{\tilde{\leftarrow}}\!$ combines Converse implication's left arrow($_{\leftarrow}\!$) with Negation's tilde($_{\sim}\!$).
• $_{Mpq}\!$: uses prefixed capital letter.
• $_{p\nleftarrow q}\!$: $_{\nleftarrow }\!$ combines Converse implication's left arrow($_{\leftarrow}\!$) denied by means of a stroke($_{/}\!$).

Natural language

Rhetorical

"not...but"

Boolean algebra

• Converse Nonimplication in a general Boolean algebra^{[1] [2]} is defined as $_{q\tilde{\leftarrow}p=q'p}\!$ ^[3][4].
• $_{r\tilde{\leftarrow}(q\tilde{\leftarrow}p)=(r\tilde{\leftarrow}q)\tilde{\leftarrow}p}\!$ iff $_{rp=0}\!$ ^[5] (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.
• $_{q\tilde{\leftarrow}p=p\tilde{\leftarrow}q\,}\!$ iff $_{q=p\,}\!$ ^[6]. Hence Converse Nonimplication is noncommutative.
• $_{0}\!$ is a left neutral element ($_{0\tilde{\leftarrow}p=p}\!$) and a right absorbing element ($_{p\tilde{\leftarrow}0=0}\!$).
• $_{1\tilde{\leftarrow}p=0}\!$, $_{p\tilde{\leftarrow}1=p'}\!$, and $_{p\tilde{\leftarrow}p=0}\!$.
• Implication $_{q \rightarrow p}\!$ is the dual of Converse Nonimplication $_{q\tilde{\leftarrow}p}\!$ ^[7].

^[1]

General Boolean Algebra Operators (ordered by decreasing precedence)
Symbol	Meaning	Operand(s)
$_{'}\!$	unary complement operator	postfix (after operand)
$_{.}\!$ or omitted	binary meet operator	infix (in between operands)
$_{+}\!$	binary join operator	infix (in between operands)
$_{\tilde{\leftarrow}}\!$	binary Converse Nonimplication operator	infix (in between operands)

^[2]

General Boolean Constants
Symbol	Meaning
$_{0}\!$	zero element
$_{1}\!$	unit element

^[3] 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.

$_{\sim x}\!$ $_{1}\!$ $_{0}\!$ $_{x}\!$ $_{0}\!$ $_{1}\!$	and	$_{y}\!$ $_{1}\!$ $_{1}\!$ $_{1}\!$ $_{0}\!$ $_{0}\!$ $_{1}\!$ $_{y_\vee x}\!$ $_{0}\!$ $_{1}\!$ $_{x}\!$	and	$_{y}\!$ $_{1}\!$ $_{0}\!$ $_{1}\!$ $_{0}\!$ $_{0}\!$ $_{0}\!$ $_{y_\wedge x}\!$ $_{0}\!$ $_{1}\!$ $_{x}\!$	then $_{y\tilde{\leftarrow}x}\!$ means	$_{y}\!$ $_{1}\!$ $_{0}\!$ $_{0}\!$ $_{0}\!$ $_{0}\!$ $_{1}\!$ $_{y\tilde{\leftarrow}x}\!$ $_{0}\!$ $_{1}\!$ $_{x}\!$
(Negation)		(Inclusive Or)		(And)		(Converse Nonimplication)

^[4] 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 $_{ ^{c}}\!$ (codivisor of 6) as complement operator, $_{_\vee}\!$ (least common multiple) as join operator and $_{_\wedge}\!$ (greatest common divisor) as meet operator, build a Boolean algebra.

$_{x^c}\!$ $_{6}\!$ $_{3}\!$ $_{2}\!$ $_{1}\!$ $_{x}\!$ $_{1}\!$ $_{2}\!$ $_{3}\!$ $_{6}\!$	and	$_{y}\!$ $_{6}\!$ $_{6}\!$ $_{6}\!$ $_{6}\!$ $_{6}\!$ $_{3}\!$ $_{3}\!$ $_{6}\!$ $_{3}\!$ $_{6}\!$ $_{2}\!$ $_{2}\!$ $_{2}\!$ $_{6}\!$ $_{6}\!$ $_{1}\!$ $_{1}\!$ $_{2}\!$ $_{3}\!$ $_{6}\!$ $_{y_\vee x}\!$ $_{1}\!$ $_{2}\!$ $_{3}\!$ $_{6}\!$ $_{x}\!$	and	$_{y}\!$ $_{6}\!$ $_{1}\!$ $_{2}\!$ $_{3}\!$ $_{6}\!$ $_{3}\!$ $_{1}\!$ $_{1}\!$ $_{3}\!$ $_{3}\!$ $_{2}\!$ $_{1}\!$ $_{2}\!$ $_{1}\!$ $_{2}\!$ $_{1}\!$ $_{1}\!$ $_{1}\!$ $_{1}\!$ $_{1}\!$ $_{y_\wedge x}\!$ $_{1}\!$ $_{2}\!$ $_{3}\!$ $_{6}\!$ $_{x}\!$	then $_{y\tilde{\leftarrow}x}\!$ means	$_{y}\!$ $_{6}\!$ $_{1}\!$ $_{1}\!$ $_{1}\!$ $_{1}\!$ $_{3}\!$ $_{1}\!$ $_{2}\!$ $_{1}\!$ $_{2}\!$ $_{2}\!$ $_{1}\!$ $_{1}\!$ $_{3}\!$ $_{3}\!$ $_{1}\!$ $_{1}\!$ $_{2}\!$ $_{3}\!$ $_{6}\!$ $_{y\tilde{\leftarrow}x}\!$ $_{1}\!$ $_{2}\!$ $_{3}\!$ $_{6}\!$ $_{x}\!$
(Codivisor 6)		(Least Common Multiple)		(Greatest Common Divisor)		(x's greatest Divisor coprime with y)

^[5]

Converse Nonimplication is nonassociative
Step	Make use of	Resulting in
$_{s.1 \,}\!$	Definition	$_{r\tilde{\leftarrow}q\,}\!$	$_{=\,}\!$	$_{r'q\,}\!$
$_{s.2 \,}\!$	$_{(s.2)\tilde{\leftarrow}p \,}\!$	$_{(r\tilde{\leftarrow}q)\tilde{\leftarrow}p\,}\!$	$_{=\,}\!$	$_{r'q\tilde{\leftarrow}p\,}\!$
$_{s.3 \,}\!$	Definition applied on $_{s.2.right\,}\!$		$_{=\,}\!$	$_{(r'q)'p\,}\!$
$_{s.4 \,}\!$	De Morgan's laws applied on $_{s.3.right \,}\!$		$_{=\,}\!$	$_{(r+q')p\,}\!$
$_{s.5 \,}\!$	Distributivity applied $_{s.4.right \,}\!$		$_{=\,}\!$	$_{rp+q'p\,}\!$
$_{s.6 \,}\!$	$_{s.5.right\,}\!$ - insert Unit element		$_{=\,}\!$	$_{r.1.p+1.q'p\,}\!$
$_{s.7 \,}\!$	$_{s.6.right\,}\!$ - expand Unit element		$_{=\,}\!$	$_{r(q+q')p+(r+r')q'p\,}\!$
$_{s.8 \,}\!$	$_{s.7.right\,}\!$ - expand expressions in brackets		$_{=\,}\!$	$_{rqp+rq'p+rq'p+r'q'p\,}\!$
$_{s.9 \,}\!$	ignore one of two equal terms in $_{s.8.right\,}\!$ (Idempotence)		$_{=\,}\!$	$_{rqp+rq'p+r'q'p\,}\!$
$_{s.10 \,}\!$	$_{s.9.right \,}\!$ - regroup common factors		$_{=\,}\!$	$_{r(q+q')p+r'q'p\,}\!$
$_{s.11 \,}\!$	$_{s.10.right \,}\!$ - join of complements equals unity		$_{=\,}\!$	$_{r.1.p+r'q'p\,}\!$
$_{s.12 \,}\!$	$_{s.11.right \,}\!$ - evaluate expression		$_{=\,}\!$	$_{rp+r'q'p\,}\!$
$_{s.13 \,}\!$	$_{s.2.left=s.12.right \,}\!$	$_{(r\tilde{\leftarrow}q)\tilde{\leftarrow}p=rp+r'q'p\,}\!$
$_{s.14 \,}\!$	Definition	$_{q\tilde{\leftarrow}p\,}\!$	$_{=\,}\!$	$_{q'p\,}\!$
$_{s.15 \,}\!$	$_{r\tilde{\leftarrow}(s.14) \,}\!$	$_{r\tilde{\leftarrow}(q\tilde{\leftarrow}p)\,}\!$	$_{=\,}\!$	$_{r\tilde{\leftarrow}(q'p)\,}\!$
$_{s.16 \,}\!$	Definition applied on $_{s.15.right\,}\!$		$_{=\,}\!$	$_{r'q'p\,}\!$
$_{s.17 \,}\!$	$_{s.15.left=s.16.right \,}\!$	$_{r\tilde{\leftarrow}(q\tilde{\leftarrow}p)=r'q'p\,}\!$
$_{s.18 \,}\!$	$_{s.13\ s.17\,}\!$	$_{(r\tilde{\leftarrow}q)\tilde{\leftarrow}p=r\tilde{\leftarrow}(q\tilde{\leftarrow}p)\,}\!$	$_{\Rightarrow\,}\!$	$_{rp+r'q'p=r'q'p\,}\!$
$_{s.19 \,}\!$	$_{(s.18).(rp)\,}\!$		$_{\Rightarrow\,}\!$	$_{(rp).(rp)+(r'q'p).(rp)=(r'q'p).(rp)\,}\!$
$_{s.20 \,}\!$	$_{s.19.right\,}\!$ - evaluate expression		$_{\Rightarrow\,}\!$	$_{rp+0=0\,}\!$
$_{s.21 \,}\!$	$_{s.20.right\,}\!$ - evaluate expression		$_{\Rightarrow\,}\!$	$_{rp=0\,}\!$
$_{s.22 \,}\!$	$_{s.18.left\ \Rightarrow\ s.21.right \,}\!$	$_{(r\tilde{\leftarrow}q)\tilde{\leftarrow}p=r\tilde{\leftarrow}(q\tilde{\leftarrow}p)\ \Rightarrow\ rp=0\,}\!$
$_{s.23 \,}\!$	$_{s.13 \,}\!$	$_{rp=0\ \Rightarrow\ (r\tilde{\leftarrow}q)\tilde{\leftarrow}p=r'q'p\,}\!$
$_{s.24 \,}\!$	$_{s.17\ s.23 \,}\!$	$_{rp=0\ \Rightarrow\ (r\tilde{\leftarrow}q)\tilde{\leftarrow}p=r\tilde{\leftarrow}(q\tilde{\leftarrow}p)\,}\!$
$_{s.25 \,}\!$	$_{s.22\ s.24 \,}\!$	$_{(r\tilde{\leftarrow}q)\tilde{\leftarrow}p=r\tilde{\leftarrow}(q\tilde{\leftarrow}p)\ \Leftrightarrow\ rp=0\,}\!$

^[6]

Converse Nonimplication is noncommutative
Step	Make use of	Resulting in
$_{s.1 \,}\!$	Definition	$_{q\tilde{\leftarrow}p=q'p\,}\!$
$_{s.2 \,}\!$	Definition	$_{p\tilde{\leftarrow}q=p'q\,}\!$
$_{s.3 \,}\!$	$_{s.1\ s.2 \,}\!$	$_{q\tilde{\leftarrow}p=p\tilde{\leftarrow}q\ \Leftrightarrow\ q'p=qp'\,}\!$
$_{s.4 \,}\!$		$_{q\,}\!$	$_{=\,}\!$	$_{q.1\,}\!$
$_{s.5 \,}\!$	$_{s.4.right\,}\!$ - expand Unit element		$_{=\,}\!$	$_{q.(p+p')\,}\!$
$_{s.6 \,}\!$	$_{s.5.right\,}\!$ - evaluate expression		$_{=\,}\!$	$_{qp+qp'\,}\!$
$_{s.7 \,}\!$	$_{s.4.left=s.6.right \,}\!$	$_{q=qp+qp'\,}\!$
$_{s.8 \,}\!$		$_{q'p=qp'\,}\!$	$_{\Rightarrow\,}\!$	$_{qp+qp'=qp+q'p\,}\!$
$_{s.9 \,}\!$	$_{s.8 \,}\!$ - regroup common factors		$_{\Rightarrow\,}\!$	$_{q.(p+p')=(q+q').p\,}\!$
$_{s.10 \,}\!$	$_{s.9 \,}\!$ - join of complements equals unity		$_{\Rightarrow\,}\!$	$_{q.1=1.p\,}\!$
$_{s.11 \,}\!$	$_{s.10.right \,}\!$ - evaluate expression		$_{\Rightarrow\,}\!$	$_{q=p\,}\!$
$_{s.12 \,}\!$	$_{s.8\ s.11\,}\!$	$_{q'p=qp'\ \Rightarrow\ q=p\,}\!$
$_{s.13 \,}\!$		$_{q=p\ \Rightarrow\ q'p=qp'\,}\!$
$_{s.14 \,}\!$	$_{s.12\ s.13 \,}\!$	$_{q=p\ \Leftrightarrow\ q'p=qp'\,}\!$
$_{s.15 \,}\!$	$_{s.3\ s.14 \,}\!$	$_{q\tilde{\leftarrow}p=p\tilde{\leftarrow}q\ \Leftrightarrow\ q=p\,}\!$

^[7]

Implication is the dual of Converse Nonimplication
Step	Make use of	Resulting in
$_{s.1 \,}\!$	Definition	$_{dual(q\tilde{\leftarrow}p)\,}\!$	$_{=\,}\!$	$_{dual(q'p)\,}\!$
$_{s.2 \,}\!$	$_{s.1.right\,}\!$ - .'s dual is +		$_{=\,}\!$	$_{q'+p\,}\!$
$_{s.3 \,}\!$	$_{s.2.right\,}\!$ - Involution complement		$_{=\,}\!$	$_{(q'+p)''\,}\!$
$_{s.4 \,}\!$	$_{s.3.right\,}\!$ - De Morgan's laws applied once		$_{=\,}\!$	$_{(qp')'\,}\!$
$_{s.5 \,}\!$	$_{s.4.right\,}\!$ - Commutative law		$_{=\,}\!$	$_{(p'q)'\,}\!$
$_{s.6 \,}\!$	$_{s.5.right\,}\!$		$_{=\,}\!$	$_{(p\tilde{\leftarrow}q)'\,}\!$
$_{s.7 \,}\!$	$_{s.6.right\,}\!$		$_{=\,}\!$	$_{p\leftarrow q\,}\!$
$_{s.8 \,}\!$	$_{s.7.right\,}\!$		$_{=\,}\!$	$_{q\rightarrow p\,}\!$
$_{s.9 \,}\!$	$_{s.1.left=s.8.right \,}\!$	$_{dual(q\tilde{\leftarrow}p)=q\rightarrow p\,}\!$