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

Grammatical

Rhetorical

"not...but"

Colloquial

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		$_{=\,}\!$	$_{(p'q)'\,}\!$
$_{s.6 \,}\!$	$_{s.5.right\,}\!$		$_{=\,}\!$	$_{(p\tilde{\leftarrow}q)'\,}\!$
$_{s.7 \,}\!$	$_{s.6.right\,}\!$		$_{=\,}\!$	$_{p\leftarrow q\,}\!$
$_{s.8 \,}\!$	$_{s.7.right\,}\!$		$_{=\,}\!$	$_{q\rightarrow p\,}\!$
$_{s.9 \,}\!$	$_{s.1.left=s.8.right \,}\!$	$_{dual(q\tilde{\leftarrow}p)=q\rightarrow p\,}\!$

Computer science

See also

This logic-related article is a stub. You can help Wikipedia by expanding it.v · Categories: Propositional calculus Logical connectives Logic stubs Wikimedia Foundation. 2010. Игры ⚽ Поможем сделать НИР Material nonimplication Soaring with the Eagles at Night, to Rise with the Pigs in the Morning Look at other dictionaries: Converse implication — is the converse of implication. That is to say; that for any two propositions P and Q, if Q implies P, then P is the converse implication of Q. It may take the following forms: p⊂q, Bpq, or p←q Contents 1 Definition 1.1 Truth table 1.2 …   Wikipedia Material nonimplication — Venn diagram of …   Wikipedia Outline of logic — The following outline is provided as an overview of and topical guide to logic: Logic – formal science of using reason, considered a branch of both philosophy and mathematics. Logic investigates and classifies the structure of statements and… …   Wikipedia Logical connective — This article is about connectives in classical logic. For connectors in natural languages, see discourse connective. For connectives and operators in other logics, see logical constant. For other logical symbols, see table of logic symbols. In… …   Wikipedia Propositional calculus — In mathematical logic, a propositional calculus or logic (also called sentential calculus or sentential logic) is a formal system in which formulas of a formal language may be interpreted as representing propositions. A system of inference rules… …   Wikipedia Logical conjunction — ∧ redirects here. For exterior product, see exterior algebra. Venn diagram of …   Wikipedia Logical biconditional — In logic and mathematics, the logical biconditional (sometimes known as the material biconditional) is the logical connective of two statements asserting p if and only if q , where q is a hypothesis (or antecedent) and p is a conclusion (or… …   Wikipedia Material conditional — The material conditional, also known as material implication, is a binary truth function, such that the compound sentence p→q (typically read if p then q or p implies q ) is logically equivalent to the negative compound: not (p and not q). A… …   Wikipedia Logical disjunction — Disjunction redirects here. For separation of chromosomes, see Meiosis. For disjunctions in distribution, see Disjunct distribution. Venn diagram of the logical disjunction of A and B …   Wikipedia Sheffer stroke — This article is about the logical NAND operator. For the electronic NAND gate, see Negated AND gate. For other uses, see Nand (disambiguation). Venn diagram of In Boolean functions and propositional calculus, the Sheffer stroke, named after …   Wikipedia 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”