In logic, converse nonimplicationLehtonen, Eero, and Poikonen, J.H. is a logical connective which is the negation of converse implication (equivalently, the negation of the converse of implication).
Definition Converse nonimplication is notated , or , and is logically equivalent to and .
Truth table The truth table of .
Notation Converse nonimplication is notated , which is the left arrow from converse implication (), negated with a stroke ().
Alternatives include , which combines converse implication's , negated with a stroke (). , which combines converse implication's left arrow () with negation's tilde (). 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 Not P, but Q.
Boolean algebra Converse nonimplication in a general Boolean algebra is defined as .
Example of a 2-element Boolean algebra: the 2 elements {0,1} with 0 as zero and 1 as unity element, operators as complement operator, as join operator and as meet operator, build the Boolean algebra of propositional logic. andandthen means(Negation)(Inclusive or)(And)(Converse nonimplication) 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 (co-divisor of 6) as complement operator, (least common multiple) as join operator and (greatest common divisor) as meet operator, build a Boolean algebra. andandthen means(Co-divisor 6)(Least common multiple)(Greatest common divisor)(x's greatest divisor coprime with y)
Properties Non-associative if and only if #s5 (In a two-element Boolean algebra the latter condition is reduced to or ). Hence in a nontrivial Boolean algebra converse nonimplication is nonassociative.
Clearly, it is associative if and only if .
Non-commutative
if and only if #s6. Hence converse nonimplication is noncommutative.
Neutral and absorbing elements
is a left neutral element () and a right absorbing element (). , , and . Implication is the dual of converse nonimplication #s7.
Converse nonimplication is noncommutative Step Make use ofResulting in s.1 Definition s.2 Definition s.3 s.1 s.2 s.4 s.5 s.4.right - expand Unit element s.6 s.5.right - evaluate expression s.7 s.4.left = s.6.right s.8 s.9 s.8 - regroup common factors s.10 s.9 - join of complements equals unity s.11 s.10.right - evaluate expression s.12 s.8 s.11 s.13 s.14s.12 s.13 s.15 s.3 s.14 Implication is the dual of converse nonimplication Step Make use ofResulting in s.1 Definition s.2s.1.right - .'s dual is + s.3 s.2.right - Involution complement s.4 s.3.right - De Morgan's laws applied once s.5 s.4.right - Commutative law s.6 s.5.right s.7 s.6.right s.8 s.7.right s.9 s.1.left = s.8.right
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.
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