De Morgan's laws


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Related to De Morgan's laws: Demorgan's theorem

De Morgan's laws

pl n
(Logic) (in formal logic and set theory) the principles that conjunction and disjunction, or union and intersection, are dual. Thus the negation of P & Q is equivalent to not-P or not-Q
[named after Augustus De Morgan (1806–71), British mathematician]
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They include (i) completion of the modus ponens rule with the modus tollens: even a simple implication a [right arrow] b, implemented by a single molecule, is reversible: given a, our system deduces b, and, given ~b, the system deduces ~a; (ii) implementation of de Morgan's laws allowing to create a valid and compact system compatible with the laws of classical logic; and (iii) implementation of proofs by contradiction which are natural in many applications of logic.
When the system has every formula rewritten in that way, it can autonomously process it and deduce new facts and formulas using modus ponens, modus tollens, and de Morgan's laws. It is also possible to ask the system whether a given value of a certain fact was deducted or not.
For example, De Morgan's laws of equivalence as well as the rules of simplification, addition, and contraposition were well known to Ockham, Burley, Buridan, and Paul of Venice, who also formulated useful strategies for coping with problems of existential import, the putative Achilles' heel of "traditional" logic.