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  "body": "\n## Statement\n\n[[De Morgan's laws|09PKJKA6J0]]: In propositional logic and Boolean algebra, **De Morgan's laws**, also known as **De Morgan's theorem**, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathematician. The rules allow the expression of [[conjunctions|025YZN3VJW]] and [[disjunctions|0F72FK8EKB]] purely in terms of each other via negation. The rules can be expressed in English as:\n\n- not (A or B) = (not A) and (not B)\n- not (A and B) = (not A) or (not B) where \"A or B\" is an \"inclusive or\" meaning *at least* one of A or B rather than an \"exclusive or\" that means *exactly* one of A or B. In formal language, the rules are written as $$\\begin{aligned} \\neg(P\\lor Q) \u0026\\iff (\\neg P) \\land (\\neg Q), \\quad \\text{ and } \\\\ \\neg(P\\land Q) \u0026\\iff (\\neg P) \\lor (\\neg Q), \\end{aligned}$$ where *P* and *Q*are [[propositions|0ZGC7ZESYN]],\n- [[$\\neg$|029JBBF693]] is the negation logic operator (NOT),\n- [[$\\land$|025YZN3VJW]] is the conjunction logic operator (AND),\n- [[$\\lor$|0F72FK8EKB]] is the disjunction logic operator (OR),\n- $\\iff$ is a metalogical symbol meaning \"can be replaced in a logical proof with\", often read as \"if and only if\". For any combination of true/false values for P and Q, the left and right sides of the arrow will hold the same truth value after evaluation.\n\nThe three Laws of thought are:\n\n- The law of identity: 'Whatever is, is.' For all a: a = a.\n- The law of non-contradiction (alternately the 'law of contradiction'): 'Nothing can both be and not be.'\n- The law of excluded middle: 'Everything must either be or not be.' In accordance with the [[law of excluded middle|0RYYK8PG4S]] or excluded third, for every proposition, either its positive or negative form is true: A[[∨|0F72FK8EKB]]¬A.\n",
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