Logical laws

Statement

De Morgan's laws: 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 and disjunctions purely in terms of each other via negation. The rules can be expressed in English as:

• not (A or B) = (not A) and (not B)
• 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 $$\begin{aligned} \neg(P\lor Q) &\iff (\neg P) \land (\neg Q), \quad \text{ and } \\ \neg(P\land Q) &\iff (\neg P) \lor (\neg Q), \end{aligned}$$ where P and Qare propositions,
• $\neg$ is the negation logic operator (NOT),
• $\land$ is the conjunction logic operator (AND),
• $\lor$ is the disjunction logic operator (OR),
• $\iff$ 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.

The three Laws of thought are:

• The law of identity: 'Whatever is, is.' For all a: a = a.
• The law of non-contradiction (alternately the 'law of contradiction'): 'Nothing can both be and not be.'
• The law of excluded middle: 'Everything must either be or not be.' In accordance with the law of excluded middle or excluded third, for every proposition, either its positive or negative form is true: A∨¬A.