---
tag: DC524NPS27
kind: statement
role: proposition
name: Logical laws
sources:
  - wikipedia: en:77662984@r1362840840
    locator: Law (mathematics), section Notable examples > Logical laws
    note: adopted
    mention: wikipedia://en/77662984#H6HD
    sections: [Statement]
inventory: [766430YD36]
---

## Statement

[[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:

- 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 *Q*are [[propositions|0ZGC7ZESYN]],
- [[$\neg$|029JBBF693]] is the negation logic operator (NOT),
- [[$\land$|025YZN3VJW]] is the conjunction logic operator (AND),
- [[$\lor$|0F72FK8EKB]] 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|0RYYK8PG4S]] or excluded third, for every proposition, either its positive or negative form is true: A[[∨|0F72FK8EKB]]¬A.