verifying the quantifier dualities in the model. Then, the quantifier dualities can be extended further to modal logic, relating the box ("necessarily") and 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: verifying the quantifier dualities in the model. Then, the quantifier dualities can be extended further to modal logic, relating the box ("necessarily") and diamond ("possibly") operators: 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: Applications of the rules include simplification of logical expressions in computer programs and digital circuit designs. De Morgan's laws are an example of a more general concept of mathematical duality. and expressed as truth-functional tautologies or theorems of propositional logic: If either A or B were true, then the disjunction of A and B would be true, making its negation false. Presented in English, this follows the logic that "since two things are both false, it is also false that either of them is true". Working in the opposite direction, the second expression asserts that A is false and B is false (or equivalently that "not A" and "not B" are true). Knowing this, a disjunction of A and B must be false also. The negation of said disjunction must thus be true, and the result is identical to the first claim. Presented in a natural language like English, it is expressed as "since it is false that two things are both true, at least one of them must be false". Working in the opposite direction again, the second expression asserts that at least one of "not A" and "not B" must be true, or equivalently that at least one of A and B must be false. Since at least one of them must be false, then their conjunction would likewise be false. Negating said conjunction thus results in a true expression, and this expression is identical to the first claim. In extensions of classical propositional logic, the duality still holds (that is, to any logical operator one can always find its dual), since in the presence of the identities governing negation, one may always introduce an operator that is the De Morgan dual of another. This leads to an important property of logics based on classical logic, namely the existence of negation normal forms: any formula is equivalent to another formula where negations only occur applied to the non-logical atoms of the formula. The existence of negation normal forms drives many applications, for example in digital circuit design, where it is used to manipulate the types of logic gates, and in formal logic, where it is needed to find the conjunctive normal form and disjunctive normal form of a formula. Computer programmers use them to simplify or properly negate complicated logical conditions. They are also often useful in computations in elementary probability theory. Let one define the dual of any propositional operator P(p, q, ...) depending on elementary propositions p, q, ... to be the operator P d {\displaystyle {\mbox{P}}^{d}} defined by verifying the quantifier dualities in the model. Then, the quantifier dualities can be extended further to modal logic, relating the box ("necessarily") and diamond ("possibly") operators: In its application to the alethic modalities of possibility and necessity, Aristotle observed this case, and in the case of normal modal logic, the relationship of these modal operators to the quantification can be understood by setting up models using Kripke semantics. The converse of the last implication does not hold in pure intuitionistic logic. That is, the failure of the joint proposition P ∧ Q {\displaystyle P\land Q} cannot necessarily be resolved to the failure of either of the two conjuncts. For example, from knowing it not to be the case that both Alice and Bob showed up to their date, it does not follow who did not show up. The latter principle is equivalent to the principle of the weak excluded middle W P E M {\displaystyle {\mathrm {WPEM} }} , This weak form can be used as a foundation for an intermediate logic. For a refined version of the failing law concerning existential statements, see the lesser limited principle of omniscience L L P O {\displaystyle {\mathrm {LLPO} }} , which however is different from W L P O {\displaystyle {\mathrm {WLPO} }} . The validity of the other three De Morgan's laws remains true if negation ¬ P {\displaystyle \neg P} is replaced by implication P → C {\displaystyle P\to C} for some arbitrary constant predicate C, meaning that the above laws are still true in minimal logic. Similarly to the above, the quantifier laws: are tautologies even in minimal logic with negation replaced with implying a fixed Q {\displaystyle Q} , while the converse of the last law does not have to be true in general. Further, one still has