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negational

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Negational might refer to
In mathematical logic, a formula is in negation normal form if the negation operator ( * * * * ¬ * * * {\displaystyle \lnot } * , not) is only applied to variables and the only other allowed Boolean operators are conjunction ( * * * * ∧ * * * {\displaystyle \land } * , and) and disjunction ( * * * * ∨ * * * {\displaystyle \lor } * , or). * Negation normal form is not a canonical form: for example, * * * * a * ∧ * ( * b * ∨ * ¬ * c * ) * * * {\displaystyle a\land (b\lor \lnot c)} * and * * * * ( * a * ∧ * b * ) * ∨ * ( * a * ∧ * ¬ * c * ) * * * {\displaystyle (a\land b)\lor (a\land \lnot c)} * are equivalent, and are both in negation normal form. * In classical logic and many modal logics, every formula can be brought into this form by replacing implications and equivalences by their definitions, using De Morgan's laws to push negation inwards, and eliminating double negations. This process can be represented using the following rewrite rules (Handbook of Automated Reasoning 1, p. 204):* * * * A * ⇒ * B * → * ¬ * A * ∨ * B * * * {\displaystyle A\Rightarrow B\to \lnot A\lor B} * * * * * * ¬ * ( * A * ∨ * B * ) * → * ¬ * A * ∧ * ¬ * B * * * {\displaystyle \lnot (A\lor B)\to \lnot A\land \lnot B} * * * * * * ¬ * ( * A * ∧ * B * ) * → * ¬ * A * ∨ * ¬ * B * * * {\displaystyle \lnot (A\land B)\to \lnot A\lor \lnot B} * * * * * * ¬ * ¬ * A * → * A * * * {\displaystyle \lnot \lnot A\to A} * * * * * * ¬ * ∃ * x * A * → * ∀ * x * ¬ * A * * * {\displaystyle \lnot \exists xA\to \forall x\lnot A} * * * * * * ¬ * ∀ * x * A * → * ∃ * x * ¬ * A * * * {\displaystyle \lnot \forall xA\to \exists x\lnot A} * [In these rules, the * * * * ⇒ * * * {\displaystyle \Rightarrow } * symbol indicates logical implication in the formula being rewritten, and * * * * → * * * {\displaystyle \to } * is the rewriting operation.] * A formula in negation normal form can be put into the stronger conjunctive normal form or disjunctive normal form by applying distributivity.

Negational might refer to

In mathematical logic, a formula is in negation normal form if the negation operator (
*
*
*
* ¬
*
*
* {\displaystyle \lnot }
* , not) is only applied to variables and the only other allowed Boolean operators are conjunction (
*
*
*
* ∧
*
*
* {\displaystyle \land }
* , and) and disjunction (
*
*
*
* ∨
*
*
* {\displaystyle \lor }
* , or).
* Negation normal form is not a canonical form: for example,
*
*
*
* a
* ∧
* (
* b
* ∨
* ¬
* c
* )
*
*
* {\displaystyle a\land (b\lor \lnot c)}
* and
*
*
*
* (
* a
* ∧
* b
* )
* ∨
* (
* a
* ∧
* ¬
* c
* )
*
*
* {\displaystyle (a\land b)\lor (a\land \lnot c)}
* are equivalent, and are both in negation normal form.
* In classical logic and many modal logics, every formula can be brought into this form by replacing implications and equivalences by their definitions, using De Morgan's laws to push negation inwards, and eliminating double negations. This process can be represented using the following rewrite rules (Handbook of Automated Reasoning 1, p. 204):*
*
*
* A
* ⇒
* B
* →
* ¬
* A
* ∨
* B
*
*
* {\displaystyle A\Rightarrow B\to \lnot A\lor B}
*
*
*
*
*
* ¬
* (
* A
* ∨
* B
* )
* →
* ¬
* A
* ∧
* ¬
* B
*
*
* {\displaystyle \lnot (A\lor B)\to \lnot A\land \lnot B}
*
*
*
*
*
* ¬
* (
* A
* ∧
* B
* )
* →
* ¬
* A
* ∨
* ¬
* B
*
*
* {\displaystyle \lnot (A\land B)\to \lnot A\lor \lnot B}
*
*
*
*
*
* ¬
* ¬
* A
* →
* A
*
*
* {\displaystyle \lnot \lnot A\to A}
*
*
*
*
*
* ¬
* ∃
* x
* A
* →
* ∀
* x
* ¬
* A
*
*
* {\displaystyle \lnot \exists xA\to \forall x\lnot A}
*
*
*
*
*
* ¬
* ∀
* x
* A
* →
* ∃
* x
* ¬
* A
*
*
* {\displaystyle \lnot \forall xA\to \exists x\lnot A}
* [In these rules, the
*
*
*
* ⇒
*
*
* {\displaystyle \Rightarrow }
* symbol indicates logical implication in the formula being rewritten, and
*
*
*
* →
*
*
* {\displaystyle \to }
* is the rewriting operation.]
* A formula in negation normal form can be put into the stronger conjunctive normal form or disjunctive normal form by applying distributivity.

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