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In logic and mathematics, statements and are said to be logically equivalent if they have the same truth value in every model. [1] The logical equivalence of p {\displaystyle p} and q {\displaystyle q} is sometimes expressed as p ≡ q {\displaystyle p\equiv q} , p :: q {\displaystyle p::q} , E p q {\displaystyle {\textsf {E}}pq} , or p q ...
The algorithm to compute a CNF-equivalent of a given propositional formula builds upon in disjunctive normal form (DNF): step 1. [ 2 ] Then ¬ ϕ D N F {\displaystyle \lnot \phi _{DNF}} is converted to ϕ C N F {\displaystyle \phi _{CNF}} by swapping ANDs with ORs and vice versa while negating all the literals.
In propositional logic, material implication [1] [2] is a valid rule of replacement that allows a conditional statement to be replaced by a disjunction in which the antecedent is negated. The rule states that P implies Q is logically equivalent to not- P {\displaystyle P} or Q {\displaystyle Q} and that either form can replace the other in ...
(As to equivalence, Howson calls it "truth-functional equivalence", while Cunningham calls it "logical equivalence".) Equivalence is symbolized with ⇔ and is a metalanguage symbol, while a biconditional is symbolized with ↔ and is a logical connective in the object language . Regardless, an equivalence or biconditional is true if, and only ...
In propositional logic and Boolean algebra, there is a duality between conjunction and disjunction, [1] [2] [3] also called the duality principle. [4] [5] [6] It is the most widely known example of duality in logic. [1] The duality consists in these metalogical theorems:
In intuitionistic logic, it is not true that every formula is logically equivalent to a prenex formula. The negation connective is one obstacle, but not the only one. The implication operator is also treated differently in intuitionistic logic than classical logic; in intuitionistic logic, it is not definable using disjunction and negation.
In classical propositional logic, they indeed coincide; the deduction theorem states that A ⊢ B if and only if ⊢ A → B. There is however a distinction worth emphasizing even in this case: the first notation describes a deduction , that is an activity of passing from sentences to sentences, whereas A → B is simply a formula made with a ...
Logical equality is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both operands are false or both operands are true. The truth table of p EQ q (also written as p = q , p ↔ q , Epq , p ≡ q , or p == q ) is as follows: