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negation: not propositional logic, Boolean algebra: The statement is true if and only if A is false. A slash placed through another operator is the same as placed in front. The prime symbol is placed after the negated thing, e.g. ′ [2]
In fact, a truth-functionally complete system, [l] in the sense that all and only the classical propositional tautologies are theorems, may be derived using only disjunction and negation (as Russell, Whitehead, and Hilbert did), or using only implication and negation (as Frege did), or using only conjunction and negation, or even using only a ...
[2] [3] For example, if is "Spot runs", then "not " is "Spot does not run". An operand of a negation is called a negand or negatum. [4] Negation is a unary logical connective. It may furthermore be applied not only to propositions, but also to notions, truth values, or semantic values more generally.
If a statement's negation is false, then the statement is true (and vice versa). If a statement (or its contrapositive) and the inverse (or the converse) are both true or both false, then it is known as a logical biconditional .
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 ...
Equivalently, if P is true or Q is true and P is false, then Q is true. The name "disjunctive syllogism" derives from its being a syllogism, a three-step argument, and the use of a logical disjunction (any "or" statement.) For example, "P or Q" is a disjunction, where P and Q are called the statement's disjuncts.
Then we have by the law of excluded middle [clarification needed] (i.e. either must be true, or must not be true). Subsequently, since P → Q {\displaystyle P\to Q} , P {\displaystyle P} can be replaced by Q {\displaystyle Q} in the statement, and thus it follows that ¬ P ∨ Q {\displaystyle \neg P\lor Q} (i.e. either Q {\displaystyle Q ...
Logical connectives can be used to link zero or more statements, so one can speak about n-ary logical connectives. The boolean constants True and False can be thought of as zero-ary operators. Negation is a unary connective, and so on.