Negation

In logic, negation, also called the logical complement, is an operation that takes a proposition P {displaystyle P} to another proposition 'not P {displaystyle P} ', written ¬ P {displaystyle eg P} , which is interpreted intuitively as being true when P {displaystyle P} is false, and false when P {displaystyle P} is true. Negation is thus a unary (single-argument) logical connective. It may be applied as an operation on notions, propositions, truth values, or semantic values more generally. In classical logic, negation is normally identified with the truth function that takes truth to falsity and vice versa. In intuitionistic logic, according to the Brouwer–Heyting–Kolmogorov interpretation, the negation of a proposition P {displaystyle P} is the proposition whose proofs are the refutations of P {displaystyle P} . No agreement exists as to the possibility of defining negation, as to its logical status, function, and meaning, as to its field of applicability..., and as to the interpretation of the negative judgment, (F.H. Heinemann 1944). Classical negation is an operation on one logical value, typically the value of a proposition, that produces a value of true when its operand is false and a value of false when its operand is true. So, if statement P {displaystyle P} is true, then ¬ P {displaystyle eg P} (pronounced 'not P') would therefore be false; and conversely, if ¬ P {displaystyle eg P} is false, then P {displaystyle P} would be true. The truth table of ¬ P {displaystyle eg P} is as follows: Negation can be defined in terms of other logical operations. For example, ¬ P {displaystyle eg P} can be defined as P → ⊥ {displaystyle P ightarrow ot } (where → {displaystyle ightarrow } is logical consequence and ⊥ {displaystyle ot } is absolute falsehood). Conversely, one can define ⊥ {displaystyle ot } as Q ∧ ¬ Q {displaystyle Qland eg Q} for any proposition Q {displaystyle Q} (where ∧ {displaystyle land } is logical conjunction). The idea here is that any contradiction is false. While these ideas work in both classical and intuitionistic logic, they do not work in paraconsistent logic, where contradictions are not necessarily false. In classical logic, we also get a further identity, P → Q {displaystyle P ightarrow Q} can be defined as ¬ P ∨ Q {displaystyle eg Plor Q} , where ∨ {displaystyle lor } is logical disjunction. Algebraically, classical negation corresponds to complementation in a Boolean algebra, and intuitionistic negation to pseudocomplementation in a Heyting algebra. These algebras provide a semantics for classical and intuitionistic logic respectively. The negation of a proposition p {displaystyle p} is notated in different ways in various contexts of discussion and fields of application. Among these variants are the following: The notation Np is Łukasiewicz notation.