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Logical negation is an operation on one logical value, typically the value of a proposition, that produces a value of true if its operand is false and a value of false if its operand is true. The truth table for NOT p (also written as ¬p , Np , Fpq , or ~p ) is as follows:
The rule states that P implies Q is logically equivalent to not-or and that either form can replace the other in logical proofs. In other words, if P {\displaystyle P} is true, then Q {\displaystyle Q} must also be true, while if Q {\displaystyle Q} is not true, then P {\displaystyle P} cannot be true either; additionally, when P {\displaystyle ...
Some of these connectives may be defined in terms of others: for instance, implication, p → q, may be defined in terms of disjunction and negation, as ¬p ∨ q; [75] and disjunction may be defined in terms of negation and conjunction, as ¬(¬p ∧ ¬q). [51]
The assertion that Q is necessary for P is colloquially equivalent to "P cannot be true unless Q is true" or "if Q is false, then P is false". [9] [1] By contraposition, this is the same thing as "whenever P is true, so is Q". The logical relation between P and Q is expressed as "if P, then Q" and denoted "P ⇒ Q" (P implies Q).
However, when a value is assigned to x, such as lava, the function then has the value true; while one assigns to x a value like ice, the function then has the value false. Propositional functions are useful in set theory for the formation of sets. For example, in 1903 Bertrand Russell wrote in The Principles of Mathematics (page 106):
In writing, phrases commonly used as alternatives to P "if and only if" Q include: Q is necessary and sufficient for P, for P it is necessary and sufficient that Q, P is equivalent (or materially equivalent) to Q (compare with material implication), P precisely if Q, P precisely (or exactly) when Q, P exactly in case Q, and P just in case Q. [3]
In Boolean logic, logical NOR, [1] non-disjunction, or joint denial [1] is a truth-functional operator which produces a result that is the negation of logical or.That is, a sentence of the form (p NOR q) is true precisely when neither p nor q is true—i.e. when both p and q are false.
The simplest case occurs when an OR formula becomes one its own inputs e.g. p = q. Begin with (p ∨ s) = q, then let p = q. Observe that q's "definition" depends on itself "q" as well as on "s" and the OR connective; this definition of q is thus impredicative. Either of two conditions can result: [25] oscillation or memory.