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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 ...
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:
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]
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):
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]
In mathematical logic, a propositional variable (also called a sentence letter, [1] sentential variable, or sentential letter) is an input variable (that can either be true or false) of a truth function. Propositional variables are the basic building-blocks of propositional formulas, used in propositional logic and higher-order logics.
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.
A Boolean-valued function (sometimes called a predicate or a proposition) is a function of the type f : X → B, where X is an arbitrary set and where B is a Boolean domain, i.e. a generic two-element set, (for example B = {0, 1}), whose elements are interpreted as logical values, for example, 0 = false and 1 = true, i.e., a single bit of information.