Search results
Results from the WOW.Com Content Network
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:
Implication alone is not functionally complete as a logical operator because one cannot form all other two-valued truth functions from it.. For example, the two-place truth function that always returns false is not definable from → and arbitrary propositional variables: any formula constructed from → and propositional variables must receive the value true when all of its variables are ...
For such a double limit to exist, this definition requires the value of f approaches L along every possible path approaching (p, q), excluding the two lines x = p and y = q. As a result, the multiple limit is a weaker notion than the ordinary limit: if the ordinary limit exists and equals L, then the multiple limit exists and also equals L. The ...
A Propositional Function, or a predicate, in a variable x is an open formula p(x) involving x that becomes a proposition when one gives x a definite value from the set of values it can take. According to Clarence Lewis, "A proposition is any expression which is either true or false; a propositional function is an expression, containing one or ...
The most thoroughly researched branch of propositional logic is classical truth-functional propositional logic, [1] in which formulas are interpreted as having precisely one of two possible truth values, the truth value of true or the truth value of false. [19]
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.
p. q.=. ~(~p v ~q) Df. where "p. q" is the logical product of p and q. 3.02. p ⊃ q ⊃ r.=. p ⊃ q. q ⊃ r Df. This definition serves merely to abbreviate proofs. Translation of the formulas into contemporary symbols: Various authors use alternate symbols, so no definitive translation can be given.
¬ p ⇔ (p → ⊥) In fact, this is the definition of negation in some systems, [8] such as intuitionistic logic, and can be proven in propositional calculi where negation is a fundamental connective. Because p → p is usually a theorem or axiom, a consequence is that the negation of false (¬ ⊥) is true.