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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 ...
A set C of attributes is a concept intent if and only if C respects all valid implications. The system of all valid implications therefore suffices for constructing the closure system of all concept intents and thereby the concept hierarchy. The system of all valid implications of a formal context is closed under the natural inference.
In propositional logic, material implication [1] [2] is a valid rule of replacement that allows a conditional statement to be replaced by a disjunction in which the antecedent is negated. The rule states that P implies Q is logically equivalent to not- P {\displaystyle P} or Q {\displaystyle Q} and that either form can replace the other in ...
The material conditional (also known as material implication) is an operation commonly used in logic. When the conditional symbol → {\displaystyle \rightarrow } is interpreted as material implication, a formula P → Q {\displaystyle P\rightarrow Q} is true unless P {\displaystyle P} is true and Q {\displaystyle Q} is false.
In mathematics, the ramification theory of valuations studies the set of extensions of a valuation v of a field K to an extension L of K. It is a generalization of the ramification theory of Dedekind domains. [1] [2] The structure of the set of extensions is known better when L/K is Galois.
An axiomatic system is a set of axioms or assumptions from which other statements (theorems) are logically derived. [97] In propositional logic, axiomatic systems define a base set of propositions considered to be self-evidently true, and theorems are proved by applying deduction rules to these axioms. [98] See § Syntactic proof via axioms.
An extension of v (to L) is a valuation w of L such that the restriction of w to K is v. The set of all such extensions is studied in the ramification theory of valuations. Let L/K be a finite extension and let w be an extension of v to L. The index of Γ v in Γ w, e(w/v) = [Γ w : Γ v], is called the reduced ramification index of w over v.
Logical consequence (also entailment or logical implication) is a fundamental concept in logic which describes the relationship between statements that hold true when one statement logically follows from one or more statements.