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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.
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 example, carrying on from the previous example, one can say that knowing that someone is called Socrates is sufficient to know that someone has a Name. A necessary and sufficient condition requires that both of the implications S ⇒ N {\displaystyle S\Rightarrow N} and N ⇒ S {\displaystyle N\Rightarrow S} (the latter of which can also be ...
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 ...
An implication A→B is simply a pair of sets A⊆M, B⊆M, where M is the set of attributes under consideration. A is the premise and B is the conclusion of the implication A→B . A set C respects the implication A→B when ¬(C⊆A) or C⊆B.
In philosophy and artificial intelligence (especially, knowledge based systems), the ramification problem is concerned with the indirect consequences of an action. It might also be posed as how to represent what happens implicitly due to an action or how to control the secondary and tertiary effects of an action.
The strict conditionals may avoid paradoxes of material implication. The following statement, for example, is not correctly formalized by material implication: If Bill Gates graduated in medicine, then Elvis never died. This condition should clearly be false: the degree of Bill Gates has nothing to do with whether Elvis is still alive.
In mathematical logic and graph theory, an implication graph is a skew-symmetric, directed graph G = (V, E) composed of vertex set V and directed edge set E. Each vertex in V represents the truth status of a Boolean literal, and each directed edge from vertex u to vertex v represents the material implication "If the literal u is true then the ...