Implicit

In mathematics and mathematical logic, the concept of logical implication encompasses a specific logical function, a specific logical relation, and the various symbols that are used to denote the function or the relation. Before the specific function, relation, and symbols are defined, it is first necessary to establish a few ideas about the connections among them.

In ordinary language, a close approximation to the concept of logical implication (also known as material conditional) is expressed by means of the following conditional form:

• If p then q.

Here p and q are propositional variables that represent propositions in a given language. In a statement of the form "if p then q", the first term, p, is called the antecedent and the second term, q, is called the consequent; and the statement as a whole is called either the conditional or the consequence. Assuming that the conditional is true, then the truth of the antecedent is a sufficient condition for the truth of the consequent, while the truth of the consequent is a necessary condition for the truth of the antecedent.