Rules of Inference for Quantifiers in Discrete Mathematics

The conditions satisfied for the existential instantiation and the universal generalization in the natural deduction system are described clear, whereas they are described rather vaguely in many textbooks on discrete mathematics. Contrasting with that the existential instantiation in some textbooks is used only for formulas without free variables, according to the condition we given, it may be also used for formulas with free variables and the ability to reason has increased. Introducing the equivalence between interpretations and the equivalence between assignments with respect to a set of formulas, it is proved that the system is sound, that is, in a proof, the conclusion is a logical consequence of the premises.