Rule of Inference

(transformation rule [in some formal system] or rule of deduction), an admissibility rule that regulates the permissible methods of proceeding from a certain collection of assertions (statements, propositions, or formulas expressing these), called premises, to a certain specific assertion (statement, proposition, or formula), called the conclusion.

Rules of inference in which the form of the premises and conclusion is clearly indicated are termed direct; these include the inference rules of the propositional calculus, which permit one to proceed from an arbitrary conjunction to one of its members or to join an arbitrary proposition to any other proposition by means of the operation of disjunction. If in the premises and conclusion only the types of derivations are indicated from one of which it is permitted to proceed to another, then we have a rule of indirect inference. A typical example of a rule of indirect inference is the deduction theorem, a rule for introducing implications in the natural-deduction propositional and predicate calculi, which permits one to proceed (within certain natural limits) from any derivation A₁, A₂, …, A_n-1, A_nǀ - B to a derivation of the form A₁, A₂,…, A_n–1ǀ–A_n ⊃ B.

Rules of inference that express methods of contensive reasoning were already partially systematized in the bounds of traditional formal logic in the forms of syllogistic modes and were subsequently absorbed, sometimes with changes, into mathematical logic; examples include the rule of modus ponens (syllogism scheme, elimination rule), which permits one to proceed from any implication and its antecedent (premise) to its consequent (conclusion). In addition, rules of inference are divided into primitive (basic, postulated) rules and derived rules (derivable from the primitive rules by means of certain metatheorems).

For the primitive inference rules of formal systems (calculi) that are, like axioms, postulates of a given system, the usual questions of consistency, completeness, and independence arise. Insofar as inference rules in one way or another express the relation of logical necessity, and since there is a close link between this relation and the operation of implication in the majority of logical calculi, the same link exists between the inference rules and theorems of any calculus, in particular between the primitive inference rules and the axioms; for example, the analogues of the inference rules of natural deduction are, respectively, the propositional-calculus axioms A & B ⊃ A, A &, B ⊃ B, A ⊃ A ∨ B, and B ⊃ A ∨ B.