Rule of Inference


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rule of inference

[′rül əv ′in·frəns]
(computer science)

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, …, An-1, Anǀ - B to a derivation of the form A1, A2,…, An–1ǀ–AnB.

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 & BA, A &, BB, AAB, and BAB.

REFERENCES

Słupecki, J., and L. Borkowski. Elementy matematicheskoi logiki i teoriia mnozhestv. Moscow, 1965. (Translated from Polish.)
Serebriannikov, O. F. Evristicheskie printsipy i logicheskie ischisleniia. Moscow, 1970.
Smirnov, V. A. Formal’nyi vyvod i logicheskie ischisleniia. Moscow, 1972.
References in periodicals archive ?
According to this theory, people should make more MP inferences than MT inferences because MP is a simple rule of inference while MT is a complex rule (reductio ad absurdum).
2005) predict that people should make more MP inferences than MT inferences in both types of conditional (except if and if not, then), because MP inference is a simple rule of inference while MT inference is a complex rule.
The inference mechanism is appropriate because the techniques of the conventional Compositional Rule of Inference are incorporated into the existing similarity-based inference.
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crisp or fuzzy) on the input universe X, a conclusion can be obtained by applying the Compositional Rule of Inference
He does not take issue with the rule of inference that controlling for a consequence of the cause is never justified and will never produce the right causal effect.
These results show that the choice of that rule of inference and of that metatheorem, for any particular axiomatic system, are not a matter of personal liking or of practical convenience, but they playa fundamental role for the extended correctness-completeness properties of the axiomatic system.
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So although the introduction of modus ponens in [sections] 14 is justified by a semantical soundness argument, the justification is of an operation on his language, not of the underlying rule of inference among Gedanken.
Fuzzy inference is based on fuzzy implication and the compositional rule of inference.
But one can achieve exactly the same effect by choosing a system with axiom "All men are mortal" and rule of inference "From All F are G and X is F infer X is G".