Rule of inference

In logic, especially in mathematical logic, a rule of inference is a scheme for constructing valid inferences. These schemes establish syntactic relations between a set of formulas called premises and an assertion called a conclusion. These syntactic relations are used in the process of inference, whereby new true assertions are arrived at from other already known ones. Rules also apply to informal logic and arguments, but the formulation is much more difficult and controversial.

Related Topics:
Logic - Mathematical logic - Inference - Syntactic - Informal logic - Arguments

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As stated, the application of a rule of inference is a purely syntactic procedure. Nevertheless it must also be valid, or more precisely validity preserving. In order for the requirement of validity preservation to make sense, some form of semantics is necessary for the assertions the rule of inference relates and the rule of inference itself. For a discussion of the interrelation between rules of inference and semantics, see the article on propositional logic.

Related Topics:
Semantics - Propositional logic

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Prominent examples of rules of inference in propositional logic are the rules of modus ponens and modus tollens. For first-order predicate logic, rules of inference are needed to deal with logical quantifiers. See also validity for more information on the informal description of such arguments. And see first-order resolution for a uniform treatment of all rules of inference as a single rule in the case of first order predicate logic.

Related Topics:
Modus ponens - Modus tollens - Predicate logic - Logical quantifier - Validity - First-order resolution

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Note that there are many different systems of formal logic each one with its own set of well-formed formulas, rules of inference and semantics. See for instance temporal logic, modal logic, or intuitionistic logic. Quantum logic is also a form of logic quite different from the ones mentioned earlier. See also proof theory. In predicate calculus, an additional inference rule is needed. It is called Generalization.

Related Topics:
Formal logic - Temporal logic - Modal logic - Intuitionistic logic - Quantum logic - Proof theory - Predicate calculus - Generalization

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In the setting of formal logic (and many related areas), rules of inference are usually given in the following standard form:

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  Premise#1

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  Premise#2

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        ...

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  Premise#n   

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  Conclusion

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This expression states, that whenever in the course of some logical derivation the given premises have been obtained, the specified conclusion can be taken for granted as well. The exact formal language that is used to describe both premises and conclusions depends on the actual context of the derivations. In a simple case, one may use logical formulae, such as in

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  A→B

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  A        

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  B

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which is just the rule modus ponens of propositional logic. Rules of inference are usually formulated as rule schemata by the use of universal variables. In the rule (schema) above, A and B can be instantiated to any element of the universe (or sometimes, by convention, some restricted subset such as propositions) to form an infinite set of inference rules.

Related Topics:
Propositions - Infinite

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A proof system is formed from a set of rules, which can be chained together to form proofs, or derivations. Any derivation has only one final conclusion, which is the statement proved or derived. If premises are left unsatisfied in the derivation, then the derivation is a proof of a hypothetical statement: "if the premises hold, then the conclusion holds."

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