Rule of inference

A rule of inference (or inference rule ) denotes a transformation rule ( transformation rule ) in a calculus of formal logic, ie a syntactic rule by which it is allowed to move from existing expressions of a formal language to new expressions. This rule-governed transition represents a conclusion

General

A valid rule of inference is to only allow the transition to such expressions, the statements and semantically from the statement of the existing expressions follows (see logical deduction ).

The exact nature of the rules of inference depends on which logical system of the calculus is set up. For the traditional and classical logic that satisfy the principle of bivalence, conclusions must be truth -preserving ( " from truths follows only truth "). Due to this property, modern statements calculi and predicate logic systems understood as proof calculi, although rules of inference per se are no rules of evidence. Final rules vary within classical logic of axioms or axiom schemata, inasmuch as they do not provide specific semantic requirements on the universe of discourse.

Modern logic calculi use the particular modus ponens, as well as introduction and elimination rules for certain logical connectives.

Five traditional rules of inference

The following five rules derived from traditional propositional logic, the tradition begins later than in the Stoa ( Megarian propositional logic ). About the slash are one or two statements, from which the statement under the dash follows.

1) Mode ponendo ponens (Latin for putting the setting end, even separation rule) is considered as the basic form of direct evidence:

If p claims may also be asserted q. Now p is asserted, ie: q. ( syntactically )

2) Mode tollendo tollens (Latin for the picking repealed ): indirect proof

3) chain end ( occasionally - actually wrong, because according to another meaning of the word " chain end " - mode called Barbara )

Called 4) Mode tollendo ponens (sometimes incorrectly disjunctive syllogism )

5 ) Indirect proof by reductio ad absurdum

Other rules of inference

Other well-known inference rules are, inter alia,

• Mode ponendo tollens

• Contraposition

• Rule of substitution
• Replacement rule

Calculi of natural deduction usually include a larger number of rules of inference; for more examples of common rules of inference, therefore, see the article in natural deduction systems.

Logical statements can be rephrased by resolution rules. In this way, certain types of conclusions can be automated as contradictory evidence.

No valid rule of inference is abduction. It is still used in artificial intelligence and knowledge representation in order to simulate " common sense ".

A rule- fair circuit, which has only one of its premises as a consequence, is a circular argument and indeed provides a conclusion, but no evidence or no argument for the conclusion is (see also begging the question ).

• Logic