View source for Rule of inference

In [[logic]], a '''rule of inference''', '''inference rule''' or '''transformation rule''' is a [[logical form]] consisting of a function which takes premises, analyzes their [[Syntax (logic)|syntax]], and returns a [[Multiple-conclusion logic|conclusion (or conclusions)]].

== Description ==

For example, the rule of inference called ''[[modus ponens]]'' takes two premises, one in the form "If p then q" and another in the form "p", and returns the conclusion "q". The rule is valid with respect to the semantics of classical logic (as well as the semantics of many other non-classical logics), in the sense that if the premises are true (under an interpretation), then so is the conclusion.

Typically, a rule of inference preserves truth, a semantic property.

In [[many-valued logic]], it preserves a general designation. But a rule of inference's action is purely syntactic, and does not need to preserve any semantic property: any function from sets of formulae to formulae counts as a rule of inference.

Usually only rules that are [[Recursion|recursive]] are important; i.e. rules such that there is an [[Effective method|effective procedure]] for determining whether any given formula is the conclusion of a given set of formulae according to the rule. An example of a rule that is not effective in this sense is the infinitary ω-rule.

Popular rules of inference in propositional logic include modus ponens, modus tollens, and contraposition. First-order predicate logic uses rules of inference to deal with [[Quantifier (logic)|logical quantifiers]].

== See also ==

* [[Immediate inference]]
* [[Inference objection]]
* [[Law of thought]]
* [[List of rules of inference]]
* [[Logical form]]
* [[Logical truth]]
* ''[[Modus ponens]]''
* [[Multiple-conclusion logic]]
* [[Quantifier (logic)]]
* [[Structural rule]]
* [[Syntax (logic)]]

== External links ==

* [https://en.wikipedia.org/wiki/Rule_of_inference Rule of inference] @ Wikipedia


[[Category:Logic]]