The Standard Form of Rules of Inference
In formal logic (and many related areas), rules of inference are usually given in the following standard form:
Premise#1
Premise#2
...
Premise#n
Conclusion
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:
A→B
A
B
This is just the modus ponens rule of propositional logic. Rules of inference are often formulated as schemata employing of metavariables. In the rule (schema) above, the metavariables 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.
A proof system is formed from a set of rules 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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