Rule of inference

In logic, a rule of inference is a function from sets of formulae to formulae. The argument is called the premise set (or simply premises) and the value the conclusion. They can also be viewed as relations holding between premises and conclusions, whereby the conclusion is said to be inferable (or derivable or deducible) from the premises. If the premise set is empty, then the conclusion is said to be a theorem or axiom of the logic.

A desirable property of a rule of inference is that it be effective in the sense of e.g. Church 1956. That is, there is an effective procedure for determining whether any given formula is inferable from any given set of formulae. A rule that is not effective is the infinitary omega rule.

A rule of inference needn't preserve any semantic property such as truth or validity. In fact, there is nothing requiring that a logic characterized purely syntactically have a semantics. A rule may preserve e.g. the property of being the conjunction of the subformula of the longest formula in the premise set. However in many systems rules of inference are used to generate theorems from each other (i.e. to prove theorems).

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. Axiom schemata can also be viewed as rules of inference with zero premises.

Note that there are many different systems of formal logic each one with its own set of well-formed formulas, rules of inference and, sometimes, 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.

In the setting of 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

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.

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."

Admissibility and Derivability

In a set of rules, an inference rule could be redundant in the sense that it is admissible or derivable. A derivable rule is one whose conclusion can be derived from its premises using the other rules. An admissible rule is one whose conclusion holds whenever the premises hold. All derivable rules are admissible. To appreciate the difference, consider the following set of rules for defining the natural numbers (the judgment $n\,\,{\mathsf {nat}}$ asserts the fact that $n$ is a natural number):

${\begin{matrix}{\frac {}{\mathbf {0} \,\,{\mathsf {nat}}}}&{\frac {n\,\,{\mathsf {nat}}}{\mathbf {s(} n\mathbf {)} \,\,{\mathsf {nat}}}}\\\end{matrix}}$

The first rule states that 0 is a natural number, and the second states that s(n) is a natural number if n is. In this proof system, the following rule demonstrating that the second successor of a natural number is also a natural number, is derivable:

${\frac {n\,\,{\mathsf {nat}}}{\mathbf {s(s(} n\mathbf {))} \,\,{\mathsf {nat}}}}$

Its derivation is just the composition of two uses of the successor rule above. The following rule for asserting the existence of a predecessor for any nonzero number is merely admissible:

${\frac {\mathbf {s(} n\mathbf {)} \,\,{\mathsf {nat}}}{n\,\,{\mathsf {nat}}}}$

This is a true fact of natural numbers, as can be proven by induction. (To prove that this rule is admissible, one would assume a derivation of the premise, and induct on it to produce a derivation of $n\,\,{\mathsf {nat}}$ .) However, it is not derivable, because it depends on the structure of the derivation of the premise. Because of this derivability is stable under additions to the proof system, whereas admissibility is not. To see the difference, suppose the following nonsense rule were added to the proof system:

${\frac {}{\mathbf {s(-3)} \,\,{\mathsf {nat}}}}$

In this new system, the double-successor rule is still derivable. However, the rule for finding the predecessor is no longer admissible, because there is no way to derive $\mathbf {-3} \,\,{\mathsf {nat}}$ . The brittleness of admissibility comes from the way it is proved: since the proof can induct on the structure of the derivations of the premises, extensions to the system add new cases to this proof, which may no longer hold.

Admissible rules can be thought of as theorems of a proof system. For instance, in a sequent calculus where cut elimination holds, the cut rule is admissible.

Other Considerations

Inference rules may also be stated in this form: (1) some (perhaps zero) premises, (2) a turnstile symbol $\vdash$ which means "infers", "proves" or "concludes", (3) a conclusion. This usually embodies the relational (as opposed to functional) view of a rule of inference, where the turnstile stands for a deducibility relation holding between premises and conclusion.

Rules of inference must be distinguished from axioms of a theory. In terms of semantics, axioms are valid assertions. Axioms are usually regarded as starting points for applying rules of inference and generating a set of conclusions. Or, in less technical terms:

Rules are statements ABOUT the system, axioms are statements IN the system. For example:

The RULE that from $\vdash p$ you can infer $\vdash Provable(p)$ is a statement that says if you've proven p, then it is provable that p is provable. This holds in Peano arithmetic, for example.
The Axiom $p\to Provable(p)$ would mean that every true statement is provable. This, however, does not hold in Peano arithmetic.

Rules of inference play a vital role in the specification of logical calculi as they are considered in proof theory, such as the sequent calculus and natural deduction.