Rules of Inference

A rule of inference is a pattern of reasoning consisting of some schemas, called premises, and one or more additional schemas, called conclusions. Rules of inference are often written as shown below. The schemas above the line are the premises, and the schemas below the line are the conclusions.

The rule in this case is called Implication Elimination (or IE), because it eliminates the implication from the first premise.

Implication Creation (IC), shown below, is another example. This rule tells us that, if a sentence ψ is true, we can infer (φ ⇒ ψ) for any φ whatsoever.

Implication Distribution (ID) tells us that implication can be distributed over other implications. If (φ ⇒ (ψ ⇒ χ)) is true, then we can infer ((φ ⇒ ψ) ⇒ (φ ⇒ χ)).

Implication Reversal (IR) allows us to infer an implication if we have an implication with the arguments reversed and negated. If we know (¬ψ ⇒ ¬φ), we can infer (φ ⇒ ψ).

An instance of a rule of inference is the rule obtained by consistently substituting sentences for the metavariables in the rule. For example, the following is an instance of Implication Elimination.

If a metavariable occurs more than once, the same expression must be used for every occurrence. For example, in the case of Implication Elimination, it would not be acceptable to replace one occurrence of φ with one expression and the other occurrence of φ with a different expression.

Note that the replacement can be an arbitrary expression so long as the result is a legal expression. For example, in the following instance of Implication Elimination, we have replaced the variables by compound sentences.

Remember that there are infinitely many sentences in our language. Even though we start with finitely many propositional constants (in a propositional vocabulary) and finitely many operators, we can combine them in arbitrarily many ways. The upshot is that there are infinitely many instances of any rule of inference involving metavariables.

A rule applies to a set of sentences if and only if there is an instance of the rule in which all of the premises are in the set. In this case, the conclusions of the instance are the results of the rule application.

For example, if we had a set of sentences containing the sentence p and the sentence (p ⇒ q), then we could apply Implication Elimination to derive q as a result. If we had a set of sentences containing the sentence (p ⇒ q) and the sentence (p ⇒ q) ⇒ (q ⇒ r), then we could apply Implication Elimination to derive (q ⇒ r) as a result.

In using rules of inference, it is important to remember that they apply only to top-level sentences, not to components of sentences. While applying to components sometimes works, it can also lead to incorrect results.

As an example of such a problem, consider the incorrect application of Implication Elimination shown below. Suppose we believe (p ⇒ q) and (p ⇒ r). We might try to apply Implication Elimination here, taking the first premise as the implication and taking the occurrence of p in the second premise as the matching condition, leading us to conclude (q ⇒ r).

Unfortunately, this is not a proper logical conclusion from the premises, as we all know from experience and as we can quickly determine by looking at the associated truth table. It is important to remember that rules of inference apply only to top-level sentences.