THE RULE OF ARROW IN

We have already looked at the simple rule of Arrow Out (→O). That rule is an elimination rule. In the simplest case, it allows us to move from two formulae, one of which is a conditional, to a formula which is not a conditional.

The rule of Arrow In (→I) is an introduction rule. It allows us to infer a formula that is itself a conditional. We need this rule to prove a sequent such as:

(1) A → B, B → C, C → D ˫ A → D

(1) is an intuitively valid sequent. But we cannot show it to be valid using only →O. Hence the need for the new rule of Arrow In:

(→I) If P (typically together with other premises or assumptions) was used to derive Q, then we can derive P → Q.

Note two conditions here: (i) in order to apply a step of → I, P must be either a premise or assumption and (ii) P must have been used in the derivation of Q. These are important conditions.

The rule of → I, together with these conditions, introduces some new ideas and techniques. In particular, we must grasp the distinction between assumptions and premises, and the corresponding need for a numbering system on the left-hand side of a proof.

Let us look at a sample proof using → I, and then we shall clarify the new ideas involved.