T5.3 p3 Conditional Introduction

T5.3 3 of 5

Subderivations

Our rule for conditional introduction requires a subderivation. Again this means that we make an assumption, asking "what if" some assumption is true. That's the motivating idea. Let's now look at the rules governing a subderivation.

First, at any time you want, you may make an assumption: in the justification field you need only write "Assumption". No line numbers need be cited. In this way, assumptions are like premises. One just makes them. However, unlike a premise, an assumption is placed to the right to form the subderivation.

Most importantly, the statements after the assumption are placed under it in the same column. That column indicates the subderivation. The only way to move back out of this subderivation, back into the column under the premises, is to use a rule (>I is all we have for now) which sanctions the termination of the subderivation.

Once a subderivation is terminated, nothing within it may be cited to justify further derivation. We say in such a case that the assumption made by the subderivation is discharged. In this case we are no longer relying on that assumption for what we derive.

Note that though one may not cite any line or lines within a terminated subderivation, one may site the whole subderivation.*

Whew! This does get a bit complicated. But it's worth having read these details. Now let's see it in action.

MouseOver the
Replay the demonstration.

From the premise '(B&S)=A' deduce 'A>S'. We will say that 'A>S' is our goal.

Helpful but Optional: One may write down the goal sentence toward the end of the derivation -- but with question marks -- so we keep in mind just where the thinking is headed.

Keeping this goal in mind should make us think of >I and the possibility of making an assumption. We assume the antecedent of the goal.

Now, we are working within a subderivation. We assumed 'A' and hopefully will be able to derive 'S'. (Why?) Toward this intermediate goal we now put 1 and 2 together...

Aha! This allows us to justify 'S' on line 4; just what we need...

...because to derive 'A>S' by >I requires only the subderivation from 'A' (line 2) to 'S' (line 4).

In fact, "If we assume 'A', then it follows that 'S' holds" is just what the subderivation means. So, it makes sense that we have now justified 'A>S'.

Here's a general strategy which should always work: When asked to derive a sentence P>Q, assume P and try to (sub)derive Q -- i.e., derive Q within the subderivation.

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