Chapter Five, Tutorial Six
SD Safeguards and Strategy

Most of the derivations we've done up until this point have not been very difficult -- we've needed to give relatively easy applications of the rules and simple presentations of the basic deductive concepts. But reasoning is often more complicated. In this tutorial we consider strategy for working on more difficult or complicated problems and safeguards against common errors. Let's begin with the latter.

Alert!

Safeguards

As derivations get more difficult you may sometimes feel the need to "cheat" on the rules just to do something...anything...to get started on the derivation! Unfortunately, this desperation leads to misapplication of rules and incorrect derivations. The solution is to learn strategies for proceeding. We will look to these strategies later in this tutorial. But we begin by looking at common pitfalls to avoid.

Consider the following argument.

[A=(B=L)] & [C=(X&(B>L))]
            A             
B>L

A mess! At first, it's hard to even begin to think about this problem. And easy to try to make a "short-cut" not sanctioned by our SD rules. So, let's look at it and see how to avoid these mistakes.

Of course, because you are attempting to show this argument valid by using a derivation of 'B>L' , you will take the two premises as lines one and two of a derivation beginning as follows.

Premise 1. [A=(B=L)]&[C=(X&(B>L))]
Premise 2.  A
??? 3. ???

But line 1 is so complicated, you may want to despair at finding something to do at line 3. But don't! Despair is the first pitfall. It makes many students want to simply assume the goal, e.g., to take line 3 to be 'B>L' justified by "Assumption". One is allowed to do this, but it is no help toward what we need: 'B>L' outside any subderivation.

However, there's no need to despair and just start making assumptions. In the pages ahead, we will discover all sorts of strategies for doing derivations. So, hang tough and you will soon be used to complicated derivations.

Now, line 3 needs to be something. The second pitfall is short-cutting a derivation by making up a rule to illicitly jump to a desired conclusion. For example...

Premise 1. [A=(B=L)]&[C=(X&(B>L))]
Premise 2.  A
1 &E 3.  B>L     (Mistake!)

Now, what's the mistake here?

  1. '&' is not the main connective of 1.
  2. The justification should be '1,2 &E'.
  3. 'B>L' is not one of the immediate components of the main connective of 1 (the first ampersand!).