Chapter Five, Tutorial Six
Putting it All Together: Safeguards and Strategies for More Complex Derivations
Reasoning is often complicated. In this tutorial we consider strategy for working on more difficult or complicated problems and safeguards against common errors. And we'll put everything we learn together to show how derivations are good tests for logical equivlance, logical truth, and of course validity of arguments.
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.
Here is a first pitfall: I've noticed that a lot of people make one simple mistake: They look at a line that says something like
Premise 1 A>(B&C)
and then type
1 MP .... A
This is wrong! The rule MP requires two inputs! Make sure you know why.
(Moral: You should know your rules very well by now.)
Consider the following argument.
[A>(B=L)] & [C=(X&(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.
But line 1 is so complicated, you may want to despair at finding something to do at line 3.
But don't! Despair-and-Assume 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. Besides, when you make an assumption, you are trying to prove it wrong, not true.
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...
|1 &E||3.||B>L (Mistake!)|
Now, what's the mistake here?