Chapter Five, Tutorial One
An Introduction to Derivations

We need to think more seriously about reasoning to conclusions. So, far we only have truth tables to test such reasoning. But truth tables are not a part of everyday life. Of course when you need to know about support for a conclusion, you usually have no time to symbolize your premises and draw up a table! And an argument's validity may depend on non-truth functional matters anyway. Instead, truth tables are good for making precise sense of the concepts of logic.

So, truth tables are not a part of anyone's regular repertoire for thinking and decision. What is? How do we normally use logic in real situations? One answer is "derivations".

Here's a bit of real-life reasoning about someone's travels. First, the premises:

I will either beat the wrap or pay the fine.


I can beat the wrap only if I both pay off the judge and don't get caught at it.

But, finally:

In this country, bribery is seriously frowned upon and not likely to succeed.

So, (because it's unlikely that I'll be able to pay off the judge and not be in deeper trouble):

I'll not be able to beat the wrap.

Thus, by the first displayed premise, the only remaining possibility is:

I'll pay the fine.

This is simple, everyday step-by-step reasoning.

In this chapter, we will develop precise rules for drawing conclusions, in step-by-step fashion, for SL arguments. But before moving to SL, think about the following two premises:

Premise 1: Agnes will attend law school only if Bob will.
Premise 2: Agnes will attend law school but Carola will not.

Which of the following conclusions can be drawn from the above two premises?

  1. Agnes WILL attend law school.
  2. Carola WILL attend law school.
  3. Bob WILL attend law school.
  4. Carola will NOT attend law school.
  5. Bob WILL attend law school but Carola will NOT.