Chapter Five, Tutorial Two
Further
Rules for SD Derivations
In the last tutorial, we presented three rules for drawing conclusions in a derivation. These are all rules of inference: they allow one to draw a conclusion (the "output") from specified "input". In this and the next two tutorials, we shall describe a system of such rules. Taken together, this collection of rules is called "SD" for "Sentence-logic Derivations".
All rules we introduce correspond to valid arguments. For example, our rule "&I" corresponds to the following argument form:
P
Q
P&Q
And all arguments of this form are trivially valid. If you think about the other two rules we've introduced, you will see that they are just as trivially valid.
To do a derivation, then, is just to make a sequence of valid inferences from the original premises. Thus, a derivation from premises to its conclusion (its final line) shows that an argument is valid. All the exercises you've just done in 5.1ex I show that arguments (from the given premises to the given conclusion) are valid. This is just a new way, the SD way, of showing an argument is valid. Most importantly, it's a method much like our normal way of thinking through to a conclusion.
We begin by looking at a rule called "reiteration". As its name implies, it allows one to repeat a sentence already on the derivation.
| R | |
|---|---|
input:  output: |
P P |
Even though you may see no immediate use for reiteration -- after all it just repeats! -- it is clear that is corresponds to a valid inference: any truth value assignment making P true, makes P true!
Now, to review. Given the single premise...
Premise 1. (A>B)&A
...which of the following sentences can you derive from one of our four rules in a single step? That is, which of the following could be line 2 of a derivation having only the premise immediately above and given only the rules so far introduced?