Chapter Five, Tutorial Five
SD Tests

We may use derivations and the set of rules we call SD in order to test for many of the properties we have defined in SL. For example -- and something you already know -- one may show an argument valid by giving a derivation. That's what we've been doing so far in this chapter.

To show an SL argument is valid using SD, simply take that argument's premises as the premises of a derivation and derive the conclusion using only the rules of SD.

Keep in mind that deriving the conclusion means deriving it outside of all subderivations

Similarly, we may say that one can derive a sentence P from some set of sentences. This simply means that if one takes the members of the set as premises, then one can derive P.

To show that a sentence P can be derived in SD from set \ of sentences, simply take the members of \ as the premises of a derivation and derive P using only the rules of SD.

These tests are straightforward. But we need to think carefully about some of the other properties before we can be sure of the SD test. Start with the concept of ...

Logical Equivalence

Question: Two members of a pair of sentences are logically equivalent if and only what?

1. the members of the pair are both true.
2. the members of the pair are both logically true.
3. it is not possible for one of the pair to be true while the other is false.