Supplementary: Why are we doing all these derivations anyway?

3/16/06

So that we can show certain arguments to be valid.

The focus of the course lies with the validity and invalidity of arguments. Now, invalidity can be established by counter-example (by producing an interpretation under which all the premises are true and the conclusion false, at the same time). But validity is a different matter. And the usual approach is to have rules of inference and to do derivations.

The rules of interference that we have (eg And Introduction, And Elimination etc.) are chosen to have certain properties. In particular they are chosen collectively to have the properties of consistency and completeness. Consistency here means that any (propositional) argument that has a derivation is valid. And completeness means that any argument that is (propositionally) valid has a (propositional) derivation. And similar results hold for predicate logic and suitable predicate rules of inference.

So, an argument is valid if and only if it has a derivation.

And we do derivations to become used to proving validity. (A typical education in Logic does not require doing hundreds and hundreds of derivations, but certainly enough need to be done to be familiar with the process of doing it. In this, Logic is similar to areas of math or stats.)

These properties of rules of inference, consistency and completeness, can be proven to hold. In more advanced Logic courses (sometimes called courses in Meta-Logic) meta-theorems like such-and-such rules of inference are consistent, or such-and-such rules of inference are complete, will be proved.