The wffs of PL

 
We are now ready to give a precise formulation of the syntax of PL, that is a definition of what counts as a grammatically correct proposition of PL. We refer to grammatically correct propositions as well-formed formulas or wffs.
 

1. Any sentence letter is a wff.
2. If a is a wff, then ~a is a wff.
3. If a is a wff and b is a wff than (a v b) is a wff.
4. If a is a wff and b is a wff than (a & b) is a wff.
5. If a is a wff and b is a wff than (a ⊃b) is a wff.
6. If a is a wff and b is a wff than (a ≡ b) is a wff.
7. Only strings formed by finite applications of rules 1-6 are wffs.

This is an example of a recursive definition. It provides a complete definition of a concept, and it is recursive because the rules can be used over and over again. If you produce wffs by following the definition, than you can produce others by applying the definition again, thus producing new wffs. For example:
 
 

In the following exercise you'll distinguish wffs from non-wffs. It's easy and fun! After you've done that, try creating your own formal system!