In some of the previous homework assignments, I have asked you to
derive formulas from scratch. These formulas have been tautologies. A
tautology is a formula that is true by its logic alone. If you wrote a
truth table for a tautology, it would be true in every row. For instance, P
<-> P is a tautology, and here is its truth table:

        P <-> P
        -------
        T  T  T
        F  T  F

        Since a tautology is always true, we should be able to derive any
tautology without the use of premises. Before we had CP, we could not do
that, for all our rules required premises. But CP does not require any
premises. CP allows us to make assumptions, which we can use as premises.
So, by using CP, we can derive any tautologous conditional. Consider, for
instance, this proof of the law of identity (P -> P):

               1.       P                   Assumption
               2.       P                   1, R
               3.   P -> P                  1-2, CP

        As you can see, this proof used no premises. Instead, it began with
an assumption. We can also use CP to derive tautologies that are not
conditionals. We can derive biconditionals by deriving two conditionals
with CP. We can derive disjunctions and conjunctions by deriving
conditionals, which we then change into disjunctions and conjunctions with
the rules of replacement. Any proof of a tautology must begin with the
proof of a conditional, but it does not have to end with the proof of a
conditional.