NEGATIONS

        As you know, a capital letter represents a statement that is either
true or false.  (For my understanding of truth, I refer you to TOP 2).  A
single letter is the simplest sort of symbolic expression we can have.  A
slightly more complicated symbolic expression is the negation.  We
represent the negation of a symbolic expression by putting a tilde, ~, in
front of it.  Thus, ~P represents the negation of P.  The truth value of a
negation is always the reverse of what it negates.  The following truth
table illustrates this point:

        P   ~P
        ------
        T   F
        F   T

        The letters beneath P represent the possible truth values for P.
Of course, T means true, and F means false.  The letters beneath the tilde
represent the truth value for ~P for each possible truth value of P.  You
should make sure that you understand the concept of a truth table, for I
will use even more complicated ones to describe the other symbols, and I
will continue to use truth tables further on.

        CONJUNCTIONS

        You may remember an old Schoolhouse Rock cartoon about
conjunctions.  It likened them to what connect trains together.  They were
words such as and, or, for, nor, but.  These words connect two thoughts in
one sentence.  In logic, conjunction takes on a stricter meaning.  A
conjunction says that two statements are both true.  Or, for, and nor do
not do that;  And and but do.  The conjunction (P & Q), for instance, says
that P and Q are both true.  That means the conjunction is true only if
both P and Q are true.  The truth table for a conjunction looks like this:

        P   Q   (P & Q)
        ---------------
        T   T      T
        T   F      F
        F   T      F
        F   F      F

        Note that this table has more rows than the previous truth table.
To get all the possible truth values, we had to get all the possible
combinations of truth values.