We start by writing
Before we go on, I want to point out a few things.
The word "show" is there to indicate that we want to prove Kd, but
we haven't proved it yet. That means we can't do anything with Kd.
If the word "show" wasn't there, Kd would be available to do things with,
so long as it isn't in a box. (I'll explain about being in boxes later. It's
not important now.) Since "availability" is important, I'll put it in a definition:
An available line is a line in the derivation that does not have
an uncancelled "show" in front of it, and is not currently inside
a box. Only whole lines can be available. Parts of lines are
never available.
Remember, line one above is not available.
Basically, you can write the word "show" followed by any formula you
like at any time. However, writing the wrong "show" line can make your
derivation impossible to finish!
Now we need rules to allow us to do things inside derivations. Since a premise
is something that is taken as true by the argument, (it may or may
not be true in real life) we should be able to write down premises. So:
Rule 1. Premise (Abbreviated by "P") If "P" is a premise of the argument being derived, then "P" may be written as a new line in the derivation. (The first six rules are conveniently listed on Logic Rules Sheet One) 
New lines are always written in by the very next free number. Given this
rule, we can add to our derivation.
The rule "premise" allows us to write down Kd, because it's a premise. We
write "P" off to the side to remind us which rule we used. But wait! Kd is
also what we're trying to show here! So we've done it. All we need
now is a rule that lets us show that we've finished.
Rule 2. Box and Cancel (Direct Proof) If the formula indicated by "show" appears as an available line below that show line, and there is no other uncancelled "show" between that show line and that formula, then that "show" may be cancelled, and a box may be drawn around that formula and any other lines there may be between it and it's show line. If the relevant show line is line #1, then the derivation is successfully finished. (The first six rules are conveniently listed on Logic Rules Sheet One) 
And we're done! (Since we don't need line 3, we can ignore it.)
I want to emphasize, as clearly as I can, that derivations only prove validity.
A successful derivation does not prove that the argument has a true conclusion.
All it proves is a sort of conditional, "whatif" kind of fact. A successful
derivation proves that if (and that's a big if) all the premises are true,
then (and only then) the conclusion cannot be false. Again, it does not prove
that the conclusion is true, it just proves the logical relationship between
the premises and the conclusion. It's a very important relationship, which
is why it has its own special name, but it's just a logical relationship.
Rule 3. Conjunction (Abbreviated
by "CJ")
If "P" and "Q" are both available lines, they can be joined together by putting a " ^ " between them to make a new sentence "P ^ Q" which may be written as a new line in the derivation. (The first six rules are conveniently listed on Logic Rules Sheet One) 
Rule 4. Simplification (Abbreviated
by "S")
If "P ^ Q" is an available line, then either "P" or "Q" may be written as a new line in the derivation. (The first six rules are conveniently listed on Logic Rules Sheet One) 








Practice 6. Use your own paper or the answer sheet at practice 06. For each of the following arguments, determine whether it is valid or invalid. If it's valid prove it valid by DERIVATION. If it's invalid prove it invalid by venn diagram or truth table.
