Chapter 5, Tutorial 5
Derived Rules: Helpful Shortcuts
In an advanced logic class, one might prove that the rules we've already given are quite sufficient to prove any valid argument as valid by derivation. (The derivations are enough, you don't need truth tables any longer; hooray!)
So, our rules all trustworthy -- none will go from true input to false output -- and they also form a set complete enough to prove any valid argument valid. Put another way, we have all the essential rules.
Still, you may still think of further rules you might want to have, rules that are trustworthy, and will allow you to short-cut derivations. These are optional rules but very helpful in practice.
Here's an obvious sort of situation illustrating how it would be nice to have more rules.
One looks at the two premises and thinks MP right away.
But they are not in exactly the right form. True, 1 says that if 'A' and 'B' have the same truth value then C is true. And 2 says that they do have the same truth value. So, one wants to conclude 'C' right away.
However, the computer won't let you. This is not MP. Close but not close enough: Premise 2 is not exactly the same thing as the antecedent of 1...it's switched around! What to do?
We could do the switching around in a long derivation like so: 
This is clearly a royal pain...2-7 just switch around to say the obvious:that if 'A' and 'B' have the same truth value, then 'B' and 'A' have the same truth value!
So, hereafter, let's just do this in one step by means of a new rule.
This one a a rule of replacment called CM:
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This new rule is not so hard to memorize. We are just commuting around the '=', so we'll call this one commutation. As a bonus, we'll give it for 'v' and '&' too. You can show later, as homework, that these are "derived" or short-cut rules too. (You don't really need any of these but they help.
CM is a rule of replacement, so it works in either direction and can be used on the components of long sentences.
Here's our new version of the derivation above, now using CM. It's much more natural.
And all you have to remember is that you can commute around '&', 'v', and '='.
Question: Why is there no commutation rule for '>'? (Hint: think about the old Fire/presence of oxygen examples, T2.1, for the answer.)
Now, think about this start to a derivation, 
and say which of the following could be on line 2: