T5.4: 2 of 4

Let's be clear on a few points before doing derivations with RD:

  1. The sidearguments are called "subderivations". Each subderivation begins with an assumption. (Just type "a", the computer will fill in the rest.) The idea is not that the assumption is true. To the contrary, we make the assumption for contradiction, in an attempt to show the assumption could not possibly be true (given the premises).
  2. Because everthing in the subderivation is suspect in the way just described, we will never cite an individual line. For example, take our one subdervation so far:
    Premise 1 A&~B  
    Assumption 2 ....what if A=B
    2 EQ 3 ....then... (A>B)&(B>A)
    3 &E 4 ....then... A>B
    1 &E 5 ....then... A
    4,5 MP 6 ....then... B
    1 &E 7 ....then... ~B
    2-7 RD 8 ~(A=B)  
  3. we have the subderivation in yellow. In the end, we prove that line 2 is false. So, we are NOT saying that these lines are true. Instead, they are all suspect. We only take them to show that 'A=B' is false. Thus, we should never take anything in yellow to be shown true. The rule is to never menation an individual line in 2-7 once we have "terminated" the subderivation.
    5,6 &I     9 A&B      MISTAKE!!!
    So, for example, if we were to extend this derivation beyond line 8, then we could not write anything like "5,6 &I" at line 9. To reiterate: We can't trust anything is the subderivation. That's the point of a subderivation...to show falsity!
  4. One trick that can help when doing subderivations is to simply rewrite something already derived. Here's the rule:


    This one is usually called "reiteration". But "rewrite" would do just as well.
  5. You're about to try a little quiz. The idea is to show that '~(A&B)' is true. Do this by assuming 'A&B' and deriving a contradiction. Good luck!

Now, let's try that little quiz...