Okay, now lets apply our three step strategy analysis to a more difficult
problem. Use derivations to show that...
may now see how the three strategy steps work for this more complicated example...
Start the Demo.
From the premise 'D>~B'
Because (step 1) our (ultimate) goal is to derive '(B&C)>~D',
and (step 2) what to do isn't obvious, step 3 suggests that we may hope
to finish the derivation by means of >I.
Thus, we assume the antecedent of our goal
Our plan to use >I provides
us with a new goal: we hope to derive the consequent '~D' by line
6 (this is step 1 again).
If it's not obvious what to do next, then step 3
again suggests the introduction rule for our goal's main connective: ~I.
This requires a second assumption and a subderivation within
Once again, we have a new goal: to derive a contradiction
within the second subderivation.
It may be obvious how to proceed at this point, but if
not move to step 3 and apply E-rules (there is no I rule to apply). We
should only use E-rules that move toward our goal of a contradiction.
line 5, we are able to terminate the second subderivation -- the subderivation
within a subderivation -- lines 3 - 5.
Finally, citing lines 2 through 6, we have shown that
if 'B&C' holds, then '~D' does too. Thus, line 7 is justified.
Before moving on, make sure you understand how the strategy steps are utilized
time and again. With a little practice they become very natural.