Finally, we should look at a problem involving ~E. This rule may be used
when an accessible line has main connective tilde. But it is often a good
idea to use other rules first. However, if no other means presents itself
if you're getting desperate! just use assume the negation
of your current goal sentence and derive any contradiction.
Often problems leading to desperation have premises which are negations of
molecular sentences. For instance, how would we show the following valid?
Start the Demo...
From the premise '~(~AvB)' deduce
Because (step 1) our ultimate goal is to derive 'A', and (step 2) what to do
isn't obvious, step 3 suggests that we may be able to profitably
apply ~E. Reason: tilde is the
main connective of our only accessible line.
Thus, we assume the negation of our goal sentence.
~E always assumes the negation of a goal!!!
Once again, we have a new goal: to derive a contradiction
But what contradiction?
Here's a hint about the contradiction to derive: from our assumption '~A' we
can derive '~AvB' just by using vI.
Compare that to line 1.
Now, it's easy to get a contradiction within the subderivation: just reiterate
This is what one can do in general: reiterate a negation for contradiction.
(There's no way to break up line 1. All we can do is reiterate it.)
Finally, we may terminate the subderivation. Line 5 is proven because we've
seen that '~A' cannot