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?

~(~AvB)
A

One last demonstration...

Start the Demo...
From the premise '~(~AvB)' deduce 'A'.

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 within the subderivation.

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 line 1.

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 be true.

Replay the demonstration.
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