T5.6 8 of 9

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

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' can*not* be true.

MouseOver
the sign to *pause* the demo.