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15.2 The Indirect Approach: ~(P&Q)

The 5-step plan works for negation around disjunction. Can we do the same thing for ~(P&Q)?

Not exactly. We’d like to do this:

1. ~(P&Q)
2. | P
3. | P&Q
4. | #
5. ~P

Premise
Assume
&Intro;2
#Intro;1,3
~Intro;2-4

But that doesn’t work. Can you see why?

No matter how inventive we get, this direct approach won’t work. That’s because the premise ~(P&Q) does not entail ~P. It only entails ~Pv~Q.

So let’s think about doing this proof:

1. ~(P&Q)
Thus,
2. ~Pv~Q

The premise is the same, and isn’t any easier to deal with. But now we can get somewhere with this conclusion.

If you want to figure it out for yourself, stop reading now and break out pen and paper.

If you keep reading, we’ll walk you through the next steps in the thought process.

Okay, that was your chance to stop reading. Since we can’t apply a rule to our premise, and nothing else tells us what to do now, we’re stuck.

Now let’s do some mental visualization. Say we do a reductio of our conclusion, which is ~Pv~Q. That gives us a new assumption to work with.

Can you tell what pattern that assumption has?  In other words, do you have an idea of what to do next?

If you want to figure it out yourself, stop now and go for it! The more you work it out yourself, the better you’ll understand and remember it.

Okay, that was your last chance. Here’s what it looks like:

1. ~(P&Q) 
2. | ~(~Pv~Q)

 ~Pv~Q

Premise
Assume

Notice that line 2 is a negation around a disjunction. You know what to do with that: the 5-step plan!

So the 5-step plan is still the key to solving this pattern. But we don’t do the 5-step plan directly on the premise. We do the 5-step plan inside a reductio.

Let’s see if you can put it all together:

Now that you’ve learned how to handle a premise with negation around a conjunction, you know how to deal with any possible scenario in a formal proof!