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14.5 Know Your Conclusions

We’ve been talking about the main connectives of premises. Since you know that premises ≠ conclusions (Section 14.3), when you face a main connective in the conclusion, it’s not the same as what you learned in the last section.

& conclusion: first prove each conjunct, and build up with &Intro.

The simplest case is a conjunction conclusion. When you need to prove P&Q, you build it up. First you prove P, and then you prove Q, and you put them together with &Intro.

v conclusion: create it from one side with vIntro.

If your conclusion is a wide-scope disjunction, you’ll probably create it with vIntro.  You first need to get just one side or the other, and then you can make the disjunction with vIntro.

Heads up: vIntro is usually in the middle of the proof, not the last line!

Warning: but there’s an important difference between vIntro and &Intro. When your conclusion is a conjunction, you’ll usually do &Intro on the final line. It’s nice and straightforward.

But when your conclusion is disjunction, and we say you’ll make the conclusion with vIntro, we don’t mean you’ll always do that last, on the final line of the proof. In fact the vIntro is rarely on the final line.

It’s not very interesting to prove PvQ when you could just prove P. More interesting is to prove PvQ when you really can’t know whether it is Pia or Quinn who is guilty, but you can still know it is one or the other.

Here’s an example.

Notice that we can’t first prove P and then add “vQ” with vIntro. Our premise doesn’t entail P! (Ditto for Q).

Our formal proofs only allow us to make valid steps, so an invalid plan will never work.

You can’t come up with an invalid proof plan, and just blindly hope it will work. Our formal proofs only allow us to make valid steps, so an invalid plan will never work.

We still created our conclusion from vIntro, but we did it in the middle of the proof, in those subproofs. And we justified the final conclusion line in some other way.

That is usually how wide-scope disjunction conclusions are: you’ll create them with vIntro, but in the middle, not at the end.

Now let’s consider the third main case: if your conclusion is a wide-scope negation.

~ conclusion: do a reductio, ~Intro!

If the conclusion is ~, do a reductio! This a key lesson: reductio is pretty much always the right plan when your conclusion is a ~, especially if it is a negation around a complex sentence, like ~(P&Q).

Do you see the difference?

Just to be clear: do not do a reductio of this: ~P&Q.

Do you see the difference? ~P&Q is a conjunction, not a negation. Remember: it is all about the main connective!

By contrast, negation is the main connective in the conclusion of this form of DeMorgan’s:

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

Hopefully you’re so tuned in to the main connectives that you also noticed the premise is a disjunction. So you know you need to do proof by cases.

Instead of not knowing what to do,  you might feel like you have too many ideas of what to do: should you do reductio first, or proof by cases first?

Reductio for a negation conclusion is such a good plan, you almost always do it first.

Short answer: reductio! Reductio for a negation conclusion is such a good plan, you almost always do it first.

Long answer: both methods will work, one is just faster than the other. You’re going to have to use reductio in this proof. If you start proof by cases first, you’ll have to do reductio twice, one in each subproof.

We teach you the fastest way to solve proofs, but any correct proof is good, even if it’s longer.

There’s a more general lesson here. In the textbook we teach you the fastest way to solve proofs, and we often ask you to figure out how we solved a proof. But that doesn’t mean there’s only one correct way to solve a proof. Any proof that properly applies the rules counts as good.

In fact, there is never only one way to do a proof. But often there is a fastest way.

Let’s complete the proof of DeMorgan’s that we’ve been talking about.

You’ll notice that even after starting with reductio, you’ll have to decide when to do &Elim. Hint: we did it after starting proof by cases. (You can tell by looking carefully at the subproof lines.)

We suggest using a pen and paper first. Remember the benefits of active learning: the more active you are, the better you will remember it!

Since there are many different ways to do a proof, we suggest using a pen and paper first, so you have total freedom to experiment.

Afterward, you can look at how the subproofs are structured in the problem in this textbook, or on your homework problem set, to see if they match up.

If they don’t match, then the first thing to consider is what order you carried out your plans in.

We told you that starting with reductio is a good idea. You also learned in Section 14.1 that you should do cases from left to right when you do proof by cases.

But nothing says you need to start proof by cases in the above proof, before doing &Elim. Sometimes there are equally good and equally fast ways to do a proof.

The key is pay attention to the subproof lines. They often tell you what order to do steps in.

As we said before, to figure out how we did it, and the answer we are looking for, the key is to pay attention to the subproof lines. The subproof lines often tell you what order to do steps in.

For example, in the proof above, line 2 starts a subproof because there is one subproof bar “|”. Line 3 then starts a second subproof, because there are two subproof bars “| |”. That means we have to start our proof by cases assumption on line 3, instead of doing &Elim.

In the next problem, we ask you to solve it in a slightly different order (but we still start with reductio).

In Section 14.7, Extra Practice, we ask you to do this same DeMorgan’s proof yet a third time, so that you can see how it would work if you started with proof by cases instead of reductio.

Don’t miss it!

Don’t miss it! If you thoroughly master this proof, and understand what it looks like to solve it in different ways, you’ll have taken major steps to mastering formal proofs.

Lastly, there’s one final thing to know about conclusions. In other textbooks you’ll often hear the advice to “work backwards” on difficult formal proofs.

The trouble is, they don’t really tell you how to work backwards. They just provide a few examples and hope you’re a quick learner.

Looking at the main connective of the conclusion basically is “working backwards.”

When we tell you to look at the main connective of the conclusion, we are basically giving you the same advice: that is working backwards.

What our advice tells you, though, is what to do with a conclusion when you work backwards: you look at its main connective to formulate a plan.