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15.3 Advanced Proof by Cases

When you have multiple disjunctions in your premises, you know you’ll have to do proof by cases. But since you have two (or more), you have to do nested proof by cases: one proof by cases inside another one.

Nested proof by cases: one proof by cases inside another one.

Let’s see what this looks like.

The classic example of this idea is this distribution law:

1. PvQ
2. PvR
Thus,
3. Pv(Q&R)

It doesn’t matter which disjunction we start with, so we’ll just begin with premise 1. We know the proof will look something like this:

1. PvQ
2. PvR
3. | P
. | …
. | Pv(Q&R)
. | Q
. | …
. | Pv(Q&R) 
. Pv(Q&R)

Premise
Premise
Assume


Assume


vElim;1,…

After line 3, we’re not sure what the lines numbers will be yet, so we left them blank.

Look carefully at those two subproofs. One of them will be easy.

Since our goal is Pv(Q&R), if we already know P, it is easy to reach our goal with vIntro.

In the second subproof, we only get to assume Q. The problem is that Pv(Q&R) doesn’t even follow from Q!

That is why this proof only works if we have both premises.

When we do nested proof by cases, it doesn’t mean every subproof will have multiple cases nested within it. Our first case doesn’t need any help.

But within our Q case, we’ll have to start subproofs for the next proof by cases.

Here’s what it will look like:

1. PvQ
2. PvR
3. | P
4. | Pv(Q&R)
5. | Q
6. | | P
. | | …
. | | Pv(Q&R)
. | | R
. | | …
. | | Pv(Q&R)
. | Pv(Q&R) 
. Pv(Q&R)

Premise
Premise
Assume
vIntro;3
Assume
Assume


Assume



vElim;1,…

The disjuncts of our second disjunction are P and R. That is why we started a subproof assuming P on line 6 and another assuming R several lines below that.

You can tell from the double subproof lines “| |” starting on line 6 that these subproofs are nested within Q’s subproof.

It might seem odd that we have to do another subproof for P, when we already did one above. But in a sense, it is just a coincidence that the two premises have a common disjunct. Since we ended our original subproof with P, and we are now considering the second case of that disjunct, Q, we are not allowed to go back and access that information.

The proof by cases rule, vElim, requires that we do subproofs for both disjuncts of premise 2, PvR, and that means we have to consider P again on line 6.

But don’t worry, it will be just as easy this time!

Now it’s your turn to see if you can complete this proof.

Now you can try this one on your own!