# Inference rules for disjunction

Resources and clarifications

Today in class, we discussed inference rules in natural deduction. There are some excellent resources out there but this chart can be helpful. The sections from the Lean theorem prover book on natural deduction for propositional and first-order predicate appear clear and useful, but I have not investigated them closely.

In any case, most of the inference rules are fairly self-explanatory, but the rules for or-elimination can be tricky.

Suppose we have $$F \vee G$$. To be very clear, let's suppose we have some sequent $$F\vee G, H \vdash F$$. How do we get $$F$$ out? Well, to do this we need two proofs: a proof of $$F$$ and a proof of $$G$$. Our scoping would look like so:

     |================================================================|
2.   | F \/ G                                                         |
3.   | H                                                              |
|----------------------------------------------------------------|
| |==============================================================|
4.   | | assume F                                                     |
| |--------------------------------------------------------------|
5.   | | F (trivially)                                                |
| |==============================================================|
| |==============================================================|
6.   | | assume G                                                     |
| |--------------------------------------------------------------|
7.   | | ...proof steps, possibly using information from (2, 3)       |
n.   | | derive F                                                     |
| |==============================================================|
n+1. | F by or-elmination (2, 5, n)                                   |
|================================================================|


Now, this particular example is trivial, but it illustrates the main issue of why you need two proofs — when we assume $$F$$, we open a new scope. We can't let our assumption that $$F$$ is true escape the scope and therefore we can't use it outside the scope. The way or-elimination works is that you must provide a proof of the goal (here, $$F$$) under each of the assumptions that $$F\vee G$$ is true, in order to use either one at will. This might be easier to understand if our goal were some new formula instead: $$F\vee G, H \vdash J$$. Then we would have:

     |================================================================|
2.   | F \/ G                                                         |
3.   | H                                                              |
|----------------------------------------------------------------|
| |==============================================================|
4.   | | assume F                                                     |
| |--------------------------------------------------------------|
5.   | | ...proof steps, possibly using information from (2, 3)       |
n.   | | J                                                            |
| |==============================================================|
| |==============================================================|
n+1  | | assume G                                                     |
| |--------------------------------------------------------------|
n+2. | | ...proof steps, possibly using information from (2, 3)       |
m.   | | J                                                            |
| |==============================================================|
m+1. | J by or-elmination (2, 5, n)                                   |
|================================================================|

Perhaps this is more intuitive because the addition of $$J$$ decouples the proof from each new assumption (4, n+1). In any case, the smallest possible value of $$J$$ is going to be one of $$F$$ or $$G$$.

There are two other observations worth meditating on:

1. We cannot discharge either of $$F$$ or $$G$$ without more information — that is, unlike and-elimination, we don't get to just pick one of the two. This is why I had included $$H$$ above.

2. The scopes of the two assumptions (lines 4 through n and lines n+1 through m+1) are non-overlapping, so none of the information inside one can be used inside the other. This is why at step n+2, we can use information from lines 2 or 3, but not 4 through n.