Metatheory - University of Cambridge

6. Completeness 46
been expanded. Note that, when we expand a disjunction by introducing new
assumptions, we must leave gaps underneath these new assumptions. We will
fill in these gaps as we continue to implement our algorithm. But, since we
do not yet know how these gaps will be filled in, we cannot yet assign definite
numbers to all the lines of our proof. So what we have written down cannot
any longer be described as a TFL-proof; rather, it is a proof-skeleton. This
is how I shall refer to it, in what follows.
We have expanded our first disjunction, thereby positioning ourselves for a
(potential) later use of ∨E with line 1. But there is another disjunction on line
2. We need to expand this disjunction too, positioning ourselves for a (potential)
later use of ∨E with line 2. At this point, however, some care is needed.
We have just made two new assumptions – ‘A’, and ‘B’, separately – and we
might be able to use ∨E with line 2 inside the scope of either assumption. So
we must continue our proof-skeleton as follows:
5 A want to use ∨E with line 1
6 ¬B want to use ∨E with line 2
¬C want to use ∨E with line 2
B want to use ∨E with line 1
¬B want to use ∨E with line 2
¬C want to use ∨E with line 2
At this point, all the disjunctions in our proof-skeleton have been expanded.
So we can move on to the next task.
Recall that we started with sentences in CNF. We first deal with all the
conjunctions (by ∧E). We then unpacked all the disjunctions. So our proofskeleton
will now have lots of lines on it that contain simply an atomic sentence,
or its negation. So the next task is to apply the rule ⊥I wherever possible. In
our example, we can apply it four times: