Metatheory - University of Cambridge

6. Completeness 51
and i+4 are both open. I choose one of these open lines to work with; it might
as well be the first, i.e. line 7. I then make a list of all the atomic sentences
and negations of atomic sentences on lines that either occur on the open line
or that occur on earlier lines (but not in a subproof that has been closed before
the open line). So, working with line 7 of my example, my list is: 2
¬C, A, B
There is, indeed, a unique valuation which makes all of these sentences true.
And this valuation will make all of my original assumptions true too. In the
case of my example, this is easily verified.
In a moment, I shall prove that this was not just good fortune, but that
SimpleSearch is guaranteed to behave in this way.
Relaxing the assumption of CNF
First, I need to explain why it was harmless to restrict our attention to sentences
in CNF. The simple reason for this is that, given any sentences, ∆, we can
always find some equivalent sentences, ∆ ∗ , in CNF.
However, we need to be a little careful here. Given any sentences ∆, the
CNF Theorem of §3.5 tells us that there are tautologically equivalent sentences,
∆ ∗ , in CNF. But, given our aims, we cannot afford to rely upon this fact. After
all, if TFL’s proof-system cannot itself prove the equivalences between ∆ and
∆ ∗ , then manipulating ∆ ∗ with TFL’s proof-system will teach us nothing about
the proof-theoretic consequences of ∆. And what we want is an algorithm which
decides whether or not ∆ are jointly contrary.
The upshot of this is that we need to prove a proof-theoretic analogue of
the CNF Theorem.
Lemma 6.3. For any sentence, there is a provably equivalent sentence in CNF.
Moreover, there is an algorithm for determining these equivalents.
Proof sketch. First, prove a proof-theoretic analogue of Lemma 2.2. That is,
show the following: If C ⊢ S, then A ⊢ A[S/C].
Then offer proof-theoretic analogues of Lemmas 3.2–3.4, noting that each
of the tautological equivalences appealed to (namely, Double Negation Elimination,
the De Morgan Laws, and the Distribution Laws) are also provable
equivalences. (Indeed, all but the Distribution Laws are derived rules of TFL’s
proof-system.)
■
⊢
So, given any sentences, we can first put them into CNF, and then apply the
steps that were sketched just earlier: apply ∧E, expand disjunctions, apply ⊥I,
and apply ∨E. Let’s make this precise.
Statement of SimpleSearch
Given any sentences ∆:
is not ‘⊥’. (ii) No instance of ∧E occurs after the line. (iii) The next line in the proof-skeleton
(if there is one) is not ‘⊥’. (iv) the next line in the proof-skeleton (if there is one) does not
depend upon all of the assumptions of the former line.
2 If I had worked instead with line i + 4, my list would have been: C, ¬A, B.
⊢