Metatheory - University of Cambridge

6. Completeness 54
Now let D be any sentence among ∆; I claim that v makes D true. To
prove this claim, I first consider the CNF sentence D ∗ generated in Step 1
of SimpleSearch, such that D ⊢ D ∗ . It is easy to show that, for every
(conjoined) disjunction in D ∗ , at least one disjunct of that disjunction appears
in Ω; it follows that each of the disjuncts is made true by v, so that D ∗ is made
true by v (I leave the details of this as an exercise). Now, since D ∗ ⊢ D, the
Soundness Theorem tells us that D ∗ ⊨ D; so v must make D true too.
Proof of (3). Suppose, for reductio, that ∆ ⊢ ⊥. Then, by the Soundness
Theorem, there is no valuation that makes all of the ∆ true. This contradicts
what we have just shown in point (2).
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⊢
And now – at long last – since we can trust the outputs of SimpleSearch, we
can show that SimpleSearch has properties (a1) and (a2):
Theorem 6.7. Given any sentences ∆:
1. If ∆ ⊢ ⊥, then SimpleSearch yields a TFL-proof which starts with ∆
as assumptions and terminates in ‘⊥’.
2. If ∆ ⊬ ⊥, then SimpleSearch shows us as much and yields a valuation
which makes all of ∆ true.
Proof. Suppose ∆ ⊢ ⊥. Then by Lemma 6.6, SimpleSearch does not output a
proof or proof-skeleton whose last line is not ‘⊥’ on no assumptions other than
∆. But by Lemma 6.4, SimpleSearch does terminate. So the last line of the
output must be ‘⊥’, on no assumptions other than ∆, which, by Lemma 6.5,
is an explicit TFL-proof that ∆ ⊢ ⊥.
Suppose ∆ ⊬ ⊥. Then by Lemma 6.5, SimpleSearch does not output a
proof-skeleton whose last line is ‘⊥’ on no assumptions other than ∆. Now
apply Lemma 6.6.
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And with this result, we have achieved what we set out to achieve! But it might
be worth reminding ourselves why we set out on this course. Our target for this
chapter was to prove the Completeness Theorem. To prove that, we needed to
prove Lemma 6.2. But Theorem 6.7 immediately entails Lemma 6.2 (just look
at the second condition). So we have proved the Completeness Theorem, and
acquired a useful little algorithm along the way!
6.3 A more abstract approach: polarising sentences
We could stop now. But I want to offer a second proof of the Completeness
Theorem. It is more abstract, more general, and in a sense, more direct.
From the perspective of proving the Completeness Theorem, it is massive
overkill to deploy SimpleSearch. For our purposes, all that matters is the
following: when ∆ ⊬ ⊥, there are some atomic sentences and negations of
atomic sentences, Ω, from which we can read off a valuation that makes all of
∆ true. From an abstract point of view, we needn’t care how SimpleSearch
generates Ω; only that Ω exists. The second proof of the Completeness Theorem
builds upon this observation.