Metatheory - University of Cambridge

6. Completeness 50
1 (A ∨ (¬B ∨ C))
2 (¬A ∨ ¬C)
3 B
4 A want to use ∨E with line 1
5 ¬A want to use ∨E with line 2
6 ⊥ ⊥I 4, 5
7 ¬C want to use ∨E with line 2
i (¬B ∨ C) want to use ∨E with line 1
i + 1 ¬A want to use ∨E with line 2
i + 2 ¬B want to use ∨E with line i
i + 3 ⊥ ⊥I 3, i + 2
i + 4 C want to use ∨E with line i
k ¬C want to use ∨E with line 2
k + 1 ¬B want to use ∨E with line i
k + 2 ⊥ ⊥I 3, k + 1
k + 3 C want to use ∨E with line i
k + 4 ⊥ ⊥I k + 3, k
k + 5 ⊥ ∨E 2, k + 1–k + 2, k + 3–k + 4
At this point, SimpleSearch has run its course. But the output is not a TFLproof
of ‘⊥’ from the initial assumptions: it is a proof-skeleton with several
gaps in it. I claim that I can infer from this that the initial assumptions are
not jointly contrary. I further claim that there is a valuation that makes all of
the initial assumptions true. Moreover, I claim that I can read off this valuation
directly from the proof-skeleton!
I shall justify all of these claims in a moment. First, though, let me explain
how to read off the valuation from the proof-skeleton. Let us say that a line in
a proof-skeleton is open iff there is gap underneath it. 1 In my example, lines 7
1 More precisely a line is open iff it meets all of the following four conditions. (i) The line