Metatheory - University of Cambridge

5. Soundness 41
i
j
A
⊥
n ¬A ¬I i–j
Let v be any valuation that makes all of ∆ n true. Note that all of ∆ n are
among ∆ i , with the possible exception of A itself. By hypothesis, line j is
shiny. But no valuation can make ‘⊥’ true, so no valuation can make all of ∆ j
true. Since the sentences ∆ i are just the sentences ∆ j , no valuation can make
all of ∆ i true. Since v makes all of ∆ n true, it must therefore make A false,
and so make ¬A true. So ∆ n ⊨ ¬A.
■
Lemma 5.11. TND is rule-sound.
Proof. Assume that every line before line n on some TFL-proof is shiny, and
that TND is used on line n. So the situation is:
i
j
A
B
k ¬A
l
B
n B TND i–j, k–l
Let v be any valuation that makes all of ∆ n true. Either v makes A true, or
it makes A false. We shall reason through these (sub)cases separately.
Case 1: v makes A true. The sentences ∆ i are just the sentences ∆ j . By
hypothesis, line j is shiny. So ∆ i ⊨ B. Furthermore, all of ∆ i are among
∆ n , with the possible exception of A. So v makes all of ∆ i true, and
hence makes B true.
Case 2: v makes A false. Then v makes ¬A true. Reasoning in exactly
the same way as above, considering lines k and l, v makes B true.
Either way, v makes B true. So ∆ n ⊨ B.
■
Lemma 5.12. →I, →E, ↔I, and ↔E are all rule-sound.
Proof. I leave these as exercises.
■
This establishes that all the basic rules of our proof system are rule-sound.
Finally, we show:
Lemma 5.13. All of the derived rules of our proof system are rule-sound.
Proof. Suppose that we used a derived rule to obtain some sentence, A, on
line n of some TFL-proof, and that every earlier line is shiny. Every use of
a derived rule can be replaced (at the cost of long-windedness) with multiple
uses of basic rules. (This was proved in fx C§30.) That is to say, we could have
used basic rules to write A on some line n + k, without introducing any further