Metatheory - University of Cambridge

2. Substitution 13
I shall prove the Claim by induction on the length of A. Suppose, for induction,
that the Claim is true of every shorter sentence. Now there are three cases:
Case 1: A is atomic. Then either A is C , in which case A[S/C] is just S;
or A[S/C] is just A. Either way, the Claim holds of A.
Case 2: A is ¬B, for some sentence B. Then, by assumption, w makes
B true iff v makes B[S/C] true. By considering the truth-table for ‘¬’, it
follows that w makes ¬B true iff v makes ¬B[S/C] true.
Case 3: A is (B ◁ C ), for some sentences B and C and some two-place
connective ◁. This is much like Case 3; I leave the details as an exercise.
This proves the Claim, by induction. It follows from the claim that w makes
all of A 1 , . . . , A n true. And, since A 1 , . . . , A n ⊨ A n+1 , we see that w makes
A n+1 true. So, appealing to the Claim again, v makes A n+1 [S/C] true. ■
Let’s see this Lemma in action. Consider these three claims:
(¬A → (A ∨ B)) ⊨ (A ∨ B)
(¬¬A → (¬A ∨ ¬B)) ⊨ (¬A ∨ ¬B)
(¬(P ↔ ¬Q) → ((P ↔ ¬Q) ∨ ¬(B ∧ (C ↔ D)))) ⊨ ((P ↔ ¬Q) ∨ ¬(B ∧ (C ↔ D)))
The first of these is easy to check using a truth table; the second is marginally
harder; and the third is a pain! But if we recognise that the second and third
are the result of uniform substitution into the first, then we can use Lemma
2.1 to save ourselves a lot of work.
There is a deeper point here. Lemma 2.1 vindicates the practice of describing
an entire argument structure as valid. For example, we can say that the
following is a valid pattern of entailment:
(¬A → (A ∨ B)) ⊨ (A ∨ B)
We know already that at least one instance of this pattern of entailment is
valid. Lemma 2.1 then tells us that the entailment will hold, regardless of what
sentences we use for A and B. Lemma 2.1 therefore underscores the purely
structural nature of entailments, in our semantics for TFL.
Lemma 2.1 concerns tautological entailment. We shall also want to have
a result to hand which concerns tautological equivalence. It will help us if we
introduce the symbol ‘ ⊨’ to symbolise tautological equivalence. So we write:
⊨
A
⊨
to symbolise that A and B are tautologically equivalent; or, in other words,
that A ⊨ B and B ⊨ A. Using this notation, we can prove a nifty little result:
⊨ B
Lemma 2.2. For any sentences A, C and S: if C
⊨
⊨ S, then A
Proof. By an induction on the length of A; I leave this as an exercise.
⊨
⊨ A[S/C].
■
In a slogan: substitution of tautological equivalents preserves tautological
equivalence. We shall rely upon this frequently, in what follows.