Computability and Logic

332 MODAL LOGIC AND PROVABILITY
The only other part of the proof that needs modification is the proof that if w |= □C,
then □C is in w. So suppose w |= □C, and let
V ={□D 1 , D 1 ,...,□D m , D m , ∼C}
where the □D i are all the formulas in w that begin with □. IfV is consistent and
v is a maximal set containing it, then w>v and v |= ∼C, which is impossible. It
follows that
□D 1 & D 1 & ··· & □D m & D m → C
□(□D 1 & D 1 & ··· & □D m & D m ) → □C
(□□D 1 & □D 1 & ··· & □□D m & □D m ) → □C
are theorems, and hence any tautologous consequence of the last of these and the
axioms □D i → □□D i is a theorem, and this includes
from which it follows that w |= □C.
(□D 1 & ··· & □D m ) → □C
Besides its use in proving decidability, the preceding theorem makes it possible
to prove syntactic results by semantic arguments. Let us give three illustrations. In
both the first and the second, A and B are arbitrary sentences, q a sentence letter not
contained in either, F(q) any sentence, and F(A) and F(B) the results of substituting
A and B respectively for any and all occurrences of q in F. In the second and third,
□ A abbreviates □A & A. In the third, •A is the result of replacing □ by □ throughout
A.
27.4 Proposition. If ⊢ K A ↔ B, then ⊢ K F(A) ↔ F(B).
27.5 Proposition. ⊢ K+(A3) □ (A ↔ B) → □ (F(A) ↔ F(B)).
27.6 Proposition. If ⊢ K+(A1)+(A3) A, then ⊢ K + (A3) •A.
Proof: For Proposition 27.4, it is easily seen (by induction on complexity of F)
that if W = (W,>,ω) and we let W ′ = (W,>,ω ′ ), where ω ′ is like ω except that for
all w
then for all w, wehave
ω ′ (w, q) = 1 if and only if W,w |= A
W,w |= F(A) if and only if W ′ ,w |= F(q).
But if ⊢ K A ↔ B, then by soundness for all w we have
and hence
W,w |= A if and only if W,w |= B
W,w |= F(B) if and only if W ′ ,w |= F(q)
W,w |= F(A) if and only if W,w |= F(B).
So by completeness we have ⊢ K F(A) ↔ F(B).