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syntactic properties of a normal modal
logic∗
CWoo†
2013-03-22 3:59:33
Recall that a normal modal logic is a logic containing all tautologies, the
schema K
(A → B) → (A → B),
and closed under modus ponens and necessitation rules. Also, the modal operator diamond is defined as
A := ¬¬A.
Let Λ be any normal modal logic. We write ` A to mean Λ ` A, or wff A ∈ Λ,
or A is a theorem of Λ. In addition, for any set ∆, ∆ ` A means there is a
finite sequence of wff’s such that each wff is either a theorem, a member of
∆, or obtained either by modus ponens or necessitation from earlier wff’s in
the sequence, and A is the last wff in the sequence. The sequence is called a
deduction (of A) from ∆.
Below are some useful meta-theorems of Λ:
1. (RM) ` A → B implies ` A → B
Proof. By assumption and by necessitation, ` (A → B), by schema K
and by modus ponens, we have the result.
2. As a result, ` A ↔ B implies ` A ↔ B.
3. (substitution theorem). If ` Bi ↔ Ci for i = 1, . . . , m, then
` A[B/p] ↔ A[C/p],
where p := (p1 , . . . , pm ) is the tuple of all the propositional variables in A
listed in order.
∗ hSyntacticPropertiesOfANormalModalLogici
created: h2013-03-2i by: hCWooi version:
h42558i Privacy setting: h1i hDefinitioni h03B45i
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Proof. For most of the proof, consult this entry for more detail. What
remains is the case when A has the form D. We do induction on the
number n of ’s in A. The case when n = 0 means that A is a wff of
PLc , and has already been proved. Now suppose A has n + 1 ’s. Then
D has n ’s, and so by induction, ` D[B/p] ↔ D[C/p], and therefore
` D[B/p] ↔ D[C/p] by 2. This means that ` A[B/p] ↔ A[C/p].
4. ` A → B implies ` A → B
Proof. By assumption, tautology ` (A → B) → (¬B → ¬A), and modus
ponens, we get ` ¬B → ¬A. By 1, ` ¬B → ¬A. By another instance
of the above tautology and modus ponens, and the definition of , we get
the result.
5. ` A ∨ B implies ` A ∨ B
Proof. Since ` A∨B ↔ (¬A → B), we have ` ¬A → B, so ` ¬A → B.
By the tautology C ↔ ¬¬C, we have ` ¬¬¬A → B, or ` ¬ A → B,
and therefore ` A ∨ B.
6. (RR) ` A ∧ B → C implies ` A ∧ B → C
Proof. By assumption and 1, ` (A ∧ B) → C. Since A ∧ B →
(A ∧ B) is a theorem (see here), we get ` A ∧ B → C by the law
of syllogism.
7. (RK) More generally, ` A1 ∧· · ·∧An → A implies ` A1 ∧· · · An → A,
where the case n = 0 is the necessitation rule.
Proof. Cases n = 1, 2 are meta-theorems 1 and 6. If ` A1 ∧ · · · ∧ An ∧
An+1 → A, or ` (A1 ∧ · · · ∧ An ) ∧ An+1 → A, then ` (A1 ∧ · · · ∧ An ) ∧
An+1 → A by 6. But ` (A1 ∧ · · · ∧ An ) ↔ A1 ∧ · · · ∧ An , the
result follows.
8. Define a function s on {¬, λ}, where λ is the empty word, such that
s(¬) = λ, the empty word, and s(λ) = ¬. Then for any wff A, and
1 , 2 ∈ {¬, λ}:
` 1 n 2 A ↔ s(1 ) n s(2 )A.
Technically speaking, this is really an infinite collection of theorem schemas.
Proof. We will check the case when 1 = ¬ and 2 = λ and leave the rest
to the reader. We do induction on n. If n = 0, then we have the tautology
¬A ↔ ¬A. Suppose ` ¬n A ↔ n ¬A. Then ` ¬n+1 A ↔ n ¬A, by
applying the induction case on wff A. Since A ↔ ¬¬A is a tautology,
` ¬n+1 A ↔ n ¬¬¬A by the substitution theorem. By the definition
of , we have ` ¬n+1 A ↔ n+1 ¬A.
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9. Let ∆ := {A | A ∈ ∆}. Then ∆ ` A implies ∆ ` A.
Proof. Induct on the length n of deduction of A from ∆. If n = 0, then
either ` A, in which case ` A by necessitation, or A ∈ ∆, in which case
A ∈ ∆. In either case, ∆ ` A. Next suppose the property holds for
all deductions of length n, and there is a deduction E of A of length n + 1.
If A is obtained from ∆ by necessitation, say A is B, where B is in E,
then a subsequence of E is a deduction of B of length ≤ n, from ∆. So by
induction, ∆ ` B, or ∆ ` A. By necessitation, ∆ ` A. Finally, if
A is obtained by modus ponens, then there is a wff B such that B, B → A
are both in E. By induction, ∆ ` B and ∆ ` (B → A), which, by
K and modus ponens, ∆ ` B → A, and as a result, ∆ ` A by
modus ponens.
Noticeably absent is the deduction theorem, for the necessitation rule says
A ` A, but this does not imply ` A → A. In fact, the wff A → A is not
a theorem in general, unless of course the logic includes the entire schema. All
we can say is the following:
10. (deduction theorem) If ∆, A ` B and B is not of the form C, then
∆ ` A → B.
Remark. It can be shown that conversely, if a modal logic obeys meta-theorem
7 above as an inference rule, then it is normal. For more detail, see here.
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