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equivalent formulations of normality∗
CWoo†
2013-03-22 4:00:29
Recall that a logic in a modal propositional language is a set of wff’s containing all the tautologies and is closed under modus ponens. Depending on
the intended meaning of the modal connective , one may extend the logic
by adding wff’s and/or imposing additional inference rules. Among the more
common modal inference rules are the following:
RK :
n≥0
RR :
(A1 ∧ · · · ∧ An ) → A
,
(A1 ∧ · · · ∧ An ) → A
RM :
(A ∧ B) → C
(A ∧ B) → C
A→B
A → B
RN (necessitation rule):
A
A
The most common modification is what is known as the normal modal logic.
It is obtained from a logic by adding the schema K:
(A → B) → (A → B)
and the rule of necessitation RN.
As it turns out, this is not the only way to formulate the notion of normality,
as the following proposition illustrates:
Proposition 1. Let L be a logic. Then the following are equivalent:
1. L is normal
2. L is closed under RK
3. L contains > and is closed under RM and RR, where > is ¬ ⊥.
4. L contains schemas (A → A), A ∧ B → (A ∧ B), and is closed
under RM
5. L contains schemas (A → A), K, and is closed under RM
∗ hEquivalentFormulationsOfNormalityi
created: h2013-03-2i by: hCWooi version:
h42565i Privacy setting: h1i hFeaturei h03B45i
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First, three quick observations:
1. If L is closed under RK, then it is closed under RR, RM, and RN.
Proof. RR, RM, and RN are RK with n = 2, 1, and 0 respectively.
2. If L is closed under RN, then L contains (A → A), and in particular
>.
Proof. Since A → A is a tautology, we have ` (A → A) by RN. Letting
A be ⊥ gives us ` >.
3. If L is closed under RM, and contains B for some wff B, then L is closed
under RN.
Proof. For any wff A in L, by modus ponens on tautology A → (B → A),
we have ` B → A, and so by RM, ` B → A. But ` B by assumption,
` A by modus ponens.
Now, we prove the proposition.
Proof. We will prove the following implications 1 ⇒ 2 ⇒ 3 ⇒ 4 ⇒ 5 ⇒ 1.
1 ⇒ 2. See this entry12558.
2 ⇒ 3. By the first observation, L is closed under RM and RR, and RN, and
therefore contains > based on the second observation.
3 ⇒ 4. Apply RR to A ∧ B → A ∧ B, we get ` A ∧ B → (A ∧ B). Apply
the third observation to >, L is closed under RN, and therefore contains
(A → A) by the second observation.
4 ⇒ 5. Apply the third observation to ` (A → A), we see that L is closed under
RN.
Next, we show that L contains K. From the tautologies A ∧ (A → B) →
A ∧ B and A ∧ B → B, we get the tautology A ∧ (A → B) → B by
the law of syllogism, so that ` (A ∧ (A → B)) → B by RM. Now,
` A ∧ (A → B) → (A ∧ (A → B)) by assumption, ` A ∧ (A →
B) → B by the law of syllogism. From the tautologies X ∧ Y ↔ Y ∧ X
and (X ∧ Y → Z) → (X → (Y → Z)), we get `K by the substitution
theorem and modus ponens.
5 ⇒ 1. Similarly, L is closed under RN. Since L contains K, it is normal.
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