In order to examine modal calculi, we need to alter the definitions from section 5.5. In particular, we add the notion of modal operators to the definitions of formulae, metaformulae etc. Aformula is defined by the grammar:
A::=P | ⊥ |F(Alist)|G(Alist)
where F ranges over propositional constructors, and G ranges over modal constructors. Metaformulae are built then constructed in an analogous way to section5.5(pg. 73).
The modal operatorsGcan also act upon metamultisets and multisets. We call a multiset of formulaemodalised if it is of the form !Γ, where ! is some modal operator.
In addition to Axioms and uniprincipal rules, which were defined for propositional con- nectives, we give definitions of two kinds of decomposable modal rule. Basic modal rulesare like uniprincipal rules, except that instead of a single propositional compound metaformula in the conclusion of the active part, there is a single modal metaformula in the conclusion of the active part.
The second kind of decomposable modal rule is a context-dependent rule (recall definition
18on page77):
Definition 20 (Modalised Context Rules) A rule is amodalised context ruleiff the active part of the rule contains elements of the form !Γ, where ! is a modality and Γ is a
metamultiset. a
Note that the modalised metamultisets could appear in the antecedent, the succedent, or both.
The rule L3, for a classical calculus extended with the S4-modalities [Troelstra and Schwichtenberg,2000] (see appendixA), is a modalised context rule:
2Γ, φ⇒3∆
2Γ,3φ,Γ0 ⇒3∆,∆0 L3 whereasR3is not:
Γ⇒∆, φ Γ⇒∆,3φ R3
Modalised context rules have large active parts, and these active parts could be further decomposed into a context, consisting of the modalised metamultisets, and what we will call theprime part of the rule. Formally:
Definition 21 (Prime Metaformulae, Prime Part) A metaformula φ is prime iff it appears in the active part of a modalised context rule.
The prime part of a modalised context rule, is the active part of the rule where all modalised metamultisets have been deleted. a
This definition is extended to prime formulae and instances in the obvious way. InL3above, the prime part of the rule is:
φ⇒
3φ⇒
As has been done in previous sections, we prove the height preserving admissibility ofWeak- ening for such sequent calculi.
Lemma 12 (Modal dp-Weakening) Let Rbe a calculus containing uniprincipal propo- sitional rules, basic modal rules, and modalised context rules. If every modalised context rule inRis an IW rule, then the rule:
Γ⇒∆ Γ,Γ0⇒∆,∆0 is height preserving admissible in R.
Proof. Let Γ ⇒∆ be an instantiation of the premiss. The proof is by induction on the heightnof the derivation of Γ⇒∆, and is standard. In the case where the last inference is a modalised context inference, we do not use the induction hypothesis, rather apply a new inference with a suitable extension of the conclusion. a
Using the definition of principal formula from section 5.5, where every metaformula in the active part of a conclusion is principal, we can keep the conditions for invertibility very close to those of section 6.2. We must be careful, however; two inferences can differ in the formulae occurring the modalised multisets. Were we to use the same conditions as in section6.2, we would encounter a problem;2A is principal on the left for many instances of the ruleL3, given above.
Lemma 13 (Right Modal Rules) Let R be a calculus containing uniprincipal proposi- tional rules, basic modal rules, and modalised context rules. The rule:
Γ⇒!(B),~ ∆ Γ,Γ0⇒∆,∆0 is height preserving admissible in Rif:
1. Γ0 ⇒∆0 is a premiss of the prime part of every modalised context inference in which !(B)~ is principal on the right.
2. Γ0⇒∆0 is the active part of a premiss of every basic modal inference in which!(B)~ is principal on the right.
3. All modalised context rules inRare IW rules.
Proof. Let Γ⇒∆⊕!(B) be an instance of the premiss of the above rule. We proceed by~ induction on the height nof the derivation. Ifn= 0, or n >0 and the last inference was a propositional rule, then we proceed as in the proof of lemma8 on page82(for, !(B) will~ never be principal for such an inference). If the last inferencerwas an instance of a modal ruleR, there are four cases:
1. !(B) is principal for~ r, andris a basic modal inference.
2. !(B) is principal for~ r, andris a modalised context inference.
3. !(B) is non-principal for~ r, andris a basic modal inference.
4. !(B) is non-principal for~ r, andris a modalised context inference.
Case 1. The result is immediate, from condition 2.
Case 2. Γ⇒∆⊕!(B) is the conclusion of a modalised context inference. Let~ •1, . . . ,•n,!1, . . . ,!m be modal operators, then for some Γ1, . . . ,Γn, ∆1, . . . ,∆m and Γ00,∆00, we can rewrite ras (using condition 1):
· · · •1Γ1,· · ·,•nΓn,Γ0 ⇒!1∆1,· · ·,!m∆m,∆0 · · ·
•1Γ1,· · ·,•nΓn,Γ00⇒!(B),~ !1∆1,· · ·,!m∆m,∆00
From condition 3 and lemma12, Weakening is height preserving admissible; weaken with Γ00 and ∆00to obtain the desired result.
Case 3. There are two further subcases; one whereris an instance of a normal rule, and one whereris an instance of an IW rule. In the former, we have, from !(B~) being non-principal for r, that !(B) appears in every premiss of the inference, as part of the context. We can~ thus apply the induction hypothesis to each premiss. To this set of premisses extended with the new context involving Γ0 and ∆0, we apply the instance of R which uses that context, and we are done. (The details are the same as in the proof of lemma8).
In the latter case, again the details are similar to the equivalent case in the proof of 8, and so are omitted.
Case 4. From condition 3, every modalised context rule is an IW rule, and therefore suppose that the root ofrwas:
•1Γ1,· · ·,•nΓn,Γ00⇒!?(D),~ !1∆1,· · ·,!m∆m,∆00
for some modal operators •1, . . . ,•n,!1, . . . ,!m and multisets Γ1, . . . ,Γn,∆1, . . . ,∆m and Γ00,∆00. Because !(B) is non-principal on the right, then !(~ B)~ ∈∆00. Therefore, let ∆∼ be
such that ∆00= ∆∼⊕!(B). We use a new instance of~ R which has Γ00+ Γ0 and ∆∼+ ∆0 as
Lemma 14 (Left Modal Rules) Let Rbe a calculus containing single-conclusion propo- sitional rules, basic modal rules, and modalised context rules. The rule:
Γ⇒!(B),~ ∆0 Γ,Γ0⇒∆,∆0 is height preserving admissible in Rif:
1. Γ0⇒∆0 is a premiss of the prime part of every modalised context inference with!(B~) principal on the left.
2. Γ0 ⇒ ∆0 is the active part of a premiss of every basic modal inference with !(B~) principal on the left.
3. Every modalised context rule inRis an IW rule.
Proof. Symmetric to the proof for right modal rules. a
It should be noted that these are not true invertibility results; we do not reconstruct the premisses of a modal rule exactly, but we derive weakened versions of the premisses of a modal rule (as in section6.2.1).
6.4.1
Examples
Consider the calculus for classical propositional logic extended with S4-modalities; in other words, G3cptogether with the four rules for theS4 modalities2 and3 (see appendixA
for all of the rules of this calculus). The rules for3were given earlier. The rules for2are: 2Γ⇒φ,3∆
2Γ,Γ0 ⇒2φ,3∆,∆0 R2
Γ, φ⇒∆ Γ,2φ⇒∆ L2
If a derivation in this system had root Γ⇒∆⊕2A, then we can use lemma 13to derive Γ⇒∆⊕A. For, the only inferences which have2Aprincipal on the right are instances of R2, and:
⇒A
is a premiss of the prime part of every such inference. However, the rule:
Γ⇒3A,∆ Γ⇒A,∆
is not height preserving admissible in this system. We cannot apply lemma 13: ∅ ⇒ A will not be a premiss of the prime part ofevery modalised context inference which has3A
principal on the right. For instance:
2Γ⇒B,3(∆, A) 2Γ⇒2B,3(∆, A)
has 3A principal on the right: it is in the premiss of the active part of the inference. However, only∅ ⇒B is a premiss of the prime part of the inference.