2.2 Sequent Systems for Modal Logic
2.2.2 A Multi-Conclusion Sequent System for CS4
In this section, we introduce a cut-free multi-conclusion variant of G3iCS4. The reason is that the multi-proponent dialogues we consider in Chapter 4 are related to multi-conclusion sequent systems and the adequateness of the dialogical sequent system DiaSeqCS4, which we introduce in 4.2, is based on this.
We call the following system G3iCS4m
. The rules are displayed in Fig- ure 2.14. Those of propositional logic are exactly the same as of G3im
(Sec- tion 2.1.3). The new rules are 2l, 2r, 3l, and3r, of which only 2l is invert- ible. Note that 3l can only be used when 3∆ is not empty.21 We show the adequateness of G3iCS4m
by proving that every G3iCS4m
-derivation can be
20M, w |=2A holds iff for all refinements w0of w, in all successors u of w0, it holdsM, u |= A.
The reference to the refinements w0 of w also occur for implicationM, w |= A ⊃ B and negation M, w, |= ¬A. For both (and only for these) the right-hand rule is critical in G3im
.
21This restriction is added here next to the premise in the rule. We do not consider it as a
ax P,Γ ⇒ ∆,P ⊥,Γ ⇒ ∆ ⊥l A,B,Γ ⇒ ∆ ∧l A∧ B,Γ ⇒ ∆ Γ ⇒ ∆,A Γ ⇒ ∆,B ∧r Γ ⇒ ∆,A∧ B A,Γ ⇒ ∆ B,Γ ⇒ ∆ ∨l A∨ B,Γ ⇒ ∆ Γ ⇒ ∆,A,B ∨r Γ ⇒ ∆,A∨ B A⊃ B,Γ ⇒ ∆,A B,Γ ⇒ ∆ ⊃l A⊃ B,Γ ⇒ ∆ A,Γ ⇒ B ⊃r Γ ⇒ A ⊃ B ¬A,Γ ⇒ ∆,A ¬l ¬A,Γ ⇒ ∆ A,Γ ⇒ ∅ ¬r Γ ⇒ ∆,¬A 2A,A,Γ ⇒ ∆ 2l 2A,Γ ⇒ ∆ 2Γ ⇒ A 2r 2Γ,Γ0 ⇒ 2A,∆ 2Γ,A⇒ 3∆ |3∆| >1 3l 2Γ,Γ0,3A⇒ 3∆,∆0 Γ ⇒ A 3r Γ ⇒3A,∆ Figure 2.14: Rules of G3iCS4m
transformed into a G3CS4-derivation and vice-versa. Soundness and com- pleteness of G3CS4 then imply soundness and completeness of G3iCS4m
.
Definition 2.1 (Deducibility (G3CS4, G3iCS4m
)). For an arbitrary G3CS4/ G3iCS4m
sequent Γ ⇒ ∆ we write for some n ∈ N: S
n Γ ⇒ ∆ iff there is a
closed G3CS4-tree with Γ ⇒ ∆ as root sequent and which has a height h 6 n with n∈ N. We write Mn Γ ⇒ ∆ iff there is a closed G3iCS4m
-tree with Γ ⇒ ∆ as root sequent and which has a height h 6 n. We also say that Γ ⇒ ∆ is deducible in n deductive steps, or that there is a derivation of height n for Γ ⇒ ∆.
Lemma 2.1 (Admissibility of Weakening (G3CS4, G3iCS4m )22). For all ϕ, Γ , ∆: 1. If Sn Γ ⇒ ∆ then Sn Γ,ϕ⇒ ∆. 2. If M n Γ ⇒ ∆ then Mn Γ,ϕ⇒ ∆. 3. If M n Γ ⇒ ∆ then Mn Γ ⇒ ϕ,∆.
The proof of this lemma is omitted here, as we discuss weakening in more detail in Chapters 3.3.1, 4.3.2, and 4.3.3.
Theorem 2.1 (G3iCS4m
Completeness). Every G3CS4 proof tree can be trans- formed into a G3iCS4m
proof tree.
Proof. We consider the G3CS4-derivation Sn Γ ⇒ C with Γ being an arbitrary
multi-set of formulas and C a single formula. We want to show that for some m and for all ∆ it holds: Mm Γ ⇒ C,∆.
We perform an induction on n.
Base Case: n = 1 — The only rule which is used in the derivation is either
ax or ⊥l.
• If the rule is ax, the root sequent has the form Γ,P ⇒ P. The rule ax
of G3iCS4m
can also be used in the sequent Γ,P ⇒ P,∆, so this is no problem.
• The case of ⊥l works accordingly.
Inductive Step: Assume we have a G3CS4-derivation Sn+1 Γ ⇒ C. We have
to perform a case analysis on the rule which is applied in the root sequent, i.e., the lowest rule application in the tree. We consider three cases here. The others are handled similarly.
1. Assume that C is a disjunction A∨ B with arbitrary formulas A and B. Also assume that A∨ B is the principal formula of the lowest rule ap- plication. Then we have either S
n Γ ⇒ A or Sn Γ ⇒ B. By hypothesis,
we obtain M
m Γ ⇒ A,∆ (or with B instead of A respectively). Then by
weakening we achieve M
m Γ ⇒ A,B,∆. Now we append the application
of ∧r below and get Mm+1 Γ ⇒ A∧ B,∆ as root.
2. Assume that S
n+1 2Γ,Γ0 ⇒ 2A and 2A is the principal. Then
Sn 2Γ ⇒ A. By hypothesis, we then get Mm 2Γ ⇒ A for ∆ = ∅. Ap-
pending2r of G3iCS4m
results in Mm+12Γ,Γ0 ⇒ 2A,∆for an arbitrary
∆.
3. Assume that S
n+1 2Γ,Γ0,3A⇒ 3C and3A is the principal of the low-
est rule application. We want to have Mm+1 2Γ,Γ0,3A ⇒ 3C,∆. We
define 3∆0 as the part of ∆ which contains 3-formulas, and let ∆00 be the rest. Then from Sn 2Γ,A⇒ 3C we get by hypothesis:
Mm 2Γ,A ⇒3C,3∆0 and adding 3l of G3iCS4 m
leads to Mm+1 2Γ,Γ0,3A⇒ 3C,3∆0,∆00.
For the transformation from multi-conclusion to single-conclusion sequents we adapt the technique by Maehara [104]. Assume that ∆ is the multi-set of formulas δ1,δ2, . . . ,δn. Then the disjunction over ∆, written W∆, is defined as
δ1∨ δ2∨ · · · ∨ δn.
We also need an additional lemma.
Lemma 2.2 (Alternative Succedents in G3CS4). For any Γ , n, and formulas A and B: if S
n Γ ⇒ A∨ B, then Sn Γ ⇒ A or Sn Γ ⇒ B.
Proof. By induction on n:
Base Case: n = 1 — In this case, the only rule application in the derivation must be ⊥l, which can also be applied in Γ ⇒ A and Γ ⇒ B.
Inductive Step: We consider the lowest rule application of the derivation tree for Sn+1 Γ ⇒ A∨ B.
• If it is an application of ∧l, ∨l, or 2l, we can simply apply the hy- pothesis on the premise of the application and append it afterwards again.
• If it is an application of ⊃l on some formula C ⊃ D, then we have Sn Γ,C⊃ D ⇒ C and Sn Γ,D ⇒ A∨ B.
By hypothesis then Sn Γ,D⇒ A or Sn Γ,D ⇒ B. Therefore, appending
⊃l again results in S
n+1 Γ,C ⊃ D ⇒ A or Sn+1 Γ,C⊃ D ⇒ B.
• The case of a ¬l-application works accordingly.
• The only other possibility is an application of ∨r, but both thinkable premises fulfil our target properties.
Theorem 2.2 (G3iCS4m
Soundness). Every G3iCS4m
proof tree can be trans- formed into a G3CS4 proof tree.
Proof. What we have to show is that for any n, Γ , and ∆: Mn Γ ⇒ ∆ implies
for some m: S
m Γ ⇒ W∆. We do this by induction on n.
Base Case: n =1 — We have two possibilities:
• The only rule application is ax in a sequent such that M1 Γ,P ⇒ P,∆. This corresponds to Sm Γ,P ⇒ P ∨ W∆. There is obviously such a
derivation: we simply make use of ∨r and ax of G3CS4 and obtain the following tree:
ax
Γ,P ⇒ P
∨r Γ,P ⇒ P∨ W∆
If P is not the left-most disjunct we simply make use of∨r several times until it is dissolved out of W∆.
• The only rule application is ⊥l. Then ⊥l can also be applied directly in the sequent of G3CS4.
Inductive Step: We have a look at three cases: 1. Assume M
n+1 Γ ⇒ A∨ B,∆ with the lowest rule application being ∨r
applied on A∨ B. Then Mn Γ ⇒ A,B,∆. By hypothesis, this corres-
ponds to Sm Γ ⇒ A∨ B ∨ W∆. This is already what we were looking
for.23
2. Assume Mn+1 Γ,A⊃ B ⇒ ∆ with the lowest rule application being ⊃l
applied on A ⊃ B. Then Mn Γ,A⊃ B ⇒ A,∆ and Mn Γ,B⇒ ∆. By
hypothesis then for some l and m: Sl Γ,A⊃ B ⇒ A∨ W∆ and
Sm Γ,B⇒ W∆. According to Lemma 2.2, the former leads us to two
cases: either S
l Γ,A ⊃ B ⇒ A or Sl Γ,A ⊃ B ⇒W∆.
• If Sl Γ,A⊃ B ⇒ A, then together with Sm Γ,B⇒ W∆, we can use ⊃l to obtain Γ,A⊃ B ⇒W∆.
• If Sl Γ,A⊃ B ⇒W∆, we are already finished. 3. Assume M
n+1 2Γ,Γ0,3A⇒ 3∆,∆0and the lowest applied rule is 3l(on
3A). Then due to the premise we get Mn 2Γ,A ⇒ 3∆ with |3∆| >1, and by hypothesis Sm 2Γ,A⇒
W3
∆ for some m.
• If W3
∆ contains only a single disjunct, i.e., is some 3C, then we can simply put3l below and appendW∆0 with rule ∨r.
• If it contains more than one disjunct, we apply Lemma 2.2 (if ne- cessary multiple times) and locate the one which is relevant for the derivation. We call it3E. Then it is again easy to build the root of the G3CS4-derivation:
23Note that this requires that associativity of ‘∨’ is admissible in the succedent of any
G3CS4-sequent. However, using Lemma 2.2 makes it possible to show easily that S
ax ⇒ Σ ; P,¬P,∆ ⇒ Σ⇒ Σ; A,B,∆ ∨ ; A∨ B,∆ ⇒ Σ; A,∆ ⇒ Σ ; B,∆ ∧ ⇒ Σ ; A∧ B,∆ ⇒ 3A,Σ; A,∆ 3 ⇒ Σ ; 3A,∆ ⇒ ∅; A,Σ 2 ⇒ 3Σ; 2A,∆ Figure 2.15: Rules of KTSU (c.f. [68]) . . . 2Γ,A ⇒ 3E 3l 2Γ,Γ0,3A⇒ 3E ∨r (multiple times) 2Γ,Γ0,3A⇒ W3 ∆ ∨r (multiple times) 2Γ,Γ0,3A⇒ W3 ∆∨ W∆0