CUT-FREE HYPERSEQUENT CALCULUS for S4.3. 1. Introduction

Home , Hypersequent, Sequent, Structural rule

Bulletin of the Section of Logic Volume 41:1/2 (2012), pp. 89–104

Andrzej Indrzejczak

CUT-FREE HYPERSEQUENT CALCULUS FOR S4.3.

Abstract Hypersequent calculi (HC) are the generalization of ordinary sequent calculi obtained by operating with sets of sequents instead of single sequents. HC, introduced independently by Pottinger and Avron, proved to be very useful in the ﬁeld of nonclassical logics. Nevertheless their application to modal logics was rather limited. We provide a cut-free hypersequent calculus HCS4.3 for modal logic of linear frames based on the idea of hypersequent formalization for G¨odel- Dummett’s logic due to Avron. Also a variant of HCS4.3 with analytic cut but with nonbranching logical rules is presented.

1. Introduction

G. Pottinger [19], and independently A. Avron [3], introduced a generalization of Gentzen’s sequent calculus called hypersequent calculus where hypersequents are (multi)sets of ordinary sequents. This simple and straight- forward modification significantly increases the expressive power of ordinary Gentzen apparatus by allowing additional transfer of information among different sequents. Hypersequents proved to be very useful in build- ing cut-free formalization of many nonclassical logics, including modal, rel- evant, multi-valued and fuzzy logics (e.g. Avron [4, 5], Baaz, Ciabattoni and Fermüller[7] and other papers of these authors). In particular, replacing ordinary sequents with hypersequents made possible obtaining different cut-free systems for S5 (Pottinger [19], Avron [4], Poggiolesi [18]) and for one of the most important intermediate (or superintuitionistic) logic Gödel- Dummett’s logic (Avron [4, 6]). 90 Andrzej Indrzejczak

The last one is of particular interest for us since it is the logic of intuitionistic relational frames with linear accessibility relation. In what follows we propose a hypersequent calculus HCS4.3 for the basic modal logic of such frames, namely, S4.3. Our calculus is based on Avron’s idea of captur- ing linearity by specific structural rule, however, differences concerning the language (nonmodal versus modal) and the logical basis (intuitionistic versus classical) of both approaches lead to some important consequences. In particular, our specific rule is not purely structural, and the completeness proof of Avron [6] for cut-free hypersequent version of Gödel-Dummett’s logic cannot be simulated directly. Nevertheless, the calculus HCS4.3 is analytic in the sense that all rules satisfy the subformula property since completeness is proved directly for cut-free version. Additionaly, we present briefly a variant of HCS3.4 which has nonbranching logical rules but requires analytic (subformula-preserving) cut. The latter system implements in hypersequent framework the idea of KE system of D’Agostino and Mon- dadori [2].

2. The Logic

Let us recall the basic facts concerning S4.3 in standard characterization, i.e., as an axiomatic system adequate to the class of linear Kripke frames. One can axiomatize S4.3 by adding to any system for classical propositional logic the following schemata: K (ϕ → ψ) → (ϕ → ψ) T ϕ → ϕ 4 ϕ → ϕ 3 (ϕ → ψ) ∨ (ψ → ϕ) Clearly, the system is closed under MP (modus ponens) and GR (Gödel’s rule). It is adequate to the class of Kripke frames where acessibility relation is reflexive, transitive and satisfies the condition of strong connectedness: ∀xyz(Rxy ∧ Rxz → Ryz ∨ Rzy) However, it may be shown that S4.3 is also characterised by the nar- rower class of linear structures, where instead of strong connectedness there holds the condition of dichotomy1:

1In fact, no axiom in standard modal language corresponds to this condition; see Goldblatt [10] for details. Cut-Free Hypersequent Calculus for S4.3. 91

∀xy(Rxy ∨ Ryx) We will prove directly the adequacy of our hypersequent calculus with respect to such a class.

3. Survey of Previous Approaches

S4.3, and other kinds of linear modal logics, was formalised by means of other calculi like natural deduction or tableaux. Also sequent calculi of different sorts were used to that aim. Most of the proposed solutions however, are based on the use of several forms of labelling which encodes some elements of relational semantics2. Before we present our solution, a brief survey of proposed approaches may appear helpful. All the systems for linear modal logics may be divided roughly according to two criteria: (a) the shape of rules and (b) the implicit strategy of linearization (of attempted falsifying model). The rules serving to express suitable conditions of connectedness, dichotomy, trichotomy, and the like, may be divided into nonbranching and branching. The former solution is rather rare, one can mention here labelled tableau system of Marx, Mikulas, Reynolds [16] and of Baldoni [8], as well as labelled natural deduction system of Indrzejczak [13]. Since construction of nonbranching rules for (strong) connectedness is based on the following form of this condition: ∀xyz(Rxy ∧ Rxz ∧ ¬Ryz → Rzy) one must have in the system sufficient resources not only for expressing that some worlds are in the relation R but also that some are not, i.e. either the apparatus of labels must be sufficiently strong (like in Baldoni’s system) or some forms of cut must be involved (in the remaining proposals). Surprisingly enough, in case of richer language of bimodal temporal logics of linear frames one can define suitable nonbranching rules for natural deduction system even without using labels (see Indrzejczak [12]). Most solutions are based on the application of branching rules. In some systems the number of branches is fixed – two in case of strong connectedness or dichotomy and three in case of weak connectedness (trichotomy).

2See, in particular, Gor´e[11] and Indrzejczak [14] Chapter 9 for the comprehensive surveys. 92 Andrzej Indrzejczak

This is obvious since suitable rules are directly modelled on semantic conditions involving disjunction. One may mention here, e.g. display calculus of Wansing [22], labelled sequent calculus of Negri [17], or nested tableau system of Kashima [15]. Both nonbranching and fixed-branching approaches realize the local strategy of linearization of attempted falsifying model. It means that we compare only two worlds at a time and put them in some order, either disjunctively (by means of branching rules) or by choosing one possibility since the other is excluded (nonbranching rules). However, there are formal systems for linear logics which realize global strategy of linearization, i.e. their rules are defined in such a way that all worlds which are generated at the same moment are immediately put in the sequence. This group consists of systems which use rules with the number of branches not fixed in advance. It depends on the number of modal formulae which are responsible for creation of new worlds in the attempted model. One may distinguish here between solutions which make it in the decreasing way (Zeman’s sequent calculus [23], Gorétableau systems [11] or increasing way (Rescher and Urquhart’s tableau system for linear temporal logics [20]). In both approaches we eventually create all possible sequences of worlds hence the number of branches is exponentially dependent on the number of world’s creating modal formulae. In this paper we provide a characterization of S4.3 in terms of hypersequent calculus which closely follow the lines of Avron’s [6] solution for Gödel-Dummett’slogic. In terms of described solutions the system should be characterised as based on fixed-branching rules realizing local strategy of searching linear model. We do not use any labels, our solution for linearization of worlds is quite similar to that of Kashima [15] but instead of nested tableaux we use hypersequents. In some sense, hypersequent calculi may be seen as being the simplest modification of ordinary Gentzen’s sequent calculi, and independent of the kind of underlying semantics. This is the main reason why this approach was successfully applied to several kinds of nonclassical logics characterised by means of different semantic structures (see Avron [5]).

4. Hypersequent Calculus

In this paper hypersequents are deﬁned as ﬁnite sets of ordinary Gentzen’s sequents. We will use the following notation: Cut-Free Hypersequent Calculus for S4.3. 93

• Γ ⇒ ∆ or s for sequents where Γ , ∆ are ﬁnite sets of formulae. • G, H for hypersequents; in particular s | G or Γ ⇒ ∆ | G stand for hypersequents with displayed sequent s (or Γ ⇒ ∆) • Γ = {ψ ∈ Γ : ψ := ϕ} • in the schema of (IW) (internal weakening) Γ ⊆ Γ 0 and ∆ ⊆ ∆0. The calculus HCS4.3 consists of the following rules:

Γ , Π ⇒ ∆ | G Π , Γ ⇒ Σ | G (AX) ϕ ⇒ ϕ (Dich) Γ ⇒ ∆ | Π ⇒ Σ | G

G Γ ⇒ ∆ | G (EW) (IW) Γ ⇒ ∆ | G Γ 0 ⇒ ∆0 | G

Γ ⇒ ∆, ϕ | G ϕ, Γ ⇒ ∆ | G (¬⇒) (⇒¬) ¬ϕ, Γ ⇒ ∆ | G Γ ⇒ ∆, ¬ϕ | G

ϕ, ψ, Γ ⇒ ∆ | G Γ ⇒ ∆, ϕ | G Γ ⇒ ∆, ψ | G (∧⇒) (⇒∧) ϕ ∧ ψ, Γ ⇒ ∆ | G Γ ⇒ ∆, ϕ ∧ ψ | G Γ ⇒ ∆, ϕ, ψ | G ϕ, Γ ⇒ ∆ | G ψ, Γ ⇒ ∆ | G (⇒∨) (∨⇒) Γ ⇒ ∆, ϕ ∨ ψ | G ϕ ∨ ψ, Γ ⇒ ∆ | G

ϕ, Γ ⇒ ∆, ψ | G Γ ⇒ ∆, ϕ | G ψ, Γ ⇒ ∆ | G (⇒→) (→⇒) Γ ⇒ ∆, ϕ → ψ | G ϕ → ψ, Γ ⇒ ∆ | G

ϕ, ϕ, Γ ⇒ ∆ | G Γ ⇒ ϕ1 | ... | Γ ⇒ ϕn | G (⇒) (⇒) ϕ, Γ ⇒ ∆ | G Γ ⇒ ∆, ϕ1, ..., ϕn | G

Note that: 1. All rules satisfy the subformula property. 2. Note that sequents are built from sets, hence in (⇒) we also need an occurrence of ϕ in the premise. This is necessary because during the proof-search performed in reverse order, i.e. from conclusion to premises, boxed formulae may be needed many times. Clearly, if sequents were made of multisets of formulae and (internal) contraction rules were present in the system such an eﬀect would be always obtainable by making a copy of ϕ by contraction before ordinary (⇒) is applied. 94 Andrzej Indrzejczak

3.( ⇒) is a one-premiss rule (non-branching) but, in contrast to ordinary sequent rules (applied also in HC’s for modal logics) introducing to the succedent, it deals with all such formulae immediately. In our system it is possible to use a simpler rule which operates only on one selected ϕi and this would be sufficient for completeness but not for confluency. It is well known that standard sequent calculi for modal logics are not confluent, i.e. proof search for provable sequent not always provides a proof and we need backtracking to check all possibilities3. Hypersequent calculi allow to overcome this difficulty, in particular, the proposed version (⇒) provides confluency of the system. Clearly, if hypersequents are meant as multisets and external contraction is a rule of the system one may still save confluency by introducing n copies of Γ ⇒ ∆, ϕ1, ..., ϕn to G and then applying ordinary (⇒) for S4, successively, to all ϕi. For hypersequents as sets our rule seems to be the best choice despite its apparent complexity. 4.( Dich) is a two-premise rule corresponding to the condition of dichotomy (which we will show formally in a moment). It is a counter- part of Avron’s [5] rule (Comm). The main difference is that (Comm) is purely structural whereas (Dich) displays boxed formulae. Clearly, without (Dich) we obtain a (confluent) HC for S4.

The proof of a hypersequent G (` G) is deﬁned in the usual way as a (binary) tree of hypersequents with G as the root and axioms as leafs. Here is the proof of an instance of axiom 3 as an example.

q ⇒ q p ⇒ p (IW) (IW) p, q, q ⇒ q q, p, p ⇒ p ( ⇒) ( ⇒) p, q ⇒ q q, p ⇒ p (Dich) p ⇒ q | q ⇒ p (⇒→) ⇒ p → q | q ⇒ p (⇒→) ⇒ p → q |⇒ q → p (⇒ ) ⇒ ( p → q), ( q → p) (⇒ ∨) ⇒ (p → q) ∨ (q → p)

3See, e.g. Chapter 7 of Indrzejczak [14]. Cut-Free Hypersequent Calculus for S4.3. 95

5. Soundness

We extend semantical notions to hypersequents in the following way:

•| = G (G is valid) iﬀ M |= G in all (S4.3-)models M

• M |= G iff ∃s∈G, M |= s • M |= s iff w s, for all w in the domain of M • w Γ ⇒ ∆ iff w ∧Γ → ∨∆, where ∧Γ (∨Γ ) stands for the conjunction (disjunction) of all elements of Γ .

Note, that as a consequence we have:

6|= G iﬀ ∃M, M 6|= G, and M 6|= G iﬀ ∀s∈G, M 6|= s which, at the end, means that ∀s∈G∃w, w 2 s. Lemma 1. (Soundness Lemma) All rules are validity-preserving.

Proof: We will show two cases: (⇒) and (Dich).

(⇒). Assume that (i) |= Γ ⇒ ϕ1 | ... | Γ ⇒ ϕn | G but (ii) 6|= Γ ⇒ ∆, ϕ1, ..., ϕn | G. Hence, there is M such that M 6|= Γ ⇒ ∆, ϕ1, ..., ϕn | G; in particular, (iii) in some w w ∧Γ and w 2 ϕi, for i ≤ n. Since all s ∈ G are falsiﬁable in some w then, from (i), at least one of Γ ⇒ ϕi must be valid. But this is impossible since by (iii), for each ϕi there is some wi accessible from w where w ∧Γ but w 2 ϕi. (Dich). Assume that (i) |= Γ , Π ⇒ ∆ | G and (ii) |= Π , Γ ⇒ Σ | G but (iii) 6|= Γ ⇒ ∆ | Π ⇒ Σ | G. Hence, there is M such that (a) M 6|= Γ ⇒ ∆,(b) M 6|= Π ⇒ Σ and (c) M 6|= G. Therefore, by (a) and 0 0 0 0 0 (b) for some w, w ,(a ) w ∧Γ , w 2 ∨∆ and (b ) w ∧Π , w 2 ∨Σ. By (c), (i) and (ii) we have (i0) |= Γ , Π ⇒ ∆ and (ii0) |= Π , Γ ⇒ Σ. Now, by dichotomy, either wRw0 or w0Rw. If the former, then (by transitivity 0 0 0 of R) w ∧Γ which, together with (ii ) and (b ) leads to contradiction. Similarly, if the latter holds we obtain contradiction by (i0) and (a0). As a consequence we obtain Theorem 1. (Soundness Theorem) If ` G, then |= G. 96 Andrzej Indrzejczak

6. Completeness

Although HCS4.3. was modelled after Avron’s calculus for G¨odel-Dummett’s logic it seems that direct simulation of his completeness proof from Avron [6] requires analytic applications of cut in case of our calculus. To avoid this inconvenience we apply the method which is based on the process of (downward) saturation in the sense of Hintikka. Definition 1. Γ ⇒ ∆ is saturated iﬀ the following holds: 1. ¬ϕ ∈ Γ implies ϕ ∈ ∆ 2. ¬ϕ ∈ ∆ implies ϕ ∈ Γ 3. ϕ ∧ ψ ∈ Γ implies ϕ ∈ Γ and ψ ∈ Γ 4. ϕ ∧ ψ ∈ ∆ implies ϕ ∈ ∆ or ψ ∈ ∆ 5. ϕ ∨ ψ ∈ Γ implies ϕ ∈ Γ or ψ ∈ Γ 6. ϕ ∨ ψ ∈ ∆ implies ϕ ∈ ∆ and ψ ∈ ∆ 7. ϕ → ψ ∈ Γ implies ϕ ∈ ∆ or ψ ∈ Γ 8. ϕ → ψ ∈ ∆ implies ϕ ∈ Γ and ψ ∈ ∆ 9. ϕ ∈ Γ implies ϕ ∈ Γ

Definition 2. s0 is a saturated extension of s iﬀ s0 is saturated and s ⊆ s0.

Lemma 2. (Saturation Lemma) if 0 s, then there is unprovable saturated extension of s. Proof: Let Γ ⇒ ∆ be unprovable but not saturated, then some of the conditions 1-9 do not hold. As an example we consider the cases of ∧ and – we proceed as follows: Assume that ϕ ∧ ψ ∈ Γ but either ϕ∈ / Γ or ψ∈ / Γ . Then add lacking formula to Γ . The obtained sequent ϕ, ψ, Γ ⇒ ∆ is unprovable, otherwise by (∧⇒), Γ ⇒ ∆ would be provable, as well. Assume that ϕ ∧ ψ ∈ ∆ but neither ϕ ∈ ∆ nor ψ ∈ ∆. Then add to ∆, one of the lacking formula, namely this one which yields unprovable sequent. At least one of them must be unprovable because if both ` Γ ⇒ ∆, ϕ and ` Γ ⇒ ∆, ψ, then by (⇒∧), Γ ⇒ ∆ would be provable, as well.

Assume that ϕ ∈ Γ but ϕ∈ / Γ , then add ϕ to Γ . The obtained sequent ϕ, Γ ⇒ ∆ is unprovable, otherwise by (⇒), Γ ⇒ ∆ would be provable too. Cut-Free Hypersequent Calculus for S4.3. 97

Consider some G, let SF (G) denote the set of all subformulae of formulae in G. Let us deﬁne some special hypersequent being its linear saturation, namely: Definition 3. LS(G) is a linear saturation of G iﬀ the following holds: 1. All elements of LS(G) are saturated 2. For each s ∈ G there is a saturated extension in LS(G) 3. If Γ ⇒ ∆ ∈ LS(G) and ϕ ∈ ∆, then saturated extension of Γ ⇒ ϕ belongs to LS(G). 4. If Γ ⇒ ∆ ∈ LS(G) and Π ⇒ Σ ∈ LS(G), then either Γ ⊆ Π or Π ⊆ Γ.

Lemma 3. (LS Lemma) If 0 G, then there is unprovable linear saturation of G.

Proof: Assume that 0 G. We build LS(G) in stages. First, by Saturation Lemma, each s ∈ G may be saturated. Let us denote by G0 the result of replacing each s ∈ G by its saturated extension s0. Clearly, 0 G0 and conditions 1, 2 of deﬁnition of LS(G) are satisﬁed. Assume that G0 is not yet an LS(G), then condition 3 or 4 is violated; we proceed as follows.

If 3 does not hold, then consider Γ ⇒ ϕ. It must be unprovable, otherwise by (⇒) also ` Γ ⇒ ∆ and consequently ` G – contradiction. Hence, by Saturation Lemma, Γ ⇒ ϕ may be extended to unprovable saturated s0. Finally add it to G0.

If 4 does not hold, then for some saturated s1 = Γ ⇒ ∆ and s2 = Π ⇒ 0 Σ we have that neither Γ ⊆ Π nor Π ⊆ Γ . Consider s1 = Γ , Π ⇒ ∆ 0 and s2 = Π , Γ ⇒ Σ. At least one of them is unprovable, otherwise, by (Dich) we obtain ` Γ ⇒ ∆ | Π ⇒ Σ and, consequently, ` G which 0 is impossible. Take unprovable si, i = 1, 2, by Saturation Lemma it has 00 00 unprovable saturated extension si . Finally, replace si by si . Clearly, the process must be repeated many times, possibly infinite. But note that SF (G) is finite and all produced sequents are made of SF (G) due to the subformula property of rules. Hence, in fact, created LS(G) (understood as a set) must be finite as well since in many cases the process of enriching our unprovable G with new sequents is redundant. However, we do not aim here in devising some smart procedure of creating finite LS(G) avoiding loops e.t.c. and providing decision procedure. 98 Andrzej Indrzejczak

Now we deﬁne a model MG for G as follows:

• W = {Γ ⇒ ∆ : Γ ⇒ ∆ ∈ LS(G)} • R(Γ ⇒ ∆, Π ⇒ Σ) iﬀ Γ ⊆ Π • V (p) = {Γ ⇒ ∆ ∈ W : p ∈ Γ }

Linearity of R, as well as transitivity or reﬂexivity follows directly from the deﬁnition of LS(G) and Γ . We need to prove:

Lemma 4. (Truth Lemma) For each ϕ ∈ SF (G) and each Γ ⇒ ∆ ∈ W it holds true:

• ϕ ∈ Γ implies Γ ⇒ ∆ ϕ • ϕ ∈ ∆ implies Γ ⇒ ∆ 2 ϕ

Proof: By induction on the complexity of formulae. The basis is obvious from the definition of V . For complex extensional formulae the result follows directly from the definition of saturated sequents and induction hypothesis. We will only show the case of . Assume that ϕ ∈ Γ . In order to show that Γ ⇒ ∆ ϕ we must show that if R(Γ ⇒ ∆, Π ⇒ Σ), then Π ⇒ Σ ϕ. Assume that Γ ⊆ Π , so ϕ ∈ Π . But then by definition of saturated sequents, it follows that ϕ ∈ Π and by induction hypothesis Π ⇒ Σ ϕ. Assume that ϕ ∈ ∆. In order to show that Γ ⇒ ∆ 2 ϕ we must show that there is some Π ⇒ Σ such that R(Γ ⇒ ∆, Π ⇒ Σ) and Π ⇒ Σ 2 ϕ. By P. 4. of definition of LS(G) and by LS Lemma in LS(G) there is saturated extension Π ⇒ Σ of Γ ⇒ ϕ. It is accessible to Γ ⇒ ∆ (because Γ ⊆ Π ). Since ϕ ∈ Σ then by induction hypothesis Π ⇒ Σ 2 ϕ. Consequently, we obtain: Theorem 2. (Completeness Theorem) If |= G, then ` G.

Proof: Assume that 0 G, then by LS Lemma exists LS(G) and 0 LS(G) too. On the basis of LS(G) we obtain a model MG which falsifies every element of LS(G). But then, by p. 2 of definition of LS(G), every element of G is falsified, as well. Therefore, 6|= G. Cut-Free Hypersequent Calculus for S4.3. 99

7. HCS4.3 with Analytic Cut

Although cut-free systems are mostly welcome, the presence of (somewhat controlled) applications of this rule is not necessarily a drawback. For example, analytically restriced forms of cut appeared already in several sequent and tableau calculi for many modal logics (see, e.g. Takano [21] and Goré[11]) and did not destroy their usefulness as tools for proof-search. In particular, if we restrict the use of cut to subformula-preserving (i.e. cut- formula is a subformula of the endsequent) we can obtain a valuable and practical system. The development of research on complexity theory has shown that even in systems where cut is completely eliminable, some controlled applications of this rule may lead to the construction of essentially shorter proofs. Hence, the absolute cut elimination is for obtaining short proofs not only dispensable but may be even troublesome. The source of the problem lies in the fact that in cut-free sequent calculi we have often too much branching with the same sequences of steps repeated in many branches. This may lead even to exponential growth of the length of proof tree. It was shown (see, e.g. D’Agostino [1]) that from the standpoint of such complexity measure (sometimes called relative proof length complexity), cut-free systems are worse, and it is also evident that exactly the possibility of cut applications is the property allowing for shorter proofs. This is not the reason to maintain that cut-free systems are generally less efficient than those with (some admissible forms of) cut. There are serious reasons to think that relative proof length complexity is not a very good measure for evaluating the practical efficiency of a system. In fact, systems producing ‘short’ proofs may be less efficient because proof search space may be significantly larger, also in case of automatization an imple- mentation of the system may be more involved. Anyway, there are serious reasons to study systems with controlled application of cut. Although it was Boolos [9] who (for the first time) paid an attention to the problem of possible drawbacks of cut-elimination the consequences of this fact were drawn by D’Agostino and Mondadori [2]. In their system KE, cut is primitive (not eliminable) and the only branching rule. Although KE is originally a tableau system working on formulae not on sequents we can easily incorporate its nonbranching rules into the context of HCS4.3. Instead of (⇒∧), (∨⇒), (→⇒) we must introduce the following pairs of nonbranching rules: 100 Andrzej Indrzejczak

ψ, Γ ⇒ ∆, ϕ | G ϕ, Γ ⇒ ∆, ψ | G (⇒∧1) (⇒∧2) ψ, Γ ⇒ ∆, ϕ ∧ ψ | G ϕ, Γ ⇒ ∆, ϕ ∧ ψ | G

ϕ, Γ ⇒ ∆, ψ | G ψ, Γ ⇒ ∆, ϕ | G (∨⇒1) (∨⇒2) ϕ ∨ ψ, Γ ⇒ ∆, ψ | G ϕ ∨ ψ, Γ ⇒ ∆, ϕ | G

ϕ, ψ, Γ ⇒ ∆ | G Γ ⇒ ∆, ϕ, ψ | G (→⇒1) (→⇒2) ϕ → ψ, ϕ, Γ ⇒ ∆ | G ϕ → ψ, Γ ⇒ ∆, ψ | G

Clearly, we cannot eliminate (Dich) and therefore in such modified system we still have two branching rules (Cut is the second). Note that in all these rules we must also have a minor premise (i.e. this main formula in the conclusion which is not compound in the schema) rewritten in the upper sequent since it may be used again during the proof-search process (performed reversely from the conclusion to premises), similarly as in (⇒) We could use rules with no rewrite at the cost of having a system with sequents being defined on multisets and with internal contraction, but from the standpoint of actual proof making it is not needed. The soundness of this system is easy to prove. Below we sketch a completeness proof for this system based on the idea of (relative) maxi- malization. Consider some unprovable G and let us define some special hypersequent being maximal extension of G, namely:

ME(G) = {Γ ⇒ ∆ : Γ ∪ ∆ ⊆ SF (G) and 0 Γ ⇒ ∆} Clearly, ME(G) exists and satisﬁes the following: Fact 1. (a) G ⊆ ME(G) (b) ME(G) is ﬁnite

(c) ∀s⊆SF (G), s ∈ ME(G) or ` s | ME(G) (a) and (c) by definition of ME(G), (b) by finiteness of SF (G). Next define the set of saturated sequents of ME(G) Sat(G) = {Γ ⇒ ∆ ∈ ME(G): Γ ∪ ∆ = SF (G)} The following two lemmata will be usefull in proving Truth Lemma. Cut-Free Hypersequent Calculus for S4.3. 101

0 Lemma 5. (Lindenbaum Lemma) ∀s∈ME(G)∃s0∈Sat(G), s ⊆ s

Proof: Let Γ ⇒ ∆ ∈ ME(G) be not saturated, then there are ϕ1, ..., ϕn ∈ SF (G) such that ϕi ∈/ Γ ∪ ∆, for n ≥ i ≥ 1. Take ϕ1, then at least one of Γ ⇒ ∆, ϕ1 , Γ , ϕ1 ⇒ ∆ is unprovable; otherwise by (Cut) ` Γ ⇒ ∆, which is impossible. By deﬁnition, unprovable extension of Γ ⇒ ∆ belongs to ME(G). Repeat this procedure to all ϕi, i ≤ n untill you obtain saturated extension of Γ ⇒ ∆. Note that we need (Cut) only in proving this version of Lindenbaum Lemma, and that we need it only in analytically restricted way, i.e. on subformulae of proved hypersequent.

Lemma 6. (Saturation Lemma) All Γ ⇒ ∆ ∈ Sat(G) are saturated in the sense of deﬁnition 1, moreover: if ϕ ∈ ∆, then Γ ⇒ ϕ ∈ ME(G) Proof: Now we do not extend sequents to their saturated extensions, as we did in the proof of Lemma 2, but show that they already have needed properties. As an example we consider the cases of ∧ and . Assume that ϕ ∧ ψ ∈ Γ but either ϕ∈ / Γ or ψ∈ / Γ . Consider ϕ∈ / Γ , then by saturation ϕ ∈ ∆, but then ` Γ ⇒ ∆ by (IW ) since ` ϕ ∧ ψ ⇒ ϕ. similarly if ψ∈ / Γ . Assume that ϕ ∧ ψ ∈ ∆ but neither ϕ ∈ ∆ nor ψ ∈ ∆. Then by saturation ϕ, ψ ∈ Γ , but then ` Γ ⇒ ∆ by (IW ) since ` ϕ, ψ ⇒ ϕ ∧ ψ.

Assume that ϕ ∈ Γ but ϕ∈ / Γ . Then by saturation ϕ ∈ ∆ and ` Γ ⇒ ∆ by (IW ) since ` ϕ ⇒ ϕ. Assume that ϕ ∈ ∆ but Γ ⇒ ϕ∈ / ME(G). By Fact 1, (c) ` Γ ⇒ ϕ | ME(G) but then ` Γ ⇒ ∆ | ME(G) by (⇒ ) so ` ME(G) since Γ ⇒ ∆ ∈ Sat(G) ⊆ ME(G) – contradiction.

Next we deﬁne a model MG for G, similarly as in the completeness proof for HCS4.3 with W being the set of these Γ ⇒ ∆ which belong to Sat(G). Clearly, we must prove that R is linear. Showing reﬂexivity and transitivity is trivial, so we provide only a proof for dichotomy.

Proof: Assume that Γ ⇒ ∆ ∈ W and Π ⇒ Σ ∈ W but neither Γ ⊆ Π nor Π ⊆ Γ . Hence, there is some ϕ, ψ such that ϕ ∈ Γ , ϕ∈ / Π 102 Andrzej Indrzejczak and ψ ∈ Π , ψ∈ / Γ . By saturation, however, ϕ ∈ Σ and ψ ∈ ∆, so ` Π , Γ ⇒ Σ and ` Γ , Π ⇒ ∆. By (Dich) we obtain ` Γ ⇒ ∆ | Π ⇒ Σ, which is impossible. The proof of Truth Lemma is the same as for HCS4.3, except for the case of ϕ ∈ ∆. So we will prove it. Proof: Assume that ϕ ∈ ∆. In order to show that Γ ⇒ ∆ 2 ϕ we must show that there is some Π ⇒ Σ such that R(Γ ⇒ ∆, Π ⇒ Σ) and Π ⇒ Σ 2 ϕ. By Saturation Lemma Γ ⇒ ϕ ∈ ME(G); hence, by Lindenbaum Lemma, there exists saturated extension Π ⇒ Σ of it which belongs to W and is related to Γ ⇒ ∆ (because Γ ⊆ Π ). Since ϕ ∈ Σ then by induction hypothesis Π ⇒ Σ 2 ϕ. Note that it is the only place so far where we need Lindenbaum Lemma (and consequently analytic cut); the other is below.

Theorem 3. (Completeness Theorem) If |= G, then ` G.

Proof: Assume that 0 G, then 0 ME(G) too. On the basis of Sat(G) we obtain a model MG which falsiﬁes every element of Sat(G), but by Lindenbaum Lemma every element of ME(G) has its extension in Sat(G) and is falsiﬁed, as well. Therefore 6|= G, since G ⊆ ME(G). Acknowledgments: I would like to thank an anonymous referee for valuable suggestions.

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Department of Logic University ofLód´z Kopcińskiego16/18 90–232Lód´z e-mail: indrzej@filozof.uni.lodz.pl