Math. Log. Quart. 39 (1993) 231 - 240

lbi a thein at i cal Logic Quarterly

@ Johann Ambrosius Bart11 199:3

Modal Sequents for Normal Modal Logics

Claudio Cerrato

Via di Bravetta 340, I-001G4 Rome, Italy')

Abstract We present sequent calculi for normal modal logics where modal and proposi- tional behaviours are separated. and we prove a cut elimination theorem for t,he basic system K, so as completeness theorems (in the new style) both for K itself and for its most popular enrichments. MSC: 03B45, 03F05. Keywords: Normal modal logic, Modal sequent, Cut-elimination, Coinplet.eness.

0. Introduction

We develop Gentzen-like calculi for normal modal logics based on nioda.1 sequents. In these sequents formulas can occur in a modal (possible or necessary) way: to denote this fact, we add to the usual modal language two metalinguistic symbols ( ) and [ 1, and we sign by these symbols those formulas occuring in a nioclal way. Modal sequents range on sets of forniulas either uiisigned or signed by ( ) or [ ] * and our calculus tranforms the modal way formulas occur into niodal operat,ors. Signed formulas, e.g. ( A ) , appear quit,e similar to t,he corresponding usual iiiodal formiilas, e.g. 0.4, but they are not really the same thing. In fact., ( A ) stresses the modal nature of the formula, and so modal rules may work on it, while OA stresses the proposit.ioiial nature, and so usual propositional rules may work on it. So, from a semantic point of view, (A4) and OA ( [A] and O A ) are the saiiie thing, while, froin a procedural point of view, they are different. In this sense the nietalinguistic signs ( ) and [ ] have nothing in common witsli the semantic metaliiiguist.ic signs used in semantic tableaus (see [2]).

Furthermore, we allow directly transforming (,4) into 0.4, but we do not. allow the opposite. In such a way we introduce an ordering in a.pplying rules (firsbly the moclal ones, then the propositional ones), without any lack in the espressive power ( in fact., Completeness also holds).

The rules of the calculi are the usual ones, plus general rules to treat. modalit.ies, plus peculiar rules, each one for each iiiodal asioni (see [l]) inst.ead of each O I I ~ for each modal system as usual (see e.g. [2], [4], [ 5 ] ) .

']I would like to gratefully acknowledge Prof. C. CELLUCCI and Prof. G. COMI for their helphtl critics and suggestions. I iun grateful to Dr. G . AMATI for the fruitful coilversation about. the topic of this work. Finally, I specially thank Prof. M. FATTOROSI-BARNABA for t.he iiiaiiy helpfill and encouraging conversation I had with hiin about this work, and the referees for t.llcir ilseful suggestions.

232 Claudio C‘errat.0

We prove the completeness of the calculi for the nornial modal logics K , KB, KD. K T (=T), K4, K5, KBD, KBT(=B), KB4, KD4, KDJ, KD45, K45, KT4 (=s4), K T 5 (=S5) (see [l]), where improperly “conipIeteness” denotes for each one of t.liose systems that tlie formulas that one can prove by the calculus of modal seql1ent.s are esactly the Hilbert-style theorems.

Finally, we exhibit a constructive proof of cut-elimination theorem for the niinimal modal system K by extending the usual proof ( [ 8 ] ) to modal sequents.

1. Modal sequents

Let L={P, A, V, 1, -, C-L, 0 , O ) be the usual modal language, and let ( ) and [ ] be two new nietalinguistic symbols. If A is a formula of L, then ( A ) and [A] are called signed formulas, and formulas and signed formulas together are called crprcssrolls. Finally, if I’ and A are (finite) sets of expressions, then r I- A is a sequcaf. We use capital italic letters for formulas of L, small greek letters for espressions aiicl caljital greek letters for sets of espressions.

A signed formula is made of two parts: (a) a formula of the language L, possibly with esplicit iiiodal operators, and (b) a nietalinguistic sign whose role is to stress the way the formula occurs; this sign has no reference to explicit modal operators occurring in the formula. Metalinguistic symbols cannot be joined together by con- nectives to form other signed formulas. The expressions are formulas logether with the indication of the way they occur: (a) in non-modal way, (b) in “possibly” way, ( c ) in “necessarily” way. In the first case, the expressions are really formulas of L, aiid we call them urisigned formulas; otherwise they are signed formulas. Act.ually, the espressions are the objects of our calculus, so the seyuents are made of espressions and rules apply to sequents of expressions.

We divide the rules in Logical rules regarding logical connectives Structural rules regarding sequents’ structure Duality rules Modal rules

regarding transformation of iiiodalities characteristic of each modal system.

Logical rules are the usual ones (see [$I, [8]), that is they affect only unsigned formulas; so e.g. the rule V I- ( “V :left” in [8]) is tlie rule

C,l”t-A D , r t - A c v ~ , r t- A *

where C, D are formulas of L and r, A are sets of espressions. We use the logical rules A I-, t- A, V t-, t- V,-+, I--, +I-, k-, 7 I-, t- l. On tlie contrary, structural rules apply to all of the expressions; so e.6. the cuf - ru le

is

r I - A . 6 6,r‘t-A’ r ,r+ A , A /

where 6 is an expression and r, I”, A, A’ are sets of espressions.

Modal Sequents for Normal Modal Logics '233

We use the structural rules Cut (see above) and the following thiniring rules TliiiiI- and I-Thin:

r ~ - 4 FI- A , a

Thiii I- I? I- A t- Thin a , r ~ 4 where a is an expressions and r, A are sets of expressions. Thinning is like weakening in [8]; furthermore, we need neither contraction nor exchange rules because we use sets (as in [7]) instead of sequences (as in [8]).

Duality rules convert metaliiiguistic signs into modal operators or t.ransfonn ''POS-

sibly" into "necessarily" and viceversa. Such rules apply only to signed formulas. We use the following duality rules:

t- 4 [A1 r I - A . , O A I-0

. . t- A, ( A ) 00 I-

[ -A] , i? I- A ( A ) , r 1 A. r I- 4, [-A] I- O0

where .4 is a formula of L and r, 4 are set.s of expressions. The notions proof and end-sequeaf are used as i n standard calcirli of seqot.1it.s (set.

e.g. [8]), an ini t ial sequent is one of tohe for111 cr I- cr, where a is a.11 espression, a forniula is called protable iff t- A is an end-sequent.

2. Normal modal calculi

Firstly, we delop our calculus for the minil1ial normal mot1a.l syst.em K . that. is, in Hilbert-style (see [l]), a niodal syst.em, i.e. a systeiii closed under t.he folloiviiig rule of inference RPL

.41 r43, . . . , A , A

where A is a tautological consequence of --II, .A?. . . . . -4,. containing the follo(ving asiom DfO

OA - -01.4, and closed under the following rule of inference RII;

.41 A . . . A 'd,, - A OAl A . . . A OA, - OA '

By I-K A we iiiean that. the forniula A is a t,lieorem of K.

rules, called K - r u l ~ s : TO develop a calculus of niodal sequent.s for K we introduce the folloiving nioclal

where A is a formula of L, r, A are set.s of formulns: of L, ant1 wllere [a] and (a) for a set o of forniulas of L are defiiieci by [a] = {[.4] : .-I E a}, (a) = { (-4) : A4 E

234 Claudio Cerrato

We call (A) ([A]) the principal formula of the rule KI- (I-K), while, as usual, the principal formula of another inference rule is the unique new expression introduced in the lower sequent, or the cut expression for tlie cut-rule.

We remark that in the K-rules the premises contain only unsigned formulas, whereas the conclusions contain only signed formulas.

The sequent calculus obtained by adding tlie K-rules to the logical, structural and duality rules is called the K-calculus. A forniula A is called K-provable, if A is sequent-style provable by the K-calculus.

T h e o r e m 1 (Soundness of the K-calculus). I f A is K-provable, f h r n I-K A. P r o o f . We translate sequents to formulas suitably adapting to the iiiodal sequent

case the Schdte franslafion [6]: Let ' be the translation from expressions to formulas of L defined by

A' = A , (A)' = OA, [A]' = CIA,

where A is a formula of L; if 0 is a set of expression, we write Q' = {4* : $ E 8) ; moreover, for a given set 0 of formulas we write

FoV. . .VFn if Q P = {Fo, . . . , F n } , V Q = { * if 0 = 0.

We extend the translation to sequents by (I' I- A)' = A ( P ) - V(A'), where r, A are sets of expressions, and to sequent-style inferences into Hilbert-style inferences by

premise,, . . . , premise, * (premiseo)*, . . . , (premise,)' ( conclusion > = (conclusion)* The last translation is well defined: in fact, each structural or logical rule is translated into a Propositional Calculus (PC) inference; translation of duality rules are simply either the rewriting of the same formula or the rewriting of an equivalent formula (by the axiom DfO); furtherniore, one can easily prove that the translations of the rules KI- and I-K are Hilbert-style inferences.

The translation ' allows us to prove tlie theorem: in fact, initial seque1it.s are clearly translated to tautologies, and sequent-style inferences are translated to Hilbert-style inferences, so that the translation of a K-provable sequent is Hilbert-style provable in K; so, if A is a K-provable formula, i.e. if 0 I- A is an end-sequent, then I-K (0 I- A)', i.e. (by tlie definition of * ) I-K A.

Now, we develop our calculus for the followiiig 15 iiornial modal systeiiis (i.e. modal systems containing K, see [I]): K, KB, KD, KT (= T), K4, K5, KBD, K B T (= B), KB4, KD4. KD5, KD45, K45, KT4 (= S4), KT5 (= S5) having beyond t.he axionis of K suitable combinations of the following formulas as proper asionis:

T . 0.4 --L A, 5 . 0-4 + OOA, 4. 0.4 - 00.4, B. .4 00.4, D. O A - 0.4.

I-,, A nieans that the formula A is a theorem of A, A being one of the above systenis. In tlie sequent calculus for tlie above nornial modal systems we use bhe follow-

iiig modal rules T, 5, 4, B, D. These rules are quite ininiediate translations of the

Modal Sequents for Normal Modal Logics 235

corresponding modal axioms into the forinalism of seyuei1t.s (this correspondence is explicitly shown in the following Theorem 2) . However, these rules respect tlie "in- troduction" nat,ure of the calculus (and in fact. expressions involved do not decrease in complesity) that could be useful to extend t.he cut-eliminat.ion theorem to all of the normal modal logics.

where A is a forinula of L and I?, A are sets of espressions. The calculus obtained e.g. by adding the rules T and 4 to the K-calculus is called

the KT4-calculus. A forniula A is called KT.l-yrouable, if it is sequent,-style provable by the KTCcalculus.

T h e o r e m 2 (Soundness of norinal-calculi). If.4 is A-protinblc, I h r n I-,, A . ir/icre A is one among the above 14 iiornial sys2em.s.

P r o o f . We can use the same proof we used for Theorem 1; we must. only prove that also the translations of the rules T, 5 , 4, B, D are Hilbert-style inferences. This is an easy check and is left to the reader.

3. Conipleteness of the normal modal calculi

T h e o r e m 3 (Completeness of the K-calculus) l f I - ~ A , f h e n '4 i s K-prornble. P r o o f . We transform each Hilbert-style K-proof into a sequenbstyle K-proof. Given a Hilbert-style K-proof, then for each forinula A proved at sonie step we

define the degree a(A) of dependence on modal inferences by

d ( A ) = 0

a(A) = n

a ( A ) = n + 1 if B I-K A by a direct use of rule RK with B ( B ) = 11.

if either I-pc .4 or A is a modal asioiii;

if Bo, . . . B, I-pc A , where each Bi is proved at. some previous step and ma~~i~s{d(Bi)} = n;

One can easily realize that this is a good definition (fised a proof) and that the degree counts the maximal number of nest,ed uses of rule RIi in the proof of A. Moreover, we can always reduce the second case to the following alternat.ives: either each Bi is a modal axiom or is proved by a direct use of FtK? or (only if B ( A ) = 0) we ca.11 reduce this case to I-pc A .

(0)

We need the following result about. PC-calculus (see e.g. [3]): if r I-pc A , then the sequent r I- A is PC-provable.

238 C!lauclio C'errat,o

Now, we transform each Hilbert-style K-proof into a sequent-style K-proof I>y an

1. 8(44) = 0. We distinguish three cases: (a) t-pc -4. Then by (0) the sequent I- A is PC-provable, and so i t is K-provable. (b) A is tlie modal axiom DfO. We prove that tlie sequent I- 0.4 - 707.4 is

induction on the degree:

K-provable:

by I- 00

b y t - 0 , t - 0

by 7 I-

by I--+

(4 I- (-4 I- [74, (.4) I- o T 4 , O A

l O 7 A I- OA

I- -0 -A - O A

aiid in a dual way one obtains I- 0.4 - - 0 4 .

(c) Bo, . . . , B, I-pc A, where each B, is a niodal axiom (in fact, 110 rise of rille RK is possible, because 8 ( A ) = 0 and tlie case I-pc A has already been treated in (a)). Then each sequent t- Bi is K-provable by ( I ) ) . Furtherinore, the sequent Bo, . . . , B, I- A is K-provable by (0). So, by s + 1 uses of cut-rule, the sequent I- .4 is K-provable.

2. We suppose the theorem is proved for & A ) 5 n aiid let 8(A) = n + 1. We distinguish two cases:

(d) B t-K A by a direct use of rule RIi, i.e. B = CO A ... A C'n - C' and -4 = OCo A . - . A OC,, - OC. Then 8(B) < 8 ( A ) = 11 + 1 (by defiiiitioii of 6). so by inductive hypothesis,

(i)

furthermore Co A . . . A C n - C, Co, . . . , C n I-pc C', so by (o),

(ii) so we obtain

the sequent I- CO A . . . A C,, - C is K-provable;

the sequeiit Co A . . . A C,, - C, CO, . . . , C',, I- C is K-provable;

cut.

oco, . . . , 0Cn t- oc logical and st.ructura1 rules

I- OCo A * * . A OC,, -+ OC

i.e. the sequent I- A is K-provable. (e) Bo, . . . , B, I-pc A, where each B, is a modal axiom or is proved by a direct

use of RK (Fpc A is not possible because 8(.4) = n + 1 > 0). If Bi is a modal axiom, then the sequent t- B, is K-provable by (h); if B, is proved by a direct use of R K then the sequent I- Bi is K-provable by (d) , recalling O( B, ) 5 B( -4) = n + 1.

Modal Sequents for Normal Modal Logics 23T

In every case, the sequents I- Bo, . . . , I- B, are K-provable, furthermore, the sequent. Bo, . . . , B, I- A is K-provable by (o), so, by s + 1 uses of cut-rule, the sequent I- .4 is K-provable.

This completes the proof of Theorem 3.

Now, we prove completeness theorems for all other 14 norinal motlal syst.ems.

T h e o r e m 4 (Completeness of noriiial modal calculi). Let A be ont n7nony K, KB, KD, KT, K4, K5, KBD, KBT, KB4, KD4, KD5, KD45, K4.X KT4, KT5. If I-A A , then .4 is A-provable.

P r o o f . W e use the same proof a s for Theoreiii 3, with a slight. modification for the case d ( A ) = 0. We must only prove that each niodal asiom aniong T. 5 , 4, B, D is provable in a calculus contaiiiing the rule T, 5 , 4. B, D. respectively. For esaiiiple we treat the cases T and 4:

by I-4

by 0 I-, I- 0

T A k . 4 4 [ A 4 1 I- [A1 by TI-

by 0 I- [A] I- [ O A ]

O A I - A OA I- OOd4

I - 0 A - . 4 4 I- 0.4 - O O A

[A1 I- A

by I-- by I--

T h e o r e m 5 (The cut-elimination theorem for K). If a seq.ucnt is K-proi:nblc, then it is K-provable uithout a cut.

P r o o f . We add t.o the usual cut.-elimiiiat.ion proof for PC (see e.g. [8]) t.1ie cases regarding duality and moda.1 rules. The rank r ( P ) of a proof P is: defioed as usual, while the grade g ( P ) of P is. as usual, the grade ~ ( c I ) of the principal formirla a of its last, inference, that, iiidicat.es t.he cotiiplesity of n with respect. t.0 tlie noit- st.ructura1 inferences, but we coii1put.e i t . wit.11 a slight motlificat.ion: i l l P C t . 1 ~ only lion-structural rules are the logical rules, each oiie iiit,rodiicing a new logical s y ~ i l ~ o l i n t,he principal formula; so the grade of n is imiiitdiately clet.rrminetl only 1 ) ~ coiiiit.iiig the number of logical coiiiiectives occuring i n c). In our case, each iion-st.riict.ural rule introduces a new logical or iiiodal or nietalinguist.ic symbol, hilt4 0 t-, t- 0. 0 t- a.ad t- 0 also delet,e a 1iiet.alinguistic symbol; so t,he glol)aI number of sy1iiI~ols does not. increase; however, only such rules int,roduce iiioc1a.l operat.ors. so t1ia.t. n e solve blie probleni assigning to each modal operat.or the grade 2 (1 for it.self plus 1 for the c.lelet.etl metalinguist~ic symbol). So y ( n ) is the iituiiber of logical contiectives orcuring in I I plus t.he number of metalinguistic symbols occuring i n (I plus the doiible of t.hc nunil)er of modal operators occuring in n. Thus. for esa.mple, we have .q( -4) < g( (-4) ) < ! / (0 -4) .

Now we consider the case that an inference upper a cut is a duality or a 1;-rule. Firstly, in t.he case of a dualit,y rule whose princil.)al formula is iiot. the cut-formula we can use t.he same proof used for logical rules \vliose principal foruiiila is not the cut-foriiiula (see e.g. [8]). So. hereafter. w e shall coiisidrr only duality nilcs nlioee p r in c i pa 1 for mu la is t.he cu t.-fonii u I a.

C a s e 1. Dua1it.y rule + duality rule. \Ye yrovr t.he t.heorem in tlie case 1 1 1 1 : c u t - formula is 0 = [-B]; the other cases (n = (-[]). (I = 0..1 autl (I = 0 . 4 ) ariw from

238 Claudio Ckrrato

this cw by suitable adaption. We must elininate the cut from the followiiig proof:

n I- A , ( B ) 00 I-

( B ) , I- A I- 00 r I- A, [431 [7B],II I- A

cut of [+I r, II* I- A*, A

where II* and A* stress (only when required from the context) that [-B] is not present.

We distinguish four cases, the first is allowed only for v ( P ) = 2, t.he others are allowed only for r ( P ) > 2:

(4 [-BI 4 A, [-I 4 II.

(c) [-B] E A, [ 4 3 ] 4 II. The proof figure is dual to that in (b).

(d) [-B] E A, [-B] E II. ( B ) , I- A*, [-B] [-B].II* I- A , ( B )

r I - A * , [ ~ B I [ - B ] , ~ * ~ - A , ( B ) ( B ) , r , I - ~ * , [ - q [ - B ] , ~ * , I - A

r ,n* I- A * , A

r, n* t- A*, A, ( B ) (B) , r ,n* I - A * , A

In every case, the thesis is proved by induct.ive hypothesis 011 t.he grade aiitl on

C a s e 2. Duality rule + K-rule. We exhibit a proof only for the rule I-K wit.11 the rule 00 I-; the other proofs are similar. The proof P we must transfornl appears as

the rank.

r I- A,-B II I- A, ( B ) I-li 00 I-

[rl, n* t- (A) , A

[-B], n I- A cut of [-B] [rl t- ( 4 1 [-I

We distinguish two cases, the first is allowed only for r( P) = 2, the s~cond is allowed only for r ( P ) > 2.

Modal Sequents for Normal Modal Logics 239

(4 [+I 4 n: B l - B

- I - I'I-A,-B -B,BI-

l" ,El -A

(B)'[rl I- (A)

s(-.B) < !?([-I) Id-

g((B)) < s([-B]) n I- A , ( B )

[ri, n t- (A), A (b) [ iB] E n:

t- A , T B l-K

rank< r ( P ) [r] l- (A), [+?I [-B], n* I- A , ( B )

PI, n* I- (A), A g((B)) < s([-l)

[rl, n* I- (A), A, ( B ) ( @ I [rl t- (A)

In both cases, the thesis is proved by inductive hypothesis on the grade and on the rank.

C a s e 3. I(-rule + K-rule. We prove the theorem only for the rule I-1; t,oget.her with the rule t-K; the other proofs are similar. The proof P we must modify appears as

r k A , A A, ll I- A , B l-K I-Ii

cut of [A] PI I- (A)* [A1 [.41* [HI I- ( N I PI

[r17 [nl I- (A) 1 (11) 1 PI We tranforni it into the following proof:

I'I-A,A A,IIl-A,B s ( A ) < s([-4,

r , n k 4 , A , B

PI 3 PI I- (A) 1 (4 I PI I-Ii

The thesis is proved by inductive hypotliesis on the grade. This conipletes the proof of Theorem 5 .

References

CHELLAS, B. F., Modal Logic: An Introduction. Cambridge Univ. Press, Cambridge 1980. FITTING, M. C., Proof hlethods for Modal and Intuitionistic Logics. D. Reidel Puhl. Comp., Dordrecht 1983. KLEENE, S. C., Introduction to Metainathemtics. North-Holland Publ. Coi~lp. , Amster- dam 1952. OHNISHI, M., and K. MATSUMOTO, Gentzen method in modal calculi. Osaka hlath. J. 9 (1957), 113-130.

240 ('lautlio C'errabo

[S] SAMBIN. G., and S. VALENTINI, A modal sequent. calculus for a fragment. of arit.linict.ic. Studia Logica 39 (1980), 245-256.

[6] SCHfiTTE, I<., Schlufiweisen-Kalkiile der Pridikat,enlogik. Math. .4nnalen 122 ( 1951), 4i-65.

[7] ~VNDHOLM, G. , Systems of deduction. In: Handbook of Philosophical Logic, volniiie 1 (D. GABBAY and F. GUENTHNER, eds.). D. Reidel Publ. Comp., Dordrecht 1983, pp. 133-188.

[g] TAKEUTI, G., Proof Theory. North-Holland Pohl. C'omp.. Anisterdani 19i5.

(Received: April 9, 1991)