Decidability of four modal logics by

KRISTER SEGERBERG (Uppsala University and Stanford University)

E. J. Lemmon has termed Kripke’s basic systemK and the fourteen logics definable by addition of one or more of the following sche- mata “the most fundamental systems of modal logic”:

D. u A + O A T. o A + A 4. o A + n o A B. O o A + A E. O o A + n A

One good reason for this judgment is that this collection contains the three best-known systems in all modal logic, viz. the Feys-von Wright system T and Lewis’s systems S4 and S5. Other celebrities include the deontic logic D and the Brouwer system. After the appearance of [2] there exist completeness results for all fifteen, and decidability results for eleven of them. In this paper we pro- pose to complete the picture by proving decidability for the four remaining systems: KE, DE, KE4, and DE4.’

This paper is to be regarded as a sequel to [3], to which the reader is referred regarding all material not explained here. Thus we use the terminology introduced there, and we also make free use of results established there, particularly lemmata 1 and 2 of section 2.

Let R be a binary relation and X a set. We adopt the following definitions:

In naming these logics we follow the convention in [2]. The general principle is that KA1 ... A, is the logic got by adding A,, as a new axiom schema to KA1 . . . A,-l, although for greater ease ’K’ may be dropped before ’D’ and ’T’. Exceptions: S4 and S5. (According t o this convention, S4 = T4 and S5 = TB4 = = TE4.)

22 KRISTER SEGERBERG

R is serial' (on X) iff Vx E X 3 y E X xRy R is euclidean' (on X ) iff v x , y , z E X[xRy & XRZ * y R z ] R is universal (on X ) iff Vx, y E X xRy We say that an element x is (R-)reflexive iff xRx.

in [2], state that these logics are determined as follows: KE -by the class of all euclidean world systems DE -by the class of all serial euclidean world systems KE4 - by the class of all euclidean transitive world systems DE4 - by the class of all serial euclidean transitive world systems

We introduce the concept of a generated model. (Cf. [ l ] p. 193.) Let '21 = (X, R, P ) be any model, and let u be any element in X . Define:

The completeness theorems for KE, DE, KE4, and DE4, given

Xu= { x E X I3n E N ~ R f i x } ~ Ru= R n (Xu x X u ) P: = p k n x u (all 12 E N)

Then W = (Xu, R", P") is called the model generated from '21 by u. A generated model is a model generated from itself by some element. A generated world system is the world system'of some generated model. The importance of these concepts resides in the fact that, with '21 as above, for all formulas A and all xEXu,

1: A iff A. This means e.g. that if A fails in any euclidean model,

A also fails in a euclidean generated model; thus KE is determined by the class of all euclidean generated world systems. And simi- larly for the other three cases we are concerned with in this paper.

L E M M A . Let %I = ( X , R ) be a euclidean world system generated by some element u. Then R is universal on X-{u}. If u is reflexive, then R is universal on all of X .

P R O O F . We begin by showing that R is an equivalence relation on X - (u } . (i) Reflexivity. Assume x E X- (u} . Then there exists some w E X such that wRx. By the euclidean condition w R x

As far as we know, this terminology goes back t o Lemmon. * By definition, uRox iff u = x , and uRn+lx iff 3y[uRny & yRx] .

DECIDABILITY OF FOUR MODAL LOGICS 23

& wRx * xRx. So xRx. (ii) Symmetry. Assume x, y EX- ( u } and xRy. By (i) xRx, and by the euclidean condition xRy & xRx * yRx. So yRx. (iii) Transitivity. Assume x, y , z E X - ( u } and xRy & yRz. By (ii) yRx, and by the euclidean condition yRx & yRz s- xRz. So xRz.

To see that R is universal on X- (u} , take any x, y EX- ( u } . Since x, y # u there are positive integers p , q such that uR% and uRqy. Because of (i) and (iii) above there exist v, w E X - ( u } such that uRvRx and uRwRy. From (ii) we conclude that xRv, and from the euclidean condition that vRw. By (iii) therefore xRy. This proves universality of R on X- (u } .

Assume finally that u is reflexive. Let x be any element in X. If x=u, then trivially uRx and xRu. If x#u, then by the same argument as in the preceding paragraph there is some v EX such that uRvRx. By the euclidean condition uRv & uRu * vRu and vRu & vRx * uRx. So uRx. Also, uRx & uRu * xRu. Thus XRU. So in case u is reflexive, R is universal on X.

T H E O R E M 1 . KE and DE are decidable.

P R O O F . To prove the theorem it is enough t o prove that KE and DE have the finite model property. Hence it will be more than enough t o prove that if L is any logic at least as strong as KE (DE), and if u is any element in the canonical model 9XL, and if Y is any set of formulas closed under subformulas, then the least filtration of 9 2 through Y will be euclidean (euclidean and serial).

Assume the hypothesis. Let CZI = (X, R, P ) be the least filtration of 9Xt through Y. Suppose 5, q, C are elements in II such that ERq and @t:. There are two cases.

(ct) There exists some y E q n (X- {u}) . If there is some z E C n n (X- (u}) , then yRLz. (This is because R,, the accessibilityrelation of 9XL, is euclidean, as is easily shown. Hence RZ is also euclidean, and so by the lemma RL is universal on X- (u} . ) Therefore qRC. If on the other hand t: n (X- {u} ) = 0, then t: = {u} . Since [ R C

For an explanation of why this is sufficient, see [3], the beginning of section 3.

24 KRISTER SEGERBERG

there exists x E E such that xRLu. Thus, since RL is euclidean, u is reflexive. By the lemma then YRLu. Hence qRC.

(p) q n ( X - { u ) ) = 0 . Then q = { u } . As before, since 31 is the least filtration there exists x E such that xRLu. So again u is reflexive. Take any z E C. By the lemma uRLz. Consequently qRC.

This proves that % is euclidean if KE C L . If also DE CL, then t21 is serial. To see this, note that RL becomes serial as soon as KDcL. For take any x in DL. If there is no y such that xRLY, then trivially oA, -A Ex, for every formula A. But by the schema D then - - A E x, which contradicts the consistency of x. Now, take any 5 in 31 and let x E E. Then there is some y in D! such that xRLY. Hence @q, where q = [ylzY. So II is serial.

T H E O R E M 2. KE4 and DE4 are decidable.

P R O O F . Let Y be any set of formulas closed under subformulas. Let L be any logic a t least as strong as KE4 and let u be any element in DL. Let a+ = ( X , R+ , P ) be the transitive closure of the least filtration of 3; through Y. From [3] we know that a+ is a filtration. We contend that a+ is euclidean; the theorem is essentially a corollary of this contention. (See the foot-note on the preceding page.)

Take any 5, q, C in 'zL+ and assume that ER+q and ER+C. There are two cases to consider.

(y) There exists some y E q n ( X - {u}) . If C n ( X - ( u } ) = 0 , then by an argument analogous to that under (a) above, qR+L If 1: n ( X - { u } ) = 0 then C = {u} . Since ER+I: we know from [3] that for some integer n 2 0 ERnC, where B is the accessibility relation of the least filtration of D; through Y. Hence there exists some w in 337: such that wRLu. It follows that u is reflexive. By the lem- ma therefore YRLu, and so qR+C. (8) q n ( X - {u} ) = 0. Then q = {u} . By the same kind of argu-

ment as under (y), u is reflexive. So for any z E C, uRLz. Hence qR'I:.

This proves 'zL+ to be euclidean if KE4 c L. If in addition DE4 G L. a+ is serial as well. The proof of this assertion parallels the corre- sponding proof in the preceding theorem.

DECIDABILITY OF FOUR MODAL LOGICS 25

References [l] E. J. LEMMON, "Algebraic semantics for modal logics II", Journal of Symbolic

[2] E. J. LEMMON & DANA SCOTT, Intensional logic. Preliminary draft of initial

[3] KRISTER SEGERBERG, "Decidability of S4.1", Theoria, 34, 1968, pp. 7-20.

Received on November 3, 1967.

Logic, 31, 1966, pp. 191-218.

chapters by E. J. L e m o n . July 1966 (mimeographed).