two simple incomplete modal logics

Two simple incomplete modal logics by

J. F. A. K. V A N BENTHEM (University of Amsterdam)

I N 1974 K. Fine and S . K. Thomason published examples of incomplete modal logics (cf. [3 ] and [9]) . These examples are rather complicated, and the purpose of this paper is to give simpler ones, in a sense to be described below.

1. Introduction

The modal formulas to be considered below belong to the modal propositional language with proposition letters p , q, r , . . ., Boolean operators 1 (negation), -+ (material implication), A (conjunction), v (disjunction), c* (material equivalence) and modal operators 0 (necessarily) and 0 (possibly). The degree d(cp) of a modal formula cp is defined inductively according to the clauses

(i) d(p) = 0, for proposition letters p , (11) 4 3) = 4 $ ) 7

(iii) d($ -+ x) = d($ A x) = d($ v x) = d($ c* x) =

(iv) d ( 0 $) = d( 0 $) = d($) + 1, for all modal formulas =maw($>, d(x)) ,

$ and x. Semantical structures are frames consisting of a non-empty domain W with a binary relation R on W (notation: 9 = ( W, R ) ) and general frames consisting of a frame 5 together with a non-empty set Wt of subsets of W which is closed under taking complements (w.r.t. W), intersections and the operation m defined by m(X) = { W E W 1 l(3 v ~ X ) R w v } . For a modal formula cp, a frame 3 and a valuationV on 9 taking the proposition letters occurring in cp to subsets of W, (9, V) +cp[w] is defined, for W E W7 by means of the usual Kripke truth definition. 9 +cp[w] means that, for all valuations V on 5,

26 J . F. A. K. VAN BENTHEM

(5, V ) cp[w]. ( 9, 719) t= cp[w] holds if, for all valuations V on 8 taking values in z1p, (8; V ) + cp[w]. It follows that, if Wis the power set of W, then (8, us) and 5 may be identified. For a set r of modal formulas, (5, V)kT[w] holds if, for all (Per, (9, V)+cp[w]. (9b T[w], (9, z1p) b T[w] are defined similarly.) (5, V) b cp means that, for all W E W, (9, V) + cp[w]. (s+ cp, $+ r, etc., are defined similarly.) The following notions of semantic consequence will be used. r + cp if, for all frames 5, %+ r only if $b cp, and r +,cp if, for all general frames (5, U), (5, W ) r only if (9, 719) b cp. Clearly, r + ,cp only if r + cp, but the converse fails, as will be seen presently.

On the syntactic side, only 1, -+ and 0 will be regarded as primi- tives (the other signs mentioned above being thought of as defined in the usual way) and a modal Zogic is, then, a set of modal formulas which (i) contains all classical tautologies, (ii) contains all formulas of the form n(cp -+ $) -, (ncp -+ n$) and is closed under (iii) the rule of detachment (“modus ponens”), (iv) prefixing of 0 (“necessitation”) and (v) the formation of substitution instances. For a set r of modal formulas, M(T) is the smallest modal logic to contain r, and r is said to axiomatize M(T). The main completeness theorem (cf. S. K. Tho- mason [ lo]) is the following:

r t= gcp if and only if cp E M(T).

For many sets r, this equivalence may be strengthened to

cp if and only if cp E M(T), r however. Such T’s are called complete. Most traditional modal logics, like S4 and S5, are complete. Sometimes even stronger equivalences hold, like for the logic Dym axiomatized by {Op + p , [7p -+ O D , O(O(P -+ UP) ‘PI -PI :

cp E Dym if and only if cp holds on all finite reflexive trees.

(Dym was named after its third axiom, which is related to “Dummett’s Formula” o(O(p -+ u p ) -+p) -+ ( O n p - p ) . ) Many results of this kind occur in K. Segerberg [8]. But there are also incomplete modal logics, like the one given in K. Fine [3], which extends S4 and has a finite axiomatization (of degree 4, with 8 proposition letters), or the

TWO SIMPLE INCOMPLETE MODAL LOGICS 27

one given in S. K. Thomason [9] which is properly contained in S4, but holds on the same frames as S4, and has a finite axiomatization (of degree 4, with 5 proposition letters).

We will give an incomplete modal logic M((cp)) where cp has only one proposition letter (and degree 4) in section 4, and another such logic M((cp}) where cp has degree 2 (and 2 proposition letters) in section 3 . There are no incomplete modal logics with only axioms of degree 1, as will be shown in section 2.

It may be of interest to note that the tense logic axiomatized by GFp FGp and H(Hp + p ) + Hp, i.e., formulas of degree 2 with only one proposition letter, is incomplete. (This is a simplified version of the incomplete tense logic in S. K. Thomason [lo].)

2. Modal formulas of degree 1

Modal formulas of degree 0 are purely propositional formulas and, therefore, uninteresting from the present point of view: M((cp}) is either the set of all formulas (if cp is not a tautology), or the smallest modal logic K (if cp is a tautology). For modal formulas of degree 1 several more interesting facts are provable. The first of these concerns definability. Call a modal formula cpfirst-order dejhable if, for some first-order formula $ = $(x) containing only = and a binary predicate constant R , it holds that

for all frames s( = ( W, R ) ) and w E W, sk q[w] iff s+ $[wI.

Well-known examples are u p + p (defined by Rxx) and n p + o o p (defined by (Vy)(Rxy + (Vz)(Ryz -+ Rxz)) ) .

( A more common notion is first-order definability of a modal formula by means of a first-order sentence $ such that

for all frames 3, $t= cp iff SF $.

If a modal formula is first-order definable in the above sense, by $(x) say, then it is defined in this new sense by (Vx)$(x). But the converse fails, as was shown in van Benthem [l]: the modal formula 0 o o n p -+ 0 O n Op is defined by (Vx)( 3y)Rxy in the second sense, but it is not definable by any first-order formula in the first


sense. This is why we prefer using the first (stronger) formulation whenever it applies.)

Now the following theorem is in van Benthem [l]:

2.1 THEOREM. order definable.

For any modal formula cp, ifd(cp) = 1, then 4" isfirst-

Once formulas of degree 2 are considered, the situation changes, however. The formulas OOp+ O o p and n ( o p + p ) -up are not first-order definable (in either sense) (cf. van Benthem [l] or R. I . Goldblatt [4]).

Theorem 2.1 is proved by noting that any modal formula cp of degree 1 whose proposition letters are p , , . . ., pn is equivalent to a conjunction of formulas of the form

( P A oQ1 A . . . A O&)-'(OR, V .. . V OR,),

where P , Ql, ..., Q k , R , , ..., R , are state descriptions of the form ( l)pl A . . . A ( ?)p, (with ( i ) p i = p i or i p i ) . Extreme cases, in which there are no P , Q's or R's, are included. It suffices to give first-order definitions for these equivalences: the conjunction of these definitions will define the original cp. Assume that no repetitions occur among Q,, . . ., Q k and R , , . . ., R, . We distinguish several cases:

(1) Some R i is some Qj: $ is x = x. (2) No R i is a Qj:

(2.1) All 2" state descriptions are among R , , . . ., R,: $ is (3y)Rxy. (2.2) Some state description is not among R , , . . ., R,: (2.2.1) P is among R , , . . ., R,: (2.2.1.1) There are no Qis : $ is Rxx. (2.2.1.2) There are Qj's: $ is i (3y,)(Rxyl A ... A (3yk)(Rxyk A

calledA It 6) (2.2.2) P is not among R , , ..., R,: (2.2.2.1) P is among Ql, ..., Q k : $ is i(3yl)(Rxy, A ... A

(2.2.2.2) P is not among Q,, . . ., Qk: (2.2.2.2.1) There are no Qj's: $ is x # x.

A n l s i + j c k Y i # y j A n , , i , , X # . Y i A 1Rxx) ...). (This $ is

A ( j yk ) (Rxy , A ~ ~ ~ i + j s k ~ i # ~ j ) . . . ) . (This $ iScalledAl6.)

TWO SIMPLE INCOMPLl3E MODAL LOGICS 29

(2.2.2.2.2) There are Q i s : $ is ?(3y,)(Rxy, A . . . A (3yk)(Rxy, A

A n l j i z j ~ k Y i # y j A \ i ~ i s k X # y i ) . . . ) . (This $ iscalled All:.)

This completes the list of possibilities, as well as the list of first-order relational properties definable by means of modal formulas of degree 1 .

Now in modal Henkin models any two w , V E W are separated by some modal cp (i.e., cp E w, cp $ v); which means that, for any distinct wl, . . ., W,E W there exist cpl, . . ., cp, such that, for all i (1 5 i 5 n), 'pi is in wi, but in no other w j ( l s j g n ) . By inspection of the proof of the above ecpivalences (between anodal cp and its defining first-orda property $), it follows that, if any modal formula cp of degree 1 occurs in the theory of some Henkin model, then its defining first-order property holds on the frame of that Henkin model, whence cp itself holds on that frame. In particular, we have proved the following theorem, originally due to D. Lewis (cf. [5]):

2.2 THEOREM. complete.

For any modal formula cp, if d(cp) = 1, then {cp} is

As an additional result we obtain the following

2.3 COROLLARY. Ifr consists of modal formulas of degree 1, then M(T) = M(A), for somefinite set A consisting of modal formulas of degree 1.

Prooj If contains some y defining x # x, then M(T) = M(Cp A i p } ) . Otherwise, drop all YET defining x = x. Replace all YET defining ( 3y)Rxy by O(p + p ) . Replace all y ET defining Rxx by u p -'p. If contains formulas defining Alti for some k , then let Y E T define Alt: for the smallest index k for which this occurs: keep this y, drop all others of this kind. (Nothing is Iost in this way, because Alti impIies Alt: for all I > k. ) The remaining two cases are treated in a similar way. It follows that A contains at most five formulas. Q.E.D.


3. The recession frame

This section is about a frame which has been studied by D. Makinson (cf. [7]) and S . K. Thomason (cf. [9]):

3.1 DEFINITION. The recession frame 35 is the frame ( W , R), where W is the set of natural numbers and Rmn holds if and only if n z m - 1. %! %g (the veiled recession frame) is the general frame (as,’@ ) in which Us is the set of finite and cofinite subsets of W.

Clearly, 39+ up +p, since R is reflexive, and also we have

3.2 Lemma

3% k U(UP + Ud ” U(U4 + UP); 3% + (OP A O ( P -+ UP)) -+Pi 3 % ~ + o o P + o o P .

Proof: The first formula is defined by the first-order property

(*) (Vy)(Rxy+(Vz)(Rxz -+((Vu)(Ryu-+ Rzu) v

v (W(Rzu -+ RW))))

(cf. van Benthem [l], chapter 1.4), which holds in any W E W. The second formula holds on 3 9 because of the “recession” Rnn-1, Rn-ln-2, etc. Finally, if V takes values in the (co)finite sets and (39, V) + OOp[w], then V(p) must becofinite, whence (3% V)+ + OOp[w]. Q.E.D.

Now let L, be the logic axiomatized by the four formulas

OP +P9

O ( 0 P + 0 4 ) v O(O4 + OP), (OP * U ( P -+UP)) +P? UOP+OOP.

Note that % % g ~ p + ~ p , whencep--+Op$L,, s i n c e 3 n g + L,. It will be shown that L, + p + U p , which means that L, is incomplete.

To state the relevant auxiliary results, we need a definition:


3.3 DEFINITION. formally,

R"wv means that v is an n-th R-successor of w, or,

R'wv stands for w = v R"+'wv stands for (3u ,+ l ) (Rnwu,+ l A R U , + ~ V ) .

(Thus, R'wv stands for ( 3ul)(R"wu, A Rulv), i.e., Rwv.)

3.4 LEMMA. If 5+ u p - p and 5k n(np-Oq) v O(Oq-Op), then 5+(Op A n ( p + n p ) ) - + p [ w ] ifand only &for any V E W such that Rwv, there is a natural number n such that R"vw holds.

Proof: The direction from left to right is easy. Let Rwv hold, but, for no natural number n, R"vw. Define V(p) as {UE W 1 for some n, R"vu}:

k p [ v ] and (5, V) k O p [ w ] , and, if U E V(p) and Ruu', then U'E V(p) ,

( o p A O(p -+ Up)) -+p has been falsified, even without using the antecedent condition of the lemma.

For the converse direction, recall that, since SF n(np -+ 0 4 ) v

v O(0q -+ Up), $k (Vx)(*). Also 5k (Vx)Rxx, because of 5 k k n p - + p . Suppose that, for some valuation V , (5, V) + Op A

A n ( p - + l J p ) [w] and (5, V) l#p[w]. Let Rwv hold with (3, v> I= + p [ v ] . By induction on n, we show that, for all n, if R"vu, then Rwu and (5, V) k p [ u ] . For n = 0 this is obvious. Next, if R"+lvu, then, for some u' with R"vu', Ru'u. By the induction hypothesis, Rwu' and <3, 0 I=p[u'l, whence <%, v> k Up[u'l--since <3, V I= t= o ( p -+ np)[w]-and (5, V) k p [ u ] . To see that Rwu, compare w and u'. Clearly, Rww ( R is reflexive) and Rwu', but not Ru'w (for

for all s with Ru's, Rws, whence Rwu. This concludes the proof, for, obviously, if R"vu, then u # w ( p holds in u, but not in w). Q.E.D.

clearly, %>, so (5, 0 t#p[wI. Now V E V W , whence (5, V) k

whence (5, V) I= p -+ Up and ( 5, 0 t= O(p -+ Up) [wl. Therefore

(5, 0 k OP[U'I and (5, V) FP[Wl). It follows, by st= (*"I, that,

Lemma 3.4 enables us to prove

3.5 LEMMA L 1 + p - + n p .

Proof: Let Bk L,. By the above, (Vx)Rxx and (Vx)(*) hold in 8, and, for all w, V E W such that Rwv, there exists an n with R"vw. Consider


any W E W. It will be shown that w has only one R-successor, viz., w itself, whence s k p --f np[w3. Define, for each n,

S,(w) = { V E W 1 Rnvw and, for no k < n, Rkvw).

(Think of these as “disjoint hulls” around w: S,(w) = {w), S,(w) = = {VE W I Rvw & v # w}, etc. Of course, S,(w) will turn out to be empty for n>0.) Define V ( p ) as Uniseven S,(w). Then (9, V ) k + OOp[w]. For, if Rwv, then, by the above, v is in some S,(w). It is to be shown that (9, v> k Op[v]. If n is even, then we are ready, for Rvv holds. If n is odd, then there is some v’ES,-,(w) such that Rvv‘. (Just take any R-sequence of length n from v to w and consider the successor of v: it must be in S,,-,(w).) Since V’E V(p) , (5, v> Op[v]. Now Op+ O n p is in L,, so (5, V) + Oup[w], i.e., there exists some VE W with Rwv and (5, V) + 17p[v]. Now Rvv, and, therefore, VE V(p) , whence v ~ S , ( w ) for some even number n. But, if n > 0, then v has an R-successor v‘ in the preceding S,- ,(w) (cf. the above argument) with v‘$ V(p) , whence (5, v> n p [ v ] . It follows that v ~ S , ( w ) , or v = w, and (9, v> + o p [ w ] . Finally, if, for any V E W, Rwv and w # v, then V E V(p) , so v ~ S , , ( w ) for some even n > 0. Say Rvv,, . . ., Rv,- , w, where v, $ V(p) (so not Rwv,). Since Rww holds (( Vx)Rxx), as well as Rwv, Rvv, and not Rwv,, (Vx)(*) implies that Rvw: a contradiction, for v$S,(w). Q.E.D.

The main result of this section has been proved:

3.6 THEOREM. L, is incomplete.

Note that L, is axiomatized by formulas of degree 5 2, containing two proposition letters in all. It follows from lemina 3.5 that L, holds on exactly those frames on which p H u p holds. Now p H up is defined by (Vy)(Rxy c-f x = y ) , so L, is an incomplete logic with a first-order definable class of frames. Moreover, L, is semantically equivalent to the “classical logic” M ( { p c-f u p ) ) , but L, is properly contained in M ( { p c-f u p } ) . In fact, there are modal logics L with L, E L M ( { p H Up)), like the logic consisting of the modal theory of Erip. This logic has been axiomatized by W. J. Blok (cf. [ 2 ] ) as L, together with the axioms:


which has the first-order definition

and

O O ( p A q) A OO(p A r ) + O p v OO(q A r ) ,

which has the first-order definition

4. Finite reflexive trees

It was stated in section 1 that Dym is complete (with respect to the class of finite reflexive trees). The logic L2 to be defined here holds on the same frames as Dym, but it is incomplete. Recall that Dym is axi-

axiomatized by u p - + p and O ( n ( p -+ U p ) -+ n n n p ) - + p . (This last formula was suggested to us by W. J. Blok.)

omatized by O P -+P? U P + O U P and O(O(P -+ U P ) -+PI -+P. L2 is

4.1 LEMMA. %Bgk L2.

Proof: Let ($3, 0 k O(O(P + U P ) + 0 0 0 P ) [ w 1 ~ V(P> cannot be

v = max(w, v' + 2). Then ( %,v> I= 0 lP[VI, so (5,v> I= O ( P -+ U P ) finite, for, in that case, let v' be the greatest member of V ( p ) and let

[v] and (3, v > t = r ~ ~ l ~ ~ p [ v ] : a contradiction, for (3, v > p p [ v ] . If V(p) is cofinite, then let v be the greatest member of W - V(p) . Suppose that v 2 w : a contradiction follows. For, (5, v> + +p[v]. Q.E.D.

It is easy to see that

I== O(P -+ o m + 31, whence (3, v> + OOOP[~ + 31 and (5, v> I=

U p -+ n n p , but we have

4.2 LEMMA. L2 I== U p + oop. 3 - Theoria I 1978


Proof: Let S+ L2 and suppose that, for some w , v, U E W, Rwv, Rvu, but not Rwu. It may be assumed that S+ (Vx)Rxx. Define, as in the proof of 3.5, s,(u) for n=O, I , 2, . . . Let ~ ( p ) = , , , W - lJn is even

Sn(u). It will be shown that (5, v> k O(U(p -+ OP) -+ OUUP>[WI, but, by the definition of V , w + V(p)-since w~S,(u)-which falsifies L,: a contradiction. Let w1 be any R-successor of w for which rlnnp does not hold. It is to be shown that (3, v> kt o ( p -+ n p ) [ w l ] . Since nnnp fails at w,, there is a w 2 ~ l J n i s e v e n S,(u) such that R3w1w2. Therefore, w1 is in some S,(u) itself, though not in So@), for Rww, and not Rwu. It then suffices to prove that, for any s in any S,(u) with

s ~ S , ( u ) with n even, then, for some S’ES,,-~(U), Rss‘ (consider any R-sequence of length n going from s to u, and let s‘ be the R-successor of s in that sequence). So it suffices to prove that, for S’ES,(U) with n odd,p A 0 1 p holds. (This yields the truth of O(p A 0 l p ) ins, but also in s’, because of reflexivity.) Now p holds in s’ by definition and, as before, s’ has an R-successor s” in Sn-l(u). But n - 1 is even, so s“+ V(p) and 0 l p holds at s‘. Q.E.D.

n>O, (5, O k t O(P-+OP)WI or (5, 0 I= O(P A O!P)[Sl. If

Lemmas 4.1 and 4.2 imply the main result of this section:

4.3 THEOREM. L2 is incomplete.

Note that, for (P, defined as n(o(p- Up) -+ 0.. . n times . . . lJp)-+p, the proof of 4.1 shows that %g* (P, for all n 2 3, and the proof of 4-2 that { q n , O P - + P ) i= UP -+ for all n L 0. SO { q n , O P ~ P ) is incomplete for all n

It remains to be shown that Dym and L2 hold on the same frames. It is obvious that n(o(p -+ Up) -+ rlr]Flp) -+p is provable in Dym, so L, is contained in Dym. On the other hand, Dyrn is contained in M(L2 U (Up -+nap)), as is shown by the following proof:

3.

(i) O(O(P --* UP) -+A -+ OO(O(P -+ UP) -+PI, (ii) OO(O(P -+ UP) ‘PI -+ O(OO(P -+ UP) -+ UP), (iii) O(ClO(P + UP) -+ UP) -+ O(U(P -+ UP) -+ UP), (iv) O(O(P -+ UP) -+ UP) -+ OO(O(P -+ UP) -+ UP),

etc., until


( 4 O(U(P + UP) +P) -+ O(O(P + UP) -+ OOOP) is

(vi) O(O(P + UP) + OOOP) +P may be applied to

(vii) O(O(P + UP) +PI -+P-

reached, and

obtain

Now the following equivalence holds, for all frames$: $& n(o(p +

+ ~ p ) + p ) + p iff (i) B+(Vx)Rxx and (ii) Bb(Vx)(Vy)(Rxy-+ -+ (Vz)(Ryz -+ Rxz)) and (iii) there is nof:w -+ W such that, for all n, Rf(n)f(n + 1) andf(n) # f ( n + 1). Clearly, the relational property on the right hand side is not first-order definable, so L2 is an incomplete logic without a first-order definable class of frames.

Finally, we turn to Dym itself. The above equivalence suggests the redundancy of the’axioms u p + p and u p -+ OUp. Now the first of these is trivially derivable from u(o(p +up) + p ) -+p, but for the second we have an open question:

1s UP -+ UUP in WU(O(P -+ UP) +PI -+PI)?

This question derives its interest from the following consideration: Incompleteness proofs show, for some r and cp, that r + cp, but not cp€M(T). I.e., cp follows from r in the semantical sense, but not mo- dally. Now the proof of ‘‘r b cp”, usually a set-theoretic one, amounts to a deduction of cp from r in some second-order system of deduction. So, there are “degrees” of incompleteness: modal deduction is inadequate with respect to system ... Now U p - + o o p follows semantically from n(n(p -+ u p ) -+p) + p by a very simple argument: By a well-known syntactic transcription of the Kripke truth definition, modal formulas may be considered as universal second-order formulas with one free individual variable x, denoting the point of evaluation. (Cf. van Benthem [l], and our definition of first-order definability in terms of $(x) in section 2.) Now substitute Rxy A x # y for subformulas of the form Py in the second-order version of o(o(p + u p ) + p ) -+p. This yields a first-order formula in R and = logically equivalent to the conjunction of (Vy)(Rxy -+ (Ryx -+ x = y)) and (Vy)(Rxy + (Vz)(Ryz -+ Rxz)). The latter formula defines u p -+

+ m o p , and indeed the second-order version of this modal formula, (VP)(( Vy)(Rxy -+ Py) -+ (Vy)(Rxy -+ (Vz)(Ryz -+ Pz))), follows from


(Vy)(Rxy 3 (Vz)(Ryz -+ Rxz)) by a simple first-order argument and universal generalization with respect to P. Thus only a minimum of second-order logic is used in this deduction: universal instantiation and generalization with respect to predicate variables. So, if the answer to the above question is negative, then modal logic is incomplete in the strongest possible sense, viz., with respect to the weakest system of second-order logic.

For the formula n ( a p - + p ) - + ~ p , which is closely related to m(o(p -+ u p ) -+p) 3 p , the analogous question has been answered in the affirmative, however: o ( 0 p + p ) -+ Up is complete with respect to the class of finite irreflexive trees. Moreover, for all frames

-+ (Vz)(Ryz-+ Rxz))[w] and (ii) there is no f : w -+ W such that f(0) = w and, for all n, Rf(n)f(n + 1). It turns out that ~ p - + o o p follows from o(op -+p) -+ u p semantically by the substitution of Rxy A (Vz)(Ryz-+Rxz) for Py. D. H. J. de Jongh and G. Sambin independently showed that also op -+ oop is in M({o([IIlp + p ) -+

-+ up)). de Jongh’s unpublished proof goes as follows (note the use

B and W E w, ,3k D(0P ’P) -+ UP[Wl iff (i) Bl= (~Y) (RxY -+

O f P A UP): 0) P ’ (O(UP A P) -+ (UP A P)); ( 4 UP -+ [7(0(UP A P> -+ (UP A

(iii) UP -+ O(UP A PI; (iv) op -+ OOP.

Sambin’s proof may be found in [6].

5. References

[l] J. F. A. K. VAN BENTHEM. Modal correspondence rheory. Dissertation. Amsterdam: Department of mathematics, University of Amsterdam, 1976.

[2] W. J . BLOK. “An axiomatization of the modal theoryoftheveiled recession frame.” Preprint. Amsterdam: Department of mathematics, University of Amsterdam, 1977.

[3] K . FINE. “An incomplete logic containing S4.” Theoria, vol. 40 (1974), pp. 23-29. [4] R. I. GOLDBLATT. “First-order definability in modal logic.” Thejournal of symbolic

[5] D. LEWIS. “Intensional logics without iterative axioms.” Journal of philosophical logic, vol. 40 (1975), pp. 35-40.

logic, vol. 3 (1974), pp. 457-466.


[6] R. MAGARI. “Representation and duality theory for diagonalizable algebras.”

[7] D. C. MAKINSON. “A normal calculus between T and S4 without the finite model

[S] K. SEGERBERG. An essay in classical modal logic. Mimeographed. Uppsala:

[9] S. K. THOMASON. “An incompleteness theorem in modal logic.” Theoria, vol. 40

[lo] S. K. THOMASON. “Semantic analysis of tense logics.” The journalof symbolic logic,

Studio logics, VOI. 34 (1975), pp. 305-313.

property.” The journal of symbolic logic, vol. 34 (1969), pp. 35-38.

Department of philosophy, University of Uppsala, 1971.

(1974), pp. 30-34.

vol. 37 (1972), pp. 150-158.

Received on June 6 , 1977. Revised version received on September 6 , 1977.

two simple incomplete modal logics

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