Blok Quasi Classical Modal

Algebraic Semantics for Quasi-Classical Modal LogicsAuthor(s): W. J. Blok and P. KhlerSource: The Journal of Symbolic Logic, Vol. 48, No. 4 (Dec., 1983), pp. 941-964Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2273660 .Accessed: 30/01/2014 10:45

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THE JOURNAL OF SYMBOLIC LOGIC Volume 48, Number 4, Dec. 1983

ALGEBRAIC SEMANTICS FOR QUASI-CLASSICAL MODAL LOGICS

W.J. BLOK AND P. KOHLER

A well-known result, going back to the twenties, states that, under some reasonable assumptions, any logic can be characterized as the set of formulas satisfied by a matrix , where St is an algebra of the appropriate type, and F a subset of the domain of X, called the set of designated elements. In particular, every quasi- classical modal logic-a set of modal formulas, containing the smallest classical modal logic E, which is closed under the inference rules of substitution and modus ponens-is characterized by such a matrix, where W now is a modal algebra, and F is a filter of W. If the modal logic is in fact normal, then we can do away with the filter; we can study normal modal logics in the setting of varieties of modal algebras. This point of view was adopted already quite explicitly in McKinsey and Tarski [8]. The observation that the lattice of normal modal logics is dually isomorphic to the lattice of subvarieties of a variety of modal algebras paved the road for an algebraic study of normal modal logics. The algebraic approach made available some general results from Universal Algebra, notably those obtained by Jonsson [6], and thereby was able to contribute new insights in the realm of normal modal logics [2], [3], [4], [10].

The requirement that a modal logic be normal is rather a severe one, however, and many of the systems which have been considered in the literature do not meet it. For instance, of the five celebrated modal systems, S1-S5, introduced by Lewis, S4 and S5 are the only normal ones, while only SI fails to be quasi-classical. The purpose of this paper is to generalize the algebraic approach so as to be applicable not just to normal modal logics, but to quasi-classical modal logics in general. To this end an algebraic theory of filtered modal algebras has to be developed. The general framework is provided by model theory; however, in order to obtain a theory comparable in strength to the algebraic theory of varieties, more subtle algebraic methods are needed.

In ? 1 we define the notion of a variety of filtered modal algebras and we show that the lattice of quasi-classical modal logics is dually isomorphic to the lattice of varieties of filtered modal algebras. We single out a special class of filtered modal algebras, which serve much the same purpose as the subdirectly irreducible algebras in the classical theory of varieties. In particular, any variety of filtered modal algebras is generated by those of its members which belong to this class.

In ?2 it is shown that Jonsson's Lemma can be generalized to the setting of filtered modal algebras. From this some interesting conclusions concerning the

Received July 17, 1981. ? 1984, Association for Symbolic Logic

0022-48 12/83/4804-0004/$03.40

941


942 W.J. BLOK AND P. KOHLER

lattice of modal logics can be drawn. One of them states that any tabular modal logic-a modal logic characterized by a finite filtered modal algebra-has only finitely many extensions.

In Makinson and Segerberg [7] a criterion was obtained for determining whether or not a given modal logic is Post complete. In ?3 we show how their result fits naturally in the theory presented here. We give a new proof of a result in Segerberg [13], [14], by explicitly constructing a certain collection of uncountably many Post complete logics.

In the next section we pursue our investigation of the lattice of modal logics in greater depth. We restrict our attention here to the extensions of K, the quasi- normal logics, in order to profit from the fact that the corresponding algebras, unlike modal algebras in general, admit of a representation by means of Kripke frames. The concept of a splitting of the lattice, so useful in the study of the lattice of normal modal logics, serves its purpose here as well. It enables us to show that the center of the lattice of quasi-normal modal logics is, quite unexpectedly, a countably infinite atomic Boolean algebra.

Finally, in ?6, we address the problem of axiomatizing a modal logic given in terms of a class of characterizing filtered modal algebras. One of the results obtained here asserts that any quasi-normal tabular modal logic is finitely axiomatizable.

The writing of this paper began while both authors were visiting the University of Manitoba and it was finished while the first was a visitor at Simon Fraser Uni- versity. Thanks are due to the organizations sponsoring these visits: the Canadian Research Council and the Deutsche Forschungsgemeinschaft, and to G. Gratzer and S.K. Thomason for their generous hospitality.

?0. Preliminaries. We follow Segerberg [12]. Modal formulas are formed in the usual way from a denumerable alphabeth of propositional variables p, q, r, the classical connectives V, A, --, -, I (falsum), T (verum), and the unary modal operator LI ("necessarily"). 0 ("possibly") is an abbreviation of -I1 --. Likewise we abbreviate (ca -p ) A (p ac) by ca 13 where ca and p are arbitrary modal formulas. Finally the iterated operators OLn and [0 are recursively defined for n < w by LI ca = c and ln+l C = LI(LnCa), E aCl = C and an 1 cx - cx A El n+1 aC. LM denotes the set of all modal formulas.

A modal logic is a set L of modal formulas containing all classical tautologies which is closed under substitution [S] and modus ponens [MP].

If L1, L2 are modal logics and L1 c L2 then L2 is an extension of L1. Given a modal logic L the set of all extensions of L ordered by inclusion forms a complete lattice A(L) with largest element LM.

A modal logic L is classical if it is also closed under the rule of preservation of equivalence:

RE: If ca 1- E L then lca x-E Lo eL. The smallest classical modal logic is denoted by E; extensions of E are called quasi- classical.

A modal logic L is normal if it contains the axiom

EJ c A FlI -2 LI(a A /)


ALGEBRAIC SEMANTICS 943

and is closed under the rules of regularity and necessity: RR: If a 3eL,then p-Oa 0i3eL. RN: If carE L then La r eL.

The smallest normal logic is denoted by K; extensions of K are called quasi-normal. Observe that E c K. Even if not explicitly mentioned all modal logics considered in this paper will be quasi-classical.

A modal algebra is an algebra W =


from a one element set with either the total or the empty relation. For more information on normal modal algebras we refer to the papers [2], [3],

[4]; a good reference for the whole subject matter is the book [9]. Note, however, that the notation there does not always match ours. In addition, the books [1], [5] serve as a background in Lattice Theory and Universal Algebra.

?1. Filtered modal algebras and quasi-classical modal logics. A filtered modal algebra is a pair where W is a modal algebra and F is

a filter of W. S4 will denote the class of all filtered modal algebras. To mention an important example: Given a modal logic L the equivalence classes of formulas of L form a filter of the Lindenbaum algebra of E; in this way we can associate with every logic L a filtered modal algebra.

If so is a modal formula in n variables and if e XF then so induces a (polynomial) function @: An A in a natural way. We will say that sp is valid on , in symbols k p, if for all a,, .. ., a. e A we have (a1, . . ., a.) e F. This validity relation gives rise to a Galois connexion between subsets of LM and subclasses of XF4: For L C LM, Y C J( we define:

Th Y = {oD e LM I V E Y I=p}, Mod L = {K Ei / IV oe L I=A}.

The importance of this notion lies in the fact that the closed subsets of LM under the associated closure operator L -~ Th Mod L are precisely the modal logics:

PROPOSITION 1.1. (1) Th Y is a modal logic for any Y a (2) If L is a modal logic then L = Th Mod L. PROOF. (1) requires a routine verification which we leave to the reader. As for

(2) it suffices to prove that Th Mod L c L. Let W be the Lindenbaum algebra of E and let F be the filter induced by L. Then obviously for all 5D E LM we have k p if and only if (p E L. It follows that L = Th. In particular, E Mod L. Thus Th Mod LC Th = L.

In the following we will see how to find the counterpart of this result on the algebraic side. To be more precise, we will exhibit a purely algebraic criterion for a class of filtered modal algebras to be the model class of some logic. Our approach will closely resemble the proof of Birkhoff's Theorem in Universal Algebra (see, e.g., [1], [5]).

To this end let us introduce the following definitions: Given , E X>kY, we will call a homomorphic image of if there exists a modal homomorphism h from W onto 5 such that h = G. will be called a relative of if there exists a modal homomorphism from 5 into W such that h-1F a G. Finally if is a family of filtered modal algebras then the direct product of this family is the algebra . Now for a class Y c XF we will denote by:

HY/-- the class of all homomorphic images of members of Y, PYI-- the class of all filtered modal algebras which are isomorphic to direct

products of members of Yl-, RY/-- the class of all relatives of members of I.

Note in this connection that an isomorphism between filtered modal algebras



and is an isomorphism h: W -S Q such that hF = G (or equivalently h-1G = F).

A class V c will be called a variety if HY/ c , RV Yc and PV a c . With these preparations we are able to state the algebraic analogue of Proposi- tion 1.1:

THEOREM 1.2. (1) Mod L is a variety for any L c LM. (2) If V is a variety then r = Mod Th -r. PROOF. (1) Exemplarily we will show that RMod L c Mod L, the other condi-

tions being verified in the same straightforward manner. So suppose that E Mod L, e. XF and h: Q -S W is a homomorphism with h-1F a G. Let p E L, (p n-ary. Then for bl, .. ., b E B we have

h(j(bi, . *, bn)) = @(h(bj), . * *, h(bj)) e F since e Mod L. Consequently (b, . . ., bn) e h-1F c G. Thus 1 (o and this shows that e Mod L.

(2) Again it suffices to show that Mod ThV c: Y. So suppose that e Mod Th Y. Let I be an index set with II = max{ A4, x0}, and let 9 be a represen- tative set of all filtered modal algebras of V with cardinality at most II. Moreover given E 9 and a map f: I -? B let O3f be the subalgebra generated by fI, and let Gf =G G Bf. Then? E R and, since V is a variety, E V. Finally let T = U{BII


LEMMA 1.3. Let r (-- X>. Then V/ = HRP/. PROOF. Any variety containing 1'f must clearly contain HRP

- - - - -

where E XF, E WI, f-1G c F, g is onto and gH = G. Let X be the pull-back off and g, i.e. D = { E A x C If(a) = g(c)} with the subalgebra structure inherited from W x (S. Let g, f be the restrictions of the pro- jections onto the respective components. Then the diagram above is commutative. Moreover let I = go1F. Then the following hold:

(1) g maps D onto A. (2) -I= F. (3) f-lH c I. To verify (1) let a E A. Since g is ontof(a) = g(c) for some c E C. Thus E

D and hence a g. (2) is an immediate consequence of (1). For (3) suppose that E f -lH, i.e. f(a) = g(c) and c E H. Consequently f(a) E gH = G and thus a Ef-'G c F. This shows that E --'F = I.

Now (1) and (2) state that E H and (3) states that E R. Putting all this together we have verified that RH WI' c HRWI". Returning to the proof of the lemma we conclude:

HHRP / = HRP /, RHRP l c HRRP Y = HRP Y,

PHRP ( c HPRP r ( HRPP / = HRP/. The inclusion RHW-I ( HRW-I we have chosen to verify is a proper one. This

can be seen from the following example: Let W be the modal algebra depicted

1

b d f C

e a

0 Uo



below with a' = c= =f= e0 = 0, 00 = a, b =I' = d' = 1. Then it is easily checked that W is simple. However S8 with B = {0, a, b, I} is a subalgebra of W having 2 as a homomorphic image. It follows from this that E HR but 0 RH.

However, denoting by Ad the class of all filtered modal algebras whose base algebra is normal, we have:

LEMMA 1.4. Let y' c-4 X>. Then RHi = HRr. PROOF. In order to establish the remaining inclusion suppose that E

HRP. Then there exist E A, E J1'Y and homomorphisms g: Q3A, hh: 3 3 such that g is onto, gG = F and h-1H c G.

h -,

-----t

Let Ho be the filter generated by h ker g, Ho = (h ker g). Then Ho is an open filter and E H; v will denote the canonical homomorphism. Now ker g c ker(v o h) and thus there is a homomorphism f: W -+ , which makes the diagram commute. We claim that f-1(H/Ho) c F, which would prove that

E R c RH c RHYr. To prove the claim, let a E A be such that f(a) e H/Ho. By assumption a = g(b) for some b E B and thus v(h(b)) = f(g(b)) = f(a) E H/Ho, i.e. h(b) E c/Ho for some c e H. Hence there exists co E Ho with h(b)- co = c * co, and again co ? h(bo) for some bo E ker g. Now

h(b + bQ) = h(b) + h(bo)' ? c-co + c6 > c and thus h(b + b6) E H. Consequently b + b6 e G and thus a = g(b) = g(b + b6) E gG = H. This finishes the proof.

To mention an illustrative example let Y be the variety generated by the one- element filtered modal algebra < 1, 1>. Then by Lemma 1.3 Y = { 1 W E X}. Clearly .1 is the smallest variety and Th ! = LM. This suggests that if E &, and F is a "large" filter, then V may be small in the sense that it is

close to the bottom in the lattice of varieties of filtered modal algebras. To make this more precise and to dispose of redundant information let us call E ,gy reduced if the only open filter contained in F is {1}. Applying this to Y we see that in fact < 1, 1> is, up to isomorphism, the only reduced algebra of S. More generally we can prove that every variety is generated by its reduced algebras. The following lemma serves as a preparation.

LEMMA 1.5. Let W be a modal algebra, F a filter of W. Then there exists a largest open filter F+ contained in F.

PROOF. Let ,F be the collection of all open filters of W which are contained in F and let F+ = V &7 where the join is taken in the filter lattice of W. However, since the open filters of 2 form a sublattice of the filter lattice it is clear that .w



is closed under finite joins. It follows that F+ = U), and hence F+ is an open filter. Thus F+ is in fact the largest open filter contained in F.

Thus, given E ca we may form its quotient . Then E H. On the other hand if VF+ denotes the natural homomorphism from W onto I/F+ then - vj(F/F+) = F since F+ ' F. This shows that also E R. In particular and generate the same variety. Moreover is reduced since every open filter of which is contained in F/F+ must be of the form GIF for some open filter G of W with F+ ' G c F. Combining all this we have proven the announced result:

PROPOSITION 1.6. Let F l c1 XF be a variety. Then < is generated by its reduced algebras.

Next suppose that , E FA where is reduced and let f: W 23 be a homomorphism with f-1G c F. Then ker f is an open filter of 2 and kerf=f-1{I} (f-1G c F. Thus ker f = {l} and hence f is 1-1. This suggests introducing the following notation: For F c: X let RiS be the subclass of Rf obtained by restricting ourselves to 1-1-homomorphisms in the definition of a relative. If, moreover, we let FR denote the reduced members of F then the observation, above can be formulated as (Ry)R ' Ri F. More importantly, however, we have the following:

LEMMA 1.7. Let F ,'yF Then (VF)R 5 HRiPF. PROOF. Let E (VT )R. Then there exist , E -&,g with

E PF such that we have a diagram

9L- f A

*-W--

where f maps 2 onto X, fG = F and g-1H c G. But thenf ker g is an open filter, f ker g c fg-1H C fG F. Since is reduced we infer f ker g = {l} or, equivalently, ker g c kerf. Thus if D = gB and i: T -6 ( is the inclusion homomorphism then there exists a unique homomorphism h: Z -- 21 making the diagram above commutative. In particular h maps Z onto W. Now let I = h-1F. Then hI = F and thus E H. Also if d E i-1H = H n gB then d = g(b) for some b E B. Now d E H implies b E g-1H c G, and consequently h(d) = h(g(b)) = f(b) E F. This finally shows d E h-1F = I. Since i is 1-1 we thus infer E Ri , which proves the inclusion.

Note that for a subclass F of S,4F an easy proof along the same lines as that of Lemma 1.4 yields that

(Vf)R ' HRiPF = RiHPF.

Finally we can narrow down the class of generators of a variety by introducing another notion of irreducibility. A filtered modal algebra will be called strongly irreducible if F is a maximal (proper) filter of W. The class of strongly



irreducible algebras in a subclass Y- of J/A will be denoted by 8'''sI. With that we have:

THiEOREM 1.8. Let 1 c; a be a nontrivial variety. Then i?< is generated by its strongly irreducible algebras, in fact V =RP(5


filtered modal algebras which will allow us, among other things, to draw the same conclusion as the one mentioned above.

If is a family in &SF and if U is a filter on the index set I, let Fu be the open filter of JJ defined by

eFu{iEIla = 1}EU.

The quotient FL IFu is denoted by FL u, and with the induced filter we have a filtered modal algebra 1Ku, the reduced product of Kli E I> with respect to the filter U. If U is an ultrafilter on Iwe will speak of an ultraproduct; if V c XiF then PuV denotes the class of all ultraproducts of families in Y. With that we can state the aforementioned generalization of Jonsson's Lemma as follows:

THEOREM 2.1. Let = c X>. Then (VO)SIR ' HRiPuV. PROOF. Let E (V Hlu

To accomplish this it will be sufficient that U satisfies the following two conditions: (1) Fu n B c ker h, (2) (G V Fu) n B c H.

Actually (1) is a consequence of (2). For if (2) holds then, in particular, Fu n B c H. Thus h(Fu n B) c F; since h(FJ n B) is an open filter of % and is reduced, we conclude h(Fu n B) = {1} and, hence, Fu n B c ker h.

Now let F be the collection of all filters V over I satisfying (2). By assumption {I} E F and thus YF # 0. A straightforward verification shows that F is closed under union of chains and thus, by Zorn's Lemma, F contains a maximal element, say U. We claim that U is an ultrafilter over I.

To see this let X, Y c I and suppose that X U Ye U. Then [X) n [Y) = [X U Y) ' U and, hence, Fu- F[x)n[y) = F[x) n Fir) or, equivalently, Fu = Fu V (Frx) n Fry)). Since U E F and the lattice of filters of a Boolean algebra is distributive, we conclude



H2 (G V Fu) n B = (G V Fu V (F[x) n Fry)) n B) = (G V Fu V Fcx)) n (G V Fu V Fry)) n B = ((G V Fu v F[x)) n B) f ((G V Fu V F[y)) n B).

Using distributivity once more we obtain

H = (H V ((G V Fu V F[x)) n B)) n (H V ((G V Fu V F[ry)) n B)). Now H is a maximal filter and thus it follows that, say, (G V Fu V F[x)) n B ' H. Let U1 be the filter over I generated by U U fX}. Then Fu, = Fu V F[x) and thus (G V Fui) n B ' H. That is U1 e F and, as U is a maximal element of F, we conclude that U1 = U, which means that X e U. Thus U is in fact an ultrafilter and the theorem is proved.

A first application of this theorem is an alternative description of the variety generated by a class of filtered modal algebras: If Y-x is a subclass of & F then by Corollary 1.9 and Theorem 2.1 V(V) = RPHRiPu'W, which in many instances is more useful than the statement of Lemma 1.3. This is, in particular, the case if Y-x is a finite set of filtered modal algebras and rests mainly on the following:

LEMMA 2.2. Let E &S be finite and suppose that e Pu. Then - .

PROOF. Suppose that = f/Fu for some ultrafilter U over L Let a = E lK. Then U {{ie IIaj b}= b I B} = Ie U, and since B is finite and U is an ultrafilter we must have {i E II a1 = b} E U for some b e B. In other words a Ej(b)!Fu for some b e B. Hence the composition IFuoI maps B onto A, where: -- H is the canonical imbedding. Clearly vFu j isS 1-1 as well and also 1Fu ojG = F. Thus )FuoI is an isomorphism and the proof is complete.

Now let , G.., be finitely many finite filtered modal algebras and suppose that E Pu{, . Q.>., } Then we may assume that

= x .. x H< n, Gn>IiEIn>lFu

and U is an ultrafilter over the disjoint union I = I, U ... U In. But then Ik e U for exactly one k E {1, . . ., n}, the restriction UJIk is an ultrafilter over Ik and = HL I i E Ik>lFUIlk. Thus by Lemma 2.2 - K3k, G>. That is, in this situation ultraproducts do not produce any new algebras. Together with Theorem 2.1 we have just proved

LEMMA 2.3. Let Y-X be afinite set offinitefiltered modal algebras. Then (V 'Y)SIR C HRj(Yx)

In particular, there are, up to isomorphism, only finitely many distinct strongly irreducible reduced filtered modal algebras in the variety generated by Y-x. This settles the question at the beginning of this section:

A tabular subvariety Y of &,F i.e. a variety generated by a finite algebra, has only finitely many subvarieties.

Let us mention here that this property does not characterize tabular varieties. In fact, in ?3 we will present nontabular varieties which are atoms of A(X4F).



Another immediate consequence of Lemma 2.3 is COROLLARY 2.4. Let , be finite strongly irreducible filtered modal

algebras. Then V = V if and only if - . Moreover, an argument similar to the one used to prove Lemma 2.3 yields that

Pu(y U #') = Puyl< U Pul'W for any two subclasses Y, Wf of . If, in particular, V, V are subvarieties of X then Y' V V = V(V U #), and, by The- orem 2. 1,

(V V Y)SIR ' HRiPu(,r U Y) = HRi(PU r U Pu#) C V U en This proves

COROLLARY 2.5. Let J, #V be subvarieties of X97. Then (< V f )SIR = "'SIR U f IRE This last observation can be looked at from a more general point of view. Having

in mind that every nontrivial variety is generated by its finitely generated strongly irreducible reduced filtered modal algebras we let S be a set containing exactly one member of each isomorphism class of finitely generated strongly irreducible reduced filtered modal algebras. Then the map so: A(X4F) -s 9(S) defined by qD(V) = fl nS is-in virtue of Corollary 2.5-a 0-1-preserving lattice imbedding. In fact, if we define an operator c on 9a(S) by putting T c =S n VT for T c S then it follows again from Corollary 2.5 that c is an additive closure operator, and clearly the image of q in M(S) is the collection of all closed sets, hence a dual Heyting algebra. Summarizing we have

THEOREM 2.6. (1) The lattice A(X4) is distributive and complete. (2) For any family and #- in A(&/F):

) v f = n. We close this section with some observations on the covering relation in A(jAF).

To this end suppose that Al', V are varieties such that -? c: #, and #- is generated by its finite members. Then there exists a finite filtered modal algebra E ~f/~~ By Corollary 2.5 and Lemma 2.3 we conclude that

()SIR - 'VSIR U HRi. It follows that there are only finitely many varieties O& such that i' c: & _: V V. c #I. At least one of those covers Y. That is, there exists a cover of Y in the interval [S, #-]. Choosing #- to be the variety &4F-note that X// is generated by its finite members-we obtain

THEOREM 2.7. Every variety Y"' c: SEA has a cover in A(G4Y). ?3. Post-complete logics. One problem which has received considerable interest

in the past is the attempt to determine all the Post-complete extensions-or at least their number-of a given logic. An account of this may be found in [7], [13], [14]. Incidentally, the recent paper [11] presents an approach to this subject matter which is-at least in spirit-rather close to ours.

A logic is called Post-complete if it has no proper consistent extension; in other words if it is a maximal element in the lattice of all extensions of E. Using the correspondence developed in ? 1 we know that a logic L is Post-complete if and only



if Mod L is an atom in the lattice of varieties of filtered modal algebras. Evidently each such variety V is generated by any algebra e Y with F : A. In particular if Y = V let Wo be the subalgebra of W generated by to, 1} and let G be a maximal filter of Wo containing Ao n F. Then e R c V and, consequently, - = V. With that we have already proved one half of the following result:

THEOREM 3. 1. Let i"' c: 'F' be a variety. Then Y-5 is an atom of A(Xf>F) if and only if V is generated by a 0-generated strongly irreducible filtered modal algebra.

PROOF. It remains to show that any such algebra generates an atom of A(X>,F). So suppose that W is a 0-generated modal algebra and F is a maximal filter of W. Now let V be a nontrivial variety contained in V. Then there exists E 1 with B : G. Let f, g be the unique homomorphisms from a0X(), the free modal algebra on 0 generators, to W, Q3, respectively. With H = f-1F we have that e H. Next let x e H. Then x = s for some variable free formula D e LM. Because of f(x) E F it follows that I= W. Consequently F q and thus g(x) e G. This shows that gH C G and hence g-1G - H. Since H is a maximal filter and G = B we must have g-1G = H. Thus E R and as a result E HR C V. Hence V = V and the proof is complete.

In case Yl- c: XA' this proof shows a little bit more. Namely assuming that is reduced as well we can use Lemma 1.4 and the observation preceding Lemma 1.7 to see that e RjH. If now is strongly irreducible and reduced then necessarily e Rj. And finally if S8 is 0-generated as well then clearly - . Thus is, up to isomorphism, the only 0-generated strongly irreducible and reduced algebra of the variety it generates. In other words:

COROLLARY 3.2 Let , be strongly irreducible and reducedfiltered normal modal algebras and suppose that W and S3 are 0-generated. Then V = V if and only if - .

These results will enable us to establish a number of conclusions in a rather straightforward way. First note that they give another indication why Segerberg in [13], [14] could restrict himself to discuss only formulas without variables: It is an easy consequence of Theorem 3.1 that any two different Post-complete modal logics can be distinguished by variable-free formulas.

Another application leads to a simple proof of [13, Theorem 1]. Let Tr be the extension of K by Z p U-4p. It is well known that Tr is Post-complete and normal, in fact Tr = Th. Moreover let D be the normal extension of K by O T. With that we have

PROPOSITION 3.3. Tr is the only Post-complete extension of D. Moreover, if L is an extension of K, the only Post-complete extension of which is Tr, then L is an extension of D.

PROOF. Let E (Mod D)SIR and suppose that W is 0-generated. Let G be the open filter of W generated by 0?' and 1?. As D is normal we must have G c F. and since is reduced we conclude G = {}. This shows that 0' = 0 as well as 1 = 1. Since W is 0-generated we must have - . The first part of the result now follows from Theorem 3.1.



On the other hand let L be a logic containing K which is not an extension of D. Then there exists n < to such that On0 TO L. Thus for some E Mod L we have (0?')?n 0 F. We may also assume that W is 0-generated. Now there exists a maximal filter G of W such that (0?')? ? G, F c G. Then e R c Mod L. By Theorem 3.1 Th is a Post-complete extension of L. Also V Lnn T and thus Th # Tr.

The only other normal Post-complete logic is A, the extension of K by Df I Again it is well known that A = Th. A similar argument as above yields that any logic L of which A is a proper extension contains a Post-complete extension different from A. However in both cases there is a smallest logic having Tr and A, respectively, as its only Post-complete extension. Interestingly, a slight modification of the proof of Proposition 3.3 yields the same result for each Post- complete logic. It should be noted that this does not remain true for the lattice of normal logics: for example there is no smallest normal logic, the only Post- complete extension of which is A.

THEOREM 3.4. Let L be a Post-complete logic. Then the set of all logics having L as their only Post-complete extension has a smallest element.

PROOF. By Theorem 3.1 L = Th where e &tFSIR and W is 0- generated. Let SL = r5DI 1= A, qD variable-free}, that is SL is the 0-ary part of L. Finally let Lo be the extension of E by SL. Clearly Lo ' L. Now suppose that e Mod Lo, Q3 0-generated and G is a maximal filter. Then essentially the argument used to prove Theorem 2.1 shows that E HR c Mod L. This shows that L is the only Post-complete extension of Lo. If, on the other hand, L1 is a logic such that qD 0 L1 for some eD G SL then there exists E Mod L, such that s 0 G. Taking the 0-generated subalgebra of Q and extending the restriction of G to a maximal filter not containing f we find a Post-complete extension of L1 which is clearly different from L.

It should also be mentioned here that the results obtained so far could have been proved by referring to [13, Lemma A], which is the equivalent of our Theorem 3.1 on the syntactical side. The major advantage of our semantical approach in this context will become clear in the following when we show that there are uncountably many Post-complete extensions of K. This result is due to Segerberg [13], but the proof given there differs from ours in that it does not explicitly construct such a set of logics.

THEOREM 3.5. K has 2Ko Post-complete extensions. PROOF. For any nonempty subset M of o)\{0} we define a frame KM=

< WM, RM> as follows:

WM = (c + 1) X {O} U M X {1}. T = , or

RM j = 1 and m < n, or ti = I and m = n.

The figure below shows a sketch of the frame for' M = {1, 4}:



.. 1, i> I.

.

*

< 1, O>

Note that RM is transitive; thus RM can be visualized as the transitive extension of the relation depicted.

Let AM be the set of all finite and cofinite subsets of WM. We first prove that WM is a subalgebra of KM. Clearly 2M is closed under the Boolean operations. Now let n < c. Then

(WM\{})? = { E WM I m < n} U {} and if n e M,

(WM\{})O = WM\t, }. Finally

(WM\{})O = WM\{f(c, 0>1. It follows that X0 e= AM for every cofinite X. Next let X be a finite subset of WM. Then there exists n < Cv such that X c S_ = (WM n ({o, .. ., n} x {0, l})) U {(c), 0>1. Now S' = (Sn\{(c), 0>)) U {1; in particular So is finite. Thus X0 is finite and all that proves the claim.

Next we show that WM is O-generated. In fact, we have for 0 < n < cv that

0={0,...,n-l} X {0}. Hence {}e f [0] for all n < c); in fact

{} = 0M+ . (0M) Also for n E M we have

{}C = {, , ..., 1 U {1, hence

{, 1 = {}c.({}c)T and thus

{1 = {}j * ({}c)'. *j}'.

This shows that {} E [0] for all n E M. Finally if n E M then

(1C = (, 1, and thus

{1 = {1C * {1'. Since M # 0 we infer that {1 e [0]. Thus all the atoms of KM+ belong to [0], consequently [0] = AM.



Now let FM be the principal filter generated by {}. Then FM is a maximal filter and by Theorem 3.1 V is an atom of A(S4&w) and Th is a Post-complete logic. Moreover, if X0 E FM for some X E AM then E X0 and hence X = WM. This shows that is reduced as well. Clearly if M # N then N, FN> and thus, by Corollary 3.2, V # VQWN, FN>. Since there are at most 2Xo logics this finishes the proof of the theorem.

Actually we have proven a little bit more: If, as usual, K4 denotes the normal extension of K by F1p --+ Etip then the transitivity of .RM implies that e Mod K4-in particular E Mod K4 for each M with 0 # M c &v\{O}. Thus K4 has likewise 28o Post-complete extensions, a result announced in [13] and proved in [14]. Also if K4Grz1 is the normal extension of K4 by

L1(Li(P -+ LiP) -*P) -+ LiP then again E Mod K4Grz1 as well as E Mod K4Grz1 for each M with 0 = M c o,)\{0}. This shows that also K4Grz1 has 28o Post-complete extensions [14, Theorem 3.6]. However, our example refutes the conjecture made in [14] that every proper normal extension of K4Grz1 has only countably many Post-complete extensions. Obviously L = n {ThI 0 # M c o\{0}} is a normal extension of K4Grz1 having 2Ro Post-complete extensions. To see that L extends K4Grz1 properly let q- 0 [L I V El . Then k for each M with 0 # M c a)\{0} showing that q e L. On the other hand

?o0K4Grzi, since e.g. P=.

?4. Splittings of the lattice of varieties. In this and the last section of the paper we will concentrate our investigations on quasi-normal logics. This could be done completely within the framework set up in ??1 and 2. We have, however, already seen that the class XF of filtered normal modal algebras has some nice properties not in general shared by &. And since the Lindenbaum algebra of K is normal we can replace X.F by YY as the base class of semantics for extensions of K.

A word of caution is appropriate here: X." is not a subvariety of fF according to our definition-XF is not closed under the operator R. When we, in spite of this, consider XA'1 as a variety then this has to be understood in such a way that we restrict the range of the operator R to the class X>. With that modification the results of ??1 and 2 remain valid, in particular those which explicitly refer to OFA. This way we obtain a dual isomorphism between the lattice of extensions of K and the lattice of subvarieties of SE.

Let us also mention how normal modal logics fit into this scheme. We denote by 4'S the class of all filtered normal modal algebras where F is an open filter. Clearly if V c AF' then Th V is a normal logic. On the other hand, if L is a normal logic then L is an extension of K and the iLindenbaum algebra of K with the filter induced by L belongs to XF'. Thus a logic L is normal if and only if L = Th Y'/ for some subclass I- of XAFK.

Reversing things we will call a subvariety y of SF normal if Th V is a normal logic. Then obviously Y7 is normal if and only if - = V(


v- = v(YF n mo) which corresponds to the normal closure of Th Y2-. On the other hand it is easy to see that if L is a normal logic and if E Mod L then also E Mod L. Since E R it follows that a variety V c: A is normal if and only if E -7 implies that E r. Moreover, for an arbitrary variety V c: SF/Y there exists a smallest normal variety containing Y, namely V+ = V{ I E r}. Correspondingly every extension of K contains a largest normal logic.

Let us now introduce the central notion of this section. A pair of elements of a lattice A is a splitting pair if it decomposes the

lattice A into two disjoint intervals, i.e. we have a ? b and for every element c E A either a < c or c < b holds. For the importance of this notion in the context of the lattice of normal modal logics we refer to [3]; see also [10]. In this section we will investigate the splittings of the lattice A((XF).

Note that any splitting pair of varieties is determined by either of its two components. Moreover, if


e(ao, a,, . . ., am) = 0on- 0 F. This shows that e se and hence 0 Mod en. Next assume that E A"7, q Mod en. We will show that then e RjH. It will follow from this that whenever & is a subvariety of SF.; such that J c Mod c, then V c A&, thus proving the theorem.

Since + ea there are elements bo, .., b. e B such that ew(bo, ..., bm) ? G. Let c = (n

-l)o2(bo, .., bm) .Oc". Note that w(bo, ..., bm)? ? 0on as well as Ot3' 2 0? and, consequently,

C? = (nul dw(bo bm 0'i 00?") = [l US(bo, bm), i=O iz1

n-1 ^n-1 ? fl Je(bo, ... , bm)0 - O12 >T f5 (bo, ..., bm) O?o = C.

Thus the filter [c) c B is open. We claim that E Rj. To this end let f: W -+ Q/[c) be defined by f(ai) = bi/[c) for 0 < i < m. Then

g(bolfc) . bn/[c)) = 5w(bo, , bm)/[c) = l1(c), and by the remark above f is a homomorphism. Since (bo, ..., bm)= c' + Oo" 0 G, it follows that c 4 0o' and, hence, f(O?'-) = 0-'/[c) + 1/[c). Now observe that [Ot-') is the smallest open filter of W not equal to {1} and con- sequentlyf is 1-1. Finally if aj E A, aj 0 F then aj < 01- and hencef(a2) < 0- 1/[c). But since c' + 0-' 0G we infer that 0O' 1/[c) 0 G/[c) and therefore f(a) 0 G/[c). Thus f-1(G/Ic)) c F and this proves our claim.

In case is a splitting algebra we will alternatively denote the splitting variety Mod e. by .i/IS, F>. Then it is an immediate consequence of the proof of Theorem 4.1 and Lemma 1.4 that

.AS~/((, F> = { E X | s RjH} = { E X.A I K 0 HRj}.

Combining this with the observations following Corollary 1.9 we can state COROLLARY 4.2. Let be a splitting algebra in X9. Then

(XJ'/K)SIR = { E XAszSIR I X Rj}. Let us note that none of the splitting algebras generate a normal variety except

. In fact, if is a splitting algebra which is not isomorphic to then V contains no nontrivial normal subvariety at all. To see this suppose that Y were a nontrivial normal subvariety of V. Then either E /-which would imply that -or E '%/., which is impossible since the formulas ZnI are valid on for all n E N and this is not the case with .

On the other hand it is quite obvious that there is a 1-1-correspondence between splitting pairs for A(XA>) and those for the lattice of all normal subvarieties of X>. In fact, if is a splitting pair for A(XA>.) then is a splitting pair for the lattice of all normal varieties. For if ? is a normal variety then either f cd or d& c W. And this in turn implies that either A+ c J& or & c: -.



On the other hand we know from [3] and the first part of the proof of Theorem 4.1 that for every splitting normal modal algebra W there is precisely one filter F of W such that is a splitting algebra in OF. This correspondence does not preserve the inclusion relation, however: If , are splitting algebras then V+ c V+ does not imply that V c V. In fact, there are countably many atoms of A(X.F) among the varieties generated by splitting algebras. To see this for n E N, let < W, R> be a frame such that W = {1, ... n} and R is a relation with {1 < i < n - 1} R c


. V Yr = X then also Mod (p V W = X. and Mod (P nYr c Mod (P n Mod 0b = Sr. Since A(XY) is distributive it follows that '%/" = Mod (n and likewise W = Mod 0b. Hence the Boolean sublattice of complemented elements of A(XA), i.e. the center of A2(X>) is at most countable. We then have

THEOREM 4.4. The center of A(XA/) is a countably infinite atomic Boolean algebra. The atoms are the varieties where is a O-generated splitting algebra.

PROOF. It follows from Theorem 4.3 and the remarks above that the center is countably infinite. Now let %/' # Y be complemented. First note that %/' = V(/s) where rs is the class of all splitting algebras in Y'/. Indeed, if Y is the complement of %/' then by Corollary 2.5 XAs ' XASIR = YI'SIR U Y SIR and hence XFs = As U 'Is. Thus V('*'s) V V(Ws) = XF and an argument as above yields that -/- = V(Q's). It follows that there is a splitting algebra E *; we may even assume that W is 0-generated. Then 0 Y and hence Y c XY/. Since 'V and XA/ are complements of each other in A(XA), we thus have Y C At

It remains to show that if is a 0-generated splitting algebra then is an atom of the center of A((XA). But that follows immediately from the fact that


XnA, however it follows from Theorem 2.1 that this is the case as soon as is strongly irreducible:

X,,F SIR C HRjPu~{ E .AKF I W E Xn The main feature of man is now that every finite strongly irreducible algebra

E X/n is a splitting algebra in mFn. The proof of this is quite similar to that of Theorem 4.1: Suppose that E XASIR and W is finite. Then by the remark following Corollary 1.9 W is a subdirectly irreducible normal modal algebra. Let e e A be such that [e) is the smallest nontrivial open filter and define

ca(p, ...5 Pam = InI dw~PamO, . ' Pam) Pe Then is a splitting pair in A(XA). Clearly k Cw. Next assume that E 4/n such that k C%. As in Theorem 1.8 and Corollary 1.9 we see that there is a strongly irreducible algebra e HR such that b Al,. We then proceed to show that e RiH exactly as in the proof of Theorem 4.1-using now the formula C instead of ai, e instead of O", the element c (n)dw(bo, . .., b,,,) and the fact that the identity (n)x = (n + 1)x holds in Q in order to show that the filter [c) c= B is open. By virtue of Lemma 1.4 we find that

e RjH c RjHHRj= RH. Thus we have proved

THEOREM 5.1. Let Y be a variety such that Y C X>" for some n < o. If E f is finite and strongly irreducible then is splitting in F and *I/ = { E Y- IK 0 RjH}.

This result enables us to infer the following lemma, a useful addition to Lemma 2.3:

LEMMA 5.2. Let ' be any set of filtered normal modal algebras such that -

cHn for some n < co. Then the finite algebras in (V1)SIR belong to HRifV.

We will now turn our attention to the following problem: Given a class Y/' of filtered normal modal algebras, find explicitly a set I of modal formulas such that VY* = Mod C, which by the results of ? 1 means that I is a basis for Th i. We will be able to solve this problem for certain reasonable collections y'/ provided they are contained in Awn for some n < co. The axiomatizations obtained will be relative to Awn-therefore we should like to find an axiomatization of ,Fn first:

LEMMA 5.3. X4n = Mod (j (F p -* In + 1 p))foralln < co. PROOF. Let I = {Ek(flp -+ In + 1 p)jk < cw}. Clearly .An 'c Mod C. On the

other hand, if e .FR satisfies I then, for all a E A, [((n)a)' + (n + I)a) generates an open filter contained in F and-since is reduced-(n)a (n + 1)a. Thus W E An and consequently E X>n. Since Mod 2= V(Mod TR) it follows that also Mod 2 c Xn. It remains to show that the for- mulaq( = *(Ep In +1 p) is equivalent to C. First note that obviously



I FK i- Also, f HKEk(p| n + 1+ p) fork =O. ., n. Suppose now that pf~_K n k([Ep +I n

u

Ip) for k = O1). . .. 1m, n I= D[aj, ..., am] then En (p +-+) (a,, . am) = 1 E F. Conversely, if e = Pj+-+ o) (a1, ..., am) E F. then-since W E Xn -[e) is an open filter contained in F and hence e = 1. This shows that p ""-(a, ..., am) 1.

(ii) If qD = F(A(x1, ..., Xm)) then, by definition, = qDa, [a, , am] if and only if p(a1, *.., am) E F.

(iii), (iv) If qD = 5b A x or p= b V x, then the result follows directly from the induction hypothesis and the fact that F is a filter.

(v) If qD = -b, then t qD[aj, . .., am] if and only if

he 0b[a1, *. ., am]. By the induction hypothesis the latter is equivalent to fnb(aj.. ., am) ? F and- since F is a maximal filter-this in turn is equivalent to

-i(Tnob) (al, ..., am)= (T1)(al..., am)' e F.

For further applications let us state the following immediate consequences:



COROLLARY 5.5. Let e A/sSIR and suppose that qD = Vx1 ... VXm b(X1, ..., Xm), where 0 is a quantifier free formula in the first order language offiltered modal algebras. Then I= qD if and only if 1= T?0.

If or is a universal sentence Vx1 * Vxmb(xi, ... . xm), where 0b(x1, *.., xm) is quantifier free, then we write Tnoi for Vx1 .. VxmTnob(x,* ..., xm).

COROLLARY 5.6. Let *- c- AYn be defined by a set I of universal sentences; moreover, suppose that (V/)SIR C Y. Then VV = Mod{Tn or I or e zi} n

PROOF. Let E (VY)SIR. Then by assumption e i and hence I= C. Since is strongly irreducible it follows from Corollary 5.5 that I= Tnor for every o e- . Thus V *- = V((VY)SIR) c Mod{Tn ci cI e- 2}. Now suppose that E XAn and 1= Tn ci for every or E 2. Then by Corollary 5.5 I= T and so E ir.

As a typical application let us consider a finite filtered normal modal algebra . Then E XFn for some n < Co. By Lemma 2.3 Y'1 = (V)SIR ' HRi and so v1 contains up to isomorphism only finitely many finite filtered modal algebras. As a result vr is an elementary class. Moreover )rl is easily seen to be closed under the formation of substructures and hence is a universal class. Let so be a universal sentence defining Air By Corollary 5.5 VV1 is axiomatized by T"(p, relative to An. Since by Lemma 5.3 XVFn is finitely axiomatizable, it follows that VV1 = V is finitely axiomatizable as well. Thus we have proved

THEOREM 5.7. Tabular subvarieties of SEF are finitely axiomatizable. For the sake of completeness we note PROPOSITION 5.8. Thefinitely axiomatizable subvarieties of XA form a sublattice

of A(X4F). PROOF. We only need to show that the join of two finitely axiomatizable sub-

varieties of XF is finitely axiomatizable again. So let , fV be finitely axiomatizable. Then there exist modal formulas sb in m variables, x in n variables such that v = Mod sb and - = Mod X. We may even assume that sb and X have no common variable. We claim that then V V #, = Mod qn where (n = sb v X. First note that V U #, C Mod so and thus f V v c Mod qo. Conversely, let be strongly irreducible such that I= p but Pb 0. Then there exist a, .. ., am E A with s(a1, .. ., am) 0 F. Now let bl, ... ., b E A. Then

0(al, ... 1am) + %(b1, ... , bn) = l(ai, .. ., am, b, .. b E F and since F is maximal this implies that 1(b, . . ., bn) E F. As a result 1 x. This shows that (Mod (O)SIR C V U MY, whence Mod D ( - V fv.

The proof shows in fact a bit more: the finitely axiomatizable subvarieties of #F form a sublattice of A(X/F). Contrast Proposition 5.8 with the known fact

that the join of two finitely based varieties of normal modal algebras need not be finitely based.

In [12] Segerberg shows that several of the familiar normal modal logics (as D, T and B. axiomatized by the formulas < > T. Clp -+ p, and < > [ll p -+ p, respectively) are not finitely axiomatizable as quasi-normal logics. We close this section with a positive counterpart to these examples.



THEOREM 5.9. Let C (-- X be a finitely based variety of normal modal algebras. If f is a subvariety of .A'n for some n < w), then V{ I 2 E A} is finitely axiomatizable as a subvariety of X>.

PROOF. By assumption there exists a modal formula qp such that W E YX if and only if W satisfies the identity &Z(x1, . . ., Xj = 1. Let Yl' = { I W ei . Then i'/ c J7n and VYV = V{ I WE E r}. Furthermore / is definable by the universal sentence Vxj ... Vxmj3(xj, . .. , xm) =1. Thus, by Corollary 5.6 Vi'/ is defined, relative to on, by i (o.

This shows that, for example, the (normal) modal logic S4 is finitely axiomatizable as a quasi-normal logic.

REFERENCES

[1] R. BALBES and P. DWINGER, Distributive lattices, University of Missouri, Columbia, 1974. [2] W.J. BLOK, The lattice of modal logics; an algebraic investigation, this JOURNAL, vol. 45

(1980), pp. 221-236. [3] , On the degree of incompleteness of modal logics and the covering relation in the lattice

of modal logics, Report 78-07, Univ. of Amsterdam, 1978. [4] , Pretabular varieties of modal algebras, Trends in Modal Logic, Stadia Logica, vol.

39, no. 2/3, (1980), pp. 101-124. [5] G. GRXTZER, Universal algebra, 2nd ed., Springer-Verlag, Berlin-Heidelberg-New York,

1979. [61 B. J6NSSON, Algebras whose congruence lattices are distributive, Mathematica Scandinavica,

vol. 21 (1967), pp. 110-121. [7] D. MAKINSON and K. SEGERBERG, Post completeness and ultrafilters, Zeitschrift fSir Mathe-

matische Logik und Grandlagen der Mathematik, vol. 20 (1974), pp. 385-388. [8] J.C.C. MCKINSEY and A. TARSKI, Some theorems about the sentential calculi of Lewis and

Heyting, this JOURNAL, vol. 13 (1948), pp. 1-15. [9] W. RAUTENBERG, Klassische and nichtklassische Aussagenlogik, Vieweg, Braunschweig-

Wiesbaden, 1979. [10] -, Der Verband der normalen und verzweigten Modallogiken, Mathematische Zeitschrift,

vol. 156 (1977), pp. 123-140. [11] G. SAMBIN and S. VALENTINI, Post completeness and free algebras, Zeitschrift fSir Mathe-

matische Logik and Grandlagen der Mathematik, vol. 26 (1980), pp. 343-347. [12] K. SEGERBERG, An essay in classical modal logic, Philosophical Studies, Uppsala, 1971. [13] , Post completeness in modal logic, this JOURNAL, vol. 37 (1972), pp. 711-715. [14] , The truth about some Post numbers, this JOURNAL, vol. 41 (1976), pp. 239-244. SIMON FRASER UNIVERSITY

BURNABY, B.C. CANADA V5A 1S6

JUSTUS-LIEBIG-UNIVERITAT 6300 GIESSEN, FEDERAL REPUBLIC OF GERMANY

Current address of W.J. Blok: University of Illinois at Chicago, Box 4348, Chicago, Illinois 60680


Article Contentsp. 941p. 942p. 943p. 944p. 945p. 946p. 947p. 948p. 949p. 950p. 951p. 952p. 953p. 954p. 955p. 956p. 957p. 958p. 959p. 960p. 961p. 962p. 963p. 964

Issue Table of ContentsThe Journal of Symbolic Logic, Vol. 48, No. 4 (Dec., 1983), pp. 913-1286+i-viiiVolume Information [pp. i - vii]Front MatterDiscontinuities of Provably Correct Operators on the Provably Recursive Real Numbers [pp. 913 - 920]Wtt-Degrees and T-Degrees of R.E. Sets [pp. 921 - 930]A Filter Lambda Model and the Completeness of Type Assignment [pp. 931 - 940]Algebraic Semantics for Quasi-Classical Modal Logics [pp. 941 - 964]Characterizing the Continuous Functionals [pp. 965 - 969]Some Model Theory of Modules. II. On Stability and Categoricity of Flat Modules [pp. 970 - 985]Modal Analysis of Generalized Rosser Sentences [pp. 986 - 999]Indecomposable Ultrafilters Over Small Large Cardinals [pp. 1000 - 1007]The Noncommutativity of Random and Generic Extensions [pp. 1008 - 1012]Forcing and Reducibilities. III. Forcing in Fragments of Set Theory [pp. 1013 - 1034]On the Ramsey Property for Sets of Reals [pp. 1035 - 1045]On a Generalization of Jensen's , and Strategic Closure of Partial Orders [pp. 1046 - 1052]Blunt and Topless End Extensions of Models of Set Theory [pp. 1053 - 1073]Degrees of Types and Independent Sequences [pp. 1074 - 1081]A Baire-Type Theorem for Cardinals [pp. 1082 - 1089]Two Further Combinatorial Theorems Equivalent to the 1-Consistency of Peano Arithmetic [pp. 1090 - 1104]Rabin's Uniformization Problem [pp. 1105 - 1119]Random Models and the Gdel Case of the Decision Problem [pp. 1120 - 1124]Model-Complete Theories of e-Free AX Fields [pp. 1125 - 1129]Model-Complete Theories of Formally Real Fields and Formally p-Adic Fields [pp. 1130 - 1139]A Tableau System of Proof for Predicate-Functor Logic with Identity [pp. 1140 - 1144]Addendum to "Logic of Reduced Power Structures" [p. 1145]Downward Transfer of Satisfiability for Sentences of L1,1 [pp. 1146 - 1150]Une Thorie de Galois Imaginaire [pp. 1151 - 1170]State-Strategies for Games in $F_{\sigma\delta} \bigcap G_{\delta\sigma}$ [pp. 1171 - 1198]Reviewsuntitled [pp. 1199 - 1201]untitled [pp. 1201 - 1203]untitled [pp. 1203 - 1204]untitled [pp. 1204 - 1206]untitled [pp. 1206 - 1207]untitled [pp. 1207 - 1209]

The Herbrand Symposium: (Marseilles July 16-July 24 1981) [pp. 1210 - 1232]Meeting of the Association for Symbolic Logic: Madison 1982 [pp. 1233 - 1239]Association for Symbolic Logic [pp. 1240 - 1272]Index of Reviews: Volumes 47, 48 [pp. 1273 - 1283]Notices [pp. 1285 - 1286]Errata: Intensional Logic in Extensional Language [p. viii]Back Matter

Blok Quasi Classical Modal

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