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All Intermediate Logics with Extra Axioms in One Variable, Except Eight, Are Not Stronglyω-CompleteAuthor(s): Camillo FiorentiniReviewed work(s):Source: The Journal of Symbolic Logic, Vol. 65, No. 4 (Dec., 2000), pp. 1576-1604Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2695065 .Accessed: 06/11/2012 11:40

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THE JOURNAL OF SYMBOLIC LOGIC

Volume 65. Number 4. Dec. 2000

ALL INTERMEDIATE LOGICS WITH EXTRA AXIOMS IN ONE VARIABLE, EXCEPT EIGHT, ARE NOT STRONGLY co-COMPLETE

CAMILLO FIORENTINI

Abstract. In [8] it is proved that all the intermediate logics axionmatizable by fornulas in one variable, except four of them, are not strongly complete. We considerably improve this result by showing that all

the intermediate logics axiomatizable by formulas in one variable, except eight of then, are not strongly

w-complete. Thus, a definitive classification of such logics with respect to the notions of canonicity, strong

completeness, co-canonicity and strong co-completeness is given.

?1. Introduction. We call logics in one variable the superintuitionistic logics obtained by adding to the propositional intuitionistic logic Int a formula containing only one variable as an axiom schema. These logics are well studied and characterized in literature. Nishimura first [1 1] introduced an effective enumeration of non intuitionistically equivalent formulas Fn in one variable; logics in one variable are obtained by adding to Int any classically formula Fn, taking care into the cases in which different formulas yield the same logic. The first significant analysis of such logics is given by Anderson in [1], where the problem of disjunction property is also treated (for more details, see [2, 3] and Section 4, where an enumeration of the formulas in one variable different from Nishimura's one will be used). Further, an important result has been obtained by Sobolev in [13], who has shown that all the logics in one variable have the finite model property, hence they are decidable and admit a semantics in terms of Kripke frames (the interested reader can find a more accessible proof in [14] or in [3]).

In this paper we prosecute the research of [8], by giving a definitive classification of such a family of logics with respect to the notions of strong completeness and strong co-completeness. We recall that canonicity and strong completeness are well-established notions in the literature of modal logics (see for instance [4]) and propositional logics (see [3]), when Kripke semantics is assumed, and have a central role as concerns the relationships between the syntactical apparatus of a logic and its semantical counterpart. The notions of co-canonicity and strong co-completeness are just the relativisation of the previous notions to finite languages (i.e., languages

Received March 4, 1999. 1991 Mathematics Subject Classification. 03B55. Key words and phrases. co-canonicity, extensive co-canonicity, strong co-completeness. I feel sincerely grateful to Prof. Pierangelo Miglioli (1946-1999) for his valuable hints and his enthu-

siastic support in this research.

(?) 2000. Association for Symbolic Logic 0022-481 2/00/6504-0007/$3.90

1576

ALL INTERMEDIATE LOGICS ... 1577

based on finite sets of propositional variables) and are less investigated in literature (see Section 2 for the precise definitions). We point out that, when dealing with concrete problems (for instance, in issues concerning Kripke completeness), the restriction to finite languages turns out to be very natural. A comprehensive comparison between these notions lacks in literature; a first work in this direction is given in [8], where a systematic treatment of the subject is proposed and, as far as we know, the first examples of complete and non strongly co-complete intermediate propositional logics are presented. In the quoted paper, a remarkable result is proved: all the logics in one variable, except four of them, are not strongly complete. In this paper we considerably improve this result and we show that all the logics in one variable, except eight of them, are not strongly co-complete (see Section 7 for the thorough classification). To get this result, we introduce some original techniques to classify co-completeness and we state a criterion for the strong co-completeness which is more general than the one in [8] (Section 3): a systematic treatment of the whole subject can be found in [7]. We also stress that, while in [8] the authors use algebraic-categorical tools, we directly act on kripkean semantics, using techniques more inspired to the classical Model Theory. Sections 5 and 6 are wholly devoted to the proof of the main result.

?2. Basic definitions. As usual, a (Kripke) frame is a pair P = ?P. <) consisting of a nonempty set P and a partial order < on P. i.e., P is a partially ordered set (poset); the elements of P are called the points of the frame P. We write a < /3 to mean that a < /3 and a z /3, we also use the notations 3 > oa and /3 > a as a synonymous of a < /3 and a < /3 respectively. A subframe of P is a frame P' = (P', <') obtained by considering a subset P' of P and the restriction <' of < to P'; the subframe is said to be a generated subframe iff P' is upward closed. If a is a point of P, the cone P-, of P is the generated subframe of P obtained by considering a and all the points greater than a. A point /3 is an immediate successor of a if a < /3 and, for all points y of P such that a < y < /3, we have either y = a or y = /3. Afinalpoint of a frame P = ?P. <) is a maximal point of P; Fin(a) denotes the set of all the final points o > a. We use the notation P = (P. <, p) to indicate a frame P with root p, where as usual the root is the minimal point of P.

A p-morphism f from P = ?P. <) onto P'= KP', <') is a surjective map f: P P' such that:

(1) f is order preserving; (2) f is open, that is, for every a E P and /3' E P', if f (a) <' /3', then there is

/P E P such that a < P and f (P) = 3'.

For a frame P, the splitting Spl(P) of P denotes the class of frames P' such that, for every generated subframe P" contained in some cone P of P', there are no p-morphisms f from P" onto P.

A Kripke model K = KP. < F), based on a frame P = (P, <), is obtained by defining the forcing relation IF between the points of P and the set of propositional variables of the language in hand, with the condition:

a < /3 and a IPp p ,/ - Vp.

The forcing relation is extended to all the formulas in the usual way so that the above property holds for all formulas. Submodels and generated submodels are defined

1578 CAMILLO FIORENTINI

similarly to subframes and generated subframes. Given a model K = (P. <, ) and oa E P, 17K(a) (or simply 7(a) if the context is clear) denotes the set of formulas forced in ae. If V is a set of propositional variables, Fv(a) (or simply Fv (a)) denotes the set of V-formulas forced in a, where a V-formula is a formula containing only propositional variables of V.

We say that a formula A is valid in K (and we write K I= A) iff a IF A for all a E P; we say that a set of formulas A is valid in K (and we write K I= A) iff K I= A for every A E A. In this case we also say that K is a model of A. We say that a formula A is valid in a frame P (and we write P I= A) iff K I= A for every Kripke model K based on P; P I= A iff P I= A for every A E A.

We recall the following fact that will be used in the sequel.

* Suppose that P' , Spl(P). Then, for every formula A, P' I= A implies P I= A.

Let F and A be two sets of formulas and let 9 be a class of frames. We say that A is a consequence of F w.r. t. A, and we write F A- A, iff, for all models K = (P, <, IF) based on frames of 9 and all a E P, it holds that:

a IF A forallA E Fz =>a I- B forsomeB E A.

An intermediate propositional logic L is a propositional logic (i.e., a set of propositional formulas closed under modus ponens and substitution of propositional variables with propositional formulas) such that Int C L C Cl, where Int is the propositional intuitionistic logic and Cl is the propositional classical logic; given a set V of propositional variables, Lv is the set of V-formulas of L.

Given two sets of formulas F and A, with F FL A we mean that there are some formulas A1, . . , A, in F and B1, . . B,,1 in A such that AI A ... A A,, BiV V Btn e FL A means A E L.

A set of formulas A is L -saturated (in the language S) if and only if:

(1) A is consistent (i.e., A VINT A A -,A); (2) A FL A implies A E A, for every formula A of Y (hence L C A); (3) AVB eAimplieseitherA eAorB e A.

If L is omitted, we mean that A is an Int-saturated set. The definition of L, V- saturated set is the relativisation of the definition of saturated set with respect to V-formulas (actually, the set of V-formulas of a L-saturated set A is a L, V- saturated set). We point out that, given a model K and a point a of K, FK (a) is a saturated set and F (a) is a V-saturated set. A saturated set A is realized in K iff = FK(a) for some point a of K (similar definition for V-saturated sets). The following lemma is well known.

LEMMA 2.1 (Inclusion-exclusion Lemma). Let L be any intermediate logic and let F and A be two sets offormulas such that F VL A. Then, there is a L-saturated set F* such that 1 C F* and 1* nmA 0. -0

Now we are ready to introduce the main notions of the paper. Let L be any intermediate propositional logic; a frame P = (P, <) is said to be a frame for L iff it belongs to the class

Fr(L) = {P = (P, <): P #L}.


Let 9 be any class of frames; then

{A :for allP- KP,<) c 9, P K=A}

is an intermediate propositional logic (called also the logic ofF). The following facts are trivial: * L CYS(Fr(L)); * FFL A #> F Fr(L) A.

The converse in general does not hold, therefore the following definitions of completeness are justified.

DEFINITION 2.2. Let L be any intermediate logic. Then: (a) L is complete (or has Kripke semantics) iff L = Y (Fr(L)). (b) L is strongly complete iff, for any two sets offormulas F and A, it holds that:

F FL A " {' F #Fr(L) A.

(c) L is strongly co-complete if, for every finite set V ofpropositional variables, for any two sets of V-formulas F V and AV, it holds that:

FV FL AV r Fv #:Fr(L) Av *]

We stress that (a) and (b) are well-established notions, while (c) is scarcely investigated in literature. The following proposition, easily provable ([7]), provides equivalent definitions.

PROPOSITION 2.3. Let L be any intermediate propositional logic. (i) L is strongly complete if and only if every L-saturated set A is realized in some

model K based on aframe for L. (ii) L is strongly co-complete if and only if, for every finite V, every L, V-saturated

set A V is realized in some model K based on a frame for L. -1

Now we introduce some definitions related to the separability of the points of a Kripke model by means of formulas (see also [8]).

DEFINITION 2.4. Let K = ?P, <H-) be any Kripke model and let V be a set of propositional variables.

(a) K is (simply) separable iff, for every a, /3 E P, FK (a) = FK () implies a =3 (b) K is (simply) V-separable iff, for every a, P3 E P, Fr (a) = F(P3) implies

ao/K =.

(c) K is well separable if, for every a, P3 E P. FK (a) C FK (P) implies a < /3 (d) K is well V-separable if, for every a, P3 E P, IF (a) C IF (P) implies a < /3. (e) K is full iff, for every a E P and every saturated set A such that FK (a) C A,

there is /3 > a such that FK (3) = A. (f) K is V-full ifffor every a E P and every V-saturated set AV such that F V(a) C

A>', there is /3 > a such that FV(fl) AV. v

We note that in literature (for instance, in [3]) separable models are also called differentiated or distinguishable, well separability is called tightness, and full separable models correspond to descriptive general frames. The following properties can be easily proved.

PROPOSITION 2.5. Let K = P k <a IF) be a Kripke model and let V be a set of propositional variables.


(i) If K is separable andfull, then K is well separable and has enough final points. (ii) If K is V-separable and V-full, then K is well V-separable and has enough final

points. (iii) If K is V-separable and V is finite, then K has finitely many final points. H

A remarkable feature of V-separable and V-full models is that of having maximal points with respect to the forcing of V-formulas in the following sense.

PROPOSITION 2.6. Let K = (P, <? H) be a V-separable and V-full Kripke model, let A be a V-formula and let a E P be such that a lJ A. Then there is a,,Y> ?a such that a,1X, I, A and, for every 3 > a,,,,.x, 3 1F A.

PROOF. Let us consider the nonempty set

= A' is a V-saturated set and F'(a) C A/ and A , A/'}.

By Zorn Lemma (see for instance [10]) applied to 9 (with respect to the partial order C), 9 has a maximal element A*. By the V-fullness of K, there is a,,7Clx > a such that F [ (afl7,j(f) = A*. Suppose that there is 5 > aMfX such that 5 I, A. Then A , I"(3), hence I"(3) E 9; on the other hand, by the V-separability of K, A* must be properly contained in F (3), in contradiction with the fact that A* is maximal in 9. Thus the proposition is proved. -

Note that it is not required that V is finite, thus the property holds also in the case that V is the set of all the variables of the language (namely, K is separable and full).

Finally, we recall the well-known notion of canonicity and its relativized counterpart (see also [8]).

DEFINITION 2.7.

(A) A logic L is said to be canonical if and only if every separable andfull model of L is based on aframe for L.

(B) A logic L is said to be co-canonical if and only if, for every finite V, every V-separable and V-full model of Lv is based on a frame for L. -1

The following facts are immediate consequences of the corresponding definitions.

PROPOSITION 2.8. Let L be any intermediate logic. Then:

(i) If L is canonical, then L is co-canonical. (ii) If L is strongly complete, then L is strongly co-complete.

(iii) If L is canonical, then L is strongly complete. (iv) If L is co-canonical, then L is strongly co-complete. A

The converses of (i) and (ii) do not hold as we will see later. We do not know whether the converses of (iii) and (iv) hold, and this seems to be a difficult open problem.

As usual, the canonical model FL = (PL, <, 1H) of a logic L is the Kripke model such that:

- PL is the set of all the L-saturated sets. - < coincides with the inclusion between sets. - For every propositional variable p and every A E PL, A H- p iff p E A.

Using the Inclusion-exclusion Lemma, one can prove that: - For every formula A and every A E PL, A IF- A if A E A.


The definition of V-canonical model FL1' of L (with V finite) is like the definition of canonical model, taking into account the L, V-saturated sets. Clearly, FL is a full model of L, even better, it contains (up to isomorphisms), as generated submodels, all the full models of L; a similar property holds for FL". Thus, denoting with PL

and P v the frames of FL and FL" respectively, we have:

* L is canonical if fL is a frame for L; * L is co-canonical if, for every finite V, PL is a frame for L.

?3. Some conditions about strong co-completeness. As a key tool in describing models, we introduce the equivalence relation -DZ, which consists in a bounded bisimulation between the points of Kripke models.

DEFINITION 3.1. Let V be afinite set of propositional variables; let K I(P. ?IF) and K' = (P', , '1 S) be (non necessarily different) Kripke models. Then the relations

f< and A' between points of K and K' are defined, inductively on n > 0, by the following conditions. Let a c P and a' E P'; then:

- a d/ a' iff, for all p E V, a IF p implies a' I' p. - a a' a iff a V a'/ and a' -, a.

a d n+ a' iffJ for all !]' E P' such that a' <' /', there is / E P such that a < /3 and/i -v /3'.

- a-, ? / a' iffa 41 a'/ and a' 4+. a.

It is easy to see that a ,' a implies a <j a' for every k < n; hence, a -' a' implies a < a' for every k < n. We remark that, for every n > ?0, is an equivalence relation havingfinitely many equivalence classes.

DEFINITION 3.2. Let A be any formula. The implicational complexity Ic(A) of A is defined, by induction on the (structural) complexity of A, by the following conditions.

- Ic(A) = 0 if A is atomic. - Ic(A) = max{Ic(B), Ic(C)} if either A = B A C or A = B V C. - Ic(B -* C) = max{Ic(B), Ic(C)} + 1. - Ic(-A) = Ic(A) + 1. ,

For a model K = (P, <, ) and a finite set V, Fr'K(a) (or simply F "' (a) if the context is clear) denotes the set of V-formulas of implicational complexity r < n forced in a.

A remarkable point is that the equivalence classes [a]<, of the relations - are characterized by formulas H ? of implicational complexity at most n (see also [9]); we give a proof of this fact in next lemma. To simplify the statement of the lemma, we assume to have T and I, intuitionistically equivalent to p -* p and -i(p -* p) respectively, as primitive atomic symbols of the language; moreover, we say that two formulas A and B essentially coincide if B is obtained from A by trivial permutations between the formulas C1 and C2 in subformulas of A of the kind C1 A C2 and C1 V C2.

LEMMA 3.3. Let K = (P, <, IF) be a Kripke model and let V be a finite set of propositional variables. For every n > 0 and every a e P, there is a V-formula Hot such that:

(1) Ic(H,) <n.


(2) For every K' = (P', <',IF') and every a' c P', if a' a' a then Hgn, essentially

coincides with Hg. (3) For every K' (P', <',1H') and every a' c P', a' IF' H' ifa , a'.

PROOF. By induction on n. Suppose n = 0. If a does not force any p c V, we can set HO = T and the lemma is satisfied (indeed, a ' a' for all a'); otherwise we take:

Ha A P. pE V:alFp

Suppose now, by the induction hypothesis, that the lemma holds for n. Let, for any a C P:

S (a) = : /3 is any point of any Kripke model and, for every 6 E P. a < 6 implies /3 t v}

S'(o) {/3 R: /3 is any point of any Kripke model and /3 $Jv a}.

If S (a) 0, then a a a' for every a' and, as before, we can set Hc,+' T. Otherwise, we define:

Ha"K A (Hf-l V H V I).

Pcs(a) ,WA

We remark that, in the above formula, we assume to take only one characteristic formula in correspondence of -d< -equivalent points in S(a) and in S'(fl) respectively. By the fact that there are finitely many non equivalent points with respect to <

and by the induction hypothesis, it immediately follows that Hgn+ is a well defined V-formula which satisfies (1) and (2). In order to prove (3), let K' = (P, ', F1-') be any Kripke model and let a' c P'. Suppose that a n a'; we prove that a' H-' H"1+. Take any 6' c P' such that:

- a' <'Y; - 6' I-' Hjr for some S(a).

By the induction hypothesis, /3v < '. Suppose now that 6' <1 P3. Then 6' -< /3 and, since a - V+ ao' and a' <' Y', there is 6 c P such that a < 6 and < - v 6'.

This implies that 6 - , against the fact that /3 E S(a); thus 6' $$ v , that is 6' c S'(fl). Since 6' I-' H4n (by the induction hypothesis, being s' < v'), we get:

oV' F'H V an V 1.

(' CS' (f)

By the generality of Y' and P3, it follows that a' IF-' HCn+

Suppose now that a $$v a'. There is P3' c P' such that:

- af' ?' /3'; - for every 6 c P such that a < 6, P' 9,v 6.

By the induction hypothesis, we have:

- /'F'Hg,;

- C E AS'(/3') implies /3' lV' Hahn (in fact, 6 $n /)

Note that S'(/3') may be empty. Therefore:

oa'I</' H,31,-* V H(fn V I.


Since, by the above assumptions, /3' E S (a), we can conclude that a' ly' H,1+. -]

Using the above lemma, it is not difficult to show that the -d< equivalences preserve the forcing of V-formulas up to the implicational complexity n, as stated in next proposition (see also [7]).

PROPOSITION 3.4. Let K = (P, <IF-) and K' = (P', <',IF') be any two Kripke models, and let V be a finite set of propositional variables. Then, for every a c P. a' C P' and every n > 0, it holds that:

(i) a n a' if and only if Fr() C F Vn(a/)

(ii) a -V a' if and only ifFr~(a) - hFl(a/) A

The relation - V assigns to each point a V-grade, according to the following definition.

DEFINITION3.5. Let K = (P, <, H) be a Kripke model, let V be a finite set of propositional variables and let a c P. We say that a has V-grade r (in K) iff r is the minimum k > 0 such that the following condition holds:

- for every / c P, /P -4 a implies/P = a. -

This means that, for k > r, the equivalence class [a]_v contains only the point a, while, for k < r, [a]_ v contains more than one point. We say that a point a of K hasfinite V-grade if it has V-grade r > 0; otherwise, a has infinite V-grade. We remark that if a has V-grade r in K, then a has V-grade r' < r in any generated submodel K' of K in which a is defined; on the other hand, a point having infinite V-grade in K may have finite V-grade in some generated submodel of K. Note also that a final point of a V-separable model K has V-grade 0 or 1 it follows that, for every 6, 6' in K, '-4 6' implies Fin(6) = Fin(6).

PROPOSITION 3.6. Let K = (P, <, p, 1-) be a V-separable and V-full Kripke model (V finite) and let K' = (P', <', P', IP') be any Kripke model such that F , (p') F (p). Then there is a map h: P' -*- P such that:

(i) h (p') p; (ii) a' <' /' implies h (a') < h(I');

(iii) if h(a') < /I and I has finite V-grade in K, there is /' E P' s.t. a' <' /' and h(')=/I

PROOF. Let a' be any point of P'; then Fv,(p') C Fv,(a'), hence Fv(p) C

Fr, (a'). Since K is V-full and V-separable, there is one and only one a E P such

that FjK(a) - Fv (a'). So we are allowed to define h: P' -*- P as follows:

h (a') = a if and only if F v(a) F v(')

(note that a' -Ev h(a') for every k > 0). Clearly h(p') p p; moreover, since K is well V-separable, (ii) immediately follows. Suppose now that h (a') < /I and /I has finite V-grade r > 0 in K. Since h(a') -rv? a', there is /' E P' such that a' <' /I' and /I -< /I'. Since /I' -v h(/'), it follows that /I -v h(/'); but /I has V-grade r in K, hence fi h(/') and (iii) is proved as well. -

We point out that h may be non-surjective; indeed, the points of K of infinite V-grade may not have any preimage in K' (while all the points of finite V-grade have at least one preimage); moreover, if K' is not V-separable, h is not infective.


As far as the V-separability of points of finite V-grade is concerned, we claim that:

LEMMA 3.7. Let K = (P, <IF-) be a Kripke model and let a be a point of K of finite V-grade in K. Then, for every /3 E P, Fr (a) = Fr (/3) implies a /3.

PROOF. If Fr(a) v (/3), then a P/ for every k > 0. Let r > 0 be the

V-grade of a; since a - /3, it holds that a /3. -

Thus, to prove the V-separability of a model, it suffices to check the points of infinite V-grade, as stated in the next proposition.

PROPOSITION 3.8. Let K = (P, <IF) be a Kripke model and let V be afinite set of propositional variables. K is V-separable if and only if, for every a, / E P of infinite V-grade, Fr(a) = Fv(P3) implies a /. A

In order to give a condition for the V-fullness, we introduce the notion of V- sequence.

DEFINITION 3.9. Let K = (P, <IF) be a Kripke model and let V be afinite set of propositional variables. A V-sequence {/3k }J/ 0 of K is a sequence of points /l3k C P such that /3k Pk+, for every k > 0.

We say that /3 E P is a limit of the V-sequence {/k } io if and only if/3 K /3k for every k > 0. -A

We stress that the point Pk of a V-sequence can be viewed as an approximation of /3 (with respect to the forcing of V-formulas) up to the V-formulas of implicational complexity k. We also remark that, if K is V-separable, then the limit is unique. The following proposition provides a necessary and sufficient condition for the V-fullness.

PROPOSITION 3. 10. Let K = (P, <, IF) be a Kripke model and let V be afinite set of propositional variables. Then K is V-full if and only if, for every cone K,, of K and every V-sequence {/3k}jI>o of Ka, {3k }v '>o has a limit in K,.

PROOF. Suppose that K is V-full; let K be a cone of K and let {/3k},j>o be a V-sequence of K.. Let us consider the set:

Al/ U FK(3k)- k>O

Then A' is a V-saturated set and Fr(a) c A'; by the V-fullness of K, there is /3 C P such that a < /3 and Fv(P3) A v. By definition of V-sequence,

Frk (ok) = Fk (o+ i) for every k, j > 0 which implies that Fk (3) = rk (/,k) for every k > 0, hence P3 O' Pk for every k > 0. We can conclude that the point 3 of

Kca is a limit of the V-sequence {3k }kv>- Conversely, let us suppose that K satisfies the condition on V-sequences; let a

be any point of K and let A'" be a V-saturated set such that F (a) C A'7. Let K* (P*, <*, pi*, I[*) be a V-full model such that Fv* (p*) = Fv (a) (for instance, we can take the cone of the V-canonical model for Int generated by the V-saturated set F v(a)); by the V-fullness of K*, there is /3* in P* such that Fv* (P*) A


Since Fv. (p ) = ' (a), it holds that:

p k oa foreveryk > 0.

By the properties of the relations -k'4, we can determine a sequence of points /k of Pa such that:

Pi* k kk foreveryk > 0.

It follows that Pk,+ h-a fP, for every k > 0, therefore {fPk,}JIv is a V-sequence of Ka. By the hypothesis of the proposition, such a V-sequence has a limit /P in Ka; this means that a < P3 and, since ,B* -v P for every k > 0, we can conclude that IF (/3*) = IF (P3), hence Fv (Pi) = Av and K is V-full. -A

We remark that the condition of Proposition 3.10 is trivially satisfied by the V- sequences { vk definitively constant. Indeed, if /k =3, for every k > n then

E P is the limit of the V-sequence; thus, in applying the proposition, we can limit ourselves to consider V-sequences which are not definitively constant.

With the tools described so far, we are able to study the co-canonicity of a logic L, since it is related to V-separable and V-full models of L. On the other hand, to treat the strong co-completeness we need knowledge of all (non necessarily V-separable or V-full) models of L. At this aim we introduce V-stable reductions.

DEFINITION 3.1 1. Let K = (P, <, p, IF-) be a Kripke model and let V be afinite set of propositional variables. We say that P' = (P', <', p') is a V-stable reduction of P (P, <, p) if and only if there is a p-morphism g from P onto P' which satisfies the following condition, for every a C P and 3' E P':

- if g(a) <' /', there is /3 E P s.t. /3 has finite V-grade in K, a < /3 and g(f/) =/'. -A

Now we show that V-stable reductions of V-separable and V-full Kripke models play a central role as far as strong co-completeness is concerned.

THEOREM 3.12 (Strong co-completeness Criterion). Let L be a strongly co-complete logic, let V be afinite set of propositional variables, let K = (P, <, p, 1F) be a V-separable and V-full model of L ' and let P' be a V-stable reduction of P. Then P' is a frame for L.

PROOF. Suppose that P' is not a frame for L. To prove the proposition, it suffices to prove that:

(1) For every K" = (P", ?",p", IF") such that F/,,(p") = Fv(p), the frame P1 = (P",?, p"/) is not a frame for L.

Indeed (1) implies that the L, V-saturated set Fv(p) cannot be realized in any Kripke model based on a frame for L, that is L is not strongly co-complete (see Proposition 2.3).

Let K" be as in (1), let h : P" -* P be as in Proposition 3.6 and let g be the p- morphism from P onto P' as in the definition of V-stable reduction. We know that h is "less than" a p-morphism, while g is "more than" a p-morphism; composing these two maps, we get:

(2) f = g o h is a p-morphism from P" onto P'

from which (1) follows.


Let e", /3" E P"; a" ?"/3" implies h (a") < h(/3"), which implies g (h(a")) <'

g(h(/3")), that is f (a") <' f(/3"). Let a" c P" and suppose f (a") <' /', that is g (h (a")) <'/ '. By definition of

g, there is /3 c P such that:

- /3 has finite V-grade in K, - h(o") < / and g(/) = /'

By definition of h, there is /3" E P" such that:

- a/" ?" /"and h(/") = .

Hence f (P") g(h(P3")) g(P3) = '. Finally, f (p") g(h(p")) g(p) = p', hence f is also surjective and (2) is proved. -A

We remark that, in previous theorem, P needs not be a frame for L. In next section we will use the machinery explained so far to classify the logics in one variable with respect to strong co-completeness (see also [7] for other applica- tions).

?4. On logics axiomatized by formulas in one variable. We recall some well known facts about logics in one variable (see for instance [1, 3, 8]). In order to describe the non intuitionistically equivalent formulas in one variable p, we consider the Int, {p}- canonical model K (Pt, <, ac, H-) defined on the frame P,, (P=o, <, a,,) of Figure 1 (straight lines represent the immediate successor relation) and with forcing relation defined on the variable p as follows:

6lVp if 6=c1.

P al a?

G3 G \/X> , C4

a5 a6

a7 a8

FIGURE 1. The {p}-canonical model K,

Let us consider the following sequence of formulas.

nfi = p

nf2 = -P

nf3= -np

nf4 = -np -) p

nfk = nfk-I -)+ nfk-3 V nfk-4 for every k > 5.


Note that:

nf5 = nf4 - nf2 V nf= (-I P -P) -- -3p VVp

nf6 = nf5 -*nf3 V nf2 = ((-P- ) -- -ip V p) V --VP

nf7 = nf6 - nf4 V nf3 =

((-1-1p -* P) -) -Vp Vp) -*-ip V -ip) -) (-1-lp -) p) V rnip.

The formulas nf,, (possibly with different enumerations) are also known in literature as Nishimura-formulas ([1 1]). The following facts are well known.

- 5 iF nfk if and only if ck <_ - am < a, implies KINT nfj2 -* nfjs - - For every {p}-formula A, there are n, m > 1 such that F-INT A +-> nfn V nf,,1.

Therefore every {p}-formula is intuitionistically equivalent to some formula of the kind:

nfk or nfk Vnfk+l withk > 1.

In correspondence, we can give the following list of superintuitionistic logics in one variable.

* Int+ (nfI Vnf2) = Int+nf4 = Cl. * Int+ (nf2 Vnf3) = Int+nf5 = Jn

(Jankov logic or Weak excluded middle logic). * NL,71 = Int + nff, for every m > 6. * NLn,n+l= Int+ (nfnV nf,+?), for everyn >3.

We point out that: * NL6 = Int + nf6 = St is also known as Scott logic ([5]); * NL7 = Int + nf7 = Ast is also known as Anti-Scott logic ([6]).

All these logics have a simple semantical characterization (see [8]). Let us call the generated subframe of Pt having root ak and the frame obtained by

the union of P,, and tk+l.

PROPOSITION 4.1. Let P be any frame. (i) P is aframe for the logic NL,1+I, for m > 3, iff P E Spl(Pt ).

(ii) P is aframefor the logic NLn+Ln+2, for n > 1, iff P E Spl(Ea,,,) . -A

As a consequence of a result due to Sobolev (see [13] or [2, 14]), the finite frames quoted in the previous proposition characterize the corresponding logics. In some cases we can describe the frames characterizing these logics without any reference to p-morphisms. For instance, the finite frames for St (i.e., the finite frames belonging to Spl(E-)) may be characterized as the finite frames P (P, <) such that, for every a E P, the points of Fin(a) are prefinally connected in P. in the sense of [5].

Even if any frame P not belonging to Spl(E ) is not a frame for NL,,+l, this fact does not prevent us from defining V-separable models K of NLv+1, with V finite, based on such a frame P. One has to guarantee that, for every p-morphism f from some generated subframe of P onto P,,, the preimages of a1 (along f ) are not separable from the preimages of 63 (along f) by means of a V-formula (in other words, there is not any V-formula which is forced exactly on the preimages of c1 ). This is discussed in next proposition.


PROPOSITION 4.2. Let K = (P, <, p, 1-) be a Kripke model and let V be afinite set of propositional variables.

(i) K is a model of NLJ11 ,for m > 5, if and only if the following condition holds:

(t) for every generated subframe P' of P, for every p-morphism f from P' onto P.,,,, for every k > 0, there are 61 and 62 in P' such that 1 -1kV 2,

f (61) = c1 and f (2) # cGI (ii) K is a model ofNLv + 1 ,+2, for n > 2, if and only if the following condition holds:

(tt) for every generated subframe P' of P, for every p-morphism f from P' onto PE+,,for every k > 0, there are 61 and 62 in P' such that 1 < 2,

f (61) = c1 and f (69) 7& cG.

PROOF. (i) Suppose that, for some m > 5, K does not satisfy (t); then there is a generated subframe P' of P and a p-morphism f from P' onto P . such that, for some n > 0 it holds that:

(*) for every 61,62 in P', s1 -, V2 and f (61) = a1 implies f (62) ,1.

Let ao1, . . . , a be a finite list of points of P' such that:

- f (ak) = a1 for every 1 < k <?; - for every /3 in P' such that f (f) =1, /3 -<~i ak for some 1 < k < j.

By Lemma 3.3, there are some V-formulas HI,. , H such that, for I < k < j, the following holds:

- for every fl in P', /3 1F Hk iffc k =$?1 P Let H be the V-formula HI V V Hi. We prove that:

(**) for every fl in P', /3 1- H iff f(/) =r1.

If /3 IF- H, then 3 IF- Hk for some 1 < k < j, hence ak /3, . By definition of 4 1'+lthere is ar in P' such that Oak < a and ac -< /3; since f (ak) l1, it follows that f (a) = a, hence, by (*), f (/3) = 1. Suppose now that f (/3) l1 and let k be such that /3 7v ak; then /3 - Hk, that is 3 HF H; thus (**) is proved.

Let s be any point of P' such that f (s) = u,,,. Since in the model K,, (defined in Section 4) al forces p, Uv) does not force the {p}-formula nf,11+? and f is a p-morphism from P' onto P,,,,, by (**) s does not force the instance nfn+?I (H) of nf,,1+i obtained by replacing p with the V-formula H; we can conclude that K is not a model of NLv'+.

Conversely, suppose that K is not a model for NLJV+1. We can assume that, for some V-formula H, the instance nfm+l (H) of the axiom schema of NL,,,+? is not valid in K. Let ar E P be such that ar IJ nf,,,+I (H), that is:

c IJ nfm(H) -* nf,1n-2(H) V nfn-3(H).

Then there is /3 > ar such that /3 IF- nfl1(H), /3 IJ nfn-2(H) and /3 IJ nf,,n-3(H). Let us define a map on the points s > /3 as follows:

- f () = Uk iff I F nfk(H), I 1Y nfk-2(H), / 1V nfk-3(H), for every 4 < k <

m; - f () = U3 iffb IF nf3(H) ands IJ nffI (H); - f() = 2 iff 1IF- nf2 (H);

-f f() = a iff IF - f I (H);


where nf I (H) coincides with H. It is easy to check that f is a p-morphism from PA onto ft . Let r > 0 be the implicational complexity of H. By definition of f, for every s1, 62 in PA it holds that:

61 TV2 '=> Of (61) = al if Of (62)- al

This means that (t) does not hold. (ii) It is proved as (i). -

Note that, if K is V-separable and k > 2, the hypothesis i I" O 2and f (61) = a implies that f (2) E {U1, U3} (indeed, Fin(bl) = Fin(62)).

?5. The canonical and co-canonical logics in one variable. We start by analyzing the eight strongly co-complete logics and, as done in [8], we introduce a finer classification about canonicity and co-canonicity. We say that a logic L is extensively canonical iff every well separable model of L having enough final points is based on a frame for L; likewise, L is extensively co-canonical iff, for every finite V, every well V-separable model of Lv is based on a frame for L. This means that in the proof of canonicity (co-canonicity) the hypothesis of fullness (V-fullness) is not required.

We recall the main result proved in [8] (see also [12], where the non strong completeness of the logic NL6 St is stated).

THEOREM 5. 1. (i) The logics Cl, Jn, NL3,4, NL4,5 are extensively canonical.

(ii) For every m > 6, n > 5, the logics NL,11 and NLnn+l are not strongly complete. A

We point out that (i) is almost immediate while (ii) requires a deep analysis of strong completeness.

THEOREM 5.2. The logic St is extensively co-canonical. PROOF. Let K = (P, <, 1-) be a well V-separable model of St', for some finite V,

and suppose that P is not a frame for St. Then there is a p-morphism f from some generated subframe P' of P onto P . Since K has finitely many final points and K is V-separable, there is a V-formula H of implicational complexity 1 such that:

- i IF H if s is final and f (b) = al; - i IF -H if s is final and f (b) = 2.

Let ar E P be such that f (a) = 5; it is easy to see that ae 1V nf6(H), which is an instance of the axiom schema of St. Since such a formula belongs to St', we get a contradiction. Thus f does not exist and St is extensively co-canonical. A

Observe that in previous proof very simple arguments are used. On the contrary, to prove the co-canonicity of the remaining logics, an essential use of the V-fullness hypothesis is required, since such logics are not extensively co-canonical (for Ast see [8], the case of NL6,7 is similar, for NL5,6 see Appendix A). This justifies the fact that next proof is rather involved, and strong mathematical principles must be used.

THEOREM 5.3. The logics Ast, NL5,6 and NL6,7 are co-canonical. PROOF. We prove the theorem only for the logic NL6,7 (the other cases are sim-

ilar'). Let K = (P, <, H-) be a V-separable and V-full model of NL6'7, for some finite V, and suppose that P is not a frame for NL6,7. Then there is a p-morphism

The case of Ast is treated in [8] in a different way.


f from some generated subframe P', contained in some cone Pe of P. onto P T5.6

Let H be the V-formula defined in the proof of Theorem 5.2; then, for every s in P', it holds that:

- 1 -ISH V --H id f (b) E {C U1U 2, U3}

(note that -iH V -i--iH has implicational complexity 3). We can apply Proposition 2.6 and we can state that:

(a) Let ae E P be such that f (a) = 6, with a E {U4, U5}. Then there is fl such that ae < /3, f (f/) = a and, for every 6 > /3, f (b) # 6.

Thus we can take ae, /3, c&', /3' > &, P0 and gMAX as follows.

- f (a) =5 and, for all s > a, f & =) - a5;

- f(&') -6;

- e< A, a < 3'and f (/)= f (') = 3; -P0= {b E P : a <b or a ? b};

- MAX{ = Y E Po: f (y) = U4and, for all 6 > y, f (b) - 4 4}.

Clearly the generated submodel K0 = (PO, <, -) of K is a V-separable and V-full model of NLv"7. We prove the following facts.

(b) For every V-sequence {Yk }I>V C 9MVA X, the limit of {Yk },(>O belongs to &MAX .

In fact, by the V-fullness of K0, {Yk}jV>o has a limit y0 E P0; since oi < y* and

y* lJ -'H V -'--'H, it follows that (y*) E {a4, U6}. Therefore there is y' such that y* < y' and y' E gMAX. Suppose that y* =/ y'; by the V-separability of K0, there is k > 3 such that y* 94 y'. Lets > Yk+? be such that -v y'; since lV --' H V --H (indeed y' I, --'H V --,H and k > 3), by the maximality of Yk+1 it holds that s = Yk+1, that is y' -Jk y *, a contradiction; hence y* y' and y* E &MAX-

(c) There is n > 0 such that, for every y E 9MAX, for every as Po s.t. s </3, it holds that y fi s.

Suppose that (c) does not hold; then, for every n > 0 there are yn E SUMIAX and ,nv /3 such that y, < fln. Starting from the points Yn, we can extract a V-

sequence {y' }I >v contained in &MAX in the following way: y' is chosen in a class containing infinitely many y, (that is, in such a class there occur infinitely many indexes n of non necessarily distinct points Yn); YJ+1 is chosen in the A<

class [y']_,, (which, inductively, contains infinitely many y,,) in such a way that

[y',]_, ''contains infinitely many points Yn as well (this is a kind of Bolzano- Weierstrass construction which can be carried out by the fact that there are only finitely many Ok' equivalence classes). Let y* be the limit of such a V-sequence; then FVk(y*) C ]Fl'k(fl) for every k > 0, hence Iv(y*) C IFv(fl) and, by the well V-separability of K0, y* < /3. This yields a contradiction, since, by (b), y* E MAfAX and f (/3) =3; therefore (c) holds. Similarly, we can prove:

(d) There is m > 0 such that, for every y E gMAx, for every s e Po s.t. s V

f/3,

it holds that a uto.

Let n and m be as in (c) and in (d) respectively, let r max(n, m, 2) and consider the set

-W = {US E Po P0 , f/ orb -1 A3}.


Since r > 2, every point s c _ is not final and Fin(G) C Fin(/3) U Fin(/3'). By (c) and (d), it holds that:

(e) For every y E gMAX and every s e A, y 5i s.

We define a map g P0 - -1%, as follows.

- g() = U ifff(U) E {i1, f 3} and, for all U's EE , as 54 a.

- g(0) =2 iff f)=2

- g ) = (3 ifff ) E {i1, 3} and there is5' E _ such that 6 < s'. - g(o) = 4 if there is y E gAvAX such that s < y and, for all 6s (E A, s 5i 6s. - g() = (5 iff = a. - 9 W = U6 if there is 6' E _ s.t. s < 6' and there is y E 9MAX s-t. as < Y

Note in particular that g(b) = (3 for every s E A; by (e), g(y) = U4 for every

YEE &MAX, g(e) = U6. Taking into account the previous statements, it is not difficult to prove that g is a p-morphism from P0 onto P

56 . Since K< is a model

of NL'7 with root &, by Condition (tt) of Proposition 4.2 there are s and s' in P0 such that as v 6', g(6) = a, and g(6') =/ al, that is g(6') = U3. By definition of g, there is /B* E _ such that hs < /P*; beings ,4Fl Y, there is s* > s such that s 'V fl*. This implies that s e A, hence g(o*) = 3, which is absurd. We can conclude that the p-morphism f cannot exist, therefore P is a frame for NL6,7 and NL6,7 is co-canonical. -

This completes the analysis of strongly co-complete logics in one variable.

?6. The non strong co-complete logics in one variable. The next step is to prove that all the other logics of the family are not strongly co-complete. Firstly we disprove the strong co-completeness of NL8 by exhibiting a model K = (P, <, ro, I-) such that, for some finite V, K is a model of NL" and P is a V-stable reduction 8 -cr7

of P = (P, <, ro). This proof can be easily extended to the infinitely many logics NL,,?1 and NL,+?, +2 for m > 9 and n > 6 (namely, the logics in one variable strictly contained in NL8), in virtue of the fact that the frame f7 is a cone of the frames Pa and Pa +l . It is left out the logic NIL9, which must be treated apart.

6.1. The logic NL8. The construction of the countermodel K for NL8 is rather complex, thus we proceed by degrees. First of all, we consider the "tower" of points ak, bk, Ck (k > 0) of root ae in Figure 2. Starting from this frame, we define

ao b co

a2 bC2

FIGURE 2. The points ak, bk, Ck, a.


the sequences of points d3, d4, d5 ... and92l . y in the following way (see Figure 3).

- The immediate successors of d, are akI 1 and Ck-2 for every k > 3. - The immediate successors of g, are ao, bo, co, fo? where fo is a new final point. - The immediate successors of g92k are g2k1 and a2k_1 for every k > 1. - The immediate successors of g2k+1 are g2k and c2k for every k > 1. - a is an immediate successor of y; moreover, for every y, y < /3 iff a < / or

/3 Jo or fl = g, for some n> 1. - a d dk, y : dk, and a : g,, for every k > 3 and n > 1.

ao ba CO fo a0 boCO

al b1 Cl 1 a, b1 c

a2 b c2 ga2 b2

a3 c3 g3 a3 b3 d3

a4 c g4 a4 d4

a a Y

FIGURE 3. The sequences of points gk and dk.

We now define the sequences e4, e5, . . ., E and h4, h5 . q (see Figure 4), which satisfy the following properties for every k > 4.

- The immediate successors of ek are d, and fo. - The immediate successors of hk are dk -1 and ek - The immediate successors of e are a and Jo. - The only immediate successor of q is e (hence q 5i dj for every j > 3; a

fortiori, q 5t el and q 5i hl for every I > 4). Finally, the root ro has, as immediate successors, all the points hk (k > 4), y and q (see Figure 5 for a global picture of K). The forcing relation is defined so that the following properties hold with respect to some finite set V of propositional variables.

* ao, bo, co, Jo, ro have V-grade 0. * The distinct equivalence classes with respect to the relation t-' having more

than one element are: -{ak, bk, Ck: k > 1} U {dk: k > 3} U {a } - {ek k > 4} U {E} - {g k > 1} U {} - {hk: k > 4} U {f}.


fo fo fo

d3 d4 d5 a

e4 e5

h4 h5 h6 Y

FIGURE 4. The sequences of points ek and hk.

f(I

FIGURE 5. The model K.


It is indeed a routine task to find such a finite V and to define the forcing such that the above properties hold. Now we show how the points of K are partitioned by the relations -v.

LEMMA 6.1. For every n > 0 the following holds.

(i) an, bn, Cn,, den gn en, h,, (when defined) have V-grade n. (ii) The distinct equivalence classes with respect to the relation 'v having more than

one element are: - {ak,bk,Ck, dk k>n+1}UfcY} - {ak e, Cd: k > n + 1} U { or} - fek :k>n+l}Uf6} - {gk :k>n+l}U{y} - {hk k > n + 1 } U {fj}.

(iii) ae, y, a, q have infinite V-grade.

PROOF. (i) and (ii) are proved, by induction on n > 0 by directly checking the definitions; (iii) is an immediate consequence of (ii). -1

Having classified the points of K, we can easily prove the V-separability and the V-fullness of K.

PROPOSITION 6.2. K is V-separable and V-full.

PROOF. To prove the V-separability, we can apply Proposition 3.8, observing that the points of infinite V-grade are pairwise V-separated since they belong to different -' classes. In order to prove the V-fullness, we apply Proposition 3.10. Let {/k}k>o be a non definitively constant V-sequence contained in some cone Kf of K. Then one of the following facts holds:

(I) {/Jk k > 0} C {ak,bk,ck : k > 0}U{dk : k > 3}U{ f} (II) {k :k > 0} C {ek k > 4} U {f}

(III) {flk : k >0} C {gC k 1} U {y} (IV) {flk :k>0}C {hk :k>4}U{ifi}. We have to show that {ffl/k}k, has a limit fi in b. Suppose that (I) holds. Then ae is the limit of {/lk }, >v; moreover, since {flk }I>o contains infinitely many distinct points, necessarily a < a, and this concludes the proof. The other cases are similar and the limits are a, y, q respectively. -1

The most delicate question lies in proving that K is a model of NL 7 since, in order to apply Proposition 4.2, we have to take into account all the possible p-morphisms from any generated subframe of P onto P7. This is treated in next lemma.

LEMMA 6.3. Let P' be a generated subframe of P and let f be a p-morphism from P' onto P 67. Then:

(i) f () 7 if and only if s 5 ro (hence P' coincides with P). (ii) f (ak) = u for every k > 0.

(iii) There is n > 3 such that f (d/() = a3 for every k > n.

PROOF. Let P' be a generated subframe of P such that there is a p-morphism f from P' onto P 7. We firstly observe that:

(*) for every a E {fo,ak,bk,ckdjgh: k > 0O j > 3,h > 1}, f () :& 95

This fact can be easily verified; as a matter of fact one can observe that the final points of the finite cone P, of P are prefinally connected in P , (in the sense of [5]),


thus Pj is a frame for St and f (s) & c5 . It follows that {ao, bo, co, fo} C Dom(f), hence {f (ao), f (bo),f (co) f (fo)} C {U1,U2}1

(1) f(ao) = f(bo) = f(co). If (1) does not hold, by the above remarks we get:

- {a2, b2 c2} C Dom(f) and f (a)) = f (b2) f (c2) =4.

This implies that a5 has not preimage in P', in fact: - if e E {ao,bocoalblcl1gl9g2}thenby(*),f(a) 95

- in all the other cases, a < a2 or a < b? or a < c2, hence f() 5

From (1), it follows that:

(2) f (ao) = f (bo) = f (co) a,1 and f (fo) = 2.

Otherwise, we would have f (ao) = f (bo) = f (co) =,2 and f(fo) = a,, hence a3 has not preimage in P'.

(3) f (hk) :& c7 for every k > 4. Suppose f (hk) = a7 for some k > 4. By (2) it holds that:

- (dk-l ) E {U1,U3};

- f(dk) E {il, 93}

Let a > hk be such that f(S) = c4; necessarily a = ek . Hence there is not a > hk

such that f (S) = a5, and (3) is proved. (4) If ae Domr(f), f (a) E {il, U3 }.

(5) PY() :& 97, f( ) :+ 97n f W + 7-

If f(y) = 7, by (*) and (4) there is not a > y such that f(S) a5 5. the same holds for E. Suppose that f(q) = a7 and let a > j be such that f(a) = a4. Necessarily a = 6, hence f (S) :& a5 for every a > q.

By (3), (4) and (5), it follows that:

(6) f(S) = a7 if and only if a = ro

and this proves (i). In particular, f is defined on all the points of P.

(7) f (gk) = a4 foreveryk > 1. In fact, gk < a0, gA < to and, for every a > f(s) f a5.

By (2) and (7), we get: (8) f (ak) = f(bk) = f (Ck) = a1 for every k > 0

and (ii) is proved. Moreover, it follows that:

(9) f (a) = al, f (y) = f (E) = f (q) = u4. (10) There is n > 3 such that f (d,,) = U3-

If (10) does not hold, then f (dk) & at, for every k > 3, hence f (dk) a a, for every k > 3. This implies that a3 has not preimage in P. which is absurd.

( 11) f (dk,) = a3 implies f (dk+l ) = vas for every k > 3. Suppose f(dk) = a3; since f(hk+l) :& a7, necessarily f(hk+l) = a5 and f(ek+l)

a5, therefore f(dk+I) = 93.

From (10) and (11), by induction on k > n, (iii) follows. Note that we have completely characterized the possible p-morphisms from P onto P,47.

By Lemma 6.3 and by the fact that ak+l <d' dk,+l for every k > 2 andj < k, K satisfies Condition (t) of Proposition 4.2; therefore:


PROPOSITION 6.4. K is a model of NLL.

Finally, we show that P%7 is a V-stable reduction of P.

THEOREM 6.5. NL8 is not strongly w-complete.

PROOF. Let us consider the following map g: P -> P

- g(ak) = g(bk) = g(ck)= g(a) = cl for every k > O. - g(fo) = 92- - g(dk ) = u3 for every k > 3.

- g(gk) = g&() = g(e) = g(i) = 4 for every k > 1. - g(ek) = g(hk) = U5 for every k > 4. - g (ro) = c7 -

Then g is a p-morphism from P onto P7 and, by definition of g, L7 is a V-stable

reduction of P. Since K is a V-separable and V-full model of NLLv and i%7 is not a frame for NL8, by the Strong w)-completeness Criterion we can conclude that NL8 is not strongly w)-complete. H

6.2. The logics NL,,+1+ (m > 9) and NLn+ln+2 (n > 6). To treat these logics, firstly we extend the countermodel K of the previous section into the models KM and Kn +1. At this aim, we consider a sequence of points t1, t2,... defined as follows (see Figure 6).

- t7 coincides with the root ro of P. - Forevery k 7, tk P. - For every k 7 7 and every j > 1, tj is an immediate successor of tk if and only

if aj is an immediate successor of Uk in ,, (defined in Section 4).

tl t2

t3 ,t4

P t5 t6

t7 ~~~~~t8

t9 / tlo

tll t12

FIGURE 6. The sequence of points tk.

The frames P,,M and Pn n+l for m > 9 and n > 6, are defined as follows.

* P,i1 = (P,,1, ?, t,,,) is the frame having as root tM1. * Pn.n+l =Pn.n+l? <, rn) is the frame such that:

- rn is the root of Pn.n+1; - the immediate successors of rn are tn and tn+ .


In order to define the models K,,1 = (Pil, ?, til, HI) and Knn+l= (P,1iix+l ?, r,1, HF), we consider a suitable increasing sequence of finite sets V,.. of propositional variables containing V (used for the logic NL8) such that the following conditions hold.

(a) a0, b0, co, fo and tk, for 1 < k < m, have V,,,-grade 0 in Kil. (b) The distinct equivalence classes with respect to the relation -'" 'in K,, having

more than one element are: - {ak, bk, Ck :k> 1}U{dk :k> 3}U{a} - {ek k > 4} U {} - {gk k > 1} U {Y} - {hk :k >4}U{1 }.

(c) aoboco, Jo, rn and tk, for 1 < k < n + 1, have Vn+I-grade 0 in Kn.1+1. (d) The distinct equivalence classes with respect to the relation '-_-'4-1 in K,2 ,I+

having more than one element are as in (b). We proceed as in previous section. PROPOSITION 6.6.

(i) Kil is Vi-separable and V,,,-fullfor every m > 9. (ii) Kn.n+I is V,+?I-separable and V?+I-fullfor every n > 6.

PROOF. We observe that the cone P of P,^ coincides with P and all the points tk

have Vil-grade 0; thus (i) follows from Proposition 6.2. In a similar way (ii) is also proved. -

LEMMA 6.7.

(i) Let P' be a generated subframe of Pil, with m > 9, and let f be a p-morphism from P' onto P . Then:

- f (ak)= ,l for every k > 0. - There is r > 3 such that f (dk) =3 for every k > r.

(ii) Let P' be a generated subframe ofP,,,n+l, with n > 6, and let f be a p-morphism from P' onto L ,P . Then:

- f (ak)= al for every k > 0. - There is r > 3 such that f (dk) = 3 for every k > r.

PROOF. (i) Suppose that there is a p-morphism f from from some generated subframe P' of PM onto Pa,. Firstly we prove that, for 2 < k < m, if f is defined on tk, then f (tk) < ak. If 2 < k < 6 or k = 8 the proof is immediate. Suppose, by absurd, that f (t7) > a7; if f (t7) > aU, then there must be s > t7 such that f (b) = o7, in contradiction with Lemma 6.3 (i). Suppose that f (t7) = a8 and let s be such that f (s) = a7. It is not the case that s < t7 or s > t8; this implies that 6= a8, but one can easily check that this is not possible; we can conclude that f (t7) < a7. If 9 < k < m, then the immediate successors of tk are tk2 and tk-3: if f is defined on tk, thenf(tk-2) < Uk-2 and f(tk-3) < Uk-3, thereforef(tk) ? Ok

On the other hand there must be a point s of P,,, such that f (s) = a...; it follows that s = t,,, hence, for all 1 < k < m - 2, f (tk) = Uk. In particular, f (t7) = U7

and, by Lemma 6.3, (i) follows. (ii) Observe that f cannot be defined on r,1. Arguing as in (i), we can prove

that f (tk) = Uk for 1 < k < m + 1; hence f (t7) = aU and, by Lemma 6.3, (ii) is proved. -1


It follows that the models KM,7 and K,1,,+l satisfy Conditions (t) and (tt) of Proposition 4.2 respectively, therefore:

PROPOSITION 6.8.

(i) K1?1 is a model of NL,>+ for every m > 9.

(ii) Kln.n+1 is a model of NL, 'K+2 for every n > 6. H

THEOREM 6.9. The logics NL,,1+I and NL,?+ I n+2, for every m > 9 and n > 6, are not strongly co-complete.

PROOF. Let g be the p-morphism from P onto P(c7 defined in Theorem 6.5; we extend g to PM,1 as follows:

- g (tk) = Uk for every 1 < k < m. By definition of g, we can assert that Pa is a V,,1-stable reduction of P ; since K is a V,,,-separable and V,,1-full model of NL1}+ and Pa is not a frame for NL,,1+, by the Strong co-completeness Criterion it follows that NL,7,+1 is not strongly co- complete.

Let P, be the frame obtained by adding to P + a root v,. We extend g to

P1.? ,+1 as follows:

- g(tk) = Uk for every 1 < k < n + 1; - g(rn) = &n.

By definition of g, Pl,,, is a Vn+I -stable reduction of P,.n +1; since K,2 ,i+l is a Vn-+?I- separable and V,,+ I -full model of NL V+ and P%, is not a frame for NL,,+1.1n2, the logic NL,,+1.,2+2 is not strongly co-complete. H

6.3. The logic NL9. It remains to analyze the logic NL9. To treat this case, we define the frame P8 (P8, ,zo) of root zo and the model K8 = (P8, <,?0, O). The

points al, bk, Ck, with k > O. d i, e, with j > 3, a, e, Jo are defined as in Section 6. 1. We introduce new sequences of points 13,14..., i4, m5 ... and n4, n5, v

in the following way.

- The immediate successors of 1Xk are akI_, Ck-2 and fo for every k > 3. - a : ik and i1k for every k > 3 (see Figure 7). - The immediate successors of mk are dk I, -1/ -I and 'k for every k > 4 - The only immediate successor of ,u is A.

- : dl, and y i /k for every k > 3 (see Figure 8). - The immediate successors of nk are ek I and ek for every k > 4. - The only immediate successor of v is 6.

- v % dk and v % 'k for every k > 3 (see Figure 9).

The immediate successors of the root zo are the points Mk and nk, for every k > 4,

,u and v.

The forcing relation is defined in such a way that the following properties hold with respect to some finite set of propositional variables W.

* ao, bo, co, fo, zo have W-grade 0. * The distinct equivalence classes with respect to the relation - having more

than one element are: - {ak, bk, Ck : k > 1} U {dk: k > 3} U {a } - {ei,.:k>3}U{e}

- {fk k > 3} U {W}


ao bo CO fo

a2 b2 C2

a33 X 3 3

a4 b4 CA 4

a

FIGURE 7. The sequence of points mk

13 14 15 fo

a

d3 d~4 md m

FIGURE 8. The sequence of points ik.

- {mk k > 4} U {fu} - {fnk :k >4}U{v}.

The proof goes on according to the lines of the previous sections.

LEMMA 6. 10. For every r > 0 the following holds.

(i) a,, b,., c,., do-, er-, I, r m,, rn, (when defined) have W-grade r. (ii) The distinct equivalence classes with respect to the relation S'HV having more

than one element are: - {akbk, Ckdk k > r+1}Ufa} - {ek k > r+ 1}U{e}

{lk:k > r+ 1}U{2} - {Mk k > r + 1} U {u}


fo fo fo fo

d3j d4 d5 a,

e3 e4 e5 6

114, fl5> n6N v /1 t,

FIGURE 9. The sequences of points ek and nk.

- {fnk: k > r + 1} U {v}. (iii) a, e, , ,u, v have infinite W-grade. H

It follows that: PROPOSITION 6.11. K8 is W-separable and W-full. Now, we study the p-morphisms on P8. LEMMA 6.12. Let P' be a generated subframe of P8 and let f be a p-morphismfrom

P' onto PU8. Then:

(i) f (ak) = GI for every k > 0. (ii) One of the followingfacts (A) and (B) holds.

(A) f (ao) = G3 (B) There is r > 3 such that f (dk) = G3 for every k > r.

PROOF. Let P' be a generated subframe of P8 such that there is a p-morphism f from P' onto P 8; we proceed as in in the proof of Lemma 6.3. We firstly observe that {ao, bo, co, fo} C Dom(f), hence {f (aO),f (bo), f (co), f (fo)} C {GU 2}.

(1) f (ao) = f (bo) = f (co)

Otherwise: - {a2, b2, C2} C Dom(f) and { f (a2), f (b2), f (C2)} C {U4, U6}.

This implies that G5 has not preimage in P', in fact: - if a E {aobocoalblclfo}, f () #0 G5;

- in the remaining cases, either a < a2 or a < b2 or ? < c2, hence f (s) z& G5

It follows that: (2) f (ao) = f (bo) = f (co) = a, and f (fo) =

By (2), we can easily prove that:

(3) f () = 8 if and only if a = z0. Therefore f is defined on all the points of P8.

(4) f (13) =G4-

If f (13) z& a4, necessarily f (13) = G5, hence f(*) = 3 for some a* e {a2, a1, bi, c1 }. Let a > z0 be such that f (s) = a4; since a < fo, it follows that a < *, which is absurd.

(5) f (lk) = G4 implies f (lk+l ) = G4, for every k > 3.

If f (lk) = 4, then either f (mk+l) = G4 or f (mk+l) = G6, therefore f (lk?+1) = G4 By (4) and (5), we can infer that:


(6) f (lk) =4 for every k > 3.

This implies that:

(7) f (ak) = (bk) = f (Ck) = a for every k > 0.

Thus (i) is proved. To prove (ii), suppose that f (a) + a3, we show that Condi- tion (B) holds. We can assert:

(8) f(a))= I, f(6)= fG)= f=(U)= f(v) = 4.

(9) There is r > 3 such that f (e,.)= 55

Otherwise, f (ek) :& a5 for every k > 3, that is f (ek) = a4 for every k > 3. It follows that a5 has not preimage in P8.

(10) f (ek) = a5 implies f (ek+I) = a5, for every k > 3.

In fact, if f (ek) = U5, necessarily f (nk+1) = U5 and f (ek+1) = U5. Therefore:

( 11) There is r > 3 such that f (ek) = a5 for every k > r.

Since f (ek) = a5 implies f (dk) = a3, (B) is proved. H

PROPOSITION 6.13. K8 is a model of NL V.

PROOF. We prove that K8 satisfies Condition (t) of Proposition 4.2. Let f be a p-morphism from some generated subframe P' of P8 onto P,8 and let k > 0. If Condition (A) of Lemma 6.12 holds, then ak+1 and a satisfy (t) of Proposition 4.2. Otherwise, let r > 3 be as in (B) and let m = max(k, r) + 1; we have that a,,, Kw d,71, f (a,?l) = a, and f (d,,)= U3, thus (t) of Proposition 4.2 holds also in this case. H

THEOREM 6.14. NL9 is not strongly co-complete.

PROOF. Let us consider the following map g: P8 -' *

- g(ak) = g(bk) = g(Ck) = g(a) = a, for every k > 0.

- g(fo) = 2-

- g (dk) = a3 for every k > 3. - g(lk) g(e) = g(;) = g(U) = g(v) = G4 for every k > 3. - g(ek) =g(nj) = a5 for every k > 3 and every j > 4. - g(Mk) = a6 for every k > 4. - g(Zo) =a8

Then g iS a p-morphism from P8 onto P,8 and, by definition of g, P8 is a W-stable reduction of P8. Since K8 is a W-separable and W-full model of NLgw and i8 iS

not a frame for NL9, by the Strong co-completeness Criterion we can conclude that NL9 is not strongly co-complete. -1

?7. Conclusion. To sum up, we give the complete classification of the logics in one variable.

* The logics Cl, Jn, NL3.4, NL4,5 are extensively canonical. * The logic NL6(= St) is extensively co-canonical and is not strongly complete. * The logics NL5.6, NL7 (= Ast) and NL6,7 are co-canonical (but not extensively

co-canonical) and are not strongly complete. * The logics NL1n and NLnn+l, for every m > 8 and n > 7, are not strongly

co-complete.


?Appendix A. Non extensive co-canonicity of NL5.6. To complete the picture, we prove that the logic NL5,6 is not extensively co-canonical by exhibiting, for some finite V, a well V-separable model K = (P, <, wo, 1) of NLV'6 based on a a frame P -(P, <, wo) which is not a frame for such a logic. The points ak, bk, Ck, di, for k > 0 and j > 3, are defined as in Section 6. 1; the points uo, wo and Vk, for k > 4, are defined as follows (see Figure 10):

- uo < i ab= ak orb = bk orb = Ck, for somek > 0, orb = ft. - The immediate successors of Vk are Vk+l, dk- I and fo for every k > 4. - The immediate successors of wo are uo and V4.

dS5

fo d4

fo d3

V5

wo

FIGURE 10. The model K of NL5.6.

The forcing relation is defined in such a way that the following properties hold with respect to some finite set V of propositional variables.

* ao, bo, co, Jo, uo, wo have V-grade 0. * The distinct equivalence classes with respect to the relation -' having more

than one element are: - {ak, bk, Ck: k > 1} U {dk: k > 3} - {Vka: k > 4}.

Thus we have:

* a,1, b1, c,1, d,1, v,, (when defined) have V-grade n. * The distinct equivalence classes with respect to the relation A-< having more

than one element are: - {ak,bk, Ck,dk :k >n+l} - {Vk :k > n+l}.

Since all the points of K have finite V-grade, the following fact is immediate.

PROPOSITION A. 1. K is a well V-separable Kripke model. H

Of course, K cannot be V-full, since there are no points of infinite V-grade. In order to obtain a V-full equivalent model, we have to add two points a and v which are the limits of the non definitively constant V-sequences contained in the sets {ak, bkCk, dJ : k > 0, ? > 3} and{ Vk : k > 4} respectively


LEMMA A.2. Let P' be a generated subframe of P and let f be a p-morphism from P' onto f745. Then:

(i) f (ak) a ifor every k > 0; (ii) f (A) =3 for infinitely many k.

PROOF. Let P' be a generated subframe of P such that there is a p-morphism J from P' onto 4 (clearly wo , Dom(f)). As in the proof of Lemma 6.3, we have:

(1) f (ao) = f (bo) = f (co) = al and f (fo)= U2.

(2) f (uO) = 4o.

If f (uO) :& o4, necessarily f (v,,) = a4 for some n > 4. It follows that f (U) = a3 if 6 = dk for some 3 < k < n - 2, hence c5 has not preimage in P', which is absurd. Therefore (2) holds and, as an immediate consequence, we have:

(3) f (ak) = f(bk) = f (Ck) = C for every k > 0 which proves (i). Let n > 4 be such that f (v,1) =a5; then:

(4) f (Vk) = U5 for every k > n.

This allows us to infer that:

(5) For every k > 3 there is mn > k such that f (dni) = a3

which is equivalent to (ii). H

It follows that K satisfies Condition (tt) of Proposition 4.2, hence K is a well V-separable (Proposition A. 1) model of NL"6. On the other hand, since there is at least a p-morphism f defined as in the proof of Lemma A.2, P is not a frame for NL '6. Thus we can conclude that:

THEOREM A.3. The logic NL5,6 is not extensively co-canonical. -

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