Normal Modal Logics Determined by Aligned Clusters

We consider the family of logics from NExt(KTB) which are determined by linear frames with reflexive and symmetric relation of accessibility. The condition of linearity in such frames was first defined in the paper [9]. We prove that the cardinality of the logics under consideration is uncountably infinite.


Introduction
Since the emergence of Kripke semantics, the semantical analysis of propositional modal logics has achieved a great success for modal logics with the transitivity axiom, or at least a weak transitivity axiom. In contrast to these rich harvests, modal logics without weak transitivity axioms seem to remain almost untouched, and a further investigation must be needed in order to open a next door of the study of modal logics.
The Brouwer logic KTB is a normal extension of the minimal normal modal logic K by adding the following axioms: Semantically, it is determined by the class of reflexive and symmetric frames (admitting non-transitivity). Hence, KTB is said to be a non-transitive logic. Adding transitivity gives us the Lewis logic S5. The feature of transitivity (or, at least weak transitivity) for frames is very desirable by modal logicians. Thus, the logics located in the interval S4-S5 are intensively studied. Also, for weak transitive logics there are known some important results mostly connected with Kripke incompleteness (see [6][7][8]11,12]). In contrast to these two families of logics, the family of non-transitive logics has not been thoroughly examined yet.
In this paper we deal with non-transitive logics and continue research initiated in the paper [9]. Actually, we extract from the whole family NExt(KTB) a sub-family of logics determined by frames having linear shape. Our motivation for such a choice has two sources. One is the logic S4.3 := S4 ⊕ (3), where: It is complete with respect to linearly quasi ordered frames (xRy or yRx for any distinct x, y ∈ W ). They are usually presented as chains of clusters. A cluster in a Kripke frame W, R is a maximal subset C ⊆ W such that for all x, y ∈ C xRy. In a reflexive and transitive frame, all clusters turn out to be disjoint. The famous results for S4.3 and its normal extensions are the following (see, for example [1]): This logic is determined by the class of reflexive and symmetric frames forming, either chains of points, or circles of points. It is proved in [2,3] that all logics from NExt(KTBAlt(3)), have also very strong properties. Theorem 1.3. (Byrd and Ullrich [2] and Byrd [3]) Every normal modal logic extending KTBAlt(3) has the finite model property and is finitely axiomatizable (and hence-decidable).
It is easily seen by the above theorem that the cardinality of the class NExt(KTBAlt(3)) is only countably infinite. This means that this is rather a nice subclass of modal logics in NExt(KTB).
It is, here, worth comparing the above result with those of Bull's and Fine's. For modal logics from NExt(S4.3), all clusters are disjoint in a frame for those logics, because of transitivity, and so every frame for them can be uniquely represented as a chain of clusters. However, in connected KTBframes, clusters are not always disjoint. Thus a representation of frames for logics in NExt(KTB) must be a little different. In a reflexive and symmetric Kripke frame, some clusters have non-empty intersection that plays a role of a link between them. In spite of this big difference, it is helpful to consider clusters in frames for logics in NExt(KTB). It has to be emphasized here that (alt 3 ) permits the existence of two-element clusters, at most. There is space here for extending their class of logics to a wider and still a gentle one.
In this paper we will consider a more general condition of linearity in reflexive and symmetric frames. We allow for the existence of n-element clusters for any n ∈ N. The appropriate requirements are defined in [9] and in [10] (see also the next section). Then, the logic determined by such a class of frames is axiomatized as follows: KTB.3 A := KTB ⊕ 3 ⊕ A where: A theorem similar to Theorems 1.1 and 1.3, is also proved for logics above KTB.3 A in [9], (see also [10]).

Theorem 1.4. Every normal modal logic extending KTB.3 A has the finite model property.
We see that all logics from those three families NExt(KTB.3 A), NExt(S4.3) and NExt(KTBAlt (3)) have the f.m.p. Thus, a question about decidability of logics from the first family arises. It depends on the answer of the following problem from [9]: Problem 1. What is the cardinality of the class NExt(KTB.3 A)?
In this paper we will solve this problem.

Preliminaries
In this section we remind the basic definitions from [9]. We apply a frametheoretic approach here.
Definition 2.1. Relation R is called a tolerance if it is reflexive and symmetric.
Note that the two notions cluster and block of tolerance coincide. But we prefer to use the second one since, in our case, clusters sometimes have non-empty intersections. Then we define: Diagram 1. A frame with linearly ordered blocks Definition 2.3. We say that a frame W, R consists of linearly ordered blocks if the following two conditions hold: Below, we give two examples.
and R is symmetric and reflexive, and additionally the following points are related (and only these points): Diagram 1). Then the tolerance has three blocks: They are linearly ordered.
Example 2.5. Suppose W := {x 1 , x 2 , x 3 , x 4 , x 5 }, R is symmetric and reflexive, and additionally the following points are related (and only these points): Diagram 2). Then the tolerance has three blocks: They are not linearly ordered since In this paper we will deal with open frames, only. We briefly recall the definition of a p-morphism between Kripke frames. Definition 2.6. Let F 1 = W 1 , R 1 and F 2 = W 2 , R 2 be Kripke frames. A map f : W 1 → W 2 is a p-morphism from F 1 to F 2 , if it satisfies the following conditions: (p1) f is from W 1 onto W 2 , (p2) for all x, y ∈ W 1 , xR 1 y implies f (x)R 2 f (y), (p3) for each x ∈ W 1 and for each a ∈ W 2 , if f (x)R 2 a then there exists y ∈ W 1 such that xR 1 y and f (y) = a.

The Existence of a Contiuum in NExt(KTB.3 A)
In this section, we show that there exists a continuum of normal modal logics in NExt (KTB.3 A). We utilize an infinite sequence S = {F k } k≥1 of Kripke frames in LOB and the characteristic formulas for such frames, to prove that the sequence {L(F k )} k≥1 of logics of the frames determines For each k ≥ 1, the frame F k := W k , R k is defined as follows: In F k , T k is a tail part, that consists of an undirected chain of k + 1 reflexive points, whereas C k is a three-point-cluster part, and these two parts are connected by a point a k ∈ T k ∩ C k . This point will play a significant role in our proof, and so we call this point a k a neck.
For each F k (k ≥ 1), we define a characteristic formula δ k . Characteristic formulas were first introduced for intuitionistic logic (and Heyting algebras) by V.Jankov [5]; for modal logics they were modified by K. Fine [4]. First of all, we prepare a finite set P k of propositional variables, that correspond to points in W k . That is, we associate p i with a point a i ∈ T k for each i (0 ≤ i ≤ k) and p k+1 for b 1 and p k+2 for b 2 . Then, the diagram Δ k of this frame F k is defined as: Then the characteristic formula δ k for the frame F k is just the conjunction of this diagram, that is, δ k := Δ k . Here we use the formula σ k := k+2 δ k ∧p 0 . The following lemma is crucial for our task.

Proof. (⇐=)
If m = n, we define a valuation V 0 on F n as: V 0 (p i ) := {a i } for 0 ≤ i ≤ m, and V 0 (p m+j ) := {b j } for j = 1, 2. Then it is obvious that σ m is satisfiable at the point a 0 in a model F n , V 0 . (=⇒) Suppose that σ m is satisfiable in F n and m > n. Formula σ m includes the following sub-formulas: p i → ♦p i+1 for i := 0, 1, 2, . . . , m + 1. The range of σ m is the whole frame F n because the frame F n consist of n + 3 points with n < m. Then obviously there is at least one point in F n , at which two distinct variables p i and p j must be true. Since σ m includes also subformulas p i → ¬p j for i = i and i, j := 0, 1, 2, . . . , m + 2 then we see that for any valuation in this case the formula σ m is not satisfiable. Then we get a contradiction. Suppose then that σ m is satisfiable in F n and m < n. One may notice that a m is the only point in F m that is related by R m to three different points except for itself, that is, b 1 , b 2 and a m−1 in F m . Therefore we find that p m is true at nowhere else but at a n in F n . Then, variables p m+1 and p m+2 can be satisfied at b 1 , b 2 in F n . For variables for the tail part in F m , p m−1 must be true at a n−1 , p m−2 must be true at a n−2 , and finally we reach the fact that p 0 must be true at the point a n−m , in the middle of the tail in F n since m < n. Hence, we see that the range of formula σ m is m + 2 in both directions from the point a n−m . Case 1. n − m ≤ m. To match the valuation in the other part of the tail we may choose for the next point a n−m−1 either p 0 or p 1 . It is because in σ m we have the sub-formulas p 0 → ♦p 0 , p 0 → ♦p 1 and p 0 → ¬♦p i , for i = 0, 1. Sub-case 1a. Suppose that we choose p 0 . Since in σ m there are also subformulas p i → ♦p i+1 , for i = 0, 1, . . . , m + 2 then at the next points a k 's with n − m − 2 ≥ k ≥ 0 we set the variables p 1 , p 2 , . . . , p n−m−1 true. At the last point a 0 in F n we valuate variable p n−m−1 . Since in this case the range of σ m is the whole frame F n then at a 0 we should have the formula p n−m−1 → ♦p n−m true. But it is impossible, so we get a contradiction. Sub-case 1b. Suppose we take p 1 and n − m < m. As above at the next points a k 's with n − m − 2 ≥ k ≥ 0 we valuate variables p 2 , p 3 , . . . , p n−m−1 . At the last point a 0 in F n we valuate variable p n−m . Again, the range of σ m is the whole frame F n . Then at a 0 we should have true the formula p n−m → ♦p n−m+1 . But it is impossible, so we get a contradiction. Sub-case 1c. Suppose we take p 1 and n − m = m. Again at the points a k 's with n − m − 2 ≥ k ≥ 0 we valuate variables p 2 , p 3 , . . . , p n−m−1 . At the last point a 0 in F n we valuate variable p n−m . But then p n−m = p m . We may notice that in σ m we have sub-formulas p m → ♦p m+1 and p m → ♦p m+2 . But at a 0 it is impossible to valuate that formulas and we get a contradiction. This axiom characterizes the class of frames in which every point can see at most n points. Here, let us take a closer look at our construction of frames in Lemma 3.1. Then it is not very hard to see that for any n, every point in the frame F n can see at most four points. Indeed, only the neck of each frame can see four points including itself. This means that all members of the class {F n } n≥1 are frames for KTBAlt(4) := KTB ⊕ alt 4 . Hence we have proved the following stronger fact. The above theorem, of course, implies that the cardinality of the class NExt(KTBAlt(4)) is uncountably infinite, which shows us a sharp boundary located between KTBAlt(3) and KTBAlt(4). In this sense, the logic KTBAlt(3) sits on a special position in the lattice NExt(KTB).
The lattice of NExt(KTB.3 A) is so intriguing that it requires further investigations. Our future work will concern the following problems: 1. Existence of splitting logics, 2. Local finiteness,

Algebraic counterpart of Kripke frames for KTB.3 A.
It would be also very interesting to generalize the axiom (3 ) together with (A) (analogously like from alt 3 to alt n ) and obtain a syntactical characterization of reflexive and symmetric frames in which each cluster is in accessibility relation with a bounded number of other clusters. Then we could investigate logics determined by frames in a shape of net.