Modal Information Logics: Axiomatizations and Decidability

The present paper studies formal properties of so-called modal information logics (MILs)—modal logics first proposed in (van Benthem 1996) as a way of using possible-worlds semantics to model a theory of information. They do so by extending the language of propositional logic with a binary modality defined in terms of being the supremum of two states. First proposed in 1996, MILs have been around for some time, yet not much is known: (van Benthem 2017, 2019) pose two central open problems, namely (1) axiomatizing the two basic MILs of suprema on preorders and posets, respectively, and (2) proving (un)decidability. The main results of the first part of this paper are solving these two problems: (1) by providing an axiomatization [with a completeness proof entailing the two logics to be the same], and (2) by proving decidability. In the proof of the latter, an emphasis is put on the method applied as a heuristic for proving decidability ‘via completeness’ for semantically introduced logics; the logics lack the FMP w.r.t. their classes of definition, but not w.r.t. a generalized class. These results are build upon to axiomatize and prove decidable the MILs attained by endowing the language with an ‘informational implication’—in doing so a link is also made to the work of (Buszkowski 2021) on the Lambek Calculus.


Introduction
This introduction is divided into two parts.First, we give a more general introduction, forwarding the logics of concern and motivating their study.Second, we break down the paper section-by-section, outlining the mathematical issues at hand and how they are solved, ending with a list of the main results achieved.

Motivation and General Introduction
Aiming to model a theory of information by using the possible-worlds semantics of modal logic, [15] introduces a modal logic of a single binary modality ' sup ' with semantics: x sup iff there exist y z such that y ; z ; and x sup y z This is motivated by construing the 'worlds' as information states; the relation as an ordering of the information states; and the supremum modality ' sup ' as providing language for speaking of 'merge' (or 'fusion') of information states.In accordance with this interpretation, modal logics with such a modality are called modal information logics (MILs).
The main focus of this paper is to study formal properties of MILs, primarily by providing axiomatizations and proving decidability results.Notably, this study also includes solutions to open axiomatization and decidability problems posed in [16-18].However, it is worth emphasizing that there is a non-technical reason for studying these logics: 1 By describing basic patterns of information in-and decrease in qualitative yet logically precise manners, MILs are (or can be viewed as) attempting to solve a major problem in the foundations of information, namely that of unifying theories of information: ranging from 'quantitative' theories (such as Fisher information, Shannon entropy, and Kolmogorov complexity) to 'qualitative' ones (more akin to our everyday usage of the word 'information'), cf.[1].
Looking at modal information logics in this light, the paper is, foremost, concerned with the following two questions: Axiomatization: What are, according to a MIL, the fundamental principles governing information?Decidability: Is there an algorithm that given any principle can tell whether it is a valid principle of information?
Now, for these questions to be well-defined, we must get clear on a principal way in which MILs can differ, namely in their notion of 'fusion': on what class of structures do we want to interpret the ' sup '-modality -what is our choice of frames?Rather general are preorders where the modality ' sup ' is defined in terms of quasi-least upper bounds; i.e., 'merges' are not unique but come in clusters.This defines the basic modal information logic, denoted MIL Pre .The informational interpretation further suggests examining the case where the relation is also anti-symmetric (resulting in posets). 2 We denote the corresponding logic as MIL Pos .
After solving the problems of axiomatization and decidability for MIL Pre and MIL Pos , we show that our techniques for doing so extend to the MILs, MIL \-Pre and MIL \-Pos , obtained by enlarging the language with the modality '\' with semantics y \ iff for all x z, if z and x sup y z , then x This extension was suggested in [18], and is motivated under the informational interpretation as an 'informational implication': an information state y 'satisfies' \ iff for all information states z and all merges x sup y z of information states y z, if z satisfies (the antecedent), then the merge x satisfies (the consequent).
It should be noted that connectives with this kind of semantics feature prominently in several logics: in fact, our informational interpretation is that of the relevance logic of [13, 14] where '\' is relevant implication; and the symbol '\' is the (left) residual of the Lambek Calculus [12] -a logic we will make a junction with.Moreover, '\' compliments ' sup ' very naturally: if, say, x sup y z , then '\' accesses this from the perspective of y (or z) while ' sup ' accesses it from the perspective of x.It is thus not surprising that the 'intensional conjunction' of [13, 14] and the 'product connective' of [12] are analogues of ' sup '.
As observed in [15-18], a final interesting aspect of MIL Pre and MIL Pos we want to mention is that, using ' sup ', the past-looking modality 'P' becomes definable, so by being modal logics of preorders and posets, they mildly extend S4.Moreover, using '\', the future-looking modality 'F' becomes definable as well.Put in this light, MIL Pre and MIL Pos (and MIL \-Pre and MIL \-Pos , respectively) are quite natural extensions of (temporal) S4 obtained by adding vocabulary for describing further structure of preorders and posets.Thus, seen from a purely mathematical angle, these MILs can be motivated by an interest in seeking a modal perspective on rather ubiquitous mathematical structures, namely preorders and posets.

Guide to Sections
Zooming in, in the order as they occur in this paper, we explain the mathematical problems we will be addressing.Starting off, we examine MIL Pre and MIL Pos , motivated by two central open problems posed by [16-18], namely (1) axiomatizing the logics and ( 2) proving (un)decidability.The first three sections of this paper are concerned with these two problems.
In Section 1, after having formally defined the logics, we, in particular, show that MIL Pre lacks the finite model property (FMP) w.r.t.preorders.This proof extends to all above mentioned MILs on their respective classes of frames as well.Although this can be taken as a (clear) indication of undecidability, we end the section by explaining why this need not be, forwarding a method for proving decidability 'via completeness' when dealing with semantically introduced logics (like MILs). 3n Section 2, we provide an axiomatization of MIL Pre and prove it to be sound and strongly complete.We do so by, given a consistent set, constructing a model for it.As the constructed models are, in fact, posets, we get as a corollary that MIL Pre MIL Pos ; thus, solving problem (1) for both logics in one go.
Following the method laid out in Section 1, in Section 3, we, first, use this axiomatization to find another class of structures for which the logic also is complete.Second, we show that on this class of structures we do, in fact, have the FMP-allowing us to deduce decidability.
Next, in Section 4, we explore the conservative extensions MIL \-Pre and MIL \-Pos obtained by adding the informational implication '\'.Combining ideas from our study of MIL Pre MIL Pos with some new ones-among which some are ours and some, more interestingly, are due to work on the Lambek Calculus of [6]-we (i) axiomatize the logics, (ii) show that MIL \-Pre MIL \-Pos , and (iii) prove them to be decidable.This crossing with the Lambek Calculus sheds one more illuminating light on modal information logics: MIL \-Pre MIL \-Pos is the Lambek Calculus (augmented with classical propositional logic) of suprema on preorders (or posets).
In summary, the main results achieved are: Axiomatizing MIL Pre and deducing MIL Pre MIL Pos .
Proving MIL Pre decidable.

Preliminaries
We start off this section by formally defining the basic modal information logics (Subsection 1.1).Then, in Subsection 1.2, we, first, show lacks of properties related to that of decidability, most notably proving that all of the logics of concern lack the finite model property w.r.t.their respective classes of definition; and then, second, sketch a general method for proving decidability in cases like ours -a method which we will employ in Sections 2 and 3.

Defining the Logics
Definition 1.1 (Language).The basic language M of modal information logic is defined using a countable set of proposition letters P and a binary modality sup .The formulas M are then given by the following BNF-grammar where p P and is the falsum constant.
Modal information logics are defined by semantical means; i.e., as sets of Mvalidities on classes of structures.The most general class of interest is that of preorders; formally, we define as follows: Definition 1.2 (Frames and models).A (Kripke) preorder frame for M is a pair W where W is a set; and is a preorder on W , i.e., reflexive and transitive.
A (Kripke) preorder model for M is a triple W V where W is a preorder frame; and V is a valuation on W , i.e., a function V P W .
For clarity, before defining the next class of structures we will be considering, we set out the basic modal information logic of preorders in full detail.Having defined the structures in which to interpret the M -formulas, we are about to define the actual semantics.In order to do so, we provide the following definition generalizing the notion of supremum from partial orders to preorders: Definition 1.3 (Supremum).Given a preorder frame W and worlds u W , we say that is a quasi-supremum (or simply supremum) of u and write sup u iff is an upper bound of u , i.e., u and ; and x for all upper bounds x of u .
In general, sup u denotes the set of quasi-suprema of u , and if this happens to be a singleton , we may write sup u .4 Definition 1.4 (Semantics).Given a preorder model W V and a world W , satisfaction of a formula M at in (written ' ' or ' ' for short) is defined using the following recursive clauses on : Notions like global truth, validity, etc. are defined as usual in possible-worlds semantics (see, e.g., [4, ch.1]).
With these notions laid out, we can define the logic as follows: Definition 1. 5 The basic modal information logic of suprema on preorders is denoted by MIL Pre , and defined as the set of M -validities on the class of all preorder frames; that is, for all preorder frames W Analogously, we denote by MIL Pos the basic modal information logic of suprema on poset frames, i.e., frames W where ' ' is a partial order (viz.an antisymmetric preorder).

Road to Decidability
Having formally set out these logics and semantics, we continue with some preliminary remarks.Objective being to get a feel for how the semantics works by stating a few minor results, and, most notably, showing that the logics lack the FMP w.r.t.their respective frames of definition; viz., for instance, MIL Pre does not have the FMP w.r.t.preorder frames.Foremost, we mention how to express the past-looking modality.

Remark 1.6
Besides the connectives ' ', ' ', ' ', ' sup ', and ' ' being definable in the standard way, the past-looking unary modality 'P' is definable as This can be seen by recalling the definition P :iff there exists such that , and observing that also sup iff there exists such that . 5ing this observation, the first contribution of our paper is to show a lack of the FMP.

Proposition 1.7 MIL Pre does not have the FMP w.r.t. preorder frames.
Proof Consider the formula N H P sup pp H P sup pp where H P is the dual of P. We claim that N only is satisfiable in infinite models. 6irst, we show that N is, indeed, satisfiable on an infinite model.Accordingly, let V where is the set of negative integers; is the less than or equal to relation on the negative integers; and V p is the set of even negative integers.
Then , clearly, is a preorder model, and for all z : Thus, for all z : z P sup pp P sup pp But then N must be globally true in ; in particular, N is satisfied in , proving the first part of the claim Second, to see that N isn't satisfiable in any finite model, observe that for any preorder, if two points are situated in the same cluster, 7 then they are suprema of the exact same (sets of) points.It follows that for any preorder model, points in the same cluster satisfy the exact same ' sup '-formulas (those are: formulas with ' sup ' as main connective).
With this in mind, it is easy to see that the satisfaction of N necessitates the existence of an infinite, strictly descending chain: if some N and some i satisfies, say, sup pp, then, in particular, there must be some i 1 i s.t.i 1 sup pp, whence i 1 must be in a cluster strictly below i .Thus, N cannot be satisfied in any finite model.
It is worth noting how the proof made essential use of the additional expressive power of our language compared to that of S4.S4 famously enjoys the FMP w.r.t.preorders-its language is, so to speak, too weak to distinguish clusters from chains.

Remark 1.8
The above proof applies to the class of posets as well since the frame was, in fact, a poset, hence neither does MIL Pos enjoy the FMP w.r.t.its class of definition.
Beyond not having the FMP, there are even more indicators of undecidability.For the purpose of this paper, these are not central, so we mention them without elaborate proof.
Remark 1.9 MIL Pre does not have the tree model property (TMP) w.r.t.preorder frames.That is, there is a formula N which is satisfiable in a preorder frame, but not in a preorder frame W where W is a reflexive and transitive tree. 7For clarity, recall that given a preorder , are said to be in the same cluster :iff . 8Consult, e.g., [4, ch. 1, def.1.7] for the definition of a tree and, in particular, a reflexive and transitive one.Additionally, note how we define the TMP in terms of the converse relation ' '; this is motivated by the way in which ' sup ' is backward-looking.Otherwise, for the case of ' ', a formula like ' p sup q p q p ' already shows the lack of 'a TMP'.

Proof
The following formula is satisfiable but not in a (converse) tree N p q sup p q p q H p q P p q H p q P p q To see that it is satisfiable, consider the four-element Boolean algebra a b with valuation V s.t. a b p q, a p q, b p q, p q. Then a b N .To see that it is not satisfiable in a (converse) tree, suppose that some W x N .Since trees, in particular, are antisymmetric, we may assume that ' ' is a partial order. 9Then x p q and there are y z s.t.x sup y z , y p q and z p q. I.e., x y z 3, so y z y.Further, by the second line of N , there must be y z s.t.y y p q and z z p q. Now if the partial order were to be a tree, we would have that x sup y z , but then by the third line of N , we would have that x p q, which would be a contradiction.
Remark 1.10 Witnessed by the same proof, not having the TMP extends to MIL Pos as well.
Observation 1.11 Our modal information logics are neither guarded nor packed (as, e.g., the guarded and packed fragments do enjoy the FMP).
At first glance, the results of this subsection might make decidability appear unlikely.But, as it turns out, there is an alternative way of proving decidability, circumventing these problems.We end this section by laying out our method for doing so.This will serve two purposes: by describing the method, we hope to (i), generally, elucidate how and when our method can work as a heuristic for proving decidability, and (ii), specifically, help the reader get a better grasp of the underlying ideas and structure of the ensuing two sections of this paper.
We (1) axiomatize the logics (and show that MIL Pre MIL Pos ), (2) use this to show the logic(s) to be complete with respect to another class of structures (where the ternary relation of sup won't necessarily be the supremum relation of a preorder, but something more general), and then (3) prove that the logic(s) enjoy the FMP on this other class of structures, from which we can deduce decidability.So to make the salient point clear: when dealing with logics introduced by a semantic definition, not having (e.g.) the FMP w.r.t. the class of definition might not be very telling.The reason being that the resulting logic can very well be complete w.r.t. to another, bigger class of structures for which it does have the FMP. 9 If we also allowed for general preorders ' ' only satisfying that their skeletal partial orders ' ' are converse trees, then the proof of no TMP would go through by changing the conjunct ' sup p q p q ' in N to the more complicated ' sup p q P p q p q P p q '.

Axiomatizing MIL Pre
While [18] obtains an axiomatization of a variant of MIL Pre extended with nominals and the global modality, the very same paper also inquires finding an axiomatization without hybrid extensions.In this section, we answer this inquiry, providing a purely modal axiomatization.In Subsection 2.1, we give a proof-theoretic description of MIL Pre , prove it to be sound, and lay some groundwork for the completeness proof of Subsection 2.2, which also allows us to conclude that MIL Pre MIL Pos .

Soundness and Preparatory Lemmas
We begin by syntactically defining a logic, suggestively called MIL Pre . 10Through a soundness and completeness proof, we then show MIL Pre exactly is an axiomatization of our semantically defined logic MIL Pre .
Definition 2.1 (Axiomatization).We define MIL Pre to be the least normal modal logic (NML) 11 in the language of M containing the following axioms: Proof Standard, tedious task checking that MIL Pre is a normal modal logic and that (Re.), ( 4), (Co.), and (Dk.) all are valid on preorder frames.
As oftentimes is the case, while proving soundness is straightforward, proving completeness is much more intricate.Our proof will be a construction using maximal consistent sets (MCSs) for which some preparatory observations and lemmas are needed.
First hurdle is that the sup -modality is in a general sense a 'logical modality': although accompanied by a ternary relation (namely the supremum relation) its interpretation is fixed given a binary relation (namely a preorder).For starters, this means that the standard construction of the canonical frame for MIL Pre won't come equipped 10 As a convention, we boldface when having 'syntactic' presentations of logics in mind and italicize when having 'semantic' presentations of logics in mind. 11Caution: As a reviewer has brought to my attention, the definition of a normal modal logic in a language with polyadic modalities given in [4, def.4.13] is wrong.The necessitation rules are too weak: each occurring ' ' should be replaced by arbitrary formulae i . 12(Re.) is short for 'Reflexivity'; (4) is the transitivity axiom; (Co.) is short for 'Commutativity'; and (Dk.) is short for 'Don't know what to call this axiom'.with a binary relation for interpreting the binary modality sup -as is the case for the preorder frames of MIL Pre -but with a ternary one.Fortunately, defining an underlying preorder from this ternary relation spells no trouble.This is summarized in the definition below.

Definition 2.3
We denote the set of all maximal consistent MIL Pre -sets by W Pre , and the ternary relation of the canonical MIL Pre -frame by C Pre . 13That is C Pre holds just in case sup From C Pre , we define the following binary relation on the canonical frame: We want to show that Pre actually is a preorder.To do so, we begin by making two observations.Observation 2.4 Since MIL Pre is an NML, we have all the usual lemmas regarding its canonical model.

Observation 2.5 The formula (T) p Pp
is derivable in MIL Pre .
Proof Some straightforward syntactical manipulations prove the claim; the key steps being (Re.) (T): uniformly substitute q for in (Re.); and (T) (Re.): use (T) to get p q p Pq and then use (Dk.).
Using these observations, in the ensuing lemma, we prove that not only is Pre a preorder, but more 'supremum-like' properties hold of the canonical relation C Pre .

Lemma 2.6
The following hold: Proof Since (Re.), ( 4), (Co.), (Dk.) all are Sahlqvist, one can prove all but (b) ii via the Sahlqvist-van Benthem algorithm (cf.next section's Lemma 3.1).As often is the case, though, a direct argument is faster; we provide such here.
(a) Let M be arbitrary.Then -by (Co.), uniform substitution (US) of MIL Pre , and closure under modus ponens (MP) of MCSs -we have sup sup which suffices to prove the claim.
(b) Right-to-left of i is immediate (using (a)).For left-to-right, suppose that C Pre for some W Pre and that .Since , we have that sup , hence sup and so we get by (Dk.) (and US and MP of MCSs) that sup -as suffices.Regarding ii , left-to-right follows by (a), while right-to-left is proven using (Dk.(d) Consequence of (a).

Completeness: Constructing our Model
Given the previous subsection's results -indicating that the canonical frame is well behaved -one might start wondering whether the canonical relation C Pre is, in fact, the supremum relation on Pre .If so, we would have completeness in our pocket.Unfortunately, this is far from being the case: not only is the canonical relation C Pre not the supremum relation on Pre , it is utterly wild. 14his forces us to make a rather complicated construction where we do not work with the canonical model per se.Instead, we construct our frame by recursively repairing so-called 'defects' and 'labeling' points of a subset of our frame with MCSs for which we prove a truth lemma.This somewhat generalized approach is useful since it (a) allows for reuse of the same MCS -i.e., different points of the frame might get labeled with the same MCS -and (b) utilizes that, in the extreme, we only need a truth lemma for one MCS, namely the one extending a given consistent set; thus, we may and will include (non-labeled) points in our construction only to ensure that other (labeled) points satisfy formulas dictated by their MCS-label.That is, we do not care what formulas these points satisfy themselves-their role is entirely auxiliary. 15o be more concrete, when recursively constructing this frame, we make sure that at each stage, its corresponding 'approximating frame' is determined by a triple l D satisfying the definition (of ) below.Specifically, in the recursive step from, say, n to n 1, we will make sure that if l n n D n then also l n 1 n 1 D n 1 . 16his is needed for the colimit construction -i.e., the structure obtained after all finite stages in the recursive construction -to be of the right form.Definition 2.7 Let W be countable set, and the set of all triples l D such that 1. l is a partial function from W to the set of all MCSs, W Pre .
2. dom l 0 , where 'dom l ' refers to the domain of l.

6
. is a partial order on dom l D, and the diagonal on W dom l D .
7. If y x then l y Pre l x (whenever x y dom l ). 17 mentioned, the recursion is carried out by repeatedly repairing 'defects'.Since our goal will be to prove a truth lemma for labeled points, any defect is, in essence, either (1) that a point x's MCS-label dictates that x satisfies some formula sup which it doesn't, i.e., sup l x but x sup ; or (2) that a point x's MCS-label dictates that x satisfies some formula sup which it doesn't, i.e., sup l x but x sup .
Although this captures the gist of what defects are, as it turns out, for the proof to work, the precise definitions must be more detailed than this.We proceed giving these., then also l n 1 n 1 D n 1 ), we give an example to convey intuition for the repairs and the general construction.
Example 2.10 Suppose l D and sup 0 0 x constitutes a sup -defect; that is, (i) x dom l , (ii) sup 0 0 l x , and there are no y z fulfilling (iii).Put crudely, the problem is that x's label l x requires x to satisfy sup 0 0 , but x is not the supremum of any y z s.t.0 l y 0 l z .To solve this, we simply add two fresh points y z immediately below x.Then using the existence lemma of the canonical model for the case sup 0 0 l x , we get two MCSs y z s.t.C Pre l x y z . 20Setting l y y and l z z , the defect has been repaired.The idea is illustrated in the top left corner of the figure below.
Further, if, say, sup 1 1 x also constitutes a sup -defect, we simply repeat the process as illustrated in the top right corner of the figure below.
While these two repairs did solve the problems they intended to, they might have created new ones.If, say, sup l x l z and l z , in solving these problems they have made sup x z z constitute a sup -defect.This is where we need the 'dummies': to repair this defect, we add a quasi-blind point d as an incomparable upper bound of z z so that x no more is the supremum of z z (cf. the bottom part of the figure). 21Since d is quasi-blind-and stays quasi-blind (viz.condition 5.)-whatever formula it satisfies is of absolutely no influence to the rest of the points: they cannot 'access' d.So, at bottom, adding dummies is a technique for altering the supremum relation without having to give second thought to what formulas the added points (the dummies) are to satisfy: they are entirely auxiliary (and, hence, do not get labeled, cf.condition 4.).And, most importantly, the alteration of a supremum relation caused by adding a dummy is sufficiently local to not mess up previously repaired defects; in this simplest of cases, we still have x sup y z sup y z after having added the dummy d.Proof Define as in the lemma by taking fresh y z and mapping them to obtained via the existence lemma for sup l x .Then the last claim is easily checked to be satisfied, and l D also clearly satisfies Proof Suppose 0 is consistent.It suffices to show that 0 is satisfiable.As previously mentioned, to show so, we will construct a model satisfying a truth lemma for labeled points by taking the colimit of a sequence of models getting ever closer to satisfying this truth lemma.We begin by extending 0 to a maximal consistent set 0 , letting 0 be the diagonal on some (any) countable set W , and setting D 0 and l 0 x 0 for some x 0 W . Then 1.-7.are satisfied, where 7. follows by reflexivity of Pre .We continue by constructing a sequence using the repair lemmas repeatedly.We do so by enumerating the set of all pairs sup x and all quadruples sup x y z . 23Then at each stage n 1 we pick the least tuple constituting a defect to l n n D n , which we repair obtaining we get that (1) l D satisfies 4.-7.; (2) l is a (partial) function from W to the set of all MCSs; and (3) l D neither has any sup -nor sup -defects.Only (3) isn't straightforward.To show this, we prove two claims, and in order to do so, we need the following observation.
Observation.Let n and x dom l n be arbitrary s.t.This is easily seen by induction, using that each l m 1 m 1 D m 1 is obtained from l m m D m using either of the repair lemmas.

Claim (a). Suppose sup
x does not constitute a defect for some l n n D n at which (i) x dom l n and (ii) sup l n x .Then this must be witnessed by some y z (cf.Definition 2.8).We show that for all m n: sup x does not constitute a defect for l m m D m , witnessed by y z.
A fortiori, neither does it for l D .
By the observation and noting that l i l i 1 for all i , it suffices to show that for all m n: x sup m y z where sup m y z sup m y z is the least upper bound of y z w.r.t. the relation m .We prove this by induction on m n.By assumption, this holds for m n.Accordingly, suppose it holds for an arbitrary m n.We show it holds for m 1.We have two cases, depending on the type of defect being repaired at stage m 1.
First, suppose the defect repaired was a sup -defect for some world s.Since the corresponding introduced dom l m 1 -worlds y s z s have no proper m 1 -predecessors, the claim follows.Reason being that, cf. the IH and the definition   Using (a) and (b), it is straightforward to see (c): If some tuple did constitute a defect at some stage n, but no longer at some later stage m n, then it didn't for all k m.
With these claims at hand, we can show (3) that l D neither has supnor sup -defects.For sup , let sup x i be an arbitrary pair in our enumeration s.t.x dom l and sup l x .Then x dom l n for some n , hence x dom l m sup l m x for all m n.If, on one hand, sup x i didn't constitute a defect to l n n D n -using claim (a) (and the observation) -we get that it wouldn't for l D either.On the other, in case it did, it would no more no later than at stage n i 1 (cf.(c)), and henceforth -by claim (c) -remain repaired.Thus, l D has no sup -defects.For sup , suppose towards contradiction that sup x y z i denotes a sup -defect.Then x y and x z, so there is some n s.t.
x n y z

If
x sup m y z for some m n, there must be some a dom l m D m s.t.
which, in particular, shows x sup y z -contradicting sup x y z i being a sup -defect.Thus, we must have x sup m y z for all m n, implying -and simultaneously contradicting -that the defect will be repaired no later than at stage n i 1 (cf.(c)).That is, there can be no sup -defects either.
Finally, setting

V p
x dom l p l x we show our truth lemma, namely that for all x dom l and all M : The proof goes by induction on the complexity of formulas.Base case is by definition and Boolean cases are straightforward.For the sup -case, we get where we in the left-to-right direction of (IH) use -apart from the induction hypothesis itself -that l D satisfies 5.-6.; i.e., in particular, neither of the witnessing y z are dummies nor in W dom l D , and so they must be in dom l .Further, left-to-right of i holds by there being no sup -defects, while right-to-left follows from there being no sup -defects.
This completes the induction, from which it follows that showing that 0 is satisfiable in a preorder model and, thus, at long last, finalizing our proof of completeness.
Proof Clearly, MIL Pre MIL Pos , and the other inclusion follows from the frame constructed in the completeness proof being a partial order.

Decidability of MIL Pre
This section consists of two parts.In Subsection 3.1, we show that MIL Pre is complete w.r.t.another class of structures .Then, in Subsection 3.2, we show that MIL Pre has the FMP w.r.t.-frames and conclude that MIL Pre (and MIL Pos ) are, after all, decidable-solving a problem posed in [16-18].

Reinterpreting sup on Generalized Structures C
Following the method laid out in Subsection 1.2, and with an axiomatization of MIL Pre at hand, we continue our road to decidability by proving completeness relative to a different class of structures.These structures will be named -frames, alluding to our denoting this class of structures as .
Before we get that far, though, the first key observation to make is that there is nothing in the syntactic definition of MIL Pre implying that the binary modality-symbol sup need be interpreted in terms of the supremum relation on a preorder.I.e., there is nothing a priori hindering us from reinterpreting MIL Pre through reinterpreting the symbol sup .
Further, MIL Pre being an NML means that there might be a canonical reinterpretation, namely the one reached through frame correspondence of MIL Pre on the class of all pairs W C where W is a set and C is an arbitrary ternary relation on W . And, indeed, that is how we proceed.

Lemma 3.1 Let W C be a frame for the modal language with a single binary modality. Then we have the following frame correspondences
Proof Standard frame correspondence proofs work, using arguments similar to the ones in the proof of Lemma 2.6(a), (b)(i), (c) and (d).Alternatively, the Sahlqvist-van Benthem algorithm also applies because the formulas are Sahlqvist.
While (Re.f), (Co.f), and (Dk.f) all match neatly with (Re.), (Co.), and (Dk.), respectively, (4f') is a slightly less elegant FO-correspondent of (4).However, as the following proposition shows, in the presence of the other axioms, the correspondence crystallizes.This corollary proven, we have arrived at the final step described in Subsection 1.2: showing the FMP of MIL Pre when reinterpreted on .Before proving this in the next subsection, we find it instructive to revisit the formula N from Proposition 1.7 and show that, although not satisfiable on a finite preorder frame, it is satisfiable on a finite -frame.We do this right after observing the following: Observation 3.5 It is not hard to prove that for any W C x W , valuation V on W C and formula , we have that

Proposition 3.3 Let W C be a frame for the modal language with a single binary modality. Then W C MIL
and hence also

The Finite Model Property
As promised, we go on proving that MIL Pre enjoys the FMP w.r.t. then use this to deduce decidability of MIL Pre .The proof of the FMP is done by employing a filtration-style argument.To this end, we define a notion extending the standard notion of a set of formulas being subformula closed.Definition 3. 7 We say that a set of M -formulas is -closed :iff (Sub) it is subformula closed; (Com) sup implies sup ; and (S-P) sup implies P .
Moreover, for any set of formulas 0 , we say that is the -closure of 0 :iff it is the least -closed set of formulas extending 0 .26 An immediate consequence of the definition is the following lemma: Lemma 3.8 Suppose 0 is a finite set of M -formulas.Then its -closure 0 is finite as well.
Less immediate is how to use this notion for a filtration-style argument of the FMP.This is the content of the following theorem, whose proof contains the actual definition of a filtration through a -closed set of formulas.where x x denotes the equivalence class on the set of worlds W defined as satisfying the same -formulas as x.

For all
x W : where V p x W x V p for all p .
We begin by proving Alternatively, below we show that this filtration indeed satisfies the homomorphic filtration condition: Cxyz C C x y z .From this and surjectivity of x x , Re f follows.On this node, it is worth (foot)noting that the culprit in hindering this inheritance argument for the three other FO-conditions are the implications in their respective definitions; e.g., for (Dk.f) we have C u C C u C , so when this implication holds by virtue of the first disjunct, namely ' C u', we cannot likewise conclude C C u .This also explains that the filtration relation and the set of formulas we are filtering through have been defined to accommodate these three axioms.As for the transitivity axiom, we have drawn inspiration from the Lemmon filtration.

MIL with Informational Implication
With MIL Pre MIL Pos axiomatized and proven decidable, this section investigates their enrichments, MIL \-Pre and MIL \-Pos , with the 'informational implication' '\'.The main goals are to provide an axiomatization and a decidability proof.
In Subsection 4.1, we formally set out the logics of concern and briefly comment on the increased expressibility.In Subsection 4.2, we, first, put forward an axiomatization and point out on an interesting junction with the Lambek Calculus.Before, second, pausing our investigation of MIL \-Pre and MIL \-Pos per se, to show that the proposed axiomatization is sound and complete w.r.t. the class .Using this result, in Subsection 4.3, we obtain soundness and completeness w.r.t.our poset frames through combining two representation results: the first achieved via an adaptation of 'bulldozing', and the second via supplementing the framework of Subsection 2.2 with an additional defect.We deduce that MIL \-Pre MIL \-Pos .Lastly, in Subsection 4.4, we modify the filtration technique of Subsection 3.2 to attain decidability of MIL \-Pre .

Augmenting with '\'
As noted, we seek to study the enrichment of the basic modal information logic(s), MIL Pre and MIL Pos , given by adding an informational implication as a binary modality.In this subsection we cover some preliminaries; specifically, some definitions followed by a few comments on expressivity.We start with supplying the following pertinent definitions: Definition 4.1 (Language).The language \-M is given by extending the basic guage of modal information logic M with a binary modality symbol '\'.
As a convention we use infix notation for '\' instead of prefix/Polish notation; that is, we write ' \ ', rather than '\ ' (as we, e.g., would do with ' sup ' and ' sup '). .We denote the modal information logic on preorders in the enriched language of \-M as MIL \-Pre , which -to be explicit -is defined as W for all preorder frames W MIL \-Pos is defined analogously.

Remark 4.4
As a minor interlude, as mentioned in the introduction, the choice of symbol '\' concurs with standard notation in the Lambek Calculus.With the semantics given, the reason becomes evident: the interpretation is the same (given a supremum relation).It is also worth pointing out that the commutativity of suprema implies that the other Lambek residual -typically denoted by ' ' -collapses into '\' in the sense that \ .Lastly, the modality ' sup ' is interpreted (again, given a supremum relation) exactly as the binary product ' ' is in the Lambek Calculus.In the next subsection, we expound this connection even further.Now, recall that the primary results we are after are (1) axiomatizing MIL \-Pre and MIL \-Pos and ( 2) showing them to be decidable.Once more, we will be following the heuristic laid out in Subsection 1.2; however, this time our completeness theorem will not be proven via model constructions but via representation results.For this to work, we, needless to say, must (a) have another class of structures for which we can prove the representation results, and (b) also already have the logic of this other class axiomatized.Regarding (a), a natural candidate arises: the -frames of the previous section.Before being able to (b) axiomatize the logic of this class (as we will in the next subsection), we must clarify how '\' is to be interpreted on -models.This is the content of the following definition:

Definition 4.5 Given a frame W C
, a valuation V on W C , a world W and a formula \ \-M with main connective '\', we let To be precise, we explicate how this generalizes our definition on preorder frames.Definition 4.6 Let Pre (resp.Pos ) be the class of pairs W S where W is a set and S W 3 is a ternary relation for which there is some preorder (resp.partial order) on W s.t. for all u W : Then the semantics of '\' on a preorder model W V comes down to where S is the supremum relation induced by .
As the last definition of this subsection, we set forth the logic of -frames in this extended language: Definition 4. 7 We write Log \ for the logic of -frames in the language \-M ; i.e., Log \ denotes the set of \-M -validities on -frames.
With these definitions out of the way, we finish up this subsection with the promised comments on expressivity.First off, we show that with the additional vocabulary provided, we are not only able to express the past-looking unary modality 'P', but also the future-looking 'F'.

Remark 4.8
The future-looking unary modality 'F' (i.e., the standard ' ') is definable as This can be seen by recalling the definition and observing that also Finally, for good measure, observe that '\' is not expressible in our simpler language M .To see this, take, e.g., a two-chain 0 1 where 0 1 and a one-chain 0 ; and let 0 p, 1 p, and 0 p.Then 0 F p while 0 F p, but for all : 0 iff 0 .

Axiomatizing Log \ (C)
Now for the promised axiomatization of Log \ , which -via the representation results of the next subsection -entails that it even is an axiomatization of MIL \-Pre and MIL \-Pos .Definition 4.9 (Axiomatization).We define MIL \-Pre to be the least set offormulas that (i) is closed under the axioms and rules of MIL Pre ; (ii) contains the K-axioms for \; 33 (iii) contains the axioms (I1) sup p p\q q, and (I2) p q\ sup pq ; 34 and (iv) is closed under the rule 32 Notice that this places us in an extension of temporal S4. 33 For reference, the K-axioms for \ are: p q \r p\r q\r and p\ q r p\q p\r . 34'(I1)' and '(I2)' are short for 'inverses': they capture how ' sup ' and '\' relate. 35'(N \ )' is short for 'necessitation'.Observe that the other necessitation rule is not validity preserving.We, e.g., have Log \ but we do not have Log \ \ .
modality and C \-Pre is defined in terms of it(s dual), the corresponding inductive step of the truth lemma goes through.Therefore, it only remains to cover the inductive step for '\'. 37To this end, the following two claims will suffice:  With these claims at our disposal, the inductive step regarding '\' in a proof of the truth lemma is immediate (the two claims cover one direction each).Since this was the last obstacle for proving the truth lemma, and we have already noted that W \-Pre C \-Pre , we can deduce strong completeness-finishing not only our proof, but also this subsection.

Bulldozing and Completeness-Via-Representation
With Log \ axiomatized, next up is showing Log \ MIL \-Pre MIL \-Pos via representation; i.e., via onto 'p-morphisms'. 38mportantly, to find the technique of onto p-morphisms in our arsenal of validitypreserving techniques, when dealing with preorder frames, we have to define the 'back'-and 'forth'-conditions in terms of the accompanying ternary (and not binary) relations.For ease of reference, let us spell this out: When dealing with preorder frames W , [\-]p-morphisms are defined in terms of the induced W S Pre . 40w to be clear, onto p-morphisms preserve validity (and, generally, consequences) of M -formulas, while onto \-p-morphisms even preserve validity (and consequences) of \-M -formulas.This means we have a formal framework for developing representation results.In this subsection, this is a substantial part of what we will be doing. 41irst up is our plighted proof that any -frame W C is the \-p-morphic image of a poset frame W S Pos , entailing that with MIL \-Pre we have achieved an axiomatization of both MIL \-Pre and MIL \-Pos .This representation is obtained by composing two other representations; the first of which generalizes 'bulldozing' from the usual unary-modality setting to our binary-modality setting.To explain how this works, we briefly observe the following: With this observed, we are ready for the first representation result, mendingframes W C so that C becomes a partial order.

Proposition 4.15 (Bulldozing). Let W C
. Then W C is a \-p-morphic image of some W C for which C is a partial order.
Proof Let W C be arbitrary.We construct W C by adapting the well-known bulldozing technique from the binary-relation setting to our ternary-relation setting.More precisely, let denote the set of maximal non-degenerate clusters of W C w.r.t. the preorder C . 42We then define the underlying set as and let the function f W W 40 These definitions extend to the notion of p-morphisms between models as well.Moreover, the notion of bisimulation for modal information logics is also defined in terms of the induced W S ; i.e., the back and forth conditions are defined in terms of the supremum relation and not in terms of the preorder ' '. 41 Regarding \-p-morphisms, it is important to have in mind that they are also required to meet ( sup -back).A notion for simply meeting (forth) and (\-back) would appear appropriate, but we will not be needing such since we do not deal with modal logics having only the modality '\'.In general, of course, the results of this section have these modal logics as special cases; e.g., our decidability proof in the next subsection. 42Recall that a cluster is non-degenerate :iff it contains at least two elements.It is maximal :iff no proper superset is a cluster.be given by To define the relation C , fix some linear order K for each K , and for all x a b W , let C xab :iff C f x f a f b and This covers all cases-completing our proof of f being an onto \-p-morphism.
Lastly, we show that (3) C is a partial order.Reflexivity and transitivity are consequences of W C . To show anti-symmetry, let x y W be arbitrary s.t.C x x y and C yyx.We have to show that x y.Going by cases we find that: If x k x z x K and y k y z y K for K K , then Ck x k x k y and Ck y k y k x so k x K -contradicting maximality of the clusters (which implies that whenever K K , we even have K K ).
If x k x z x K k y z y y, then x y follows by anti-symmetry of our lexicographical ordering (since the ordering of the integers is linear and so is K ).Thus, we've shown C to be anti-symmetric, which completes our proof of (3) C being a partial order, thus finalizing our bulldozing proof.
Using this representation, we continue further mending -frames W C into real poset frames (i.e., frames whose ternary relation is the supremum relation of a partial order).We do so through another representation, which is obtained by adopting the framework of the completeness proof of Section 2 (2.13).In brief, in the proof to come, we will also be constructing a poset frame recursively by repairing defects.However, this time, the defects will be determined by an onto function, which we iteratively extend seeking to make it an onto \-p-morphism.And, although the supand sup -defects only need minor revision, we do need to include a third kind of defect corresponding to (\-back).
Many of the arguments will be almost identical to the ones of the completeness proof of Section 2, and so will be omitted or only hinted at.But -although the general set-up is very similar -since there are some differences, it is worth spelling out.We proceed doing so.

Definition 4.16 Given any W C
, we let E be some set disjoint from W of cardinality max W 0 , and W C be the set of all quadruples f D X such that 1.' f is an onto function from W D X to W ; 2.' D X E ; 3.' D X E; 4.' D X ; 6.' is a partial order on W D X ; and 7.' if y x then f y C f x . 43xt, we define the revised versions of the sup -and sup -defects and their complementary revised repair lemmas, before subsequently stating and proving the last defect/repair pair.Definition 4.17 Then, witnessed by y and z , x y z does not constitute a ( sup -back)-defect of f D X .
Proof Defining as described, the proof of f D X W C resembles the one of Lemma 2.11: 1.'-6.' are obvious, and 7.' is shown using C f x f y f z and W C .Moreover, the latter claim is immediate.Proof Extending to f D X as described, it follows similarly to the proof of Lemma 2.12 that f D X satisfies 1.'-7.' and x sup y z .Only two things are worth mentioning: (1) for proving 7.', we use that if u d then u x , hence f u f u C f x f d , and (2) for proving x sup y z , we need that is a partial order (this is where we use bulldozing). 44Notice the similarity between ( sup -back)-defects and sup -defects (2.8). 45And between (forth)-defects and sup -defects (2.9). 46Now 'd ' is no longer short for 'dummy', but for 'duplicate' (of x ): f d f x .We stress: this is key.(However, this is only a good intuition for the D-worlds introduced in this repair lemma-not for those in the next.) Our third and last defect, naturally, bears much resemblance to the ( sup -back)defect.It is defined as follows: Then, witnessed by x and z , x y z does not constitute a \-back defect of f D X .
Proof A matter of going over the definition.
Employing these repairs, we are ready to prove the desired representation result.

Proposition 4.23 Every W C
for which C is a partial order, is a \-p-morphic image of a poset frame.
Proof Let W C be arbitrary s.t.C is a partial order.For the sake of simplicity, assume W is countable: as oftentimes is the case, the adjustments of the ensuing proof needed for the case where W 0 are conceptually insignificant but notationally taxing. 47Besides, by a 'standard translation' and the Löwenheim-Skolem Theorem, has the countable model property w.r.t.-formulas, so, for instance, starting with a countable frame, we can bulldoze it into a countable whose underlying preorder is a partial order.
As in the completeness proof of Section 2, using the repair lemmas repeatedly, we will be constructing a sequence The adjustments in case W 0 are doing transfinite recursion and induction instead.
We begin the sequence by setting  This follows by an easy induction, using that each f m 1 D m 1 X m 1 m 1 is obtained from f m D m X m m using one of the repair lemmas.
Claim (a').Let n and x y z W D n X n be arbitrary s.Then for all m n: x sup m y z a fortiori, x sup y z .We prove the claim by induction.By assumption, it holds for m n, so assume it holds for some m n.We show it holds for m 1.This time we have three cases, depending on the type of defect being repaired at stage m 1.The cases of a ( sup -back)-repair and (forth)-repair are the exact same as in Theorem 2.13.Consequently, suppose stage m 1 was obtained by (\-back)-repairing some s y s z s through introducing the worlds s z s .Then s is the only possible counterexample to x sup m 1 y z , so assume y m 1 s m 1 z .Then we must have y m y s m z , so by the IH x m y s , hence x m 1 s .n with no change concerning the cases of ( sup -back)-repairs and (forth)-repairs.Therefore, assume l m 1 m 1 D m 1 was obtained by (\-back)-repairing some x y z by introducing x z .Then there is no change in predecessors of a, which suffices for the claim.

Claim (b')
Finally, from these claims we likewise get (c): some tuple did constitute a defect at some stage n, but no longer at some later stage m n, then it didn't for all k m.
Noteworthy is the overlap between our definitions of ( sup -back)-defects and (\back)-defects, which assures that claim (a') applies to both types of defects.And using (c) along with claim (a') and (b') in an analogous manner to what we did in the completeness proof, we get that f D X neither has (forth)-, ( sup -back)nor (\-back)-defects.
Lastly, the fact that there are no defects, entails that f is a \-p-morphism from W D X to W C , so since f also is onto, we've shown the desired.
At long last, combining the two representations, we can deduce that we have achieved the axiomatization we were seeking.Consequently, all that remains to be proven is (F2'). 51 Using this, we can conclude that the basic modal information logic of preorders (or posets) endowed with the informational implication is decidable.

Corollary 4.28 MIL \-Pre is decidable (and so is MIL \-Pos ).
Proof We have shown that MIL \-Pre MIL \-Pre Log \ F so since MIL \-Pre is finitely axiomatizeable and complete w.r.t. a recursively enumerable (r.e.) class of finite frames [simply check for satisfaction of the first-order formulas (Re.f), (4f), (Co.f), and (Dk.f)], we obtain decidability of MIL \-Pre .
Closing off this section, we state the following corollary:

Corollary 4.29 Let
-M be the extension of the basic language M the unary modality ' ', and let the semantics for ' ' be the usual one, namely those of the forward-looking modality 'F' given in Remark 4.8.Then letting MIL -Pre and MIL -Pos be the MILs of this language on preorders and posets, respectively, we get that both are decidable.
Proof A decision procedure is given as follows: For any -M -formula , translate it into a formula t \-M in accordance with Remark 4.8, and then use the decision procedure of the preceding corollary.

Conclusion and Future Work
This paper's exploration of modal information logics has come to an end.We summarize this inquiry, clarify where it leaves us, and point to future lines of research.
First, we examined the basic modal information logics of suprema on preorders and posets, namely MIL Pre and MIL Pos .We showed that -even if they do not enjoy the FMP w.r.t.their frames of definition -they are decidable.This was shown 'via completeness' by (1) axiomatizing them; (2) deducing that they are one and the same logic; and (3) obtaining another class of frames complete w.r.t. the logic(s), which, importantly, did enjoy the FMP. 50Recall that '\' is a ' -ed' modality; therefore, this presentation of the second filtration clause. 51The proof of Lemma 2 in [6] pertains to showing the satisfaction of (F2') in our present setting, so the ensuing argument is only given for the sake of completeness of the current proof-we claim no originality whatsoever.

8
We continue by making this basic intuition rigorous -starting with providing the repair lemmas.

Lemma 2 .
11 ( sup -repair lemma).Suppose sup x is a sup -defect of some l D .Then we can extend to l D by taking distinct y z W dom l D s.t.l l y z y u z u x u D D C Pre l x and y z witness that sup x does not constitute a sup -defect of l D .
z s u s m u y s and z s are the only possible counterexamples to the claim.Second, suppose l m 1 m 1 D m 1 was obtained via sup -repairing some s y s z s by introducing the dummy d s .Notice that, by IH and the definition m 1 m u d s d s u m y s m z s the only possible counterexample to the claim is d s .Accordingly, suppose d s m 1 y z.Going by cases, we prove that this implies d s m 1 x: If y s m y z, then, by IH, y s m x so d s m 1 x.If z s m y z, then as above.If y s m y and z s m z, then, by the observation, either (a) y s y or (b) y s m x or (c) m y s m x .If (b), then d s m 1 x.And if (c), then note that as s is a m -upper bound of y s z s , it must also be a m -upper bound of y z , hence, by IH, x m s -contradicting m y s m x .Thus, we may assume (a) y s y; and, analogously, z s z.But then s sup m y s z s sup m y z x, hence s y s z s x y z couldn't have constituted a sup -defect because C Pre l m x l m y l m z .If z s m y and y s m z, then as above.This exhausts all cases, showing d s m 1 x, which completes the induction.
dom l n D n are s.t. a n b.Then for all m n, we have that a m b.A fortiori, a b.Follows by induction on m, noting that if l m 1 m 1 D m 1 was obtained by sup -repairing some x by introducing some y z, we would have m 1 m y u z u x u that is, there is no change in successors of b.Likewise, if l m 1 m 1 D m 1 was obtained by sup -repairing some x y z by introducing a dummy d, there is no change in predecessors of a.This exhausts the cases, hence proves the claim.

Definition 4 . 2 (
Semantics).Given a preorder model W V , a world W and a formula \ \-M with main connective '\' , which precisely shows the claim.

Definition 4 . 13
Given any two frames W C W C , a function f W W is denoted a p-morphism if it satisfies the following conditions: (forth) if C x y z , then C f x f y f z ; and ( sup -back) if C f x yz, then there exist y z W s.t.f y y f z z and C x y z .If f additionally satisfies (\-back) if C x f y z, then there exist x z W s.t.f x x f z z and C x y z , we denote it a \-p-morphism. 3938 Another commonly used term for 'p-morphism' is 'bounded morphism'.

Observation 4 . 14
For any W C , let C and C be given as follows: C y x C x x y C y x z C x yz C xzy Then, by definition of the class , it is not too hard to see that (a) C C , and (b) C is a preorder on W .Moreover, if C happened to be the supremum relation of some preorder , i.e., if C x yz iff x sup y z , then C .

K , pick any a f 1 a and b f 1 ba a z x 1 and b b z x 1 .
a b k b z b z x z a or z x z a and k x K k a z x z b or z x z b and k x K k b We claim that (1) W C ; (2) W C is a \-p-morphic image of W C witnessed by f ; and (3) C is a partial order.We begin by proving (1) W C .We have that (Re.f) is satisfied because (a) W C Re f by assumption and (b) for all K: K is, as a (weak) linear order, in particular, reflexive;(4f) can be seen to be satisfied by a straightforward, but tedious check using W C 4 f .Only non-trivial case is when C x xa by virtue of (iii): there one must observe that if C aab then f b cannot be in the same cluster as f x by maximality of clusters K ; (Co.f) is satisfied because (a) W C Co f and (b) the definition of C is symmetrical in the two last arguments; and (Dk.f) is satisfied because (a) W C Dk f and (b) if C xab holds by virtue of (i), then C x xa holds by virtue of (i); if C xab holds by virtue of (ii) or (iv), then C x xa holds by virtue of (iii); if C xab holds by virtue of (iii), then C x xa holds by virtue of (v); and if C xab holds by virtue of (v), then C x xa holds by virtue of (v).Having proven (1), we continue by proving (2).f is clearly (a) surjective and (b) a homomorphism.Therefore, it remains to show that (c) the back conditions are satisfied.Beginning with ( sup -back), suppose C f x a b for arbitrary x W , a b W .We then have to find a b W s.t.C xab, f a a , and f b b .We go by cases: (i) If x W K using surjectivity of f .(ii) If x k z K and a b K , pick any a f 1 a and b f 1 b .This exhausts all cases, hence f satisfies the ( sup -back) condition, thus is a pmorphism.Continuing with (\-back), suppose C x f a b for some a W and x b W . Again, we go by cases: begin by picking any x f 1 x .Then C f x f x b by (Dk.f) and (Co.f) of W C and because C x f a b, so by the just proved sup -back condition and the definition of C , we can find a b W s.t.C x x b and f b b.We claim that C x a b .To see this, first recall that C x f a b, f x x and f b b.Second, notice that C x x b must hold by virtue of (i), (iii) or (v).If by virtue of (i), then so does C x a b ; if by virtue of (iii), then C x a b holds by virtue of (ii) (since, by assumption, either a W ; and if by virtue of (v), then C x a b holds by virtue of (iv).ii And if a k a z a K and x K , then setting x x z a 1 , we, again by (Dk.f) and (Co.f), get that C f x f x b, hence by the sup -back condition we can find a b W s.t.C x x b and f b b.Now because (1) C x f a b and (2) C x x b must hold by virtue of (iii) or (v), we get that C x a b likewise holds by virtue of either (iii) or (v) since z a 1 z a .

K
, then C x x y and Cyyx by definition of f and C , so since W K K contains no non-degenerate clusters by definition, we must have x y.

Lemma 4 .
20 ((forth)-repair lemma).Suppose x y z is a (forth)-defect of some f D X W C .Then we can extend to f D X W C by (a Claim (a') Claim (b').Let n and suppose that a b W D n X n are s.t. a n b.Then for all m n, we have that a m b.A fortiori, a b.Once again by induction on m

Theorem 4 .
24 Every W C is a \-p-morphic image of a poset frame.Thus, MIL \-Pre is sound and strongly complete w.r.t.preorder frames, and, in particular, MIL \-Pre MIL \-Pos MIL \-Pre Additionally, as a special case, we get another proof of MIL Pre being sound and strongly complete w.r.t.preorder frames, and, particularly MIL Pre MIL Pos MIL Pre Proof The first assertion follows from propositions 4.15 and 4.23 because onto \-pmorphisms are closed under composition.Soundness and strong completeness is the upshot of onto \-p-morphisms preserving the consequence relation of a frame and the fact that Pos Pre ; so also, in particular MIL \-Pre MIL \-Pos MIL \-PreLastly, since \-p-morphisms are p-morphisms and \-M , this also restricts to the special case of the basic modal information language.

MIL Pre . Theorem 2.2 (Soundness). MIL Pre MIL Pre .
With these defects defined, next up is repairing them.Before providing the actual repair lemmas demonstrating how to coherently repair each of the defects (making sure that if l n n D n Pre l u follows by transitivity of Pre .With all of these preliminaries out of the way, we are finally in a position to construct the needed frame and prove completeness.MIL Pre is strongly complete w.r.t.MIL Pre .So, in particular, MIL Pre MIL Pre . y y l y Pre l y follows by Pre being a preorder, hence reflexive, cf.Lemma 2.6 (c).y x l y Pre l x follows by Lemma 2.6 (d).y u For u x, l y d x To see this, observe that if x y, since z x, we would have by 7. that l z Pre l x hence (cf.Lemma 2.6 (b)) C Pre l x l y l z

Theorem 3.9 MIL Pre admits
filtration w.r.t. the class .Thus, MIL Pre Log F where Log F denotes the NML of the class of finite -frames.
, using z P , then y P , and in turn x P .This completes our proof of 1.For proving 2., it suffices to show that W C V is a filtration of W C V through .That is, we need to check two conditions, namely (F2) follows by definition of our filtration relation.For (F1), suppose C x yz and sup .Then the only non-trivial part is to show that 28 Corollary 3.10 MIL Pre is decidable (and so is MIL Pos ).
satisfies 1.' and 3.'-7.',and(2) f D X has no defects whatsoever.Again, only (2) is not straightforward, and, again, for proving (2) two claims and an observation are helpful.Observation'.Let n and x W D n X n be arbitrary s.t.