Merging and Evolution of Data Bases: A Many-Logic Approach

Merging and Evolution of Data Bases: A Many-Logic Approach Manuel A. Martins

Abilio Rodrigues

CIDMA & Dep. of Mathematics University of Aveiro martins@ua.pt

Federal University of Minas Gerais Minas Gerais, Brazil abilio.rodrigues@gmail.com

A new approach to information-based logics using many-logic modal structures (MLMSs) was recently proposed by Martins et al. In such a setting, it is possible to have worlds with different underlying logics within the same frame, all anchored to a base lattice that contains the semantics of each logic as a down-complete sublattice. In this paper, we take the lattice L6, defined by the six-valued semantics of the logic of evidence and truth LETK+ , as our base lattice. We also consider four-, three-, and two-valued formal systems associated with down-complete sublattices of L6. These logics are capable of representing paracomplete, paraconsistent, and classical contexts with six, four, three, and two informational scenarios, respectively. As applications of MLMSs, we will discuss the merging and temporal evolution of databases whose underlying logics are two-, three-, four-, and six-valued. We will also investigate inconsistency measures in this context.

1

Introduction

An information-based logic is a logic suitable for processing information in the sense of taking a database as a set of premises and drawing conclusions from these premises in a sensible way. Nuel Belnap in [1] proposed an information-based interpretation of the logic of first-degree entailment (FDE ), motivated by the idea of computer dealing with possibly inconsistent and incomplete databases, that is, situations in which there may be conflicting information or no information at all about a sentence A. In the former scenario, explosion fails; in the latter, excluded middle fails. Such situations are not exceptional, but rather a common reality, often resulting from noise, uncertainty, and inaccuracies in the process of collecting information. The computer conceived by Belnap ‘modifies its current set-up’ when it gets new information [1, p. 36]. A ‘set-up’, thus, is a specific configuration of the computer over time, and these configurations can be seen as information states represented by worlds in a Kripke model. The many-logic modal structures (MLMS) proposed in [7] and [12] are well-suited for representing the connections between information states viewed as databases with different underlying logics, the evolution of databases over time, and databases accessed by users with different privileges. This framework considers frames whose worlds may be associated with different formal systems. Each of these systems is represented by a down-complete sublattice (dc-sublattice, for short) of a common base lattice. Modal relationships are then established through this shared algebraic structure, which embeds the semantics of the individual logics and provides a unified setting in which they can interact. As a consequence, connections between worlds can be understood as interactions between their underlying logics. In such a setting, the modalities are interpreted in a way that differs from the traditional Kripkean approach. A. Indrzejczak, T. Jarmużek, M. Klonowski, & M. Zawidzki (Eds.): Twelfth International Conference on Non-Classical Logics. Theory and Applications (NCL’26) EPTCS 455, 2026, pp. 91–104, doi:10.4204/EPTCS.455.8

© M. Martins & A. Rodrigues This work is licensed under the Creative Commons Attribution License.

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In addition to introducing the general formal machinery of MLMSs1 , we will investigate a number of applications based on the lattice L6, which provides the six-valued semantics for the logic of evidence and truth LETK+ , introduced by Coniglio and Rodrigues in [2]. Logics of evidence and truth (LETs) are equipped with a classicality operator ◦ that recovers classical logic for sentences in its scope. Thus, when ◦A holds, the sentence A is subjected to classical logic, otherwise the underlying logic of A is FDE, or an extension of FDE. LET s can be interpreted in terms of information, and in this case a sentence ◦A means that the information conveyed by A, positive or negative, is reliable. The connective ◦, thus, can be thought of as a kind of certification. L6 extends the well-known logical lattice L4, defined by the four-valued semantics of FDE [1, p. 42], by adding a new top and a new bottom, which correspond to two additional semantic values intuitively interpreted as reliable positive information and reliable negative information (cf. [2, Sect. 4.1]). LETK+ , as LET s in general, is able to represent six scenarios: the four scenarios of FDE plus two more scenarios of reliable information. A detailed study of logics defined by dc-sublattices of L6 can be found in [12]. The aim of this article is to explore how dynamic interactions between distinct logical contexts can be formalized in richer computational settings, as a step toward extending the original proposal. In particular, we explore mechanisms for information merging and temporalization, enabling the representation not only of static accessibility relations between worlds, but also of the evolution, synchronization, and reconciliation of information states over time. By incorporating temporal dimensions into MLMSs, we allow logical structures to capture successive stages of knowledge acquisition, revision, and propagation, while merge operations provide formal tools for combining information originating from different sources. We introduce generalized modal and temporal operators capable of modeling transitions between paraconsistent, paracomplete, and classical reasoning environments within a unified semantic architecture. These extensions reinforce the applicability of MLMSs to distributed databases, evolving knowledge systems, and multi-agent environments, where the continuous integration and temporal management of conflicting information are essential for coherent and reliable reasoning. The remainder of this paper is structured as follows. Section 2 presents the general machinery of many-logic modal structures. In Section 3, we present the matrix logics considered in this paper. These include LETK+ and some well-known two-, three-, and four-valued logics, which are capable of representing subsets of the six scenarios characterized by LETK+ . The lattices associated with these logics are down-complete sublattices of L6. In Section 4, the main section of this paper, we propose further developments of MLMSs, namely: temporalization of MLMSs that allow to define temporal operators past, future, and next; merging of MLMSs, that intends to represent the merging of databases; analysis of inconsistencies and certified data in databases.

2

Many-logic modal structures

In this section, we recall the key definition of many-logic modal structures as presented in [12, Sect. 2]. We start by introducing some notions related to lattices that will be important in the 1 The development of MLMSs has recently emerged as a framework for extending modal reasoning beyond the classical setting. The foundational proposal was introduced by Freire and Martins in [7], who developed the notion of modality across different logical systems. The algebraic basis of the framework was later investigated by Freire, Madeira, and Martins in [6], who introduced lattice-based constructions relating the lattices associated with distinct logics. Further developments include the study of regular non-normal modal systems ([8]) and the institutional foundations of many-logic modal logic ([9]).

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definition of the main concept of this section. Definition 2.1. Let L = ⟨L, ·, +⟩ be a complete lattice with associated partial order ≤. A downcomplete sublattice (dc-sublattice) of L is a complete lattice L′ = ⟨L′ , ·′ , +′ ⟩ with associated partial order ≤′ such that: 1. L′ ⊆ L; 2. ≤′ = ≤ ∩ (L′ × L′ ), i.e.: for every x, y ∈ L′ , x ≤′ y iff x ≤ y. We denote by DsubLat(⟨L, ·, +⟩) the set of all dc-sublattices of ⟨L, ·, +⟩. Definition 2.2 (The down-interpretation). Let L = ⟨L, ·, +⟩ be a complete lattice with associated partial order ≤, and let L′ = ⟨L′ , ·′ , +′ ⟩ be a dc-sublattice of L. The down-interpretation of value x ∈ L in L′ is defined as follows: ′

xL =

{y ∈ L′ : y ≤ x}.

_ L′

Definition 2.3 (Filtered L -lattices). Let a filtered L -lattice be a matrix logic ⟨L, D⟩ such that L is an algebra over Σ with domain L and D ⊆ L, where the ΣL -reduct ⟨L, ·, +⟩ is a lattice. Consider a propositional language L , defined over a signature Σ containing ΣL = ⟨∧, ∨⟩, and a set of sentential letters Var. Definition 2.4 (Many-logic modal structures, MLMS). Let LAT ⊆ DsubLat(⟨L, ·, +⟩), such that ⟨L, ·, +⟩ ∈ LAT, and L □ the language obtained by extending L with the unary connective □. A many-logic modal structure M over LAT and L □ is a tuple ⟨W, R, I, v⟩ such that W ̸= 0/ and: 1. L is a set of matrix logics ⟨L′ , D′ ⟩ over Σ such that ΣL -reduct ⟨L′ , ·, +⟩ of L′ belongs to LAT. 2. I : W → L assigns a matrix logic ⟨Lw , Dw ⟩ to each world w ∈ W . By simplicity, and when there is no risk of confusion, we may write I(w) = Lw , where Lw is the domain of Lw . 3. R is a relation from worlds to worlds, i.e. R ⊆ W ×W . 4. v is a valuation function from the set Var of sentential letters to values in I(w), i.e. for each world w ∈ W and p ∈ Var, v : W ×Var −→ L such that v(w, p) ∈ Lw . In each world w a valuation vw (:= v(w, −)) is extended to all formulas of L based on the matrix logic ⟨Lw , Dw ⟩ related to the lattice assigned to w, given by I(w). To extend a valuation vw to all formulas of L □ (i.e. to assign semantic values to formulas □A) we interpret in a complete sublattice L′ ⊆ L elements not necessarily in L′ according to the down interpretation. Let w and w′ be worlds, Lw and Lw′ the corresponding lattices, and vw (A) and vw′ (A) the values assigned to a formula A in w and w′ by a valuation v. We denote by (vw (A))Lw′ and (vw′ (A))Lw the values vw (A) and vw′ (A) ‘seen’ from the perspective of the worlds w′ and w respectively, according to the down-interpretation. The semantic value of a formula □A in a world w is the infimum of the set of values of A in all the worlds accessed from w as seen from the perspective of the local lattice Lw . A clause for the connective □ is thus defined as follows: Definition 2.5 (Semantic clause for □). vw (□A) =

V Lw

{(vw′ (A))Lw : w′ ∈ W ∧ wRw′ }.

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According to the intuitive interpretation we propose, vw (□A) = x means ‘A has the semantic value x for an agent accessing the information available in the acessible worlds w’. We say that A holds in w in a structure M, denoted by M, w ⊨ A, if vw (A) ∈ D. It is to be observed that in the framework adopted here, saying that a formula has a specific value (vw (A) = x) in a model M and a world w is generally more informative than saying that it holds (or does not hold). This is due to the multivalued semantics of the logics we will see in Section 3 below, which may involve multiple designated (or non-designated) values. Examples of manylogic modal structures based on these logics, together with their corresponding lattices, can be found in Section 5 of [12].

3

The logics

In this paper we consider the following matrix logics: LETK+ = ⟨L6, {T, T0 , b}⟩, FDE⊥→ = ⟨L4w , {T0 , b}⟩ LJ4 = ⟨L4s , {T, b}⟩ w LP→ ⊥ = ⟨B3 , {T0 , b}⟩ J3 = ⟨B3s , {T, b}⟩

K3→ = ⟨N3w , {T0 }⟩ L3 = ⟨N3s , {T }⟩ CLw = ⟨C2w , {T0 }⟩ CLs = ⟨C2s , {T }⟩ CLs = ⟨C2s , {T }⟩

Let the set LAT = {L6, L4w , L4s , B3w , B3s , N3w , N3s ,C2w ,C2s } be the collection of algebras underlying the logics above. For the semantic characterization of these logics in terms of down-complete sublattices of the lattice L6, as well as for a detailed presentation of the corresponding lattices, their connection with the scenarios represented by each logic – which are subsets of the six scenarios represented by LETK+ – and their characterization as information-based logics, we refer the reader to Sections 3 and 4 of [12]. Here, we only note that J3 is the three-valued paraconsistent logic introduced in D’Ottaviano et al. [5], L3 is the Lukasiewicz’ three-valued paracomplete logic, FDE⊥→ and LP→ ⊥ extend respectively FDE and the logic of paradox LP with a material implication and a bottom particle. Moreover, K3→ extends Kleene’s K3 with a material implication, and LJ4 may be viewed either as a paraconsistent version of L3 or as a paracomplete version of J3.

4

Operations on information structures

In this section, we introduce operations on MLMSs, the evolution-structures that represent their development over time, and the tools for measuring inconsistency. Let a generalized literal be a formula of the form p, ¬p, or ◦p, for a variable p 2 . A manylogic modal structure can be viewed as a W -family Kw (w ∈ W ) of generalized literals defined as the smallest set satisfying the following conditions:

2 An explanation of the notion of extended literals p, ¬p, and ◦p can be found in [2] and [13], where six valued logics of evidence and truth (LET s) are investigated. Belnap-Dunn logic works with two primitive notions, positive information and negative information, represented by literals p and ¬p. The six-valued semantics of LET s adds one more primitive notion, reliable information, expressed by ◦p. Once again, we refer the reader in particular to the paper [13], where these concepts are explained in detail.

M. Martins & A. Rodrigues For any propositional variable p: - if vw (p) = T , then p, ◦p ∈ Kw , - if vw (p) = F, then ◦p, ¬p ∈ Kw , - if vw (p) = b, then p, ¬p ∈ Kw ,

95 - if vw (p) = n, then ◦p ∈ Kw , - if vw (p) = n, then ◦p ∈ Kw , - if vw (p) = T0 , then p ∈ Kw , - if vw (p) = F0 , then ¬p ∈ Kw .

An information state is thought of here as the set of formulas that hold in a world w, which is the set Kw of generalized literals closed under the notion of logical consequence related to the lattice Lw .

4.1

Merging many-logic modal structures

The merging of two compatible many-logic modal structures can be defined in a natural way. The general idea is as follows: given two structures M and M ′ and a world w that belongs to both M and M ′ , the merging yields a structure M ′′ in which w contains the generalized literals that belong to w in M and M ′′ . This operation, of course, cannot yield a trivial world. Thus, M and M ′ have to be compatible in the sense that in the resulting structure there cannot be a world with literals p, ¬p and ◦p. Definition 4.1. Two many-logic modal structure M = ⟨W, R, I, v⟩ and M ′ = ⟨W ′ , R′ , I ′ , v′ ⟩ are said to be incompatible if for some w ∈ W ∩ W ′ there is a sentential letter p such that one of the following conditions hold: - vw (p) = T and (v′w (p) = b or v′w (p) = F0 or v′w (p) = F) - vw (p) = T0 and v′w (p) = F - vw (p) = b and (v′w (p) = T or v′w (p) = F) - vw (p) = F0 and v′w (p) = T - vw (p) = F and (v′w (p) = T or v′w (p) = T0 or v′w (p) = b) When they are not incompatible, they are called compatible. The definition above ensures that there is no world in which for a sentential letter p, {p, ¬p, ◦p} ⊆ Kw ∪ Kw′ . Note that two weak worlds (C2w , B3w , N3w , and L4w ) are always compatible, since there is no formula ◦p in the respective sets Kw . Besides, it is not required that a world w ∈ W ∩W ′ be assigned the same lattice in M and M ′ . The merging of two compatible structures M and M ′ results in the many-logic modal structure M//M ′ as defined below. Definition 4.2. Given two compatible many-logic modal structure M = ⟨W, R, I, v⟩ and M ′ = ⟨W ′ , R′ , I ′ , v′ ⟩, the merging of M with M ′ is the many-logic modal structure M//M ′ = (W ∪W ′ , R ∪ ¯ v̄) where: R′ , I, ¯ - I(w) is defined using I(w) and I ′ (w) according to the Table 1 below. - v̄ is defined as follows: - For w ∈ W ∩ W ′ : v̄w (p) is defined using v(p) and v′ (p) as defined in Table 2 below, where the X indicates that the merging cannot be done. - For w ∈ W \W ′ : v̄w (p) = vw (p) - For w ∈ W ′ \W : v̄w (p) = v′w (p)

96

Merging and Evolution of Data Bases I(w)\I ′ (w) L6 L4s L4w B3s B3w N3s N3w C2s C2w

L6 L6 L6 L6 B3s L6 L6 L6 C2s L6

L4s L6 L4s L6 B3s L6 L4s L6 C2s L6

L4w L6 L6 L4w B3s B3w L6 L4w C2s B3w

B3s B3s B3s B3s B3s B3s B3s B3s C2s B3s

B3w L6 L6 B3w B3s B3w L6 B3w C2s B3w

N3s L6 L4s L6 B3s L6 N3s L6 C2s L6

N3w L6 L6 L4w B3s B3w L6 L4w C2s B3w

C2s C2s C2s C2s C2s C2s C2s C2s C2s C2s

C2w L6 L6 L4w B3s B3w L6 C2w C2s B3w

Table 1: Merging lattices vw \v′w T T0 b n F0 F

T T T X T X X

T0 T T0 b T0 b X

b X b b b b X

n T T0 b n F0 F

F0 X b b F0 F0 F

F X X X F F F

Table 2: Merging valuations The definition of merging, alternatively, can be done in terms of the W -family Kw : - For w ∈ W ∩W ′ : KwM//M′ = KwM ∪ KwM′ - For w ∈ W \W ′ : KwM//M′ = KwM - For w ∈ W ′ \W : KwM//M′ = KwM′ For each world w ∈ W ∪ W ′ , the set Kw defines the valuation v̄ according to the Table 2. For worlds in W ∩W ′ , it is not required that the associated lattice be the same. The rationale for the Table 1 is as follows. Two weak worlds w and w′ are always compatible, but when merged the result may be a lattice different from Lw and Lw′ . Merging C2w with C2w yields a B3w : there cannot be any incompleteness in the resulting world, but there may be contradictions. Merging N3w with N3w yields a L4w : there may be incompleteness as well as contradictions in the resulting world. Merging B3w with N3w yields a B3w : there cannot be any incompleteness in the resulting world, but there may be contradictions. And so on. Merging L6 always yields L6, except in the case of C2s and B3s . In the former, for every p, either both ◦p and p or both ◦p and ¬p belong to w; in the case of B3s , either ◦p and p, or ◦p and ¬p, or p and ¬p belong to w. So, clearly the resulting world must be C2s and B3s , respectively. It is unlikely, however, that worlds B3s and C2s will be compatible with L6. Another case that is worth mentioning is the merging of N3s with L4w . The resulting world may have incompleteness from both N3s and L4w , and contradictions from L4w , as well as certified information from N3s and non-certified information from L4w . Thus, such a merging yields L6.

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Example 4.1. Consider the structures M1 and M2 : M1

M2 I(w1 ) = L4w , I(w2 ) = C2w vw1 (p) = T0 , vw2 (p) = F0 R2 = {⟨w1 , w2 ⟩, ⟨w2 , w1 ⟩, ⟨w1 , w1 ⟩, ⟨w2 , w2 ⟩}

I(w1 ) = L6, I(w2 ) = L4w , I(w3 ) = B3s vw1 (p) = T, vw2 (p) = T0 , vw3 (p) = T R1 = {⟨w1 , w2 ⟩, ⟨w1 , w3 ⟩, ⟨w1 , w1 ⟩, ⟨w2 , w2 ⟩, ⟨w3 , w3 ⟩}

w2

w2

C2w

w1

L4w

w3 B3s

L4w w1

L6

The merging of M1 and M2 yields the structure M1 //M2 below: ¯ 1 ) = L6, I(w ¯ 2 ) = B3w , I(w ¯ 3 ) = B3s I(w v̄w1 (p) = T0 , v̄w2 (p) = b, v̄w3 (p) = T R1 ∪ R2 = {⟨w1 , w2 ⟩, ⟨w1 , w3 ⟩, ⟨w2 , w1 ⟩, ⟨w1 , w1 ⟩, ⟨w2 , w2 ⟩, ⟨w3 , w3 ⟩}

w2

w3 B3s

B3w w1

L6

When applying the merging operation in a real-world situation, names may be assigned to databases so as to indicate which of them are to be merged. Databases sharing the same name are intended to be merged into a single database. In the example above, the idea was to keep the B3s -database of M1 unchanged while merging the L6-database of M1 with the L4w -database of M2 , and the L4w -database of M1 with the C2w -database of M2 . The names assigned to the worlds reflect the intended merging process: databases given the same name are merged together. Proposition 4.1. (i) Let M = ⟨W, R, I, v⟩ and M ′ = ⟨W, R, I, v′ ⟩ be two many-logic modal structures. Then for any w ∈ W and any formula A, vw (A) ∈ {b, T0 , T } or v′w (A) ∈ {b, T0 , T } imply v̄w (A) ∈ {b, T0 , T }. (ii) The merging operation is commutative and associative, but not idempotent. Proof. Regarding item (i), it is easy to see that if A is assigned designated value by vw or v′w , which is a value represented by a snapshot (z1 , z2 , z3 ) such that z1 = 1, the value assigned to A by v̄w will also be designated. As for (ii), it is clear to see that commutativity follows from the fact that ∪ and ∩ are commutative and the Table 1 is commutative. Associativity follows from the fact that ∪ and ∩ are associative and the Table 1 in each case the resulting value depends

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on Kw ∪ Kw′ . To see that idempotency does not hold, consider the merging of two C2w -worlds, which yields a B3w -world.

4.2

Temporalising many-logic modal structures

In Example 5.4 of [12], we showed how the temporal evolution of information states over time can be represented using only the tools already available in the definition of MLMSs. However, the temporal evolution of information states may also be represented by means of what we call here T-evolution structures. Consider now the language L T , obtained by extending the language L □ with the unary operators [P] and [F] with the following meanings: [P](A): the semantic value of A in the past, [F](A): the semantic value of A in the future. The semantic values of A in the past and in the future are given, respectively, by the infimum of the values that A assumes in the past and by the infimum of the values that A assumes in the future. Definition 4.3. A pointed many-logic modal structure is a pair (M, w0 ), where M = ⟨W, R, I, v⟩ is a many-logic modal structure and w0 ∈ W . w0 is called the reference world of M. A pointed many-logic modal structure is a many-logic modal structure together with a distinguished world. The purpose of introducing pointed structures is to make explicit the world at which formulas are evaluated in T -evolution structures. Let PAT be a set of pointed many-logic modal structures ⟨M, w0 ⟩. Definition 4.4 (T -evolution structures). A T -evolution structure M T is a triple ⟨T, <, g⟩, where ⟨T, <⟩ is a flow of time (that is, T is a non-empty set and < is a partial order on T ), and g : T → PAT is a function mapping each time point in T to a structure M in PAT . The value τ of a formula A in a structure M T at a time point t ∈ T is determined by a mapping τ(A,t) → L6, defined as follows: 1. τ(A,t) = v(A, w0 ), where w0 is the reference world of g(t), V 2. τ([P]A,t) = {τ(A, s) : s < t}, V 3. τ([F]A,t) = {τ(A, s) : t < s}, 4. τ(¬A,t) = ¬τ(A,t), 5. τ(A ∧ B,t) = τ(A,t) ∧ τ(B,t), 6. τ(A ∨ B,t) = τ(A,t) ∨ τ(B,t), 7. τ(A → B,t) = τ(A,t) → τ(B,t). A path in a T-evolution structure M T is a subset T ′ ⊆ T such that the elements of T ′ are linearly ordered with respect to the ordering defined on T . Some remarks are in order here: (i) If (T, <) is linear, discrete, and without end point, a next operator [X] can be added with the following semantic clause: τ([X]A,t) = τ(A,t ′ ), where t ′ is the time point immediately after t.

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(ii) A path can be any sequence of worlds linearly ordered, and a path can be contained in another path. Paths will be used below to define increasing and decreasing of consistency and certification (Definition 4.8). (iii) The order < applies only to the reference worlds of the structures in PAT . In each structure M, the reference world w0 can intuitively be interpreted as a sort of ‘central database’ that accesses every other database. (iv) The value τ(□A,t) in a structure M T is the value of □A at the reference world of the structure assigned to t in M T . The values of ([F]□A,t) and ([P]□A,t) are given by the infimum of the values of □A after and before t respectively. Example 4.2. Consider the set PAT = {⟨M1 , w1 ⟩, ⟨M2 , w3 ⟩, ⟨M3 , w5 ⟩, ⟨M4 , w7 ⟩} of pointed manylogic structures.

s

s

w2 N3

w4 N3

w6 N3

w1 L6

w3 L6

w5 L6

s

w8 N3

s

w7 L6

⟨M1 , w1 ⟩

⟨M2 , w3 ⟩

⟨M3 , w5 ⟩

⟨M4 , w7 ⟩

W = {w1 , w2 } I(w1 ) = L6, I(w2 ) = N3s vw1 (p) = T0 , vw2 (p) = n vw1 (q) = F0 , vw2 (q) = n vw1 (□p) = n vw1 (□q) = F0

W = {w3 , w4 } I(w3 ) = L6, I(w4 ) = N3s vw3 (p) = T0 , vw4 (p) = F vw3 (q) = F0 , vw4 (q) = T vw3 (□p) = F vw3 (□q) = F0

W = {w5 , w6 } I(w5 ) = L6, I(w6 ) = N3s vw5 (p) = T, vw6 (p) = T vw5 (q) = F, vw6 (q) = F vw5 (□p) = T vw4 (□q) = F

W = {w7 , w8 } I(w7 ) = L6, I(w8 ) = N3s vw5 (p) = n, vw6 (p) = n vw5 (q) = n, vw6 (q) = n vw5 (□p) = n vw4 (□q) = n

Consider the T -evolution structure M T : t2

t4

t1 t3 T = {1, 2, 3, 4}, g(t1 ) = M1 , g(t2 ) = M2 , g(t3 ) = M3 , g(t4 ) = M4 . τ([P](□p,t1 )) = n τ([P](□p,t2 )) = F τ([P](□p,t3 )) = F τ([P](□p,t4 )) = F

τ([F](□p,t1 )) = F τ([F](□p,t2 )) = F τ([F](□p,t3 )) = T τ([F](□p,t4 )) = n

τ([P](□q,t1 )) = F τ([P](□q,t2 )) = F0 τ([P](□q,t3 )) = F τ([P](□q,t4 )) = n

τ([F](□q,t1 )) = n τ([F](□q,t2 )) = F τ([F](□q,t3 )) = n τ([F](□q,t4 )) = F

The intuitive interpretation can be as follows. The path t1 ,t3 represents a circumstance in which all the information available in the reference world w1 of the structure M1 has been refined.

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The resulting database, which corresponds to reference world w5 of the structure M3 , has certified information that is the same in the worlds w5 and w6 . The uncertainty has decreased. The path t1 ,t2 ,t4 , on the other hand, represents an inverse circumstance in which the uncertainty has increased. Different information came from the database represented by the worlds w4 in the stages t2 . This uncertainty is represented in the stage t4 by the value n in both worlds.

4.3

Measuring inconsistency

The quality of information in a database depends on both its inconsistencies and the reliability of the information it contains. The logic LETK+ distinguishes reliable from unreliable information by means of the connective ◦, which is intuitively understood as a form of data certification. In this section, we define measures of inconsistency and certification within a structure M, based on the number of occurrences of pairs of sentences p, ¬p as well as instances of ◦p in the respective W -family Kw (cf. [3, 4, 10, 11]). Although these measures are related, they remain independent in the sense that, except in the logics J3 and LJ4, it is possible to have no contradiction regarding p, yet ◦p does not hold. We further extend these concepts to T -evolution structures and introduce the notions of consistency improvement and certification improvement over time. Definition 4.5. Let M = ⟨W, R, I, v⟩ be a many-logic modal structure where |W | is finite. (i) The inconsistencies at w ∈ W , inc(w), is the number of variables p such that p, ¬p are in the database, that is: inc(w) = |{p : vw (p) = b}|, or alternatively: inc(w) = |{p : p ∈ Kw and ¬p ∈ Kw }|. (ii) The certificated data at w, cert(w), is the number of variables p such that ◦p is in the database, that is: cert(w) = |{p : vw (p) = T or vw (p) = F}|, or alternatively: cert(w) = |{p : ◦p ∈ Kw }|. (iii) The inconsistencies in M, inc(M), is the sum of the inconsistencies in the worlds w such that w ∈ W , that is: inc(M) = ∑ inc(w) w∈W

(iv) The certificated data in M, cert(M), is the sum of the certified data in the worlds w such that w ∈ W , that is: cert(M) = ∑ cert(w) w∈W

(v) The relative inconsistency of M, rinc(M), is the sum of the inconsistencies in M divided by the cardinality of W , that is: rinc(M) =

inc(M) |W |

(vi) The relative certified data in M, rcert(M), is the sum of the certified data in M divided by the cardinality of W , that is: rcert(M) =

cert(M) |W |

M. Martins & A. Rodrigues

101

In particular cases data do not have the same relevance and we must consider this in the previous definition. Let Λ ⊂ N be a scale of relevance for data and µ : Var → Λ. Definition 4.6. Let M = ⟨W, R, I, v⟩ be a many-logic modal structure where |W | is finite and µ : Var → Λ ⊂ N. (i) The inconsistencies at w ∈ W modulo µ, incµ (w), is the µ-weighted sum of the number of variables p such that p, ¬p are in the database, that is: incµ (w) =

∑ µ(p), vw (p)=b

(ii) The certificated data at w modulo µ, cert µ (w), is the µ-weighted sum of the number of variables p such that ◦p is in the database, that is: cert µ (w) =

∑

µ(p),

vw (p)=T or vw (p)=F

The other definitions can also be extended with weights for the proposition symbols. The point of the notions of relative inconsistency and certification is to obtain a normalized result proportional to the size of W . Note that the Definition 4.5 above imposes no restrictions on the cardinality of the sets {p : vw (p) = b}, {p : vw (p) = T }, and {p : vw (p) = F}. However, if they are not finite, the values of rcertM and rincM will be ω. Proposition 4.2. Let M = ⟨W, R, I, v⟩ and M ′ = ⟨W ′ , R′ , I ′ , v′ ⟩ be two compatible many-logic modal structure. Then: (i) cert(M) ≤ cert(M//M ′ ), (ii) inc(M) ≤ inc(M//M ′ ), (iii) If W ∩W ′ = 0/ then inc(M//M ′ ) = inc(M) + inc(M ′ ) and cert(M//M ′ ) = cert(M) + cert(M ′ ). Proof. Item (i) follows from the fact that |{p : ◦p ∈ Kw }| ≤ |{p : ◦p ∈ Kw ∪ Kw′ }|. The remaining cases are left to the reader. Below we extend these ideas to T -evolution structures. Definition 4.7. Let (T, <) be a flow of time and M T a T -evolution structure, where |Wt | is finite for every t ∈ T . (i) The relative inconsistent measure at t is the function rincMT : T → [0, 1] defined as rincMT (t) = rinc(g(t)) (ii) The certification ratio is the function rcertMT : T → [0, 1] rcertMT (t) = rinc(g(t)) Some useful properties of T -evolution structures may be defined.

102

Merging and Evolution of Data Bases

Definition 4.8. Let T ′ be a path in a T -evolution structure. We say that: (i) Consistency increases along T ′ if and only if for any i, j ∈ T ′ , i < j implies rincMT (i) > rincMT ( j), (ii) Consistency decreases along T ′ if and only if for any i, j ∈ T ′ , i < j implies rincMT (i) < rincMT ( j), (iii) Certification increases along T ′ if and only if for any i, j ∈ T ′ , i < j implies rcertMT (i) < rcertMT ( j), (iv) Certification decreases along T ′ if and only if for any i, j ∈ T ′ , i < j implies rcertMT (i) > rcertMT ( j). We say that a T -evolution structure: (v) Improves consistency if and only if for any i, j ∈ T , i < j implies rinc(i) > rinc( j), (vi) Improves certification if and only if for any i, j ∈ T , i < j implies rcertMT (i) < rcertMT ( j). In the example below we illustrate the measures of inconsistency, certification, relative inconsistency and certified data in a T -evolution structure. Example 4.3. Consider the set PAT = {⟨M1 , w1 ⟩, ⟨M2 , w3 ⟩, ⟨M3 , w5 ⟩} s

s

w2 B3

w4 B3

w6 B3

w1 L6

w3 L6

w5 L6

s

M1

M2

M3

I(w1 ) = L6, I(w2 ) = B3s Kw1 = {◦p, p, q} Kw2 = {p, ¬p, q, ¬q} inc(w1 ) = 0, inc(w2 ) = 2 cert(w1 ) = 1, cert(w2 ) = 0 inc(M1 ) = 2, cert(M1 ) = 1 rinc(M1 ) = 1, rcert(M1 ) = 0.5

I(w3 ) = L6, I(w4 ) = B3s Kw3 = {p, ¬p, q} Kw4 = {p, ¬p, q, ¬q} inc(w3 ) = 1, inc(w4 ) = 2 cert(w3 ) = 0, cert(w4 ) = 0 inc(M2 ) = 3, cert(M2 ) = 0 rinc(M2 ) = 1.5, rcert(M2 ) = 0

I(w3 ) = L6, I(w4 ) = B3s Kw1 = {◦p, p, ◦q, q} Kw2 = {◦p, p, ◦q, q} inc(w1 ) = 0, inc(w2 ) = 0 cert(w1 ) = 2, cert(w2 ) = 2 inc(M3 ) = 0, cert(M3 ) = 4 rinc(M3 ) = 0, rcert(M3 ) = 2

Now consider the T-evolution structure M T where T = {1, 2, 3}, g(t1 ) = M1 , g(t2 ) = M2 , g(t3 ) = M3 . t2 t1 t3

M. Martins & A. Rodrigues rinc(t1 ) = 1 rinc(t2 ) = 1.5 rinc(t3 ) = 0

103 rcert(t1 ) = 0.5 rcert(t2 ) = 0 rcert(t3 ) = 2.

Consistency increases and certification decreases along the path T ′ = {t1 ,t2 }, while consistency decreases and certification increases along the path T ′′ = {t1 ,t3 }. However, with regard to the T-structure M T , neither consistency nor certification increases or decreases.

Final remarks In this paper, we discussed mechanisms for database merging and temporal evolution, emphasizing how informational refinement may be modeled through changes in the underlying lattices. In this setting, inconsistency measures play an important role in characterizing the informational state of databases and in guiding their evolution toward more consistent and certified forms of knowledge. Future research includes the investigation of refinement processes capable of capturing database evolution in terms of certification and inconsistency levels through (T)-evolution structures. Another relevant direction concerns the extension of the lattice (L6) by iteratively introducing new maximal and minimal elements, allowing the representation of finer degrees of certification. Determining the logics generated by the corresponding filters in these extended structures remains an open and promising problem. Acknowledgments. This work is supported by CIDMA (https://ror.org/05pm2mw36) through the Portuguese Foundation for Science and Technology (FCT, https://ror.org/00snfqn58), under grant UID/04106/2025 (https://doi.org/10.54499/UID/04106/2025) and by Project Banksy with reference COMPETE2030-FEDER-00892000. The second author acknowledges support from the National Council for Scientific and Technological Development (CNPq), Brazil, research grant 307889/2025-4.

References [1] N. Belnap (1977): How a computer should think. In G. Ryle, editor: Contemporary Aspects of Philosophy, Oriel Press. Reprinted in New Essays on Belnap-Dunn Logic, Springer, 2019. [2] M. Coniglio & A. Rodrigues (2024): From Belnap-Dunn Four-Valued Logic to Six-Valued Logics of Evidence and Truth. Studia Logica 112(3), pp. 561–606. Available at https: //doi.org/10.1007/s11225-023-10062-5. [3] D. Costa & M. A. Martins (2017): Paraconsistency in hybrid logic. J. Log. Comput. 27(6), pp. 1825–1852. Available at https://doi.org/10.1093/logcom/exw027. [4] D. Costa & M. A. Martins (2018): Inconsistency measures in hybrid logics. In: Measuring inconsistency in information, London: College Publications, pp. 169–194. [5] I. D’Ottaviano & N. da Costa (1970): Sur un probléme de Jaśkowski. Comptes Rendus de l’Académie des Sciences 270(A), pp. 1349–1353. Available at https://gallica.bnf.fr/ ark:/12148/bpt6k480298g/f1359.item.

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[6] A. Freire, A. Madeira & M. A. Martins (2025): Lattices for many-logics modal logic: constructions and representations. In: Artificial intelligence logic and applications. The 4th international conference, AILA 2024, Lanzhou, China, August 10–11, 2024. Proceedings, Singapore: Springer, pp. 32–46. Available at https://doi.org/10.1007/ 978-981-96-0354-1_3. [7] A. Freire & M. A. Martins (2025): Modality across different logics. Logic Journal of the IGPL 33(3), p. jzae082. Available at https://doi.org/10.1093/jigpal/jzae082. [8] A. Freire & M. A. Martins (2026): Regular non-normal modal classicalities. In print. Available at https://doi.org/10.48550/arXiv.2602.20287. [9] A. Freire, M. A. Martins & A. Madeira (2024): The Institution of Many-Logics Modal Logic. In: Recent Trends in Algebraic Development Techniques: 27th IFIP WG 1.3 International Workshop, WADT 2024, Enschede, The Netherlands, July 8, 2024, Revised Selected Papers, Springer-Verlag, Berlin, Heidelberg, p. 94–110. Available at https://doi.org/10.1007/ 978-3-031-88930-1_5. [10] J. Grant & A. Hunter (2006): Measuring inconsistency in knowledgebases. Journal of Intelligent Information Systems 27(2), p. 159–184. Available at https://doi.org/10. 1007/s10844-006-2974-4. [11] J. Grant & M.V. Martinez, editors (2018): Measuring inconsistency in information. 73, London: College Publications. [12] M. A. Martins, A. Rodrigues, M. E. Coniglio & A. R. Freire (2026): On Many-logic Modal Structures and Information-based Logics. Journal of Logic, Language and Information 35, p. 203–231. Available at https://doi.org/10.1007/s10849-026-09467-x. [13] A. Rodrigues & M. Coniglio (2026): Positive, Negative, and Reliable Information in a FirstOrder Logic of Evidence and Truth. arXiv preprint arXiv:2604.18526. arXiv:2604.18526.

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