On Many-logic modal structures and information-based logics

Abstract: This paper proposes an approach to information-based logics using many-logic modal structures (MLMS). These structures can express accessibility relations between worlds with different underlying logics by anchoring them to a base lattice, which contains the semantics of each logic as a down-complete sublattice. MLMS are suitable for representing connections between information states (i.e., configurations of databases) and the evolution of information states over time. We will illustrate the application of MLMS by means of the six-valued logic of evidence and truth LET+K , related to the lattice L6, and some four-, three-, and two-valued logics related to down-complete sublattices of L6. These logics are capable of representing paracomplete, paraconsistent, and classical contexts with six-, four-, three-, and two-valued scenarios.

• The paper develops a framework using down-complete sublattices within a base lattice to support multiple logics within a single Kripke-style semantics, addressing varying underlying logics in different layers.
• Demonstrates nine information-based logics, including examples like LET$_K$ and BS4, enhanced with twist structures and bivaluation semantics to integrate classical reasoning scenarios with certification.
• Validities are characterized on frames structured by logic assignments that align with failing necessitation, holding of axiom (K), and the need for modified frame axioms for Euclidean structures.

This paper develops a framework for information-based logics built on many-logic modal structures (MLMS), extending prior work by Freire and Martins. The central technical innovation is the replacement of ordinary sublattices with down-complete sublattices (dc-sublattices) of a base lattice, which broadens the class of logics that can coexist within a single Kripke-style semantics. The framework is instantiated using the six-valued logic of evidence and truth LET$_K$ and its associated lattice $L6$, together with a family of four-, three-, and two-valued logics defined on dc-sublattices of $L6$ (2605.07898).

Motivation and background

The paper situates itself in the Belnap–Dunn tradition, in which FDE is read as an information-based logic for a question-answering system whose database may be inconsistent or incomplete. The authors' guiding questions are: how can information be transferred between worlds governed by different logics, and how should $\Box$ be interpreted when worlds have heterogeneous underlying logics? Their answer anchors all local logics to a common base lattice: each world's logic has as its semantic domain a dc-sublattice of the base lattice, and values from other worlds are "seen" through the down-interpretation, which maps a value $x$ to the join in the sublattice of all sublattice values below $x$.

Many-logic modal structures

An MLMS over a set $\LAT$ of dc-sublattices of a complete lattice $\mathsf{L}$ is a tuple $\langle W, R, I, v\rangle$, where $I$ assigns to each world $L6$0 a matrix logic whose domain is a member of $L6$1, and $L6$2 assigns sentential letters values in $L6$3. The modal clause is non-standard:

$L6$4

i.e., the infimum, computed in the local lattice $L6$5, of the values of $L6$6 at accessible worlds after down-interpretation into $L6$7. The reading offered is that $L6$8 means "$L6$9 has value $L6$0 for an agent accessing the information available from $L6$1."

Two lattice-theoretic results support the machinery: Proposition 3 shows that joins and meets commute appropriately with the down-interpretation under inequalities, and Proposition 4 shows that for $L6$2-lattices, $L6$3. Since $L6$4 is finite and distributive, these yield the simplification that $L6$5 equals the down-interpretation of the $L6$6-infimum of the accessed values — so the modal computation can be performed globally and then projected locally.

From four to six scenarios

The paper extends Belnap's four scenarios ($L6$7, $L6$8, $L6$9, $\Box$0: told true, told false, both, neither) by adding two "strong" values $\Box$1 and $\Box$2, representing certified (reliable) positive and negative information. This yields the six scenarios of LET$\Box$3, whose classicality operator $\Box$4 recovers classical reasoning for sentences in its scope; $\Box$5 functions as a certification mark.

The family of logics considered, all extensions of FDE with material implication validating modus ponens and the deduction theorem, includes:

Logic	Lattice	Designated values	Character
LET$\Box$6	$\Box$7	$\Box$8	paraconsistent + paracomplete
FDE$\Box$9	$x$0	$x$1	FDE + implication + bottom
BS4	$x$2	$x$3	strong FDE variant
LP$x$4	$x$5	$x$6	paraconsistent only
J3 / LFI1	$x$7	$x$8	paraconsistent, certified
K3$x$9	$x$0	$x$1	paracomplete only
N3$x$2	$x$3	$x$4	paracomplete, certified
CL$x$5, CL$x$6	$x$7, $x$8	$x$9, $\LAT$0	classical

The weak lattices use $\LAT$1 (non-certified), the strong ones $\LAT$2 (certified). Notably, the authors introduce a new four-valued logic obtained by dropping either excluded middle or explosion from the three-valued members, and they observe that the matrices of the three- and two-valued logics arise as submatrices of the six-valued tables — though neither BS4 nor FDE$\LAT$3 is a submatrix of LET$\LAT$4's tables.

Semantics via twist structures

All nine logics receive uniform presentations through non-deterministic two-valued bivaluation semantics and deterministic twist structures over triples $\LAT$5 recording $\LAT$6, $\LAT$7, and $\LAT$8. The six values correspond exactly to admissible snapshots; the triples $\LAT$9 and $\mathsf{L}$0 are excluded because paracomplete and paraconsistent scenarios cannot carry certification. Designated values are those with $\mathsf{L}$1. In CL$\mathsf{L}$2, $\mathsf{L}$3 becomes a top particle; in CL$\mathsf{L}$4, a bottom particle. The equivalence between the bivaluation consequence relations and the matrix/twist consequence relations is asserted via the snapshot correspondence, though the paper does not present full soundness/completeness proofs for each logic individually.

Worked examples

Four examples illustrate the intended applications. In one, an $\mathsf{L}$5-world with global access sees a contradictory $\mathsf{L}$6 at an accessible $\mathsf{L}$7-database, forcing $\mathsf{L}$8 despite $\mathsf{L}$9 holding everywhere accessible — a direct illustration of failure of necessitation-like behavior. Another example models users with different privileges viewing a single $\langle W, R, I, v\rangle$0 database "through $\langle W, R, I, v\rangle$1-glasses" (which map $\langle W, R, I, v\rangle$2 to $\langle W, R, I, v\rangle$3 and everything below $\langle W, R, I, v\rangle$4 to $\langle W, R, I, v\rangle$5) versus "$\langle W, R, I, v\rangle$6-glasses" (which certify definite information). A final example uses a transitive frame across three temporal stages to model the evolution of a database resolving a contradiction: $\langle W, R, I, v\rangle$7 moves from $\langle W, R, I, v\rangle$8 through $\langle W, R, I, v\rangle$9 to $I$0 as the inconsistent world is repaired.

Modal validities and frame characterization

The paper defines frames as triples $I$1, since the logic assignment is part of the frame structure. The results here are mixed and informative:

• Failure of validity transfer: if $I$2 holds at all accessible worlds, $I$3 need not hold at $I$4, because a designated value at $I$5 may be down-interpreted to a non-designated value at $I$6 (e.g., $I$7 seen from an $I$8-world becomes $I$9). Consequently, the necessitation rule fails, even for formulas valid in every logic of $L6$00.
• Axiom (K) holds in all frames.
• Reflexive frames satisfy (T) and transitive frames satisfy (4); the proofs rely only on upward closure of designated sets and the property that $L6$01 iff $L6$02 and $L6$03, suggesting generalizability beyond $L6$04.
• Euclidean frames are not characterized by (5): a counterexample with three mutually accessing worlds of types FDE$L6$05, K3$L6$06, and LP$L6$07 refutes it. Following Freire and Coniglio's strategy, the modified formula $L6$08 is proved to characterize Euclidean frames exactly, exploiting the fact that $L6$09 takes values only in $L6$10.
• Duality: defining $L6$11 by the dual supremum clause does not generally give $L6$12 across mixed-logic structures. The paper introduces an up-interpretation $L6$13 and proves that with the clause $L6$14, the duality $L6$15 holds at every world of every structure. This restoration of duality is the strongest structural result in the modal section.

The proof of the Euclidean characterization theorem notes explicitly that the appeal to classical negation suggests Euclideanness is tied more closely to classical valuations than other frame properties, but the nature of this connection is left unclarified.

Limitations and open questions

Several limitations are acknowledged directly. First, the application of these logics to actual database management systems is stated to fall outside the scope of the paper; the interpretations remain conceptual. Second, the soundness and completeness of the twist-structure semantics relative to the bivaluation semantics are presented via construction rather than proved in detail for each logic. Third, the behavior of axioms (5), (B), and (D) depends on the choice among at least four candidate definitions of $L6$16, none of which is canonically privileged; the paper shows the alternatives diverge (e.g., $L6$17 vs. $L6$18 under the plain supremum clause). Fourth, the characterization results for reflexivity and transitivity are conjectured to generalize beyond $L6$19 and $L6$20, but no general theorem is established. Open problems named include: a merging operation on databases that provably avoids trivial worlds; temporal modalities (past, future, next, until, since) for evolving structures; and quantitative measures of inconsistency and certification based on counts of contradictory or certified variables.

Conclusion

The paper delivers a technically coherent extension of MLMS semantics to dc-sublattices, a uniform twist-structure presentation of nine information-based logics spanning classical, paracomplete, paraconsistent, and certified scenarios, and a first map of which standard modal axioms survive when worlds carry different logics. Its most substantive findings are the failure of necessitation, the exact characterization of Euclidean frames by $L6$21, and the recovery of $L6$22/$L6$23 duality via the up-interpretation. Whether the framework supports effective implementation in database systems, and whether the frame-characterization results extend to arbitrary base lattices, remain open.