Strictly Kripke-Complete Logics

• Strictly Kripke-complete logics are uniquely determined by their validating Kripke frames within a logic lattice, ensuring a degree of completeness equal to one.
• They are characterized by iterated splittings in tense-logical frameworks, distinguishing them from ordinary Kripke semantics that require generalized or enriched models.
• The framework extends to strictly positive, fuzzy, and quantified settings, clarifying boundaries where Kripke completeness meets complexity and semantic variations.

Searching arXiv for recent and foundational papers on strictly Kripke-complete logics and related completeness notions. Strictly Kripke-complete logics are logics that are uniquely determined by their validating Kripke frames within a specified lattice of logics. In the tense-logical setting, this uniqueness is formalized by the degree of Kripke-incompleteness: a logic $L$ is strictly Kripke-complete in a lattice $\mathcal L$ iff exactly one logic in $\mathcal L$ has the same class of Kripke frames as $L$, i.e. iff its degree is $1$ (Chen, 6 Jul 2025). Across adjacent literatures, the same topic also appears as a distinction between completeness for ordinary Kripke semantics—standard relational frames with Boolean forcing—and completeness only for enriched or generalized semantics, such as many-valued Kripke-style models, neighborhood frames, algebraic semantics, or topological structures (Lewis-Smith, 2023). The subject is therefore both a lattice-theoretic classification problem and a semantic boundary problem.

1. Definition and semantic scope

In the tense-logical framework, if $L_0$ is a base logic and $L\in \mathsf{NExt}(L_0)$, the degree of Kripke-incompleteness is

$$\mathsf{deg}_{\mathsf{NExt}(L_0)}(L)=\left|\left\{L'\in \mathsf{NExt}(L_0):\mathsf{Fr}(L')=\mathsf{Fr}(L)\right\}\right|.$$

Strict Kripke-completeness is the special case $\mathsf{deg}(L)=1$ (Chen, 6 Jul 2025). This is stronger than mere Kripke-completeness, because it requires not only that $L$ be determined by some frame class, but also that no distinct logic in the ambient lattice share that same frame class.

A second, semantically stricter contrast arises in work on nonclassical logics. There, the relevant question is often whether a logic is complete for ordinary Kripke frames—frames $\mathcal L$0 or $\mathcal L$1 with Boolean truth and the standard implication clause—or only for a generalized Kripke-style semantics with extra algebraic structure, many-valued truth at worlds, or nonstandard forcing clauses (Lewis-Smith, 2023). This broader usage does not reduce to the lattice-theoretic degree notion, but it marks the same conceptual boundary: whether ordinary Kripke semantics is exact, or only an approximation to a richer semantic reality.

2. Lattice-theoretic characterization in tense logics

The most precise recent characterization is given for the lattices $\mathcal L$2, $\mathcal L$3, and $\mathcal L$4. In all three cases, the strictly Kripke-complete logics are exactly the iterated splittings (Chen, 6 Jul 2025). An iterated splitting has the form

$\mathcal L$5

where each $\mathcal L$6 splits the extension lattice obtained at the previous stage. This extends the older modal picture in which union-splittings captured the degree-$\mathcal L$7 logics.

The sharpest difference appears between the transitive and reflexive-transitive cases. In $\mathcal L$8 and $\mathcal L$9, iterated splittings collapse to union-splittings, and strict Kripke-completeness coincides with that simpler notion. In $\mathcal L$0, by contrast, union-splittings are too narrow: the iterated splittings are exactly

$\mathcal L$1

whereas the union-splittings are only $\mathcal L$2 (Chen, 6 Jul 2025). This establishes that iterated splitting, rather than union-splitting, is the correct general notion for tense logics.

The same paper proves a Blok-style dichotomy in all three lattices: $\mathcal L$3 so every logic is either strictly Kripke-complete or shares its frame class with continuum many distinct logics (Chen, 6 Jul 2025). This dichotomy turns strict Kripke-completeness into a classification of the degree-$\mathcal L$4 points of the lattice.

3. Strictly positive implication logics

For strictly positive implication logics, the relevant notion is eq-completeness: for a set $\mathcal L$5 of strictly positive implications,

$\mathcal L$6

for every strictly positive implication $\mathcal L$7, where $\mathcal L$8 is relational consequence over Kripke frames and $\mathcal L$9 is algebraic consequence over meet-semilattices with monotone operators (Kikot et al., 2017). In this setting, eq-completeness is the exact formalization of strict Kripke-completeness for the strictly positive fragment.

A stronger property is complexity: every validating semilattice with operators embeds into the strictly positive reduct of the full complex algebra of a validating Kripke frame. Complexity implies eq-completeness, but not conversely (Kikot et al., 2017). This distinction is central. The paper proves that $L$0 is complex and hence eq-complete, and then develops two general proof methods: embedding into complex algebras and the method of syntactic proxies.

The positive examples are structurally broad. Spi-logics axiomatized by rooted tree-profile Horn implications are complex and therefore eq-complete; so are leapfrog implications and large families of grammar-style existential implications. The paper also proves that $L$1, where $L$2 corresponds to functionality, is complex and thus eq-complete. At the same time, there are natural eq-complete logics that are not complex. The paradigmatic cases are $L$3 for $L$4, the bounded-cluster $L$5-fragments $L$6, and the strictly positive fragment of linear quasiorders $L$7: all are eq-complete, but the corresponding non-complexity theorems show that strict Kripke-completeness is strictly broader than embeddability into complex algebras (Kikot et al., 2017).

The negative side is equally strong. The simple implication $L$8 yields an incomplete spi-logic, as does $L$9. More generally, Horn correspondence does not by itself guarantee completeness, and for non-rooted tree-profile Horn implications the paper proves a generic incompleteness theorem. Finally, it is undecidable, given a finite set $1$0 of strictly positive implications, whether $1$1 is eq-complete or complex (Kikot et al., 2017).

4. Fuzzy and many-valued boundaries

The clearest boundary result for ordinary Kripke completeness in fuzzy logic is that, among propositional fuzzy logics extending Hájek’s $1$2, the only ones that are sound and strongly complete with respect to classes of ordinary Kripke frames or models are the extensions of Gödel logic (Safari et al., 2016). The paper establishes this by an axiom-by-axiom analysis of $1$3 under standard Kripke semantics. The result is exclusionary: $1$4 itself is not characterized by any class of ordinary Kripke frames/models, while Gödel–Dummett logic is sound and strongly complete for reflexive, transitive, connected, persistent Kripke models (Safari et al., 2016).

A later development gives $1$5 a Kripke-style semantics, but not in the ordinary strict sense. The semantics is based on Linear Bova–Montagna structures: worlds form a linear order, atomic valuations are sloping functions into the standard MV-chain $1$6, and implication is interpreted by a nonstandard $1$7-construction over pointwise MV-residua. The resulting semantics is sound and complete for $1$8, and it specializes to ordinary linearly ordered Kripke semantics for Gödel–Dummett logic when values are restricted to $1$9 (Lewis-Smith, 2023). The crucial point is negative: this does not show that $L_0$0 is strictly Kripke-complete in the ordinary sense, because truth at worlds is many-valued, valuations are sloping rather than arbitrary persistent sets, and validity is defined by algebraic inequalities rather than Boolean forcing (Lewis-Smith, 2023).

The finite-model problem for modal Gödel logics displays a related phenomenon. The standard modal Gödel systems are not complete with respect to finite Gödel-Kripke models, and the natural candidate extensions do not restore completeness. The paper resolves the resulting 15-year open problem by introducing new axiomatizations that are complete for witnessed, hence finite, Gödel-Kripke semantics: for crisp models, $L_0$1 and its mono-modal fragments; for valued models, $L_0$2 (Vidal et al., 15 May 2026). Here strictness appears as a finite-semantic strengthening: the intended finite Gödel-Kripke semantics determines a stronger logic than the older axiomatizations.

5. Generalized Kripke-style semantics beyond strictness

Several important logics admit only generalized or nonstandard Kripke-style semantics. For classical first-order logic, one can define classical Kripke models with a preorder of worlds, monotone domains, a primitive relation of strong refutation on atomic formulas, and a monotone predicate of exploding worlds. Forcing and refutation are then defined by orthogonality: $L_0$3 This yields constructive soundness and cut-free completeness for classical logic, but only in an extended technical sense: it is not completeness for ordinary intuitionistic Kripke semantics with standard forcing (0904.0071).

A different example is the logic $L_0$4, a Lewis-style strict implication logic in which strict equivalence is forced to behave as propositional identity. The semantic contribution there is algebraic: a model is a Boolean-algebra-like structure with a modal operator $L_0$5, and the central theorem is

$L_0$6

The paper proves strong algebraic completeness for $L_0$7, extends the framework to $L_0$8, $L_0$9, and $L\in \mathsf{NExt}(L_0)$0, and explicitly does not provide a Kripke completeness theorem for $L\in \mathsf{NExt}(L_0)$1; it states only that $L\in \mathsf{NExt}(L_0)$2 and $L\in \mathsf{NExt}(L_0)$3 have Kripke semantics, whereas for $L\in \mathsf{NExt}(L_0)$4 no known natural Kripke semantics is available (Lewitzka, 2013).

Strictly positive provability logic yields both a positive and a negative case. The reflection calculus $L\in \mathsf{NExt}(L_0)$5 is presented as complete with respect to a natural class of finite Kripke frames. Its extension $L\in \mathsf{NExt}(L_0)$6, however, is not Kripke complete as a full logic: the paper proves that the sequent

$L\in \mathsf{NExt}(L_0)$7

is valid on every Kripke frame validating $L\in \mathsf{NExt}(L_0)$8, but is not derivable in $L\in \mathsf{NExt}(L_0)$9 and is arithmetically invalid (Beklemishev, 2017). The positive residue is fragmentary: the variable-free fragment of $$\mathsf{deg}_{\mathsf{NExt}(L_0)}(L)=\left|\left\{L'\in \mathsf{NExt}(L_0):\mathsf{Fr}(L')=\mathsf{Fr}(L)\right\}\right|.$$0 is complete with respect to the universal Kripke frame $$\mathsf{deg}_{\mathsf{NExt}(L_0)}(L)=\left|\left\{L'\in \mathsf{NExt}(L_0):\mathsf{Fr}(L')=\mathsf{Fr}(L)\right\}\right|.$$1 built from Ignatiev’s frame, and its Lindenbaum algebra is represented by the Ignatiev $$\mathsf{deg}_{\mathsf{NExt}(L_0)}(L)=\left|\left\{L'\in \mathsf{NExt}(L_0):\mathsf{Fr}(L')=\mathsf{Fr}(L)\right\}\right|.$$2-algebra (Beklemishev, 2017).

6. Quantified, transfer, and extension phenomena

In propositionally quantified modal logic, the first completeness level is often quantifiable general frames, not full Kripke frames. Any normal propositionally quantified modal logic containing the Barcan scheme is strongly complete with respect to the class of quantifiable general frames validating it (Ding et al., 2024). A further theorem then gives a sufficient condition for genuine Kripke completeness: if $$\mathsf{deg}_{\mathsf{NExt}(L_0)}(L)=\left|\left\{L'\in \mathsf{NExt}(L_0):\mathsf{Fr}(L')=\mathsf{Fr}(L)\right\}\right|.$$3 is a set of Sahlqvist formulas whose Kripke-frame class has finite diversity, then

$$\mathsf{deg}_{\mathsf{NExt}(L_0)}(L)=\left|\left\{L'\in \mathsf{NExt}(L_0):\mathsf{Fr}(L')=\mathsf{Fr}(L)\right\}\right|.$$4

is sound and strongly complete for the corresponding Kripke frames (Ding et al., 2024). As a special case, the logic of Euclidean Kripke frames receives the finite axiomatization

$$\mathsf{deg}_{\mathsf{NExt}(L_0)}(L)=\left|\left\{L'\in \mathsf{NExt}(L_0):\mathsf{Fr}(L')=\mathsf{Fr}(L)\right\}\right|.$$5

(Ding et al., 2024). This is a direct example of upgrading general-frame completeness to genuine Kripke completeness by additional structural principles.

A different transfer principle is established for modal logics with transitive closure. If a logic $$\mathsf{deg}_{\mathsf{NExt}(L_0)}(L)=\left|\left\{L'\in \mathsf{NExt}(L_0):\mathsf{Fr}(L')=\mathsf{Fr}(L)\right\}\right|.$$6 admits definable filtration (ADF), then its transitive-closure expansion $$\mathsf{deg}_{\mathsf{NExt}(L_0)}(L)=\left|\left\{L'\in \mathsf{NExt}(L_0):\mathsf{Fr}(L')=\mathsf{Fr}(L)\right\}\right|.$$7 also admits definable filtration; hence $$\mathsf{deg}_{\mathsf{NExt}(L_0)}(L)=\left|\left\{L'\in \mathsf{NExt}(L_0):\mathsf{Fr}(L')=\mathsf{Fr}(L)\right\}\right|.$$8 has the finite model property and is Kripke complete (Kikot et al., 2020). The construction iterates to PDL-like expansions with union, composition, and repeated closure. The significance is methodological: Kripke completeness can be preserved under substantial modal enrichment without relying on canonicity.

The modal embedding $$\mathsf{deg}_{\mathsf{NExt}(L_0)}(L)=\left|\left\{L'\in \mathsf{NExt}(L_0):\mathsf{Fr}(L')=\mathsf{Fr}(L)\right\}\right|.$$9 gives a parallel transfer method for intermediate logics. For propositional $\mathsf{deg}(L)=1$0,

$\mathsf{deg}(L)=1$1

Using Kripke semantics for $\mathsf{deg}(L)=1$2, the paper recovers sharp frame completeness theorems for $\mathsf{deg}(L)=1$3, Gödel–Dummett logic $\mathsf{deg}(L)=1$4, and $\mathsf{deg}(L)=1$5: frames with at most two worlds, linearly ordered frames, and frames with a single maximal world, respectively (Lewitzka, 2015). The paper does not claim a general classification of strictly Kripke-complete intermediate logics, but it provides a uniform method for deriving strong frame characterizations.

Quantified modal logic also motivates movement away from ordinary Kripke semantics. For one-way pretransitive Horn modal logics, the quantified logic $\mathsf{deg}(L)=1$6 is already known to be complete with respect to expanding-domain Kripke frames, and the paper proves that it is also complete with respect to constant-domain neighborhood frames (Kudinov, 2021). This is not a separation theorem between neighborhood and Kripke completeness, but it shows that alternative semantics can preserve completeness while changing the domain behavior that ordinary Kripke semantics would force.

7. Boundary examples and related completeness phenomena

Several complete logics sit on sharp semantic or finitary boundaries without themselves furnishing new degree-$\mathsf{deg}(L)=1$7 classifications. In product modal logic, if $\mathsf{deg}(L)=1$8 and $\mathsf{deg}(L)=1$9 are Kripke complete and consistent, then local tabularity of $L$0 requires local tabularity of both factors, but that is not sufficient. The central example is

$L$1

which is Kripke complete as a product logic, not locally tabular, yet every proper extension is locally tabular (Shapirovsky et al., 2024). By contrast, $L$2 is not prelocally tabular (Shapirovsky et al., 2024). These examples do not redefine strict Kripke-completeness, but they show that complete logics can lie exactly at the boundary where stronger properties collapse.

A comparable boundary appears in intuitionistic modal logic with topological semantics. A family including $L$3, $L$4, $L$5, $L$6, $L$7, $L$8, $L$9, $\mathcal L$00, $\mathcal L$01, and $\mathcal L$02 is strongly complete with respect to birelational Kripke semantics (Aguilera et al., 25 Apr 2026). On the topological side, however, the $\mathcal L$03-family is strongly complete for Alexandroff bitopological models, whereas the $\mathcal L$04-family requires Alexandroff tritopological derivative models. A key counterexample shows that naive bitopological derivative semantics validates

$\mathcal L$05

which is not derivable in the intended $\mathcal L$06-logics (Aguilera et al., 25 Apr 2026). Thus the paper does not exhibit non-Kripke-complete logics; rather, it shows that topological completeness can require semantic structure richer than the most obvious Kripke-inspired topology.

Taken together, these developments give a layered picture of strict Kripke-completeness. In its exact lattice-theoretic form, it is now characterized by iterated splittings in major tense-logical lattices (Chen, 6 Jul 2025). In the strictly positive setting, it is captured by eq-completeness and separated from the stronger notion of complexity (Kikot et al., 2017). In fuzzy, truth-theoretic, classical, quantified, and topological settings, the main lesson is often negative or qualified: ordinary Kripke semantics may be too strong, too weak, or simply not the right target, and generalized Kripke-style semantics may recover completeness without delivering strict ordinary Kripke-completeness [(Lewis-Smith, 2023); (0904.0071); (Lewitzka, 2013)]. The topic is therefore best understood not as a single theorem schema, but as a family of classification and boundary problems centered on when frame semantics determines a logic exactly, uniquely, and without semantic enrichment.