Degree of Kripke-Incompleteness

• Degree of Kripke-incompleteness is defined as the number of logics in a lattice that share exactly the same class of Kripke frames, resulting in either a singleton or continuum (2^aleph0) outcome.
• The analysis employs Blok-style dichotomies and splitting techniques to distinguish strictly Kripke-complete logics from those with a continuum of variants in tense-logical lattices.
• Reflective unfolding and tense-adapted Jankov formulas are key methodologies used to construct continuum families, clarify modal distinctions, and understand lattice structures in tense logics.

Searching arXiv for the cited work and closely related papers on Kripke incompleteness and Blok-style dichotomies. The degree of Kripke-incompleteness of a logic $L$ in a lattice $\mathcal{L}$ of logics is the cardinality of the set of logics in $\mathcal{L}$ that validate exactly the same class of Kripke frames as $L$. In the tense-logical setting, this notion is formulated against a background in which completeness is handled via general frames, while the degree itself is defined from the Kripke-frame class $\mathrm{Fr}(L)$. The recent systematic treatment for tense logics establishes a Blok-style dichotomy for three lattices—$\mathfrak{K}=\mathrm{NExt}(K_t)$, $\mathfrak{LT}=\mathrm{NExt}(K4_t)$, and $\mathrm{NExt}(S4_t)$—showing that every logic in each of these lattices has degree of Kripke-incompleteness either $1$ or $2^{\aleph_0}$, and identifying the degree-$\mathcal{L}$0 logics with iterated splittings (Chen, 6 Jul 2025).

1. Definition and semantic framework

Let $\mathcal{L}$1 be a lattice of logics, for example $\mathcal{L}$2. For $\mathcal{L}$3, the degree of Kripke-incompleteness is defined by

$\mathcal{L}$4

Here $\mathcal{L}$5 denotes the class of Kripke frames validating $\mathcal{L}$6, with equality of frame classes taken up to isomorphism, and isomorphic frames identified (Chen, 6 Jul 2025). A logic is strictly Kripke-complete in $\mathcal{L}$7 iff $\mathcal{L}$8 (Chen, 6 Jul 2025).

For tense logics, the language has a denumerable set of propositional variables, Boolean connectives, and two unary modal operators $\mathcal{L}$9 and $\mathcal{L}$0, interpreted respectively as future and past necessity. Their duals are $\mathcal{L}$1 and $\mathcal{L}$2 (Chen, 6 Jul 2025). A Kripke frame is a pair $\mathcal{L}$3, with converse relation $\mathcal{L}$4, while a general frame is $\mathcal{L}$5 where $\mathcal{L}$6 is closed under Boolean operations and under $\mathcal{L}$7 and $\mathcal{L}$8 (Chen, 6 Jul 2025). Valuations extend by

$\mathcal{L}$9

$L$0

These semantic clauses are part of the basic setup for the tense-logical results (Chen, 6 Jul 2025).

A central subtlety is that the paper proves completeness using general frames, but defines the degree using Kripke frames. The constructions are arranged so that added general-frame constraints do not produce new Kripke frames, and an immediate inequality is

$L$1

where $L$2 is the degree of the finite model property, i.e. the number of logics in $L$3 sharing the same $L$4 (Chen, 6 Jul 2025). In the three tense lattices studied, one in fact has $L$5 (Chen, 6 Jul 2025).

The general-frame completeness theorem used throughout states that for any tense logic $L$6,

$L$7

where $L$8 is the class of all general frames validating $L$9 and $\mathrm{Fr}(L)$0 its rooted subframes (Chen, 6 Jul 2025). Rootedness is defined by $\mathrm{Fr}(L)$1 for some $\mathrm{Fr}(L)$2, with $\mathrm{Fr}(L)$3 generated from forward and backward steps (Chen, 6 Jul 2025). This rooted general-frame perspective is structurally important in the tense proofs.

2. Modal background and the emergence of the degree notion

The classical point of reference is Blok’s dichotomy theorem for normal modal logics: for every $\mathrm{Fr}(L)$4, the degree of Kripke-incompleteness is either $\mathrm{Fr}(L)$5 or $\mathrm{Fr}(L)$6 (Chen, 6 Jul 2025). In the modal case, the proof relies on splitting elements in $\mathrm{Fr}(L)$7, on Jankov–Fine formulas isolating finite frames, and on canonical frame constructions that preserve splitting characterizations (Chen, 6 Jul 2025). The tense-logical results are presented explicitly as a generalization of this theorem (Chen, 6 Jul 2025).

The broader literature on incompleteness supplies important context, but does not always formalize the same cardinal-valued degree. Litak’s “A continuum of incomplete intermediate logics” constructs $\mathrm{Fr}(L)$8 pairwise distinct Kripke-incomplete intermediate logics by setting

$\mathrm{Fr}(L)$9

for $\mathfrak{K}=\mathrm{NExt}(K_t)$0, where the $\mathfrak{K}=\mathrm{NExt}(K_t)$1 are independent Jankov formulas (Litak, 2018). That paper explicitly notes that no formal numeric “degree of Kripke-incompleteness” is defined there, but that the continuum construction naturally motivates gradations by the cardinality and structure of $\mathfrak{K}=\mathrm{NExt}(K_t)$2, by witness complexity, and by lattice position (Litak, 2018). This suggests that the degree notion later used for tense logics can be viewed as a precise lattice-theoretic refinement of a more qualitative incompleteness landscape.

A different comparison point is Thomason’s logic $\mathfrak{K}=\mathrm{NExt}(K_t)$3, a normal modal logic strictly between $\mathfrak{K}=\mathrm{NExt}(K_t)$4 and $\mathfrak{K}=\mathrm{NExt}(K_t)$5, shown to be Kripke-incomplete and in fact incomplete with respect to any class of complete Boolean algebras with operators (Vosmaer, 2012). That work interprets “degree” only informally, as the robustness with which a logic resists semantic representation, and identifies Thomason’s example as “completely incomplete” in the sense of BAO-incompleteness (Vosmaer, 2012). This does not coincide with the lattice-theoretic degree $\mathfrak{K}=\mathrm{NExt}(K_t)$6, but it shows that incompleteness admits several distinct but related measures.

3. The tense-logical lattices and the dichotomy theorems

The tense-logical results are stated for three lattices. The base systems are $\mathfrak{K}=\mathrm{NExt}(K_t)$7, the least normal tense logic; $\mathfrak{K}=\mathrm{NExt}(K_t)$8, obtained from $\mathfrak{K}=\mathrm{NExt}(K_t)$9 by adding transitivity axioms for future and past; and $\mathfrak{LT}=\mathrm{NExt}(K4_t)$0, obtained from $\mathfrak{LT}=\mathrm{NExt}(K4_t)$1 by adding the $\mathfrak{LT}=\mathrm{NExt}(K4_t)$2-axioms for reflexivity (Chen, 6 Jul 2025). The paper also notes that frames validating $\mathfrak{LT}=\mathrm{NExt}(K4_t)$3 are non-degenerate clusters (Chen, 6 Jul 2025).

The three lattices and their degree-$\mathfrak{LT}=\mathrm{NExt}(K4_t)$4 logics can be summarized as follows.

Lattice	Degree-$\mathfrak{LT}=\mathrm{NExt}(K4_t)$5 logics	Splitting characterization
$\mathfrak{LT}=\mathrm{NExt}(K4_t)$6	$\mathfrak{LT}=\mathrm{NExt}(K4_t)$7, $\mathfrak{LT}=\mathrm{NExt}(K4_t)$8	iterated splittings = union-splittings
$\mathfrak{LT}=\mathrm{NExt}(K4_t)$9	$\mathrm{NExt}(S4_t)$0, $\mathrm{NExt}(S4_t)$1	iterated splittings = union-splittings
$\mathrm{NExt}(S4_t)$2	$\mathrm{NExt}(S4_t)$3 and every extension of $\mathrm{NExt}(S4_t)$4	iterated splittings = strictly Kripke-complete

For $\mathrm{NExt}(S4_t)$5, every logic $\mathrm{NExt}(S4_t)$6 satisfies

$\mathrm{NExt}(S4_t)$7

and in fact

$\mathrm{NExt}(S4_t)$8

The strictly Kripke-complete logics in $\mathrm{NExt}(S4_t)$9 are exactly the iterated splittings, which in this lattice coincide with the union-splittings, namely $1$0, where

$1$1

Here $1$2 is the one-point frame with empty relation (Chen, 6 Jul 2025).

For $1$3, one likewise has

$1$4

Again the strictly Kripke-complete logics are exactly the iterated splittings, and in this lattice they are the union-splittings $1$5 with

$1$6

The same axiom excludes the dead-end frame $1$7 in rooted transitive frames (Chen, 6 Jul 2025).

For $1$8, every logic $1$9 satisfies

$2^{\aleph_0}$0

and again

$2^{\aleph_0}$1

In this lattice, iterated splittings are exactly the strictly Kripke-complete logics, but they do not coincide with union-splittings. The iterated splittings are precisely

$2^{\aleph_0}$2

that is, $2^{\aleph_0}$3 itself and every extension of $2^{\aleph_0}$4 (Chen, 6 Jul 2025).

4. Strict Kripke-completeness, splittings, and lattice structure

A splitting pair in a lattice $2^{\aleph_0}$5 is a pair $2^{\aleph_0}$6 such that for every logic $2^{\aleph_0}$7 in the lattice, exactly one of $2^{\aleph_0}$8 and $2^{\aleph_0}$9 holds (Chen, 6 Jul 2025). Equivalently, $\mathcal{L}$00 is completely meet-prime and its complement is completely join-prime (Chen, 6 Jul 2025). A union-splitting is a logic of the form $\mathcal{L}$01 for some family of splittings, and an iterated splitting is a logic obtained by successive splitting inside sublattices above the previous split, with the base logic counted as depth $\mathcal{L}$02 (Chen, 6 Jul 2025).

In $\mathcal{L}$03 and $\mathcal{L}$04 there is, by Kracht’s result, a unique splitting pair, namely $\mathcal{L}$05 (Chen, 6 Jul 2025). The paper proves that in both lattices

$\mathcal{L}$06

and these are exactly the strictly Kripke-complete logics (Chen, 6 Jul 2025). Thus degree $\mathcal{L}$07 is exceptionally sparse in the non-reflexive and transitive tense settings considered.

The $\mathcal{L}$08 case is structurally different. Kracht’s theorem gives exactly two splitting pairs in $\mathcal{L}$09:

$\mathcal{L}$10

Consequently, union-splittings are fewer than iterated splittings (Chen, 6 Jul 2025). The paper proves that the iterated splittings are precisely $\mathcal{L}$11, and that these coincide with the strictly Kripke-complete logics (Chen, 6 Jul 2025). The chain $\mathcal{L}$12 is described as isomorphic to $\mathcal{L}$13, with elements $\mathcal{L}$14, the cluster logics (Chen, 6 Jul 2025).

This distinction between $\mathcal{L}$15 and $\mathcal{L}$16 on the one hand and $\mathcal{L}$17 on the other is one of the main structural outcomes. It reveals that the characterization “strictly Kripke-complete iff union-splitting,” familiar from the modal setting, persists in the first two tense lattices but fails in the reflexive-transitive tense lattice (Chen, 6 Jul 2025). A plausible implication is that reflexive-transitive tense semantics supports a finer interaction between lattice-theoretic primeness and frame-theoretic completeness than the $\mathcal{L}$18 and $\mathcal{L}$19 settings do.

5. Proof methods and continuum constructions

The main techniques adapt modal incompleteness methods to the tense language and, in several places, replace modal canonical-frame arguments by tense-specific constructions (Chen, 6 Jul 2025). One key device is reflective unfolding. Given frames $\mathcal{L}$20 and $\mathcal{L}$21 with designated points $\mathcal{L}$22 and $\mathcal{L}$23, one forms $\mathcal{L}$24 by disjoint union with identification and added forward/backward edges, and takes a transitive closure $\mathcal{L}$25 when needed (Chen, 6 Jul 2025). Iterating this yields “book” frames $\mathcal{L}$26 whose reachability degree $\mathcal{L}$27 grows linearly with $\mathcal{L}$28, while a surjective $\mathcal{L}$29-morphism back to $\mathcal{L}$30 remains available (Chen, 6 Jul 2025). This is used to obtain, for any $\mathcal{L}$31, a finite rooted frame refuting a given formula and with $\mathcal{L}$32 (Chen, 6 Jul 2025).

A second device is the family of master modalities $\mathcal{L}$33, defined inductively by

$\mathcal{L}$34

These express reachability of depth $\mathcal{L}$35 by alternating future and past steps (Chen, 6 Jul 2025). The formula

$\mathcal{L}$36

forces finite reachability degree at most $\mathcal{L}$37 (Chen, 6 Jul 2025). The paper treats these as a tense analogue of “pre-transitivity” behavior (Chen, 6 Jul 2025).

A third ingredient is a tense-adapted Jankov formula construction. For a finite rooted frame $\mathcal{L}$38, the formula $\mathcal{L}$39 is defined so that for any $\mathcal{L}$40-transitive rooted general frame $\mathcal{L}$41,

$\mathcal{L}$42

This replaces the modal use of canonical frames in isolating finite frames (Chen, 6 Jul 2025).

The central step in the continuum construction uses the intersection lemma for rooted classes:

$\mathcal{L}$43

for rooted classes $\mathcal{L}$44 and tense logics $\mathcal{L}$45 (Chen, 6 Jul 2025). Starting with a formula $\mathcal{L}$46 refuted by a finite rooted frame $\mathcal{L}$47 with sufficiently large $\mathcal{L}$48, the paper constructs, for each $\mathcal{L}$49, a general frame $\mathcal{L}$50 by combining $\mathcal{L}$51 with a specially designed frame $\mathcal{L}$52 that syntactically encodes $\mathcal{L}$53 (Chen, 6 Jul 2025). One then sets

$\mathcal{L}$54

The proof shows that $\mathcal{L}$55 while the $\mathcal{L}$56 are pairwise distinct, thereby obtaining continuum many different logics with the same Kripke-frame class (Chen, 6 Jul 2025).

For $\mathcal{L}$57 and $\mathcal{L}$58 the encoding uses $\mathcal{L}$59-chains with distinguished points $\mathcal{L}$60 and separator formulas $\mathcal{L}$61 and $\mathcal{L}$62, together with

$\mathcal{L}$63

Then $\mathcal{L}$64 iff $\mathcal{L}$65 (Chen, 6 Jul 2025). For $\mathcal{L}$66 the construction is more delicate: it uses a “Nishimura–Rieger-like ladder” with two interleaved chains $\mathcal{L}$67 and $\mathcal{L}$68, special points $\mathcal{L}$69, and separator formulas $\mathcal{L}$70, with

$\mathcal{L}$71

again distinguishing the $\mathcal{L}$72 (Chen, 6 Jul 2025).

6. Examples, comparisons, and open directions

Concrete degree-$\mathcal{L}$73 examples are explicitly listed. In $\mathcal{L}$74 the strictly Kripke-complete logics are $\mathcal{L}$75 and

$\mathcal{L}$76

the logic asserting that everywhere there is either a successor or a predecessor (Chen, 6 Jul 2025). In $\mathcal{L}$77 the same pattern holds with $\mathcal{L}$78 in place of $\mathcal{L}$79 (Chen, 6 Jul 2025). In $\mathcal{L}$80 the degree-$\mathcal{L}$81 logics are $\mathcal{L}$82 itself and every extension of $\mathcal{L}$83; for instance, $\mathcal{L}$84 is an iterated splitting and strictly Kripke-complete (Chen, 6 Jul 2025).

The degree-$\mathcal{L}$85 cases are equally explicit. In $\mathcal{L}$86, any logic other than $\mathcal{L}$87 or $\mathcal{L}$88 has degree $\mathcal{L}$89 (Chen, 6 Jul 2025). In $\mathcal{L}$90, any logic other than $\mathcal{L}$91 or $\mathcal{L}$92 has degree $\mathcal{L}$93 (Chen, 6 Jul 2025). In $\mathcal{L}$94, any logic not in $\mathcal{L}$95 has degree $\mathcal{L}$96 (Chen, 6 Jul 2025). Thus in each of the three lattices the dichotomy is exhaustive.

The comparison with modal and intermediate logics is instructive. In the modal case, Blok’s dichotomy states $\mathcal{L}$97 for every $\mathcal{L}$98, and the strictly Kripke-complete logics are precisely union-splittings (Chen, 6 Jul 2025). The tense results recover this pattern for $\mathcal{L}$99 and $\mathcal{L}$00, but not for $\mathcal{L}$01, where strict Kripke-completeness matches iterated splitting rather than union-splitting (Chen, 6 Jul 2025). By contrast, Litak’s continuum of incomplete intermediate logics shows that a continuum phenomenon also appears in the superintuitionistic setting, although there it is not packaged as a formal cardinal-valued degree (Litak, 2018). Thomason’s logic, finally, illustrates a different axis: not multiplicity of logics sharing a frame class, but semantic resistance across Kripke, neighborhood, and complete BAO semantics (Vosmaer, 2012).

The tense paper also records several consequences and open problems. For all three lattices,

$\mathcal{L}$02

so the degree of Kripke-incompleteness coincides with the degree of the finite model property, and the same dichotomy holds for $\mathcal{L}$03 (Chen, 6 Jul 2025). The constructions for $\mathcal{L}$04 and $\mathcal{L}$05 rely on finite transitivity via the master modalities $\mathcal{L}$06, whereas $\mathcal{L}$07 is not finitely transitive, requiring the special $\mathcal{L}$08 constructions together with bounds such as $\mathcal{L}$09, $\mathcal{L}$10, $\mathcal{L}$11, and the Grzegorczyk-like formulas $\mathcal{L}$12 (Chen, 6 Jul 2025). Extending the dichotomy to other finitely transitive base systems such as $\mathcal{L}$13 and $\mathcal{L}$14 is identified as an open direction, and the text notes that Kracht showed infinitely many splittings in such lattices, suggesting more complex behavior (Chen, 6 Jul 2025). It also remains open whether there exists a tense base logic $\mathcal{L}$15 such that $\mathcal{L}$16 exhibits anti-dichotomy for the finite model property (Chen, 6 Jul 2025).

Taken together, these results place the degree of Kripke-incompleteness at the intersection of frame semantics, lattice theory, and incompleteness constructions. In the tense setting studied in (Chen, 6 Jul 2025), the invariant is sharply dichotomous, tightly linked to splitting structure, and technically governed by reflective unfolding, tense-adapted Jankov formulas, and separator-based continuum constructions. This suggests that, at least for the lattices considered there, Kripke incompleteness is not graded by a rich spectrum of intermediate cardinalities, but by a rigid alternative between strict uniqueness and continuum-sized ambiguity.