---
title: Degree of Kripke-Incompleteness
url: https://www.emergentmind.com/topics/degree-of-kripke-incompleteness
type: topic
---

# Degree of Kripke-Incompleteness

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-$1$ logics with iterated splittings [2507.04533].

## 1. Definition and semantic framework

Let $\mathcal{L}$ be a lattice of logics, for example $\mathcal{L}=\mathrm{NExt}(K_t)$. For $L\in\mathcal{L}$, the degree of Kripke-incompleteness is defined by
$$
\deg_{\mathcal{L}}(L):=\left|\{L'\in\mathcal{L}:\mathrm{Fr}(L')=\mathrm{Fr}(L)\}\right|.
$$
Here $\mathrm{Fr}(L)$ denotes the class of Kripke frames validating $L$, with equality of frame classes taken up to isomorphism, and isomorphic frames identified [2507.04533]. A logic is strictly Kripke-complete in $\mathcal{L}$ iff $\deg_{\mathcal{L}}(L)=1$ [2507.04533].

For tense logics, the language has a denumerable set of propositional variables, Boolean connectives, and two unary modal operators $G$ and $H$, interpreted respectively as future and past necessity. Their duals are $F\varphi:=\neg G\neg\varphi$ and $P\varphi:=\neg H\neg\varphi$ [2507.04533]. A Kripke frame is a pair $F=(X,R)$, with converse relation $\check{R}:=\{(y,x):R(x,y)\}$, while a general frame is $(X,R,A)$ where $A\subseteq\mathcal{P}(X)$ is closed under Boolean operations and under $R[\cdot]$ and $\check{R}[\cdot]$ [2507.04533]. Valuations extend by
$$
V(G\varphi)=R[V(\varphi)],\qquad V(H\varphi)=\check{R}[V(\varphi)],
$$
$$
V(F\varphi)=X\setminus \check{R}[X\setminus V(\varphi)],\qquad V(P\varphi)=X\setminus R[X\setminus V(\varphi)].
$$
These semantic clauses are part of the basic setup for the tense-logical results [2507.04533].

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
$$
\deg_{\mathcal{L}}(L)\leq df_{\mathcal{L}}(L),
$$
where $df_{\mathcal{L}}(L)$ is the degree of the finite model property, i.e. the number of logics in $\mathcal{L}$ sharing the same $\mathrm{Fin}(L)$ [2507.04533]. In the three tense lattices studied, one in fact has $\deg_{\mathcal{L}}(L)=df_{\mathcal{L}}(L)$ [2507.04533].

The general-frame completeness theorem used throughout states that for any tense logic $L$,
$$
L=\mathrm{Log}(\mathrm{GF}(L))=\mathrm{Log}(\mathrm{GF}_r(L)),
$$
where $\mathrm{GF}(L)$ is the class of all general frames validating $L$ and $\mathrm{GF}_r(L)$ its rooted subframes [2507.04533]. Rootedness is defined by $X=R^{\#\omega}[x]$ for some $x\in X$, with $R^{\#}$ generated from forward and backward steps [2507.04533]. 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 $L\in\mathrm{NExt}(K)$, the degree of Kripke-incompleteness is either $1$ or $2^{\aleph_0}$ [2507.04533]. In the modal case, the proof relies on splitting elements in $\mathrm{NExt}(K)$, on Jankov–Fine formulas isolating finite frames, and on canonical frame constructions that preserve splitting characterizations [2507.04533]. The tense-logical results are presented explicitly as a generalization of this theorem [2507.04533].

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 $2^{\aleph_0}$ pairwise distinct Kripke-incomplete intermediate logics by setting
$$
L_X:=\mathrm{IPC}+\{\delta,\kappa,bb_2\}\cup\{J(F_n):n\in X\},
$$
for $X\subseteq\omega$, where the $J(F_n)$ are independent Jankov formulas [1808.06284]. 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 $X$, by witness complexity, and by lattice position [1808.06284]. 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 $L$, a normal modal logic strictly between $T$ and $S4$, shown to be Kripke-incomplete and in fact incomplete with respect to any class of complete Boolean algebras with operators [1202.3268]. 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 [1202.3268]. This does not coincide with the lattice-theoretic degree $\deg_{\mathcal{L}}(L)$, 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 $K_t$, the least normal tense logic; $K4_t$, obtained from $K_t$ by adding transitivity axioms for future and past; and $S4_t$, obtained from $K4_t$ by adding the $T$-axioms for reflexivity [2507.04533]. The paper also notes that frames validating $S5_t$ are non-degenerate clusters [2507.04533].

The three lattices and their degree-$1$ logics can be summarized as follows.

| Lattice | Degree-$1$ logics | Splitting characterization |
|---|---|---|
| $\mathfrak{K}=\mathrm{NExt}(K_t)$ | $K_t$, $L^*$ | iterated splittings = union-splittings |
| $\mathfrak{LT}=\mathrm{NExt}(K4_t)$ | $K4_t$, $L^*$ | iterated splittings = union-splittings |
| $\mathrm{NExt}(S4_t)$ | $S4_t$ and every extension of $S5_t$ | iterated splittings = strictly Kripke-complete |

For $\mathfrak{K}=\mathrm{NExt}(K_t)$, every logic $L$ satisfies
$$
\deg_{\mathfrak{K}}(L)\in\{1,2^{\aleph_0}\},
$$
and in fact
$$
\deg_{\mathfrak{K}}(L)=df_{\mathfrak{K}}(L)\in\{1,2^{\aleph_0}\}.
$$
The strictly Kripke-complete logics in $\mathfrak{K}$ are exactly the iterated splittings, which in this lattice coincide with the union-splittings, namely $\{K_t,L^*\}$, where
$$
L^*:=K_t/\mathrm{Log}(\bullet)=K_t\oplus(\langle F\rangle\top\vee \langle P\rangle\top).
$$
Here $\bullet$ is the one-point frame with empty relation [2507.04533].

For $\mathfrak{LT}=\mathrm{NExt}(K4_t)$, one likewise has
$$
\deg_{\mathfrak{LT}}(L)\in\{1,2^{\aleph_0}\},\qquad
\deg_{\mathfrak{LT}}(L)=df_{\mathfrak{LT}}(L)\in\{1,2^{\aleph_0}\}.
$$
Again the strictly Kripke-complete logics are exactly the iterated splittings, and in this lattice they are the union-splittings $\{K4_t,L^*\}$ with
$$
L^*:=K4_t/\mathrm{Log}(\bullet)=K4_t\oplus(\langle F\rangle\top\vee \langle P\rangle\top).
$$
The same axiom excludes the dead-end frame $\bullet$ in rooted transitive frames [2507.04533].

For $\mathrm{NExt}(S4_t)$, every logic $L$ satisfies
$$
\deg_{\mathrm{NExt}(S4_t)}(L)\in\{1,2^{\aleph_0}\},
$$
and again
$$
\deg_{\mathrm{NExt}(S4_t)}(L)=df_{\mathrm{NExt}(S4_t)}(L)\in\{1,2^{\aleph_0}\}.
$$
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
$$
\mathrm{NExt}(S5_t)\cup\{S4_t\},
$$
that is, $S4_t$ itself and every extension of $S5_t$ [2507.04533].

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

A splitting pair in a lattice $\mathrm{NExt}(L_0)$ is a pair $(L_1,L_0/L_1)$ such that for every logic $L$ in the lattice, exactly one of $L\subseteq L_1$ and $L\supseteq L_0/L_1$ holds [2507.04533]. Equivalently, $L_1$ is completely meet-prime and its complement is completely join-prime [2507.04533]. A union-splitting is a logic of the form $L=\oplus_i L_i$ 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 $0$ [2507.04533].

In $\mathrm{NExt}(K_t)$ and $\mathrm{NExt}(K4_t)$ there is, by Kracht’s result, a unique splitting pair, namely $(\{\mathrm{Log}(\bullet)\},L^*)$ [2507.04533]. The paper proves that in both lattices
$$
\text{iterated splittings}=\text{union-splittings}=\{\text{base},L^*\},
$$
and these are exactly the strictly Kripke-complete logics [2507.04533]. Thus degree $1$ is exceptionally sparse in the non-reflexive and transitive tense settings considered.

The $S4_t$ case is structurally different. Kracht’s theorem gives exactly two splitting pairs in $\mathrm{NExt}(S4_t)$:
$$
(\mathrm{Log}(Ch_2),S5_t)\quad\text{and}\quad (\mathrm{Log}(Ch_1),S4_t/\mathrm{Log}(Ch_1)).
$$
Consequently, union-splittings are fewer than iterated splittings [2507.04533]. The paper proves that the iterated splittings are precisely $\mathrm{NExt}(S5_t)\cup\{S4_t\}$, and that these coincide with the strictly Kripke-complete logics [2507.04533]. The chain $\mathrm{NExt}(S5_t)$ is described as isomorphic to $(\omega,\geq)$, with elements $\mathrm{Log}(Cl_n)$, the cluster logics [2507.04533].

This distinction between $\mathfrak{K}$ and $\mathfrak{LT}$ on the one hand and $\mathrm{NExt}(S4_t)$ 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 [2507.04533]. A plausible implication is that reflexive-transitive tense semantics supports a finer interaction between lattice-theoretic primeness and frame-theoretic completeness than the $K_t$ and $K4_t$ 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 [2507.04533]. One key device is reflective unfolding. Given frames $F$ and $G$ with designated points $w\in F$ and $u\in G$, one forms $F\{w+u\}G$ by disjoint union with identification and added forward/backward edges, and takes a transitive closure $F\{w+^t u\}G$ when needed [2507.04533]. Iterating this yields “book” frames $F^n_{w,u}$ whose reachability degree $\mathrm{rdg}$ grows linearly with $n$, while a surjective $t$-morphism back to $F$ remains available [2507.04533]. This is used to obtain, for any $n$, a finite rooted frame refuting a given formula and with $\mathrm{rdg}\geq n$ [2507.04533].

A second device is the family of master modalities $\Delta^n$, defined inductively by
$$
\Delta^0\psi:=(\psi\wedge \top),\qquad
\Delta^{k+1}\psi:=\Delta^k\psi\vee (\langle F\rangle\psi\wedge \Delta^k\psi)\vee (\langle P\rangle\psi\wedge \Delta^k\psi).
$$
These express reachability of depth $n$ by alternating future and past steps [2507.04533]. The formula
$$
bz_n:\Delta^{n+1}p\to \Delta^n p
$$
forces finite reachability degree at most $n$ [2507.04533]. The paper treats these as a tense analogue of “pre-transitivity” behavior [2507.04533].

A third ingredient is a tense-adapted Jankov formula construction. For a finite rooted frame $G$, the formula $J^k(G)$ is defined so that for any $k$-transitive rooted general frame $F$,
$$
F\nvDash \neg J^k(G)\quad\text{iff}\quad
\text{there is a surjective $t$-morphism from a generated $k$-transitive subframe of }F\text{ onto }G.
$$
This replaces the modal use of canonical frames in isolating finite frames [2507.04533].

The central step in the continuum construction uses the intersection lemma for rooted classes:
$$
K(L_1\cap L_2)=K(L_1)\cup K(L_2),
$$
for rooted classes $K$ and tense logics $L_1,L_2$ [2507.04533]. Starting with a formula $\varphi_L\in L\setminus L^*$ refuted by a finite rooted frame $F_L$ with sufficiently large $\mathrm{rdg}$, the paper constructs, for each $I\subseteq\mathbb{N}$, a general frame $F_I$ by combining $F_L$ with a specially designed frame $F'_I$ that syntactically encodes $I$ [2507.04533]. One then sets
$$
L_I:=L\cap \mathrm{Log}(F_I).
$$
The proof shows that $\mathrm{Fr}(L_I)=\mathrm{Fr}(L)$ while the $L_I$ are pairwise distinct, thereby obtaining continuum many different logics with the same Kripke-frame class [2507.04533].

For $\mathrm{NExt}(K_t)$ and $\mathrm{NExt}(K4_t)$ the encoding uses $\omega$-chains with distinguished points $i^*$ and separator formulas $\gamma_n$ and $\gamma_m^*$, together with
$$
\chi_i:=\neg\varphi_L\to \Delta^k\gamma_i^*.
$$
Then $\chi_i\in L_I$ iff $i\in I$ [2507.04533]. For $\mathrm{NExt}(S4_t)$ the construction is more delicate: it uses a “Nishimura–Rieger-like ladder” with two interleaved chains $A$ and $B$, special points $x_0,x_1,x_2,y_0,y_1,r_0,r_1,r'$, and separator formulas ${}_{c_n}$, with
$$
\chi_i:=({}_{c_i}\to \Delta^k\varphi_L),
$$
again distinguishing the $L_I$ [2507.04533].

## 6. Examples, comparisons, and open directions

Concrete degree-$1$ examples are explicitly listed. In $\mathfrak{K}$ the strictly Kripke-complete logics are $K_t$ and
$$
L^*=K_t\oplus(\langle F\rangle\top\vee \langle P\rangle\top),
$$
the logic asserting that everywhere there is either a successor or a predecessor [2507.04533]. In $\mathfrak{LT}$ the same pattern holds with $K4_t$ in place of $K_t$ [2507.04533]. In $\mathrm{NExt}(S4_t)$ the degree-$1$ logics are $S4_t$ itself and every extension of $S5_t$; for instance, $\mathrm{Log}(Cl_n)$ is an iterated splitting and strictly Kripke-complete [2507.04533].

The degree-$2^{\aleph_0}$ cases are equally explicit. In $\mathfrak{K}$, any logic other than $K_t$ or $L^*$ has degree $2^{\aleph_0}$ [2507.04533]. In $\mathfrak{LT}$, any logic other than $K4_t$ or $L^*$ has degree $2^{\aleph_0}$ [2507.04533]. In $\mathrm{NExt}(S4_t)$, any logic not in $\mathrm{NExt}(S5_t)\cup\{S4_t\}$ has degree $2^{\aleph_0}$ [2507.04533]. 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 $\deg(L)\in\{1,2^{\aleph_0}\}$ for every $L\in\mathrm{NExt}(K)$, and the strictly Kripke-complete logics are precisely union-splittings [2507.04533]. The tense results recover this pattern for $\mathrm{NExt}(K_t)$ and $\mathrm{NExt}(K4_t)$, but not for $\mathrm{NExt}(S4_t)$, where strict Kripke-completeness matches iterated splitting rather than union-splitting [2507.04533]. 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 [1808.06284]. 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 [1202.3268].

The tense paper also records several consequences and open problems. For all three lattices,
$$
\deg(L)=df(L),
$$
so the degree of Kripke-incompleteness coincides with the degree of the finite model property, and the same dichotomy holds for $df(L)$ [2507.04533]. The constructions for $K_t$ and $K4_t$ rely on finite transitivity via the master modalities $\Delta^n$, whereas $S4_t$ is not finitely transitive, requiring the special $F_I$ constructions together with bounds such as $alt_n^\pm$, $bw_n^\pm$, $bd_n$, and the Grzegorczyk-like formulas $grz^\pm$ [2507.04533]. Extending the dichotomy to other finitely transitive base systems such as $S4.2_t$ and $S4.3_t$ is identified as an open direction, and the text notes that Kracht showed infinitely many splittings in such lattices, suggesting more complex behavior [2507.04533]. It also remains open whether there exists a tense base logic $L$ such that $\mathrm{NExt}(L)$ exhibits anti-dichotomy for the finite model property [2507.04533].

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 [2507.04533], 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.

Source: https://www.emergentmind.com/topics/degree-of-kripke-incompleteness