Pre-Uniform Locally Tabular Logic

• Pre-uniformly locally tabular logic is a class of superintuitionistic logics that are locally tabular but not uniformly so until extended, refining the landscape above IPC.
• The methodology integrates implication-depth, Kripke frame criteria, and Heyting algebra properties to distinguish between local and uniform tabularity.
• The logic Box exemplifies maximal non-uniformity, serving as a critical tool in classifying and axiomatizing locally tabular logics.

Pre-uniformly locally tabular logic arises within the study of superintuitionistic logics extending intuitionistic propositional logic (IPC), characterizing those logics that are locally tabular but fail to be uniformly locally tabular, while every proper extension admits uniform local tabularity. The distinction is rigorously formalized via implication-depth, Kripke frame conditions, and algebraic properties of Heyting algebras, culminating in the explicit example of the intermediate logic “Box,” which serves as a maximal non-uniformly locally tabular system. This notion refines the landscape of tabularity in superintuitionistic logic and has critical implications for the classification and axiomatization of such logics (Almeida, 16 Jan 2026).

1. Implication-Depth and Uniform Local Tabularity

The measure of implication-nesting in propositional formulas sets the basis for uniform local tabularity. Specifically, the implication-depth $d(\phi)$ is defined recursively by: $d(p)=d(\perp)=d(\top)=0$ for atoms and constants, $$d(\phi\wedge\psi)=d(\phi\vee\psi)=\max(d(\phi),d(\psi))$$ for conjunction and disjunction, and $d(\phi\rightarrow\psi)=\max(d(\phi),d(\psi))+1$ for implication. An intermediate logic $L\supseteq \mathrm{IPC}$ is called $n$-uniformly locally tabular ($n$-uniform) if every formula $\phi$ in $L$ is provably equivalent to a formula $\psi$ of implication-depth at most $n$. Uniform local tabularity (ULTab) holds for $L$ if $L$ is $n$-uniform for some finite $n$. This syntactic constraint organizes logics within a hierarchy based on the maximum nesting of implication allowed for provable equivalence.

2. Algebraic and Kripke Frame Criteria

Extensions $L$ of $\mathrm{IPC}$ are associated with their classes of Heyting-algebraic models $\operatorname{Alg}(L)$; $L$ is $n$-uniform if and only if every finitely generated subalgebra $A$ of any $H\in \operatorname{Alg}(L)$ is already generated by its elements under all terms of implication-depth $\leq n$. In terms of Kripke frames, let $\mathbb{K}$ be a class of finite posets, with $L=\operatorname{Log}(\mathbb{K})$ the corresponding logic. $L$ is $n$-uniform if for every pair of finite $\mathbb{K}$-models $M,N$ over the same finite set of propositional variables: if $M$ and $N$ are $n$-bisimilar then they are fully bisimilar. This bisimulation characterization provides a semantic analogue to the syntactic and algebraic definitions.

3. Pre-Uniform Locally Tabular Logics: Formalization

Pre-uniformly locally tabular logics refine the classification of locally tabular superintuitionistic systems. Such a logic $L$ satisfies two conditions: (a) $L$ is not uniformly locally tabular itself ($L\notin$ ULTab); (b) every proper extension $L' \supsetneq L$ is uniformly locally tabular ($L'\in$ ULTab). This property generalizes the “pre-locally tabular” behavior known from modal logic extensions of $\mathsf{S4}$ to the superintuitionistic setting, highlighting that local tabularity and uniform local tabularity can diverge above $\mathrm{IPC}$.

4. The Logic Box: Frame and Axiomatic Presentation

Almeida’s principal example of a pre-uniformly locally tabular logic is Box. Box is constructed as follows:

• Frame-theoretic definition: For each $n\in\mathbb{N}$, define a finite rooted frame $S_n$ by taking the projective-cascade frame $N_n^{n-2}$ and adjoining a new maximal point. Box is then given by $Box = \operatorname{Log}(\{S_n : n\in\mathbb{N}\})$ (Proposition 5.2).
• Axiomatic presentation: Equivalently, $Box = wPL \oplus bw_2 \oplus \neg(p\vee\neg\neg p)$, where:
  • $$wPL = \mathrm{IPC} \oplus (q\rightarrow p) \vee (((p\rightarrow q)\rightarrow p)\rightarrow p)$$ (weak Peirce’s law), characterizing finite cascade (Boolean-sum) posets.
  • $bw_2$ restricts width two, forbidding antichains of size three.
  • $\neg(p\vee\neg\neg p)$ ensures every finite model possesses a top element.

These ingredients together construct Box as a distinctive boundary in the lattice of locally tabular logics.

5. Structural Features and Proof of Pre-Uniform Local Tabularity

Box is locally tabular because $wPL$ is known to be locally tabular (Mardaev’s theorem, Theorem 3.9), and the additional axioms only further restrict finite frames. However, Box fails to be uniformly locally tabular: one constructs two infinite families of finite Box-frames, $M_n^k$ and $N_n^{k-1}$ for $n>2$, $k<n-1$, which admit $k$-bisimilarity but not $(k+1)$-bisimilarity. By the bisimulation characterization, Box is not $m$-uniform for any $m<\omega$, i.e., $Box\notin$ ULTab (Lemma 5.3).

Every proper extension $L'\supsetneq Box$ must impose either a finite bound on the height of its frames or on the stack-depth of their cascade components. Using bounded bisimulation techniques (Proposition 3.14 and Lemma 5.4), one establishes that $L'$ is $n$-uniform for some $n$, and thus $L'\in$ ULTab. Box thereby exemplifies the pre-uniform locally tabular phenomenon.

6. Impact and Classification Significance

The existence of Box resolves a question posed by Shehtman, demonstrating that, unlike modal extensions of $\mathsf{S4}$ where local and uniform tabularity coincide, locally tabular logic above $\mathrm{IPC}$ can fail uniformity, and a maximal non-uniformly locally tabular system exists. Box serves as a unique pre-uniformly locally tabular logic above the Kuznetsov–Gerciu logic (KG), and checks for uniform local tabularity in extensions of KG reduce to containment of the axiomatization of Box.

The Box logic inaugurates a systematic study of the $n$-uniform hierarchy inside locally tabular logics (LTab), substantiating that it is strictly finer. Full axiomatizations are provided for $n\leq 2$ (CPC, Sme, 2Uni, LC, BD$_2$), and for $n=3$ significant examples (LFC, logic of finite combs) are exhibited, yet Box demonstrates that the union of all $n$-uniformly locally tabular classes is still strictly contained in the set of locally tabular logics (Almeida, 16 Jan 2026).

7. Context, Extensions, and Future Directions

Pre-uniform local tabularity parallels classical phenomena in modal logic but reveals novel structure in superintuitionistic landscapes. The detailed classification of locally tabular, $n$-uniform, and pre-uniformly locally tabular logics clarifies the granularity of logical varieties concerning implication-depth constraints and finite frame properties. The approach foregrounds the interplay between syntactic axiomatizations, algebraic reformulations via Heyting algebras, and Kripke frame semantics.

A plausible implication is that further refinement and exploration of the $n$-uniform hierarchy may uncover additional distinguished boundary points or provide new challenges in the axiomatization of higher-uniform superintuitionistic logics. The pre-uniform phenomenon, as embodied by Box, also offers new tools and criteria for analyzing the structural order of logics above IPC, particularly in relation to decision procedures and finite model constraints.