--- title: Pre-Uniform Locally Tabular Logic url: https://www.emergentmind.com/topics/pre-uniformly-locally-tabular-logic type: topic --- # Pre-Uniform Locally Tabular Logic 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 [2601.11208]. ## 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