Unification types and union splittings in intermediate logics

TL;DR

This paper classifies the unification types of various intermediate logics, identifying maximal and minimal cases, and explores properties like tabularity, projectivity, and unification finiteness in these logics.

Contribution

It provides a detailed characterization of unification types in intermediate logics, extending previous Kripke model characterizations to include new classifications and properties.

Findings

Four maximal logics with nullary unification are tabular.

Two minimal logics with hereditary finitary unification are locally tabular.

No locally tabular intermediate logic has infinitary unification.

Abstract

Following a characterization [10] of locally tabular logics with finitary (or unitary) unification by their Kripke models we determine the unification types of some intermediate logics (extensions of {\sf INT}). There are exactly four maximal logics with nullary unification ${\mathsf L}(\mathfrak R_{2}+)$, ${\mathsf L}(\mathfrak R_{2})\cap{\mathsf L}(\mathfrak F_{2})$, ${\mathsf L}(\mathfrak G_{3})$ and ${\mathsf L}(\mathfrak G_{3}+)$ and they are tabular. There are only two minimal logics with hereditary finitary unification: ${\mathsf L}$($\mathbf F_{un}$), the least logic with hereditary unitary unification, and ${\mathsf L}$( $\mathbf F_{pr}$) the least logic with hereditary projective approximation; they are locally tabular. Unitary and non-projective logics need additional variables for mgu's of some unifiable formulas, and unitary logics with projective approximation are exactly projective. None of locally tabular intermediate logics has infinitary unification. Logics with finitary, but not hereditary finitary, unification are rare and scattered among the majority of those with nullary unification, see the example of $\mathsf H_3\mathsf B_2$ and its extensions.