Degrees of FMP in extensions of bi-intuitionistic logic
Anton Chernev
Abstract:
This thesis contributes to the study of degrees of the finite model property (FMP), initiated by G. Bezhanishvili, N. Bezhanishvili and T. Moraschini (2022). We investigate degrees of FMP in extensions of bi-intuitionistic logic through the lens of universal algebra. Motivated by the characterisation of degrees of FMP for intuitionistic logic, which utilises the Kuznetsov-Gerčiu variety KG, we define its bi-intuitionistic counterpart bi-KG. We obtain a description of the subdirectly irreducible members of bi-KG and, as a result, we prove that they are simple algebras. This enables us to develop a method of comparing subvarieties of bi-KG using local embeddability properties of their finitely generated simple members.
As an application of this method, we provide a full description of subvarieties of bi-KG with the FMP. Consequently, bi-KG turns out to enjoy the FMP, while its least subvariety containing all 1-generated Heyting algebras lacks the FMP. Our main result is a dichotomy-style theorem characterising degrees of FMP of subvarieties of bi-KG, meaning that the only two degrees of FMP are 1 and 2א0 . This is in sharp contrast with (intuitionistic) KG, where all cardinals κ with κ ≤ א0 occur as FMP degrees relative to KG. Finally, we translate the statement into logical terms to obtain a corresponding result about degrees of FMP relative to the extension of bi-intuitionistic logic algebrised by bi-KG.