Abstract model theory for non-classical logics: seeing the bigger picture behind the growing diversity

• Applicant: Dr. Grigory Olkhovikov
• Subject Area: Theoretical Philosophy
• Term: from 2018 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 407399494

• Final Report Year: 2022

Final Report Abstract

The project has led to a significant extension of the body of known results in the (hitherto barely existing) domain of abstract model theory of intuitionistically flavored logics. The main success in this direction is the first series of Lindström results for several variants of intuitionistic first-order logic. Another key change resulting from the project research is a sizeable increase in our knowledge about bi-intuitionistic logic. We now know more both about the propositional and about the first-order variant of this system, and we even know more about its history. Both of these changes have already had their continuation within the current research efforts of the logic community. The two main surprises uncovered by the work on the project are (1) how much more difficult is to extend the classical setting of abstract model theory to intuitionistic first-order logic as compared to modal first-order logic, and (2) how difficult is to improve on the main result in a well-rounded fashion. These surprises shaped both the path taken within the project and the results obtained. Most of the delays that occurred during the realization of the project can be traced, in one way or another, to these two major sources of difficulties. The results of the project have been presented on several international workshops and conferences including the Australasian Association for Logic Conference, the Paris-Bochum-Moscow Workshop on Mathematical Philosophy, and the Logic Supergroup, and were met with a genuine interest by the audience.

Publications

• A Lindström Theorem for Intuitionistic Propositional Logic. Notre Dame Journal of Formal Logic, 61(1).
  Badia, Guillermo & Olkhovikov, Grigory
  
• A Lindström theorem in many-valued modal logic over a finite MTL-chain. Fuzzy Sets and Systems, 388, 26-37.
  Badia, Guillermo & Olkhovikov, Grigory
  
• Maximality of bi-intuitionistic propositional logic. Journal of Logic and Computation, 32(1), 1-31.
  Olkhovikov, Grigory & Badia, Guillermo
  
• Questions to Michael Dunn. Logical Investigations, 27(1), 9-19.
  Wansing, Heinrich; Olkhovikov, Grigory & Omori, Hitoshi
  
• CRAIG INTERPOLATION THEOREM FAILS IN BI-INTUITIONISTIC PREDICATE LOGIC. The Review of Symbolic Logic, 17(2), 611-633.
  Olkhovikov, Grigory K. & Badia, Guillermo
  
• A Lindström theorem for intuitionistic first-order logic. Annals of Pure and Applied Logic, 174(10), 103346.
  Olkhovikov, Grigory; Badia, Guillermo & Zoghifard, Reihane