Periodic Reporting for period 2 - SYSMICS (Syntax Meets Semantics: Methods, Interactions, and Connections in Substructural logics.)
Reporting period: 2018-03-01 to 2019-02-28
The acronym SYSMICS stands for: Syntax meets Semantics – Methods, Interactions, and Connections in Substructural logics.Substructural logics are formal reasoning systems that refine classical logic by weakening the structural rules in Gentzen sequent calculus. While classical logic generally formalises the notion of truth, substructural logics allow to handle more precisely resources, vagueness, meaning, and language syntax; these systems are motivated by studies in computer science, epistemology, economy, and linguistics. In addition, from a theoretical point of view, substructural logics provide a refined perspective of classical logic, since the former often exhibit features which are either absent or trivialised in the classical case.
Traditionally, substructural logics have been investigated following three main approaches: proof theoretic, algebraic and abstract study. Although some connections among these approaches were observed long ago, in large part these practices developed in independence. As a result, the research directions, tools and motivations for each approach developed in relative isolation.
The main objective of this project is to establish a network of collaborations between the experts of these diverse methods to investigate substructural logics in a cohesive fashion, taking into account these three distinct yet complementary points of view. This combined perspective on substructural logics might have a deep impact on the field and this project will provide a stable basis of cooperation for a large, international community of algebraists, logicians and theoretical computer scientists. To reach this goal, the workloadhas been organized in work packages as follows:
WP1 – Managements tasks and practicalities.
WP2 – Organisation of the two conferences. Press conferences. Organisation of four workshops. Preparation of two broad audience-oriented papers.
WP3 – Organisationof two schools.
WP4 – Developing a theory of translations between logics, establishing connections between deductive theorems and algebraic filters, study finite and infinite Beth’s definability for abstract logics, and to develop a resource-conscious Abstract Algebraic Logic
WP5 – Developinganalytic calculi for substructural logics, and deploy them toprove meta-logical theorems.
WP6 – Developing the algebraic machinery needed for a deeper understanding ofsubstructural logics.
WP7 – Export the method of canonical formulas to substructural logics. Providealgebraic methods for verifying the robustness of proof systems.
WP8 – Dual semantics for substructural logics and canonical extensions. Comprehensivestudy of completions and their applications to decidability problems.
WP9 – Algebraic semantics of modal extensions of substructural logics. Investigationof relations between truth preserving consequence relations andparaconsistency.
- the first conference was successfully held in Barcelona, from the 4th to the 9th of September 2016,
- the first school was held in Olomouc from the 20th to the 24th of June 2017,
- the first workshop was held at the campus of the University of Salerno (Italy) from the 5th to the 8th of September 2017,
- the second workshop within the project was held in Vienna at the Faculty of Mathematics, University of Vienna, from February 26th to February 28th 2018,
- the third SYSMICS workshop has been held from June 11th to June 13th at the University of Cagliari, Italy,
- the second school took place in Les Diablerets (Switzerland), from 22 to 26 August 2018,
- the fourth and last SYSMICS workshop took place in Orange (California, USA) from 14th to 17th of September 2018,
- the final conference has been organized by the ILLC of the University of Amsterdam and took place from 21st to 25th of January 2019.
WP4 –The participants involved in this work package are proceeding in the systematic organization of the state-of-art knowledge on abstract algebraic logic and undertook the endeavour of writing a textbook on the subject. Several important results have been obtained for each task of the WP.
WP5 – Analytic calculi for different substructural logics have been defined and the interaction of already known systems with proof assistants (such as Isabelle or HOL) have been analysed.
WP6 –The book “Residuated structures in algebra and logic”, being written by some project participants, has seen significant progress.Original results on varieties of residuated algebras have been found, in some case with specific emphasis on the computational aspects of the problem.
WP7 – The method of canonical formulas has been extended to several modal and substructural logics and progress has been made in the axiomatisation of a modal logic for Abelian lattice-ordered groups.
WP8 – A duality for the class of residuated lattices have been conceived in a way that could subsume most of the existing approaches and give a rigorous account of their connections. The characterisation of separable finitely presented MV-algebras has been investigated.
WP9 –The progress covers several different aspects of the theory of residuated lattices and they revolve around the idea of axiomatising and investigate special logics that have particular residuated lattices as models. Moreover, modal expansions of well known fuzzy logics have been investigated.
Both of the project conferences have included public lectures, which were very successful and included participants from outside the academia.