SHORE
SHORE

SHORE
SHORE
 : 
The Shape of Reasoning: Many-Valued Logics and Uncertainty
The Shape of Reasoning: Many-Valued Logics and Uncertainty

A Project coordinated by IIIA.

Web page:

Principal investigator: 

Collaborating organisations:

Universitat de Barcelona (UB)

Universitat de Barcelona (UB)

Funding entity:

Ministerio de Ciencia e Inovación
Ministerio de Ciencia e Inovación

Funding call:

Funding call URL:

Project #:

PID2022-141529NB-C22
PID2022-141529NB-C22

Total funding amount:

68.625,00€
68.625,00€

IIIA funding amount:

Duration:

01/Sep/2023
01/Sep/2023
31/Aug/2027
31/Aug/2027

Extension date:

Logic can be broadly understood as the science of correct reasoning. While classical logic formalizes this idea under the assumption that statements are either true or false independently from our knowledge about them or capability to justify them constructively, the 20th century saw the birth of many nonclassical systems motivated by a variety of aspects of reasoning for which this assumption seemed either inadequate or unsatisfactory.

Among the wide plethora of motivations for extending, or even deviating, from classical logic, we recall the necessity of interpreting modal and intensional discourse which led to the introduction of modal logic, as well as that of dealing with imprecise, gradual, or uncertain information, which did the same for many-valued logics in a broad sense.
A striking feature of these nonclassical systems is that they admit a geometric interpretation which, in turn, makes them amenable to the visual intuitions on shapes and deformations. For modal logic this is made possible by the variants of Priestley duality, known as Jónsson- Tarski duality, which allow us to cross the mirror between modal algebraic and Kripke frames. On the other hand, in the case of many- valued logics the geometric interpretation takes the form of a semantics based on rational polyhedra. These latter allows us to think in terms of faces, vertexes, and edges. Furthermore, classical uncertainty theories such as probability theory, belief function theory or possibility theory, together with their subjectivist foundations, allow for a geometric-based treatment relying on convex and finitely generated objects, i.e., convex polyhedra (aka polytopes).

The present proposal aims at developing a uniform logical, geometric, and algebraic representation for uncertainty theories. More specifically, for subjective uncertainty theories, such uniform geometric treatment is crucial both at the level of theory and to that of applications as this will allow a comprehensive and general treatment of uncertainty.

Logic can be broadly understood as the science of correct reasoning. While classical logic formalizes this idea under the assumption that statements are either true or false independently from our knowledge about them or capability to justify them constructively, the 20th century saw the birth of many nonclassical systems motivated by a variety of aspects of reasoning for which this assumption seemed either inadequate or unsatisfactory.

Among the wide plethora of motivations for extending, or even deviating, from classical logic, we recall the necessity of interpreting modal and intensional discourse which led to the introduction of modal logic, as well as that of dealing with imprecise, gradual, or uncertain information, which did the same for many-valued logics in a broad sense.
A striking feature of these nonclassical systems is that they admit a geometric interpretation which, in turn, makes them amenable to the visual intuitions on shapes and deformations. For modal logic this is made possible by the variants of Priestley duality, known as Jónsson- Tarski duality, which allow us to cross the mirror between modal algebraic and Kripke frames. On the other hand, in the case of many- valued logics the geometric interpretation takes the form of a semantics based on rational polyhedra. These latter allows us to think in terms of faces, vertexes, and edges. Furthermore, classical uncertainty theories such as probability theory, belief function theory or possibility theory, together with their subjectivist foundations, allow for a geometric-based treatment relying on convex and finitely generated objects, i.e., convex polyhedra (aka polytopes).

The present proposal aims at developing a uniform logical, geometric, and algebraic representation for uncertainty theories. More specifically, for subjective uncertainty theories, such uniform geometric treatment is crucial both at the level of theory and to that of applications as this will allow a comprehensive and general treatment of uncertainty.

In Press
Tommaso Flaminio,  Lluís Godo,  & Giuliano Rosella (In Press). Conditionals based on selection functions, modal operators and probabilities. Kai Sauerwald, & Matthias Thimm (Eds.), A. Bjorndahl (Ed.): Theoretical Aspects of Rationality and Knowledge 2025 (TARK 2025) . EPTCS. [BibTeX]
Tommaso Flaminio,  Lluís Godo,  & Giuliano Rosella (In Press). On measuring the possibility of selection function-based conditionals, general updates, and qualitative capacities. Kai Sauerwald, & Matthias Thimm (Eds.), Proc. of ECSQARU 2025 . Springer, Cham. [BibTeX]
Lydia Castronovo,  Tommaso Flaminio,  Lluís Godo,  & Giuseppe Sanfilippo (In Press). Towards an algebraic and probabilistic setting for iterated Boolean conditionals. Kai Sauerwald, & Matthias Thimm (Eds.), Proc. of ECSQARU 2025 . Springer, Cham. [BibTeX]
2025
Valeria Giustarini,  & Sara Ugolini (2025). Amalgamation failures in MTL-algebras. The Logic Algebra and Truth Degrees (LATD) 2025 - Abstract Booklet (pp. 104-107). Università degli Studi di Siena, DIISM. [BibTeX]
Tommaso Flaminio,  Francesco Manfucci,  & Sara Ugolini (2025). Conditionals as quotients in Boolean algebras. The Logic Algebra and Truth Degrees (LATD) 2025 - Abstract Booklet (pp. 150-153). Università degli Studi di Siena, DIISM. [BibTeX]
Sara Ugolini (2025). Generalization of terms in Łukasiewicz logic. International Workshop on Topological Methods in Logic VII -ToLo 2025, Invited talk . [BibTeX]
Tommaso Flaminio,  & Sara Ugolini (2025). Generalization of terms up to equational theory. The Logic Algebra and Truth Degrees (LATD) 2025 - Abstract Booklet (pp. 237-242). Università degli Studi di Siena, DIISM. [BibTeX]
Tommaso Flaminio,  Lluís Godo,  Ramón Pino Pérez,  & Lluis Subirana (2025). On Lockean Beliefs that are Deductively Closed and Minimal Change. Giovanni Casini, Besik Dundua, & Temur Kutsia (Eds.), Logics in Artificial Intelligence - 19th European Conference, {JELIA}2025, Kutaisi, Georgia, September 1-4, 2025, Proceedings, Part {II} (pp. 28--42). Springer. https://doi.org/10.1007/978-3-032-04590-4\_3. [BibTeX]
Tommaso Flaminio,  & Lluis Subirana (2025). Quantitative Lockean Thesis and its Logical Representation. M. Baczynski, B. De Baets, M. Holcapek, V. Kreinovich, & J. Medina (Eds.), Advances in Fuzzy Logic and Technology (pp. 347--358). Springer Nature Switzerland. [BibTeX]
Giuliano Rosella,  & Sara Ugolini (2025). THE ALGEBRAS OF LEWIS’S COUNTERFACTUALS: AXIOMATIZATIONS AND ALGEBRAIZABILITY. The Review of Symbolic Logic, 18, 563–588. https://doi.org/10.1017/S1755020324000303. [BibTeX]
Tommaso Flaminio,  Serafina Lapenta,  & Sebastiano Napolitano (2025). Unimodular triangulations in Łukasiewicz logic: Complexity bounds of probabilistic coherence. International Journal of Approximate Reasoning, 187, 109565. https://doi.org/10.1016/j.ijar.2025.109565. [BibTeX]
2024
Nick Galatos,  Valeria Giustarini,  & Sara Ugolini (2024). Constructing conical and perfect residuated lattices. Booklet of Abstracts, TACL 2024 (pp. 133-134). [BibTeX]
Sara Ugolini (2024). Equational anti-unification. Association of Symbolic Logic Annual Meeting 2024, Special Session in Algebraic Logic, Iowa State University (USA), Invited talk . [BibTeX]
Tommaso Flaminio,  Lluís Godo,  & Giuliano Rosella (2024). Possibility of Conditionals and Conditional Possibilities: From a Triviality Result to Possibilistic Imaging. Pierre Marquis, Magdalena Ortiz, & Maurice Pagnucco (Eds.), Proc. of the 21st International Conference on Principles of Knowledge Representation and Reasoning - Main Track (KR 2024) (pp. 372-382). https://doi.org/10.24963/kr.2024/35. [BibTeX]  [PDF]
Sara Ugolini (2024). Projectivity in quasivarieties of logic. Topology, Algebra and Categories in Logic - TACL 2024, Invited talk . [BibTeX]
Tommaso Flaminio
Tenured Scientist
Phone Ext. 431841

Valeria Giustarini
PhD Student
Sara Ugolini
Contract Researcher
Amanda Vidal
Contract Researcher