## Research

### Algebraic Logic

Algebraic Logic is the discipline that studies "Bridge Theorems" that allow to cross the mirror between Logic and Algebra by associating a purely semantic interpretation (such as the amalgamation property) with a given metalogical property (such as the interpolation property). This allows to study metalogical phenomena through the lenses of their semantic counterparts, which are typically amenable to the powerful methods of Universal Algebra, Lattice Theory and Category Theory. This perspective proved to be very fruitful both in the study of concrete logical systems such as Fuzzy, Modal and Intuitionistic Logics as well as in its most general formulation known as Abstract Algebraic Logic.

### Mathematical Fuzzy Logic

Mathematical Fuzzy Logic is the part of Mathematical Logic that studies logical systems which extend the two-valued semantics of classical logic by allowing formulas to take value in the real unit interval. These formalisms provide a more faithful representation of those properties and predicates which, for their own nature, are perceived as graded. Several points of contact with other areas of mathematics such as functional analysis, probability theory, universal algebra, real convex geometry, have been explored and are a fruitful ongoing line of research.

### Modal and Intuitionistic Logic

Modal and Intuitionistic Logics are some of the main non-classical logics. Modal Logic is known for extending the expressive power of classical (and many-valued) logic, while preserving many of its desirable computational features. On the other hand, Intuitionistic Logic is the deductive system that governs the constructive aspects of mathematics. Both these logics have a natural semantics consisting of topological Kripke frames, which proved to be a versatile tool in the modellization of problems ranging from Theoretical Computer Science and Artificial Intelligence to Philosophy and Metaphysics. In view of Duality Theory, this topological semantics is dually equivalent to a purely algebraic one which, moreover, is amenable to the methods of Algebraic Logic.

### Formal Logics for Artificial Inteligence

The development of formal logics capable to simulate human-like reasoning is one of the central objective around which the logical community of Artificial Intelligence moved its first steps. Besides being a fundamental tool for analysis and the unquestionable ground for knowledge representation and reasoning, tools arising from the study of formal logic are retained to play a key role in the future of AI and, in particular for the integration between symbolic and sub-symbolic AI. This line of research aims at investigating formal logical methods for non-monotonic, causal, uncertain, preferencial reasoning and to lift them to more general and abstract levels.

## Seminar on Non-classical logics

### The seminar is back!

This website might not be up to date.
If you are interested in receiving the news on the seminar, please contact our memeber Joan Gispert.

#### Next Session:

April 17th at 10:30 at the IMUB room (second floor) of the Facultat de Matemàtiques i Informàtica of Universitat de Barcelona.
Structure theory and amalgamation failures in integral residuated chains, , by Valeria Giustarini , from the IIIA - CSIC.
Abstract: Residuated structures play an important role in the field of algebraic logic since they constitute the equivalent algebraic semantics, in the sense of Blok and Pigozzi, of substructural logics. The algebraic investigation of residuated lattices is a powerful tool in the systematic and comparative study of such logics. Thus, the development of constructions that allow one to obtain new structures from known ones is of utter importance for the understanding of both residuated lattices and substructural logics.
One of the most well known constructions that allows one to get a new algebra starting from two known ones is the ordinal sum construction. Intuitively, we stack one algebra on top of the other gluing the top element. This construction preserves both products and divisions of the initial algebras. This is the main difference with the gluing construction introduced by Galatos and Ugolini, where the products are preserved but some of the divisions are redefined. In this work, we propose two possible iterations of this construction, which allows us to understand how some residuated chains can be decomposed and how their different parts interact. Moreover, we provide an axiomatization for the varieties generated by n-potent, archimedean chains that can be constructed via such iterated partial gluing.
Algebraic constructions of this kind are also important for the study of logical properties of the associated substructural logics. One of the most well-known bridge theorems is the connection between the amalgamation property in a variety of algebras and the Robinson property, or in some relevant cases the deductive interpolation property, of the corresponding logic.
In this work we focus on the study of amalgamation in classes of totally ordered residuated lattices, solving some open problems: most importantly, we establish that semilinear commutative (integral) residuated lattices and their pointed versions do not have the amalgamation property.
This is a joint work with Sara Ugolini.

#### Previous Sessions:

• March 20th at 10:30 at the IMUB room (second floor) of the Facultat de Matemàtiques i Informàtica of Universitat de Barcelona.
AExtending Blok-Esakia Theorem to the monadic setting, , by Luca Carai, from the University of University of Milan.
Abstract: Intuitionistic and modal logic are strictly connected: the intuitionistic propositional calculus IPC can be faithfully translated into the propositional modal logic S4 via the Gödel translation. This allows us to determine whether a formula is a theorem of IPC by checking the validity of its translation in S4. A normal extension M of S4 is said to be a modal companion of an extension L of IPC if L can be faithfully translated into M via the Gödel translation. By Esakia's Theorem, the largest modal companion of IPC is the Grzegorczyk modal logic Grz. The study of modal companions culminated into the celebrated Blok-Esakia Theorem which states that mapping an extension of IPC to its largest modal companion gives rise to an isomorphism between the lattices of extensions of IPC and of normal extensions of Grz.
In this talk we will examine the challenges one has to face to study modal companions of extensions of the monadic fragment, also known as the one-variable fragment, of the predicate intuitionistic logic. We will see that the natural generalizations of Esakia's and Blok-Esakia Theorems fail in the monadic setting. The talk will end on a positive note by showing that adding the monadic Casari's axiom allows to restore Esakia's Theorem. This talk is based on a joint work with G. Bezhanishvili
• March 13th at 10:30 at the IMUB room (second floor) of the Facultat de Matemàtiques i Informàtica of Universitat de Barcelona.
Epimorphisms between finitely generated algebras. Part 2, , by Tommaso Moraschini, from the University of Barcelona.
Abstract: This series of talks is based on joint work with Luca Carai and Miriam Kurtzhals [3]. A quasivariety K is said to have the weak epimorphism surjectivity property (weak ES property, for short) when all the epimorphisms between finitely generated members of K are surjective [4]. From a logical standpoint, the interest of the weak ES property is motivated as follows: when a quasivariety K algebraizes a logic L, the former has the weak ES property iff the latter has the Beth definability property (which, roughly, means that every implicit definition in L can be turned explicit) [1]. In general, the task of determining whether a quasivariety has the weak ES property is nontrivial (see, e.g., [2]) and our results facilitate the detection of failures of the weak ES property.
In the first talk of this series, we will provide some motivating examples and present a more tangible characterization of the weak ES property, based on the new notion of a full subalgebra. As a consequence, we will obtain a purely algebraic proof of a classic result of Kreisel (see [5]), stating that every variety of Heyting algebras has the weak ES property (or, equivalently, that every superintuitionistic logic has the Beth definability property).
Bibliography:
[1] W.J. Blok and E. Hoogland. The Beth property in Algebraic Logic. Studia Logica, 83(1–3):49–90, 2006.
[2] M.A. Campercholi. Dominions and primitive positive functions. J. Symb. Log., 83(1):40–54, 2018.
[3] L. Carai, M. Kurtzhals, and T. Moraschini. Epimorphisms between finitely generated algebras. Submitted. Available online on the ArXiv, 2024.
[4] L. Henkin, J.D. Monk, and A. Tarski. Cylindric Algebras. Part II. North-Holland, Amsterdam, 1985.
• March 6th at 10:30 at the IMUB room (second floor) of the Facultat de Matemàtiques i Informàtica of Universitat de Barcelona.
Epimorphisms between finitely generated algebras. Part 1 , , by Miriam Kurtzhals, from the University of Barcelona.
This series of talks is based on joint work with Luca Carai and Tommaso Moraschini [3]. A quasivariety K is said to have the weak epimorphism surjectivity property (weak ES property, for short) when all the epimorphisms between finitely generated members of K are surjective [4]. From a logical standpoint, the interest of the weak ES property is motivated as follows: when a quasivariety K algebraizes a logic L, the former has the weak ES property iff the latter has the Beth definability property (which, roughly, means that every implicit definition in L can be turned explicit) [1]. In general, the task of determining whether a quasivariety has the weak ES property is nontrivial (see, e.g., [2]) and our results facilitate the detection of failures of the weak ES property.
In the first talk of this series, we will provide some motivating examples and present a more tangible characterization of the weak ES property, based on the new notion of a full subalgebra. As a consequence, we will obtain a purely algebraic proof of a classic result of Kreisel (see [5]), stating that every variety of Heyting algebras has the weak ES property (or, equivalently, that every superintuitionistic logic has the Beth definability property).
Bibliography:
[1] W.J. Blok and E. Hoogland. The Beth property in Algebraic Logic. Studia Logica, 83(1–3):49–90, 2006.
[2] M.A. Campercholi. Dominions and primitive positive functions. J. Symb. Log., 83(1):40–54, 2018.
[3] L. Carai, M. Kurtzhals, and T. Moraschini. Epimorphisms between finitely generated algebras. Submitted. Available online on the ArXiv, 2024.
[4] L. Henkin, J.D. Monk, and A. Tarski. Cylindric Algebras. Part II. North-Holland, Amsterdam, 1985.
[5] G. Kreisel. Explicit definability in intuitionistic logic.J. Symb. Log.,25:389-390,1960
• February 21st at 10:30 at the IMUB room (second floor) of the Facultat de Matemàtiques i Informàtica of Universitat de Barcelona.
Algebraic and coalgebraic analysis of some many-valued modal logics, , by Wolfgang Poiger, from the University of Luxembourg.
Abstract: In this talk, we identify a class of algebraically 'well-behaved' many-valued logics, and study their modal extensions from an algebraic and coalgebraic perspective. These 'well-behaved' logics are the ones whose algebras of truth-degrees are semi-primal, for example, finitely-valued Łukasiewicz logic.
In more detail, in this talk we show how to lift algebra/coalgebra dualities (e.g., Jónsson-Tarski duality), as well as coalgebraic logics (e.g., normal or non-normal classical modal logic) to the semi-primal level. This involves a canonical way to lift endofunctors on the variety of Boolean algebras to endofunctors on the variety generated by a semi-primal algebra. We show that completeness, expressivity and finite axiomatizability of the corresponding classical logics are preserved under this lifting, and show how some of the resulting varieties of many-valued modal algebras may be axiomatized.
• February 9th at 10:15 at room B6 (ground floor) of the Facultat de Matemàtiques i Informàtica of Universitat de Barcelona.
Pointed lattice subreducts of varieties of residuated lattices, , by Adam Prenosil, UB.
Abstract: We shall discuss the lattice and pointed lattice subreducts of varieties of residuated lattices (RLs). While it is easy to observe that every lattice is a subreduct of an integral commutative RL, the problem of describing the pointed lattice subreducts (that is, subreducts in the lattice signature expanded by the constant 1 for the multiplicative unit) is more subtle. We axiomatize the quasivarieties of pointed lattice subreducts of integral and of semiconic commutative RLs, and show that they coincide with the subreducts of integral and of semiconic RLs. We also show that every lattice is a subreduct of a commutative cancellative RL, thus settling a basic open problem about this variety of RLs. The talk will cover some new results as well as some older ones, and list some remaining open problems.

## People

### Permanent Staff

#### Pilar Dellunde

Full Professor at the Department of Philosophy of the Autonomous University of Barcelona.

Logics for Artificial Intelligence. Probabilistic Argumentation Frameworks in AI. Explainable AI.

#### Francesc Esteva

Adjunct Professor Ad Honorem at the Artificial Inteligence Research Institute (IIIA) of the CSIC.

Mathematical Fuzzy Logic. Approximate and Uncertain Reasoning and Soft computing. Algebraic logic. Modal logic.

#### Tommaso Flaminio

Tenured Scientist at the Artificial Inteligence Research Institute (IIIA) of the CSIC.

Mathematical Fuzzy Logic. Probability logic. Uncertain Reasoning. Algebraic logic. Modal logic.

#### Joan Gispert

Associate Professor at the Department of Mathematics and Computer Science of the University of Barcelona.

Many Valued and Fuzzy Logic. Algebraic Logic. Universal Algebra.

#### Lluís Godo

Research Professor at the Artificial Intelligence Research Institute (IIIA) of the CSIC.

Possibilistic logic. Mathematical Fuzzy Logic. Similarity-based reasoning. Argumentation systems. Multi-Agent systems.

#### Tommaso Moraschini

Associate Professor at the Department of Philosophy of the University of Barcelona.

Algebraic Logic. Modal and Intuitionistic Logic. Duality Theory. Universal Algebra.

### Postdocs

Beatriu de Pinos fellow at the Department of Philosophy of the University of Barcelona.

Algebraic logic. Universal algebra. Residuated lattices. Paraconsistent Logic.

#### Sara Ugolini

Ramon y Cajal Fellow (tenure track) at the Artificial Inteligence Research Institute of the CSIC.

Algebraic Logic. Mathematical Fuzzy Logic. Universal Algebra. Uncertain Reasoning.

#### Amanda Vidal

Marie Curie Fellow at the Artificial Inteligence Research Institute of the CSIC.

Formal logic in AI. Mathematical Fuzzy Logic. Modal Logic. Complexity theory in non-classical logics.

### Students

#### Damiano Fornasiere

PhD Fellow of J. Gispert and T. Moraschini at the UB.

Algebraic Logic. Modal and Intuitionistic Logic. Duality Theory.

#### Valeria Giustarini,

PhD fellow of T. Moraschini and S. Ugolini at the IIIA - CSIC and the UB.

Algebraic Logic, Non-classical Logic, Universal Algebra.

#### Miriam kurtzhals

PhD fellow of L. Carai (Milano U.) and T. Moraschini at the UB.

Algebraic Logic. Modal and Intuitionistic Logic. Duality Theory. Universal Algebra.

#### Francesco Manfucci

Master student of S. Ugolini at the University of Siena and JAE intro fellow at the IIIA- CSIC.

Algebraic Logic. Mathematical Fuzzy Logic. Universal Algebra.

#### Miguel Martins

PhD fellow of J. Gispert and T. Moraschini and the UB.

Algebraic logic. Modal and (bi-)intuitionistic logic. Duality theory.

## Publications

### Indexed Proceedings & Book Chapters

• E. A. Corsi, T. Flaminio, H. Hosni. Towards a unified view on logics for un- certainty. In Proceedings of SUM’22. Lecture Notes in Artificial Intelligence. 2022.e Forthcoming
• T. Flaminio, A. Gilio, L. Godo, G. Sanfilippo. Compound conditionals as random quantities and Boolean algebras. Proceedings of the 19th International Conference on Principles of Knowledge Representation and Reasoning – KR’22: 141–151, 2022.
• T. Flaminio, A. Gilio, L. Godo, G. Sanfilippo. Canonical extensions of conditional probabilities and compound conditionals. Proceedings of IPMU’22. Communications in Computer and Information Science 1602: 584–597, 2022.
• T. Flaminio, L. Godo, P. Menchón, R. O. Rodríguez (In Press). Algebras and relational frames for Gödel modal logic and some of its extensions. M. Coniglio, E. Koubychkina, D. Zaitsev (Eds.), Many-valued Semantics and Modal Logics: Essays in Honour of Yuriy Vasilievich Ivlev. Springer (Also as CoRR, abs/2110.02528). Forthcoming.
• T. Flaminio, L. Godo, P. Menchón, R.O. Rodriguez. Rotations of Gö del algebras with modal operators. Proceedings of IPMU’22. Communications in Computer and Information Science 1601: 676–688, 2022.
• T. Flaminio, L. Godo, S. Ugolini. An approach to inconsistency-tolerant reasoning about probability based on Łukasiewicz logic. In Proceedings of SUM’22. Lecture Notes in Artificial Intelligence. 2022. Forthcoming.

## Projects

### Running projects

• #### The Geometry of Non-Classical Logics

I+D+i research project funded by the Ministry of Science and Innovation of Spain.
2020 - 2023
Principal Investigator: Tommaso Moraschini.

• #### ISINC: Inference Systems for Inconsistent Information: logical foundations

I+D+i research project funded by the Ministry of Science and Innovation of Spain.
2020 - 2023
Principal Investigator: Lluís Godo.

• #### SuMoL. Substructural Modal Logics for Knowledge Representation

I-Link project funded by the Spanish National Research Council (CSIC).
2020 - 2022
Principal Invertigator: Tommaso Flaminio.

• #### Mosaic. Modalities in Substructural Logics: Theory, Methods and Applications

Marie Skłodowska-Curie RISE project funded by the Horizon 2020 of the European Union.
2021 - 2024
Principal Invertigator: Tommaso Flaminio.

• #### Non-Classical Logics Research Group

Funded by the Agency for Management of University and Research Grants of the Government of Catalonia.
2017 - 2021.
Principal Invertigator: Joan Gispert.

### Previous projects

• #### Applied Philosophy for the Value-Based Design of Social Network Apps

Project RECERCAIXA, funded by La Caixa Foundation.
2019 - 2021
Principal Investigator: Pilar Dellunde.

• #### RASO: Razonamiento, Satisfacción y Optimización

I+D+i research project funded by the Ministry of Science and Innovation of Spain.
2016 - 2020
Principal Investigator: Lluís Godo.

• #### SYSMICS. Syntax Meets Semantics: Methods, Interactions, and Connections in Substructural logics

Marie Skłodowska-Curie RISE project funded by the Horizon 2020 of the European Union.
2016 - 2019.
Coordinating researcher at the University of Barcelona: Ramon Jansana.
Coordinating researcher at the Artificial Inteligence Research Institute (CSIC): Lluis Godo.
Coordinating researcher at the Autonomous University of Barcelona: Pilar Dellunde.

• #### Algebraic Logic and Non-Classical Logics

I+D+i research project funded by the Ministry of Economy and Competitiveness of Spain.
2017 - 2019.
Principal Invertigator: Ramon Jansana.

## Funding

There is a number of Master, PhD, postdoctoral and tenure track schemes available for researchers interested in coming to work with our group, including the following.

### Master

CSIC JAE-ICU and Master grants at the UB. To apply, please contact in advance a member of our research group from the IIIA-CSIC or the UB.

### PhD

You can find general information on PhDs the UB here . There are grants at national (FPU), regional (FI) and university (APIF) level, as well as grants associated to research projects (FPI) which will be publicized here when available.

### Postdocs

Funding opportunities include the Juan de la Cierva national scheme (for young postdocs, 2 years), the Beatriu de Pinos Catalan scheme cofunded by the European Union (3 years), and Marie Skłodowska-Curie Actions funded by the European Union (2 years).

### Tenure Traks

Funding opportunities include the Ramon y Cajal national scheme (4 years), the Beatriz Galindo national scheme (4 years), and the Serra Hunter Catalan scheme (4 years).

## Teaching

The Barcino research group gravitates around Bachelor, Master and PhD programmes devoted to logic and its applications to computer science.

The Bachelor in Artificial Intelligence of the Autonomous University of Barcelona offers training in the rapidly evolving field of AI where logic finds a natural application as a tool for explaining algorithmic based decisions.

The Master in Pure and Applied Logic of the University of Barcelona centers on some of the most influential areas of mathematical logics including Set Theory, Model Theory, Non-Classical Logics, Algebraic Logic, Computability Theory and Proof Theory.

Lastly, the Univeristy of Barcelona offers a PhD programme in Mathematical Logic. For information, please check the PhD Program in Mathematics and Computer Science.

## Recent & Upcoming Events

### 2024

#### "Topology, Algebra and Categories in Logic", or "Topology, Algebra and Categories in Logic (mirror)"

We are excited to announce the group is organizing the 2024 edition of the TACL conference, to be held in Barcelona in July 2024.

#### "Conditionals 2024"

Members of the group are organizing the first edition of the Conditionals meeting, to be held in Barcelona in October 2024.

### 2022

#### "Non-classical logic days"

During October 11th and 14th, 2022, Barcino is organizing a workshop in non-classical logics, where colleagues from the Institute of Computer Science from the Czech Academy of Sciences, and from the Universidad Nacional de La Plata (Argentina) will talk. The programme can be consulted here.

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100.00