It is well known that in non-classical logics there are two main directions: logics with additional operators (modal logics, temporal logics, epistemic logics, etc.); and logics with non-classical implications (essentially these ones where φ → ψ is not equivalent to ¬φ ∨ ψ). We want combine both kinds of approaches by taking an underlying logic with non-classical implication and extender it with modal op- erators. In particular, because we would like the modal notions of necessity and possibility could be interdefinible, we will consider an underlying non-classical logic which accept the law of double negation (¬¬φ → φ) and reject the law of excluded middle (φ ∨ ¬φ) which correspond to the constructive logic with strong negation originally suggested by D. Nelson and A. A. Markov. From the point of view of this kind of constructive logic, it is not equivalent to refute a sentence φ by reduction ad absurd um and by construction of a counter-example of φ. The first one to formalize this idea was Vorob’ev who gave an axiomatization for this logic. However, it is not evident from these axioms the connection between strong negation and refutations by means of counter-examples. Nevertheless, Vakarelov introduces a special class of lattices whose elements and operations may be interpreted in a way very closed to the intuitive notion of a counter-example. Without saying, this special class of lattices is given by a twist constructions in the sense of Kalman. Currently, it is known as N3-lattices and it was also studied independently by Fidel. Moreover, a full representation – in fact, a categorical equivalence – was finally obtained by Sendlewski. We follow this line of research and work over one of the most challenging research trends in non-classical logic: the attempt to combine different non-classical approaches, in our case Nelson logic and modal logic. This combination offers the skills of deal- ing with modal notions like belief, knowledge, and obligations in interaction with other aspects of reasoning that can be best handled using many-valued logics, for instance, vagueness, incompleteness, and uncertainty. The study that we are going to introduce could be especially interesting from the point of view of Theoretical Computer Science and Artificial Intelligence. In this first part of the seminar we will introduce the main constructions of Nelson Algebras. 

Ricardo Oscar Rodríguez is a doctor in computer science with specialization in artificial intelligence. He is a professor associated with exclusive dedication in the Computer Science Department, Faculty of Exact and Natural-UBA Sciences and researcher at the Institute of Computer Science (UBA-Conicet). His scientific works enroll in the development of logical models for reasoning under incompleteness and uncertainty. It has been Cochair and Finacial Chair de Ijcai2015-Buenos Aires (the most important conference in AI). He has organized an ethics symposium in the CLMPST2023. He is a member of the program committee of the most important lectures of AI and is a reviewer of magazine articles such as Artificial Intelligence Journal, International Journal of Approximate Reasoning, Journal of Symbolic Logic, Fuzzy Set and Systems. It also actively participates in the Stop Killer Robots campaign, being the author of several reports on the issue of autonomous weapons. Dictates courses and seminars on artificial ethics and intelligence. He is a member of the Argentine Computer Society (Sadio) and the Argentine Society of Philosophical Analysis (Sadaf). He has been a technical evaluator of projects from different Fonsooft programs from 2006 to 2018, especially in AI issues. He also is coordinator of the Technical Advisory Committee of the University of Buenos Aires that evaluates research projects.