Model Theory in a Paraconsistent Environment

The purpose of this thesis is to develop a paraconsistent Model Theory. The basis for such a theory was launched by Walter Carnielli, Marcelo Esteban Coniglio, Rodrigo Podiack, and Tarcísio Rodrigues in the article ‘On the Way to a Wider Model Theory: Completeness Theorems for First-Order Logics of Formal Inconsistency’ [The Review of Symbolic Logic, vol. 7 (2014)].Naturally, a complete theory cannot be fully developed in a single work. Indeed, the goal of this work is to show that a paraconsistent Model Theory is a sound and worthy possibility. The pursuit of this goal is divided in three tasks: The first one is to give the theory a philosophical meaning. The second one is to transpose as many results from the classical theory to the new one as possible. The third one is to show an application of the theory to practical science.The response to the first task is a Paraconsistent Reasoning System. The start point is that paraconsistency is an epistemological concept. The pursuit of a deeper understanding of the phenomenon of paraconsistency from this point of view leads to a reasoning system based on the Logics of Formal Inconsistency. Models are regarded as states of knowledge and the concept of isomorphism is reformulated so as to give raise to a new concept that preserves a portion of the whole knowledge of each state. Based on this, a notion of refinement is created which may occur from inside or from outside the state.In order to respond to the second task, two important classical results, namely the Omitting Types Theorem and Craig’s Interpolation Theorem are shown to hold in the new system and it is also shown that, if classical results in general are to hold in a paraconsistent system, then such a system should be in essence how it was developed here.Finally, the response to the third task is a proposal of what a Paraconsistent Logic Programming may be. For that, the basis for a paraconsistent PROLOG is settled in the light of the ideas developed so far.Abstract prepared by Bruno Costa Coscarelli.E-mail: [email protected]: http://repositorio.unicamp.br/jspui/handle/REPOSIP/331697

Paraconsistent Reasoning in Cops and Robber Game with Uncertain Information: A Simulation-Based Analysis

We apply a non-classical four-valued logic in the process of reasoning regarding strategies for cops in a modified game of “Cops and Robber” played on a graph. We extend the game by introducing uncertainty in a form of random failures of detecting devices. This is realized by allowing that a robber can be detected in a node only with the given probability PA. Additionally, with the probability PF, cops can be given a false-positive, i.e., they are informed that the robber is located at some node, whereas it is located somewhere else. Consequently, non-zero PFintroduces a measurement noise into the system. All the cops have access to information provided by the detectors and can communicate with each other, so they can coordinate the search. By adjusting the number of detectors, PA, and PFwe can achieve a smooth transition between the two well-known variants of the game: “with fully visible robber” and “with invisible robber”. We compare a simple probabilistic strategy for cops with the non-parametric strategy based on reasoning with a four-valued paraconsistent logic. It is shown that this novel approach leads to a good performance, as measured by the required mean catch-time. We conclude that this type of reasoning can be applied in real-world applications where there is no knowledge about the underlying source of errors which is particularly useful in robotics.

PROSE

The study of paraconsistent reasoning with ontologies is especially important for the Semantic Web since knowledge is not always perfect within it. However, classical OWL reasoners cannot support reasoning with inconsistent ontologies. In this article, the authors present a plugin-based framework called prose to provide rich paraconsistent reasoning services for OWL ontologies, whose architecture contains the three following parts: a classical OWL reasoner, a multi-valued transformer, and an OWL API connecting with them. Within the proposed framework prose, they implement different multi-valued paraconsistent reasoning in the OWL. Moreover, they select three popular classical OWL reasoners and two typical kinds of reasoning services for users. As the authors excepted, prose does exactly enable current classical OWL reasoners to tolerate inconsistency in a simple and convenient way. Finally, they evaluate the three reasoners in a united framework (prose) and, as a result, those results can amend the analysis of the three reasoners on inconsistent ontologies.

Paraconsistent Measurement of the Circle

A theorem from Archimedes on the area of a circle is proved in a setting where some inconsistency is permissible, by using paraconsistent reasoning. The new proof emphasizes that the famous method of exhaustion gives approximations of areas closer than any consistent quantity. This is equivalent to the classical theorem in a classical context, but not in a context where it is possible that there are inconsistent innitesimals. The area of the circle is taken 'up to inconsistency'. The fact that the core of Archimedes's proof still works in a weaker logic is evidence that the integral calculus and analysis more generally are still practicable even in the event of inconsistency.

Resolving conflicts in knowledge for ambient intelligence

Ambient intelligence (AmI) proposes pervasive information systems composed of autonomous agents embedded within the environment who, in orchestration, complement human activity in an intelligent manner. As such, it is an interesting and challenging application area for many computer science fields and approaches. A critical issue in such application scenarios is that the agents must be able to acquire, exchange, and evaluate knowledge about the environment, its users, and their activities. Knowledge populated between the agents in such systems may be contextually dependent, ambiguous, and incomplete. Conflicts may thus naturally arise, that need to be dealt with by the agents in an autonomous way. In this survey, we relate AmI to the area of knowledge representation and reasoning (KR), where conflict resolution has been studied for a long time. We take a look at a number of KR approaches that may be applied: context modelling, multi-context systems, belief revision, ontology evolution and debugging, argumentation, preferences, and paraconsistent reasoning. Our main goal is to describe the state of the art in these fields, and to draw attention of researchers to important theoretical issues and practical challenges that still need to be resolved, in order to reuse the results from KR in AmI systems or similar complex and demanding applications.