Paraconsistent Probabilities: Consistency, Contradictions and Bayes’ Theorem

Juliana Bueno-Soler,Walter Carnielli

doi:10.3390/e18090325

Abstract

This paper represents the first steps towards constructing a paraconsistent theory of probability based on the Logics of Formal Inconsistency (LFIs). We show that LFIs encode very naturally an extension of the notion of probability able to express sophisticated probabilistic reasoning under contradictions employing appropriate notions of conditional probability and paraconsistent updating, via a version of Bayes’ theorem for conditionalization. We argue that the dissimilarity between the notions of inconsistency and contradiction, one of the pillars of LFIs, plays a central role in our extended notion of probability. Some critical historical and conceptual points about probability theory are also reviewed.

Highlights

IntroductionThis paper represents the first steps towards constructing a paraconsistent theory of probability based on the Logics of Formal Inconsistency (LFIs)
Faculty of Technology, State University of Campinas –UNICAMP, Campinas 13484-332, Brazil; Centre for Logic, Epistemology and the History of Science and Department of Philosophy, State University of Campinas—UNICAMP, Campinas 13083-859, Brazil
What happens is that each semantics expresses logical truth and logical consequence in its own way, and we show that, quite surprisingly, the equivalence between such two distinct semantics holds for the paraconsistent probability theory based on Ci

Summary

Introduction

This paper represents the first steps towards constructing a paraconsistent theory of probability based on the Logics of Formal Inconsistency (LFIs). We argue that the dissimilarity between the notions of inconsistency and contradiction, one of the pillars of LFIs, plays a central role in our extended notion of probability. Even an irrelevant contradiction in traditional logic obliges a reasoner that follows such a logic to derive anything from a pair of contradictory statements {α, ¬α}, as a result of the so-called Principle of Explosion (PEx): α, ¬α, ` β, for arbitrary β. Recognizes that some contradictions may be intolerable, and those would destroy the very act of reasoning (that is, lead to trivialization). This amounts to admitting that not all contradictions are equivalent. The Logics of Formal Inconsistency (LFIs), a family of paraconsistent logics designed to express the notion of consistency (and inconsistency, as well) within the object language by employing a connective ◦ (reading ◦α as “α is consistent”) realizes such an intuition

Objectives

Findings

Discussion

Conclusion