Sesión Lógica y ComputabilidadProbabilities with gaps, gluts and reliability
Marcelo Esteban Coniglio
CLE, Universidade Estadual de Campinas (UNICAMP), Brazil - Esta dirección de correo electrónico está siendo protegida contra los robots de spam. Necesita tener JavaScript habilitado para poder verlo.
In this talk we present a study of a non-classical probability framework that allows us to deal with information with gaps and gluts, while also managing its reliability. It is built upon the logic of evidence and truth \(LET_{K}^{+}\), a six-valued paradefinite logic introduced in [1] that extends the Belnap-Dunn logic First-Degree Entailment (FDE). \(LET_{K}^{+}\) handles paraconsistent and paracomplete scenarios by introducing a recovery (or reliability) operator \(\circ\) to restore classical principles like excluded middle and explosion for certain formulas.
First, we define twist models \(M = \langle X, v \rangle\) based on a particular twist structure for \(LET_{K}^{+}\) over \(\wp(X)\). In these models, a valuation \(v\) assigns to each formula \(\varphi\) a triple of sets \((|\varphi|^+, |\varphi|^-, |\varphi|^\circ)\), which is interpreted as three states representing positive, negative, and reliable evidence. This triple partitions the state space of \(\varphi\) in \(M\) into six regions, namely, \(B^\circ_\varphi\) (reliable belief), \(B_\varphi\) (unreliable belief), \(D_\varphi\) (unreliable disbelief), \(D_\varphi^\circ\) (reliable disbelief), \(C_\varphi\) (conflict), and \(U_\varphi\) (uncertainty). This generalizes the four-regions partition proposed in [2] using FDE.
Based on this semantical framework, three kinds of probability functions for formulas of \(LET_{K}^{+}\) are introduced by means of a measure \(\mu\):
(1) A one-dimensional function \(p(\varphi) = \mu(|\varphi|^+)\).
(2) A three-dimensional “twist” function \(\widetilde{p}(\varphi) = (\mu(|\varphi|^+), \mu(|\varphi|^-), \mu(|\varphi|^\circ))\).
(3) A six-dimensional function \(\widehat{p}(\varphi)\) that assigns probabilities by applying the measure directly to each of the six evidence regions.
We present three main results: the proof of equivalence among these three approaches, the soundness and completeness of their syntactic axiomatizations, and a generalization of classic update rules within the \(LET_{K}^{+}\) framework. Regarding the last contribution, we obtain a one-dimensional Jeffrey update over the partition \(\{\varphi, {\sim}\varphi\}\) (where \(\sim\) is the classical negation definable in \(LET_{K}^{+}\)) and a finer-grained six-dimensional Jeffrey update over the six evidence regions.
This research was financed by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP, Brazil), Thematic Project “Rationality, logic and probability – RatioLog", grant 2020/16353-3, and by the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq, Brazil), grant 309830/2023-0.
Trabajo en conjunto con: Verónica Borja Macías (Universidad Tecnológica de la Mixteca, México) y Alejandro Hernández-Tello (Universidad Tecnológica de la Mixteca, México).
Referencias
[1] Marcelo E. Coniglio and Abilio Rodrigues. From Belnap-Dunn four-valued logic to six-valued logics of evidence and truth. Studia Logica, 112(3):561-606, 2024.
[2] Dominik Klein, Ondrej Majer, and Soroush Rafiee Rad. Probabilities with gaps and gluts. Journal of Philosophical Logic, 50(5):1107-1141, 2021.