(Joint work with Verónica Borja Macías and Alejandro Hernández-Tello)

In this talk, we introduce a non-classical probabilistic framework capable of dealing with information that may contain gaps or gluts, while also accounting for its reliability. It is built upon the logic LETK+, a six-valued paradefinite logic that expands Belnap-Dunn's First-Degree Entailment (FDE) by adding a 'classicality' recovery operator o that allows us to locally restore the missing principles of excluded middle and explosion [1].

First, we define non-standard twist models M = (X, v) based on a particular twist structure semantics for LETK+. In these models, a valuation v assigns to each formula φ a triple (|φ|+, |φ|-, |φ|o), which is interpreted as sets of states (subsets of X) representing positive, negative, and reliable evidence. This triple structure partitions the state space into six distinct regions. This generalizes to LETK+ the four-region proposal for FDE in [2].

From this, three kinds of probability functions are introduced by means of a measure μ:

(1) A one-dimensional function p(φ) = μ(|φ|+).

(2) A three-dimensional "twist" function p3(φ) = (μ(|φ|+), μ(|φ|-), μ(|φ|o)).

(3) A six-dimensional function p6(φ) that assigns probabilities applying the measure μ directly to each of the six evidence regions.

The main results to be presented are the following:

(i) the three notions of probability are equivalent

(ii) each of them can be defined in an abstract (axiomatic) way

(iii) the axiomatization of each notion is sound and complete w.r.t. the original definition

(iv) Jeffrey update can be generalized to a more granular, six-valued update based on the LETK+ probabilistic framework. This finer update allows for belief revision based on detailed changes in the nature of evidence, such as its reliability.

References:

[1] M. E. Coniglio and A. Rodrigues. From Belnap-Dunn four-valued logic to six-valued logics of evidence and truth. Studia Logica, 112(3):561–606, 2024.

[2] D. Klein, O. Majer, and S. Rafiee Rad. Probabilities with gaps and gluts. Journal of Philosophical Logic, 50(5):1107-1141, 2021.