A probability language for imperfect information
A new formal framework gives LET+K, a six-valued paradefinite logic, a way to represent gaps, gluts and event reliability or classicality with probability functions. The result is a formal construction, with its central claims expressed through mathematical definitions and proofs rather than measurements from observed data.
LET+K is modeled with six truth values, written T, t, b, n, f and F. Three of them, T, t and b, are designated, meaning that the matrix treats them as the values that count for designation. For general readers, the important point is that the framework has more than a single true-or-false slot for each formula, which is how it can preserve distinctions such as gaps and gluts.
The construction then gives those distinctions a state space. It starts with a set V of propositional variables, considers all assignments from V to the six truth values, and uses a finite assignment set X in the completeness construction. In the twist-model semantics, each formula partitions X into exactly six disjoint regions.
Turning the partition into probability
That partition is what makes probability possible. A probability model induces functions with one, three or six components, and the six-valued function assigns a measure, or probability mass, to each formula's six regions. Those regions are described as reliable belief, belief, conflict, disbelief, reliable disbelief and uncertainty.
The paper says the one-, three- and six-component representations can be translated into one another. They are therefore different ways of expressing the same formal probability framework, with a tradeoff between greater expressivity and greater simplicity.
In its one-dimensional form, the framework is defined by axioms. They require normalization, zero probability for formulas that entail every formula, monotonicity under derivability, and an inclusion-exclusion rule. In ordinary language, the last rule says that the probabilities of two formulas added separately equal the probability of their conjunction plus the probability of their disjunction.
What the formal results establish
The paper reports soundness and completeness for the axiomatic and semantic definitions. For a general reader, that means it gives formal results linking the rules stated about probability functions with the functions generated by its models. These results concern the formal construction, not empirical performance or application validation.
The same comparison is made for updating beliefs. In the single-valued case, the semantic and syntactic Jeffrey updates are shown to produce the same function. The six-valued version is reported to have the same equivalence under the paper's admissibility conditions.
But six-valued updating is not commutative. If two updates are reversed, they can yield different functions, so the order in which information is incorporated matters within this construction.
The paper also treats positive and negative Bayesian updating as special cases of Jeffrey updating, with the relevant evidence assigned probability one or zero.
A formal result with practical questions still open
Those results should be read within the limits of the paper's design. The completeness construction uses a finite assignment space, and the supplied evidence does not include empirical performance or application validation. The work therefore establishes a formal probability framework and formal update correspondences, but it does not report how the approach performs in an application.
The document is an arXiv version 2 preprint dated 19 September 2026. Its contribution at this stage is a vocabulary for separating belief, conflict, disbelief, reliability and uncertainty in a six-region model, together with formal links among several representations and update procedures.
Paper data and sources
Original title: Probabilities beyond Belnap-Dunn logic: dealing with gaps, gluts and reliability
Authors: Verónica Borja Macias, Marcelo E. Coniglio, Alejandro Hernández-Tello
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text