  This paper introduces the Quantified Boolean Bayesian Network (QBBN), which
provides a unified view of logical and probabilistic reasoning. The QBBN is
meant to address a central problem with the Large Language Model (LLM), which
has become extremely popular in Information Retrieval, which is that the LLM
hallucinates. A Bayesian Network, by construction, cannot hallucinate, because
it can only return answers that it can explain. We show how a Bayesian Network
over an unbounded number of boolean variables can be configured to represent
the logical reasoning underlying human language. We do this by creating a
key-value version of the First-Order Calculus, for which we can prove
consistency and completeness. We show that the model is trivially trained over
fully observed data, but that inference is non-trivial. Exact inference in a
Bayesian Network is intractable (i.e. $\Omega(2^N)$ for $N$ variables). For
inference, we investigate the use of Loopy Belief Propagation (LBP), which is
not guaranteed to converge, but which has been shown to often converge in
practice. Our experiments show that LBP indeed does converge very reliably, and
our analysis shows that a round of LBP takes time $O(N2^n)$, where $N$ bounds
the number of variables considered, and $n$ bounds the number of incoming
connections to any factor, and further improvements may be possible. Our
network is specifically designed to alternate between AND and OR gates in a
Boolean Algebra, which connects more closely to logical reasoning, allowing a
completeness proof for an expanded version of our network, and also allows
inference to follow specific but adequate pathways, that turn out to be fast.
