Statistical Parsing for Logical Information Retrieval

TL;DR

This work advances formal NL reasoning by extending the Quantified Boolean Bayesian Network into a Logical Bayesian Network that supports negation and contrapositive reasoning via a new NEG factor, enabling complete forward natural deduction reasoning. It introduces a typed three-tier logical language and a deterministic typed slot grammar that maps NL to precise logical forms, with LLMs providing preprocessing and reranking to handle ambiguity. The system achieves 44/44 correctness on inference tests and 33/33 zero-ambiguity parses across 12 grammar patterns, demonstrating a viable pipeline from text to verifiable probabilistic logic. By combining LLM-assisted annotation with a verifiable symbolic core, the approach embodies a semi-automatic bridge that aligns with the Bitter Lesson, offering scalable, interpretable, and checkable reasoning for open-domain language understanding. Open-source code and data support replication and future scaling toward end-to-end autonomous reasoning.

Abstract

In previous work (Coppola, 2024) we introduced the Quantified Boolean Bayesian Network (QBBN), a logical graphical model that implements the forward fragment of natural deduction (Prawitz, 1965) as a probabilistic factor graph. That work left two gaps: no negation/backward reasoning, and no parser for natural language. This paper addresses both gaps across inference, semantics, and syntax. For inference, we extend the QBBN with NEG factors enforcing P(x) + P(neg x) = 1, enabling contrapositive reasoning (modus tollens) via backward lambda messages, completing Prawitz's simple elimination rules. The engine handles 44/44 test cases spanning 22 reasoning patterns. For semantics, we present a typed logical language with role-labeled predicates, modal quantifiers, and three tiers of expressiveness following Prawitz: first-order quantification, propositions as arguments, and predicate quantification via lambda abstraction. For syntax, we present a typed slot grammar that deterministically compiles sentences to logical form (33/33 correct, zero ambiguity). LLMs handle disambiguation (95% PP attachment accuracy) but cannot produce structured parses directly (12.4% UAS), confirming grammars are necessary. The architecture: LLM preprocesses, grammar parses, LLM reranks, QBBN infers. We argue this reconciles formal semantics with Sutton's "bitter lesson" (2019): LLMs eliminate the annotation bottleneck that killed formal NLP, serving as annotator while the QBBN serves as verifier. Code: https://github.com/gregorycoppola/world