# Statistical Parsing for Logical Information Retrieval

arXiv: https://arxiv.org/abs/2602.12170 · HTML: https://mlsanity.com/paper/2602.12170/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 36 references that are in the map.
- First posted on arXiv 12 Feb 2026.

## Builds on (most cited first, 8 of 8)

- [Neural Probabilistic Logic Programming in DeepProbLog](https://mlsanity.com/paper/1907.08194/) (2019)
- [Colorless green recurrent networks dream hierarchically](https://mlsanity.com/paper/1803.11138/) (2018)
- [End-to-End Differentiable Proving](https://mlsanity.com/paper/1705.11040/) (2017)
- [Loopy Belief Propagation for Approximate Inference: An Empirical Study](https://mlsanity.com/paper/1301.6725/) (2013)
- [Large Linguistic Models: Investigating LLMs' metalinguistic abilities](https://mlsanity.com/paper/2305.00948/) (2023)
- ["You Are An Expert Linguistic Annotator": Limits of LLMs as Analyzers of Abstract Meaning Representation](https://mlsanity.com/paper/2310.17793/) (2023)
- [Approximation-Aware Dependency Parsing by Belief Propagation](https://mlsanity.com/paper/1508.02375/) (2015)
- [The Quantified Boolean Bayesian Network: Theory and Experiments with a Logical Graphical Model](https://mlsanity.com/paper/2402.06557/) (2024)

## Built on by (most cited first, 1 of 1)

- [Transformers are Bayesian Networks](https://mlsanity.com/paper/2603.17063/) (2026)

## Abstract (arXiv)

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