Modal Logical Neural Networks for Financial AI

Antonin Sulc

cs.LGMar 12, 2026

Modal Logical Neural Networks for Financial AI

Authors: Antonin Sulc

Organizations: Lawrence Berkeley National Lab, Berkeley, CA, USA

Abstract

The financial industry faces a critical dichotomy in AI adoption: deep learning often delivers strong empirical performance, while symbolic logic offers interpretability and rule adherence expected in regulated settings. We use Modal Logical Neural Networks (MLNNs) as a bridge between these worlds, integrating Kripke semantics into neural architectures to enable differentiable reasoning about necessity, possibility, time, and knowledge. We illustrate MLNNs as a differentiable ``Logic Layer'' for finance by mapping core components, Necessity Neurons ( $\Box$ ) and Learnable Accessibility ( $A_θ$ ), to regulatory guardrails, market stress testing, and collusion detection. Four case studies show how MLNN-style constraints can promote compliance in trading agents, help recover latent trust networks for market surveillance, encourage robustness under stress scenarios, and distinguish statistical belief from verified knowledge to help mitigate robo-advisory hallucinations.

Figures & tables

Panel of Figure 1 — Figure 1: Modal Logical Neural Networks for Finance. Left: Kripke semantics with possible worlds (temporal: $t,t+1$ ; stress: crash world $w_{c}$ ). Middle: Two learning modes—(A) Deductive: fixed accessibility $R$ (e.g., time flow), (B) Inductive: learned $A_{\theta}$ (e.g., trust networks). Modal operators $\Box$ (necessity: all worlds) and $\Diamond$ (possibility: $\geq$ 1 world) enable differentiable logical reasoning. Right: Four financial applications leveraging these modes for compliance, surveillance, risk management, and interpretability.

Table 2

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Category	Count	Explanation
Verified Safe	475	“Safe across all risk worlds.”
Trap Detected	155	“Title standard, but clause opens risk world.”
Uncertain	19	“Moderate risk. Human review recommended.”

Appendix

Table 3: MLNN Explanation Categories on CUAD Test Set

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Modal Logic Neural Networks

Antonin Sulc, Noor Naddour

Lawrence Berkeley National Laboratory, Berkeley, CA, USA · The University of Queensland, Brisbane, QLD, Australia

Logical Reasoning Differentiable Logic Neuro-Symbolic Reasoning

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Neural Networks are indispensable to natural sciences and society. Their impact extends from applications in public health to workforce productivity. Here, we introduce Modal Logic Neural Networks (MLNNs) -- an end-to-end differentiable logical neural network realisation of modal logic which evaluates a learnable truth function across possible-world semantics. This neural architecture handles para-consistency and inconsistency via a learnable world accessibility relation and valuation function. Because the modality is fixed by which frame axioms the relation satisfies rather than by the operator, one differentiable engine covers the epistemic, doxastic, deontic and temporal readings, with applications from verification of reactive and distributed systems to legal discourse and microeconomic utility models. In this paper, we introduce a model of differentiable Kripke semantics, and establish their soundness, convergence, and structural guarantees. We show four applications, in which the learned relation reads as a trust matrix, an operating-regime embedding with safety bounds, a temporal precedence order, and a recovered constraint graph.

Abstract

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