A Multimodal AI System: comparing LLMs and Theorem Proving Systems

This paper discusses a multimodal AI system applied to legal reasoning for tax law. The results given here are very general and apply to systems developed for other areas besides tax law. A central goal of this work is to gain a better understanding of the relationships between LLMs (Large Language Models) and automated theorem-proving methodologies. To do this, we suppose (1) two cases for the theorem-proving system: one where it has a countable number of total meanings for its countable number of atoms and the other is where it has an uncountable number of total meanings for its countable number of atoms, and (2) LLMs can have an uncountable number of token meanings. With this in mind, the results given in this paper use the downward and upward Löwenheim–Skolem theorems and logical model theory to contrast these two AI modalities. One modality focuses on syntactic proofs and the other focuses on logical semantics based on LLMs. Particularly, one modality uses a rule-based first-order logic theorem-proving system to perform legal reasoning. The objective of this theorem-proving system is to provide proofs as evidence of valid legal reasoning when enacted laws are applied to particular situations. These proofs are syntactic structures that can be presented in the form of narrative explanations of how the answer to the legal question was determined. The second modality uses LLMs to analyze and transform a user's tax query so this query can be sent to a first-order logic theorem-proving system to perform its legal reasoning function. The main goal of our application of LLMs is to enhance and simplify user input and output for the theorem-proving system. Using logical model theory, we show how there can exist an equivalence between laws represented in logic of the theorem-proving system, fixed in time when the theorem-proving system was set up, and new semantics given by LLMs. These results are based on logical model theory and Löwenheim–Skolem theorems.