Inference of Abstraction for Grounded Predicate Logic

TL;DR

This paper proposes a novel approach to logical reasoning in AI by deriving full joint distributions over models from data, offering insights into abstraction, grounding, and limitations of predicate logic.

Contribution

It introduces a new method for combining probabilistic reasoning with symbolic logic, revisiting foundational principles and addressing longstanding limitations.

Findings

Derived full joint distributions over models from data.

Provided a new perspective on predicate logic limitations.

Reproduced theoretical proofs demonstrating the approach.

Abstract

An important open question in AI is what simple and natural principle enables a machine to reason logically for meaningful abstraction with grounded symbols. This paper explores a conceptually new approach to combining probabilistic reasoning and predicative symbolic reasoning over data. We return to the era of reasoning with a full joint distribution before the advent of Bayesian networks. We then discuss that a full joint distribution over models of exponential size in propositional logic and of infinite size in predicate logic should be simply derived from a full joint distribution over data of linear size. We show that the same process is not only enough to generalise the logical consequence relation of predicate logic but also to provide a new perspective to rethink well-known limitations such as the undecidability of predicate logic, the symbol grounding problem and the principle of explosion. The reproducibility of this theoretical work is fully demonstrated by the included proofs.