Inference of Abstraction for a Unified Account of Symbolic Reasoning from Data

TL;DR

The paper addresses unifying symbolic reasoning within a Bayesian, data-driven framework by modeling data, language, and world-models in a single generative distribution $p(L,M,D;\mu)$. A tunable parameter $\mu$ governs whether reasoning adheres to classical logic or accommodates empirical and paraconsistent forms, with constructs like maximal consistent and maximal possible subsets enabling robust inference from inconsistent or impossible premises. By deriving reasoning modes (logical, empirical, paraconsistent, parapossible) from this shared model, the approach provides a principled path for inference grounding and scalable probabilistic reasoning over symbolic knowledge. The framework highlights the computational trade-offs in posterior inference and offers a unified perspective on how data give rise to symbolic conclusions.

Abstract

Inspired by empirical work in neuroscience for Bayesian approaches to brain function, we give a unified probabilistic account of various types of symbolic reasoning from data. We characterise them in terms of formal logic using the classical consequence relation, an empirical consequence relation, maximal consistent sets, maximal possible sets and maximum likelihood estimation. The theory gives new insights into reasoning towards human-like machine intelligence.

Figures (7)

• Figure 1: Pellicano:12 (A) The Kanizsa triangle illusion, (B) The hollow-face illusion, and (C) Shepard's table illusion.
• Figure 2: A schematic of how the probability distribution over data determines the probability distribution over logical formulas. For simplicity, an arrow is omitted if the formula at the end of the arrow is false in the model at the start of the arrow and if the model at the end of the arrow is not supported by the data at the start of the arrow.
• Figure 3: The left two graphs illustrate reasoning of $\alpha\in L$ from $\Delta\subseteq L$ using $p(L,M,D;\mu=1)$. The leftmost shows the assumptions of $[\![\Delta]\!]=[\![\![\Delta]\!]\!]$ and $[\![\Delta]\!]\neq\emptyset$. Each arrow from a datum to model, denoted respectively by a black circle on the top layer and a cell on the middle layer, represents that the datum supports the model. Each model with an incoming arrow thus has a non-zero probability. A model is coloured in green (resp. blue) if all the formulas in $\Delta$ are (resp. $\alpha$) true in the model. The second shows the assumption of $[\![\![\Delta]\!]\!]\neq\emptyset$. The right two graphs illustrate reasoning of $\alpha\in L$ from $\Delta\subseteq L$ using $p(L,M,D;\mu\to1)$. The third shows the assumption of $(\!(\Delta)\!)=(\!(\!(\Delta)\!)\!)$. $\Delta_{1}$, $\Delta_{2}$ and $\Delta_{3}$ are the cardinality-maximal consistent subsets of $\Delta$. The rightmost shows no assumption. $\Delta_{1}$ and $\Delta_{2}$ are the cardinality-maximal possible subsets of $\Delta$.
• Figure 4: Three examples of reasoning from inconsistency. The probability versus $\mu$.
• Figure 5: The left can fit with any full joint distribution. The right can fit with any full joint distribution with the conditional independence, $p(X_{3}|X_{2},X_{1})=p(X_{3}|X_{1})$, that rarely holds without data modification.

Theorems & Definitions (35)

• Proposition 1
• proof
• Example 1
• Definition 1: Possibility
• Theorem 1
• proof
• Corollary 1
• proof
• Example 2
• Theorem 2