Statistical learning and logical reasoning are two major fields of AI expected to be unified for human-like machine intelligence. Most existing work considers how to combine existing logical and statistical systems. However, there is no theory of inference so far explaining how basic approaches to statistical learning and logical reasoning stem from a common principle. Inspired by the fact that much empirical work in neuroscience suggests Bayesian (or probabilistic generative) approaches to brain function including learning and reasoning, we here propose a simple Bayesian model of logical reasoning and statistical learning. The theory is statistically correct as it satisfies Kolmogorov's axioms, is consistent with both Fenstad's representation theorem and maximum likelihood estimation and performs exact Bayesian inference with a linear-time complexity. The theory is logically correct as it is a data-driven generalisation of uncertain reasoning from consistency, possibility, inconsistency and impossibility. The theory is correct in terms of machine learning as its solution to generation and prediction tasks on the MNIST dataset is not only empirically reasonable but also theoretically correct against the K nearest neighbour method. We simply model how data causes symbolic knowledge in terms of its satisfiability in formal logic. Symbolic reasoning emerges as a result of the process of going the causality forwards and backwards. The forward and backward processes correspond to an interpretation and inverse interpretation in formal logic, respectively. The inverse interpretation differentiates our work from the mainstream often referred to as inverse entailment, inverse deduction or inverse resolution. The perspective gives new insights into learning and reasoning towards human-like machine intelligence.

翻译：统计学习与逻辑推理是人工智能的两大核心领域，二者的融合被认为是实现类人机器智能的关键。现有研究主要关注如何组合现有逻辑系统与统计系统，但尚未有推理理论能够阐明统计学习与逻辑推理的基本方法源自同一原理。受神经科学领域大量实证研究启发——这些研究表明贝叶斯（或概率生成）方法可解释包括学习与推理在内的大脑功能——本文提出一个简洁的贝叶斯逻辑推理与统计学习模型。该理论在统计上具有正确性：它满足柯尔莫哥洛夫公理体系，同时符合芬斯塔特表示定理与最大似然估计，并以线性时间复杂度实现精确贝叶斯推理。该理论在逻辑上具有正确性：它从一致性、可能性、不一致性与不可能性出发，通过数据驱动方式泛化不确定性推理。该理论在机器学习层面亦具正确性：针对MNIST数据集的生成与预测任务，其解决方案不仅具有经验合理性，且在理论上比K近邻方法更优。我们通过符号逻辑中的可满足性概念，简洁建模了数据如何引发符号知识。符号推理作为因果前向与反向过程的结果涌现。前向过程与反向过程分别对应形式逻辑中的解释与逆解释。其中，逆解释过程使本研究区别于主流方法（常被称为逆蕴涵、逆演绎或逆归结）。该视角为迈向类人机器智能的学习与推理研究提供了全新洞见。