# A Formal Approach to the Problem of Logical Non-Omniscience

arXiv: https://arxiv.org/abs/1707.08747 · HTML: https://mlsanity.com/paper/1707.08747/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Jul 2017.

## Built on by (most cited first, 2 of 2)

- [A Theory of Bounded Inductive Rationality](https://mlsanity.com/paper/2307.05068/) (2023)
- [Understanding Generalization Requires Universal Induction](https://mlsanity.com/paper/2609.34458/) (2026)

## Abstract (arXiv)

We present the logical induction criterion for computable algorithms that assign probabilities to every logical statement in a given formal language, and refine those probabilities over time. The criterion is motivated by a series of stock trading analogies. Roughly speaking, each logical sentence phi is associated with a stock that is worth 1 per share if phi is true and nothing otherwise, and we interpret the belief-state of a logically uncertain reasoner as a set of market prices, where ptN(phi)=50% means that on day N, shares of phi may be bought or sold from the reasoner for 50%. A market is then called a logical inductor if (very roughly) there is no polynomial-time computable trading strategy with finite risk tolerance that earns unbounded profits in that market over time. We then describe how this single criterion implies a number of desirable properties of bounded reasoners; for example, logical inductors outpace their underlying deductive process, perform universal empirical induction given enough time to think, and place strong trust in their own reasoning process.