We present the logical induction criterion for computable algorithms thatassign probabilities to every logical statement in a given formal language, andrefine those probabilities over time. The criterion is motivated by a series ofstock trading analogies. Roughly speaking, each logical sentence phi isassociated with a stock that is worth $1 per share if phi is true and nothingotherwise, and we interpret the belief-state of a logically uncertain reasoneras a set of market prices, where pt_N(phi)=50% means that on day N, shares ofphi may be bought or sold from the reasoner for 50%. A market is then called alogical inductor if (very roughly) there is no polynomial-time computabletrading strategy with finite risk tolerance that earns unbounded profits inthat market over time. We then describe how this single criterion implies anumber of desirable properties of bounded reasoners; for example, logicalinductors outpace their underlying deductive process, perform universalempirical induction given enough time to think, and place strong trust in theirown reasoning process.
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