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 pt_N(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.Comment: In Proceedings TARK 2017, arXiv:1707.0825
We give an algorithm AL,T which assigns probabilities to logical sentences. For any simple infinite sequence {φs n } of sentences whose truthvalues appear indistinguishable from a biased coin that outputs "true" with probability p, we have limn→∞ AL,T (sn) = p.
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