  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.
