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Logical induction is defined in terms of logical sentences and theories, but its principles are applicable in much greater generality and abstraction. Indeed, one such generalization was studied under the name "universal induction." We proposed a slightly different generalization in order to model reasoning with incomplete models. Here, we describe a formalism that includes all these cases and many more, using the language of measure theory. This provides the following advantages:

The formalism is applicable to event spaces substantially different from truth assignments or bit sequences, e.g. we can consider sequences of real numbers.
The formalism treats probabilities and expectations on the same footing, rather than constructing expectations as in section 4.8 of original paper. We consider this more convenient.
In our opinion, this language is more mathematically natural than the original formalism, at least for applications unrelated to formal logic.

On the other hand, we ignore all computational considerations. Obviously these are often important, but in the study of purely information-theoretic questions the use of numerical approximations only serves to obscure.

All proofs are in the Appendix.

##Results

Fix $X$ a compact Polish space. For example, $X$ might be $O^{ω}$ or the space of propositionally consistent truth assignments in some language or $[0, 1]^{ω}$ . The role of "pricings" is served by $P (X)$ : the space of probability measures on $X$ . A market is thus a sequence ${μ_{n} \in P (X)}_{n \in N}$ . The "deductive process" is replaced by a sequence of closed sets $X = X_{0} \supseteq X_{1} \supseteq X_{2} \supseteq \dots$ A trading strategy is a continuous function $τ : P (X) \to C (X)$ , were $P (X)$ is equipped with the weak* topology (as before) and $C (X)$ is the space of continuous functions from $X$ to $R$ equipped with the uniform convergence topology. Here, we should think of $τ (μ)$ as the share portfolio acquired by the strategy given pricing $μ$ , where the cost of the acquisition is understood to be $E_{μ} [τ (μ)]$ while the ultimate value of the portfolio is $τ (μ) (x)$ (in the following we will use the less clumsy notation $τ (μ, x)$ ; in fact, we can equivalently define $T (X)$ as the set of continuous functions from $P (X) \times X$ to $R$ ) for some $x \in ⋂_{n} X_{n}$ . We denote the set of trading stra

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Logical Induction