Betting on what is neither verifiable nor falsifiable
This addresses a theoretical gap in prediction markets for researchers in mathematical logic, philosophy, and AI, offering an incremental extension to handle non-verifiable or non-falsifiable events.
The paper tackles the problem of applying prediction markets to questions without fixed resolution criteria, such as logical or algorithmic uncertainty, by proposing an approach using options or a verification-falsification game. This provides an alternative to existing frameworks like Garrabrant induction and relates to constructivism in philosophy and mathematics.
Prediction markets are useful for estimating probabilities of claims whose truth will be revealed at some fixed time -- this includes questions about the values of real-world events (i.e. statistical uncertainty), and questions about the values of primitive recursive functions (i.e. logical or algorithmic uncertainty). However, they cannot be directly applied to questions without a fixed resolution criterion, and real-world applications of prediction markets to such questions often amount to predicting not whether a sentence is true, but whether it will be proven. Such questions could be represented by countable unions or intersections of more basic events, or as First-Order-Logic sentences on the Arithmetical Hierarchy (or even beyond FOL, as hyperarithmetical sentences). In this paper, we propose an approach to betting on such events via options, or equivalently as bets on the outcome of a "verification-falsification game". Our work thus acts as an alternative to the existing framework of Garrabrant induction for logical uncertainty, and relates to the stance known as constructivism in the philosophy of mathematics; furthermore it has broader implications for philosophy and mathematical logic.