Arbitrage Opportunities Correspond to Probability Inequality Identities

We consider a fixed-odds gambling market over arbitrary logical propositions, where participants trade bets involving conjunctions, disjunctions, and negations. In this setting, we establish a three-way correspondence between the financial feasibility of trading strategies, the validity of universal probability inequalities, and the solutions to bounded Maximum Satisfiability (MaxSAT) problems. The central result demonstrates that proving a trading strategy constitutes an arbitrage opportunity (i.e., guaranteeing a risk-free profit regardless of the outcome) is equivalent to proving a specific inequality identity holds for all probability functions, and is computationally equivalent to establishing a lower bound on a corresponding MaxSAT instance. Dually, we show that checking the coherence of a strategy (i.e., ensuring it does not guarantee a loss) also corresponds to verifying a probability identity and bounding a MaxSAT problem from above.