 And it can further be shown that any function \(P_{\alpha}\) that expresses such betting-related belief-strengths on all statements in agent \(\alpha\)’s language must satisfy axioms for unconditional probabilities analogous to axioms  1–5.[4]  Moreover, it can be shown that any function \(P_{\beta}\) that satisfies these axioms is a possible rational belief function for some ideally rational agent \(\beta\). These relationships between belief-strengths and the desirability of outcomes (e.g., gaining money or goods on bets) are at the core of subjectivist Bayesian decision theory. Subjectivist Bayesians usually take inductive probability to just be this notion of probabilistic belief-strength.