 Axiom 2: Suppose that an agent’s betting quotient for a tautology (or a logical or necessary truth) \(H\) is not equal to 1. If \(Q(H) \gt 1\) the bookie sells the bet that pays $1 if \(H\) is true and 0 otherwise, for \(Q(H)\). In the case where \(Q(H) \lt 1\), the bookie will buy the bet in which the agent pays the bookie $1 if \(H\) is true and nothing if \(H\) is false, for \(Q(H)\). In this case, the payoff table for the agent will again be as above for axiom 1. Notice that since \(H\) is a tautology (or logical or necessary truth) it must be true, which means that at the conclusion of the bet, the agent will have lost [\(1-Q(H)\)].