A logic for propositions with indefinite truth values

In the first part of this paper a logic is defined for propositions whose probability of being true may not be known. A speaker's beliefs about which propositions are true are still interesting in this case. The meaning of propositions is determined by the consequences of asserting them: in this logic there are debates which incur certain costs for the protagonists.The second part of the paper describes the mathematics of the resulting logic which displays several novel features.