Reasoning about chance:Proof theory for aleatoric logic

Aleatoric Logic is a logic of chance, where the interpretation of propositional atoms is modelled as the flip of a coin, or a marble drawn from an urn. Propositions correspond to these chance events and consequently their interpretation is inherently probabilistic. Propositions are not evaluated as true or false, but instead are evaluated as likelihoods, giving a many-valued logic similar to fuzzy logics. An equational theory is given through an equivalence relation that associates two propositions with identical likelihood: for example the chance of a coin landing heads and then tails, will be the same as the chance of the coin landing tails and then heads, regardless of the coin s bias. In this paper we present a refined syntax for aleatoric logic and provide a proof system for the equational theory.