Modeling Uncertain Self-Referential Semantics with Infinite-Order Probabilities

The notion of a “second-order probability” (a probability distribution defined over a space of probability distributions), while not exactly commonplace, has become less obscure in recent decades due to the advent of imprecise probability theory as developed by Peter Walley [1] and others [2]. It is straightforward to generalize the notion further to thirdorder probabilities and so forth; and the reason these notions are rarely discussed is not so much that there are mathematical difficulties, but rather that no one has seen much use for them. However, so far as we have been able to tell, the notion of infinite-order probabilities has not been previously discussed. Intuitively, the notion of an infinite-order probability is simply “a probability distribution of probability distributions of probability distributions of ... of probability distributions.” The goal of this paper is to briefly introduce these novel mathematical objects; and then to explain their apparent importance in the domain of semantics, via their ability to synthesize uncertain semantics with self-referential semantics.