  In this paper, we consider the problem of learning a first-order theorem
prover that uses a representation of beliefs in mathematical claims to
construct proofs. The inspiration for doing so comes from the practices of
human mathematicians where "plausible reasoning" is applied in addition to
deductive reasoning to find proofs.
  Towards this end, we introduce a representation of beliefs that assigns
probabilities to the exhaustive and mutually exclusive first-order
possibilities found in Hintikka's theory of distributive normal forms. The
representation supports Bayesian update, induces a distribution on statements
that does not enforce that logically equivalent statements are assigned the
same probability, and suggests an embedding of statements into an associated
Hilbert space.
  We then examine conjecturing as model selection and an alternating-turn game
of determining consistency. The game is amenable (in principle) to self-play
training to learn beliefs and derive a prover that is complete when logical
omniscience is attained and sound when beliefs are reasonable. The
representation has super-exponential space requirements as a function of
quantifier depth so the ideas in this paper should be taken as theoretical. We
will comment on how abstractions can be used to control the space requirements
at the cost of completeness.
