  Jakob Bernoulli, working in the late 17th century, identified a gap in
contemporary probability theory. He cautioned that it was inadequate to specify
force of proof (probability of provability) for some kinds of uncertain
arguments. After 300 years, this gap remains in present-day probability theory.
We present axioms analogous to Kolmogorov's axioms for probability, specifying
uncertainty that lies in an argument's inference/implication itself rather than
in its premise and conclusion. The axioms focus on arguments spanning two
Boolean algebras, but generalize the obligatory: "force of proof of A implies B
is the probability of B or not A" in the case that the Boolean algebras are
identical. We propose a categorical framework that relies on generalized
probabilities (objects) to express uncertainty in premises, to mix with
arguments (morphisms) to express uncertainty embedded directly in
inference/implication. There is a direct application to Shafer's evidence
theory (Dempster-Shafer theory), greatly expanding its scope for applications.
Therefore, we can offer this framework not only as an optimal solution to a
difficult historical puzzle, but also to advance the frontiers of contemporary
artificial intelligence.
  Keywords: force of proof, probability of provability, Ars Conjectandi, non
additive probabilities, evidence theory.
