I'm probably explaining it poorly in the post. P0 is not just a function of statements in F. P0 is a probability measure on the space of truth assignments i.e. functions {statement in F} -> {truth, false}. This probability measure is defined by making the truth value of each statement an independent random variable with 50/50 distribution.

PD is produced from P0 by imposing the condition "there is no contradiction of length <= D" on the truth assignment, i.e. we set the probability of all truth assignments that violate the condition to 0 and renormalize the probabilities of all other assignments. In other words P_D(s) = # {D-consistent truth assignments in which s is assigned true} / # {D-consistent truth assignments}.

Technicality: There is an infinite number of statements so there is an infinite number of truth assignments. However there is only a finite number of statements that can figure in contradictions of length <= D. Therefore all the other statements can be ignored (i.e. assumed to have independent probabilities of 1/2 like in P_0). More formally, the sigma-algebra of measurable sets on the space of truth assignments is generated by sets of the form {truth assignment T | T(s) = true} and {truth assignment T | T(s) = false}. The set of D-consistent truth assignments is in this sigma algebra and has positive probability w.r.t. our measure (as long as F is D-consistent) so we can use this set to form a conditional probability measure.