The probabilistic satisfiability of a logical expression is a fundamental concept known as the partition function in statistical physics and field theory, an evaluation of a related graph's Tutte polynomial in mathematics, and the Moore-Shannon network reliability of that graph in engineering. 
It is the crucial element for decision-making under uncertainty.
Not surprisingly, it is provably hard to compute exactly or even to approximate.
Many of these applications are concerned only with a subset of problems for which the solutions are monotonic functions.