The most important problems for society are describable only in vague terms, dependent on subjective positions, and missing highly relevant data. This thesis is intended to revive and further develop the view that giving non-trivial, rigorous deductive arguments concerning such problems -without eliminating the complications of vagueness, subjectivity, and uncertainty- is, though very difficult, not problematic in principle, does not require the invention of new logics (classical first-order logic will do), and is something that more mathematically-inclined people should be pursuing. The framework of interpreted formal proofs is presented for formalizing and criticizing rigorous deductive arguments about vague, subjective, and uncertain issues, and its adequacy is supported largely by a number of major examples. This thesis also documents progress towards a web system for collaboratively authoring and criticizing such arguments, which is the ultimate goal of this project. ; Ph.D.