This dissertation presents some formal results in Bayesian confirmation theory with novel applications to topics in contemporary epistemology, political science, and legal theory. The primary theme in each chapter is that confirmation, not probability, is the central formal notion one should adopt in approaching various topics in these three disciplines. In each chapter, I argue for this by highlighting some wide-ranging and important consequences that result from a confirmation-theoretic approach. Chapter 1 provides an introduction to the machinery of Bayesian confirmation theory and, with a case study, demonstrates the significance of moving away from probability to confirmation. Chapter 2 addresses the question of whether or not an epistemic closure principle is correct. Most contemporary epistemologists think that epistemic closure is obviously correct, in need of little defense or argument. I show that by shifting our focus from probability to confirmation, and exploiting the different formal properties exhibited by probability functions and confirmation functions, we are able to develop a novel case against closure. Here I also prove some new formal results that bear on the debate over the adequacy of Bayesian measures of confirmation. Chapter 3 discusses a particular form of voting--supermajority voting--from a Bayesian perspective. The standard route taken to motivate supermajority voting is via the Condorcet framework and its well-known "jury theorem". I show that a Bayesian confirmation-theoretic approach provides a much more general and powerful approach to supermajority voting. The basic idea is that supermajority voting provides superior evidence to simple majority voting. From an epistemological point of view, I argue this makes supermajority voting preferable to simple majority voting. This chapter also presents some new mathematical results that improve upon the contemporary Bayesian literature on evidential support from multiple pieces of evidence. Chapter 4 is an essay in Bayesian jurisprudence and legal epistemology. In Anglo-American criminal jurisprudence, one important and frequently used standard of criminal proof is proof beyond reasonable doubt. Recent work on defining the notion of proof beyond reasonable doubt, however, is rather vague and often confusing. In this chapter, I propose a new and precise account proof beyond reasonable doubt. My approach will make central use of Bayesian confirmation theory, and the guiding idea will be that proof beyond reasonable doubt is established when a certain level of confirmatory support is reached. I will also discuss the "presumption of innocence" doctrine, since it plays an important role in my account of proof beyond reasonable doubt. My discussion will make some important philosophical and mathematical advances on our understanding of both these doctrines.