In 1988, the US Nuclear Regulatory Commission approved an amendment that allowed the use of best-estimate methods. This led to an increased development, and application of Best Estimate Plus Uncertainty (BEPU) safety analyses. However, a greater burden was placed on the licensee to justify all uncertainty estimates. A review of the current state of the BEPU methods indicate that there exists a number of significant criticisms, which limits the BEPU methods from reaching its full potential as a comprehensive licensing basis. The most significant criticism relates to the lack of a formal framework for distinguishing between aleatory and epistemic uncertainties. This has led to a prevalent belief that such separation of uncertainties is for convenience, rather than one out of necessity. In this thesis, we address the above concerns by developing a statistically rigorous framework to characterize the different uncertainty types. This framework is grounded on the philosophical concepts of knowledge. Considering the Plato problem, we explore the use of probability as a means to gain knowledge, which allows us to relate the inherent distinctness in knowledge with the different uncertaintytypesforanycomplexphysicalsystem. Thisframeworkis demonstrated using nuclear analysis problems, and we show through the use of structural models that the separation of these uncertainties leads to more accurate tolerance limits relative to existing BEPU methods. In existing BEPU methods, where such a distinction is not applied, the total uncertainty is essentially treated as the aleatory uncertainty. Thus, the resulting estimated percentile is much larger than the actual (true) percentile of the system's response. Our results support the premise that the separation of these two distinct uncertainty types is necessary and leads to more accurate estimates of the reactor safety margins. / Thesis / Doctor of Philosophy (PhD)