CHA’s focus on exploration, description, and theorizing does not mean that questions of causal inference are ignored. CHA values exploration just as much as testing, on the grounds that only an equal weighing of them will translate results into actual answers. Its approach to methodology is more heterodox than that of methodologies that concentrate narrowly on questions of causal inference. This heterodoxy reflects an older, traditional understanding of methodology as a series of research cycles. Each cycle has an exploratory stage that involves exploration, description, conceptualization, and theorizing and a confirmatory stage that involves data collection, data analysis, and replications. In addition to having those stages, research cycles are iterative and constantly update our knowledge. CHA employs those elements of the research cycle in nonlinear fashion and thus engages in a kind methodological bricolage or what is sometimes called abduction. The goal of bricolage is to align the ontological presuppositions of methods with the ontological attributes of questions. Methods thus differ in their appropriateness rather than in their sophistication.

Credibility models were first proposed in the beginning of the twentieth century to update predictions of insurance losses in light of recently available data of insurance claims. The oldest approach is the limited-fluctuation credibility method, also called the classical approach, which proposes to update the loss prediction as a weighted average of the prediction based purely on the recent data and the rate in the insurance manual. Full credibility is achieved if the amount of recent data is sufficient, in which case the updated prediction will be based on the recent data only. If, however, the amount of recent data is insufficient, only partial credibility is attributed to the data, and the updated prediction depends on the manual rate as well.

Chapter 14: Every square complex matrix is unitarily similar to an upper triangular matrix, but which matrices are unitarily similar to a diagonal matrix? The answer is the main result of this chapter: the spectral theorem for normal matrices. Hermitian, skew-Hermitian, unitary, and circulant matrices are unitarily diagonalizable. As a consequence, they have special properties that we investigate in this and following chapters.

Few models in theoretical physics have been studied for as long, or in as much detail, as the Ising model. It’s the simplest model to display a nontrivial phase transition, and as such it plays a unique role in theoretical physics. In addition, the Ising model can be applied to a wide range of physical systems, from magnets and binary liquid mixtures, to adsorbed monolayers and superfluids, to name just a few. In this chapter, we present some of the background material that sets the stage for a detailed study of the Ising model in the chapters to come.

The conclusion revisits the complementarities between CHA and other methodologies. It emphasizes that exploration and testing are equally important for translating results into answers. It discusses the historical origins of the logical positivists’ distinction between domains of discovery and confirmation, in order to underscore its artificiality and misrepresentation of the actual scientific process. It highlights four contributions that CHA makes to social inquiry more broadly: the centrality of historical thinking across all methods, the beneifts of greater ontological transparency, the importance of paying attention to testworthiness of hypotheses, and the inescapability of methodological heterodoxy.

This chapter covers three areas of international law that, though distinct, are interconnected. The law of statehood defines which entities qualify as states, the most important actors on the world stage and the international legal persons with the most rights, duties and functions. The law of self-determination accords rights to ‘peoples’ rather than to states, but in limited circumstances it enables peoples to create new states for themselves. That can be a way for a people to seize control of its own destiny, especially if it has been subject to colonialism or oppression. Finally, the law of territory determines the geographical borders within which many of the rights, duties and functions of states are applicable, including how those borders can change.