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This book consists in very large part of an exposition of a mathematical theory — the theory (or a theory) of sets. This exposition has at its core a sequence of proofs designed to establish theorems. One central element in the exposition is explicit definitions to explain our use of various words and symbols. It is a requirement of such a definition that it should be formally eliminable, so that every occurrence of the word defined could, in principle, be replaced by the phrase that defines it without affecting the correctness of the proof. This chapter discusses the axiomatic method, the use of various canons of logical reasoning in deducing the consequences of axioms, axiom schemes, and the distinction between first- and second-order logic.

Chapter I begins by describing the salient parts of a formal science of economics that the author has put together in this book and in earlier works (B.P. Stigum 1990 and 2003). The parts comprise a unitary methodological basis for a science, an explication of the meaning of facts and fiction in econometrics, and a confrontation of the methods of present-day applied econometrics with the methods that a formal science advocates. Next the chapter discusses interesting ways in which the axiomatic method and the model-theoretic method of developing theories are used in mathematical economics and statistics, and describes the underlying ideas of a formal theory-data confrontation. Thereafter follows a discussion of three controversial aspects of formal econometrics: (1) the essence of an economic theory; (2) the double role of theory in applied econometrics and the need for bridge principles; and (3) why theory is required both for the design of an empirical analysis and for the interpretation of its results. The chapter ends with a brief description of the contents of the remaining nine chapters.

This chapter introduces the notation and basic concepts that are used throughout the book. The chapter has five sections. First it reviews unidimensional poverty measurement with particular attention to the well-known Foster-Greer-Thorbecke measures of income poverty, as many methods presented in Chapter 3, as well as the measure presented in Chapters 5-9, are based on these measures. The second section introduces the notation and basic concepts for multidimensional poverty measurement that are used in subsequent chapters. Third, it defines indicators’ scales of measurement, and fourth, it addresses issues of comparability across people and dimensions. The fifth section systematically explains different properties that have been proposed in axiomatic approaches to multidimensional poverty measurement, which enable the analyst to understand the ethical principles embodied in a measure and to be aware of the direction of change they will exhibit under certain transformations.

In this chapter, the Bolzanian system of inferences, i.e. of propositions asserting the subsistence of the relation of derivability among other propositions, is developed in detail. A particularly interesting and noteworthy trait of the method adopted by Bolzano in this treatment is that this method is not an axiomatic one, but proceeds as the production of a catalogue, ordered according to the old principle: from the simpler to the more complex.

The question “When is C a cause of E?” is well-studied in philosophy—much more than the equally important issue of quantifying the causal strength between C and E. In this chapter, we transfer methods from Bayesian Confirmation Theory to the problem of explicating causal strength. We develop axiomatic foundations for a probabilistic theory of causal strength as difference-making and proceed in three steps: First, we motivate causal Bayesian networks as an adequate framework for defining and comparing measures of causal strength. Second, we demonstrate how specific causal strength measures can be derived from a set of plausible adequacy conditions (method of representation theorems). Third, we use these results to argue for a specific measure of causal strength: the difference that interventions on the cause make for the probability of the effect. An application to outcome measures in medicine and discussion of possible objections concludes the chapter.

Confirmation of scientific theories by empirical evidence is an important element of scientific reasoning and a central topic in philosophy of science. Bayesian Confirmation Theory—the analysis of confirmation in terms of degree of belief—is the most popular model of inductive reasoning. It comes in two varieties: confirmation as firmness (of belief), and confirmation as increase in firmness. We show why increase in firmness is a particularly fruitful explication of degree of confirmation, and how it resolves longstanding paradoxes of inductive inference (e.g., the paradox of the ravens, the tacking paradoxes and the grue paradox). Finally, we give an axiomatic characterization of various confirmation measures and we discuss the question of whether there is a single adequate measure of confirmation or whether a pluralist position is more promising