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The prior four chapters focused on developing tools for constructing and morally ranking an outcome set: tools that include an account of well-being, principles for measuring well-being using a set U of utility functions that operate on life-histories, and a continuous prioritarian SWF. This chapter turns to the topic of moral bridge principles. It argues that a continuous prioritarian SWF should be merged with a (refined version of) expected utility theory so as to generate a ranking of choices—notwithstanding violations of the ex ante Pareto and Pigou–Dalton principles. While the SWF framework defended here satisfies the Pareto and Pigou–Dalton principles in terms of the ranking of outcomes, the ex ante versions of these principles constitute an additional requirement which, on balance, should be rejected. The dilemmas that arise in specifying norms of fair distribution under conditions of uncertainty have been discussed by philosophers and social choice theorists.

This chapter begins an account of the normative sources of evaluative‐practical reasoning, and its distinctive concepts. One such source is a principle that can be called the ‘Bridge’ principle, since it is the bridge by which thought crosses from evaluative reasons to practical reasons. The concept of a person's good and the concept of personal ideals both belong within this sentimentalist territory.

A division is drawn between empirical sentences, pure mathematical sentences, and mixed sentences, including bridge principles. Mixed sentences are made-true relative to a calculus when derivable from that calculus plus empirical truths. The metaphysical content of mixed sentences in general has two sources, empirical truths and pure mathematics. Conservative extension results are needed if an applied calculus is to be of any use; such results are pieces of applied mathematics themselves. The regress which ensues is the usual one in semantics (which is not to say it is not objectionable). Two general accounts of theoretical language are contrasted: a Quinean holistic empiricism and a ‘molecular’ position compatible with neo-formalism. The latter insists on an independent grasp of mathematics as part of an explanation of how we can enlarge our conceptual reach beyond the empirical. Quinean platonism and Fieldian fictionalism, in which mathematics is not independently grasped, cannot, it is argued, explain this enlargement; both lead to an instrumentalistic, relativistic, anti-realism.

This chapter is about formal models, norms, and their relations. It first discusses bridge principles, which relate a formal model to the domain it is meant to model. For example, bridge principles relate the probability axioms in a Bayesian model to rational requirements on degrees of belief. Next, the chapter lays out the elements needed to engage in formal modeling: the modeling framework itself, bridge principles, and modeling rules to tell you when and how a framework is used. These elements are illustrated by considering deduction systems in logic as formal modeling frameworks. Finally, the chapter discusses the modeling methodology of this book and its advantages, especially in how counterexamples are treated and how a modeling outlook encourages one to be explicit about the choices made in building a model.

The chapter begins with a discussion of the status of bridge principles in empirical analyses in which the data generating process does not appear. Then it contrasts the ideas of encompassing and congruence in present-day econometrics and formal econometrics, and explicates how the notions of encompassing and congruence can be used to determine the status of bridge principles in formal econometrics. The chapter concludes with a discussion of present-day and formal econometric prescriptions for the use of theory in the book’s case studies, and with a commentary on three problematic aspects of present-day econometric methodology. The results of a case study with the methods of present-day econometrics differ from the results which the same case study yields with the methods of formal econometrics. The chapter attributes the differences to the respective researchers’ different understanding of theory, and describes how a researcher’s understanding of theory affects the inference about social reality that he can draw from statistical analyses. The three methodological problems concern analysis of positively valued time series, empirical irrelevance of qualitative response models, and the search for theoretically meaningful cointegrating relations. The discussion suggests solutions to problems and good reasons for concern about a misguided search for cointegrating relations.

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 reviews and evaluates the modeling methodology that has been applied over the course of the book. It examines how the book’s modeling approach was crucial to constructing and defending the Certainty-Loss Framework (CLF), a modeling framework capable of representing rational requirements on agents in stories involving memory loss and context-sensitivity. A modeling methodology also has general advantages: It forces us to be explicit about the choices that go into building a model; it focuses our attention on the principles bridging a model with the modeled world; and it allows us to narrow the domain of potential counterexamples to our view. In the best case, a modeling project can also generate new insights and lead us to a deeper understanding of the phenomena being modeled.

This chapter focuses on what exactly models in the Certainty-Loss Framework (CLF) say about agents’ degrees of belief. Developing the standard interpretation of CLF given in the previous chapter, it lays down explicit bridge principles relating CLF’s formal models to norms. Under this intepretation, CLF models indicate necessary but not sufficient evaluative requirements for rational consistency. The chapter considers objections to these evaluative requirements and exactly how idealized the notion of rationality involved needs to be. Finally, the chapter provides a model theory for CLF’s derivation system and considers alternative interpretations for the framework, such as interpretations on which CLF models represent imprecise probabilities (also known as “ranged” degrees of belief).

The book's introduction postulates that the time is ripe to reflect on the critical importance of bridges today. Some bridges are political and historical, others moral, aesthetic, and psychological, their variety informing the topics of the chapters to follow. They include bridges both connecting and disconnecting our globalized world, sea-bridges of migration, bridges joining death to life and self to community. Bridges of language, war, and music are also issues to follow. Some bridges resemble the work of angels, others would seem to link Earth to Hell. The overarching question of this introduction is just how to read these bridge-forms, these figures and symbols of human life. A key point of departure is the classic essay of Georg Simmel, “Bridge and Door.” A summary of this leads to a distillation of principles attached to this archetypal image of the bridge, including reciprocity and two-way direction, transcendence, commitment to destination, and third spaces of union, some intimated briefly by sayings of Friedrich Nietzsche, Henry Thoreau, and György Lukács.

This chapter is motivated by the following methodological question: under what circumstances should we call two structures the same? We start by surveying several formal notions of sameness of structure: definitional equivalence, interpretability, mutual interpretability, and biinterpretability. We then discuss the notions of sameness of theory which arise by considering the class of models of a theory, and some new notions of sameness of theory (such as faithful interpretability). We close the chapter by examining three ambitious claims concerning sameness of theory: that it preserves truth-value, that it preserves arithmetical consequence, and that it delivers two proofs for the price of one. We argue that the more philosophically ambitious versions of these preservation-theses fail, in general.