This chapter provides examples of the central concepts of the book: initial algebras and terminal coalgebras. These are mainly for polynomial endofunctors on the category of sets, but also for such functors on sorted sets, nominal sets, and presheaves. We discuss connections to induction and recursion and to their duals, coinduction and corecursion, and also to bisimulation. We present Lambek’s Lemma, a result used throughout the book.

We motivate the book based on categorical formulations of recursion and induction. We also discuss the background that readers should have and preview many of the topics in the book.

A more expressive axiomatic theory of syntax is presented. It is shown that this theory generalises the theory of chapter 5 and allows the derivation of many natural properties of syntax.

Most intelligence models include an array of processes and mechanisms that enable experts to generalize their knowledge and processes during career transitions and produce flexibility in cognitive structures that enable individuals to overcome limitations in applying expert knowledge and processes across domains and functional areas. These processes have been described variously as insightful thinking, induction, eduction, elaborating and mapping, novelty and metaphorical capacity, inductive inference, divergent production abilities, analogy, flexibility of use, and closure. They are discussed in light of the retrospective interviews with twenty-four elite performers in three domains (business, sports, and music) who successfully and repeatedly transitioned to higher positions within their field.

In ‘Aristotle on the Stages of Cognitive Development’, Thomas Kjeller Johansen examines Aristotle’s contributions to our thinking about concepts from a different perspective, namely in connection to Aristotle’s psychology. He revisits Aristotle’s account of how we acquire universal concepts mainly on the basis of Metaphysics A.1, Posterior Analytics 1.31 and 2.19, and the De Anima. The chapter begins by articulating the following puzzle. On the one hand, Aristotle points out (An. Post. 1.31, 2.19) that we perceive the universal in the particular. On the other, he suggests (Metaph. A.1) that it is only when we have craft and science that we grasp the universal, while perception, memory, and experience all are concerned with the particular. Building on the widespread view that, according to Aristotle, the universal grasped in craft and science is the universal cause, Johansen argues that we should understand perception, memory, and experience teleologically, as stages in the ordering of perceptual information that allows this causal concept to emerge.

This chapter begins by differentiating qualitative and quantitative research. While some have argued that these approaches are incommensurable paradigms, this chapter argues that they are commensurable but suited to answering different research questions. It introduces a typology of research questions, with six types of question – three qualitative (describing phenomena, theoretical framing, and generating explanations) and three quantitative (measuring phenomena, testing theory, and exploring explanations). The chapter ends by reviewing heuristics to help researchers generate novel and productive research questions.

Equality and equal treatment are the principal purpose of WTO law. However, that purpose is accomplished in varying conditions, which make it difficult to regularly attain the consistency and coherence that an egalitarian and obligatory conception of the law assumes. Consequently, this chapter proceeds to demonstrate how WTO law is focused secondarily on fairness and corrective justice and how this focus begets a subordinate emphasis in law on rights, which in turn gives rise to a contractual structure that is retrospectively oriented and reasoned inductively. It also demonstrates how various features of WTO law like the non-violation cause of action and implementation are consistent with such a rights-based ethos.

This chapter details the practical, theoretical, and philosophical aspects of experimental science. It discusses how one chooses a project, performs experiments, interprets the resulting data, makes inferences, and develops and tests theories. It then asks the question, "are our theories accurate representations of the natural world, that is, do they reflect reality?" Surprisingly, this is not an easy question to answer. Scientists assume so, but are they warranted in this assumption? Realists say "yes," but anti-realists argue that realism is simply a mental representation of the world as we perceive it, that is, metaphysical in nature. Regardless of one's sense of reality, the fact remains that science has been and continues to be of tremendous practical value. It would have to be a miracle if our knowledge and manipulation of the nature were not real. Even if they were, how do we know they are true in an absolute sense, not just relative to our own experience? This is a thorny philosophical question, the answer to which depends on the context in which it is asked. The take-home message for the practicing scientist is "never assume your results are true."

The digitalization of business organizations and of society in general has opened up the possibility of researching behaviours using large volumes of digital traces and electronic texts that capture behaviours and attitudes in a broad range of natural settings. How is the availability of such data changing the nature of qualitative, specifically interpretive, research and are computational approaches becoming the essence of such research? This chapter briefly examines this issue by considering the potential impacts of digital data on key themes associated with research, those of induction, deduction and meaning. It highlights some of the ‘nascent myths’ associated with the digitalization of qualitative research. The chapter concludes that while the changes in the nature of data present exciting opportunities for qualitative, interpretive researchers to engage with computational approaches in the form of mixed-methods studies, it is not believed they will become the sine qua non of qualitative information systems research in the foreseeable future.

Inductive reasoning involves generalizing from samples of evidence to novel cases. Previous work in this field has focused on how sample contents guide the inductive process. This chapter reviews a more recent and complementary line of research that emphasizes the role of the sampling process in induction. In line with a Bayesian model of induction, beliefs about how a sample was generated are shown to have a profound effect on the inferences that people draw. This is first illustrated in research on beliefs about sampling intentions: was the sample generated to illustrate a concept or was it generated randomly? A related body of work examines the effects of sampling frames: beliefs about selection mechanisms that cause some instances to appear in a sample and others to be excluded. The chapter describes key empirical findings from these research programs and highlights emerging issues such as the effect of timing of information about sample generation (i.e., whether it comes before or after the observed sample) and individual differences in inductive reasoning. The concluding section examines how this work can be extended to more complex reasoning problems where observed data are subject to selection biases.

Chapter 12 criticizes Karl Poppers and Imre Lakatos’ views on theory appraisal, which have been particularly influential among writers on economic methodology, although their influence has waned. Popperian critics of economics are right to claim that economists seldom practice the falsificationism that many preach, but, in contrast to authors such as Mark Blaug (), I argue that the problem is with the preaching, not with the practice: falsificationism is not a feasible methodology. Although Lakatos provides more resources with which to defend economics than does Popper, his views are also inadequate and for a similar reason. Both Popper and Lakatos deny that there is ever reason to believe that scientific statements are close to the truth or likely to be true, and neither provides a viable construal of tendencies. In denying that such reasons to accept generalizations have a role in either engineering or in theoretical science, Popper and Lakatos are implicitly calling for a radical and destructive transformation of human practices.

Chapter 10 considers what conditions must be met if one is to have good reason to accept tendency claims or inexact laws, and it presents an interpretation of J. S. Mills views on confirmation, which still appear to dominate methodological practice in economics. It begins in §10.1 by discussing well-known Bayesian, hypothetico-deductive, and likelihood approaches to confirmation, before focusing on an indirect inductive method, which Mill calls “the method a priori” or “the deductive method.” §10.2 lays out the broad outlines of Mill’s deductive method. §10.3 expands upon Mill’s method a posteriori – his direct inductive method – to address the question of how economists can know whether their fundamental generalizations express inexact laws or tendencies. §10.4 examines in detail what Mill has to say about his deductive method, while §10.5 lays out the implicit algorithm that Mill offers for testing the implications of theoretical hypotheses in an inexact and separate science.