This chapter argues that Aristotle’s syllogistic emerged from a dialectical matrix as well as from considerations pertaining to scientific demonstration and demonstration in mathematics. This means that, even early on, non-dialogical components motivated and were integrated into theories and practices of deduction. The chapter also briefly discusses two other formidable ancient intellectual traditions and their reflections on logic and reasoning, namely the Indian tradition and the Chinese tradition. It is argued that, while these were indeed highly sophisticated, fully-fledged theories of deduction are not to be found in classical Indian or classical Chinese thought.

This chapter defines and introduces the explanandum of the book, i.e. the phenomenon (or phenomena) that it is about: deductive reasoning and argumentation. It presents deduction as having three main characteristics: necessary truth-preservation – which is perhaps the most central one, distinguishing deduction from other forms of inference and argument such as induction and abduction – perspicuity, and belief-bracketing. It also discusses a number of puzzling features of deduction, i.e. philosophical issues pertaining to deduction that remain open questions, as they have not yet been adequately ‘solved.’ These are: the range and scope of deductive reasoning and argumentation, the nature of deductive necessity, and the function(s) of deduction.

This chapter critically discusses the prominent dialogical accounts of logic and deduction proposed by Lorenzen, Hintikka, and Lakatos. It is argued that, while they contain valuable insights, Lorenzen’s dialogical logic and Hintikka’s game-theoretical semantics ultimately both fail to provide a satisfactory philosophical account of logic and deduction in dialogical terms. This critical evaluation then leads to a precise formulation of the dialogical model defended in the book, the Prover–Skeptic model, which is by and large inspired by Lakatos’ ‘proofs and refutations’ model, but with some important modifications.

This chapter retraces the genealogical development of deduction in the Latin and Arabic medieval traditions and in the early modern period, and finally the emergence of mathematical logic in the nineteenth century. It is shown that dialogical conceptions of logic remained pervasive in the Latin medieval tradition, but that they coexisted with other, non-dialogical conceptualizations, in part because of the influence of Arabic logic. In the modern period, however, mentalistic conceptions of logic and deduction became increasingly prominent. The chapter thus explains why we (i.e. twenty-first-century philosophers) have by and large forgotten the dialogical roots of deduction.

This chapter returns to the three main features of deduction defined in Chapter 1 from a cognitive, empirically informed perspective: necessary truth-preservation, perspicuity, and belief-bracketing. It discusses experimental findings that lend support to the dialogical conceptualization of these three features presented in Chapter 4. It also discusses the notion of internalization as formulated by Lev Vygotsky, which allows for an explanation of how deductive practices can also take place in purely mono-agent situations: as an intrapersonal enactment of interpersonal dialogues. The upshot is that framing deductive practices dialogically provides cognitive scaffolding that facilitates the ontogenetic development of deductive reasoning in an individual.

In this chapter, it is argued that what is needed to make progress on the issues described in Chapter 1 is a ‘roots’ approach, i.e. going back to the roots of deduction. The distinction between phylogenetic, ontogenetic, and historical roots is introduced, and it is argued that all three perspectives must be taken into account. The chapter further briefly presents the four main senses in which deduction has dialogical roots treated in this book: philosophical roots, historical roots, cognitive roots, and with respect to mathematical practices.

This chapter presents an overview of experimental work on deductive reasoning, which has shown that human reasoners do not seem to reason spontaneously according to the deduction canons. However, there are also experimental results suggesting that, when tackling deductive tasks in groups, performance comes much closer to the canons. These findings offer a partial vindication of the dialogical conception of deduction insofar as they show that, when given the opportunity to engage in dialogues with others, humans become better deductive reasoners.

This chapter presents a dialogical rationale based on the Prover–Skeptic model for the three main features of deduction identified in Chapter 1: necessary truth-preservation, perspicuity, and belief-bracketing. Moreover, it addresses four important ongoing debates in the philosophy of logic: the normativity of logic, logical pluralism, logical paradoxes, and logical consequence. It is shown that the Prover–Skeptic model provides a promising vantage point to address the questions raised in these debates.

This chapter is a tutorial about some of the key issues in semantics of the first-order aspects of probabilistic programming languages for statistical modelling – languages such as Church, Anglican, Venture and WebPPL. We argue that s-finite measures and s-finite kernels provide a good semantic basis.

Reasoning about probabilistic programs is hard because it compounds the difficulty of classic program analysis with sometimes subtle questions of probability theory. Having precise mathematical models, or semantics, describing their behaviour is therefore particularly important. In this chapter, we review two probabilistic semantics. First, an operational semantics which models the local, step-by-step, behaviour of programs, then a denotational semantics describing global behaviour as an operator transforming probability distributions over memory states.