This chapter is intended to introduce some basic notions of quantum theory needed in the subsequent chapters for the reader who is not familiar with them. Quantum mechanics is a fundamental physics subject that studies the phenomena at the atomic and subatomic scales. This chapter introduces the required mathematical tools and presents the postulates mainly through their mathematical formalisms. The physics interpretation of these is only very briefly discussed.

This chapter is devoted to studying a class of even more complex quantum systems modelled as so-called super-operator-valued Markov chains (SVMCs). This new model is particularly useful in modelling the high-level structure of quantum programs and quantum communication protocols. Several algorithms for checking SVMCs are presented in this chapter.

This is the concluding chapter of the book. It briefly discusses several possible directions for the further development, including the problem of state space explosion in model checking quantum systems, possible applications in verification and analysis of quantum circuits, quantum cryptographic protocols, and more generally, quantum programs.

This chapter investigates deductive practices in what is arguably their main current instantiation, namely practices of mathematical proofs. The dialogical hypothesis delivers a compelling account of a number of features of these practices; indeed, the fictive characters Prover and Skeptic can be viewed as embodied by real-life mathematicians. The chapter includes a discussion of the ontological status of proofs, the functions of proofs, practices of mathematicians such as peer review and collaboration, and a brief discussion of probabilistic and computational proofs. It also discusses three case studies: the reception of Gödel’s incompleteness results, a failed proof of the inconsistency of Peano Arithmetic, and a purported proof of the ABC conjecture.

Throughout this book, deduction has been examined and discussed from many angles and perspectives. However, one question has remained conspicuously unaddressed until now: Is deduction a correct, reliable method for reasoning? In other words, is deduction justified (Dummett, 1978)?

This investigation has focused extensively on the social conditions and factors influencing the emergence of deduction, both historically and ontogenetically. It is thus reasonable to ask whether it offers a social constructivist account of deduction, which in turn has implications for the justification problem. Indeed, on at least some versions of social constructivism, the very question of the correctness of deductive reasoning as a scientific method, understood in absolute terms, is seen as misguided.

This chapter examines the historical roots of deduction in Ancient Greek philosophy and mathematics. It relies extensively on the work of G.E.R. Lloyd and Reviel Netz to argue that dialogical debating practices in a democratic city-state like Athens were causally instrumental for the emergence of the axiomatic-deductive method in mathematics. The same sociocultural political background was decisive for the emergence of practices of dialectic, the kinds of dialogical interactions famously portrayed in Plato’s dialogues. In turn, dialectic provided the background for the emergence of the first fully-fledged theory of deduction in history, namely Aristotle’s syllogistic.

This chapter focuses on the ‘phylogeny’ of deduction, i.e. how deductive reasoning may have emerged given the genetically endowed cognitive apparatus of humans. It discusses reasoning in non-human animals, Mercier and Sperber’s account of the evolution of reasoning, Heyes’ concept of cognitive gadgets, and neurological studies of deductive reasoning. It is argued that the emergence of deduction should not be viewed as genetically encoded in humans but rather as a product of cultural processes, roughly as described by the cognitive gadgets model.

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.