This chapter examines Wilfrid Sellars’s idea of material inference – the view that there are valid inferences expressed in non-logical language that are irreducible to logical principles. Material inference can be characterised by three features: it is non-enthymemic, non-logical, and non-monotonic. However, the exact sense in which material inferences are valid remains unclear.After exploring various attempts to define the validity of material inference – such as through truth-preservation, analytic entailment, or reduction to general principles – I argue that material inference is not valid in the strict sense. Typically, validity refers to deductive validity, where the truth of the premises guarantees the truth of the conclusion.Nonetheless, pragmatists do not apply an external standard to our reasoning practices. Material inference can be understood not as valid, but as a legitimate and rationally authoritative form of reasoning. The key concept here is pragmatic propriety, which designates the social-normative entitlement to infer – rendering an inference either legitimate or illegitimate.

The history of astronomy and the history of philosophy are inextricably fused. This chapter highlights some of the dramatic episodes when astronomy confronted philosophical issues in metaphysics – questions regarding the objects astronomy studies; in epistemology – what is the most fruitful way to acquire knowledge in astronomy? And finally the ultimate existential question – what kind of a science is astronomy? Is it even a natural science at all?

This chapter examines Wittgenstein’s Tractatus Logico-Philosophicus and its model of logical context. Against readings that see it as purely anti-contextualist, the chapter shows how logic functions as a form of context in early Wittgenstein. Through biographical and historical context, it demonstrates how the Tractatus emerged from and responded to specific intellectual environments, while setting up the book’s broader argument about parallel developments in anthropology and philosophy.

I draw a contrast between what I call a ‘presentational-phenomenological model of apprehension and ultimate evidence’ – which I attribute to Descartes, Locke and Hume – and Leibniz’s ‘logical-conceptual model of apprehension and ultimate evidence’. Although there are different uses of the word ‘concept’, Leibniz’s model, in my view, illustrates the strictest and best notion of the concept of concepts: one that centrally relies on logic.

This chapter presents a portrait of study and teaching at the Faculty of Arts in Paris during the first half-century of the university's existence: from enrolment under a master to obtaining a licence, entering the corporation of the Magistri Artium and, eventually, enrolment in one of the higher faculties (theology, canon law or medicine).

This chapter discusses the work of twelfth-century theologians in Paris who laid the foundations for the development of theology as a discipline in the university. These thinkers explored the characteristics and limits of the discourse on God in theological treatises and summae, which employed increasingly sophisticated technical terminology drawn in part from grammar, logic, and rhetoric.

This chapter traces the development of a number of Trinitarian issues throughout the second half of the twelfth century - the classification of theological language, the debate about why we can say 'God begot God' but not 'essence begot essence', and the definition of the personal properties - and show how they shape Lateran IV and continue thereafter. Finally, the chapter indicates some new approaches and areas of focus among theologians writing after the council.

A variety of digital devices and circuits are introduced. The use of binary numbers in digital electronics is discussed. The AND, OR, XOR, NOT, NAND, NOR, XNOR, and buffer logic gates are presented, followed by a discussion of implementing logical functions. The Karnaugh map and Boolean algebra are introduced. Different ways of constructing logic gates are presented. Half- and full-adder circuits are developed. Several types of flip-flops are discussed. Building on this foundation, we introduce counters, decoders, shift registers, D/A and A/D converters, multiplexers, demultiplexers, memory arrays, automated processing, programmable logic devices, and digital EM communications.

This section offers a review and frequently asked questions regarding ethnography in international business.

This chapter provides a discussion on multivariate random variables, which are collections of univariate random variables. The chapter discusses how the presence of multiple random variables gives rise to concepts of covariance and correlation, which capture relationships that can arise between variables. The chapter also discussed the multivariate Gaussian model, which is widely used in applications.

In this introduction, we first describe the contents of the Summa Logicae in some detail, situating the work in the larger context of medieval logical texts of the thirteen and fourteenth centuries and explaining why it occupies pride of place in Ockham’s philosophical project. Second, we argue that the Summa Logicae was most likely composed in Avignon between 1324 and 1328 contrary to the accepted view that Ockham wrote it in London over the summer of 1323. Third, we trace the legacy of the Summa Logicae from its first reception in Oxford and Paris in the 1330s, into the Parisian controversies of the 1330s and 1340s, and its dissemination further into Europe over the course of the next century or so. We end this history by noting the 1974 publication of the modern critical edition of the Summa Logicae, which was an enormously significant landmark in Ockham studies.

I first present a model of Ockham’s semantics that puts modality front and center (it is a presheaf semantics over a branching timeline). I then show what kinds of statements about language Ockham’s semantics supports. Finally, I discuss how Ockham’s semantics fares in light of Tarski’s and Montague’s paradoxes.