This chapter examines the issue of the “logic” underlying Hegel’s exposition of the “concept of nature.” Given the systematic structure of Hegel’s Philosophy of Nature, which is positioned between the Logic and the Philosophy of Spirit, the problem of the logic guiding the immanent development of nature’s forms as well as the development of the philosophical cognition of them is, in Hegel’s view, a particularly relevant one. At the center of this chapter is the question of whether the “logic of nature” is the same logic presented in the first systematic division of Hegel’s philosophy or rather a modified variant of such a logic. The logic of nature, it argues, combines the determinations of pure speculative thinking (or the determinations of the “absolute idea”) with the specific conditions offered by the concept and by the reality of nature. Crucial to this logic is, first, Hegel’s famous definition of nature as the “idea in the form of otherness.” Such a definition obtains from the end of the Logic, which is be examined in detail, and followed through some crucial passages from the Philosophy of Nature. This chapter follows the development of the logic of nature between two extremes – the “absolute idea” and “spirit” – and concludes with a brief examination of the three syllogisms with which Hegel crowns his encyclopedic system.

Hegel has commonly been ridiculed for views expressed in his 1801 dissertation, On the Orbits of the Planets, in the final pages of which he had adopted a series of numbers from Plato’s Timaeus – a cosmological text earlier taken seriously by Kepler – to account for the ratios of the distances from the sun of the then known six planets of the solar system. While defenders of Hegel have usually toned down the extent of these claims, this chapter argues that Hegel’s reference to Plato’s Pythagorean cosmology must be taken seriously – not as cosmology, however, but as instantiating the logic appropriate for empirically based science. Hegel’s allusion to Plato’s mythologically expressed “syllogism” is consistent with his idea that logic as Plato conceived it allowed its application to the empirical world but that this applicability had been compromised by Aristotle adaptation of it. With the proper grasp of logic’s utilization of the category of “singularity” in its difference to “particularity” – available to Plato but not Aristotle – we can appreciate how, while Kepler’s Laws were empirically based, Newton’s were not as they relied on abstract entities that could not be justified empirically.

This chapter argues that Hegel’s aim in his philosophy of nature is not to compete with natural science but to show that there is reason in nature – reason that science cannot see but that works through the causal processes discovered by science. It considers first the transition from Hegel’s logic to his philosophy of nature and argues that the latter continues the project of the former, starting with reason, or the “absolute idea”, as nature, as sheer externality. It then argues that Hegel derives nature’s categories logically – a priori – from the idea-as-externality, and subsequently matches them with empirical phenomena (rather than constructing categories to fit the latter). It provides an abridged account of Hegel’s physics in order to show how the categories of physical (as opposed to mechanical or organic) nature are derived from one another and how they are embodied in physical phenomena, such as sound, heat, and magnetism. It then concludes by arguing that, contrary to appearances, Hegel’s conception of light complements, and is not simply at odds with, that presented by quantum physics.

This chapter argues that Collingwood’s “logic of question and answer” (LQA) can best be understood in the light of contemporary argumentation theory. Even if Collingwood quite often describes LQA in terms of inner thinking and reasoning, as was still usual in his time, his insistence on the normative (“criteriological”) character of LQA, paired with his attack on the pretensions of psychologists to describe logic (as well as other normative endeavours) in a purely empirical manner, makes clear that LQA has the same aspirations as the rising discipline of formal (mathematical) logic. The concise exposition of the form, content, and application of LQA is supported by references to all the relevant passages in Collingwood’s oeuvre as well as illustrated by means of a concrete example of his way of doing history. Although a recent and still developing discipline, contemporary argumentation theory was born as an attempt to describe and analyze argumentative texts as guided by norms constitutive of our argumentative practices in a way that completely escapes formal logic. It thus provides a place for LQA that has so far been lacking.

The process of identifying and interpreting norms of customary international law, while appearing to be primarily based on an inductive analysis of state practice and opinio juris, is sometimes a deductive exercise based on logic and reason. Logic permeates every decision in international law. Logic manifests itself inherently throughout the process and can be identified in all steps of reasoning in identifying, interpreting and applying customary international law. Logic, however, can constitute the application of either an inductive or deductive inference. This chapter focuses on situations in which the International Court of Justice (ICJ) and the Permanent Court of International Justice (PCIJ) applied a deductive approach, identifying or interpreting norms of customary international law without seeming to consult state practice and opinio juris. Specifically, it considers whether norms that can be reasonably inferred or deduced from existing rules, or that are simply logical for the operation of the international legal system, can be identified as norms of customary international law under a complementary, supplementary or distinctive interpretive approach.

We briefly offer the reader a sense of what “logic” is supposed to be: its scope, its goals, and the kind of tools logicians use. We discuss the relationship between logic and the rest of mathematics, outline various conceptions of logic and ways it has been applied, and offer a concrete example of the kind of reasoning one might wish to “formalize” and how this might look.

This chapter discusses Wittgenstein’s remarks on mathematics in sections 6.02–6.031 and 6.2–6.241. These remarks are limited to arithmetic, with definitions of the natural numbers (6.02) and of multiplication (6.241). In the first part, we discuss Wittgenstein’s criticism of the theory of types involved in Russell’s rival ‘logicist’ account, with specific criticisms of Russell’s axioms of infinity (5.535) and reducibility (6.1232). The second part presents Wittgenstein’s positive account of natural numbers in terms of ‘repeated applications of an operation’ (based on his remarks on operations at 5.2–5.254) and of arithmetical calculations. Limited parallels with the lambda calculus are brought to the fore, while explanations that presuppose a scheme of primitive recursion are criticized. The third section discusses related philosophical remarks about the centrality of the ‘method of substitution’ (6.24), arithmetical equations as ‘pseudo-propositions’ (6.241 & 6.2), and the claim that the identity of the two sides of an equation is merely perceived, not assertable (6.2322). Looking ahead, the fourth and final part discusses briefly Wittgenstein’s reasons for abandoning this approach in the early ‘middle period’.

The Introduction provides an overview of the book’s argument about how novels in nineteenth-century Britain (by George Eliot, Wilkie Collins, William Thackeray, and Thomas Hardy) represented modes of thinking, judging, and acting in the face of uncertainty. It also offers a synopsis of key intellectual contexts: (1) the history of probability in logic and mathematics into the Victorian era, the parallel rise of statistics, and the novelistic importance of probability as a dual concept, geared to both the aleatory and the epistemic, to objective frequencies and subjective degrees of belief; (2) the school of thought known as associationism, which was related to mathematical probability and remained influential in the nineteenth century, underwriting the embodied account of mental function and volition in physiological psychology, and representations of deliberation and action in novels; (3) the place of uncertainty in treatises of rhetoric, law, and grammar, where considerations of evidence were inflected by probability’s epistemological transformation; and (4) the resultant shifts in literary probability (and related concepts like mimesis and verisimilitude) from Victorian novel theory to structuralist narratology, where understandings of probability as a dual concept were tacitly incorporated.

The essay studies four aspects of Buridan’s use of principles in his logic. (1) Some principles, such as being said of all or of none, are taken over from the tradition but applied with some qualifications: for example, certain modal syllogisms governed by this principle are said to be “quasi-perfect.” (2) Some principles not used by Buridan’s realist predecessors are introduced, for example, Things that are identical with one and the same thing are identical with each other. (3) Some principles familiar from the tradition are put to powerful new uses, for example, In every pair of contradictories, one is true and the other false, and it is impossible for both to be true together or for both to be false together; again, every proposition is true or false, and it is impossible for the same proposition to be true and false together. (4) Some principles, taken from Porphyry, the Categories, and the Posterior Analytics, which were used by realist logicians to connect the theory of genus and species with the theory of modal syllogisms, are no longer used in that way. It seems that (1)–(3) have the effect of making the theory of consequences (and thus the syllogistic) more “scientific,” while (4) has the effect of separating the theory of consequences from the logical theories contained in the books of the Organon other than the Prior Analytics. If this is so, then Buridan’s logic marks an important stage in the emergence of modern conceptions of logic. A further question is how all of this relates to Buridan’s professed humanistic conception of logic as an art – not a science – “just as the leader is the savior of the army, so reasoning with learning is the leader in human life, whether that life be contemplative, namely, speculative, or active.”