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This chapter takes up the proofs of Theorems 8.1 and 8.2. For each prime to p integer n, we have the n'th power homomorphism [n] : G → G. Formation of the direct image is an exact functor from Perv to itself, which maps Neg to itself, in Ƿ to itself, and which (because a homomorphism) is compatible with middle convolution. So for a given object N in G_arith, [n]_* allows us to view 〈N〉_arith as a Tannakian subcategory of 〈[n]_*N〉_arith, and 〈N〉_geom as a Tannakian subcategory of 〈[n]_*N〉_geom.

The creation and conceptual nature of full-blown we-mode we-intentions and joint intentions is discussed in this chapter. Intending jointly as a group (= in the we-mode) is taken to entail the individual participants' shared we-intentions in the we-mode. A detailed account of we-intentions is given, and an alleged circularity problem concerning the circularity of accounting for joint intentions in terms of we-intentions is dissolved. It is also argued that we-mode joint intentions—and we-mode attitudes and actions in general—are irreducible to their I-mode counterparts. This is largely because of the group-reason requirement that the we-mode involves.

This chapter examines the implications of post‐reductionist philosophy of science for dualism and the status of the explanatory gaps. The primary argument for dualism is based on a Negative Explanation‐to‐Metaphysics Connection Principle (“Negative EMC”), to the effect that if A is not reducible to B, then B→A is not metaphysically necessary and A is not metaphysically supervenient upon B. But if other special sciences are not reducible to physics either, the dualist is faced with a dilemma. Either she must give up Negative EMC, and with it the principal argument for dualism, or she must draw similar conclusions with respect to other irreducible phenomena, the result being not a dualism but a pluralism of higher ordinality. Dualism can be reconciled with explanatory pluralism only by producing a reason to think that only the mind‐body gap implies a failure of supervenience.

Starting from the premise that Aristotelian explanation proceeds via the concepts of natures and potentials (and not, say, laws), this chapter argues that for Aristotle the generation of a living organism is not the actualization of the natures and potentials of the materials from which organisms develop, but rather the actualization of a primitive, irreducible potential to produce out of the appropriate materials an organism of a certain form. Each of the major texts in which Aristotle defends his natural teleology is shown to argue for or presuppose that material causes are insufficient to produce such an end. Aristotle's teleology is thus is an empirical thesis and not an a priori one brought to nature. A Postscript articulates various aspects of this view more precisely, showing that a part is for the sake of something only if it has come to be for the sake of something.

This chapter identifies three categories of questions answers to which are required if one is to have a full understanding of Aristotle's teleology: Analysis, Basis, Extent. It then focuses on the ‘Basis’ questions and presents a typology of recent interpretations of the ontological basis of Aristotelian natural teleology, which it labels (i) Strong irreducibility (e.g., Gotthelf); (ii) Regulative/pragmatic (e.g., Nussbaum and Sorabji); (iii) limited irreducibility (e.g., Charles); (iv) weak irreduciblilty (e.g., M. Bradie and F.D. Miller, Jr.); (v) intrinsic cause/eliminativism (e.g., S. S. Meyer). Views (iv) and (v) are assessed at some length. Meyer, it is argued, confuses Aristotle's grounds for rejecting his opponents’ view (eliminativism) with what he takes to be the basis of his own view (anti‐reductionism). In addressing view (v) it is it is proposed that, in the face of contemporary science, Aristotle would have retreated to something like the contemporary etiological view of biological teleology.

This chapter takes a close look at the account across Generation of Animals I–II of the efficient cause of animal generation. The aim is to understand better the relationship of material‐efficient causation to the teleological causation Aristotle insists is central to the coming to be of animals. Special attention is given to Aristotle's actual account of embryogenesis — the sequential development of the parts of the embryo — in GA II.6. This examination shows that there is no evidence that Aristotle thought material necessity by itself was causally sufficient for embryogenesis; rather material‐efficient causation does the bulk of its work as a ‘tool’ that the budding organism's formal nature — its ‘irreducible potential for form’ — makes use of in achieving its inherent ends. In the process the chapter provides significant insight into the essential flow of argument across GA I–II, for those unfamiliar with the treatise.

This chapter investigates the question which begings as follows. Suppose we have a geometrically irreducible middle extension sheaf G on 𝔾ₘ/k which is pure of weight zero, such that the object N := G(1/2)[1] ɛ G_arith is “dimension” n and has Gsubscript geom,N = G_arith,N = GL(n). Suppose in addition we are given s ≤ 2 distinct characters χ‎ᵢ of kˣ.

This chapter attempts to answer the central objection that many philosophers will raise against the idea (which was defended in the previous chapter) that there are irreducible normative facts. According to a plausible naturalistic conception of the world, the natural facts in some sense determine the fundamental nature of the world. But how can this idea of irreducible normative facts be reconciled with such a naturalistic conception of the world? A modest form of naturalism implies only that all contingent facts are realized in (and so also supervene on) natural facts. It is argued here that this view can be reconciled with the thesis that there are irreducible normative facts and properties (although the reconciliation requires some far-reaching reflections on the nature of the concepts of ‘essence’ and ‘metaphysical necessity’, and even on the question of which is the right modal logic for metaphysical necessity).

It is shown that powers are both conceptually and ontologically irreducible to non-power (i.e. categorical) properties. This involves both rejecting the three most influential conceptual analyses of power-ascriptions – the simple conditional analysis, the causal conditional analysis, and Lewis' reformed causal conditional analysis – and the view that instantiations of powers are identical to instantiations of non-power properties. In the last section, it is shown that accepting powers as genuine and irreducible properties does not commit one to the doctrines of essentialism or ‘natural necessity’.

This chapter presents a map of the territory in which the varieties of realism and antirealism are located. Topics covered include realism, the connection between realism and truth, presuppositional fulfilment, mind-independence, irreducibility, and causal networking. An overview of the subsequent chapters is presented.

This chapter presents a promising way of defending the determination of value by nature, while resisting reduction. Recent developments in both property theory and value theory help clarify a thesis about the relation between irreducibility and multiple realizability which has not been as clearly articulated as it can be. This in turn will help to establish and illustrate the possibility of supervenience without reduction.

There are two challenges for the supporters of emergence. The first is to show that emergent properties do not succumb to the threat of epiphenomenalism, and that emergent phenomena can have causal powers vis-a-vis physical phenomena. This must be done without violating the causal/explanatory closure of the physical domain. The second challenge is to give a positive characterization of emergence that goes beyond supervenience and irreducibility. This chapter argues that unless this is done, the thesis that minds emerge from bodies remains uninteresting and without much content. Saying that they are not reducible to bodies speaks little about their relationship.

Having attacked a number of features that some views have regarded as constitutive of the concept of ‘the self’, Davidson now discusses those features that he thinks should replace the traditional ones. He especially emphasizes the special authority that our propositional attitudes are equipped with and the irreducible role of indexical sentences. In Davidson's opinion, the thoughts expressed by indexical sentences relate us to the world around us and thereby form a constitutive component of our concept of the self.

The problem of consciousness is not that there are a number of competing explanatory theories, for which we lack the evidence that would enable us to choose between them; it is that we have no sense of what a possible explanation would even look like. Two senses in which consciousness might be mysterious to us are distinguished: an ontological or metaphysical sense in which consciousness is thought to have an occult non-natural nature; and an epistemological sense, in which we do not or cannot understand the nature of consciousness. It is then argued that consciousness is mysterious in the latter sense but not in the former. Consciousness exists and has an explanation, but its nature is deeply and intractably hidden from us owing to the limitations of our cognitive powers. Accepting these theoretical limitations removes the temptation to postulate a set of variously unacceptable solutions to the mind-body problem, which display a typical form (the ‘DIME shape’): deflationary reductionism, outright irreducibility, the eerily magical, ontological elimination.

In his early work (1747‐56), Kant is Leibnizian in his commitment to a distinction between things in themselves and phenomena, grounded on a contrast between intrinsic and relational properties. He is anti‐Leibnizian in his argument that there is Receptivity, since there is real causal influence; and anti‐Leibnizian in his argument that there is Irreducibility, since relations fail to supervene on intrinsic properties. The latter argument is of considerable interest, and open to interpretation: whether it moves illicitly from unilateral to bilateral reducibility; which notions of intrinsicness are appropriate; whether it concerns relations in general, or causal power (specifically attraction and impenetrability in a pioneering field theory). Irreducibility here yields a doctrine of superadded force: since ‘substance never has the power, through its own intrinsic properties, to determine others’, such power is added by God.

Humility does follow from Receptivity—given Irreducibility. For Leibniz, Reducibility is the mirror that allows relations to be taken as intrinsic properties, phenomena to be taken as monads: reducibility permits (confused) acquaintance of things in themselves, since through perceptual access to phenomenal relations, we gain access to intrinsic properties. When Kant denies reducibility, that mirror is broken: Receptivity says we have knowledge only of what affects us; Distinction says that what affects us are the relational, causal powers of substance; Irreducibility says that it is ‘not through its own intrinsic properties’ that substance affects us. Humility follows: ‘we have no insight whatsoever into the intrinsic nature of things’. Kant's commitment to Irreducibility persisted beyond early work, extending throughout his philosophical career.

This chapter discusses the concept of consciousness, which is the most evident barrier to a comprehensive naturalism that relies only on resources of physical science. It reveals that the existence of consciousness appears to imply that the universe's physical description is only a part of the truth. It studies conceptual behaviorism, a strategy that can be used to place the mental into the physical world picture. It then considers the failure of reductionism in the philosophy of mind, which has implications that extend far beyond the mind-body problem, and the basic intelligibility of the universe. This chapter also suggests several additions to a standard evolutionary explanation, describes the difference between emergent answers and reductive answers to the constitutive question, and emphasizes the irreducibility of conscious experience to the physical.

This chapter probes Barry Stroud's steadfast opposition to metaphysical subjectivism about value. Stroud argues in his work that global subjectivism about evaluative matters is literally untenable: the chapter shows how this starting conclusion emerges from three key aspects of evaluative thought pressed by Stroud. The cognitivism, irreducibility, and indispensability of evaluative thought seem together to rule out noncognitivist, error-theoretic, and reductive response-dependent construals of the evaluative domain, thereby closing off a wide variety of routes to value subjectivism. The chapter suggests, however, that the conviction that evaluative matters somehow constitutively depend on us may survive Stroud's determined attempts at excision. The chapter also considers the relation of Stroud's work to mainstream contemporary metaethics, noting that—contrary to what one might suppose—Stroud does not see his arguments as supporting moral realism. In fact, Stroud's negative conclusion about value subjectivism is liable to unsettle both poles of contemporary metaethical debate.