It is shown that the theory of fields with an automorphism has a decidable model companion. Quantifier-elimination is established in a natural language. The theory is intimately connected to Ax's theory of pseudofinite fields, and analogues are obtained for most of Ax's classical results. Some indication is given of the connection to nonstandard Frobenius maps.

We extend the field of Laurent series over the reals in a canonical way to an ordered differential field of “logarithmic-exponential series” , which is equipped with a well behaved exponentiation. We show that the LE-series with derivative 0 are exactly the real constants, and we invert operators to show that each LE-series has a formal integral. We give evidence for the conjecture that the field of LE-series is a universal domain for ordered differential algebra in Hardy fields. We define (...) composition of LE-series and establish its basic properties, including the existence of compositional inverses. Various interesting subfields of the field of LE-series are also considered. (shrink)

I see model theory as becoming increasingly detached from set theory, and the Tarskian notion of set-theoretic model being no longer central to model theory. In much of modern mathematics, the set-theoretic component is of minor interest, and basic notions are geometric or category-theoretic. In algebraic geometry, schemes or algebraic spaces are the basic notions, with the older “sets of points in affine or projective space” no more than restrictive special cases. The basic notions may be given sheaf-theoretically, or functorially. (...) To understand in depth the historically important affine cases, one does best to work with more general schemes. The resulting relativization and “transfer of structure” is incomparably more flexible and powerful than anything yet known in “set-theoretic model theory”.It seems to me now uncontroversial to see the fine structure of definitions as becoming the central concern of model theory, to the extent that one can easily imagine the subject being called “Definability Theory” in the near future.Tarski's set-theoretic foundational formulations are still favoured by the majority of model-theorists, and evolution towards a more suggestive language has been perplexingly slow. None of the main texts uses in any nontrivial way the language of category theory, far less sheaf theory or topos theory. Given that the most notable interactions of model theory with geometry are in areas of geometry where the language of sheaves is almost indispensable, this is a curious situation, and I find it hard to imagine that it will not change soon, and rapidly. (shrink)

A nontrivial ring with unit eliminates imaginaries just in case its complete theory has the following property: every definable m-ary equivalence relation E may be defined by a formula f = f, where f is an m-ary definable function. We show that for certain natural expansions of the field of p-adic numbers, elimination of imaginaries fails or is independent of ZPC. Similar results hold for certain fields of formal power series.

We give foundational results for the model theory of AfinK, the ring of finite adeles over a number field, construed as a restricted product of local fields. In contrast to Weispfenning we work in the language of ring theory, and various sortings interpretable therein. In particular we give a systematic treatment of the product valuation and the valuation monoid. Deeper results are given for the adelic version of Krasner's hyperfields, relating them to the Basarab–Kuhlmann formalism.

Let n be a positive integer and FAℓ be the free abelian lattice-ordered group on n generators. We prove that FAℓ and FAℓ do not satisfy the same first-order sentences in the language if m≠n. We also show that is decidable iff n{1,2}. Finally, we apply a similar analysis and get analogous results for the free finitely generated vector lattices.

Gauss used quadratic forms in his second proof of quadratic reciprocity. In this paper we begin to develop a theory of binary quadratic forms over weak fragments of Peano Arithmetic, with a view to reproducing Gauss’ proof in this setting.

Gauss used quadratic forms in his second proof of quadratic reciprocity. In this paper we begin to develop a theory of binary quadratic forms over weak fragments of Peano Arithmetic, with a view to reproducing Gauss’ proof in this setting.

We construct, by model-theoretic methods, several exponentiations on the universal enveloping algebra U of the Lie algebra.

In this second paper of the series we do a local analysis of quadratic forms over completions of a non-standard model of IΔ0+Ω1.

We give a definition, in the ring language, of Zp inside Qp and of Fp[[t]] inside Fp), which works uniformly for all p and all finite field extensions of these fields, and in many other Henselian valued fields as well. The formula can be taken existential-universal in the ring language, and in fact existential in a modification of the language of Macintyre. Furthermore, we show the negative result that in the language of rings there does not exist a uniform (...) definition by an existential formula and neither by a universal formula for the valuation rings of all the finite extensions of a given Henselian valued field. We also show that there is no existential formula of the ring language defining Zp inside Qp uniformly for all p. For any fixed finite extension of Qp, we give an existential formula and a universal formula in the ring language which define the valuation ring. (shrink)

Background:The debate over the ethical implications of care robots has raised a range of concerns, including the possibility that such technologies could disrupt caregiving as a core human moral activity. At the same time, academics in information ethics have argued that we should extend our ideas of moral agency and rights to include intelligent machines.Research objectives:This article explores issues of the moral status and limitations of machines in the context of care.Design:A conceptual argument is developed, through a four-part scheme derived (...) from the work of Alasdair MacIntyre. No empirical data are used.Ethical consideration:No primary data were gathered for this study. Secondary sources and authorship have been acknowledged throughout.Findings / discussion:Certain kinds of social experience, including the narrative unity of a life, and the giving and receiving of care, are essential for moral development. Machines, no matter how advanced, cannot participate in such experiences in key respects, and thus cannot develop as practical reasoners.Conclusion:It follows that they cannot be moral agents and that they cannot care. There are, it seems, no such things as care robots. In view of the institutional power of tech companies and commissioning bodies, care practitioners need to take more of a lead in developing new assistive technologies which are appropriate to their practice. (shrink)

Part I: The 'explanatory gap'. 1. Why take morality to be objective? -- 2. The gap opens: evolution and our capacity for moral knowledge -- Part II: Secular responses. 3. Alternatives to realism: Simon Blackburn and Allan Gibbard -- 4. Procedures and reasons: Tim Scanlon and Christine Korsgaard -- 5. Natural goodness: Philippa Foot's moral objectivism -- 6. Natural goodness and 'second nature': John McDowell and David Wiggin -- Part III: Theism. 7. From goodness to God: closing the explanatory gap (...) -- 8. Purpose without theism? Axiarchism and Neoplatonism. (shrink)

It is argued that reliance on the testimony of others cannot be viewed as reliance on a kind of evidence. Speech being essentially voluntary, the speaker cannot see his own choice of words as evidence of their truth, and so cannot honestly offer them to others as such. Rather, in taking responsibility for the truth of what he says, the speaker offers a guarantee or assurance of its truth, and in believing him the hearer accepts this assurance. I argue that, (...) contrary to appearances, this account is compatible with the hearer acquiring knowledge, and in fact throws interesting light on the idea of knowledge. -/- . (shrink)

This a general account of the school of Mo-tzu, its social basis as a movement of craftsmen, its isolated place in the Chinese tradition, and the nature of its later contributions to logic, ethics, and science. It assesses the relation of Mohist thinking to the structure of the Chinese language, and grapples with the textual dynamics of later Mohist writings, particularly in regard to grammar and style, technical terminology, the use and significance of stock examples, and overall organization. Includes edited (...) and annotated Chinese text with an English translation and commentary, a glossary, and a photographic reproduction of the unemended text from the Taoist Patrology. (shrink)

Despite its central position in the history of European and Christian thought on the protection of human dignity, the virtue of mercy is currently a problematic and under-developed concept in business ethics, compared to related ideas of care, compassion or philanthropy. The aim of this article is to argue for its revival as a core principle of ethical business practice. The article is conceptual in method. An overview is provided of the scope of contemporary business ethics research on related topics (...) and clarifies some of the similarities and differences between mercy and popular terms, such as compassion and prosocial behaviour. The question is then explored as to why mercy has so little traction in business ethics. Some of the history of the idea of mercy in European and Anglophone philosophy is discussed, from Anselm and Aquinas to the present day, showing how discourse on mercy came to be split into a wider concept of the prevention and alleviation of suffering (_misericordia_) and a narrower one of clemency or leniency. Aquinas’s wide concept of the virtue of _misericordia_ is developed as the basis for a principle of mercy, which is applicable directly to corporations. The practical implications of this for standards of corporate behaviour towards employees and other stakeholders are then considered through the introduction of a ‘vulnerability grid’. The grid offers a critical contrast to other analysis tools, such as stakeholder power-interest matrices. Cases drawn from empirical studies are used to illustrate application of the grid and to challenge some common assumptions of stakeholder theory. (shrink)