We prove the Main Gap for the class of a -models (sufficiently saturated models) of an arbitrary stable 1-based theory T . We (i) prove a strong structure theorem for a -models, assuming NDOP, and (ii) roughly compute the number of a -models of T in any given cardinality. The analysis uses heavily group existence theorems in 1-based theories.

We present a structure theorem for superstable quasi-varieties without DOP. We show that every algebra in such a quasi-variety weakly decomposes as the product of an affine algebra and a combinational algebra, that is, it is bi-interpretable with a two sorted structure where one sort is an affine algebra, the other sort is a combinatorial algebra and the only non-trivial polynomials between the two sorts are certain actions of the affine sort on the combinatorial sort.

We study perfectly trivial theories, 1-based theories, stable varieties, and their mutual interaction. We give a structure theorem for the models of a complete perfectly trivial stable theory without DOP: any model is the algebraic closure of a nonforking regular tree of elements. We also give a structure theorem for stable varieties, all of whose completions have NDOP. Such a variety is a varietal product of an affine variety and a combinatorial variety of an especially simple form.

Shepherdson [14] showed that for a discrete ordered ring I, I is a model of IOpen iff I is an integer part of a real closed ordered field. In this paper, we consider integer parts satisfying PA. We show that if a real closed ordered field R has an integer part I that is a nonstandard model of PA (or even IΣ₄), then R must be recursively saturated. In particular, the real closure of I, RC (I), is recursively saturated. We (...) also show that if R is a countable recursively saturated real closed ordered field, then there is an integer part I such that R = RC(I) and I is a nonstandard model of PA. (shrink)

Let M = 〈M, +, <, 0, {λ}λ∈D〉 be an ordered vector space over an ordered division ring D, and G = 〈G, ⊕, eG〉 an n-dimensional group definable in M. We show that if G is definably compact and definably connected with respect to the t-topology, then it is definably isomorphic to a 'definable quotient group' U/L, for some convex V-definable subgroup U of 〈Mⁿ, +〉 and a lattice L of rank n. As two consequences, we derive Pillay's conjecture (...) for a saturated M as above and we show that the o-minimal fundamental group of G is isomorphic to L. (shrink)

§1. Introduction. By and large, definitions of a differentiable structure on a set involve two ingredients, topology and algebra. However, in some cases, partial information on one or both of these is sufficient. A very simple example is that of the field ℝ where algebra alone determines the ordering and hence the topology of the field:In the case of the field ℂ, the algebraic structure is insufficient to determine the Euclidean topology; another topology, Zariski, is associated with the ield but (...) this will be too coarse to give a diferentiable structure.A celebrated example of how partial algebraic and topological data determines a differentiable structure is Hilbert's 5th problem and its solution by Montgomery-Zippin and Gleason.The main result which we discuss here is of a similar flavor: we recover an algebraic and later differentiable structure from a topological data. We begin with a linearly ordered set ⟨M, <⟩, equipped with the order topology, and its cartesian products with the product topologies. We then consider the collection of definable subsets of Mn, n = 1, 2, …, in some first order expansion ℳ of ⟨M, <⟩. (shrink)

We prove in particular that, in a large class of dp-minimal theories including the p-adics, definable types are dense amongst nonforking types.

We investigate the group of definable homomorphisms between two definable abelian groups A and B, in an o-minimal structure . We prove the existence of a “large”, definable subgroup of . If contains an infinite definable set of homomorphisms then some definable subgroup of B admits a definable multiplication, making it into a field. As we show, all of this can be carried out not only in the underlying structure but also in any structure definable in.

We consider VC-minimal theories admitting unpackable generating families, and show that in such theories, forking of formulae over a model M is equivalent to containment in global types definable over M, generalizing a result of Dolich on o-minimal theories in [4].

We prove a generalization of the Elekes–Szabó theorem [G. Elekes and E. Szabó, How to find groups?, Combinatorica 32 537–571 ] for relations defina...

Let ℛ be an o-minimal expansion of a real closed field R. We continue here the investigation we began in [11] of differentiability with respect to the algebraically closed field [Formula: see text]. We develop the basic theory of such K-differentiability for definable functions of several variables, proving theorems on removable singularities as well as analogues of the Weierstrass preparation and division theorems for definable functions. We consider also definably meromorphic functions and prove that every definable function which is meromorphic (...) on Kn is necessarily a rational function. We finally discuss definable analogues of complex analytic manifolds, with possible connections to the model theoretic work on compact complex manifolds, and present two examples of "nonstandard manifolds" in our setting. (shrink)

This volume gathers together for the first time are all the key texts in a crucial debate in modern philosophy, centered on Leibniz's famous 1695 essay, the "New System of the Nature of Substances and their Communication," in which he introduced his strikingly original theory of metaphysics. His "system" became increasingly famous and drew him into discussion and development of these ideas, both in public and in private, with a variety of thinkers, most notably the great French philosopher Pierre Bayle. (...) Woolhouse's and Francks's new English edition gives the only full representation of this debate, and will therefore be essential reading for anyone who wishes to gain a proper understanding of Leibniz's philosophy and its intelletual context. All the texts are newly translated and extensively annotated; many appear in English for the first time. (shrink)

We study the Vapnik–Chervonenkis density of definable families in certain stable first-order theories. In particular, we obtain uniform bounds on the VC density of definable families in finite $\mathrm {U}$-rank theories without the finite cover property, and we characterize those abelian groups for which there exist uniform bounds on the VC density of definable families.

Zoé Chatzidakis , David Marker , Amador Martin-Pizarro , Rahim Moosa , Sergei Starchenko Source: Notre Dame J. Formal Logic, Volume 54, Number 3-4, 277--277.

Plato's dialogues frequently criticize traditional Greek myth, yet Plato also integrates myth with his writing. Daniel S. Werner confronts this paradox through an in-depth analysis of the Phaedrus, Plato's most mythical dialogue. Werner argues that the myths of the Phaedrus serve several complex functions: they bring nonphilosophers into the philosophical life; they offer a starting point for philosophical inquiry; they unify the dialogue as a literary and dramatic whole; they draw attention to the limits of language and the limits of (...) knowledge; and they allow Plato to co-opt cultural authority as a way of defining and legitimating the practice of philosophy. Platonic myth, as a species of traditional tale, is thus both distinct from philosophical dialectic and similar to it. Ultimately, the most powerful effect of Platonic myth is the way in which it leads readers to participate in Plato's dialogues and to engage in a process of self-examination. (shrink)

t. 1. Apo ton prōto skeptomeno anthrōpo mechri ton 6. aiōna M.Ch. -- t. 2. Apo ton 7. aiōna M.Ch. mechri to sēmera.

This article is the guest editors’ introduction to the special issue in Business & Society on “SMEs and CSR in Developing Countries.” The special issue includes four original research articles by Hamann, Smith, Tashman, and Marshall; Allet; Egels-Zandén; and Puppim de Oliveira and Jabbour on various aspects of the relationship of small and medium enterprises to corporate social responsibility in developing countries.

We propose a formal framework for interpretable machine learning. Combining elements from statistical learning, causal interventionism, and decision theory, we design an idealised explanation game in which players collaborate to find the best explanation for a given algorithmic prediction. Through an iterative procedure of questions and answers, the players establish a three-dimensional Pareto frontier that describes the optimal trade-offs between explanatory accuracy, simplicity, and relevance. Multiple rounds are played at different levels of abstraction, allowing the players to explore overlapping causal (...) patterns of variable granularity and scope. We characterise the conditions under which such a game is almost surely guaranteed to converge on a optimal explanation surface in polynomial time, and highlight obstacles that will tend to prevent the players from advancing beyond certain explanatory thresholds. The game serves a descriptive and a normative function, establishing a conceptual space in which to analyse and compare existing proposals, as well as design new and improved solutions. (shrink)

Since the launch of applications such as DALL-E, Midjourney, and Stable Diffusion, generative artificial intelligence has been controversial as a tool for creating artwork. While some have presented longtermist worries about these technologies as harbingers of fully automated futures to come, more pressing is the impact of generative AI on creative labour in the present. Already, business leaders have begun replacing human artistic labour with AI-generated images. In response, the artistic community has launched a protest movement, which argues that AI (...) image generation is a kind of theft. This paper analyzes, substantiates, and critiques these arguments, concluding that AI image generators involve an unethical kind of labour theft. If correct, many other AI applications also rely upon theft. (shrink)

In Seeing Things as They Are (Searle 2015), Searle developed a direct realist’s theory of perception. According to direct realism, physical objects are directly and immediately perceived. Searle claims that Berkeley’s theory of perception goes against direct realism. For Searle, Berkeley’s theory suggests that only subjective experiences (ideas) are directly and immediately perceived, not physical objects. Contrary to Searle, G. S. Pappas claims that Berkeley’s theory of perception is consistent with the view that physical objects are immediately perceivable (Pappas 1982; (...) Pappas 1987). Given the conflicting interpretations of Berkeley’s theory of perception, this paper attempts to defend Searle’s position on Berkeley’s theory of perception. It refutes Pappas’ interpretation by arguing that he is inconsistent with Berkeley’s theses of ontological phenomenalism and heterogeneity thesis. (shrink)

Capital punishment—the legally authorized killing of a criminal offender by an agent of the state for the commission of a crime—stands in special need of moral justification. This is because execution is a particularly severe punishment. Execution is different in kind from monetary and custodial penalties in an obvious way: execution causes the death of an offender. While fines and incarceration set back some of one’s interests, death eliminates the possibility of setting and pursuing ends. While fines and incarceration narrow (...) one’s routes to happiness, death eliminates its possibility. Not surprisingly, there is much debate about the moral permissibility of capital punishment. This entry maps the terrain of this debate. The first section discusses justifications of the death penalty as they appear in major theories of punishment. The second section surveys moral objections to execution that apply to most justifications. The third addresses procedural criticisms, which do not target the morality of execution so much as the justice of its implementation. (shrink)

Many people are worried about the harmful effects of climate change but nevertheless enjoy some activities that contribute to the emission of greenhouse gas (driving, flying, eating meat, etc.), the main cause of climate change. How should such people make choices between engaging in and refraining from enjoyable greenhouse-gas-emitting activities? In this chapter, we look at the answer provided by decision theory. Some scholars think that the right answer is given by interactive decision theory, or game theory; and moreover think (...) that since private climate decisions are instances of the prisoner’s dilemma, one rationally should engage in these activities provided that one enjoys them. Others think that the right answer is given by expected utility theory, the best-known version of individual decision theory under risk and uncertainty. In this chapter, we review these different answers, with a special focus on the latter answer and the debate it has generated. (shrink)

Critics of the computational connectionism of the last decade suggest that it shares undesirable features with earlier empiricist or associationist approaches, and with behaviourist theories of learning. To assess the accuracy of this charge the works of earlier writers are examined for the presence of such features, and brief accounts of those found are given for Herbert Spencer, William James and the learning theorists Thorndike, Pavlov and Hull. The idea that cognition depends on associative connections among large networks of neurons (...) is indeed one with precedents, although the implications of this for psychological issues have been interpreted variously — not all versions of connectionism are alike. (shrink)

The practical goal of preventing premature death seems uncontroversial. But the term ‘premature death’ is vague with several, sometimes conflicting definitions. This ambiguity results in several conceptions with which not all will agree. Moreover, the normative rationale behind the goal of preventing premature deaths is masked by the operational definition of existing measures. In this article, we argue that ‘premature death’ should be recognized as a normative concept. We propose that normative theories should be used to justify measures of premature (...) death to provide them with normative validity and public legitimacy. (shrink)