We explore Shelah's model‐theoretic dividing line methodology. In particular, we discuss how problems in model theory motivated new techniques in model theory, for example classifying theories by their potential (consistently with Zermelo–Fraenkel set theory with the axiom of choice (ZFC)) spectrum of cardinals in which there is a universal model. Two other examples are the study (with Malliaris) of the Keisler order leading to a new ZFC result on cardinal invariants and attempts to clarify the “main gap” by (...) reducing the dependence of certain versions on (highly independent) cardinal arithmetic. (shrink)

We generalise the properties $\mathsf {OP}$, $\mathsf {IP}$, k- $\mathsf {TP}$, $\mathsf {TP}_{1}$, k- $\mathsf {TP}_{2}$, $\mathsf {SOP}_{1}$, $\mathsf {SOP}_{2}$, and $\mathsf {SOP}_{3}$ to positive logic, and prove various implications and equivalences between them. We also provide a characterisation of stability in positive logic in analogy with the one in full first-order logic, both on the level of formulas and on the level of theories. For simple theories there are the classically equivalent definitions of not having $\mathsf {TP}$ and (...) class='Hi'>dividing having local character, which we prove to be equivalent in positive logic as well. Finally, we show that a thick theory T has $\mathsf {OP}$ iff it has $\mathsf {IP}$ or $\mathsf {SOP}_{1}$ and that T has $\mathsf {TP}$ iff it has $\mathsf {SOP}_{1}$ or $\mathsf {TP}_{2}$, analogous to the well-known results in full first-order logic where $\mathsf {SOP}_{1}$ is replaced by $\mathsf {SOP}$ in the former and by $\mathsf {TP}_{1}$ in the latter. Our proofs of these final two theorems are new and make use of Kim-independence. (shrink)

In this article, we want to reply to the recent article by Buturovic, to be able to correct some statements and allegations about this combined procedure. Organ donation after euthanasia is an extremely difficult procedure from an ethical point of view. On the one hand, we see a suffering patient who wants to die but who also wants to make an altruistic effort to donate his organs. On the other hand, we visualise a patient in need of an organ but (...) who is wary of the fact that someone else needs to die in order to potentially receive a transplant organ. Healthcare professionals seem to walk a tightrope when balancing between the interests of the patients at these two extremes: while facilitating the dying patient’s last wish on the one hand and abiding by all regulations regarding donation and transplantation on the other. Yet, these physicians, nurses and transplant coordinators do their utmost best to keep a strict line between euthanasia and organ donation, to avoid any external pressure on the patient, and to respect his autonomy. They really make an utmost attempt to make the process bearable for the donating patient. However, undeniably the patient who is about to undergo organ donation after euthanasia is nevertheless confronted with dozens of feelings and thoughts. However, this does not imply that procedural safeguards are failing to disentangle organ donation from euthanasia. (shrink)

It may seem strange, in view of the spate of recent literature on the subject, that yet another article should be forthcoming on what is certainly the most familiar, as well as the most vexed, of all Platonic passages. But it is precisely this spate of literature that has impelled me to write. The time seems to have come for an article which, rather than seeking desperately for something new, sets out instead to reaffirm those facts and conclusions that even (...) the most resolutely original of scholars could hardly venture to dispute. This article will therefore be based, not on any of the numerous modern interpretations, but on what Plato himself actually wrote. It will contain a number of tentative suggestions which, to the best of my knowledge, have not been made before. They can, and probably will, be rejected. But its primary purpose remains to restate, whenever possible in Plato's own words, a number of important facts that are at once so simple and so obvious that they seem repeatedly, and especially in recent years, to have been quite forgotten. (shrink)

In this essy I shall enter into the vexing question of Plato’s „theory of intermediates“, and the relation of this theory to the Sun, Line and Cave section of Republic VI and VII. My thesis is that in the last 75 years or so scholarly opinion has reached a complete impasse, having veered from one extreme to another, rather than in the fashion of an Hegelian thesis and antithesis; this conflict of opinion desperately requires a sythesizing „third“, and in (...) the conclusion of the paper I try to supply this „third“. (shrink)

We give a new characterization of _SOP_ (the strict order property) in terms of the behaviour of formulas in any model of the theory as opposed to having to look at the behaviour of indiscernible sequences inside saturated ones. We refine a theorem of Shelah, namely a theory has _OP_ (the order property) if and only if it has _IP_ (the independence property) or _SOP_, in several ways by characterizing various notions in functional analytic style. We point out some connections (...) between dividing lines in first order theories and subclasses of Baire 1 functions, and give new characterizations of some classes and new classes of first order theories. (shrink)

At the end of the Sixth Book of the Republic Plato explains the Idea of Good by means of the Figure of the Sun. As the sun is the cause both of the becoming of that which is subject to becoming and of our apprehension of it and of its changes through the eye, so the idea of good is the cause of the being of that which is and also of our knowledge of it. As the sun is beyond (...) γxs22EFνεσις, so the Idea of Good is beyond Being. Glaucon says he does not understand. The simile is further elucidated by means of a line, divided into two parts, of which one stands for the νιητxs22EFν γxs22EFνος τε καxs22EF τxs22EFπος, where the Idea of Good bears rule, the other for the xs22EFρατxs22EFν γxs22EFνος τε καxs22EF τxs22EFπος, over which the sun is lord. The line is to be divided unequally , and subdivided in the same proportions. Thus we get a line consisting of four parts in the ratio 4 : 6 : : 6 : 9. Let us call the four parts A B C D respectively, A being the smallest, D the greatest, B and C necessarily equal. A stands for εxs1F31κxs22EFνες, shadows, images in water and on polished surfaces, and the like: B stands for animals, plants, and the creations of human industry: C for the objects of that enquiry in which the objects denoted by B are treated as images, i.e. mathematical enquiries: D for the objects apprehended by dialectic, the Ideas themselves. The first equation asserted is—The objects of opinion : objects of knowledge : : representation : original . There follows an explanation of the inferiority of mathematical to philosophical reasoning, and an explanation of the statement that the objects denoted by B are used as images or symbols by the enquiry concerned with C; as a result of which Glaucon perceives that the general distinction between C and D is that between the τxs22EFχναι , i.e. those sciences in which the Guardians were to be educated, and Philosophy or Dialectic. Finally a special πxs22EFθημα or affection of the soul is allotted to each of the four divisions of the line, to A εxs1F31κασxs22EFα, to B πxs22EFστις, to C διxs22EFνοια, to D νxs22EFησις, each πxs22EFθημα being clear in the same degree in which the objects it is concerned with are true. (shrink)

The second highest level of the divided line in Plato’s Republic (510b-511a) appears to be about the entities of mathematics—entities such as particular (though non-physical) triangles. It differs from the highest level in two respects. It involves reasoning from hypotheses, and it uses visible images. This article defends the traditional view that the passage is indeed about these mathematical ‘intermediates’; and tries to show how the apparently different features of the second level are related, by focussing on Plato’s need (...) to give an account of how we can speak of many particulars of the same kind, without assuming that they are imperfect copies, in the way sensible things can be imperfect copies, of forms. (shrink)

This paper takes a detailed look at the Republic’s Divided Line analogy and considers how we should respond to its most contentious implication: that pistis and dianoia have the same degree of ‘clarity’ (σαφήνεια). It argues that we must take this implication at face value and that doing so allows us to better understand both the analogy and the nature of dianoia.

We discuss dividing lines (in model theory) and some test questions, mainly the universality spectrum. So there is much on conjectures, problems and old results, mainly of the author and also on some recent results.

From the care Plato takes in describing the excellence of the doctor in book 3 to the characterization of various pathological elements in the regimes he describes in book 8, the Republic teems with references to medical terms and concepts. The following investigates the breadth of the influence of medicine on the Republic. I argue that a medical vocabulary proves indispensable to indicating the relationship between philosophy and politics that the Republic envisages. In order to do so, this paper examines (...) the confluence of medicine and metaphysics revealed by a comparison between the discussion of the divided line and ancient characterizations of diagnosis. I then conclude with a reading of the Glaucus image in book 10 that emphasizes its self-diagnostic character. (shrink)

The "divided line" passage seems to compare the four states of the soul and their objects to the four segments of a line, somewhat as follows.

the famous passage at the end of the sixth book of the Republic, in which Socrates compares our cognitive states to different sections of a single line, is often read as a "map" of Plato's epistemology. Symbolizing the different levels of cognition, the divided line may be taken to present a forceful image of the various stages that the soul must traverse in its quest for true knowledge. Its reputation as a snapshot of the epistemology of Plato's mature (...) dialogues makes the divided line, despite, or perhaps because of, persisting disagreements over its interpretation, one of the most vexed passages of the Platonic corpus.In the ancient readings of the divided line, many of the modern exegetical debates... (shrink)

We introduce the notion of positive local combinatorial dividing-lines in model theory. We show these are equivalently characterized by indecomposable algebraically trivial Fraïssé classes and by complete prime filter classes. We exhibit the relationship between this and collapse-of-indiscernibles dividing-lines. We examine several test cases, including those arising from various classes of hypergraphs.

The directions for constructing the figure are to take a line cut into two unequal parts, and cut each part in the same ratio. The proportions of the lengths of segments to one another will then represent the "relative clarity" of each of four kinds of knowledge, and Book vi. closes with a summary of these proportions. If we letter the four segments from top to bottom a, b, c, and d, their relation is a:b :: c:d. From the (...) context, it is quite clear that these four segments are unequal. However, if any line is cut in a ratio m/n, and its segments subdivided in that same ratio, two of the resulting segments will be equal, and their ratio 1:1. However, this is the construction explicitly given in the directions. (shrink)

This essay argues that there is an important analogy between the hierarchically ordered divisions of the divided line in Republic Book VI and the hierarchy of constitutions described in Books VIII-IX. Imagination corresponds to tyranny, belief to democracy, mathematical understanding to oligarchy, and dialectical reason to timocracy. The unhypothetical principle disclosed through the activity of dialectic, the idea of the Good itself, corresponds to the aristocratic rule of philosopher kings.

Plato believed that behind everything in the universe lie mathematical principles. Plato was inspired by Pythagoras (571 BCE), who developed a school of mathematics at Crotona that studied sacred geometry as a form of religion. The school’s motto was “God is number,” or “All is Number”. Pythagoras believed that numbers represented God in pattern, symmetry, and infinity. When one of its students, Hippasus told the world the secret of the existence of irrational numbers, Greek geometry was born and Pythagoras’ idea (...) of divinity in numbers died because how could God not be perfect and symmetrical? In Plato’s Republic he discusses something called The Divided Line, which is a map, of sorts, for reaching what he calls the highest Good, which is the ultimate truth where one realizes the true state of the universe and can see the world for what it really is. Many mathematicians have attempted to plot Plato’s Divided Line only to come across a litany of problems and conundrums. Some have said that it the Divided Line cannot be plotted and is merely an allegory not meant to be plotted. This paper discusses some of the conundrums preventing the plotting of Plato’s Divided Line (not an exhaustive list), including Whole ‘vs’ Separate, Equality ‘vs’ Ontological Dissimilarity, Linear ‘vs’ Non-linear, and Infinity ‘vs’ Finite. This paper also explores a new understanding of the Allegory of the Cave in light of ‘the problem of the irrational.’ In exploring the link between the Divided Line and the ‘the problem of the irrational,’ I was able to plot it. It was found that the Divided Line is not a line in the linear sense, but a spiral, the Golden Ratio! This paper is an example of a new category of scholarly inquiry I call “Math Theory” based on scholarly mathematical axioms in theory, rather than including actual maths. In my papers I use existing mathematical equations and place them in an encompassing theory, rather than finding new formulae to fit an existing theory. Keywords: Pythagoras, divided line, math theory, highest good, all is number. (shrink)

The old question whether or no the doctrine of ‘intermediate mathematical objects’ ascribed to Plato by Aristotle is to be found in the Divided Line of Republic vi, has been recently raised again in a careful and lucid discussion by Mr. W. F. R. Hardie. I may clear the ground by saying at once that I agree with that part of Mr. Hardie's chapter which deals with those criticisms of the traditional view that have been put forward by Prof. (...) Ferguson and the late Prof. Stocks; nevertheless I do not believe that the objects of the third segment are intermediate between Forms and Sensibles: I propose to argue that they, like the objects of the highest segment, are Forms. (shrink)

The old question whether or no the doctrine of ‘intermediate mathematical objects’ ascribed to Plato by Aristotle is to be found in the Divided Line of Republic vi, has been recently raised again in a careful and lucid discussion by Mr. W. F. R. Hardie. I may clear the ground by saying at once that I agree with that part of Mr. Hardie's chapter which deals with those criticisms of the traditional view that have been put forward by Prof. (...) Ferguson and the late Prof. Stocks; nevertheless I do not believe that the objects of the third segment are intermediate between Forms and Sensibles: I propose to argue that they, like the objects of the highest segment, are Forms. (shrink)

In Republic VI, Plato’s Socrates attempts to explain the nature of human understanding by means of a simile of a line divided into four unequal segments. Socrates directs Glaucon to accept as names for the four states ‘rational knowledge’ for the highest, ‘understanding’ for the second, ‘belief’ for the third, and for the last, ‘perception of images.’ He then directs Glaucon to arrange the four states in a proportion, ‘considering that they participate in saphēneia in the same degree to (...) which their objects participate in reality.’ Although saphêneia here is usually translated as ‘clarity, ’I argue that neither ‘clarity’ nor ‘truth’ nor ‘knowledge’ provides an entirely satisfactory translation. Rather, in this context saphêneia is best understood as ‘a full, accurate, and sure awareness.’ Plato’s simile, properly understood, provides not only an explanation of his rationalist theory of knowledge but also a coherent line of argument in support of that theory. (shrink)

"TAKE A LINE cut in two unequal sections, one for the kind that is seen, the other for the kind that is thought, and go on and cut each section in the same ratio". In order to follow this request, not only must one know geometry, which treats linear magnitudes; one must also know the relations between geometry and the art which treats kinds. The problem of the first cut in the line is the problem of determining what (...) ratio of two parts of a line is the same as the ratio of the two kinds: visible and thinkable. In order to solve this problem, a procedure seems to be needed for deciding when a ratio of magnitudes, a geometric ratio, is the same as a ratio of kinds, or at least what ratio of magnitudes is the same as the ratio of visible to thinkable. (shrink)

"TAKE A LINE cut in two unequal sections, one for the kind that is seen, the other for the kind that is thought, and go on and cut each section in the same ratio". In order to follow this request, not only must one know geometry, which treats linear magnitudes; one must also know the relations between geometry and the art which treats kinds. The problem of the first cut in the line is the problem of determining what (...) ratio of two parts of a line is the same as the ratio of the two kinds: visible and thinkable. In order to solve this problem, a procedure seems to be needed for deciding when a ratio of magnitudes, a geometric ratio, is the same as a ratio of kinds, or at least what ratio of magnitudes is the same as the ratio of visible to thinkable. (shrink)

Este trabalho, uma das primeiras obras dedicadas ao significado do Tratado de Lisboa (2007) nas suas vertentes política, histórica e jurídica, constitui não apenas uma reflexão clara e bem fundamentada sobre o seu objecto (o que já seria notável dada a sua complexidade e ainda relativa novidade) mas serve ainda ao leitor, especialista ou não nas matérias da União Europeia, como um guia para a análise do Tratado de Lisboa e do processo que lhe deu origem, a falhada «Constituição Europeia», (...) a d... (shrink)

This paper presents a new interpretation of the objects of dianoia in Plato’s divided line, contending that they are mental images of the Forms hypothesized by the dianoetic reasoner. The paper is divided into two parts. A survey of the contemporary debate over the identity of the objects of dianoia yields three criteria a successful interpretation should meet. Then, it is argued that the mental images interpretation, in addition to proving consistent with key passages in the middle books of (...) the Republic, better meets those criteria than do any of the three main positions. (shrink)

Elaborating the analogy between the sun and the good, Plato's Socrates tells Glaucon to divide a line αβ into two unequal segments at γ. The result is that αγ represents what is intelligible and γβ what is visible.1 Then Glaucon is to divide each of the two segments by the same ratio as he used in the original division.2 Whatever proportion he used to make the cuts γ, δ, and ε in the divided line, generating its four segments, (...) the geometrical implication is that the two middle segments must be equal in length. As both Nicholas D. Smith and Richard Foley have emphasized, when Socrates reiterates the characteristics of the line at 534a3–5, transposing δγ and γε, there should be no doubt that Plato... (shrink)

Is inheritable genetic modification the new dividing line in gene therapy? The editors of this searching investigation, representing clinical medicine, public health and biomedical ethics, have established a distinguished team of scientists and scholars to address the issues from the perspectives of biological and social science, law and ethics, including an intriguing Foreword from Peter Singer. Their purpose is to consider how society might deal with the ethical concerns raised by inheritable genetic modification, and to re-examine prevailing views (...) about whether these procedures will ever be ethically and socially justifiable. The book also provides background to define the field, and discusses the biological and technological potential for inheritable genetic modification, its limitations, and its connection with gene therapy, cloning, and other reproductive interventions. For scientists, bioethicists, clinicians, counsellors and public commentators, this is an essential contribution to one of the critical debates in current genetics. (shrink)

In general, scholars have viewed the mathematical detail of Plato’s Divided Line discussion in Republic VI-VII as irrelevant to the substance of his epistemology.Against this stance this essay argues that this detail serves a serious and instructive purpose and makes manifest some central features of Plato’s account of human knowledge.