In this paper the author presents a new form of hexagon and the solution of the open problem of classifying plane hexagons. In particular are illustrated the forms of crossed and simple n -gons for n = 3, 4, 5, 6 and also the forms of simple ones for n = 7, 8, 9. A graphic way to construct new forms of polygons is illustrated.

Logic in Action/Doron Avital Nothing is more difficult, and therefore more precious, than to be able to decide (Napoleon Bonaparte) Introduction -/- This book was born on the battlefield and in nights of secretive special operations all around the Middle East, as well as in the corridors and lecture halls of Western Academia best schools. As a young boy, I was always mesmerized by stories of great men and women of action at fateful cross-roads of decision-making. Then, like as today, (...) I felt the full weight of the moment of truth as it confronts the individual, the man or woman of action, with the imminent necessity to decide. Alongside, as I was climbing up the ladder of education, I found myself just as well captivated by the great intellectual stories of our times, in particular 20th century analytic philosophy and, at its core, the machinery of modern logic. How do then the cool contemplative modes of the philosophical and analytic outlooks, that are better fitting of the philosopher's armchair at the fireplace, hang together with the heat of the battlefield or the angst of the ticking clock of time left to the completion of a daring special operation? Growing up I was pondering exactly that. Not accidently, and in fact as consequence, I have chosen a course in life that challenged me to visit the poles of both theory and practice. I set myself on a mission to unravel the tension between the contemplative inquiry, supposedly theoretical, aiming at the logical structure of things, and the uncompromising test of life and action. The puzzle of the gap between theory and practice has become ever since the guiding question of my life. -/- Not long ago, I met two friends, Avner Shor, a veteran combat soldier of the special forces unit I eventually commanded, and Aviram Halevi, a fellow officer with whom I served. Why will we not put into writing the ideas that guided us during our years of military action, we wondered. At that time, I used to deliver the keynote lectures to the senior officers' course of Israel central intelligence organization, the Mossad. For this I already had to put some of these ideas in writing. Avner was quick to introduce me with a notable publisher and even helped me write the first chapter. With Aviram I sat to many meetings, where he would challenge me with key questions, and we followed them with memories of operations and battles, aiming to draw the required lessons. This served me well when I finally sat down to write the first draft of the book. The result was more extensive than me and my friends had expected. I had to add a particularly long chapter dealing with the story of modern philosophy and logic. This was required, I felt, as no decision-making process lives in an intellectual vacuum. Knowingly or subconsciously, the intellectual climate of the time guides us for better or worse with the decisions we finally take. The initial draft was naturally cumbersome and inaccessible. But here I had the privilege of meeting a particularly talented editor, Rami Rothholtz. I found in Rami both an editor and a critical reader. As an infantry combat soldier at youth and now a senior journalist and editor, Rami insisted on simplifying and refining my ideas. Had it not been for his work, this book would have been literally unreadable. *** I opened my book with a brief description of a single night of a special operation deep at the Lebanon Bekaa valley, far away from Israel northern border. The mission was to kidnap a senior terror leader known to be holding in captivity an Israeli pilot. This was also the last special operation I commanded in my military career. A single special operation night, but what had to precede it? For sure, many months of intense and skillful preparations. What is Planning? This is the question with which I open up chapter (2) of the book. I confront in this chapter two approaches to planning. The first is the one I saw as a mission to myself to cultivate in my many years of military service. The second is the one I saw as representing a deep-rooted fallacy that holds a firm grip on today's conventional wisdom about planning. To further examine the logic of the competing models, I was required in chapter (3) to analyze the concept of "The Standard". I argue that the false model of planning has its roots in an attitude of an uncompromising adherence to the power of literal obedience to standards and rules. The standard, or what is outlined by a rule as prescribing a future course of action, is seen as an answer to the fundamental tension between theory (planning) and practice (execution). I argue in contrast that the standard does not produce an answer but rather what I call: "A Well-Defined Question". In essence, it is a meticulously-well-planned invitation to exercise a fresh judgment. A judgement that has as an anchor the default answer that the standard produces but is open to extension as it confronts the new frame of reference of the test of reality. In chapter (4), I present a new tool for planning and execution: "The Polygon of Risks" or "The Polygon of Execution". The Polygon is a geometric illustration of the underlying tension that exists between all dimensions of the operational project. The construction and conceptualization of the trade-offs that exist between the various dimensions - segments of the polygon - is the first topic on the planner's agenda. I position the concepts of "Daring" and "Taking Proactive and Well-Informed Risks" as necessary logical constituents of the polygon and show how in contrast the false model of planning is responsible for an operational culture that is risk-averse. It encourages, in fact, the piling up of false securities at the segments' level that eventually lead to the collapse of the polygon. The necessary trade-offs relationships that exist between the segments of the polygon express the fundamental tension between partnership and competition: partnership in a joint goal and competition over resources, both material and abstract. When this tension is not managed well, that is, when the players' local optimizations overshadow the global optimization required for the success of the project, the execution polygon collapses. In this scenario, we may find the setting of the bar of "High Standards" in a players' spheres of responsibility to conceal a hidden agenda of personal insurances taken against a possible failure. Once the overall risk is not mitigated by way of collaborative joint executional dialogue, the process that leads to a collapse of the polygon is literally unstoppable. *** In chapter (5), I discuss "(Fateful) Decisions". I examine the way in which we should translate the polygon into sequential critical decisions on the timeline. The execution polygon is not a rigid scaffolding but a dynamic conceptual illustration of the -/- project. It will need now to be converted into successive decision junctions. We will find the content of a decision and its timing on the timeline to be inseparable. The right decision is the one that is taken at the right time. The moment of decision is in fact the last moment of deliberation, the first moment available for action, i.e. the first possible "Can" is also the latest possible "Must". Deciding too soon is analog to the "Jumping of the Gun". The decision maker is forcing a pre-conceived answer to a state of affairs not ripe for an answer, e.g. the target in the shooting range is not yet erected. Late to decide and the decision maker may envision what he or she should have done but reality is no longer available for them to harness it right, e.g. consider a tennis player's frustration when reviewing a too-late-to-react move in the video replay. I review in this chapter many fateful decisions, both historical, from Napoleon's Waterloo, to a Bridge Too Far of WWII, to the failed rescue attempt of American hostages at the time of the Iranian Revolution, as well as fateful decisions I had to take in the course of my military life, both in combat and in special operations, and show the sense in which they all answer to our analysis. The theme of chapter (6) is "The Moment of Truth". This is the moment when everything is on the line and from which there is no turning back. I discuss the formative repetitive structure of the schooling and simulation phase and contrast it with the "One Shot, One Opportunity" nature of the moment of truth. This is extremely helpful when we analyze the logical structure of failures as unwarranted repetitions of past lessons. The schooling curriculum indeed consists of important abstractions drawn from lessons of the past. However, it is when we project them in a fashion that is literal or mechanical that we cross path with the possibility of a failure. The paradox of education is that what we must repeat in school and simulation must be extended and not repeated in real life. What is required of us so we do not fall into the fallacy of repetition in face of the moment of truth is the core subject of the discussion of this chapter. In chapter (7), I took a lengthy philosophical detour surveying the intellectual drama of modern logic, and in general, the story of 20th century analytic philosophy. Readers who choose to bypass the somehow demanding narrative of this drama can do so and still keep the argumentative thread of the book. However, even by way of skipping the more technical parts, I would still urge the readers to delve into the discussion in this chapter as I believe this would grant them a deeper understanding of the overall conceptual framework. I explore here the work of two key figures: the logician Kurt Gödel and the philosopher Ludwig Wittgenstein, both were crowned in 2000 by TIME Magazine, as the leading Logician and Philosopher, respectively, of the 20th Century. I review the fascinating logical result of Gödel's Incompleteness Theorem and examine its relevance as setting a theoretical limit to the power of formalization. This amazing logical result - a result that shakes the logical foundations of 20th century thought in as much as the discovery of irrational numbers shook the foundations of the Pythagorean program of the ancient Greeks - bears significantly on the question of action and the blind trust we place in the power of formalization in the form of protocols and rules. It is here that Wittgenstein's philosophy enters in full force. We follow Wittgenstein as he dismantles our conception of rules as if there were "Railroads to Infinite", i.e. as prescribing for us in-advance their proper employments in all possible -/- future eventualities. Instead we propose that the dilemma of action demands of us an extension of rules rather than a mechanical or blind repetition of their literal imperative. In turn, this extension can serve as a new lesson added to the curriculum of school and simulation. The theme of chapter (8), the concluding chapter, is Strategy, Logic, and Ethics. The chapter comes to place the ethical dilemma in the conceptual map we presented in the book. I discuss here the tension between strategy as a process of setting goals and deriving backward as it were the steps that are necessary for achieving these goals, to the inescapable dictation of the ethical imperative. This unavoidable tension is particularly important against the backdrop of contemporary culture that is guided by a vision of personal success and achievement that may threaten our ability to do what is right. Most of all, I am troubled in this chapter, with the role logic may play, first as a liberating force enabling a change that is tuned to the ethical imperative, and second, as a force designed to justify on logical grounds, as it were, the existing power structure. It is in this junction that brave individuals will be called to challenge the governing logic of the zeitgeist - logic that operates now as an oppressive force that serves the vested interest of beneficials of the existing order. When success is favored over doing what is right, as we must admit is the prevailing cultural mood of our times, that the words of Emanuel Kant carry their full weight: "Do the Right thing and leave the Consequences to God". For this we need heroes and heroines that challenge the time by bearing the full weight of responsibility that comes with the doing of what is Right. *** This book was written more than anything from the perspective of the heroes and heroines of action. Sometimes they may be missing the exact words or the full insight into the solid logical complexity that stands between them and the doing of what is right. They feel what is right to do on the battlefield of life, whether in real battle, in political or economic life or whether in the life of creativity and the quest of the intellect. They do need, in analogy, proper logical and intellectual artillery. I hope they will find the logical cannons they require as they travel through the pages of this book. The rest is in their hands at the Moment of Truth. -/- Doron Avital, Tel Aviv 2011 -/- . (shrink)

Richard Cross provides the first full study of Duns Scotus's theory of cognition, examining his account of the processes involved in cognition, from sensation, through intuition and abstraction, to conceptual thought. Cross places Scotus's thought clearly within the context of 13th-century study on the mind, and of his intellectual forebears.

I show that given Jonathan Bennett's theory of 'even if,' the following statement is logically true iff the principle of conditional excluded is valid: (SE) If Q and if P wouldn't rule out Q, then Q even if P. Hence whatever intuitions support the validity of (SE) support the validity of Conditional Excluded Middle, too. Finally I show that Bennett's objection to John Bigelow's theory of the conditional can be turned into a (perhaps) more telling one, viz. that on Bigelow's (...) theory 'if P then Q' and 'if P and Q then R' do not jointly entail 'if P then R'. (shrink)

European Journal of Philosophy, Volume 30, Issue 3, Page 1110-1124, September 2022.

The Christian doctrine of the Incarnation maintains that the second person of the Trinity became a human being, retaining all attributes necessary for being divine and gaining all attributes necessary for being human. As usually understood, the doctrine involves the claim that the second person of the Trinity is the subject of the attributes of Jesus Christ, the first-century Jew whose deeds are reported in various ways in the New Testament. The fundamental philosophical problem specific to the doctrine is this: (...) how is it that one and the same thing could be both divine and human. Most of the philosophical effort devoted to the specifics of the doctrine has centered on this issue, which is the focus of this article. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reply to Jiwei CiBen Cross (bio)I am very grateful to Jiwei Ci for his thoughtful and considerate response to my article. I think my understanding of Weber, legitimacy, political realism, and Ci's own work has benefited greatly as a result.I. What's the Problem with Performance Legitimacy?Ci and I both accept that legitimacy is best understood in terms of Weber's "descriptively de jure legitimacy." Put simply, we both think that (...) a state is legitimate if and only if a large enough number of its constituents think it is legitimate. According to this view, therefore, a source of legitimacy is any type of belief—or narrative, as many political realists tend to prefer—that constituents take to give them sufficient reason for regarding the state as legitimate.On the surface, understanding legitimacy in this way seems open to the possibility of a very wide variety of sources of legitimacy. All that matters is whether the source of legitimacy can persuade people to regard the state as legitimate. Weber's study of legitimacy focuses on three particular sources: [End Page 165] legal, traditional, and charismatic. However, as Ci (2024, pp. 149–150) rightly points out, Weber can allow for other sources of legitimacy as well, provided that they have the requisite persuasive power.1The question that Ci and I are interested in here is whether the state's good performance can be a viable source of legitimacy. In my view, it might be possible for a government to secure its legitimacy in this fashion if two conditions are met:(1). the state can reliably maintain good performance—or at least can reliably convince its constituents that it maintains good performance.(2). the state has a sufficiently large constituency of people who possess a shared interest that is served by good performance.I previously expressed doubts about whether the second condition can be reliably met (Cross 2024, pp. 139–143). Hence, I am somewhat skeptical about the possibility of performance legitimacy, but this skepticism does not stem from a categorical distinction between performance and legitimacy. Nor does this skepticism stem from doubts about the ability of any state to maintain good performance. Rather, it stems from doubts about whether the existence of a shared interest is a reliable basis for identifying the constituency of legitimacy.Ci, on the other hand, does think that good performance and legitimacy are categorically distinct, albeit interrelated. He holds that good performance can enhance an existing source of legitimacy. He also holds that it can offer some compensation for the absence of an existing source of legitimacy. However, he denies that good performance can ever count as a source of legitimacy in its own right. Furthermore, Ci's arguments for this claim, as I understand them, are not reducible to the further claim that it is unlikely that any state can reliably sustain the kind of unwaveringly good performance that reliably serves the shared interests of its constituency. That is, although he appears highly skeptical of the state's capacity to satisfy (1) and (2), this skepticism is not, strictly speaking, essential to his arguments.Our main disagreement, then, concerns the correct basis for skepticism about performance legitimacy. Although I think it would be fair to say that Ci's skepticism is stronger and more general than my own, this appears to stem from our different reasons for this skepticism. In what follows, I will try to briefly summarize Ci's objections to performance legitimacy and the extent of my disagreements with them.II. The Genealogical ArgumentI previously suggested that Ci's distinction between performance-enhancing legitimacy and performance that ameliorates a lack of legitimacy may be a distinction without a difference (Cross 2024, pp. 132–135). What reason do rulers have to prefer the former to the latter? [End Page 166]Ci's response helpfully clarifies what he takes to be at stake in this distinction. Legitimacy, he argues, is meant to provide a justification specifically for the state's monopoly of violence. As Ci rightly notes, I did not address the significance of this point. I did not distinguish between a justification for the state's monopoly of... (shrink)

[IMPORTANT CORRECTION - See end of abstract.] In Syntactical Treatments of Modality, with Corollaries on Reflexion Principles and Finite Axiomatizability, Acta Philosophica Fennica 16 (1963), 153–167, Richard Montague shows that the use of a single syntactic predicate (with a context-independent semantic value) to represent modalities of alethic necessity and idealized knowledge leads to inconsistency. In A Note on Syntactical Treatments of Modality, Synthese 44 (1980), 391–395, Richmond Thomason obtains a similar impossibility result for idealized belief: under a syntactical treatment of (...) belief, the assumption that idealized belief is deductively closed, together with certain other plausible conditions on idealized belief, imply that an ideal believer with consistent beliefs cannot believe the truth of Robinson's Arithmetic. In this essay I assert an impossibility result similar to Thomason's but which does not assume that belief is deductively closed or ideal in any other way. There are technical mistakes in the formulations of Lemmas 1-4 and Theorem 2, however. Corrections can be found in the Erratum section of the author's website. (shrink)

In this essay I present a new version of the Paradox of the Knower and show that this new paradox vitiates a certain argument against epistemic closure. I then prove a theorem that relates the new paradox to epistemological scepticism. I conclude by assessing the use of the Knower in arguments against syntactical treatments of knowledge.

In a recent article, J. Angelo Corlett criticises what he takes to be the ‘offensiphobic’ practices characteristic of many universities. The ‘offensiphobe’, according to Corlett, believes that offensive speech ought to be censured precisely because it offends. We argue that there are three serious problems with Corlett’s discussion. First, his criticism of ‘offensiphobia’ misrepresents the kinds of censorship practiced by universities; many universities may in some way censure speech which they regard as offensive, but this is seldom if ever a (...) manifestation of ‘offensiphobia’. Second, we attempt to reconstruct Corlett’s criticism of ‘offensiphobia’ as a criticism of the practice of censuring hate speech, and show that this argument is unsuccessful. Third, we offer some brief reflections on how labelling universities as ‘offensiphobic’ is especially problematic in light of the current climate of political interference in university research and teaching. (shrink)

Corporate gifts – from philanthropic donations to individual reward schemes – attract considerable attention from scholars for the kinds of moral, economic and political logics that motivate them. This article considers the gifts that transnational corporations give to producers and draws from Marilyn Strathern’s writings on exchange and personhood in order to reverse dominant analyses. Focused on the gifting of gold coins to industrial workers at a global manufacturing unit in India, it brings together field-based observations with a diverse field (...) of literature on the gift in anthropology. Against an analysis that sees the corporate gift harnessed directly to a corporate bottom line, this article proposes an alternative accounting that uses Strathern’s notions of ‘elicitation’, ‘revelation’ and ‘detachment’ to explore the contours of knowledge, personhood and relationality in the transaction. If corporate gifts have powerful effects, the article argues, it is because they establish difference between the person of the giver and the person of the recipient and because they materialize actions, desires and capacities that accrue to and transform the recipients rather than simply because they are vessels for the interests of global capital. As social theory confronts the political economy of corporate giving, Strathern’s writings prompt provocative questions about agency and power that challenge the hegemonic status of the modern corporation. (shrink)

Despite the results of David Lewis, Peter Gärdenfors, and others, showing that imaging and classical conditionalization coincide only in the most trivial probabilistic models of belief revision, it turns out that imaging on a proposition A can always be described via Popper function conditionalization on a proposition that entails A. This result generalizes to any method of belief revision meeting certain minimal requirements. The proof is illustrated by an application of imaging in the context of the Monty Hall Problem.

This article argues that Rawlsian liberal political institutions are incapable of ensuring that the basic welfare needs of the worst off are met. This argument consists of two steps. First, I show that institutions are incapable of ensuring that the basic needs of the worst off are met without pursuing certain non-taxation-based courses of action that are designed to alter the work choices of citizens. Second, I argue that such actions are not permissible for Rawlsian institutions. It follows that a (...) state governed by a Rawls-type political conception of justice is parasitic on the willingness of citizens to be sufficiently motivated by concern for the worst off, but it cannot require or even encourage them to be so motivated. (shrink)

Recent empirical literature suggests that some of the most prominent environmental policies that the Chinese government has pursued have involved at least some measure of participation from citizen...

The ‘consensus’ model of public reason, associated with John Rawls’s political liberalism, has been criticised for excluding certain reasons from receiving consideration where the justification of the constitutional essentials is concerned. One limitation of these criticisms is that they typically focus on the exclusion of reasons political liberals are committed to excluding, notably reasons based on religious and comprehensive views. I argue that public reason excludes some reasons, central to the interests of many oppressed groups, that public reason advocates will (...) agree should not be excluded.L’idée de raison publique comme «consensus», associée au libéralisme politique de John Rawls, a été attaquée du fait de sa supposée incapacité à permettre la prise en compte de certaines raisons dans les cas où la substance même de la constitution est en question. Une limitation de ces critiques est qu’elles se concentrent habituellement sur l’exclusion de ces raisons que les libéraux eux-mêmes s’efforcent d’exclure, en particulier des raisons fondées sur des vues globales ou religieuses. Je défends l’idée selon laquelle la raison publique exclut de fait certaines raisons, essentielles aux intérêts de nombreux groupes opprimés, que les tenants de la raison publique eux-mêmes considèreraient ne pas devoir être exclues. (shrink)

In this article I defend a new argument against moralist theories of legitimacy and in favour of realist theories. Moralist theories, I argue, are vulnerable to ideological and wishful thinking because they do not connect the demands of legitimacy with the interests of rulers. Realist theories, however, generally do manage to make this connection. This is because satisfying the usual realist criteria for legitimacy – the creation of a stable political order that transcends brute coercion – is usually necessary for (...) rulers to preserve their rule. (shrink)

I argue that, if it is thought desirable to avoid the collapse of disability into generic social disadvantage, it is necessary to draw a distinction between impairment (a bodily configuration) and disability (the way in which the environment prevents someone with an impairment from undertaking certain kinds of activities), as in social models of disability. I show how to draw such a distinction by utilizing a distinction between intrinsic and extrinsic properties. I argue further that, using this distinction, it is (...) possible to define ‘impairment’ in ways that do not appeal to notions of the normal, and to define ‘disability’ in terms of ‘impairment.’. (shrink)

Most realist theories of legitimacy are internalist theories, meaning that they regard legitimacy as a function of how subjects view their own rulers. However, some realists seek to qualify their internalism by holding that legitimacy is not simply a matter of whether subjects accept their rulers’ exercise of power. According to one such view, legitimacy requires that rulers’ power be ‘acceptable’ to subjects, in the sense that it can be justified on the basis of values that they accept. Call this (...) acceptability internalism. In this article, I argue that realists should reject acceptability internalism. I first argue that acceptability internalism has the disadvantage of separating the concept of legitimacy from the interests of rulers. I then consider two arguments in favour of acceptability internalism, and argue that both should be rejected. (shrink)