Humans possess two nonverbal systems capable of representing numbers, both limited in their representational power: the first one represents numbers in an approximate fashion, and the second one conveys information about small numbers only. Conception of exact large numbers has therefore been thought to arise from the manipulation of exact numerical symbols. Here, we focus on two fundamental properties of the exact numbers as prerequisites to the concept of EXACT NUMBERS : the fact that all numbers (...) can be generated by a successor function and the fact that equality between numbers can be defined in an exact fashion. We discuss some recent findings assessing how speakers of Munduruc (an Amazonian language), and young Western children (3-4 years old) understand these fundamental properties of numbers. (shrink)

Despite its computational elegancy, Jackendoff's proposal to reconcile competing approaches by postulating a parallel architecture for phonological, syntactic, and semantic modules is disappointing. We argue that it is a pragmatic version of the leading module which Jackendoff would probably prefer, but which he does not explicitly acknowledge. This internal conflict leads to several shortcomings and even distortions of information presented in the book.

Although thinkers of the past might have started from presumptions of fundamental difference and inequality between (say) the genders, or people of different races, this is no longer the case. At least in mainstream political philosophy, we are all now presumed to be, in some fundamental sense, basic equals. Of course, what follows from this putative fact of basic equality remains enormously controversial: liberals, libertarians, conservatives, Marxists, republicans, and so on, continue to disagree vigorously with each other, despite all (...) presupposing some kind of basic equality. They may argue about who gets what, how much, and why, but the starting point - that all people are in some sense deserving of prima facie equal consideration - has become an axiom of our moral and political thinking. But why? Why are we basic equals? The trouble is that as soon as one asks for an explanation of this foundational premise, it begins to look shaky. After all, on any conceivable metric human beings are notably unequal, and often to striking degrees. Philosophers in this area tend to talk of equal worth, but often without trying to specify what exactly that means. Philosophically, basic equality is neither acceptable nor rejectable. It is not rejectable because we appear to be, as a matter of fact, profoundly committed to the truth of the claim that we are all one another's basic equals. But to the extent that there appear to be no philosophical arguments for believing in the truth of basic equality, it is not acceptable either. The aim of this book is to try and show why basic equality is acceptable. To do so, however, it will also contend that we need to approach the question rather differently to how it has mostly been handled so far. What's required is an exercise in what Bernard Williams called 'impure philosophy': it must include insights from other areas of human intellectual endeavour, whether psychology and history or even what we learn from how we go about practicing basic equality in our collective lives. (shrink)

Understanding what numbers are means knowing several things. It means knowing how counting relates to numbers (called the cardinal principle or cardinality); it means knowing that each number is generated by adding one to the previous number (called the successor function or succession), and it means knowing that all and only sets whose members can be placed in one-to-one correspondence have the same number of items (called exact equality or equinumerosity). A previous study (Sarnecka & Carey, 2008) linked (...) children's understanding of cardinality to their understanding of succession for the numbers five and six. This study investigates the link between cardinality and equinumerosity for these numbers, finding that children either understand both cardinality and equinumerosity or they understand neither. This suggests that cardinality and equinumerosity (along with succession) are interrelated facets of the concepts five and six, the acquisition of which is an important conceptual achievement of early childhood. (shrink)

Egalitarians think that equality in the distribution of goods somehow matters. But what exactly is egalitarianism? This article argues for a characterization based on novel principles essentially involving risk. The characterization is then used to resolve disputed questions about egalitarianism. These include: the way egalitarianism is concerned with patterns, in particular its relationship to strong separability; the relationship between egalitarianism and other distributive views, such as concerns with fairness and with giving priority to the worse off; and the relationship (...) between egalitarianism and evaluative measurement. But egalitarianism is subject to a particularly severe form of the levelling-down objection, and is claimed to be false. (shrink)

This article focuses on how young children acquire concepts for exact, cardinal numbers. I believe that exact numbers are a conceptual structure that was invented by people, and that most children acquire gradually, over a period of months or years during early childhood. This article reviews studies that explore children’s number knowledge at various points during this acquisition process. Most of these studies were done in my own lab, and assume the theoretical framework proposed by Carey. In this (...) framework, the counting list and the counting routine form a placeholder structure. Over time, the placeholder structure is gradually filled in with meaning to become a conceptual structure that allows the child to represent exact numbers A number system is a socially shared, structured set of symbols that pose a learning challenge for children. But once children have acquired a number system, it allows them to represent information that they had no way of representing before. (shrink)

Social inequality may obstruct the generation of knowledge, as the rich and powerful may bring about social acceptance of skewed views that suit their interests. Epistemic equality in the context of justification is a means of preventing such obstruction. Drawing on social epistemology and theories of equality and distributive justice, we provide an account of epistemic equality. We regard participation in, and influence over a knowledge-generating discourse in an epistemic community as a limited good that needs to (...) be justly distributed among putative members of the community. We argue that rather than trying to operationally formulate an exact criterion for distributing this good, epistemic equality may be realized by insisting on active participation of members of three groups in addition to credited experts: relevant disempowered groups, relevant uncredited experts, and relevant stakeholders. Meeting these conditions fulfills the political, moral, and epistemic aims of epistemic equality. (shrink)

In an attempt to give a unified account of common properties of various resource bounded reducibilities, we introduce conditions on a binary relation ≤r between subsets of the natural numbers, where ≤r is meant as a resource bounded reducibility. The conditions are a formalization of basic features shared by most resource bounded reducibilities which can be found in the literature. As our main technical result, we show that these conditions imply a result about exact pairs which has been previously (...) shown by Ambos-Spies [2] in a setting of polynomial time bounds: given some recursively presentable ≤r-ideal I and some recursive ≤r-hard set B for I which is not contained in I, there is some recursive set C, where B and C are an exact pair for I, that is, I is equal to the intersection of the lower ≤r-cones of B and C, where C is not in I. In particular, if the relation ≤r is in addition transitive and there are least sets, then every recursive set which is not in the least degree is half of a minimal pair of recursive sets. (shrink)

Equality and identity. Bulletin of Symbolic Logic. 19 (2013) 255-6. (Coauthor: Anthony Ramnauth) Also see https://www.academia.edu/s/a6bf02aaab This article uses ‘equals’ [‘is equal to’] and ‘is’ [‘is identical to’, ‘is one and the same as’] as they are used in ordinary exact English. In a logically perfect language the oxymoron ‘the numbers 3 and 2+1 are the same number’ could not be said. Likewise, ‘the number 3 and the number 2+1 are one number’ is just as bad from a (...) logical point of view. In normal English these two sentences are idiomatically taken to express the true proposition that ‘the number 3 is the number 2+1’. Another idiomatic convention that interferes with clarity about equality and identity occurs in discussion of numbers: it is usual to write ‘3 equals 2+1’ when “3 is 2+1” is meant. When ‘3 equals 2+1’ is written there is a suggestion that 3 is not exactly the same number as 2+1 but that they merely have the same value. This becomes clear when we say that two of the sides of a triangle are equal if the two angles they subtend are equal or have the same measure. -/- Acknowledgements: Robert Barnes, Mark Brown, Jack Foran, Ivor Grattan-Guinness, Forest Hansen, David Hitchcock, Spaulding Hoffman, Calvin Jongsma, Justin Legault, Joaquin Miller, Tania Miller, and Wyman Park. -/- ► JOHN CORCORAN AND ANTHONY RAMNAUTH, Equality and identity. Philosophy, University at Buffalo, Buffalo, NY 14260-4150, USA E-mail: [email protected] The two halves of one line are equal but not identical [one and the same]. Otherwise the line would have only one half! Every line equals infinitely many other lines, but no line is [identical to] any other line—taking ‘identical’ strictly here and below. Knowing that two lines equaling a third are equal is useful; the condition “two lines equaling a third” often holds. In fact any two sides of an equilateral triangle is equal to the remaining side! But could knowing that two lines being [identical to] a third are identical be useful? The antecedent condition “two things identical to a third” never holds, nor does the consequent condition “two things being identical”. If two things were identical to a third, they would be the third and thus not be two things but only one. The plural predicate ‘are equal’ as in ‘All diameters of a given circle are equal’ is useful and natural. ‘Are identical’ as in ‘All centers of a given circle are identical’ is awkward or worse; it suggests that a circle has multiple centers. Substituting equals for equals [replacing one of two equals by the other] makes sense. Substituting identicals for identicals is empty—a thing is identical only to itself; substituting one thing for itself leaves that thing alone, does nothing. There are as many types of equality as magnitudes: angles, lines, planes, solids, times, etc. Each admits unit magnitudes. And each such equality analyzes as identity of magnitude: two lines are equal [in length] if the one’s length is identical to the other’s. Tarski [1] hardly mentioned equality-identity distinctions (pp. 54-63). His discussion begins: -/- Among the logical concepts […], the concept of IDENTITY or EQUALITY […] has the greatest importance. -/- Not until page 62 is there an equality-identity distinction. His only “notion of equality”, if such it is, is geometrical congruence—having the same size and shape—an equivalence relation not admitting any unit. Does anyone but Tarski ever say ‘this triangle is equal to that’ to mean that the first is congruent to that? What would motivate him to say such a thing? This lecture treats the history and philosophy of equality-identity distinctions. [1] ALFRED TARSKI, Introduction to Logic, Dover, New York, 1995. [This is expanded from the printed abstract.] . (shrink)

In this paper we steer a course between two views of the value of equality that are usually understood as diametrically opposed to one another: on the one hand, the view that equality has intrinsic value; on the other, the view that equality is a normatively redundant notion. We proceed by analysing the different ways in which the equal possession of certain relevant properties justifies distributive equality. We then present an account of ‘basic equality’ that (...) serves to single out where this equal possession is really doing normative work. Our account departs from some common assumptions about basic equality but is nevertheless plausible in as much as it assigns to that concept a precise and normatively significant role in grounding the non-instrumental value of distributive equality. By showing when, exactly, equality is basic, we are able to explain both what is right and what is wrong in each of the two opposed views of the value of equality. The argument proceeds at a fairly high level of abstraction and is therefore applicable to a number of rival egalitarian theories. (shrink)

This essay took one of the “externalities” of education, equality of opportunity, and suggested that the more extensive state intervention argued for by “the left” was not needed for its provision. This was argued by saying that what “the left” wanted from equality of opportunity was not equality as such, but merely “adequate resources” for all. An “adequate” education it was suggested, could be provided without extensive state intervention for the great majority of children. Only for those (...) children who were too poor, or whose parents were too irresponsible, would it be necessary for the state to intervene to ensure the provision of education, and then only to ensure that these children had access to sufficient funds to enable them to share in the educational experiences provided for other children. Several details in this argument need to be worked out in more detail, including the important matter of exactly what an “adequate education” might comprise. However, if these details could be worked out satisfactorily, the argument could serve to challenge the left's anxieties about moves towards a market in educational provision. The argument suggests that such moves need not lead to the sacrifice of those externalities which “the left” holds dear. (shrink)

Alan Patten | : Equal Recognition seeks to restate the case in favour of liberal multiculturalism in a manner that is responsive to major objections that have been advanced by critics in recent years. The book engages, among other questions, with two central unresolved problems. First, how should ideas of culture and cultural preservation be understood, given widespread suspicion that these ideas rely on an unavowed, but objectionable, form of essentialism? And, second, what exactly is the normative basis of cultural (...) rights claims, and what are the limits on those claims? My four commentators advance a variety of different criticisms of the book’s answers to these questions. I offer replies to each of their main challenges. | : Equal Recognition réaffirme l’argument en faveur du multiculturalisme libéral de façon à répondre aux objections majeures qui lui ont été adressées ces dernières années. Le livre se penche, entre autres, sur deux problèmes centraux, non résolus. Premièrement, comment les idées de culture et de préservation culturelle devraient être comprises, étant donné la suspicion répandue selon laquelle ces idées se fondent sur une forme d’essentialisme aussi inavoué que répréhensible? Deuxièmement, quelle est exactement la base normative des requêtes de droits culturels, et quelles sont les limites à ces requêtes? Mes quatre commentateurs avancent une variété de critiques différentes quant aux réponses que ce livre apporte à ces questions. Je réponds ici à chacune d’elles. (shrink)

The concepts of freedom and equality lie at the heart of much contemporary political debate. But how, exactly, are these concepts to be understood? And do they really represent desirable political values? Norman begins from the premise that freedom and equality are rooted in human experience, and thus have a real and objective content. He then argues that the attempt to clarify these concepts is therefore not just a matter of idle philosophical speculation, but also a matter of (...) practical politics, for the philosophical conclusions we reach also have important implications for the day-to-day world of political action. Touching on the work of such influential thinkers as Mill, Berlin, Hayek, Nozick, Rawls, and Williams, this book serves as a valuable introduction to the central issues in political philosophy. (shrink)

Some argue that same-sex marriage is not an equal rights issue because, where same-sex marriage is illegal, heterosexuals and homosexuals have the exact same right to marry—i.e., the right to marry one adult of the opposite sex. I dispute this argument by pointing out that while societies that prohibit same-sex marriage equally permit individual heterosexuals and homosexuals to marry one adult of the opposite sex, same-sex couples in such societies are denied an important right that opposite-sex couples enjoy—i.e., the (...) right to marry. I argue that the right to marry is fundamentally, not an individual right, but a couple’s collective right, analogous to assembly rights. (shrink)

How, exactly, must we strike the balance between security and equality? Must we insist, out of respect for the equality of persons, that the police refrain from using ethnic profiling and opt for some other strategy in their pursuit of terrorists, or must we allow the police to continue with this policy, which seems to sacrifice equality for the sake of security? This paper assesses the ethical status of ethnic profiling from the perspective of the ideal of (...) equality. The paper shows how the ethical status of ethnic profiling changes depending on how exactly we specify the egalitarian ideal. Furthermore, it argues that on a plausible interpretation of the ideal of equality, ethnic profiling is not in principle objectionable. (shrink)

Historically, critics of capitalism have had a great deal to say about the defects and social ills that afflict capitalist society and correspondingly little to say about how alternative institutional arrangements might solve these problems. One can only speculate about why this has been so. One reason might be a simple matter of priorities. Bertolt Brecht once said that when a man's house is on fire, one does not inquire too closely into alternative arrangements for shelter. The analogy between capitalism (...) and a burning house may seem overwrought today, but in the dark days of the Depression of the 1930s it probably seemed more apt. Another explanation for the scant attention paid to alternatives to capitalism has to do with both the factual and ideological beliefs of capitalism's critics. If one believes that the existing order would be destroyed by a mass movement, that new institutions would be constructed by the people in a democratic spirit, and that furthermore all of this would be a good thing, it would be unwise and counterproductive to try to spell out exactly where history is headed. After all, a genuine mass movement has little use for self-proclaimed prophets of history. Finally, men and women of modest intellectual pretensions might be humbled by the prospects of trying to spell out in any detail social institutions that should exist or might exist but are not as yet found anywhere in the world. (shrink)

The principle that each individual is entitled to an equal opportunity to benefit from any public health care system, and that this entitlement is proportionate neither to the size of their chance of benefitting, nor to the quality of the benefit, nor to the length of lifetime remaining in which that benefit may be enjoyed, runs counter to most current thinking about the allocation of resources for health care. It is my contention that any system of prioritisation of the resources (...) available for healthcare or of rationing such resources must be governed by this principle. -/- This can have apparently paradoxical conclusions in that it can seem wasteful to give someone with a very slim chance of a lifesaving treatment the same priority as someone with a much better chance. In an important and thoughtful recent paper, Julian Savulescu has concentrated on this apparent weakness and has argued for a particular conception of the good or benefit to be achieved by a healthcare system which purports to demonstrate the inadequacies of an equal opportunities approach to prioritisation and to replace it with an altogether better account. This paper will show that a rational ‘reasons based consequentialism’ is more in line with the equal opportunities approach, which I defended some time ago in these pages, than with that of Savulescu. I shall then examine more closely the conception of equal opportunities in health care and show that if we give weight to an individual's reasons, and what is expected to be good for them, we will opt for exactly the equality based account of distributive justice that I have recommended. (shrink)

It has been argued that there is a genuine conflict between the views of geometry defended by Hume in the Treatise and in the Enquiry: while the former work attributes to geometry a different status from that of arithmetic and algebra, the latter attempts to restore its status as an exact and certain science. A closer reading of Hume shows that, in fact, there is no conflict between the two works with respect to geometry. The key to understanding Hume's (...) view of geometry is the distinction he draws between two standards of equality in extension. (shrink)

The four-valued semantics of Belnap–Dunn logic, consisting of the truth values True, False, Neither, and Both, gives rise to several nonclassical logics depending on which feature of propositions we wish to preserve: truth, nonfalsity, or exact truth. Interpreting equality of truth values in this semantics as material equivalence of propositions, we can moreover see the equational consequence relation of this four-element algebra as a logic of material equivalence. In this paper, we axiomatize all combinations of these four-valued logics, (...) for example, the logic of truth and exact truth or the logic of truth and material equivalence. These combined systems are consequence relations which allow us to express implications involving more than one of these features of propositions. (shrink)

Where exactly should we place Adam Smith in the cannon of classical liberalism? Smith's advocacy of free market economics and defence of religious liberty in The Wealth of Nations suffice for including him somewhere in that tradition.1 The nature and extent of Smith's liberalism, however, remain up for debate. One recent trend has been to characterise Smith as a proponent of social liberalism. This includes those like Stephen Darwall, Samuel Fleischacker and Charles Griswold, who have drawn attention to a kind (...) of descriptive moral egalitarianism in Smith.2 Humans, Smith seems to hold, are naturally disposed to valuing one another under a conception of equality. But that is not all these scholars suggest. They have also hinted at something more contentious ? the idea that, according to Smith, we value one another in a way resonant with contemporary notions of human dignity, conceived as the inherent value of persons grounding certain rights to, or restrictions on, treatment by others.3 In saying so, these scholars have hit upon something remarkable. However, I also think their arguments in this respect are both indirect and incomplete. Consequently, the full import of Smith's view remains obscure. This essay aims to bring some clarity. 1I intend this historically. I grant there are good reasons to be sceptical about the ultimate fate of liberty in capitalist society (e.g. Marxist reasons and reasons based on various postmodern critiques of enlightenment ideology). Also, the designation ?free market? should be understood loosely, as most scholars now agree it is a mistake to identify Smith with thoroughgoing laissez-faire economics. 2Darwall, S., ?Sympathetic Liberalism: Recent Work on Adam Smith?, Philosophy Fleischacker, S., A Third Concept of Liberty (Princeton: Princeton University Press, 1999) and On Adam Smith's Wealth of Nations (Princeton: Princeton University Press, 2004); and Griswold, C., Adam Smith and the Virtues of Enlightenment (New York: Cambridge University Press, 1999). Other major commentators holding some version of this view might include Raphael, D. D. The Impartial Spectator (New York: Oxford University Press, 2007) and Vivienne Brown, Adam Smith's Discourse: Canonicity, Commerce and Conscience (London: Routledge, 1994). 3See e.g. Fleischacker (2004), 205; Darwall at 142, 156 and Griswold at 235?239. However, one must read Fleischacker carefully, for he also uses the adjectival ?dignified? to express Smith's concern with what is ?honourable? or ?respectable? about persons, which use does not obviously match up with the notion of inherent value (see e.g. p. 207). Darwall's argument includes by far the most explicit discussion of ?dignity? as I've defined it. But as Darwall's article is ostensibly a book review (albeit a substantive one that addresses three books at once, including Griswold's), it cannot be called a direct inquiry. Griswold never explicitly puts his interpretation in terms of ?dignity?, but that is clearly what he is after. Thus Darwall also reads him that way. (shrink)

T.H. Green argues that rights cannot exist, save for in a society in which people recognise each other as “ ἴσοι καὶ ὅμοιοι.” Green leaves the phrase untranslated, and there is a certain ambiguity about what exactly Green meant and a lack of attention to this stipulation in the secondary literature. This essay argues that equality and sameness in Green must be understood as two categorically different criteria, both of which must be satisfied in order to make rights recognition (...) possible. Further, this essay examines a tension this distinction between equality and sameness reveals in Green’s work: Green’s commitment to patriotism in his Lectures on the Principles of Political Obligation may seem at first to be at odds with his more cosmopolitan tendencies in the Prolegomena to Ethics. However, a way of resolving this tension is suggested: re-conceiving what patriotism and “sameness” means by reading Green’s work as less dependent on the concept of “nation” than that of subsequent British idealists such as Bernard Bosanquet demonstrates that sameness and equality, though different, complement each other. (shrink)

Democracy has been justified as the political system whose citizens are sovereign, which is to say most free or most equal in their political experience, participation or consent, and most likely to be benefited by economic freedoms. Most importantly, democracy is recommended as that form of government which gets things more right than any other form of government. But this traditional view, and also more recent qualifications of this view, is simply inadequate, refuted and rendered nonsensical by very real electoral, (...) wealth, income and power inequalities in democratic societies. Nevertheless, it is this kind of hierarchic democracy, like those of the United States and the United Kingdom, whose systems of government are exactly not true to the idea that two heads are better than one and more heads better than two, which reaches to judgements about Palestine, 9/11, Iraq, 7/7 and about all that is to come after those things. (shrink)

Democracy has been justified as the political system whose citizens are sovereign, which is to say most free or most equal in their political experience, participation or consent, and most likely to be benefited by economic freedoms. Most importantly, democracy is recommended as that form of government which gets things more right than any other form of government. But this traditional view, and also more recent qualifications of this view, is simply inadequate, refuted and rendered nonsensical by very real electoral, (...) wealth, income and power inequalities in democratic societies. Nevertheless, it is this kind of hierarchic democracy, like those of the United States and the United Kingdom, whose systems of government are exactly not true to the idea that two heads are better than one and more heads better than two, which reaches to judgements about Palestine, 9/11, Iraq, 7/7 and about all that is to come after those things. (shrink)

Tom Regan's seafaring dog that is justifiably thrown out of the lifeboat built for four to save the lives of four humans has been the topic of much discussion. Critics have argued in a variety of ways that this dog nips at Regan's Achilles heel. Without reviewing previous discussions, with much of which I certainly agree, this article develops an unexplored approach to exposing the vulnerability of the position that Regan takes on sacrificing the dog to save the humans. It (...) argues that when dealing with the seafaring dog, Regan abandons his own principles, and that this is exactly what he should do. Regan should abandon his view that all subjects-of-a-life have equal inherent worth. (shrink)

In a previous paper we have shown that in quantum chromodynamics the gluon propagator vanishes in the infrared limit, while the ghost propagator is more singular than a simple pole. These results were obtained after angular averaging, but in the current paper we go beyond this approximation and perform an exact calculation of the angular integrals. The powers of the infrared behaviour of the propagators are changed substantially. We find the very intriguing result that the gluon propagator vanishes in (...) the infrared exactly like p2, whilst the ghost propagator is exactly as singular as 1/p4. We also find that the value of the infrared fixed point of the QCD coupling is much decreased from the y-max estimate: it is now equal to 4π/3. Following a recent study by von Smekal et al.[1], we analyzed in Ref.[2] the coupled Dyson- Schwinger equations for the gluon and ghost form factors F and G. The approximations were two-fold: firstly the vertices were taken bare, and secondly angular averaging was introduced (the so-called y-max approximation). Deferring to later work an improvement of the vertices. (shrink)

This is a defense of Rawls against recent criticism, ironically my own, though it is also a critique insofar as it addresses a problem that Rawls never does. As a defense, it is not a retraction of the original charges. As a critique, it is not more of the same op-position. In either capacity, it is not an afterthought. The charges were conceived from the outset with a specific solution in mind, which would have been too distracting to pursue in (...) the same article. This is that solution. It also highlights the problem. The original charges were that Rawls’s decision procedure for ethics does not justify his own moral principles, namely his principles of justice, and that the underlying problem may well keep the decision procedure from justifying any moral principles whatsoever, or at least any normatively useful ones. The underlying problem was, and still is, the model’s inherent universalism, which is built into the decision procedure through design specifications precluding relativism, yet only at the cost of limiting the relevant moral principles to generalities that are already widely accepted, thereby render-ing the procedure at best redundant and very likely vacuous as an ethical justification model. These difficulties are manifested in the work of Rawls as the dogmatism of champi-oning a distinctive conception of justice, a liberal one as he himself calls it, through a justification model that is too universalistic to permit such a bias and possibly also too universalistic to permit any substantive conclusions at all. The solution contemplated here is to position the decision procedure as a dynamic justification model responsive to moral progress, as opposed to a static one indifferent to such progress and equally open to all moral input, thus removing the inconsistency between the universalistic design and any distinctive or controversial principles, including the ones Rawls himself recommends, so long as they are consistent with moral progress. (shrink)

We show that the usual axiom system of quasi polyadic equality algebras is strongly redundant. Then, so called non‐commutative quasi‐polyadic equality algebras are introduced (), in which, among others, the commutativity of cylindrifications is dropped. As is known, quasi‐polyadic equality algebras are not representable in the classical sense, but we prove that algebras in are representable by quasi‐polyadic relativized set algebras, or more exactly by algebras in.

At least since Benjamin Constant gave a speech on the subject in 1819 at the Athenee Royal in Paris, there has been occasional debate over the exact character of ancient democracy. This debate lives on today in a spirited and lively exchange going on largely among ancient historians over the character of Athenian democracy, particularly on its political and theoretical articulations. The purpose of this paper is to investigate two specific aspects of this debate, namely the understanding Athenian citizens (...) - and particularly democratically inclined Athenian citizens - had of equality and liberty of speech, and their linkage in the Athenian mind. It is argued that the Athenian notion of equality and free speech were linked with liberty, and are not entirely dissimilar to modern understandings of the same concepts. (shrink)

It is often supposed that, given two potential objects of choice X and Y, a specific set of circumstances, and a specific choosing agent, one of the following must be true: (1) opting for X is a better choice than opting for Y, (2) opting for Y is a better choice than opting for X, or (3) opting for X and opting for Y are exactly equally good choices. My aim in this paper is to show how some philosophical insights (...) concerning color perception can illuminate the possibility of two options, X and Y, being such that, even given a specific set of circumstances and a specific choosing agent, none of (1), (2), or (3) holds, but X and Y are positively related to one another as “on a par.”. (shrink)

This book argues that it can be both reasonable and appropriate to adopt a certain kind of misanthropy. The author defends a cognitivist version of misanthropy, an attitude whose central feature is the judgment that humanity is morally bad. Misanthropy is often dismissed on moral grounds. Many people hold that malice toward human persons is problematic and vulnerable to moral objections. In this book, the author advocates for cognitivist misanthropy. He defends an Asymmetry Thesis, according to which a morally bad (...) deed carries more weight than a morally good deed, even if the harm of the former is exactly equal to the benefit of the latter. He makes the case that being misanthropic in the cognitivist sense is morally permissible and compatible with a broad range of moral reasons for action. He also considers the role of misanthropy in environmental thought, arguing that charges of misanthropy against certain "non-anthropocentric" views do not have the force they are typically thought to carry. Finally, the author investigates the practical implications of adopting cognitivist misanthropy, asking what living with such an attitude would involve. A Philosophical Defense of Misanthropy will appeal to researchers and advanced students working in ethics and the philosophy of human nature. (shrink)

This chapter gives a detailed study of diagram-based reasoning in Euclidean plane geometry (Books I, III), as well as an exploration how to characterise a geometric practice. First, an account is given of diagram attribution: basic geometrical claims are classified as exact (equalities, proportionalities) or co-exact (containments, contiguities); exact claims may only be inferred from prior entries in the demonstration text, but co-exact claims may be asserted based on what is seen in the diagram. Diagram control (...) by constructions is necessary for this to work. Case-branching occurs when a construction renders a diagram un-representative. The roles of diagrams in reductio arguments, and of objection in probing a demonstration, are discussed. (shrink)

Four important arguments for probabilism—the Dutch Book, representation theorem, calibration, and gradational accuracy arguments—have a strikingly similar structure. Each begins with a mathematical theorem, a conditional with an existentially quantified consequent, of the general form: if your credences are not probabilities, then there is a way in which your rationality is impugned. Each argument concludes that rationality requires your credences to be probabilities. I contend that each argument is invalid as formulated. In each case there is a mirror-image theorem and (...) a corresponding argument of exactly equal strength that concludes that rationality requires your credences not to be probabilities. Some further consideration is needed to break this symmetry in favour of probabilism. (shrink)

Four important arguments for probabilism—the Dutch Book, representation theorem, calibration, and gradational accuracy arguments—have a strikingly similar structure. Each begins with a mathematical theorem, a conditional with an existentially quantified consequent, of the general form: if your credences are not probabilities, then there is a way in which your rationality is impugned.Each argument concludes that rationality requires your credences to be probabilities.I contend that each argument is invalid as formulated. In each case there is a mirror-image theorem and a corresponding (...) argument of exactly equal strength that concludes that rationality requires your credences not to be probabilities. Some further consideration is needed to break this symmetry in favour of probabilism. I discuss the extent to which the original arguments can be buttressed.1. Introduction2. The Dutch Book Argument2.1. Saving the Dutch Book argument2.2. ‘The Dutch Book argument merely dramatizes an inconsistency in the attitudes of an agent whose credences violate probability theory’3. Representation Theorem-based Arguments4. The Calibration Argument5. The Gradational Accuracy Argument6. Conclusion. (shrink)

Number concepts must support arithmetic inference. Using this principle, it can be argued that the integer concept of exactly ONE is a necessary part of the psychological foundations of number, as is the notion of the exact equality - that is, perfect substitutability. The inability to support reasoning involving exact equality is a shortcoming in current theories about the development of numerical reasoning. A simple innate basis for the natural number concepts can be proposed that embodies (...) the arithmetic principle, supports exact equality and also enables computational compatibility with real- or rational-valued mental magnitudes. (shrink)

James Griffin (1986, 1997, 2000) and Ruth Chang (1997) have argued that alternatives (and values) can be comparable when it is neither true that one is better than the other, nor true that they are exactly equal in value. The relation which holds between them has gone under various names: the alternatives are (Griffin) or (Chang). In this paper, I give a formal analysis of this relation. This analysis allows us to distinguish between two slightly different notions of . It (...) is argued that the distinction between these notions is important for discussions of rationality, as is the distinction between or and incomparability. (shrink)