We reexamine some of the classic problems connected with the use of cardinal utility functions in decision theory, and discuss Patrick Suppes's contributions to this field in light of a reinterpretation we propose for these problems. We analytically decompose the doctrine of ordinalism, which only accepts ordinal utility functions, and dis- tinguish between several doctrines of cardinalism, depending on what components of ordinalism they specifically reject. We identify Suppes's doctrine with the major deviation from ordinalism that conceives of utility functions (...) as representing preference di¤erences, while being non- etheless empirically related to choices. We highlight the originality, promises and limits of this choice-based cardinalism. (shrink)
Creationism about fictional entities requires a principle connecting what fictions say exist with which fictional entities really exist. The most natural way of spelling out such a principle yields inconsistent verdicts about how many fictional entities are generated by certain inconsistent fictions. Avoiding inconsistency without compromising the attractions of creationism will not be easy.
As a representative of the papacy Bellarmine was an extremely moderate one. In fact Sixtus V in 1590 had the first volume of his Disputations placed on the Index because it contained so cautious a theory of papal power, denying the Pope temporal hegemony. Bellarmine did not represent all that Hobbes required of him either. On the contrary, he proved the argument of those who championed the temporal powers of the Pope faulty. As a Jesuit he tended to maintain the (...) relative autonomy of the state, denying the temporal powers ascribed by radical papalists and Augustinians. Their argument was generally framed as a syllogism: Christ, who possessed direct temporal power as both God and man, exercised it on earth; the Pope is the vicar of Christ; therefore the Pope possesses and may exercise direct temporal jurisdiction. Bellarmine simply denied that Christ had exercised the temporal power, which as God, it is true, he possessed. Moreover, he drew up and circulated a list of patristic passages collected under the title De Regno Christi quale sit, to prove to the Pope the orthodoxy of his position. (shrink)
Permissivist metaontology proposes answering customary existence questions in the affirmative. Many of the existence questions addressed by ontologists concern the existence of theoretical entities which admit precise formal specification. This causes trouble for the permissivist, since individually consistent formal theories can make pairwise inconsistent demands on the cardinality of the universe. We deploy a result of Gabriel Uzquiano’s to show that this possibility is realised in the case of two prominent existence debates and propose rejecting permissivism in favour of (...) substantive ontology conducted on a cost–benefit basis. (shrink)
The cardinal role that notions of respect and self-respect play in Rawls’s A Theory of Justice has already been abundantly examined in the literature. However, it has hardly been noticed that these notions are also central for Michael Walzer’s Spheres of Justice. Respect and self-respect are not only central topics of his chapter on “recognition”, but constitute a central aim of his whole theory of justice. This paper substantiates this thesis and elucidates Walzer’s criticism of Rawls’s that we need to (...) distinguish between “self-respect” and “self-esteem”. (shrink)
The cardinal role that notions of respect and self-respect play in Rawls’s A Theory of Justice has already been abundantly examined in the literature. In contrast, it has hardly been noticed that these notions are also central to Michael Walzer’s Spheres of Justice. Respect and self-respect are not only central topics of his chapter “Recognition”, but constitute a central aim of a “complex egalitarian society” and of Walzer’s theory of justice. This paper substantiates this thesis and elucidates Walzer’s criticism of (...) Rawls that we need to distinguish between “self-respect” and “self-esteem”. (shrink)
In this article, a possible generalization of the Löb’s theorem is considered. Main result is: let κ be an inaccessible cardinal, then ¬Con( ZFC +∃κ) .
The independence phenomenon in set theory, while pervasive, can be partially addressed through the use of large cardinal axioms. A commonly assumed idea is that large cardinal axioms are species of maximality principles. In this paper, I argue that whether or not large cardinal axioms count as maximality principles depends on prior commitments concerning the richness of the subset forming operation. In particular I argue that there is a conception of maximality through absoluteness, on which large cardinal axioms are restrictive. (...) I argue, however, that large cardinals are still important axioms of set theory and can play many of their usual foundational roles. (shrink)
An overview of what Frege accomplishes in Part II of Grundgesetze, which contains proofs of axioms for arithmetic and several additional results concerning the finite, the infinite, and the relationship between these notions. One might think of this paper as an extremely compressed form of Part II of my book Reading Frege's Grundgesetze.
There are long-standing doubts about whether data from subjective scales—for instance, self-reports of happiness—are cardinally comparable. It is unclear how to assess whether these doubts are justified without first addressing two unresolved theoretical questions: how do people interpret subjective scales? Which assumptions are required for cardinal comparability? This paper offers answers to both. It proposes an explanation for scale interpretation derived from philosophy of language and game theory. In short: conversation is a cooperative endeavour governed by various maxims (Grice 1989); (...) because subjective scales are vague and individuals want to make themselves understood, scale interpretation is a search for a focal point (Schelling 1960). A specific focal point it hypothesised; if this hypothesis is correct, subjective data will be cardinally comparable. Four individually necessary and jointly sufficient conditions for cardinal comparability are specified. The paper then argues this hypothesis can be empirically be tested, makes an initial attempt to do so using subjective well-being data, and concludes it is supported. Numerous areas for further research are identified including, at the end of the paper, how certain tests could be used to ‘correct’ subjective data if they are not cardinal. (shrink)
In this article we derived an important example of the inconsistent countable set in second order ZFC (ZFC_2) with the full second-order semantics. Main results: (i) :~Con(ZFC2_); (ii) let k be an inaccessible cardinal, V is an standard model of ZFC (ZFC_2) and H_k is a set of all sets having hereditary size less then k; then : ~Con(ZFC + E(V)(V = Hk)):.
This article reintroduces Fr. Zeferino González, OP (1831-1894) to scholars of Church history, philosophy, and cultural heritage. He was an alumnus of the University of Santo Tomás in Manila, a Cardinal, and a champion of the revival of Catholic Philosophy that led to the promulgation of Leo XIII’s encyclical Aeterni Patris. Specifically, this essay presents, firstly, the Cardinal’s biography in the context of his experience as a missionary in the Far East; secondly, the intellectual tradition in Santo Tomás in Manila, (...) which he carried with him until his death; and lastly, some reasons for his once-radiant memory to slip into an undeserved forgetfulness. (shrink)
An analysis of Cardinal Joseph Ratzinger's statements regarding relativism in his 2005 homily to the conclave meeting to elect the new pope in the context of the charge of "relativism" in 20th-century philosophy. Parts of this essay are adapted from Barbara Herrnstein Smith,"Pre-Post-Modern Relativism," in *Scandalous Knowledge: Science, Truth and the Human* (Edinburgh: Edinburgh University Press, 2005; Durham, NC: Duke University Press, 2006), 18 – 45.
According to the so-called strong variant of Composition as Identity (CAI), the Principle of Indiscernibility of Identicals can be extended to composition, by resorting to broadly Fregean relativizations of cardinality ascriptions. In this paper we analyze various ways in which this relativization could be achieved. According to one broad variety of relativization, cardinality ascriptions are about objects, while concepts occupy an additional argument place. It should be possible to paraphrase the cardinality ascriptions in plural logic and, as (...) a consequence, relative counting requires the relativization either of quantifiers, or of identity, or of the is one of relation. However, some of these relativizations do not deliver the expected results, and others rely on problematic assumptions. In another broad variety of relativization, cardinality ascriptions are about concepts or sets. The most promising development of this approach is prima facie connected with a violation of the so-called Coreferentiality Constraint, according to which an identity statement is true only if its terms have the same referent. Moreover - even provided that the problem with coreferentiality can be fixed - the resulting analysis of cardinality ascriptions meets several difficulties. (shrink)
The Overgeneration Argument is a prominent objection against the model-theoretic account of logical consequence for second-order languages. In previous work we have offered a reconstruction of this argument which locates its source in the conflict between the neutrality of second-order logic and its alleged entanglement with mathematics. Some cases of this conflict concern small large cardinals. In this article, we show that in these cases the conflict can be resolved by moving from a set-theoretic implementation of the model-theoretic account to (...) one which uses higher-order resources. (shrink)
I consider the first-order modal logic which counts as valid those sentences which are true on every interpretation of the non-logical constants. Based on the assumptions that it is necessary what individuals there are and that it is necessary which propositions are necessary, Timothy Williamson has tentatively suggested an argument for the claim that this logic is determined by a possible world structure consisting of an infinite set of individuals and an infinite set of worlds. He notes that only the (...) cardinalities of these sets matters, and that not all pairs of infinite sets determine the same logic. I use so-called two-cardinal theorems from model theory to investigate the space of logics and consequence relations determined by pairs of infinite sets, and show how to eliminate the assumption that worlds are individuals from Williamson’s argument. (shrink)
In this paper I present two new arguments against the possibility of an omniscient being. My new arguments invoke considerations of cardinality and resemble several arguments originally presented by Patrick Grim. Like Grim, I give reasons to believe that there must be more objects in the universe than there are beliefs. However, my arguments will rely on certain mereological claims, namely that Classical Extensional Mereology is necessarily true of the part-whole relation. My first argument is an instance of a (...) problem first noted by Gideon Rosen and requires an additional assumption about the mereological structure of certain beliefs. That assumption is that an omniscient being’s beliefs are mereological simples. However, this assumption is dropped when I present my second argument. Thus, I hope to show that if Classical Extensional Mereology is true of the part-whole relation, there cannot be an omniscient being. (shrink)
If $U$ is a normal ultrafilter on a measurable cardinal $\kappa$, then the intersection of the $\omega$ first iterated ultrapowers of the universe by $U$ is a Prikry generic extension of the $\omega$th iterated ultrapower.
This paper presents and defends an argument that the continuum hypothesis is false, based on considerations about objective chance and an old theorem due to Banach and Kuratowski. More specifically, I argue that the probabilistic inductive methods standardly used in science presuppose that every proposition about the outcome of a chancy process has a certain chance between 0 and 1. I also argue in favour of the standard view that chances are countably additive. Since it is possible to randomly pick (...) out a point on a continuum, for instance using a roulette wheel or by flipping a countable infinity of fair coins, it follows, given the axioms of ZFC, that there are many different cardinalities between countable infinity and the cardinality of the continuum. (shrink)
According to Cantor (Mathematische Annalen 21:545–586, 1883 ; Cantor’s letter to Dedekind, 1899 ) a set is any multitude which can be thought of as one (“jedes Viele, welches sich als Eines denken läßt”) without contradiction—a consistent multitude. Other multitudes are inconsistent or paradoxical. Set theoretical paradoxes have common root—lack of understanding why some multitudes are not sets. Why some multitudes of objects of thought cannot themselves be objects of thought? Moreover, it is a logical truth that such multitudes do (...) exist. However we do not understand this logical truth so well as we understand, for example, the logical truth $${\forall x \, x = x}$$ . In this paper we formulate a logical truth which we call the productivity principle. Rusell (Proc Lond Math Soc 4(2):29–53, 1906 ) was the first one to formulate this principle, but in a restricted form and with a different purpose. The principle explicates a logical mechanism that lies behind paradoxical multitudes, and is understandable as well as any simple logical truth. However, it does not explain the concept of set. It only sets logical bounds of the concept within the framework of the classical two valued $${\in}$$ -language. The principle behaves as a logical regulator of any theory we formulate to explain and describe sets. It provides tools to identify paradoxical classes inside the theory. We show how the known paradoxical classes follow from the productivity principle and how the principle gives us a uniform way to generate new paradoxical classes. In the case of ZFC set theory the productivity principle shows that the limitation of size principles are of a restrictive nature and that they do not explain which classes are sets. The productivity principle, as a logical regulator, can have a definite heuristic role in the development of a consistent set theory. We sketch such a theory—the cumulative cardinal theory of sets. The theory is based on the idea of cardinality of collecting objects into sets. Its development is guided by means of the productivity principle in such a way that its consistency seems plausible. Moreover, the theory inherits good properties from cardinal conception and from cumulative conception of sets. Because of the cardinality principle it can easily justify the replacement axiom, and because of the cumulative property it can easily justify the power set axiom and the union axiom. It would be possible to prove that the cumulative cardinal theory of sets is equivalent to the Morse–Kelley set theory. In this way we provide a natural and plausibly consistent axiomatization for the Morse–Kelley set theory. (shrink)
In this article we proved so-called strong reflection principles corresponding to formal theories Th which has omega-models or nonstandard model with standard part. An posible generalization of Lob’s theorem is considered.Main results are: (i) ConZFC Mst ZFC, (ii) ConZF V L, (iii) ConNF Mst NF, (iv) ConZFC2, (v) let k be inaccessible cardinal then ConZFC .
In this paper we view the first order set theory ZFC under the canonical frst order semantics and the second order set theory ZFC_2 under the Henkin semantics. Main results are: (i) Let M_st^ZFC be a standard model of ZFC, then ¬Con(ZFC + ∃M_st^ZFC ). (ii) Let M_stZFC_2 be a standard model of ZFC2 with Henkin semantics, then ¬Con(ZFC_2 +∃M_stZFC_2). (iii) Let k be inaccessible cardinal then ¬Con(ZFC + ∃κ). In order to obtain the statements (i) and (ii) examples of (...) the inconsistent countable set in a set theory ZFC + ∃M_stZFC and in a set theory ZFC2 + ∃M_st^ZFC_2 were derived. It is widely believed that ZFC + ∃M_stZFC and ZFC_2 + ∃M_st^ZFC_2 are consistent, i.e. ZFC and ZFC_2 have a standard models. Unfortunately this belief is wrong. Book. Advances in Mathematics and Computer Science Vol. 1 Chapter 3 There is No Standard Model of ZFC and ZFC2 ISBN-13 (15) 978-81-934224-1-0 See Part II of this paper DOI: 10.4236/apm.2019.99034 . (shrink)
Are there any things that are such that any things whatsoever are among them. I argue that there are not. My thesis follows from these three premises: (1) There are two or more things; (2) for any things, there is a unique thing that corresponds to those things; (3) for any two or more things, there are fewer of them than there are pluralities of them.
Vigorous debate over the moral propriety of cognitive enhancement exists, but the views of the public have been largely absent from the discussion. To address this gap in our knowledge, four experiments were carried out with contrastive vignettes in order to obtain quantitative data on public attitudes towards cognitive enhancement. The data collected suggest that the public is sensitive to and capable of understanding the four cardinal concerns identified by neuroethicists, and tend to cautiously accept cognitive enhancement even as they (...) recognize its potential perils. The public is biopolitically moderate, endorses both meritocratic principles and the intrinsic value of hard work, and appears to be sensitive to the salient moral issues raised in the debate. Taken together, these data suggest that public attitudes toward enhancement are sufficiently sophisticated to merit inclusion in policy deliberations, especially if we seek to align public sentiment and policy. (shrink)
This article explores the main similarities and differences between Derek Parfit’s notion of imprecise comparability and a related notion I have proposed of parity. I argue that the main difference between imprecise comparability and parity can be understood by reference to ‘the standard view’. The standard view claims that 1) differences between cardinally ranked items can always be measured by a scale of units of the relevant value, and 2) all rankings proceed in terms of the trichotomy of ‘better than’, (...) ‘worse than’, and ‘equally good’. Imprecise comparability, which can be understood in terms of the more familiar notions of cardinality and incommensurability, rejects only the first claim while parity rejects both claims of the standard view. -/- I then argue that insofar as those attracted to imprecise comparability assume that all rankings are trichotomous, as Parfit appears to, the view should be rejected. This is because imprecise equality is not a form of equality but is a sui generis ‘fourth’ basic way in which items can be ranked. We should, I argue, understand imprecise equality as parity, and imprecise comparability as entailing ‘tetrachotomy’ – that if two items are comparable, one must better than, worse than, equal to, or on a par with the other. Thus those attracted to the idea that cardinality can be imprecise should abandon trichotomy and accept parity and tetrachotomy instead. -/- Finally, I illustrate the difference between Parfit’s trichotomous notion of imprecise comparability and parity by examining how each notion might be employed in different solutions to the problem posed by the Repugnant Conclusion in population ethics. I suggest that parity provides the arguably more ecumenical solution to the problem. (shrink)
What is it to know more? By what metric should the quantity of one's knowledge be measured? I start by examining and arguing against a very natural approach to the measure of knowledge, one on which how much is a matter of how many. I then turn to the quasi-spatial notion of counterfactual distance and show how a model that appeals to distance avoids the problems that plague appeals to cardinality. But such a model faces fatal problems of its (...) own. Reflection on what the distance model gets right and where it goes wrong motivates a third approach, which appeals not to cardinality, nor to counterfactual distance, but to similarity. I close the paper by advocating this model and briefly discussing some of its significance for epistemic normativity. In particular, I argue that the 'trivial truths' objection to the view that truth is the goal of inquiry rests on an unstated, but false, assumption about the measure of knowledge, and suggest that a similarity model preserves truth as the aim of belief in an intuitively satisfying way. (shrink)
Are there different sizes of infinity? That is, are there infinite sets of different sizes? This is one of the most natural questions that one can ask about the infinite. But it is of course generally taken to be settled by mathematical results, such as Cantor’s theorem, to the effect that there are infinite sets without bijections between them. These results settle the question, given an almost universally accepted principle relating size to the existence of functions. The principle is: for (...) any sets A and B, if A is the same size as B, then there is a bijection from A to B. The aim of the paper, however, is to argue that this question is in fact wide open: to argue that we are not in a position to know the answer, because we are not in one to know the principle. The aim, that is, is to argue that for all we know there is only one size of infinity. (shrink)
Despite the proliferation of studies on corporate social responsibility, there is a lack of consensus and a cardinal methodological base for research on the quality of CSR communication. Over the decades, studies in this space have remained conflicting, unintegrated, and sometimes overlapping. Drawing on semiotics—a linguistic-based theoretical and analytical tool, our article explores an alternative perspective to evaluating the quality and reliability of sustainability reports. Our article advances CSR communication research by introducing a theoretical-cum-methodological perspective which provides unique insights into (...) how to evaluate the quality of CSR communication. Particularly, we illustrate the application of our proposed methodology on selected U.K. FTSE 100 companies. Our two-phased analysis employed the Greimas Canonical Narrative Schema and the Semiotic Square of Veridiction in drawing meanings from selected sustainability/csr reports. In addition, we present a distinctive CSR report quality model capable of guiding policy makers and firms in designing sustainability/csr reporting standards. (shrink)
The iterative conception of set is typically considered to provide the intuitive underpinnings for ZFCU (ZFC+Urelements). It is an easy theorem of ZFCU that all sets have a definite cardinality. But the iterative conception seems to be entirely consistent with the existence of “wide” sets, sets (of, in particular, urelements) that are larger than any cardinal. This paper diagnoses the source of the apparent disconnect here and proposes modifications of the Replacement and Powerset axioms so as to allow for (...) the existence of wide sets. Drawing upon Cantor’s notion of the absolute infinite, the paper argues that the modifications are warranted and preserve a robust iterative conception of set. The resulting theory is proved consistent relative to ZFC + “there exists an inaccessible cardinal number.”. (shrink)
This paper is concerned with learners who aim to learn patterns in infinite binary sequences: shown longer and longer initial segments of a binary sequence, they either attempt to predict whether the next bit will be a 0 or will be a 1 or they issue forecast probabilities for these events. Several variants of this problem are considered. In each case, a no-free-lunch result of the following form is established: the problem of learning is a formidably difficult one, in that (...) no matter what method is pursued, failure is incomparably more common that success; and difficult choices must be faced in choosing a method of learning, since no approach dominates all others in its range of success. In the simplest case, the comparison of the set of situations in which a method fails and the set of situations in which it succeeds is a matter of cardinality (countable vs. uncountable); in other cases, it is a topological matter (meagre vs. co-meagre) or a hybrid computational-topological matter (effectively meagre vs. effectively co-meagre). (shrink)
We examine some of Connes’ criticisms of Robinson’s infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes’ own earlier work in functional analysis. Connes described the hyperreals as both a “virtual theory” and a “chimera”, yet acknowledged that his argument relies on the transfer principle. We analyze Connes’ “dart-throwing” thought experiment, but reach an opposite conclusion. In S , all definable sets of reals are (...) Lebesgue measurable, suggesting that Connes views a theory as being “virtual” if it is not definable in a suitable model of ZFC. If so, Connes’ claim that a theory of the hyperreals is “virtual” is refuted by the existence of a definable model of the hyperreal field due to Kanovei and Shelah. Free ultrafilters aren’t definable, yet Connes exploited such ultrafilters both in his own earlier work on the classification of factors in the 1970s and 80s, and in Noncommutative Geometry, raising the question whether the latter may not be vulnerable to Connes’ criticism of virtuality. We analyze the philosophical underpinnings of Connes’ argument based on Gödel’s incompleteness theorem, and detect an apparent circularity in Connes’ logic. We document the reliance on non-constructive foundational material, and specifically on the Dixmier trace −∫ (featured on the front cover of Connes’ magnum opus) and the Hahn–Banach theorem, in Connes’ own framework. We also note an inaccuracy in Machover’s critique of infinitesimal-based pedagogy. (shrink)
A problem for Aristotelian realist accounts of universals (neither Platonist nor nominalist) is the status of those universals that happen not to be realised in the physical (or any other) world. They perhaps include uninstantiated shades of blue and huge infinite cardinals. Should they be altogether excluded (as in D.M. Armstrong's theory of universals) or accorded some sort of reality? Surely truths about ratios are true even of ratios that are too big to be instantiated - what is the truthmaker (...) of such truths? It is argued that Aristotelianism can answer the question, but only a semi-Platonist form of it. (shrink)
An enduring puzzle in philosophy and developmental psychology is how young children acquire number concepts, in particular the concept of natural number. Most solutions to this problem conceptualize young learners as lone mathematicians who individually reconstruct the successor function and other sophisticated mathematical ideas. In this chapter, I argue for a crucial role of testimony in children’s acquisition of number concepts, both in the transfer of propositional knowledge (e.g., the cardinality concept), and in knowledge-how (e.g., the counting routine).
Three theories contend as explanations of perpetrator behavior in the Holocaust as well as other cases of genocide: structural, intentional, and situational. Structural explanations emphasize the sense in which no single individual or choice accounts for the course of events. In opposition, intentional/cutltural accounts insist upon the genocides as intended outcomes, for how can one explain situations in which people ‘step up’ and repeatedly kill defenseless others in large numbers over sustained periods of time as anything other than a choice? (...) Situational explanations offer a type of behavioral account; this is how people act in certain environments. Critical to the situational account as I discuss it is the ‘Asch paradigm’, i.e. experimentally attested conditions for eliciting conformityof behavior regardlesss of available evidence of prior beliefs. In what follows, I defend what I term above a version of situational explanations of perpetrator behavior. Moreover, I maintain that the factors that explain provide an understanding as well. While not committed to the complete irrelevance or exclusion of cultural or structural factors, nonetheless situational analyses can account both for what happened and why. A cardinal virtue of this version of situational explanations consists in showing how shallow the problem of understanding turns out to be for such cases. (shrink)
I defend the idea that a liberal commitment to value neutrality is best honoured by maintaining a pure cardinality component in our rankings of opportunity or liberty sets. I consider two challenges to this idea. The first holds that cardinality rankings are unnecessary for neutrality, because what is valuable about a set of liberties from a liberal point of view is not its size but rather its variety. The second holds that pure cardinality metrics are insufficient for (...) neutrality, because liberties cannot be individuated into countable entities without presupposing some relevantly partisan evaluative perspective. I argue that a clear understanding of the liberal basis for valuing liberty shows the way to satisfying responses to both challenges. (shrink)
“There are no gaps in logical space,” David Lewis writes, giving voice to sentiment shared by many philosophers. But different natural ways of trying to make this sentiment precise turn out to conflict with one another. One is a *pattern* idea: “Any pattern of instantiation is metaphysically possible.” Another is a *cut and paste* idea: “For any objects in any worlds, there exists a world that contains any number of duplicates of all of those objects.” We use resources from model (...) theory to show the inconsistency of certain packages of combinatorial principles and the consistency of others. (shrink)
Two expressive limitations of an infinitary higher-order modal language interpreted on models for higher-order contingentism – the thesis that it is contingent what propositions, properties and relations there are – are established: First, the inexpressibility of certain relations, which leads to the fact that certain model-theoretic existence conditions for relations cannot equivalently be reformulated in terms of being expressible in such a language. Second, the inexpressibility of certain modalized cardinality claims, which shows that in such a language, higher-order contingentists (...) cannot express what is communicated using various instances of talk of ‘possible things’, such as ‘there are uncountably many possible stars’. (shrink)
Each of the degenerating constitutions in Book VIII of Plato's Republic is the result of the disappearance of one of the four cardinal virtues. The failure of wisdom creates a timocracy; the failure of courage, an oligarchy; the failure of moderation, a democracy; the failure of justice, a tyranny. The degeneration shows that the disunited virtues are imperfect, though they have some power to stave off vice. Thus Book VIII implies a unity of the virtues thesis according to which perfect (...) virtues can only exist in a united state, but imperfect simulacra of virtue can exist in a disunited state. -/- Published 2011 in Apeiron: A Journal for Ancient Philosophy and Science. (Please note that the pagination in the uploaded document is not the same as the pagination in the published edition.). (shrink)
In this paper, we discuss some rather puzzling facts concerning the semantics of Warlpiri expressions of cardinality, i.e. the Warlpiri counterparts of English expressions like one,two, many, how many. The morphosyntactic evidence, discussed in section 1, suggests that the corresponding expressions in Warlpiri are nominal, just like the Warlpiri counterparts of prototypical nouns, eg. child. We also argue that Warlpiri has no articles or any other items of the syntactic category D(eterminer). In section 2, we describe three types of (...) readings— "definite", "indefinite" and "predicative"—which are generally found with Warlpiri nouns, including those which correspond to English common nouns and cardinality expressions. A partial analysis of these readings is sketched i n section 3. Since Warlpiri has no determiner system, we hypothesize that the source of (in)definiteness in this language is semantic. More specifically, we suggest that Warlpiri nominals are basically interpreted as individual terms or predicates of individuals and that their three readings arise as a consequence of the interaction of their basic meanings, which are specific to Warlpiri, with certain semantic operations, such as type shifting (Rooth and Partee 1982, Partee and Rooth 1983, Partee 1986, 1987), which universally can or must apply in the process of compositional semantic interpretation. (shrink)
Nils-Frederic Wagner takes issue with my argument that influential critics of “transplant” thought experiments make two cardinal mistakes. He responds that the mistakes I identify are not mistakes at all. The mistakes are rather on my part, in that I have not taken into account the conceptual genesis of personhood, that my view of thought experiments is idiosyncratic and possibly self-defeating, and in that I have ignored important empirical evidence about the relationship between brains and minds. I argue that my (...) case still stands and that transplant thought experiments can do damage to rivals of a psychological continuity theory of personal identity like Marya Schechtman’s Person Life View. (shrink)
In a single framework, I address the question of the informational basis for evaluating social states. I particularly focus on information about individual welfare, individual preferences and individual (moral) judgments, but the model is also open to any other informational input deemed relevant, e.g. sources of welfare and motivations behind preferences. In addition to proving some possibility and impossibility results, I discuss objections against using information about only one aspect (e.g. using only preference information). These objections suggest a multi-aspect informational (...) basis for aggregation. However, the multi-aspect approach faces an impossibility result created by a lack of inter-aspect comparability. The impossibility could be overcome by measuring information on non-cardinal scales. (shrink)
In "Truth and Method" Hans Georg Gadamer revealed hermeneutics as one of the foundational epistemological elements of history, in contrast to scientific method, which, with empiricism, constitutes natural sciences’ epistemology. This important step solved a number of long-standing arguments over the ontology of history, which had become increasingly bitter in the twentieth century. But perhaps Gadamer’s most important contribution was that he annulled history’s supposed inferiority to the natural sciences by showing that the knowledge it offers, though different in nature (...) from science, is of equal import. By showing history’s arrant independence from the natural sciences, the former was furnished with a new-found importance, and thrust on an equal footing with the latter—even in a distinctly scientific age such as ours. This essay intends to show that the idea of history’s discrete ontology from science was prefigured almost a century earlier by Benedetto Croce. Croce and Gadamer show compelling points of contact in their philosophies, notwithstanding that they did not confer equal consequence to what may be identified as Gadamer’s principal substantiation of history’s epistemology—hermeneutics. Of course this essay does not aspire to be exhaustive: the thought of both philosophers is far too dense. Nevertheless, the main points of contact shall be outlined, and, though concise, this essay seeks to point out the striking similarities of these two cardinal philosophers of history. (shrink)
In his 1927 Analysis of Matter and elsewhere, Russell argued that we can successfully infer the structure of the external world from that of our explanatory schemes. While nothing guarantees that the intrinsic qualities of experiences are shared by their objects, he held that the relations tying together those relata perforce mirror relations that actually obtain (these being expressible in the formal idiom of the Principia Mathematica). This claim was subsequently criticized by the Cambridge mathematician Max Newman as true but (...) trivial, insofar as from a closed body of observations (or “Ramsey sentence”) one can always generate other equally-satisfactory networks of relations, provided they respect the original set’s cardinality. Since any model thus generated will be empirically adequate, “[t]he defence is therefore driven back from the fairly safe fictitious-real classification to the much less tenable ‘trivial’ and ‘important’” (Newman 1928). Given the definitional rigour afforded by the initial appeal to isomorphism (via one-to-one correspondences in extension), the received assessment, shared by Russell himself, is that retreating to a pragmatic adjudication would betoken a fatal blow. However, I suggest that reliance on “importance” can be avoided if we incorporate an impersonal criterion of diachronic precedence. When collecting observations, an ordinality emerges alongside the cardinality which gives that underived structure an irrevocable epistemological privilege. Hence, I argue that, all other things being equal, any construct parasitic on an antecedent theory ought to be regarded as inferior/dispensable, since it was generated by an algorithm lacking the world-involving pedigree of its host structure. (shrink)
More often than not Cusanus is interpreted in a theological way, under strong theological presuppositions and within the range of religion. This may be quite understandable since he was a cardinal and had important functions in the Papal States. But what are the philosophical implications if some of his texts are neither meant to assert a belief nor to search for reasons for it, but only to reflect upon the presuppositions of this belief and its different traditions? – A word-for-word (...) interpretation of the first proposition, which follows the dialogue „De non aliud“ : definitio quae se et omnia definit, ea est, quae per omnem mentem quaeritur, gives us a hint to the shift in the concept of definitio during the dialogue. Cusanus begins in quite a traditional manner and ends in a supremely abstract and speculative intuition. The not-other determines itself in a vision and by this puts every thing in its proper place; we as human beings aspire to repeat this vision in our mental life in contact with the world. In this way Cusanus does what all great philosophers do: he reflects in a given set of opinions what is the meaning of „to be“. (shrink)
Introduction to mathematical logic, part 2.Textbook for students in mathematical logic and foundations of mathematics. Platonism, Intuition, Formalism. Axiomatic set theory. Around the Continuum Problem. Axiom of Determinacy. Large Cardinal Axioms. Ackermann's Set Theory. First order arithmetic. Hilbert's 10th problem. Incompleteness theorems. Consequences. Connected results: double incompleteness theorem, unsolvability of reasoning, theorem on the size of proofs, diophantine incompleteness, Loeb's theorem, consistent universal statements are provable, Berry's paradox, incompleteness and Chaitin's theorem. Around Ramsey's theorem.
By pure calculus of names we mean a quantifier-free theory, based on the classical propositional calculus, which defines predicates known from Aristotle’s syllogistic and Leśniewski’s Ontology. For a large fragment of the theory decision procedures, defined by a combination of simple syntactic operations and models in two-membered domains, can be used. We compare the system which employs `ε’ as the only specific term with the system enriched with functors of Syllogistic. In the former, we do not need an empty name (...) in the model, so we are able to construct a 3-valued matrix, while for the latter, for which an empty name is necessary, the respective matrices are 4-valued. (shrink)
Here, we analyse some recent applications of set theory to topology and argue that set theory is not only the closed domain where mathematics is usually founded, but also a flexible framework where imperfect intuitions can be precisely formalized and technically elaborated before they possibly migrate toward other branches. This apparently new role is mostly reminiscent of the one played by other external fields like theoretical physics, and we think that it could contribute to revitalize the interest in set theory (...) in the future. (shrink)
In this paper I explore humility as a paradigm, with reference to recent debates over the morality and rationality of emotions, and to the relation between religion and emotion. In Ancient Greek ethics, humility did not yet play a role; with the rise of Christianity, however, it becomes one of the cardinal virtues -- only to disappear again with the onset of modernity. Against a culture-pessimistic interpretation of this development, this article begins by characterising the relation between virtue and emotion, (...) before reconstructing the inner rationality of humility and showing how it can be traced through several transformations to a modern ethics of responsibility. Against this background, possible manifestations of the humble attitude in the present are made plausible. (shrink)
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