We consider five ordinal notation systems of ε0 which are all well-known and of interest in proof-theoretic analysis of Peano arithmetic: Cantor’s system, systems based on binary trees and on countable tree-ordinals, and the systems due to Schütte and Simpson, and to Beklemishev. The main point of this paper is to demonstrate that the systems except the system based on binary trees are equivalent as structured systems, in spite of the fact that they have their origins in different views (...) and trials in proof theory. This is true while Weiermann’s results based on Friedman-style miniaturization indicate that the system based on binary trees is of different character than the others. (shrink)

The standard method of generating countable ordinals from uncountable ordinals can be replaced by a use of fixed point extractors available in the term calculus of Howard’s system. This gives a notion of the intrinsic complexity of an ordinal analogous to the intrinsic complexity of a function described in Gödel’s T.

This article provides a detailed comparison between two systems of collapsing functions. These functions play a crucial role in proof theory, in the analysis of patterns of resemblance, and the analysis of maximal order types of well partial orders. The exact correspondence given here serves as a starting point for far reaching extensions of current results on patterns and well partial orders. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Transitive extensional well founded relations provide an intuitionistic notion of ordinals which admits transfinite induction. However these ordinals are not directed and their successor operation is poorly behaved, leading to problems of functoriality. We show how to make the successor monotone by introducing plumpness, which strengthens transitivity. This clarifies the traditional development of successors and unions, making it intuitionistic; even the (classical) proof of trichotomy is made simpler. The definition is, however, recursive, and, as their name suggests, the plump ordinals (...) grow very rapidly. Directedness must be defined hereditarily. It is orthogonal to the other four conditions, and the lower powerdomain construction is shown to be the universal way of imposing it. We treat ordinals as order-types, and develop a corresponding set theory similar to Osius' transitive set objects. This presents Mostowski's theorem as a reflection of categories, and set-theoretic union is a corollary of the adjoint functor theorem. Mostowski's theorem and the rank for some of the notions of ordinal are formulated and proved without the axiom of replacement, but this seems to be unavoidable for the plump rank. The comparison between sets and toposes is developed as far as the identification of replacement with completeness, and there are some suggestions for further work in this area. Each notion of set or ordinal defines a free algebra for one of the theories discussed by Joyal and Moerdijk, namely joins of a family of arities together with an operation s satisfying conditions such as x ≤ sx, monotonicity or s(x ∨ y) ≤ sx ∨ sy. Finally we discuss the fixed point theorem for a monotone endofunction s of a poset with least element and directed joins. This may be proved under each of a variety of additional hypotheses. We explain why it is unlikely that any notion of ordinal obeying the induction scheme for arbitrary predicates will prove the pure result. (shrink)

There is a widely held view on measurement inferences, that goes back to Stevens’s ([1946]) theory of measurement scales and ‘permissible statistics’. This view defends the following prohibition: you should not make inferences from averages taken with ordinal scales (versus quantitative scales: interval or ratio). This prohibition is general—it applies to all ordinal scales—and it is sometimes endorsed without qualification. Adhering to it dramatically limits the research that the social and biomedical sciences can conduct. I provide a Bayesian (...) analysis of this inferential problem, determining when measurements from ordinal scales can be used to confirm hypotheses about relative group averages. The prohibition, I conclude, cannot be upheld, even in a qualified sense. The beliefs needed to make average comparisons are less demanding than those appropriate for quantitative scales. I illustrate with the alleged paradigm ordinal scale, Mohs’ scale of mineral hardness, arguing that the literature has mischaracterized it. (shrink)

Visual imagery is a form of sensory imagination, involving subjective experiences typically described as similar to perception, but which occur in the absence of corresponding external stimuli. We used the Activation Likelihood Estimation algorithm (ALE) to identify regions consistently activated by visual imagery across 40 neuroimaging studies, the first such meta-analysis. We also employed a recently developed multi-modal parcellation of the human brain to attribute stereotactic co-ordinates to one of 180 anatomical regions, the first time this approach has been combined (...) with the ALE algorithm. We identified a total 634 foci, based on measurements from 464 participants. Our overall comparison identified activation in the superior parietal lobule, particularly in the left hemisphere, consistent with the proposed ‘top-down’ role for this brain region in imagery. Inferior premotor areas and the inferior frontal sulcus were reliably activated, a finding consistent with the prominent semantic demands made by many visual imagery tasks. We observed bilateral activation in several areas associated with the integration of eye movements and visual information, including the supplementary and cingulate eye fields (SCEFs) and the frontal eye fields (FEFs), suggesting that enactive processes are important in visual imagery. V1 was typically activated during visual imagery, even when participants have their eyes closed, consistent with influential depictive theories of visual imagery. Temporal lobe activation was restricted to area PH and regions of the fusiform gyrus, adjacent to the fusiform face complex (FFC). These results provide a secure foundation for future work to characterise in greater detail the functional contributions of specific areas to visual imagery. (shrink)

Emergency and exception are central concepts in Schmitt’s theory of decisionism. The present article explores the distinction between the two by focusing on their emergence, i.e. the process by which in times of crisis a potentially alternative or-der comes into existence and becomes visible. The primary aim of the comparison is to provide a more detailed and less conventional account of Schmitt’s excep-tionalist decisionism. In order to achieve this aim, three relevant questions must be raised: How does the sovereign succeed (...) in gaining acceptance, by exploiting what situation, and by convincing whom? Based on the different answers provid-ed by Schmitt over the years, it is possible to distinguish three different phases of his thought: the sovereign decisionism of the early 1920s, the concrete-order thinking of the 1930s, and the jurisprudential institutionalism advanced from 1950s onwards. (shrink)

A significant open problem in inner model theory is the analysis of HODL[x] as a strategy premouse, for a Turing cone of reals x. We describe here an obstacle to such an analysis. Assuming sufficient large cardinals, for a Turing cone of reals x there are proper class 1-small premice M,N, with Woodin cardinals δ,ε, respectively, such that M|δ and N|ε are in L[x], M and N are countable in L[x], and the pseudo-comparison of M with N succeeds, is in (...) L[x], and lasts exactly ω1L[x] stages. Moreover, we can take M=M1, the minimal iterable proper class inner model with a Woodin cardinal, and take N to be M1-like and short-tree-iterable. (shrink)

The article deals with the problem of the dispersion of ordinal variables. At first, it specifies the very concept of dispersion for this type of scale. Then some of the most known measures that fit to the concept of ordinal variation are recalled. They are constructed with two different types of statistical models: using loss functions and using distance functions. Finally, a new approach, which is the use of an axiomatic method for the construction of a dispersion measure, (...) is proposed. Some relations and comparisons between different measures and between different approaches are shown. (shrink)

In 1986 David Gauthier proposed an arbitration scheme for two player cardinal bargaining games based on interpersonal comparisons of players’ relative concessions. In Gauthier’s original arbitration scheme, players’ relative concessions are defined in terms of Raiffa-normalized cardinal utility gains, and so it cannot be directly applied to ordinal bargaining problems. In this paper I propose a relative benefit equilibrating bargaining solution for two and n-player ordinal and quasiconvex ordinal bargaining problems with finite sets of feasible basic (...) agreements based on the measure of players’ ordinal relative individual advantage gains. I provide an axiomatic characterization of this bargaining solution and discuss the conceptual relationship between RBEBS and ordinal egalitarian bargaining solution proposed by Conley and Wilkie. I show the relationship between the measurement procedure for ordinal relative individual advantage gains and the measurement procedure for players’ ordinal relative concessions, and argue that the proposed arbitration scheme for ordinal games can be interpreted as an ordinal version of Gauthier’s arbitration scheme. (shrink)

Nakata (Theory Decis 71:559–574, 2011) presents a model of acquisition of information where the agent does not know what pieces of information she is missing. In this note, we point out some technical problems in a few of Nakata’s results and show how to correct them.

Metanormativists hold that moral uncertainty can affect how we ought, in some morally authoritative sense, to act. Many metanormativists aim to generalize expected utility theory for normative uncertainty. Such accounts face the “easy problem of intertheoretic comparisons”: the worry that distinct theories’ assessments of choiceworthiness are incomparable. The easy problem may well be resolvable, but another problem looms: while some moral theories assign cardinal degrees of choiceworthiness, other theories’ choiceworthiness assignments are merely ordinal. Expected choiceworthiness over such theories (...) is undefined. Call this the “hard problem of intertheoretic comparisons.” This paper argues that to solve the hard problem, we should model moral theories with imprecise choiceworthiness. Imprecise choiceworthiness assignments can model incomplete cardinal information about choiceworthiness, with precise cardinal choiceworthiness and merely ordinal choiceworthiness as limiting cases. Generalizing familiar decision theories for imprecise choiceworthiness to the case of moral uncertainty generates puzzles, however: natural generalizations seem to require reifying parts of the model that don’t correspond to anything in normative reality. I discuss three ways of addressing this problem: by demystifying the reified elements by using them as promiscuously as possible; by constructing alternative decision theories that don’t require the troublesome elements; and by employing an alternative model of metanormative decision problems, and of moral uncertainty generally. (shrink)

In this highly original study, Nuccio Ordine uses the figure of the ass as a lens through which to focus on the thought and writings of the great Renaissance humanist philosopher Giordano Bruno.

“A little masterpiece of originality and clarity.”—George Steiner “A necessary book.”—Roberto Saviano “A wonderful little book that will delight you.”—François Busnel International Best Seller / Now in English for the First Time In this thought-provoking and extremely timely work, Nuccio Ordine convincingly argues for the utility of useless knowledge and against the contemporary fixation on utilitarianism—for the fundamental importance of the liberal arts and against the damage caused by their neglect. Inspired by the reflections of great philosophers and writers (e.g., (...) Plato, Dante, Montaigne, Shakespeare, Borges, and Calvino), Ordine reveals how the obsession for material goods and the cult of utility ultimately wither the spirit, jeopardizing not only schools and universities, art, and creativity, but also our most fundamental values—human dignity, love, and truth. Also included is Abraham Flexner’s 1939 essay “The Usefulness of Useless Knowledge,” which originally prompted Ordine to write this book. Flexner—a founder and the first director of the Institute for Advanced Study in Princeton—offers an impassioned defense of curiosity-driven research and learning. (shrink)

Traditionally, the investigation of truth has been anchored in a priori reasoning. Cognitive science deviates from this tradition by adding empirical data on how people understand and use concepts. Building on psychophysics and machine learning methods, we introduce conceptual scaling, an approach to map people's understanding of abstract concepts. This approach, allows computing participant-specific conceptual maps from obtained ordinal comparison data, thereby quantifying perceived similarities among abstract concepts. Using this approach, we investigated individual's alignment with philosophical theories on truth (...) and the predictive capacity of conceptual maps. The obtained results indicated that, while people's understanding of truth is multifaceted and encapsulates notions of coherence and authenticity, alignment is best for the correspondence theory of truth. Furthermore, conceptual maps allowed predicting individual outcomes with an accuracy of roughly 70%. This research demonstrates that conceptual scaling offers accurate descriptions of individual's understanding of abstract concepts, behavioral predictions, and quantification of alignment with theoretical perspectives. (shrink)

Pensar en el cine como ocasión y potencia del pensamiento significa en efecto, en primer lugar, sustraer la imagen contemporánea del dominio de los estudios especializados y, en segundo lugar, pensar a partir de aquello que el cine crea (como recordaba Deleuze): esto es, las imágenes mismas. El cine redescubre su potencia a través de sus propias formas. Entrevistas de la revista italiana Fata Morgana con: Jacques Rancière, Roberto Esposito, Jean-luc Nancy, Slavoj Zizek, Julia Kristeva, Werner Herzog, Raúl Ruiz, Yervant (...) Gianikian, Ángela Ricci Lucchi, Edgar Reitz, Paul Schraoer, Georges Didi-Huberman, Jean-Louis Comolli, Nuccio Ordine. (shrink)

According to comparativism, degrees of belief are reducible to a system of purely ordinal comparisons of relative confidence. (For example, being more confident that P than that Q, or being equally confident that P and that Q.) In this paper, I raise several general challenges for comparativism, relating to (i) its capacity to illuminate apparently meaningful claims regarding intervals and ratios of strengths of belief, (ii) its capacity to draw enough intuitively meaningful and theoretically relevant distinctions between doxastic (...) states, and (iii) its capacity to handle common instances of irrationality. (shrink)

We present a theory of decision by sampling (DbS) in which, in contrast with traditional models, there are no underlying psychoeconomic scales. Instead, we assume that an attribute’s subjective value is constructed from a series of binary, ordinal comparisons to a sample of attribute values drawn from memory and is its rank within the sample. We assume that the sample reflects both the immediate distribution of attribute values from the current decision’s context and also the background, real-world distribution (...) of attribute values. DbS accounts for concave utility functions; losses looming larger than gains; hyperbolic temporal discounting; and the overestimation of small probabilities and the underestimation of large probabilities. Ó 2005 Elsevier Inc. All rights reserved. (shrink)

This paper is a survey of how economists and philosophers approach the issue of comparisons. More precisely, it is about what formal representation is appropriate whenever our ability to compare things breaks down. We restrict our attention to failures that arise with ordinal comparisons. We consider a number of formal approaches to this problem including one based on the idea of parity. We also consider the claim that the failure to compare things is a consequence of vagueness. (...) We contrast two theories of vagueness; fuzzy set theory and supervaluation theory. Some applications of these theories are described. (shrink)

In “The Irrelevance of the Diachronic Money-Pump Argument for Acyclicity,” The Journal of Philosophy CX, 8 (August 2013), 460-464, Johan E. Gustafsson contends that if Davidson, McKinsey and Suppes’ diachronic money-pump argument in their "Outlines of a Formal Theory of Value, I," Philosophy of Science 22 (1955), 140-160 is valid, so is the synchronic argument Gustafsson himself offers. He concludes that the latter renders irrelevant diachronic choice considerations in general, and the two best-known diachronic solutions to the money pump problem (...) in particular. I argue here that this reasoning is incorrect, and that Gustafsson’s synchronic argument is faulty on independent grounds. Specifically, it is based on a false analogy between the derivation of a synchronic ordinal ranking from a transitive series of pairwise comparisons, and the putative derivation of such a ranking from an intransitive series. The latter is not possible under the assumption of revealed preference theory, and is highly improbable even if that assumption is rejected. Moreover, Gustafsson’s argument raises issues of fidelity to the historical texts that must be addressed. I conclude that the money pump, and cyclical choice more generally, are necessarily diachronic; and therefore that the two best-known diachronic solutions to the money pump problem remain relevant. (shrink)

The paper deals with a method of hierarchization of alternatives in a multicriteria environment by means of statistical nonparametric procedures. For each criterion, alternatives are disposed on an ordinal scale. After that a procedure similar to ANOVA is activated on the data. The differences relating to the average ranks of each action are used to build the hierarchical algorithm. The concept of outranking, in a probabilistic meaning, is reached in this way. Thereafter, we arrive at the concept of dominance (...) using pairwise comparisons. A case study is presented which implements the methodological suggestion. (shrink)

Measurement practices are central to most sciences. In the human sciences, however, it remains controversial whether the measurement of human attributes—depression, happiness, intelligence, etc.—has been successful. Are, say, widely used depression questionnaires valid measuring instruments? Can we trust self-reported happiness scales to deliver quantitative measurements as it is sometimes claimed? These and related questions are till today hotly disputed. There are two main frameworks under which human measurements are studied and criticized. One is the so-called construct validity framework. Here, criticisms (...) to human measurements are typically of the form “this instrument is not valid: it does not actually measure the attribute we set out to measure”. The second framework is the standard typology of measurement scales (nominal, ordinal, interval, and ratio). Human measurements are commonly challenged for being merely ordinal—not quantitative: interval or ratio—despite the fact that many researchers use measurement results as if they were quantitative. The first part of the dissertation studies how adequate these frameworks are for evaluating measuring instruments and the inferences they afford. Regarding the concept of validity, it is commonly understood unconditionally, that is, without restricting validity judgments about measuring instruments to context-specific situations. Instruments are said to be (in)valid simpliciter. In Chapter 2, I argue against this conception of validity and in favor of a contextual one. Regarding the second main framework, the standard typology of scales is typically linked to a set of prescriptions regarding (un)justified measurement inferences. I call this classification-cum-prescriptions the “received view” on measurement scales. In chapters 3 and 4, I question the idea that the received view is an impeccable guide for clarifying the kinds of inferences we are licensed to make from measurement results in human science contexts. I motivate general doubts about the adequacy of the received view in Chapter 3, and I articulate these doubts in detail for the specific case of average group comparisons in Chapter 4. The upshot of this first part of the dissertation is a deeper awareness of the complexity surrounding which inferences can legitimately be made from measuring instruments. The first part of the dissertation is largely framed under the assumption that some human attributes are indeed quantitative, even if they are not currently being measured quantitatively. The second part addresses two important issues raised by that assumption. Chapter 5 tackles the so-called “quantity objection”: that human science attributes are themselves not quantitative, thus they cannot be quantitatively measured. This objection has been deployed to argue against optimistic positions regarding human quantification. I argue that the quantity objection is not successful in this sense—it begs the question to these optimistic human scientists that, just like their colleagues in the physical sciences have done, postulate theoretical quantitative attributes as working hypotheses. But what does treating human attributes quantitatively as working hypotheses amount to? And what are, or can be, “amounts” of depression, happiness, etc.? Chapter 6 argues that there is not one but various approaches for quantitatively conceiving of human attributes, each with its own success conditions. This chapter offers a conceptual framework for making progress on debates about controversial human measurements. (shrink)

We are often unsure about what we ought to do. This can be because we lack empirical knowledge, such as the extent to which future generations will be harmed by climate change. It can also be because we lack normative knowledge, such as the relative moral importance of the interests of present people and the interests of future people. However, though the question of how one ought to act under empirical uncertainty has been addressed extensively by both economists and philosophers---with (...) expected utility theory providing the standard formal framework---the question of how one ought to act under normative uncertainty is comparatively neglected. My thesis attempts to address this gap. -/- In my thesis I develop a view that I call metanormativism: that there are second-order norms that govern action that are relative to a decision-maker's uncertainty about first-order norms. -/- In the first part of the thesis, I defend one specific metanormative view: that under normative uncertainty decision-makers should maximise expected choice-worthiness, treating normative uncertainty analogously with empirical uncertainty. Drawing on the analogy between decision-making under normative uncertainty and social choice theory, I defend this view at length in response to the problem of merely ordinal theories and the problem of intertheoretic value comparisons. -/- In the second part of the thesis, I explore the implications of metanormativism for other philosophical issues. I argue that it has important consequences regarding the theory of rational action in the face of incomparable values, the causal/evidential debate in decision-theory, and our assessment of the value of moral philosophy. (shrink)

Current controversies about the ordination of women have shown the need for a re‐examination of what the Christian Church means by priesthood. This article looks at the Epistle to the Hebrews’ contribution to our understanding. To that end it focuses on the institution of priesthood in its first‐century Jewish context and shows the use made of it by the author of Hebrews in his presentation of Christian faith.Section 1 emphasizes some all‐important differences between the NT’s use of the language of (...) priesthood and ours. Not least, it nowhere uses “priest” to designate Christian ministers. All the more striking, therefore, is Hebrews’ depiction of Jesus as “high priest”.Section 2 discusses Judaism’ Day of Atonement ceremonies – Hebrews’ dominant cultic model. In the comparison drawn between Christ’s death and exaltation with these rites, he becomes not only the high priest but also the expiatory victim.As far as Judaism’ cultic institutions are concerned, however, Jesus was not and never could be a priest, since he was of the tribe of Judah rather than Levi. Hence Hebrews appeals to Melchizedek. How this non‐Israelite model is used by Hebrews to subvert the whole notion of priesthood as caste is discussed in section 3.Finally, in section 4 Isaacs concludes that for Hebrews there is no longer a role for an ongoing priesthood, since Jesus has definitively achieved access to God, which was its raison d’être. Melchizedekian high priesthood is unique to Christ; neither inherited nor transmitted. Hence, unlike other NT authors, for Hebrews even the church as a corporate body is not a priesthood. As its closing chapters show, in this Epistle the cultic model gives way to the more inclusive one of pilgrimage. (shrink)

Let T(Ω) be the ordinal notation system from Buchholz-Schütte (1988). [The order type of the countable segmentT(Ω)0 is — by Rathjen (1988) — the proof-theoretic ordinal the proof-theoretic ordinal ofACA 0 + (Π 1 l −TR).] In particular let ↦Ω a denote the enumeration function of the infinite cardinals and leta ↦ ψ0 a denote the partial collapsing operation on T(Ω) which maps ordinals of T(Ω) into the countable segment TΩ 0 of T(Ω). Assume that the (fast (...) growing) extended Grzegorczyk hierarchy $(F_a )_{a \in T(\Omega )_0 }$ and the slow growing hierarchy $(G_a )_{a \in T(\Omega )_0 }$ are defined with respect to the natural system of distinguished fundamental sequences of Buchholz and Schütte (1988) in the following way: $$\begin{array}{*{20}c} {G_0 (n): = 0,} & {F_0 (n): = (n + 1)^2 ,} \\ {\begin{array}{*{20}c} {G_{a + 1} (n): = G_a (n) + 1,} \\ {G_l (n): = G_{l[n]} (n),} \\ \end{array} } & {\begin{array}{*{20}c} {F_{a + 1} (n): = \underbrace {F_a (...F_a }_{n + 1 - times}(n)...),} \\ {F_l (n): = F_{l[n]} (n),} \\ \end{array} } \\ \end{array}$$ wherel is a countable limit ordinal (term) and (l[n]) n ∈N is the distinguished fundamental sequence assigned tol. For each ordinal (term)a in T(Ω) and each natural numbern letC n (a) be the formal term which results from the ordinal terma by successively replacing every occurence ofψ a by $\psi _{ - 1 + C_n (a)}$ whereψ −1 is considered as a defined function symbol, namely $\psi _{ - 1} b: = F_{\psi _0 b + 1} (n + 1)$ . (Note thatψ a 0=Ω a ) In this article it is shown that for each ordinal termψ 0 a in T(Ω) there exists a natural numbern 0 such thatψ 0 C n (a) ∈ T(Ω) and $G_{\psi _0 a} (n) \leqslant F_{\psi _0 C_n (a) + 1} (n + 1)$ holds for alln≥n 0. This hierarchy comparison theorem yields a plethora of new results on nontrivial lower bounds for the slow growing ordinals — i.e. ordinals for which the slow growing hierarchy yields a classification of the provably total functions of the theory in question — of various theories of iterated inductive definitions (and subsystems ofKPi) and on the number and size of the subrecursively inaccessible ordinals — i.e. ordinals at which the extended Grzegorczyk hierarchy and the slow growing hierarchy catch up — below the proof-theoretic ordinal ofACA 0+(Π 1 l −TR). In particular these subrecursively inaccessibles ordinals are necessarily of the form $\psi _0 \Omega ..._{\Omega _\omega }$. (shrink)

This book presents and survey of the foundations of mathematics. The emphasis is on a mathematical comparison of systems rather than on any exhaustive development of analysis within a single system. Nevertheless, for most systems considered, enough details are given for the development of arithmetic, and the method of constructing the other notions of analysis is indicated. The elements of the general theory of cardinal and ordinal numbers are also furnished in the course of this work.

Human aesthetic practices show a sensitivity to the ways that the appearance of an artefact manifests skills and other qualities of the maker. We investigate a possible origin for this kind of sensibility, locating it in the need for co-ordination of skill-transmission in the Acheulean stone tool culture. We argue that our narrative supports the idea that Acheulian agents were aesthetic agents. In line with this we offer what may seem an absurd comparison: between the Acheulian and the Quattrocento. In (...) making it we display some hidden richness in what counts as an aesthetic response to an artefact. We conclude with a brief review of rival explanations—biological and/or cultural—of how this skills-based sensibility became a regular feature of human aesthetic practices. (shrink)

. Consistent valuation and societal prioritization of risks presupposes comparability among risks, that is, in order to rank risks in order of severity, and allocate risk preventative resources accordingly, we must be able to determine whether one risk is better or worse than another, and by how much. It is often claimed, however, that some risks are not amenable to this kind of comparison because they are incommensurable, which roughly means that they are not comparable with respect to a common (...) cardinal measure. The aim of this thesis is to i) consider what it means to say that two risks are incommensurable, ii) explore if incomparability - comparison failure with respect to a common ordinal scale - ever occurs, and how to model it if it does. Essay I is a critical examination of the most prominent argument for incomparability, the so-called small improvement argument. It is argued that the argument fails because it conflates incomparability and a kind of evaluative indeterminacy. Essay II outlines so-called margin of error principles for comparative value judgements. They are based on the idea that if a proposition concerning the value relation between two value-bearing options is true, but there are sufficiently similar cases in which it is false, it is not available to be known. The usefulness of these principles is demonstrated by utilizing them in an epistemological case against SIA. Essay III presents a novel account of incomplete preference orderings which acknowledges that incomparability can vary in degrees. This is achieved by means of a probabilistic analysis of preferences. (shrink)

In this paper, we identify a paradigm of metalinguistic comparatives in Greek headed by the preposition para. Para clauses are lexically distinct from other comparatives clauses in Greek (headed by apo, apoti). Building on earlier intuitions, we propose a semantics of metalinguistic MORE as a contrast between two propositions in terms of how appropriate of preferred they are by some individual. Syntactically, metalinguistic comparison appears to behave like a co-ordinate structure with ellipsis in the para-clause. Our account is extended to (...) metalinguistic negation, lexicalized by oxi in Greek, which, on a par with metalinguistic comparison, is defined as a binary operator, also contrasting two propositions. (shrink)

Propositional attitude verbs, such as think and want, have long held interest for both theoretical linguists and language acquisitionists because their syntactic, semantic, and pragmatic properties display complex interactions that have proven difficult to fully capture from either perspective. This paper explores the granularity with which these verbs’ semantic and pragmatic properties are recoverable from their syntactic distributions, using three behavioral experiments aimed at explicitly quantifying the relationship between these two sets of properties. Experiment 1 gathers a measure of 30 (...) propositional attitude verbs’ syntactic distributions using an acceptability judgment task. Experiments 2a and 2b gather measures of semantic similarity between those same verbs using a generalized semantic discrimination task and an ordinal scale task, respectively. Two kinds of analyses are conducted on the data from these experiments. The first compares both the acceptability judgments and the semantic similarity judgments to previous classifications derived from the syntax and semantics literature. The second kind compares the acceptability judgments to the semantic similarity judgments directly. Through these comparisons, we show that there is quite fine-grained information about propositional attitude verbs’ semantics carried in their syntactic distributions—whether one considers the sorts of discrete qualitative classifications that linguists traditionally work with or the sorts of continuous quantitative classifications that can be derived experimentally. (shrink)

The dictum ‘“ought” implies “can”’ has a status in moral philosophy in some respects like that of ‘a good player needs good co-ordination’ in talk about ball-games. Clearly, you say something important but not conclusive about proficiency in playing a ball-game when you say that it requires good co-ordination: similarly, you say something important but not conclusive about obligation when you say that it implies a certain possibility or power or ability. Each dictum is a reminder: the one about such (...) courses of physical instruction, the other about such exhortations to duty, as are worth persevering with. It would be hopeless to keep on teaching a boy the moves and tricks of rugby football if he could never co-ordinate well enough to get his eye in, so to speak. Correspondingly, it would be meaningless to recommend that someone ought to do something the specification of which involved a contradiction, and pointless to suggest that he ought to do something which, for quite general reasons, was not, and was certain to remain not, within his power. So each dictum expresses a bluff, no-nonsense wisdom which we should count on before involving ourselves in certain more detailed commitments. But probably this is as far as the comparison between the two sayings can well be taken. (shrink)

Psychologists debate whether mental attributes can be quantified or whether they admit only qualitative comparisons of more and less. Their disagreement is not merely terminological, for it bears upon the permissibility of various statistical techniques. This article contributes to the discussion in two stages. First it explains how temperature, which was originally a qualitative concept, came to occupy its position as an unquestionably quantitative concept (§§1–4). Specifically, it lays out the circumstances in which thermometers, which register quantitative (or cardinal) (...) differences, became distinguishable from thermoscopes, which register merely qualitative (or ordinal) differences. I argue that this distinction became possible thanks to the work of Joseph Black, ca. 1760. Second, the article contends that the model implicit in temperature’s quantitative status offers a better way for thinking about the quantitative status of mental attributes than models from measurement theory (§§5–6). (shrink)

This paper provides a characterization of a new class of ordinal poverty measures that are defined by means of the aggregate generalized poverty gap. To be precise, we propose to use the sum of the differences between the transformed fixed poverty line and the transformed level of income of each person below the line as our measure. If the transformation is strictly concave, the resulting measure is strictly inequality averse with respect to the incomes of the poor. In analogy (...) to some existing results on inequality measurement, we show that the only relative (scale-invariant) members of our class are based on strictly concave power functions or the natural logarithm. Moreover, we show that our measures allow for a useful decomposition that is akin to those examined in some earlier contributions. In an empirical analysis, we compare the logarithmic variant of our index to two well-established alternative orderings. Unlike numerous indices that appear in the earlier literature, ours do not explicitly depend on the number of poor or on the total population size, thereby ruling out any direct influence of the head-count ratio on poverty comparisons. (shrink)

The COVID-19 pandemic rapidly led to changes in the mode of teaching, learning and assessments in most tertiary institutions worldwide. Notably, non-invigilated summative assessments became predominant. These changes heightened anxiety and depression, especially among individuals with less resilient coping mechanism. We explored the perceptions and experiences of mental health difficulties of students in tertiary education regarding non-invigilated alternative assessments in comparison to invigilated assessments. A pragmatic, mixed method cross sectional design was conducted online via Qualtrics. Thematic analysis of text was (...) carried out using NVivo 12. In the quantitative analysis, univariable and multivariable ordinal logistic models were used to examine the potential factors for preference among students in higher education. A total of 380 Nursing and Social Science students responded to the survey. Approximately 77% of students perceived non-invigilated assessments to be less stressful compared to invigilated exams. Age, course of study, stage of studies, and number of units enrolled per semester were identified as significant drivers for students’ perceived preference for non-invigilated assessments. There was an inverse relationship between the perception of stress associated with invigilated exams and the age of students. For instance, students aged between 18-24 were 5 times more likely to prefer non-invigilated exams compared to those aged 55 or more. Comparatively, students in early stages of studies had higher preference for non-invigilated assessments. However, there was a preference reversal for students enrolled in 2 or less units per semester. Social sciences students were two times more likely to prefer non-invigilated examinations to invigilated examinations compared to nursing students. The findings reinforce the use of alternative assessments in higher education as a mitigating agency to lessen the mental health burden of tertiary students in post COVID-19 era. (shrink)

This book offers a plurality of perspectives on the historical origins of logicism and on contemporary developments of logicist insights in philosophy of mathematics. It uniquely provides up-to-date research and novel interpretations on a variety of intertwined themes and historical figures related to different versions of logicism. The essays, written by prominent scholars, are divided into three thematic sections. The first section focuses on major authors like Frege, Dedekind, and Russell, providing a historical and theoretical exploration of such figures in (...) the philosophical and mathematical milieu in which logicist views were first expounded. Section II sheds new light on the interconnections between these founding figures and a number of influential other traditions, represented by authors like Hilbert, Husserl, and Peano, as well as on the reconsideration of logicism by Carnap and the logical empiricists. Finally, the third section assesses the legacy of such authors and of logicist themes for contemporary philosophy of mathematics, offering new perspectives on highly debated topics--neo-logicism and its extension to accounts of ordinal numbers and set-theory, the comparison between neo-Fregean and neo-Dedekindian varieties of logicism, and the relation between logicist foundational issues and empirical research on numerical cognition--which define the prospects of logicism in the years to come. This book represents a comprehensive account of the development of logicism and its contemporary relevance for the logico-philosophical foundations of mathematics. It will be of interest to graduate students and researchers working in philosophy of mathematics, philosophy of logic, and the history of analytic philosophy. (shrink)

Il seguente saggio è un'interpretazione speculativa di Padri, scolastici e noi stessi di Hans Urs von Balthasar. L'articolo propone che l'analogia possa essere usata per comprendere le somiglianze e le differenze tra stili teologici in epoche diverse. L'analogia può essere uno strumento euristico per riconoscere la forma strutturale che rimane coerente attraverso i secoli, nonostante le espressioni singole di questa forma secondo la situazione concreta e storica delle persone a cui si rivolge il messaggio del Vangelo. Per affermare ciò, questo (...) saggio traccia un confronto tra il modo in cui si sviluppano la teologia e la dottrina Cattolica. In entrambi i casi, troviamo un paradigma di omogeneità sostanziale all'interno di una differenza proposizionale o stilistica. La differenza non esclude l'unità in un ordine superiore. Diversi stili in teologia sono analogicamente uniti nella forma strutturale generale e immutabile della verità rivelata in Cristo. The following essay is a tentative interpretation of Hans Urs von Balthasar’s Fathers, Scholastics and Ourselves. It proposes that analogy can be used to understand the similarities and differences between theological styles in different epochs. It can be a heuristic tool to recognize the structural form that remains consistent throughout the ages, despite the unique expressions of this form according to the concrete and historical situation of the people being addressed by the message of the Gospel. This essay draws a comparison between the way Catholic theology and doctrine develop. In both cases, we find a paradigm of substantial homogeneity within a propositional or stylistic difference. The difference does not rule out a unity in a higher order. Different styles in theology are analogically united in an overarching and unchanging form of the truth revealed in Christ. (shrink)

We present some new decision and comparison problems of unusually high computational complexity. Most of the problems are strictly combinatorial in nature; others involve basic logical notions. Their complexities range from iterated exponential time completeness to (0 time completeness to ((((,0) time completeness to ((((,,0) time completeness. These three ordinals are well known ordinals from proof theory, and their associated com- plexity classes represent new levels of computational complexity for natural decision problems. Proofs will appear in an extended version of (...) this manuscript to be published elsewhere. (shrink)

Since Sen's insightful analysis of Arrow's Impossibility Theorem (Sen, 1970/1979), Arrow's theorem is often interpreted as a consequence of the exclusion of interpersonal information from Arrow's framework. Interpersonal comparability of either welfare levels or welfare units is known to be sufficient for circumventing Arrow's impossibility result (e.g. Sen, 1970/1979, 1982; Roberts, 1980; d'Aspremont, 1985). But it is less well known whether one of these types of comparability is also necessary or whether Arrow's conditions can already be satisfied in much narrower (...) informational frameworks. This note explores such a framework: the assumption of (ONC+0), ordinal measurability of welfare with the additional measurability of a 'zero-line', is shown to point towards new, albeit limited, escape-routes from Arrow's theorem. Some existence and classification results are established, using the condition that social orderings be transitive as well as the condition that social orderings be quasi-transitive. (shrink)

This paper proposes a revised Theory of Moves (TOM) to analyze matrix games between two players when payoffs are given as ordinals. The games are analyzed when a given player i must make the first move, when there is a finite limit n on the total number of moves, and when the game starts at a given initial state S. Games end when either both players pass in succession or else a total of n moves have been made. Studies are (...) made of the influence of i, n, and S on the outcomes. It is proved that these outcomes ultimately become periodic in n and therefore exhibit long-term predictable behavior. Efficient algorithms are given to calculate these ultimate outcomes by computer. Frequently the ultimate outcomes are independent of i, S, and n when n is sufficiently large; in this situation this common ultimate outcome is proved to be Pareto-optimal. The use of ultimate outcomes gives rise to a concept of stable points, which are analogous to Nash equilibria but consider long-term effects. If the initial state is a stable point, then no player has an incentive to move from that state, under the assumption that any initial move could be followed by a long series of moves and countermoves. The concept may be broadened to that of a stable set. It is proved that every game has a minimal stable set, and any two distinct minimal stable sets are disjoint. Comparisons are made with the results of standard TOM. (shrink)

There can be no relevant cardinal assessment of the welfares of individuals that would allow traditional comparisons of average and total welfare of whole societies to be made. Given that cardinally additive welfare measures are unavailable, I work out some of the implications of an ordinal utilitarian analysis of international distributional issues. I first address the general problem of utilitarian comparisons between aggregates, then the nature of ordinal transfers between groups or nations, and then the complications (...) that population growth in impoverished nations entails for such comparisons. I conclude with remarks on the difficulties and the benefits of thinking ordinally in general. (shrink)

Human aesthetic practices show a sensitivity to the ways that the appearance of an artefact manifests skills and other qualities of the maker. We investigate a possible origin for this kind of sensibility, locating it in the need for co-ordination of skill-transmission in the Acheulean stone tool culture. We argue that our narrative supports the idea that Acheulean agents were aesthetic agents. In line with this we offer what may seem an absurd comparison: between the Acheulean and the Quattrocento. In (...) making it we display some hidden complexity in human aesthetic responses to an artefact. We conclude with a brief review of rival explanations—biological and/or cultural—of how this skills-based sensibility became a regular feature of human aesthetic practices. (shrink)

Measuring psychological attributes, such as motivation, typically involves rating scales, assuming that an attribute can be ordered, and that ratings represent this order. Previously, only the first assumption had been tested, albeit limited. First, we checked the ordinal structure of motivation, looking at whether people can establish transitive relations between motivation levels in pairwise comparisons; and we found different ordering patterns: strict transitive, weak transitive, changing order, and intransitivity. The rate of intransitivity was similar to that found previously (...) and was somewhat higher than we obtained when we asked participants to compare definitely quantitative attributes. Second, we checked if specific ordering patterns were related to individual interpretations of the statements that deviated from expected motivation types. Indeed, about a third of participants miscategorized statements, and these deviant interpretations were related to intransitivity as well as weak transitivity. Third, we checked whether Likert ratings represent the order of motives obtained from pairwise comparisons. We found rather homomorphic representation: ratings correlated with the order, but they did not differentiate between different ordering patterns and hierarchies of motives. We conclude that the Likert rating scale provides less information about respondents than pairwise ordering. The findings question the mainstream practice of using rating scales without testing underlying assumptions. (shrink)