It is widely held that under ordinal utility, utility differences are ill-defined. Allegedly, for these to be well-defined (without turning to choice under risk or the like), one should adopt as a new kind of primitive quaternary relations, instead of the traditional binary relations underlying ordinal utility functions. Correlatively, it is also widely held that the key structural properties of quaternary relations are entirely arbitrary from an ordinal point of view. These properties would be, in a nutshell, the hallmark of (...) cardinal utility. While much is obviously true in these two tenets, this note explains why, as stated, they should be abandoned. Any ordinal utility function induces a rich quaternary relation. There is such a thing as ordinal utility differences. Furthermore, this induced quaternary relation respects, apart from completeness, the most standard structural properties of quaternary relations. These properties are, from an ordinal point of view, anything but arbitrary; from a quaternary perspective only completeness should be considered the hallmark—if any—of cardinal utility. These facts are explained to be especially relevant to the critical appreciation of the ordinalist methodology. (shrink)

Higher-order logic, with its type-theoretic apparatus known as the simple theory of types (STT), has increasingly come to be employed in theorizing about properties, relations, and states of affairs—or ‘intensional entities’ for short. This paper argues against this employment of STT and offers an alternative: ordinal type theory (OTT). Very roughly, STT and OTT can be regarded as complementary simplifications of the ‘ramified theory of types’ outlined in the Introduction to Principia Mathematica (on a realist reading). While STT, understood as (...) a theory of intensional entities, retains the Fregean division of properties and relations into a multiplicity of categories according to their adicities and ‘input types’ and discards the division of intensional entities into different ‘orders’, OTT takes the opposite approach: it retains the hierarchy of orders (though with some modifications) and discards the categorisation of properties and relations according to their adicities and input types. In contrast to STT, this latter approach avoids intensional counterparts of the Epimenides and related paradoxes. Fundamental intensional entities lie at the base of the proposed hierarchy and are also given a prominent part to play in the individuation of non-fundamental intensional entities. (shrink)

There are multiple formal characterizations of the natural numbers available. Despite being inter-derivable, they plausibly codify different possible applications of the naturals – doing basic arithmetic, counting, and ordering – as well as different philosophical conceptions of those numbers: structuralist, cardinal, and ordinal. Nevertheless, some influential philosophers of mathematics have argued for a non-egalitarian attitude according to which one of those characterizations is more “legitmate” in virtue of being “more basic” or “more fundamental”. This paper addresses two related issues. First, (...) we review some of these non-egalitarian arguments, lay out a laundry list of different, legitimate, notions of relative priority, and suggest that these arguments plausibly employ different such notions. Secondly, we argue that given a metaphysical-cum-epistemological gloss suggested by Frege’s foundationalist epistomology, the ordinals are plausibly more basic than the cardinals. This is just one orientation to relative priority one could take, however. Ultimately, we subscribe to an egalitarian attitude towards these formal characterizations: they are, in some sense, equally “legitimate”. (shrink)

Transfinite ordinal numbers enter mathematical practice mainly via the method of definition by transfinite recursion. Outside of axiomatic set theory, there is a significant mathematical tradition in works recasting proofs by transfinite recursion in other terms, mostly with the intention of eliminating the ordinals from the proofs. Leaving aside the different motivations which lead each specific case, we investigate the mathematics of this action of proof transforming and we address the problem of formalising the philosophical notion of elimination which characterises (...) this move. (shrink)

Georg Cantor's absolute infinity, the paradoxical Burali-Forti class Ω of all ordinals, is a monstrous non-entity for which being called a "class" is an undeserved dignity. This must be the ultimate vexation for mathematical philosophers who hold on to some residual sense of realism in set theory. By careful use of Ω, we can rescue Georg Cantor's 1899 "proof" sketch of the Well-Ordering Theorem––being generous, considering his declining health. We take the contrapositive of Cantor's suggestion and add Zermelo's choice function. (...) This results in a concise and uncomplicated proof of the Well-Ordering Theorem. (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)

The main objective of this PhD Thesis is to present a method of obtaining strong normalization via natural ordinal, which is applicable to natural deduction systems and typed lambda calculus. The method includes (a) the definition of a numerical assignment that associates each derivation (or lambda term) to a natural number and (b) the proof that this assignment decreases with reductions of maximal formulas (or redex). Besides, because the numerical assignment used coincide with the length of a specific sequence of (...) reduction - the worst reduction sequence - it is the lowest upper bound on the length of reduction sequences. The main commitment of the introduced method is that it is constructive and elementary, produced only through analyzing structural and combinatorial properties of derivations and lambda terms, without appeal to any sophisticated mathematical tool. Together with the exposition of the method, it is presented a comparative study of some articles in the literature that also get strong normalization by means we can identify with the natural ordinal methods. Among them we highlight Howard[1968], which performs an ordinal analysis of Godel’s Dialectica interpretation for intuitionistic first order arithmetic. We reveal a fact about this article not noted by the author himself: a syntactic proof of strong normalization theorem for the system of typified lambda calculus λ⊃ is a consequence of its results. This would be the first strong normalization proof in the literature. (written in Portuguese). (shrink)

ABSTRACT (ENG) One of the concerns of Greek philosophy centred on the question of how a manifold and ordered universe arose out of the primitive state of things. From the mythical accounts dating around the seventh century B.C. to the cosmologies of the Classical period in Ancient Greece, many theories have been proposed in order to answer to this question. How these theories differ in positing a “something” that pre-existed the ordered cosmos has been widely discussed. However, scholars have rarely (...) made explicit how they differ in style of thought. In the span of four centuries the first deductive arguments of the Eleatic philosophers culminated in the emergence of logical proof and a form of explanation of natural phenomena, which consisted of searching for the simplest and fewest premises and deducting implications. In this paper it will be discussed how, at distinct stages of its development, the deductive thinking informed the solutions proposed to solve the chaos-order problem, that of how an ordered universe has been possible. -/- ABSTRACT (ITA) Una delle più importanti questioni della filosofia greca è stata quella di comprendere come sia stato possibile un universo ordinato a partire da uno stato primordiale. Dalle teogonie del VII secolo a.C. fino alle cosmologie dei filosofi dell’età classica, sono state proposte diverse teorie per dare risposta a questa domanda. Come esse differiscano nel postulare l’esistenza di un “qualcosa” di primordiale che preesisteva all’ordine del cosmo è stato molto discusso. Pochi studiosi, però, le hanno esaminate sullo sfondo della lenta evoluzione del pensiero deduttivo, culminata nella dimostrazione in geometria e in una forma di spiegazione dei fenomeni che consisteva nel cercare semplici premesse e inferire conclusioni. In questo articolo si mostrerà come il lento affermarsi della spiegazione razionale prima, dell’argomento deduttivo e della dimostrazione in geometria poi, abbiano dato forma alle diverse risposte al problema caos-ordine, e in particolare alla domanda su come sia sorto un universo ordinato. (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)

Darwiche and Pearl’s seminal 1997 article outlined a number of baseline principles for a logic of iterated belief revision. These principles, the DP postulates, have been supplemented in a number of alternative ways. Most suggestions have resulted in a form of ‘reductionism’ that identifies belief states with orderings of worlds. However, this position has recently been criticised as being unacceptably strong. Other proposals, such as the popular principle (P), aka ‘Independence’, characteristic of ‘admissible’ operators, remain commendably more modest. In this (...) paper, we supplement the DP postulates and (P) with a number of novel conditions. While the DP postulates constrain the relation between a prior and a posterior conditional belief set, our new principles notably govern the relation between two posterior conditional belief sets obtained from a common prior by different revisions. We show that operators from the resulting family, which subsumes both lexicographic and restrained revision, can be represented as relating belief states associated with a ‘proper ordinal interval’ (POI) assignment, a structure more fine-grained than a simple ordering of worlds. We close the paper by noting that these operators satisfy iterated versions of many AGM era postulates, including Superexpansion, that are not sound for admissible operators in general. (shrink)

Gauthier and Hobbes reduce Prisoners Dilemmas to co-ordination problems (CPs). Many think rational, face-to-face agents can solve any CP by agreed fiat. But though an agent can rationally use a symmetry-breaking technique (ST) to decide between equal options, groups cannot unless their members' STs luckily converge. Failing this, the CP is escapable only by one agent's non-rational stubbornness, or by the group's "conquest" by an outside force. Implications: one's strategic rationality is group-relative; there are some optimums groups in principle (...) cannot rationally choose; thus justice cannot always be a rationally contracted optimum. Howard Sobel provides the point of departure. (shrink)

The symbolic argument against women's ordination supposes that the theological significance of Christ's sex is his saving relationship to the Church, which takes the form of that of a bridegroom and his bride. It infers that a male priest alone is fit to represent Christ in his capacity as the Saviour of the Church, and thus that only men should be ordained. Since the emergence of the symbolic argument, however, scholars have rediscovered a long tradition of understanding Christ's saving (...) relationship to the Church in maternal terms. While remaining neutral on whether women ultimately ought to be ordained or not, I argue that the kind of reasoning in the symbolic argument, if updated with the Jesus as Mother tradition, would suggest that it is fitting for a female priest to represent Jesus as Mother, just as it is fitting for a male priest to represent Christ the Bridegroom. Other critics of the symbolic argument tend to contest what is seen as its ‘literalist’ or ‘essentialist’ or overly gendered-valanced assumptions about priestly representation. I show that even if we grant to the symbolic argument more gendered and ‘essentialist’ views on each of the major points of disagreement in these debates, women might still be fit to represent Christ because of the maternal ways that Christ has traditionally been thought to relate to the Church. As a result, the symbolic argument may be repurposed to support women's ordination rather than undermine it. (shrink)

For wise men and academics, the order of the world had been an implicit truth for a long time. Nobody had doubted it. The purpose of philosophy and theological discourse has always been to record and contemplate the mundane and celestial world. Because it has a Creator, the universe (and the human being) are in complete order (id est they are in orderliness), in a rigorous hierarchy and, at the same time, they have a meaning, a purpose, they are part (...) of a meaningful whole. In an orderly world, life instantly acquires meaning. This venerable certainty has become, in time, a simple hypothesis. Philosophers are no longer sure that the world is orderly. The following essays comment on the decay of a certainty and, in the end, on its loss. Borges and Georges Perec dared to record (with bitter irony) this loss. And this tripartite essay will speak about them. (shrink)

Dopo mezzo secolo di oblio, la natura della coscienza è ora l'argomento più caldo nelle scienze comportamentali e nella filosofia. A partire dal lavoro pionieristico di Ludwig Wittgenstein negli anni '30 (i libri blu e brown) e dagli anni '50 ad oggi dal suo logico successore John Searle, ho creato la seguente tabella come euristica per promuovere questo studio. Le righe mostrano vari aspetti o modi di studio e le colonne mostrano i processi involontari e i comportamenti volontari che comprendono (...) i due sistemi (processi duali) della struttura logica della coscienza (LSC), che possono anche essere considerati come la struttura logica della razionalità (LSR-Searle), del comportamento (LSB), della personalità (LSP), della realtà (LSOR), dell'intenzionalità (LSI) -il termine filosofico, filosofico la Psicologia Descrittiva della Coscienza (DPC) , la Psicologia descrittiva del Pensiero (DPT) –o meglio, il Linguaggio della Psicologia Descrittiva del Pensiero (LDPT), termini introdotti qui e nei miei altri scritti molto recenti. Coloro che desiderano un quadro aggiornato completo per il comportamento umano dalla moderna vista a due systems possono consultare il mio libro 'La struttura logica dellafilosofia, psicologia, Mind e il linguaggio in Ludwig Wittgenstein e John Searle' 2nd ed (2019). Coloro che sono interessati a più dei miei scritti possono vedere 'Talking Monkeys--Filosofia, Psicologia, Scienza, Religione e Politica su un Pianeta Condannato--Articoli e Recensioni 2006-2019 3rd ed (2019) e Suicidal Utopian Delusions nel 21st Century 4th ed (2019). (shrink)

Seguendo l’esposizione data in (Orsi 2012), riguardante una comparazione fra alcuni aspetti dell’opera di Carl Schmitt e di Jürgen Habermas in filosofia politica, centrata sulla nozione di ordine ed inquadrata, nelle sue basi, entro la sociologia delle religioni di Max Weber, sarà possibile, oltre l’individuazione in essa di un comune punto di convergenza fra il pensiero dei questi autori nella nozione di ordine, portare avanti, su un piano teoretico di livello superiore, un ulteriore raffronto più orientato verso la metodologia della (...) ricerca filosofica così come intesa da Martin Heidegger, la quale permetterà tra l’altro di vedere sia Schmitt che Habermas da un altro possibile punto di vista prospettico. (shrink)

The editor's introduction discusses Clarence I. Lewis's conceptual pragmatism when compared with post-empiricist epistemology and argues that several Cartesian assumptions play a major role in the work, not unlike those of Logical Positivism. The suggestion is made that the Cartesian legacy still hidden in Logical Positivism turns out to be a rather heavy ballast for Lewis’s project of restructuring epistemology in a pragmatist key. More in detail, the sore point is the nature of inter-subjectivity. For Lewis, no less than for (...) the Logical Positivists at the time of the Protocols Controversy and Husserl in the Cartesian Meditations, this is a problem without a solution. The reason is that all these philosophers are apparently unable to realize that the existence of a plurality of knowing subjects cannot be treated at once both as a speculative problem and a methodological one. Lewis, thanks to his pragmatist approach both comes closer to the right answer and offers an even more naïve unsatisfactory solution to the pseudo-problem under discussion. The fact that he has clear in mind that inter-subjectivity means not only a plurality of linguistic utterances but also a co-existence of different kinds of practical behaviour. Eventually, the very idea of mind, the key-idea in the book, suffers from the above mentioned tension. In fact, if inter-subjective communication and action is considered at a methodological level, the very idea of mind would not need an analysis, and no kind of ‘reflexive’ analysis. Methodology might be limited to a ‘naïve’ level where the existence of the world and a plurality of subjects be taken as a bedrock of uncritically accepted evidence. Philosophical reflection on ultimate evidence, instead, would take a different approach, maybe the one Wittgenstein was putting into practice in the same years when Mind and the world order was written, namely it would be bound to question the very meaning of the idea of ‘mind’ as an undue fiction – the same carried out by Descartes – when he assumed the Cogito to be at once a body of self-evident truths and a thing or substance, the familiar Platonic idea of psyche or soul. (shrink)

Till just few decades ago, scholars used to use the label ‘Ockhamism’ to mark a turning-point in the history of mediaeval philosophy, above all in the history of natural philosophy. That turning-point was exemplified by the once so-called ‘Buridanian school’, today known simply as ‘Parisian school of natural philosophy’, whose leading representative was for sure John Buridan. But looking carefully at some crucial points of the Picard master’s idea of ‘nature’, concerning specifically the relationship between God and secondary causes on (...) one side, and the role played by free will, both the divine and the human one, on the other side, one will find out that Buridan’s ‘Ockhamism’ is at least questionable: Scotus’ modal logic is the conceptual framework, within which he redetermines the boundaries of natural laws, but the ‘pure ockhamist’ idea of the prevalence of the will on the intellect in God’s nature is the source of an image of the world in which voluntary behaviours and natural laws are intertwined. Therefore, Buridan’s natural philosophy results as a peculiar synthesis of two of the main divergent lines of thinking of late mediaeval philosophy. (shrink)

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 distinguish 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 differences, while being non- etheless empirically related to choices. We highlight the originality, promises and limits of this choice-based cardinalism. (shrink)

We define a notion of the intelligence level of an idealized mechanical knowing agent. This is motivated by efforts within artificial intelligence research to define real-number intelligence levels of compli- cated intelligent systems. Our agents are more idealized, which allows us to define a much simpler measure of intelligence level for them. In short, we define the intelligence level of a mechanical knowing agent to be the supremum of the computable ordinals that have codes the agent knows to be codes (...) of computable ordinals. We prove that if one agent knows certain things about another agent, then the former necessarily has a higher intelligence level than the latter. This allows our intelligence no- tion to serve as a stepping stone to obtain results which, by themselves, are not stated in terms of our intelligence notion (results of potential in- terest even to readers totally skeptical that our notion correctly captures intelligence). As an application, we argue that these results comprise evidence against the possibility of intelligence explosion (that is, the no- tion that sufficiently intelligent machines will eventually be capable of designing even more intelligent machines, which can then design even more intelligent machines, and so on). (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)

This quantitative study utilized the descriptive method of research that aimed to assess the implementation of environmental conservation practices in Baybay City Leyte with 92 officials and 115 zone residents of the 23 barangays in the city. Data were gathered through online and pen-paper surveys using frequency mean and Chi-square test. Barangay officials & Zone residents showed the same level of awareness of the environmental conservation practices implemented in Baybay City. Based on the results, the highest average weighted mean environmental (...) conservation practices were the Ordinance on Solid Waste Management. Whereas, the Ordinance on the Use of Plastics Bags has the lowest average weighted mean. There is no significant difference in the level of awareness of the environmental conservation practices between the barangay officials and zone residents. The findings on the level of implementation of the environmental conservation practices, barangay officials asserted a high level of implementation while zone residents claim to have implemented of all the (3) ordinances. For barangay officials, the Ordinance on Solid Waste Management had the highest weighted mean. While the Ordinance on the Use of Plastics Bags had the lowest weighted mean. Whereas, for the zone residents the Ordinance on Regulating the Use of Plastics Bags had the highest weighted mean. While the Ordinance on Anti-Littering and Prohibiting Persons from Throwing any Refuse or Garbage, Spitting, or Improper Urinating into Streets and Canals had the lowest weighted mean. There is a significant difference in the level of implementation of the environmental conservation practices between the barangay officials and zone residents on the Ordinance on Anti-Littering and Prohibiting Persons from Throwing any Refuse or Garbage, Spitting, Improper Urinating into Streets and Canal and Ordinance on Solid Waste Management. However, there is no significant difference in the level of implementation of environmental conservation practices in the Ordinance of the Use of Plastics Bags. (shrink)

From classical mechanics, in particular the motion in a straight line, together set theory and ordinal number theory, we prove a not-classical behaviour, a discontinuous motion and emission.

Ordinal polytheism is motivated by the cosmological and design arguments. It is also motivated by Leibnizian–Lewisian modal realism. Just as there are many universes, so there are many gods. Gods are necessary concrete grounds of universes. The god-universe relation is one-to-one. Ordinal polytheism argues for a hierarchy of ranks of ever more perfect gods, one rank for every ordinal number. Since there are no maximally perfect gods, ordinal polytheism avoids many of the familiar problems of monotheism. It links theology with (...) counterpart theory, mathematics and computer science. And it entails that the system of universes has an attractive axiological structure. (shrink)

This paper develops co-ordinated multiple-domain supervenience relations to model determination and dependence relations between complex entities and their constituents by appealing to R-related pairs and by making use of associated isomorphisms. Supervenience relations are devised for order-sensitive and repetition-sensitive mereologies, for mereological systems that make room for many-many composition relations, as well as for hierarchical mereologies that incorporate compositional and hylomorphic structure. Finally, mappings are provided for theories that consider wholes to be prior to their parts.

The contemporary revival of Eucharistic ecclesiology has occurred alongside a new understanding of the episcopacy as a distinct grade of holy orders. Both of these developments make possible a new synthetic understanding of the presbyterate, building on classical theological approaches to orders that incorporate both of these perspectives. In this essay, I will attempt to show how the theology of the presbyterate articulated by Thomas Aquinas might help supplement and be supplemented by that of John Zizioulas. The synthesis I propose (...) has the merit of corresponding with much of the classical approach to orders, while advancing a new thesis on the Eucharistic role of the presbyterate that establishes a clearer connection between contemporary ecclesiology and the theology of holy orders. (shrink)

Decision-making under normative uncertainty requires an agent to aggregate the assessments of options given by rival normative theories into a single assessment that tells her what to do in light of her uncertainty. But what if the assessments of rival theories differ not just in their content but in their structure -- e.g., some are merely ordinal while others are cardinal? This paper describes and evaluates three general approaches to this "problem of structural diversity": structural enrichment, structural depletion, and multi-stage (...) aggregation. All three approaches have notable drawbacks, but I tentatively defend multi-stage aggregation as least bad of the three. (shrink)

in Oct-14-1998 ordinance INDESP-IO4 established the federal software certification and verification requirements for gaming machines in Brazil. The authors present the rationale behind these criteria, whose basic principles can find applications in several other software authentication applications.

The quantum information introduced by quantum mechanics is equivalent to that generalization of the classical information from finite to infinite series or collections. The quantity of information is the quantity of choices measured in the units of elementary choice. The qubit can be interpreted as that generalization of bit, which is a choice among a continuum of alternatives. The axiom of choice is necessary for quantum information. The coherent state is transformed into a well-ordered series of results in time after (...) measurement. The quantity of quantum information is the ordinal corresponding to the infinity series in question. Number and being (by the meditation of time), the natural and artificial turn out to be not more than different hypostases of a single common essence. This implies some kind of neo-Pythagorean ontology making related mathematics, physics, and technics immediately, by an explicit mathematical structure. (shrink)

The goal of this note is to bring into wider attention the often neglected important work by Bertrand Russell on the foundations of physics published in the late 1920s. In particular, we emphasize how the book The Analysis of Matter can be considered the earliest systematic attempt to unify the modern quantum theory, just emerging by that time, with general relativity. More importantly, it is argued that the idea of what I call Russell space, introduced in Part III of that (...) book, is more fundamental than quantum theory, general relativity, and quantum gravity since the topological ordinal space proposed by Russell would naturally incorporate into its very fabric the emergent nature of spacetime by deploying event assemblages, and not spacetime or particles, as the fundamental building blocks of the world. (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 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)

One influential approach seeks to capture the idea of society by characterising social action, or interaction, in terms of the particular kinds of awareness it involves. Another approach focuses on social order, seeing it as a form of order that arises spontaneously when rational and mutually aware individuals succeed in solving co-ordination problems. Yet another approach focuses on the role played by communication in achieving collective agreement on the way the world is to be classified and understood, where this (...) is seen as a pre-condition of co-ordination and co-operation. This encyclopedia entry provides a brief account of each approach and offers an assessment of their stengths and limitations. (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)

Adelson & Pentland (Chapter 11) use an engaging metaphor to illustrate their position on scene analysis: interpretations are produced by a workshop that employs a set of specialists, each concerned with a single aspect of the scene. The authors argue that it is too expensive to have a supervisor co-ordinate the specialists and that it is too expensive to let them operate independently. They then show that a careful sequencing of the specialists leads to solutions of minimum cost, at least (...) for their world of Mondrian panels. (shrink)

Gignac and Zajenkowski (2020) find that “the degree to which people mispredicted their objectively measured intelligence was equal across the whole spectrum of objectively measured intelligence”. This Comment shows that Gignac and Zajenkowski’s (2020) finding of homoscedasticity is likely the result of a recoding choice by the experimenters and does not in fact indicate that the Dunning-Kruger Effect is a mere statistical artifact. Specifically, Gignac and Zajenkowski (2020) recoded test subjects’ responses to a question regarding self-assessed comparative IQ onto a (...) linear IQ scale when a normal IQ scale would likely have been more appropriate. More generally, researchers studying self-assessed intelligence should be aware of potential measurement problems that may arise when transforming an ordinal scale onto an interval scale. (shrink)

One theoretical charge (of Optimality Theory in its early conception) must have been to retain that sense of qualitative particularity as affecting as constraining theory relevant to a proscribed field when clearly a motivation was to divine in circumscriptions operational consequences conceived on a deferred abstractive level. An attraction of the theory's embodying results of constraint interactions as responsive to theory-internal qualitative implementation, as being in fact supplementarily transparent to co-ordinations of variously language specific implementations, qualitative identifications, was apparent naturalistic (...) coordination of defined features taken here—in OT—at one fell-swoop. Differently put, the uses of analyses of the relevant types of spaces (‘primitive from the viewpoint of linguistic theories’) will (latterly) involve a period of time-stretch or compression over a previously normal (I take it) process of embeddings in a new theorised and recursively enacted invocation/re-invocation of base as digestively present in results of corrected theories. Or as Prince and Smolensky have it -/- 'What we have done, in essence, is to replace the iterative procedure (act/evaluate, act/evaluate, . . . ) with a recursive scheme: collect the results of all possible actions, then sort recursively.' OT, 16. (shrink)

The paper considers the symmetries of a bit of information corresponding to one, two or three qubits of quantum information and identifiable as the three basic symmetries of the Standard model, U(1), SU(2), and SU(3) accordingly. They refer to “empty qubits” (or the free variable of quantum information), i.e. those in which no point is chosen (recorded). The choice of a certain point violates those symmetries. It can be represented furthermore as the choice of a privileged reference frame (e.g. that (...) of the Big Bang), which can be described exhaustively by means of 16 numbers (4 for position, 4 for velocity, and 8 for acceleration) independently of time, but in space-time continuum, and still one, 17th number is necessary for the mass of rest of the observer in it. The same 17 numbers describing exhaustively a privileged reference frame thus granted to be “zero”, respectively a certain violation of all the three symmetries of the Standard model or the “record” in a qubit in general, can be represented as 17 elementary wave functions (or classes of wave functions) after the bijection of natural and transfinite natural (ordinal) numbers in Hilbert arithmetic and further identified as those corresponding to the 17 elementary of particles of the Standard model. Two generalizations of the relevant concepts of general relativity are introduced: (1) “discrete reference frame” to the class of all arbitrarily accelerated reference frame constituting a smooth manifold; (2) a still more general principle of relativity to the general principle of relativity, and meaning the conservation of quantum information as to all discrete reference frames as to the smooth manifold of all reference frames of general relativity. Then, the bijective transition from an accelerated reference frame to the 17 elementary wave functions of the Standard model can be interpreted by the still more general principle of relativity as the equivalent redescription of a privileged reference frame: smooth into a discrete one. The conservation of quantum information related to the generalization of the concept of reference frame can be interpreted as restoring the concept of the ether, an absolutely immovable medium and reference frame in Newtonian mechanics, to which the relative motion can be interpreted as an absolute one, or logically: the relations, as properties. The new ether is to consist of qubits (or quantum information). One can track the conceptual pathway of the “ether” from Newtonian mechanics via special relativity, via general relativity, via quantum mechanics to the theory of quantum information (or “quantum mechanics and information”). The identification of entanglement and gravity can be considered also as a ‘byproduct” implied by the transition from the smooth “ether of special and general relativity’ to the “flat” ether of quantum mechanics and information. The qubit ether is out of the “temporal screen” in general and is depicted on it as both matter and energy, both dark and visible. (shrink)

Quantum information is discussed as the universal substance of the world. It is interpreted as that generalization of classical information, which includes both finite and transfinite ordinal numbers. On the other hand, any wave function and thus any state of any quantum system is just one value of quantum information. Information and its generalization as quantum information are considered as quantities of elementary choices. Their units are correspondingly a bit and a qubit. The course of time is what generates choices (...) by itself, thus quantum information and any item in the world in final analysis. The course of time generates necessarily choices so: The future is absolutely unorderable in principle while the past is always well-ordered and thus unchangeable. The present as the mediation between them needs the well-ordered theorem equivalent to the axiom of choice. The latter guarantees the choice even among the elements of an infinite set, which is the case of quantum information. The concrete and abstract objects share information as their common base, which is quantum as to the formers and classical as to the latter. The general quantities of matter in physics, mass and energy can be considered as particular cases of quantum information. The link between choice and abstraction in set theory allows of “Hume’s principle” to be interpreted in terms of quantum mechanics as equivalence of “many” and “much” underlying quantum information. Quantum information as the universal substance of the world calls for the unity of physics and mathematics rather than that of the concrete and abstract objects and thus for a form of quantum neo-Pythagoreanism in final analysis. (shrink)

Multidimensional concepts are everywhere, and they are important. Examples include moral value, welfare, scientific confirmation, democracy, and biodiversity. How, if at all, can we aggregate the underlying dimensions of a multidimensional concept F to yield verdicts about which things are Fer than which overall? Social choice theory can be used to model and investigate this aggregation problem. Here, we focus on a particularly thorny problem made salient by this social choice-theoretic framework: the underlying dimensions of a given concept might be (...) measurable on different types of scales—e.g., some ordinal and some cardinal. An underappreciated impossibility theorem due to Anna Khmelnitskaya shows that seemingly plausible constraints on aggregation across scale types are inconsistent. This impossibility threatens to render the notion of overall Fness incoherent. We attempt to defuse this threat, arguing that the impossibility depends on an overly restrictive conception of measurement and of how measurement constrains aggregation. Adopting a more flexible—and, we think, more perspicuous—conception of measurement opens an array of possibilities for aggregation across disparate scale types. (shrink)

Our visual experience of the world is one of diverse objects and events, each with particular colors, shapes, and motions. This experience is so coherent, so immediate, and so effortless that it seems to result from a single system that lets us experience everything in our field of view. But however appealing, this belief is mistaken: there are severe limits on what can be visually experienced. -/- For example, in a display for air-traffic control it is important to track all (...) moving items. For a single item, this can be done without problem. Three or four can also be tracked, although some degree of effort may be needed. As the number is increased further, accurate tracking becomes more and more difficult—and eventually, impossible. Performance is evidently affected by a factor within the observer which enables certain kinds of perception to occur, but is limited in some way. This factor is generally referred to as attention. -/- At various times, attention has been associated with clarity of perception, intensity of perception, consciousness, selection, or the allocation of a limited “resource” enabling various operations (see Hatfield, 1998). During the past several decades, considerable progress has been achieved by focusing on the idea of selection (Broadbent, 1982). In particular, attention can be productively viewed as contingently selective processing. This can be embodied in various ways by various processes—there need not be a single quantity identified with all forms of attention, or a single site where it operates (Allport, 1993; Tsotsos, 2011). Although “paying attention” is often considered to be a unitary operation, it may simply refer to the control of one or more selective processes, ideally in a co-ordinated way. While this view has some cost in terms of conceptual simplicity, it can help make sense of a large set of phenomena. -/- This article surveys several of the major issues in our understanding of attention and how it relates to perception. It focuses on vision, since many—if not all—considerations are similar for all sensory modalities, and the level of understanding achieved in this domain is currently the most advanced. (shrink)

Linnebo in 2018 argues that abstract objects like numbers are “thin” because they are only required to be referents of singular terms in abstraction principles, such as Hume's principle. As the specification of existence claims made by analytic truths (the abstraction principles), their existence does not make any substantial demands of the world; however, as Linnebo notes, there is a potential counter-argument concerning infinite regress against introducing objects this way. Against this, he argues that vicious regress is avoided in the (...) account of arithmetic based on Hume's principle because we are specifying numbers in terms of the concept of equinumerosity, or its ordinal equivalent. But far from being only a matter for philosophy, this implies a distinct empirical prediction: in cognitive development, the principle of equinumerosity is primary to number concepts. However, by analysing and expanding on the bootstrapping theory of Carey in 2009, I argue in this paper that there are good reasons to think that the development could be the other way around: possessing numerosity concepts may precede grasping the principle of equinumerosity. I propose that this analysis of early numerical cognition can also help us understand what numbers as thin objects are like, moving away from Platonist interpretations. (shrink)

A number of Christian theologians and philosophers have been critical of overly moralizing approaches to the doctrine of sin, but nearly all Christian thinkers maintain that moral fault is necessary or sufficient for sin to obtain. Call this the “Moral Consensus.” I begin by clarifying the relevance of impurities to the biblical cataloguing of sins. I then present four extensional problems for the Moral Consensus on sin, based on the biblical catalogue of sins: (1) moral over-demandingness, (2) agential unfairness, (3) (...) moral repugnance, and (4) moral atrocity. Next, I survey several partial solutions to these problems, suggested by the recent philosophical literature. Then I evaluate two largely unexplored solutions: (a) genuine sin dilemmas and (b) defeasible sinfulness. I argue that (a) creates more problems than it solves and that, while (b) is well-motivated and solves or eases each of the above problems, (b) leaves many biblical ordinances about sin morally misleading, creating (5) a pedagogical problem of evil. I conclude by arguing that (5) places hefty explanatory burdens on those who would appeal to (b) to resolve the four extensional problems discussed in this paper. So Christian thinkers may need to consider a more radical separation of sin and moral fault. (shrink)

It is standard in set theory to assume that Cantor's Theorem establishes that the continuum is an uncountable set. A challenge for this position comes from the observation that through forcing one can collapse any cardinal to the countable and that the continuum can be made arbitrarily large. In this paper, we present a different take on the relationship between Cantor's Theorem and extensions of universes, arguing that they can be seen as showing that every set is countable and that (...) the continuum is a proper class. We examine several principles based on maximality considerations in this framework, and show how some (namely Ordinal Inner Model Hypotheses) enable us to incorporate standard set theories (including ZFC with large cardinals added). We conclude that the systems considered raise questions concerning the foundational purposes of set theory. (shrink)

I follow standard mathematical practice and theory to argue that the natural numbers are the finite von Neumann ordinals. I present the reasons standardly given for identifying the natural numbers with the finite von Neumann's (e.g., recursiveness; well-ordering principles; continuity at transfinite limits; minimality; and identification of n with the set of all numbers less than n). I give a detailed mathematical demonstration that 0 is { } and for every natural number n, n is the set of all natural (...) numbers less than n. Natural numbers are sets. They are the finite von Neumann ordinals. (shrink)

It is well known that the following features hold of AR + T under the strong Kleene scheme, regardless of the way the language is Gödel numbered: 1. There exist sentences that are neither paradoxical nor grounded. 2. There are 2ℵ0 fixed points. 3. In the minimal fixed point the weakly definable sets (i.e., sets definable as {n∣ A(n) is true in the minimal fixed point where A(x) is a formula of AR + T) are precisely the Π1 1 sets. (...) 4. In the minimal fixed point the totally defined sets (sets weakly defined by formulae all of whose instances are true or false) are precisely the ▵1 1 sets. 5. The closure ordinal for Kripke's construction of the minimal fixed point is ωCK 1. In contrast, we show that under the weak Kleene scheme, depending on the way the Gödel numbering is chosen: 1. There may or may not exist nonparadoxical, ungrounded sentences. 2. The number of fixed points may be any positive finite number, ℵ0, or 2ℵ0 . 3. In the minimal fixed point, the sets that are weakly definable may range from a subclass of the sets 1-1 reducible to the truth set of AR to the Π1 1 sets, including intermediate cases. 4. Similarly, the totally definable sets in the minimal fixed point range from precisely the arithmetical sets up to precisely the ▵1 1 sets. 5. The closure ordinal for the construction of the minimal fixed point may be ω, ωCK 1, or any successor limit ordinal in between. In addition we suggest how one may supplement AR + T with a function symbol interpreted by a certain primitive recursive function so that, irrespective of the choice of the Godel numbering, the resulting language based on the weak Kleene scheme has the five features noted above for the strong Kleene language. (shrink)

According to contextualist theories in metaethics, when you use a moral term in a context, the context plays an ineliminable part in determining what natural property will be the semantic value of the term. Furthermore, on subjectivist and relativist versions of these views, it is either the speaker's own moral code or her moral community's moral code that constitutes the reference-fixing context. One standard objection to views of this type is that they fail to enable us to disagree in ordinary (...) conversations. In this chapter, I develop a new response to this objection on the basis of Kai von Fintel and Anthony Gillies' notion of proposition clouds. I argue that, because we live in a multicultural society, the conversational contexts we face will fail to disambiguate between all the things we could mean. This is why we can at best put into play proposition clouds when we make moral utterances. All the propositions in such clouds are then available for rejection and acceptance on the behalf of our audiences. The norms of conversation then guide us to make informative contributions to the conversation - accept and reject propositions in a way that leads to co-ordination of action and choice. (shrink)

We argue that a semantics for counterfactual conditionals in terms of comparative overall similarity faces a formal limitation due to Arrow’s impossibility theorem from social choice theory. According to Lewis’s account, the truth-conditions for counterfactual conditionals are given in terms of the comparative overall similarity between possible worlds, which is in turn determined by various aspects of similarity between possible worlds. We argue that a function from aspects of similarity to overall similarity should satisfy certain plausible constraints while Arrow’s impossibility (...) theorem rules out that such a function satisfies all the constraints simultaneously. We argue that a way out of this impasse is to represent aspectual similarity in terms of ranking functions instead of representing it in a purely ordinal fashion. Further, we argue against the claim that the determination of overall similarity by aspects of similarity faces a difficulty in addition to the Arrovian limitation, namely the incommensurability of different aspects of similarity. The phenomena that have been cited as evidence for such incommensurability are best explained by ordinary vagueness. (shrink)

Gaisi Takeuti extended Gentzen's work to higher-order case in 1950's–1960's and proved the consistency of impredicative subsystems of analysis. He has been chiefly known as a successor of Hilbert's school, but we pointed out in the previous paper that Takeuti's aimed to investigate the relationships between "minds" by carrying out his proof-theoretic project rather than proving the "reliability" of such impredicative subsystems of analysis. Moreover, as briefly explained there, his philosophical ideas can be traced back to Nishida's philosophy in Kyoto's (...) school. For the proving the consistency of such systems, it is crucial to prove the well-foundedness of ordinals called "ordinal diagrams" developed for it. Takeuti presented such arguments several times in order to show that they are admitted in his stand point. As a starting point of investigating his finitist stand point, we formulate the system of ordinal notations up to ε0 and reconstruct the well-foundedness arguments of them. (shrink)

In this paper I will compare Hertz and Wittgenstein in order to bring forth a new interpretation of the Tractatus. I shall look at Wittgenstein’s ideas about simple objects and compare them with Hertz’s material particles. I shall then claim that if one understands Hertz’s material particles as logical entities that are more co-ordinate like than physical, one can reach an interpretation of the Tractatus that is deliberately silent about the nature of reality, therefore escaping both objectivist and subjectivist interpretations (...) of the book. This, I think, is just one of the great results of re-thinking Wittgenstein’s ideas through Hertz’s. (shrink)

In this article we raise a problem, and we offer two practical contributions to its solution. The problem is that academic communities interested in digital publishing do not have adequate tools to help them in choosing a publishing model that suits their needs. We believe that excessive focus on Open Access (OA) has obscured some important issues; moreover exclusive emphasis on increasing openness has contributed to an agenda and to policies that show clear practical shortcomings. We believe that academic communities (...) have different needs and priorities; therefore there cannot be a ranking of publishing models that fits all and is based on only one criterion or value. We thus believe that two things are needed. First, communities need help in working out what they want from their digital publications. Their needs and desiderata should be made explicit and their relative importance estimated. This exercise leads to the formulation and ordering of their objectives. Second, available publishing models should be assessed on the basis of these objectives, so as to choose one that satisfies them well. Accordingly we have developed a framework that assists communities in going through these two steps. The framework can be used informally, as a guide to the collection and systematic organization of the information needed to make an informed choice of publishing model. In order to do so it maps the values that should be weighed and the technical features that embed them. Building on our framework, we also offer a method to produce ordinal and cardinal scores of publishing models. When these techniques are applied the framework becomes a formal decision–making tool. Finally, the framework stresses that, while the OA movement tackles important issues in digital publishing, it cannot incorporate the whole range of values and interests that are at the core of academic publishing. Therefore the framework suggests a broader agenda that is relevant in making better policy decisions around academic publishing and OA. (shrink)