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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

Daniel Isaacson has advanced an epistemic notion of arithmetical truth according to which the latter is the set of truths that we grasp on the basis of our understanding of the structure of natural numbers alone. Isaacson’s thesis is then the claim that Peano Arithmetic (PA) is the theory of finite mathematics, in the sense that it proves all and only arithmetical truths thus understood. In this paper, we raise a challenge for the thesis and show how it can be (...) overcome. We introduce the concept of purity for theories of arithmetic: a theory of arithmetic is pure when it only proves arithmetical truths. Then, we argue that, under Isaacson’s thesis, some PA-provable truths—including transfinite induction claims for infinite ordinals and some consistency statements—are seemingly not arithmetical in Isaacson’s sense, and hence that Isaacson’s thesis might entail the impurity of PA. Nonetheless, we conjecture that the advocate of Isaacson’s thesis can avoid this undesirable consequence: the arithmetical nature, as understood by Isaacson, of all contentious PA-provable statements can be justified. As a case study, we explore how this is done for transfinite induction claims with infinite ordinals below ε0. To this end, we show that the PA-proof of such claims employs exclusively resources from finite mathematics, and that ordinals below ε0 are finitary objects despite being infinite. (shrink)

Renaissance philosopher, mathematician, and theologian Nicholas of Cusa (1401-1464) said that there is no proportion between the finite mind and the infinite. He is fond of saying reason cannot fully comprehend the infinite. That our best hope for attaining a vision and understanding of infinite things is by mathematics and by the use of contemplating symbols, which help us grasp "the absolute infinite". By the late 19th century, there is a decisive intervention in mathematics and its philosophy: the philosophical mathematician (...) Georg Cantor (1845-1918) says that between the realm of the finite and the absolute infinite, there is an intermediate realm partaking in properties in a certain sense of both the finite and the infinite: the transfinite realm. Like the finite, the transfinite realm is a realm of mathematical objects, numbers, and knowledge. Like the absolute infinite, the transfinite is a form of infinity insofar as transfinite sets and numbers transcend any finite number. Echoing Cusanus and neo-Platonism, Cantor says that the transfinite sequence of all ordinals is a symbol of the absolutely infinite, that is, God. This paper considers how the doctrine of symbolism and the philosophers' different commitments to the laws of logic, especially the Law of Non-Contradiction (LNC), enables these thinkers to articulate a transcendental apophatic approach to divinity. (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.

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.

Pubblicato agli inizi degli anni Cinquanta del Novecento da Ernst Bloch (1885-1977), Gli strati della categoria della possibilità diventò in seguito il diciottesimo capitolo del suo Principio speranza. Vi è presentata un’indagine teorica e storica su che cosa sia la possibilità. Da Aristotele a Marx, passando per Averroè, Giordano Bruno e Hegel, lettrici e lettori ripercorreranno le proposte avanzate nella storia della filosofia per affrontare la questione del possibile, fino a scoprire la teoria della possibilità reale. Grazie alla sua particolare (...) struttura, che sviluppa l’argomentazione facendo emergere in ordine di complessità crescente i principali elementi dell’ontologia del non essere ancora, questo testo è una preziosa chiave per orientarsi nella lettura della filosofia di Bloch. (shrink)

In this short paper I will introduce an idea which, I will argue, presents a fundamental additional challenge to the machine consciousness community. The idea takes the questions surrounding phenomenology, qualia and phenomenality one step further into the realm of intersubjectivity but with a twist, and the twist is this: that an agent’s intersubjective experience is deeply felt and necessarily co-affective; it is enkinaesthetic, and only through enkinaesthetic awareness can we establish the affective enfolding which enables first the perturbation, and (...) then the balance and counter-balance, the attunement and co-ordination of whole-body interaction through reciprocal adaptation. (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)

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)

This paper suggests that time could have a much richer mathematical structure than that of the real numbers. Clark & Read (1984) argue that a hypertask (uncountably many tasks done in a finite length of time) cannot be performed. Assuming that time takes values in the real numbers, we give a trivial proof of this. If we instead take the surreal numbers as a model of time, then not only are hypertasks possible but so is an ultratask (a sequence which (...) includes one task done for each ordinal number—thus a proper class of them). We argue that the surreal numbers are in some respects a better model of the temporal continuum than the real numbers as defined in mainstream mathematics, and that surreal time and hypertasks are mathematically possible. (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)

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)

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)

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)

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)

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)

Information and communication technologies (ICTs) place a crucial emphasis on accountability, intellectual property rights, neutrality, openness, privacy, transparency, and trust; they provide a platform or infrastructure of social norms and expectations. Developing the concept of infraethics, this paper argues that all societies need rules for effective co-ordination and collaboration of their infrastructures, and that their design and maintenance is one of the crucial challenges for our own world today. -/- .

Despre necesitatea şi utilitatea dezvoltării unei filosofii specifice tehnologiei blockchain, accentuând pe aspectele ontologice. După o Introducere în care evidenţiez principalele direcţii filosofice pentru această tehnologie emergentă, în Tehnologia blockchain explicitez modul de funcţionare al blockchain, punând în discuţie direcţiile ontologice de dezvoltare în Proiectarea şi Modelarea acestei tehnologii. Următoarea secţiune este dedicată principalei aplicaţii a tehnologiei blockchain, Bitcoin, cu implicaţiile sociale ale acestei criptovalute. Urmează o secţiune de Filosofie în care identific tehnologia blockchain cu conceptul de heterotopie dezvoltat de (...) Michel Foucault şi o interpretez în lumina tehnologiei notaţionale dezvoltată de Nelson Goodman ca sistem notaţional. În secţiunea Ontologii dezvolt două direcţii de dezvoltare considerate de mine ca importante: Ontologia narativă, bazată pe ideea de ordine şi structură a istoriei transmise prin naraţia istoriei a lui Paul Ricoeur, şi un sistem ontologic bazat pe Ontologiile de intreprindere, bazat pe conceptele şi modelele de intreprindere specifice webului semantic, şi pe care eu îl consider cel mai bine dezvoltat şi cel care va deveni, probabil, sistemul ontologic formal, cel puţin în ceea ce priveşte aspectele economice şi juridice ale tehnologiei blockchain. În Concluzii vorbesc despre direcţiile viitoare de dezvoltare a filosofiei tehnologiei blockchain în general ca o teorie explicativă şi robustă din punct de vedere fenomenologic, consistentă şi care să permită testabilitatea, şi a ontologiilor în special, argumentând pentru necesitatea unei adopţii globale a unui sistem ontologic pentru a dezvolta soluţii transversale şi a rentabiliza tehnologia. -/- Cuvinte cheie: filosofie, blockchain, ontologie, bitcoin, proiectare, modele -/- CUPRINS -/- Abstract Introducere Tehnologia blockchain - - - Proiectare - - - Modele Bitcoin Filosofia Ontologii - - - Ontologii narative - - - Ontologii de intreprindere Concluzii Note Bibliografie -/- DOI: 10.13140/RG.2.2.25492.35204. (shrink)

Incommensurability may be regarded as driving specialisation, on the one hand, and as posing some problems to interdisciplinarity, on the other hand. It may be argued, however, that incommensurability plays no role in either specialisation or interdisciplinarity. Scientific specialties could be defined as simply 'different' (that is, about different things), rather than 'incommensurable' (that is, competing for the explanation of the same phenomena). Interdisciplinarity could be viewed as the co- ordinated effort of scientists possessing complemetary and interlocking skills, and not (...) as the overcoming of some sort of incommensurable divide. This article provides a comprehensive evaluative examination of the relations between specialisation, interdisciplinarity, and incommensurability. Its aim is to defend the relevance of incommensurability to both specialisation and interdisciplinarity. At the same time, it aims at correcting the tendency, common among many philosophers, to regard incommensurability in a restrictive manner - such as, for example, as an almost purely semantic issue. (shrink)

A god is a cosmic designer-creator. Atheism says the number of gods is 0. But it is hard to defeat the minimal thesis that some possible universe is actualized by some possible god. Monotheists say the number of gods is 1. Yet no degree of perfection can be coherently assigned to any unique god. Lewis says the number of gods is at least the second beth number. Yet polytheists cannot defend an arbitrary plural number of gods. An alternative is that, (...) for every ordinal, there is a god whose perfection is proportional to it. The n -th god actualizes the best universe(s) in the n -th level of an axiological hierarchy of possible universes. Despite its unorthodoxy, ordinal polytheism has many metaphysically attractive features and merits more serious study. (shrink)

This paper introduces the special issue on the Concept of God of the Journal of Applied Logics (College Publications). The issue contains the following articles: Logic and the Concept of God, by Stanisław Krajewski and Ricardo Silvestre; Mathematical Models in Theology. A Buber-inspired Model of God and its Application to “Shema Israel”, by Stanisław Krajewski; Gödel’s God-like Essence, by Talia Leven; A Logical Solution to the Paradox of the Stone, by Héctor Hernández Ortiz and Victor Cantero; No New Solutions to (...) the Logical Problem of the Trinity, by Beau Branson; What Means ‘Tri-’ in ‘Trinity’ ? An Eastern Patristic Approach to the ‘Quasi-Ordinals’, by Basil Lourié; The Éminence Grise of Christology: Porphyry’s Logical Teaching as a Cornerstone of Argumentation in Christological Debates of the Fifth and Sixth Centruies, by Anna Zhyrkova; The Problem of Universals in Late Patristic Theology, by Dirk Krasmüller; Intuitionist Reasoning in the Tri-unitrian Theology of Nicolas of Cues, by Antonino Drago. (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)

In the 1980s and 1990s, a series of attempts were made to put into U.S. law a civil rights ordinance that would make it possible to sue the makers and distributors of pornography for doing so (under certain conditions). One defence of such legislation has come to be called "the free speech argument against pornography." Philosophers Rae Langton, Jennifer Hornsby and Caroline West have supposed that this defence of the legislation can function as a liberal defence of the legislation: in (...) particular, a defence of the legislation based on the value of women's liberty. This would be somewhat unexpected given MacKinnon's own antipathy toward liberalism. In this paper, I argue that the free speech argument against pornography cannot be used as a liberal defence of the ordinances. The legislation is, to some extent, self-defeating insofar as it understood in terms acceptable to a fairly standard kind of liberal. This becomes apparent when we consider the value pornography can have for women, which we can see if we consider what female makers, distributors and consumers of pornography have to say about why they make, distribute and consume it. (shrink)

Elementary patterns of resemblance notate ordinals up to the ordinal of Pi^1_1-CA_0. We provide ordinal multiplication and exponentiation algorithms using these notations.

Reinhardt’s conjecture, a formalization of the statement that a truthful knowing machine can know its own truthfulness and mechanicalness, was proved by Carlson using sophisticated structural results about the ordinals and transfinite induction just beyond the first epsilon number. We prove a weaker version of the conjecture, by elementary methods and transfinite induction up to a smaller ordinal.