A union ultrafilter is an ultrafilter over the finite subsets of ω that has a base of sets of the form ${\text{FU}}\left$, where X is an infinite pairwise disjoint family and ${\text{FU}} = \left\{ {\bigcup {F|F} \in [X]^{ < \omega } \setminus \{ \emptyset \} } \right\}$. The existence of these ultrafilters is not provable from the $ZFC$ axioms, but is known to follow from the assumption that ${\text{cov}}\left = \mathfrak{c}$. In this article we obtain various models of $ZFC$ that (...) satisfy the existence of union ultrafilters while at the same time ${\text{cov}}\left = \mathfrak{c}$. (shrink)

In the context of [Formula: see text], we analyze a version of Hindman’s finite unions theorem on infinite sets, which normally requires the Axiom of Choice to be proved. We establish the implication relations between this statement and various classical weak choice principles, thus precisely locating the strength of the statement as a weak form of the [Formula: see text].

Political fact-checkers evaluate the truthfulness of politicians’ claims. This paper contributes to an emerging scholarly debate on whether fact-checkers treat political parties differently in a systematic manner depending on their ideology (bias). We first examine the available approaches to analyze bias and then present a new approach in two steps. First, we propose a logistic regression model to analyze the outcomes of fact-checks and calculate how likely each political party will obtain a truth score. We test our model with a (...) sample of fact-checks from Newtral, a major Spanish fact-checker. Our model would signal bias under two assumptions: a) all political parties are on average equally accurate in their statements; b) the verification method gives precise instructions and is implemented systematically. We investigate this second assumption with a series of interviews with Newtral fact-checkers. We show that standard verification protocols are so loosely implemented that fact-checks reflect a set of journalistic decisions, rather than a bias in the statistical sense. We call for a more rigorous definition of verification methods as a pre-requisite for an unbiased assessment of politician’s claims. (shrink)

Dynamic topological logic combines topological and temporal modalities to express asymptotic properties of dynamic systems on topological spaces. A dynamic topological model is a triple 〈X ,f , V 〉, where X is a topological space, f : X → X a continuous function and V a truth valuation assigning subsets of X to propositional variables. Valid formulas are those that are true in every model, independently of X or f. A natural problem that arises is to identify the logics (...) obtained on familiar spaces, such as . It [9] it was shown that any satisfiable formula could be satisfied in some for n large enough, but the question of how the logic varies with n remained open.In this paper we prove that any fragment of DTL that is complete for locally finite Kripke frames is complete for . This includes DTL○; it also includes some larger fragments, such as DTL1, where “henceforth” may not appear in the scope of a topological operator. We show that satisfiability of any formula of our language in a locally finite Kripke frame implies satisfiability in by constructing continuous, open maps from the plane into arbitrary locally finite Kripke frames, which give us a type of bisimulation. We also show that the results cannot be extended to arbitrary formulas of DTL by exhibiting a formula which is valid in but not in arbitrary topological spaces. (shrink)

This article explores the relationship between the philosophy of the limit of Eugenio Trías and the sufism of Ibn ʿArabī. Firstly, it explains the function of the philosophy of religion in the triasian system and why the andalusian master has a privileged position. Secondly, it presents some essential aspects of the akbarian doctrine obtured by the philosophy of limit, as the declaration of the unity of Being, the path of servanthood, the transit from the sudden passion of love to the (...) spiritual love, the spiritual knighthood and the station of no-station. All of them share a common element: the trascendence of the egoic and the interest on the other as other. Finally, the article defends that Trias obturates the andalusian master doctrine due to the influence of Henry Corbin and mainly to the own limitations of its own philosophical project. The absent Ibn ʿArabī will be its shadow. (shrink)

El presente artículo es un estudio sobre las diferenciaciones (aqsām) del amor, uno de los puntos más importantes del principal escrito que Ibn ʿArabī dedicó a la cuestión amorosa, el capítulo 178 de Las Iluminaciones de La Meca (al-Futūḥāt al-Makkiyya). A través de un juego de oscilación y equilibrio entre perspectivas ontológicas y epistemológicas aparentemente enfrentadas –incomparabilidad/similaridad, oculto/manifiesto, unidad/multiplicidad, espíritu/cuerpo– y un recurrente manejo del lenguaje de las alusiones (išāra), el maestro andalusí distingue entre amor divino (ilāhī), espiritual (rūḥānī), natural (...) (ṭabī‘ī) y elemental (ʿunṣurī). Se ofrece un análisis y comentario del texto akbarí que pretende aclarar la especificidad de cada una de estas nociones, así como su íntima relación con el discurso amoroso del autor. (shrink)

En su gran texto dedicado al amor — el capítulo 178 de _Las Iluminaciones de La Meca_—, Ibn ʿArabī comenta siete cualidades (_ṣ__ifāt_) que, según el Corán, suscitan el amor de Dios hacia los seres humanos. A lo largo de nuestro trabajo analizaremos el comentario akbarí y mostraremos cómo la clave para actualizar dichas cualidades consiste en_ hacer del propio ser un regalo de __amor_. O lo que es lo mismo: realizar un movimiento análogo al de Dios en cuanto que (...) El Omni-Compasivo (_al-Ra__ḥ__mān_), Quien a cada instante, de modo incondicional y sin esperar nada a cambio, asiste amorosamente a todas y cada una de Sus criaturas. (shrink)

A growing number of recreational races are being held in different locations, drawing many local and visiting runners. This study examined the relationships between quality, value, satisfaction, and loyalty among runners in a recreational race and examines potential differences in relationships between these constructs based on the runners’ experience. The participants were 985 runners with a mean age of 40.74±9.41years. Validated, reliable ad hoc instruments were used. A multi-group analysis was performed to ascertain the existence of relationships between the constructs (...) and differences in the relationships between the different study groups. The results show that quality is a direct antecedent of value and satisfaction. Value is directly related to satisfaction and indirectly related to loyalty. Satisfaction is related to the loyalty of participants in the race. Differences in the relationship between satisfaction and loyalty were dependent upon the runners’ experience. Loyalty to a race depends primarily on satisfaction and is modified by the runners’ experience. (shrink)

We show that given a finite, transitive and reflexive Kripke model 〈 W , ≼, ⟦ ⋅ ⟧ 〉 and $${w \in W}$$ , the property of being simulated by w (i.e., lying on the image of a literalpreserving relation satisfying the ‘forth’ condition of bisimulation) is modally undefinable within the class of S4 Kripke models. Note the contrast to the fact that lying in the image of w under a bi simulation is definable in the standard modal language even (...) over the class of K4 models, a fairly standard result for which we also provide a proof. We then propose a minor extension of the language adding a sequent operator $${\natural}$$ (‘tangle’) which can be interpreted over Kripke models as well as over topological spaces. Over finite Kripke models it indicates the existence of clusters satisfying a specified set of formulas, very similar to an operator introduced by Dawar and Otto. In the extended language $${{\sf L}^+ = {\sf L}^{\square\natural}}$$ , being simulated by a point on a finite transitive Kripke model becomes definable, both over the class of (arbitrary) Kripke models and over the class of topological S4 models. As a consequence of this we obtain the result that any class of finite, transitive models over finitely many propositional variables which is closed under simulability is also definable in L + , as well as Boolean combinations of these classes. From this it follows that the μ -calculus interpreted over any such class of models is decidable. (shrink)

Given the pervasiveness of AI systems and their potential negative effects on people’s lives (especially among already marginalised groups), it becomes imperative to comprehend what goes on when an AI system generates a result, and based on what reasons, it is achieved. There are consistent technical efforts for making systems more “explainable” by reducing their opaqueness and increasing their interpretability and explainability. In this paper, we explore an alternative non-technical approach towards explainability that complement existing ones. Leaving aside technical, statistical, (...) or data-related issues, we focus on the very conceptual underpinnings of the design decisions made by developers and other stakeholders during the lifecycle of a machine learning project. For instance, the design and development of an app to track snoring to detect possible health risks presuppose some picture or another of “health”, which is a key notion that conceptually underpins the project. We take it as a premise that these key concepts are necessarily present during design and development, albeit perhaps tacitly. We argue that by providing “justificatory explanations” about how the team understands the relevant key concepts behind its design decisions, interested parties could gain valuable insights and make better sense of the workings and outcomes of systems. Using the concept of “health”, we illustrate how a particular understanding of it might influence decisions during the design and development stages of a machine learning project, and how making this explicit by incorporating it into ex-post explanations might increase the explanatory and justificatory power of these explanations. We posit that a greater conceptual awareness of the key concepts that underpin design and development decisions may be beneficial to any attempt to develop explainability methods. We recommend that “justificatory explanations” are provided as technical documentation. These are declarative statements that contain at its simplest: (1) a high-level account of the understanding of the relevant key concepts a team possess related to a project’s main domain, (2) how these understandings drive decision-making during the life-cycle stages, and (3) it gives reasons (which could be implicit in the account) that the person or persons doing the explanation consider to have plausible justificatory power for the decisions that were made during the project. (shrink)

Every human community creates its own representation of its surrounding world and of the men who constitute that world. It sets out in an orderly fashion the raison d’être of social and cultural organisation, it ritualises the ties between men and their relationship with their environment. Man creates the world while the world creates man, through a relationship which varies with each society; ethnography shows us innumerable versions. Human cultures consist of symbols. It is always a matter of reducing the (...) world to the human factor, but in line with a social imaginary quantity specific to a particular group, which is itself the tributary of its past and the possible influence of other groups. (shrink)

Representations of the body depend on a social framework, a vision of the world and a definition of the person. The body is a symbolic construction and not a reality in its own right. A priori, its characterization seems to be self-evident, but ultimately nothing is less comprehensible. Far from being unanimously accepted by human societies, making the body stand out as a reality in some way distinct from man seems an uneasy effort, contradictory between one time and place and (...) another. Many societies do not retain it as part of their vision of the world. They do not detach man from his body in the dualist fashion so common to Western man. We might recall here the incident recounted by Maurice Leenhardt, who asked an elderly Kanak what the West had contributed to Melanesia. The answer surprised him. “What you brought us is the body”. With the intrusion of cultural and social values and forms from the Western world that tend to individualism, there came an awareness of the body as a barrier and a boundary distinguishing each person from every other person. In the societies to which we refer, the components of a person include the flesh without setting this off separately. The body itself is an abstraction. On the phenomenological level, only a person whose body gives him a face and establishes his presence in the world can exist. Man is indiscernible from the flesh that models him. (shrink)

In 1866, six Inuits were taken to the United States for the purpose of serving as specimens to American scientists at the Natural History Museum. Shortly after their arrival in New York, four of them had died. One of the survivors returned to the Arctic, while the sixth, Minik, now alone, fought to make possible the return of the remains of his dead companions to their village. Since the latter were being exhibited, as was then often the case (and happens (...) even today in many museums), in order to offer visitors examples of the Inuk people, Minik protested in vain. In 1909 he returned to Greenland when the scientists denied that the remains of his friends still existed. Several years later, still pursuing the issue, he went back to the United States to take up the fight against the bureaucracy for the repatriation of the bodies. He died in the United States in 1918. It was only in 1993 that he won out and that the remains of the four were returned to the Arctic homeland. (shrink)

Nothing is more mysterious for man than the substance of his own body. Every society has attempted in its way to give a particular answer to this primary enigma in which man has his roots. Innumerable theories of the body that have followed each other during the course of history or that still coexist today are directly connected to the world views of these different societies. Even more, they are dependent on the conceptions of the person. The modern view of (...) the body, that which anatomo-physiology incarnates, is a direct function of the emergence and development of individualism within the European societies of the Renaissance, especially in the 17th century, that marks a crystallization, very clear at the social level, of this tendency. Moreover, the explosion of the present knowledge of the body that makes anatomophysiology one theory among others, even though it is dominant, denotes another stage in individualism, a yet stronger falling back on the ego: the emergence of a society in which the atomization of individuals has become an important fact, an atomization submitted to or desired, according to the case, which does not appear in contradiction to present research in new ways of socializing, new forms of tribalism and so on, as is clearly indicated by what we agree to call the associative phenomenon. This is a characteristic of societies in which individualism is a structural fact: the development of an infinitely plural and polyphonic character. In these societies, in fact, the initiative is assumed by individuals or groups more than it is in a culture that tends to become a simple formal framework. (shrink)

Many amateur sportsmen in the West, have today started undertaking long and intensive ordeals where their personal capacity to withstand increasing suffering is the prime objective. Running, jogging, the triathlon and trekking are the sorts of ordeal where people without any particular ability are not pitting themselves against others but are committed to testing their own capacity to withstand increasing pain. Constantly called upon to prove themselves in a society where reference points are both countless and contradictory and where values (...) are in crisis, people are now seeking a one-to-one relationship by redical means, testing their strength of character, their courage and their personal resources. Going right on to the end of a self-imposed ordeal gives a legitimacy to life and provides a symbolic plank that supports them. Performance itself is of secondary significance; it has a value only to the individual. There is no struggle against a third party, only a method for reinforcing personal will-power and over-coming suffering by going right to the limit of a personally imposed demand. The physical limit has come to replace the moral limit that present-day society is failing to provide. Overcoming suffering tempers the individual, providing a renewed significance and value to his life. (shrink)

Adolescents are said to be, figuratively speaking, thin-skinned. But their thin-skinnedness is also real: both ambivalent and ambiguous, the border between self and other is, for many young people, a source of constant turmoil. The recourse to bodily self-harm is a means of dealing with this turmoil and the feelings of powerlessness it generates. Drawing on extensive semi-structured interviews conducted over the course of the last twenty years, this article explores the experiences of adolescents who engage in self-cutting. A deliberate (...) and controlled use of pain, this ‘symbolic homeopathy’ – that is, harming oneself to feel less pain – acts as a defence against externally imposed suffering. Far from being destructive, self-harm practices can paradoxically be understood as survival techniques. Part of a long-term, ongoing project investigating adolescent risk-taking, this article seeks to better understand the experiences of teens who injure themselves through skin-cutting. (shrink)

The face embodies for the individual the sense of identity, that is to say, precisely the place where someone recognizes himself and where others recognize him. From the outset the face is meaning, translating in a living and enigmatic form the absoluteness yet minuteness of individual difference. Any alteration to the face puts at stake the sense of identity. Disfigurement destroys the sense of identity of an individual who can no longer recognize himself or be recognized by others. Disfigurement places (...) a mask on the face. The goal of a facial transplant consists of restoring an individual’s place in the world and reviving his taste for life, returning to him his ‘human shape’. Facial transplants raise essential anthropological questions such as ‘Who am I?’ and ‘To whom belongs this face that henceforth is mine?’. (shrink)

The notion of information puts the human, the animal and the vegetable all on the same plane, and tends to dissolve the previous specificities of these categories. DNA, in this way, is fetishized. Also, the notion of information, and of the gene, has moved from the domain of expert or technical culture to become a part of mass culture: a development that has important social consequences. The human body is seen as a prototype that needs to be tested or rectified (...) (a consequence of the notion of ‘bad’ genes). From prenatal testing to the courtroom, it is implicitly around the notion a ‘life worth living’ (that is to say, of good morphological or genetic quality) that intense debates have arisen concerning the status of the body, of illness and of disability. Genetics is becoming part of the social environment, and it is setting itself up as a form of secular religion, with fundamentalist overtones. This article will analyse the anthropological shifts that have grown up around these new definitions of the human. (shrink)

Risk is most often associated with danger and perceived as a harmful aspect of life, as an insidious and unwelcome threat that should be avoided. Risk-taking, however, is sometimes a singular passion, a source of pleasure that becomes a way of life. When freely pursued as a valorised activity, it can be a path to self-fulfilment, an opportunity to confront new situations, and a means for redefining one’s self, testing personal abilities, increasing self-esteem or gaining recognition. Deliberate risk-taking is a (...) form of character building. It accommodates life’s intensities. It is extremely popular in high-risk physical activities and sports and postmodern forms of adventure. (shrink)

Dans un contexte de crise existentielle chez les jeunes générations, si les autres modes de symbolisation ont échoué, échapper à la mort, réussir l'épreuve, administrent la preuve ultime qu'une garantie règne sur son existence. Ces épreuves sont des rites intimes, privés, autoréférentiels, insus, détachés de toute croyance, et tournant le dos à une société qui cherche à les prévenir. Parfois elles provoquent un sentiment de renaissance personnelle, elles se muent en formes d'auto-initiation.In an existential crisis among the younger generations, if (...) other modes of symbolization have failed to escape death, pass the test, administering the ultimate proof that security prevails on its existence. These tests are rites intimate, private, self-referential insus, detached from any belief, and turning his back to a society that seeks to prevent. Sometimes they cause a sense of personal rebirth, they are transformed into forms of self-initiation. (shrink)

Risk-taking behaviours often reflect ambivalent ways of calling for the help of one’s close friends and family – those who count. It is an ultimate means of finding meaning and a system of values; and it is a sign of the adolescent’s active resistance and his attempts to find his place in the world again. It contrasts with the far more insidious risk of depression and the radical collapse of meaning. In spite of the suffering it engenders, risk-taking nevertheless has (...) a positive side, fostering independence in adolescents and a search for reference points, it leads to a better self-image and is a means of developing one’s identity. It is nonetheless painful in terms of its possible repercussions: injuries, death or addiction. But let us not forget that the suffering is upstream, perpetuated by a complex relation between a society, a family structure and a life history. Paradoxically, for some young people who are suffering, the risk is rather that they will remain immured in their world-weariness, with a potentially radical outcome (i.e. suicide). The turbulence caused by risk-taking behaviours illustrates a determination to be rid of one’s suffering and to fight on so that life can, at last, be lived. (shrink)

Résumé L’évaluation devient aujourd’hui une nouvelle tyrannie, mais sous une forme intuitive, elle est au cœur de toutes les activités humaines. Elle est au cœur des activités physiques et sportives à risque où un enjeu de vie ou de mort est toujours présent, surtout dans le contexte de l’alpinisme solitaire où une part d’imprévisible demeure toujours, mais contribue à donner son sel à l’action.

In this work, we shall study structural abduction and how ways of searching for solutions to the corresponding abductive problems could be modeled. Specifically, we shall define modal semantic tableaux for normal modal systems and study its applications to structural abduction. This method even makes structural abduction clearer and, as it shall be seen, when a radical change of logic is epistemologically required, the corresponding tableau will have pertinent information to suggest it.

We introduce frame-equivalence games tailored for reasoning about the size, modal depth, number of occurrences of symbols and number of different propositional variables of modal formulae defining a given frame property. Using these games, we prove lower bounds on the above measures for a number of well-known modal axioms; what is more, for some of the axioms, we show that they are optimal among the formulae defining the respective class of frames.

We introduce the logics GLP Λ, a generalization of Japaridze’s polymodal provability logic GLP ω where Λ is any linearly ordered set representing a hierarchy of provability operators of increasing strength. We shall provide a reduction of these logics to GLP ω yielding among other things a finitary proof of the normal form theorem for the variable-free fragment of GLP Λ and the decidability of GLP Λ for recursive orderings Λ. Further, we give a restricted axiomatization of the variable-free fragment (...) of GLP Λ. (shrink)

The paper presents a new logic for reasoning about the formation of beliefs through perception or through inference in non-omniscient resource-bounded agents. The logic distinguishes the concept of explicit belief from the concept of background knowledge. This distinction is reflected in its formal semantics and axiomatics: we use a non-standard semantics putting together a neighborhood semantics for explicit beliefs and relational semantics for background knowledge, and we have specific axioms in the logic highlighting the relationship between the two concepts. Mental (...) operations of perceptive type and inferential type, having effects on epistemic states of agents, are primitives in the object language of the logic. At the semantic level, they are modelled as special kinds of model-update operations, in the style of dynamic epistemic logic. Results about axiomatization, decidability and complexity for the logic are given in the paper. (shrink)

Let T be a second-order arithmetical theory, Λ a well-order, λ < Λ and X ⊆ ℕ. We use $[\lambda |X]_T^{\rm{\Lambda }}\varphi$ as a formalization of “φ is provable from T and an oracle for the set X, using ω-rules of nesting depth at most λ”.For a set of formulas Γ, define predicative oracle reflection for T over Γ ) to be the schema that asserts that, if X ⊆ ℕ, Λ is a well-order and φ ∈ Γ, then$$\forall \,\lambda (...) < {\rm{\Lambda }}\,.$$In particular, define predicative oracle consistency ) as Pred–O–RFN{0=1}.Our main result is as follows. Let ATR0 be the second-order theory of Arithmetical Transfinite Recursion, ${\rm{RCA}}_0^{\rm{*}}$ be Weakened Recursive Comprehension and ACA be Arithmetical Comprehension with Full Induction. Then,$${\rm{ATR}}_0 \equiv {\rm{RCA}}_0^{\rm{*}} + {\rm{Pred - O - Cons\ }}\left \equiv {\rm{RCA}}_0^{\rm{*}} + \,{\rm{Pred - O - Cons\ }}\left \equiv {\rm{RCA}}_0^{\rm{*}} + \,{\rm{Pred - O - RFN}}\,_{{\bf{\Pi }}_2^1 } \left.$$We may even replace ${\rm{RCA}}_0^{\rm{*}}$ by the weaker ECA0, the second-order analogue of Elementary Arithmetic.Thus we characterize ATR0, a theory often considered to embody Predicative Reductionism, in terms of strong reflection and consistency principles. (shrink)

AimThe functional prognosis of patients after coma following either cardiac arrest or acute structural brain injury is often uncertain. These patients are associated with high mortality and disability. N20 and N70 somatosensory evoked potentials are used to predict prognosis. We evaluated the utility of SSEP as an early indicator of long-term prognosis in these patients.MethodsThis was a retrospective cohort study of patients admitted to the intensive care unit with a diagnosis of coma after CA or ABI. An SSEP study was (...) performed, including N20 and N70 at 24–72 h, after coma onset. Functional recovery was assessed 6–12 months later using the modified Glasgow scale. The study was approved by our local research ethics committee.ResultsIn the CA and ABI groups, the absence of N20 or N70 was a strong indicator of poor outcome. Conversely, the presence of N70 was an indicator of a good outcome.ConclusionSomatosensory evoked potentials are useful early prognostic markers with high specificity and sensitivity. Moreover, N70 has additional potential value for improving the prediction of good long-term functional outcomes.Clinical Trial Registration:[https://clinicaltrials.gov/], identifier [2018/01/001]. (shrink)

Este artículo explora la relación entre la filosofía del límite de Eugenio Trías y el sufismo de Ibn ʿArabī. En primer lugar, pretende explicar la función de la filosofía de la religión en el sistema triasiano y por qué el maestro andalusí ocupa un lugar privilegiado en ella. Segundo, se ocupa de algunos aspectos esenciales de la doctrina akbarí que la filosofía del límite obtura, como la declaración de la unidad del Ser (tawḥīd), la conjugación de lo exotérico y lo (...) esotérico, la superación de la propia creencia, el camino de la servidumbre (ʿubūdiyya), el tránsito de la pasión súbita de amor (hawā) al amor espiritual (ḥubb), la caballería espiritual (futuwwa) y la morada de la no-morada (maqām lā maqām), los cuales comparten un nexo común: el trascendimiento de lo egoico y el interés en el otro en tanto otro. Finalmente, se defenderá que la obturación triasiana se debe tanto al influjo de Henry Corbin, como sobre todo, a las propias limitaciones de su proyecto filosófico, para el que el Ibn ʿArabī ausente constituye una suerte de sombra. (shrink)

The language of linear temporal logic can be interpreted on the class of dynamic topological systems, giving rise to the intuitionistic temporal logic ${\sf ITL}^{\sf c}_{\Diamond \forall }$, recently shown to be decidable by Fernández-Duque. In this article we axiomatize this logic, some fragments, and prove completeness for several familiar spaces.

For any ordinal $\Lambda$, we can define a polymodal logic $\mathsf{GLP}_\Lambda$, with a modality $[\xi]$ for each $\xi < \Lambda$. These represent provability predicates of increasing strength. Although $\mathsf{GLP}_\Lambda$ has no Kripke models, Ignatiev showed that indeed one can construct a Kripke model of the variable-free fragment with natural number modalities, denoted $\mathsf{GLP}^0_\omega$. Later, Icard defined a topological model for $\mathsf{GLP}^0_\omega$ which is very closely related to Ignatiev's. In this paper we show how to extend these constructions for arbitrary $\Lambda$. (...) More generally, for each $\Theta,\Lambda$ we build a Kripke model $\mathfrak I^\Theta_\Lambda$ and a topological model $\mathfrak T^\Theta_\Lambda$, and show that $\mathsf{GLP}^0_\Lambda$ is sound for both of these structures, as well as complete, provided $\Theta$ is large enough. (shrink)

Ordinal functions may be iterated transfinitely in a natural way by taking pointwise limits at limit stages. However, this has disadvantages, especially when working in the class of normal functions, as pointwise limits do not preserve normality. To this end we present an alternative method to assign to each normal function f a family of normal functions Hyp[f]=〈fξ〉ξ∈OnHyp[f]=〈fξ〉ξ∈On, called its hyperation, in such a way that f0=idf0=id, f1=ff1=f and fα+β=fα∘fβfα+β=fα∘fβ for all α, β.Hyperations are a refinement of the Veblen hierarchy (...) of f. Moreover, if f is normal and has a well-behaved left-inverse g called a left adjoint, then g can be assigned a cohyperationcoH[g]=〈gξ〉ξ∈OncoH[g]=〈gξ〉ξ∈On, which is a family of initial functions such that gξgξ is a left adjoint to fξfξ for all ξ. (shrink)