There are close similarities between the Weihrauch lattice and the zoo of axiom systems in reverse mathematics. Following these similarities has often allowed researchers to translate results from one setting to the other. However, amongst the big five axiom systems from reverse mathematics, so far $\mathrm {ATR}_0$ has no identified counterpart in the Weihrauch degrees. We explore and evaluate several candidates, and conclude that the situation is complicated.

We study the reverse mathematics of interval orders. We establish the logical strength of the implications among various definitions of the notion of interval order. We also consider the strength of different versions of the characterization theorem for interval orders: a partial order is an interval order if and only if it does not contain 2 \oplus 2. We also study proper interval orders and their characterization theorem: a partial order is a proper interval order if and only if it (...) contains neither 2 \oplus 2 nor 3 \oplus 1. (shrink)

We study the computability-theoretic complexity and proof-theoretic strength of the following statements: (1) "If X is a well-ordering, then so is ε X ", and (2) "If X is a well-ordering, then so is φ(α, X)", where α is a fixed computable ordinal and φ represents the two-placed Veblen function. For the former statement, we show that ω iterations of the Turing jump are necessary in the proof and that the statement is equivalent to ${\mathrm{A}\mathrm{C}\mathrm{A}}_{0}^{+}$ over RCA₀. To prove the (...) latter statement we need to use ω α iterations of the Turing jump, and we show that the statement is equivalent to ${\mathrm{\Pi }}_{{\mathrm{\omega }}^{\mathrm{\alpha }}}^{0}{-\mathrm{C}\mathrm{A}}_{0}$ . Our proofs are purely computability-theoretic. We also give a new proof of a result of Friedman: the statement "if x is a well-ordering, then so is φ(x, 0)" is equivalent to ATR₀ over RCA₀. (shrink)

We prove that the maximal order type of the wqo of linear orders of finite Hausdorff rank under embeddability is φ2, the first fixed point of the ε-function. We then show that Fraïssé’s conjecture restricted to linear orders of finite Hausdorff rank is provable in +“φ2 is well-ordered” and, over , implies +“φ2 is well-ordered”.

We show that the maximal linear extension theorem for well partial orders is equivalent over RCA0 to ATR0. Analogously, the maximal chain theorem for well partial orders is equivalent to ATR0 over RCA0.

We investigate the uniform computational content of the open and clopen Ramsey theorems in the Weihrauch lattice. While they are known to be equivalent to $\mathrm {ATR_0}$ from the point of view of reverse mathematics, there is not a canonical way to phrase them as multivalued functions. We identify eight different multivalued functions and study their degree from the point of view of Weihrauch, strong Weihrauch, and arithmetic Weihrauch reducibility. In particular one of our functions turns out to be strictly (...) stronger than any previously studied multivalued functions arising from statements around $\mathrm {ATR}_0$. (shrink)

We analyze the axiomatic strength of the following theorem due to Rival and Sands [28] in the style of reverse mathematics. Every infinite partial order P of finite width contains an infinite chain C such that every element of P is either comparable with no element of C or with infinitely many elements of C. Our main results are the following. The Rival–Sands theorem for infinite partial orders of arbitrary finite width is equivalent to $\mathsf {I}\Sigma ^0_{2} + \mathsf {ADS}$ (...) over $\mathsf {RCA}_0$. For each fixed $k \geq 3$, the Rival–Sands theorem for infinite partial orders of width $\leq \!k$ is equivalent to $\mathsf {ADS}$ over $\mathsf {RCA}_0$. The Rival–Sands theorem for infinite partial orders that are decomposable into the union of two chains is equivalent to $\mathsf {SADS}$ over $\mathsf {RCA}_0$. Here $\mathsf {RCA}_0$ denotes the recursive comprehension axiomatic system, $\mathsf {I}\Sigma ^0_{2}$ denotes the $\Sigma ^0_2$ induction scheme, $\mathsf {ADS}$ denotes the ascending/descending sequence principle, and $\mathsf {SADS}$ denotes the stable ascending/descending sequence principle. To the best of our knowledge, these versions of the Rival–Sands theorem for partial orders are the first examples of theorems from the general mathematics literature whose strength is exactly characterized by $\mathsf {I}\Sigma ^0_{2} + \mathsf {ADS}$, by $\mathsf {ADS}$, and by $\mathsf {SADS}$. Furthermore, we give a new purely combinatorial result by extending the Rival–Sands theorem to infinite partial orders that do not have infinite antichains, and we show that this extension is equivalent to arithmetical comprehension over $\mathsf {RCA}_0$. (shrink)

We study the provability in subsystems of second-order arithmetic of two theorems of Harrington and Shelah [6] about Borel quasi-orderings of the reals. These theorems turn out to be provable in ATR0, thus giving further evidence to the observation that ATR0 is the minimal subsystem of second-order arithmetic in which significant portion of descriptive set theory can be developed. As in [6] considering the lightface versions of the theorems will be instrumental in their proof and the main techniques employed will (...) be the reflection principles and Gandy forcing. (shrink)

We show that the quasi-order of continuous embeddability between finitely branching dendrites (a natural class of fairly simple compacta) is $\Sigma_1^1$ -complete. We also show that embeddability between countable linear orders with infinitely many colors is $\Sigma_1^1$ -complete.

In (Fund Math 60:175–186 1967), Wolk proved that every well partial order (wpo) has a maximal chain; that is a chain of maximal order type. (Note that all chains in a wpo are well-ordered.) We prove that such maximal chain cannot be found computably, not even hyperarithmetically: No hyperarithmetic set can compute maximal chains in all computable wpos. However, we prove that almost every set, in the sense of category, can compute maximal chains in all computable wpos. Wolk’s original result (...) actually shows that every wpo has a strongly maximal chain, which we define below. We show that a set computes strongly maximal chains in all computable wpo if and only if it computes all hyperarithmetic sets. (shrink)

In this paper we give a proof of the II12-completeness of the set of countable better quasi orderings . This result was conjectured by Clote in [2] and proved by the author in his Ph.d. thesis [6] . Here we prove it using Simpson's definition of better quasi ordering and as little bqo theory as possible.

We determine the computational complexity of the Hahn-Banach Extension Theorem. To do so, we investigate some basic connections between reverse mathematics and computable analysis. In particular, we use Weak König's Lemma within the framework of computable analysis to classify incomputable functions of low complexity. By defining the multivalued function Sep and a natural notion of reducibility for multivalued functions, we obtain a computational counterpart of the subsystem of second-order arithmetic WKL0. We study analogies and differences between WKL0 and the class (...) of Sep-computable multivalued functions. Extending work of Brattka, we show that a natural multivalued function associated with the Hahn-Banach Extension Theorem is Sep-complete. (shrink)

We study Lebesgue and Atsuji spaces within subsystems of second order arithmetic. The former spaces are those such that every open covering has a Lebesgue number, while the latter are those such that every continuous function defined on them is uniformly continuous. The main results we obtain are the following: the statement “every compact space is Lebesgue” is equivalent to $\hbox{\sf WKL}_0$ ; the statements “every perfect Lebesgue space is compact” and “every perfect Atsuji space is compact” are equivalent to (...) $\hbox{\sf ACA}_0$ ; the statement “every Lebesgue space is Atsuji” is provable in $\hbox{\sf RCA}_0$ ; the statement “every Atsuji space is Lebesgue” is provable in $\hbox{\sf ACA}_0$ . We also prove that the statement “the distance from a closed set is a continuous function” is equivalent to $\Pi^1_1\hbox{-\sf CA}_0$. (shrink)

We introduce the notion of τ-like partial order, where τ is one of the linear order types ω, ω*, ω + ω*, and ζ. For example, being ω-like means that every element has finitely many predecessors, while being ζ-like means that every interval is finite. We consider statements of the form “any τ-like partial order has a τ-like linear extension” and “any τ-like partial order is embeddable into τ” . Working in the framework of reverse mathematics, we show that these (...) statements are equivalent either to equation image or to equation image over the usual base system equation image. (shrink)

In this paper we study the reverse mathematics of two theorems by Bonnet about partial orders. These results concern the structure and cardinality of the collection of initial intervals. The first theorem states that a partial order has no infinite antichains if and only if its initial intervals are finite unions of ideals. The second one asserts that a countable partial order is scattered and does not contain infinite antichains if and only if it has countably many initial intervals. We (...) show that the left to right directions of these theorems are equivalent to ACA0 and ATR0, respectively. On the other hand, the opposite directions are both provable in WKL0, but not in RCA0. We also prove the equivalence with ACA0 of the following result of Erdös and Tarski: a partial order with no infinite strong antichains has no arbitrarily large finite strong antichains. (shrink)

A quasi-order Q induces two natural quasi-orders on $${\mathcal{P}(Q)}$$, but if Q is a well-quasi-order, then these quasi-orders need not necessarily be well-quasi-orders. Nevertheless, Goubault-Larrecq (Proceedings of the 22nd Annual IEEE Symposium 4 on Logic in Computer Science (LICS’07), pp. 453–462, 2007) showed that moving from a well-quasi-order Q to the quasi-orders on $${\mathcal{P}(Q)}$$ preserves well-quasi-orderedness in a topological sense. Specifically, Goubault-Larrecq proved that the upper topologies of the induced quasi-orders on $${\mathcal{P}(Q)}$$ are Noetherian, which means that they contain no (...) infinite strictly descending sequences of closed sets. We analyze various theorems of the form “if Q is a well-quasi-order then a certain topology on (a subset of) $${\mathcal{P}(Q)}$$ is Noetherian” in the style of reverse mathematics, proving that these theorems are equivalent to ACA 0 over RCA 0. To state these theorems in RCA 0 we introduce a new framework for dealing with second-countable topological spaces. (shrink)

When a linear order has an order preserving surjection onto each of its suborders, we say that it is strongly surjective. We prove that the set of countable strongly surjective linear orders is a [Formula: see text]-complete set. Using hypotheses beyond ZFC, we prove the existence of uncountable strongly surjective orders.

Let ⪯R be the preorder of embeddability between countable linear orders colored with elements of Rado's partial order . We show that ⪯R has fairly high complexity with respect to Borel reducibility , although its exact classification remains open.

We study Polish spaces for which a set of possible distances $A \subseteq R^+$ is fixed in advance. We determine, depending on the properties of A, the complexity of the collection of all Polish metric spaces with distances in A, obtaining also example of sets in some Wadge classes where not many natural examples are known. Moreover we describe the properties that A must have in order that all Polish spaces with distances in that set belong to a given class, (...) such as zero-dimensional, locally compact, etc. These results lead us to give a fairly complete description of the complexity, with respect to Borel reducibility and again depending on the properties of A, of the relations of isometry and isometric embeddability between these Polish spaces. (shrink)

Resumen La argumentación es una forma de discurso en la que un hablante defiende una postura en una discusión. El presente trabajo se enfoca en la complejidad argumentativa y los recursos evaluativos y evidenciales que caracterizan las disputas de niños de diverso nivel socioeconómico. Se analizaron 94 disputas entre niños de 4 años y otros niños (5 a 10 años) en situaciones de juego espontáneo registradas en hogares de sectores socioeconómicos medio y bajo. El análisis identificó diferencias marginales en la (...) complejidad argumentativa entre los grupos. Por su parte se hallaron diferencias significativas en recursos de evaluativos de emocionalidad y en los evidenciales deónticos y exclamativos a favor de los sectores de nivel socioeconómico bajo. Los sectores medios registraron un mayor empleo de expresiones de causalidad y conectores. Los resultados se discuten a la luz de los hallazgos de investigaciones previas. Abstract Argumentation is a type of discourse in which a speaker defends a claim in a discussion. The present work focuses on the argumentative complexity and the evaluative and evidential resources that characterize the disputes of children of different socioeconomic levels. We analyzed 94 disputes between 4-year-old children and other children (5 to 10 years old) in situations of spontaneous play registered in homes of middle and low socioeconomic status (SES). The analysis identified marginal differences in argumentative complexity between the groups. On the other hand, significant differences were found in evaluative expressions of emotion and in deontic and exclamatory evidential markers, more frequently used by the low SES group. The mid SES group tends to use a higher number of expressions of causality and connectors. The results are discussed with reference to the findings of previous research. (shrink)

El objetivo de este artículo es clarificar la noción de «vida auténtica» tal y como es descrita por Joan Roura-Parella en su última obra Tema y variaciones de la personalidad (1950). Por «vida auténtica» se entiende una vida autogobernada, que permite al individuo realizar su propia singularidad y, por tanto, preservar su autonomía y dignidad como persona. Una vida dominada por factores ajenos al propio individuo, sean cuales sean éstos, es una vida alienada, que no permite el desarrollo del individuo (...) en tanto la persona que es. Se señala la relevancia filosófica de la propuesta de Roura-Parella, que es la de ofrecer una justificación no-evidencial para una forma de vida cristiana, agapística. De acuerdo con Roura-Parella, una forma de vida basada en la entrega amorosa para con el prójimo está justificada en tanto que ésta es el resultado último de nuestra propia condición natural, constituyendo así un acto de afirmación personal y siendo, por tanto, condición necesaria para gozar de una vida auténtica — «la vida alcanza su hondo sentido al salirse de sí misma para darse a los demás» (Roura-Parella, Tema y variaciones de la personalidad, 1950, p. 229). Se señalan las diferencias y semejanzas de la propuesta de Roura-Parella con respecto a las de Miguel de Unamuno y Joaquim Xirau, autores que fueron cercanos a Roura-Parella y quienes, aunque partiendo de razonamientos distintos a los de Roura-Parella, también concibieron la entrega amorosa para con el prójimo como un acto de afirmación personal. (shrink)

In this paper, I will argue that the main contrast between religious fictionalism and other recently developed fictionalist positions in other non-religious fields of enquiry is the sort of personal and affective relationship said to be felt by the religious person between them and God, the feeling of being in a loving and personal communion with God. I will argue that a realist, non-Meingonian artifactual fictionalist understanding of God, along the lines that philosophers such as Schiffer and Thomasson have already (...) defended on non-religious grounds regarding fictional characters, seems to be the most direct way of preserving the possibility of the religious person standing in an actual relation to God while also maintaining a fictionalist understanding of religious faith. Last, I will argue that conceiving of God in these realist artifactual terms, and despite allowing the possibility of the religious person standing in an actual relation to God, fails to preserve a genuine personal relationship between the concrete religious person and God. (shrink)

In this paper I argue for a syncretistic theory of depiction, which combines the merits of the main paradigms which have hitherto faced themselves on this issue, namely the perceptualist and semioticist approaches. The syncretistic theory indeed takes from the former its stress on experiential factors and from the latter its stress on conventional factors. But the theory is even more syncretistic than this, for the way it accounts for the experiential factor vindicates several claims defended by different perceptualist theories. (...) In a nutshell, according to the syncretistic theory a picture depicts its subject iff i) it is transformed into an entity-cum-meaning and ii) one has the twofold experience of seeing that subject in the picture qua noninterpreted entity, the image, just in case one consciously misrecognizes it in consciously seeing that image, for that subject resembles the image in some grouping properties (originally labelled Gestalt-qualities in psychology). By appealing to objective resemblance in grouping properties, the theory can vindicate what are nowadays taken to be the most neglected doctrines in the perceptualist camp: objective resemblance theories. By appealing, moreover, to conscious misrecognition, the theory not only squares with both the seeing-in and the recognition theories of depiction, but it also shows the grain of truth in illusion theories of depiction, since conscious misrecognition is a kind of perceptual illusion. (shrink)

El objetivo de este artículo es explorar la manera en que Joaquim Xirau y Miguel de Unamuno concibieron el amor. A pesar de que no hay motivos suficientes para concluir una influencia directa y filosóficamente relevante de Unamuno en la obra de Xirau, sí podemos concluir que la forma en que ambos autores concibieron el amor guarda ciertas similitudes. Así, ambos coinciden en concebir la entrega amorosa como un ejercicio de afirmación individual que trae consigo un incremento de la propia (...) singularidad. También, tanto Unamuno como Xirau conciben la entrega amorosa como un acto que, aun surgiendo del propio individuo y no de como es de hecho el mundo, personifica aquello amado, dejando de ser así visto como un mero objeto y mostrándose bajo una forma personal. Y ambos, aunque de un modo distinto, hacen del relato bíblico acerca de la vida y Resurrección de Cristo un aspecto central de sus maneras de entender el amor. (shrink)

This chapter contains sections titled: Introduction Argentina Mexico Brazil Chile and Puerto Rico Peru Other Centers Concluding Remarks References Further Reading.

The aim of this chapter is to analyse the concept of love —understood in the broad and Christian-inspired sense of love as agape-charis love— in relation to the animes Dragon Ball and Dragon Ball Z. I first comment on the character of Piccolo —and how his friendship with Son Gohan— leads to him losing all his original villainous traits. I argue that the evolution of the character of Piccolo through his friendship with Son Gohan illustrates the philosophical claim that a (...) loving and giving of oneself to the other —far from being a sign of weakness— is a courageous and self-affirming exercise. The result may move the individual to an inner transformative experience that leads to the enjoyment of a renewed and authentic life that is worth living. I then argue that Son Goku’s respectful attitude towards his opponents is ultimately a loving attitude. Last, I argue that it is the inspiring and loving example of Son Goku that ultimately explains why characters such as Vegeta —who are at first presented as villains— end up losing their villainous traits after fighting against Son Goku. (shrink)

The purpose of this succinct work is to present N. Machiavelli's classic republican view from his proposition of an inevitable paradox, the founding of an expansionist republic, difficult to govern, or the founding of a stable, but small and weak republic. Such a paradox, according to Machiavelli, should direct and condition all the constitutive devices of the republic when choosing what will be its destiny as a political body. The model of republic preferred by the Florentine will be the expansionist (...) model of Rome, leading him to assume all the devices that gave this republic its power. From this presentation of the Machiavellian proposition, we will analyse the assimilation of republican thought in England from the Elizabethan period, as well as the political-social scenario that exists there. This itinerary will allow us to understand, in general, why classical republicanism was received on English soil from the perspective of establishing a mixed, stable government, thus favouring the spread of the Venice myth as a serene republic and delaying the use, even that mitigated, of the republican presuppositions expressed in the Machiavellian work that directed towards a Roman model. (shrink)

Neste breve artigo, argumentaremos que a pandemia da Covid-19, como outras na história humana, colocou em desnudo as bases da estrutura político-econômica moderna. Defenderemos que tais bases apoiam-se em dois esteios, uma noção política de ‘Estado’, ápice da organização humana, e uma concepção racionalista, técnica e produtiva, que promulga a possibilidade de resposta a qualquer crise surgida na sociedade. Como contrapondo a esta estrutura, apresentaremos a teoria maquiaveliana, que aponta para uma relação indelével entre política e fator natural, natureza e (...) sociedade. Nesta teoria, com uma concepção antropológica própria, as necessidades impostas pela natureza frente a ação humana são determinantes para se pensar a própria construção política. Como resultado, consideraremos ser necessário repensar a estrutura político-econômica moderna em seus pretensos mitos de incontestável avanço político, por meio do ‘Estado’, e inabalável base racional, principalmente técnico-produtiva, retomando o lugar de importância do fator natural para constituição de uma sociedade mais ‘humana’. (shrink)

Tencionamos, neste artigo, demonstrar os meandros constitutivos do que seja um formalismo moral em Immanuel Kant, tendo como pressuposto teórico sua compreensão de uma teoria da vontade. Este movimento de análise e demonstração teórica terá como fio condutor a ideia de autonomia que permeia a construção da vontade no formalismo kantiano, ou seja, a pressuposição de uma possível autonomia formal da razão é a base na qual se assenta a teoria moral deste pensador. Para execução de nossa pretensão será necessário (...) reconstituir os pontos nevrálgicos do que venha a ser uma teoria da vontade para em seguida analisar os pressupostos kantianos que realçam a centralidade da autonomia da razão na compreensão de uma vontade livre. Desta maneira, o que se pretende por em desnudo é a originalidade kantiana, em relação à tradição que promulga a vontade como marca da liberdade humana, a partir de seu pressuposto de autonomia formal na constituição moral do indivíduo humano. (shrink)

Neste artigo intentamos empreender uma leitura do judaísmo antigo, ou veterotestamentário, em alguns de seus aspectos principais, à contraluz da visão maquiaveliana do papel da religião na organização e expansão do Estado. O primeiro passo desta empresa será dado ao expormos os traços fundantes e delineadores da tradição judaica antiga, principalmente nos livros que compõe o chamado velho testamento, demonstrando o uso da religião na ordenação e expansão do Estado, bem como na formação do indivíduo. Em um segundo momento demonstraremos (...) como se cria uma nova tradição judaica, influenciada e influenciadora do mundo grego antigo, logo no início da era cristã a qual será assimilada e explorada em seu viés místico no humanismo renascentista. Esta tradição judaico helenístico será a mais conhecida e assimilada nos círculos eruditos europeus da renascença. Por fim, buscaremos demonstrar como os traços constitutivos do judaísmo antigo, em detrimento ao judaísmo helenístico, podem ser lidos à luz da descrição maquiaveliana do papel da religião na organização e expansão do Estado e na formação do indivíduo. (shrink)

Our objective in this brief article is guided by the demonstration of the existence of a theory of political action in Machiavelli's republican thought, with such a theory having its own character that directs it to highlight the action of individuals in the social context. In addition to this objective, we hope to support the thesis that such a theory of political action has a republican scope, not just “republicanist”, in keeping with the Machiavellian preference for institutions that impress on (...) individuals a civic sense based above all on the materiality of political action in the body social. From this assertion, we indicate that our itinerary will be guided by the demonstration of the search for the valorization of political action in Machiavelli's theory, the materiality of such action, to the detriment of its pure intention, the central focus of Florentine's work. This disposition of the centrality of political action in Machiavelli republicanism will underscore its appreciation for outlining the political functions of the search for recognition, glory and especially the benefit of the political body as a whole. (shrink)

In this brief article we will seek to support the thesis that there is in the thought of Coluccio Salutati, 14th century Florentine chancellor and prominent humanist, a clear republicanism that turns to the issue of the freedom of the republic and the active life of the individuals participating in it. However, in connection with this demonstration, we will also maintain that such republicanism has strong traces of Augustinian influence, mainly in view of the disposition of laws in the political (...) arrangement, as well as the relationship between active life and the search for a human being’s own good. In view of this, we intend to demonstrate that, in general, Salutati’s Renaissance republicanism turns to the defence of the republic, its freedom and its characteristic of the proper scenario for the active life of individuals, however, it still maintains an ethical-political perspective typical of the medieval period, centrally influenced by Augustinian thought. (shrink)

Tencionamos, neste artigo, tratar da questão da imitação no pensamento maquiaveliano. Tomaremos como balizas suas postulações acerca da natureza do homem, as possibilidades de formatação de uma república profícua em vista desta natureza e o papel dos exemplos dos indivíduos que agem em prol do bem da república. Sendo a questão da imitação o fio condutor, elegeremos ainda um tripé para demonstrar nossa hipótese, a saber, o fator antropológico; o fator educacional; o fator político. Por este movimento, pretende-se demonstrar que (...) a aparente negatividade das postulações maquiavelianas sobre a natureza humana não inviabilizam a promulgação de uma república estável, tendo os bons exemplos políticos um papel preponderante neste propósito. (shrink)

En este artículo expongo y analizo la propuesta berkeleyana del conocimiento nocional, que representa entre otras cosas el intento del irlandés por conocer a la mente o espíritu, esto es, a aquella cosa pensante y activa que por su propia actividad resulta irrepresentable como idea. Como el conocimiento nocional ya se menciona en los Comentarios Filosófi cos me remitiré a ellos para conocer los orígenes del mismo; sin embargo, como tal conocimiento aparece con mayor detalle en obras posteriores me serviré (...) sobre todo del Tratado para abordar la compleja doctrina nocional. // In this paper I expose and analyze the berkeleian proposal of notional knowledge. Among other things, this proposal represents Berkeley ́s attempt to know the mind or spirit, that is, the thinking and active thing that by its own activity results unrepresentable as idea. As such knowledge is already mentioned in the Philosophical Commentaries, I will refer to them in order to know the origins of that proposal. However, as notional knowledge appears in more detail in later works, I will make use especially of the Treatise to address the complex notional doctrine. (shrink)

Lorenzo Sacconi and his coauthors have put forward the hypothesis that impartial agreements on distributive rules may generate a conditional preference for conformity. The observable effect of this preference would be compliance with fair distributive rules chosen behind a veil of ignorance, even in the absence of external coercion. This paper uses a Dictator Game with production and taking option to compare two ways in which the device of the veil of ignorance may be thought to generate a motivation for, (...) and compliance with a fair distributive rule: individually-as a thought experiment that should work as a moral cue- and collectively-as an actual process of agreement among subjects. The main result is that actual agreement proves to be necessary for agents to be led towards a fair distributive principle and to generate a significant amount of compliance in absence of external authority. This conclusion vindicates the role of actual agreements in generating motivational power in correspondence with fair distributive rules. (shrink)