In lieu of an abstract, here is a brief excerpt of the content:Sense and Sensibility: IARPT’s Four Existential OrientationsWilliam David Hart (bio)I. Introduction: IARPT’s Liberal HorizonThe concerns of the Institute of American Religious and Philosophical Thought are worlds apart from the preoccupations that animate the characters in Jane Austen’s novels. This is not to say that IARPT is disinterested in romance, love, and heartbreak. It is to say, rather, that Sense and Sensibility, the title of Austen’s 1811 novel, (...) is a good cover term for the matters that I wish to address. I say this despite the fact that many characters in the novel, especially Fanny Dashwood, are insufferable members of a British ruling class that, along with its wealth-hording, patriarchy-sustaining institution of primogeniture, needed to be expropriated by the working class. I say this despite the fact that the sensibility of the British aristocracy, as is true of all aristocracies, is off-putting. With this proviso, it is precisely the sense and sensibility of the Institute (the love, romance and, perhaps, tragedy) that I wish to survey through a focus on what I take to be its animating existential concerns. I call these orientations existential rather than merely intellectual because they go beyond ideas, professional commitments, and implicate the entire range of life dispositions. These orientations are habit-forming.Constitutive of these orientations are anxieties regarding the meaning and significance of those things for which we live and die. The sheer fact of consciousness, of conscious matter, of matter conscious of its consciousness and the wonder and awe this inspires as we consider the very existence of things is part of our existential ensemble too. For some of us wonder and awe before the groundless gratuity of existence, let’s call this ontological curiosity, compete with death anxiety and its intimations for preeminence in how we imagine the meaning and significance of our life. This is not a binary matter. For some people, perhaps for most, the joy of existence larded as it is with terrible things exceeds, as the case make be, the quiet desperation, sorrow, or terror, and the emptiness at the center of all things. My exploration focuses on the anxieties rather than the curiosity. I take it for granted that the deep source of anxiety is mortality and its various intimations. Our quest for lives of meaning and significance is threatened by the “horrid d’s”1 of death, disease, dread, despair, [End Page 5] and disappointment. Various thinkers have theorized how we cope with this anxiety. Some argue that denial is a dominant way of dealing with death, which they take to be foundational to the other meaning and significance-threatening phenomena.2 Where the answers to these threats provided by traditional religions are found lacking or otherwise no longer credible, denial expresses itself in the form of various immortality projects: typically, the puritanical, maniacal, and methodical pursuit of health and wealth, in consumerism of all kinds, from the hottest new diet, to a Peloton exercise bike, to a Tesla. And then there is the pursuit of fame and celebrity, which many people can no longer distinguish, the hope, captured so well in the theme song for the television series, Fame. “Fame / I’m going to live forever / Baby, remember my name.”3 We want to be remembered. Whether by future generations or, inside the digital clouds of Instagram or TikTok, or, as some would have it, within the consequent nature of God. Of course, for some populations, death is not so easily denied. Death has an in their face immediacy, insistence, and oppugnancy that rips through flesh and leaves its subject not merely anxious but dead. I speak, for example, of the mind-numbing repetitiveness with which police kill Black people. Or the myriad ways, including environmental racism, in which Blacks are subject to premature death. Some thinkers even theorize that death is the existential horizon for Black people in an ontologically distinctive way.4 Thus the March 2020 police-raid killing of Breonna Taylor in her home may be a statistical anomaly but is experienced as a monstrous revelation of the nature of things, of Black social life as social death. This is not a... (shrink)

The article discusses the link between freedom, crime and punishment. According to some theorists, crime does not only cause a person to have less freedom; it constitutes, _in and of itself_, a breach of the freedom of others. Punishment does not only cause people to have more freedom, for instance by preventing crimes; it constitutes, _in and of itself_, respect for mutual freedom. If the latter claims are true, crime and punishment must have certain _meanings_ that make them denials/affirmations of (...) freedom irrespective of their consequences. My aim is to show that such an immanent connection between crime/punishment and freedom exists. I do so by explicating the “natural meaning” of crime and punishment. This way of addressing the topic is inspired by Jean Hampton’s use of H. P. Grice’s concept of natural meaning. Expanding on Hampton’s theory, drawing on both H. L. A. Hart and Kant, I argue that crime has the natural meaning of denying freedom, and punishment has the natural meaning of affirming freedom. The paper presents an ideal theory, not a justification for actual criminal justice practices, which in most countries unfortunately fail to instantiate the value of mutual freedom. (shrink)

As traditionally conceived, the creation of a new rule of customary international law requires that states believe the law to already require the conduct specified in the rule. Distinguishing the process whereby a customary rule comes to exist from the process whereby that customary rule becomes law dissolves this chronological paradox. Creation of a customary rule requires only that states come to believe that there exists a normative standard to which they ought to adhere, not that this standard is law. (...) What makes the customary rule law is adherence by officials in the international legal system to a rule of recognition that treats custom as a source of valid law. Confusion over this distinction arises because in the international legal system the same agents whose beliefs give rise to a customary rule are the legal officials whose adherence to the rule of recognition leads them to deem that rule legally valid. The proposed solution to the chronological paradox employs H.L.A. Hart’s analysis of the concepts of law and a legal system, and in particular, the idea of a rule of recognition. Yet Hart famously denies the existence of a rule of recognition for international law. Hart’s denial rests on a failure to distinguish between the ontological and authoritative resolution functions of a rule of recognition, however. Once such a distinction is drawn, it can be argued that customary international law rests on a rule of recognition that serves the ontological function of making customary norms legal, though not the authoritative resolution function of settling disputes over the alleged legality of particular norms. (shrink)

For [Formula: see text], the coarse similarity class of A, denoted by [Formula: see text], is the set of all [Formula: see text] such that the symmetric difference of A and B has asymptotic density 0. There is a natural metric [Formula: see text] on the space [Formula: see text] of coarse similarity classes defined by letting [Formula: see text] be the upper density of the symmetric difference of A and B. We study the metric space of coarse similarity classes (...) under this metric, and show in particular that between any two distinct points in this space there are continuum many geodesic paths. We also study subspaces of the form [Formula: see text] where [Formula: see text] is closed under Turing equivalence, and show that there is a tight connection between topological properties of such a space and computability-theoretic properties of [Formula: see text]. We then define a distance between Turing degrees based on Hausdorff distance in the metric space [Formula: see text]. We adapt a proof of Monin to show that the Hausdorff distances between Turing degrees that occur are exactly 0, [Formula: see text], and 1, and study which of these values occur most frequently in the senses of Lebesgue measure and Baire category. We define a degree a to be attractive if the class of all degrees at distance [Formula: see text] from a has measure 1, and dispersive otherwise. In particular, we study the distribution of attractive and dispersive degrees. We also study some properties of the metric space of Turing degrees under this Hausdorff distance, in particular the question of which countable metric spaces are isometrically embeddable in it, giving a graph-theoretic sufficient condition for embeddability. Motivated by a couple of issues arising in the above work, we also study the computability-theoretic and reverse-mathematical aspects of a Ramsey-theoretic theorem due to Mycielski, which in particular implies that there is a perfect set whose elements are mutually 1-random, as well as a perfect set whose elements are mutually 1-generic. Finally, we study the completeness of [Formula: see text] from the perspectives of computability theory and reverse mathematics. (shrink)

Abstract:Spirituality is an undeniable human need and is thus the subject of increasing interest among management scholars and practitioners. In this article, we propose using archetypal psychology as a framework for understanding the human need for spirituality more clearly because it provides important insights into spirituality and organizational life. Because most spiritual needs reside in the deepest aspects of the self, an archetypal approach helps us recognize not only that we have spiritual needs but alsowhywe have them. We present three (...) common archetypes and their implications in a management context. That is followed by an application of the archetypal approach to some of the more spiritually corrosive aspects of organizational life and a discussion of the implications of archetypes as a source of motivation. (shrink)

We investigate the complexity of various combinatorial theorems about linear and partial orders, from the points of view of computability theory and reverse mathematics. We focus in particular on the principles ADS (Ascending or Descending Sequence), which states that every infinite linear order has either an infinite descending sequence or an infinite ascending sequence, and CAC (Chain-AntiChain), which states that every infinite partial order has either an infinite chain or an infinite antichain. It is well-known that Ramsey's Theorem for pairs (...) (RT₂²) splits into a stable version (SRT₂²) and a cohesive principle (COH). We show that the same is true of ADS and CAC, and that in their cases the stable versions are strictly weaker than the full ones (which is not known to be the case for RT₂² and SRT₂²). We also analyze the relationships between these principles and other systems and principles previously studied by reverse mathematics, such as WKL₀, DNR, and BΣ₂. We show, for instance, that WKL₀ is incomparable with all of the systems we study. We also prove computability-theoretic and conservation results for them. Among these results are a strengthening of the fact, proved by Cholak, Jockusch, and Slaman, that COH is Π₁¹-conservative over the base system RCA₀. We also prove that CAC does not imply DNR which, combined with a recent result of Hirschfeldt, Jockusch, Kjos-Hanssen, Lempp, and Slaman, shows that CAC does not imply SRT₂² (and so does not imply RT₂²). This answers a question of Cholak, Jockusch, and Slaman. Our proofs suggest that the essential distinction between ADS and CAC on the one hand and RT₂² on the other is that the colorings needed for our analysis are in some way transitive. We formalize this intuition as the notions of transitive and semitransitive colorings and show that the existence of homogeneous sets for such colorings is equivalent to ADS and CAC, respectively. We finish with several open questions. (shrink)

Whenever a structure with a particularly interesting computability-theoretic property is found, it is natural to ask whether similar examples can be found within well-known classes of algebraic structures, such as groups, rings, lattices, and so forth. One way to give positive answers to this question is to adapt the original proof to the new setting. However, this can be an unnecessary duplication of effort, and lacks generality. Another method is to code the original structure into a structure in the given (...) class in a way that is effective enough to preserve the property in which we are interested. In this paper, we show how to transfer a number of computability-theoretic properties from directed graphs to structures in the following classes: symmetric, irreflexive graphs; partial orderings; lattices; rings ; integral domains of arbitrary characteristic; commutative semigroups; and 2-step nilpotent groups. This allows us to show that several theorems about degree spectra of relations on computable structures, nonpreservation of computable categoricity, and degree spectra of structures remain true when we restrict our attention to structures in any of the classes on this list. The codings we present are general enough to be viewed as establishing that the theories mentioned above are computably complete in the sense that, for a wide range of computability-theoretic nonstructure type properties, if there are any examples of structures with such properties then there are such examples that are models of each of these theories. (shrink)

Several notions of computability-theoretic reducibility between [Formula: see text] principles have been studied. This paper contributes to the program of analyzing the behavior of versions of Ramsey’s Theorem and related principles under these notions. Among other results, we show that for each [Formula: see text], there is an instance of RT[Formula: see text] all of whose solutions have PA degree over [Formula: see text] and use this to show that König’s Lemma lies strictly between RT[Formula: see text] and RT[Formula: see (...) text] under one of these notions. We also answer two questions raised by Dorais, Dzhafarov, Hirst, Mileti, and Shafer on comparing versions of Ramsey’s Theorem and of the Thin Set Theorem with the same exponent but different numbers of colors. Still on the topic of the effect of the number of colors on the computable aspects of Ramsey-theoretic properties, we show that for each [Formula: see text], there is an [Formula: see text]-coloring [Formula: see text] of [Formula: see text] such that every [Formula: see text]-coloring of [Formula: see text] has an infinite homogeneous set that does not compute any infinite homogeneous set for [Formula: see text], and connect this result with the notion of infinite information reducibility introduced by Dzhafarov and Igusa. Next, we introduce and study a new notion that provides a uniform version of the idea of implication with respect to [Formula: see text]-models of RCA0, and related notions that allow us to count how many applications of a principle [Formula: see text] are needed to reduce another principle to [Formula: see text]. Finally, we fill in a gap in the proof of Theorem 12.2 in Cholak, Jockusch, and Slaman. (shrink)

We build an א₁-categorical but not א₀-categorical theory whose only computably presentable model is the saturated one. As a tool, we introduce a notion related to limitwise monotonic functions.

Cholak, Goncharov, Khoussainov, and Shore [1] showed that for each k > 0 there is a computably categorical structure whose expansion by a constant has computable dimension k. We show that the same is true with k replaced by ω. Our proof uses a version of Goncharov's method of left and right operations.

We give some new examples of possible degree spectra of invariant relations on Δ 0 2 -categorical computable structures, which demonstrate that such spectra can be fairly complicated. On the other hand, we show that there are nontrivial restrictions on the sets of degrees that can be realized as degree spectra of such relations. In particular, we give a sufficient condition for a relation to have infinite degree spectrum that implies that every invariant computable relation on a Δ 0 2 (...) -categorical computable structure is either intrinsically computable or has infinite degree spectrum. This condition also allows us to use the proof of a result of Moses [23] to establish the same result for computable relations on computable linear orderings. We also place our results in the context of the study of what types of degree-theoretic constructions can be carried out within the degree spectrum of a relation on a computable structure, given some restrictions on the relation or the structure. From this point of view we consider the cases of Δ 0 2 -categorical structures, linear orderings, and 1-decidable structures, in the last case using the proof of a result of Ash and Nerode [3] to extend results of Harizanov [14]. (shrink)

We show that for every c.e. degree a > 0 there exists an intrinsically c.e. relation on the domain of a computable structure whose degree spectrum is {0, a}. This result can be extended in two directions. First we show that for every uniformly c.e. collection of sets S there exists an intrinsically c.e. relation on the domain of a computable structure whose degree spectrum is the set of degrees of elements of S. Then we show that if α ∈ (...) ω∪{ω } then for any α-c.e. degree a > 0 there exists an intrinsically α-c.e. relation on the domain of a computable structure whose degree spectrum is {0, a}. All of these results also hold for m-degree spectra of relations. (shrink)

There has been increasing interest over the last few decades in the study of the effective content of Mathematics. One field whose effective content has been the subject of a large body of work, dating back at least to the early 1960s, is model theory. Several different notions of effectiveness of model-theoretic structures have been investigated. This communication is concerned withcomputablestructures, that is, structures with computable domains whose constants, functions, and relations are uniformly computable.In model theory, we identify isomorphic structures. (...) From the point of view of computable model theory, however, two isomorphic structures might be very different. For example, under the standard ordering of ω the success or relation is computable, but it is not hard to construct a computable linear ordering of type ω in which the successor relation is not computable. In fact, for every computably enumerable degree a, we can construct a computable linear ordering of type ω in which the successor relation has degree a. It is also possible to build two isomorphic computable groups, only one of which has a computable center, or two isomorphic Boolean algebras, only one of which has a computable set of atoms. Thus, for the purposes of computable model theory, studying structures up to isomorphism is not enough. (shrink)

We give a straightforward computable-model-theoretic definition of a property of \Delta^0_2 sets called order-computability. We then prove various results about these sets which suggest that, simple though the definition is, the property defies any easy characterization in pure computability theory. The most striking example is the construction of two computably isomorphic c.e. sets, one of which is order-computable and the other not.

We show that for every computably enumerable degree a > 0 there is an intrinsically c.e. relation on the domain of a computable structure of computable dimension 2 whose degree spectrum is { 0 , a } , thus answering a question of Goncharov and Khoussainov 55–57). We also show that this theorem remains true with α -c.e. in place of c.e. for any α∈ω∪{ω} . A modification of the proof of this result similar to what was done in (...) class='Hi'>Hirschfeldt shows that for any α∈ω∪{ω} and any α -c.e. degrees a 0 ,…, a n there is an intrinsically α -c.e. relation on the domain of a computable structure of computable dimension n+1 whose degree spectrum is { a 0 ,…, a n } . These results also hold for m-degree spectra of relations. (shrink)

We construct a class of relations on computable structures whose degree spectra form natural classes of degrees. Given any computable ordinal and reducibility r stronger than or equal to m-reducibility, we show how to construct a structure with an intrinsically invariant relation whose degree spectrum consists of all nontrivial r-degrees. We extend this construction to show that can be replaced by either or.

1. Setting off: An introduction. 1.1. A measure of motivation. 1.2. Computable mathematics. 1.3. Reverse mathematics. 1.4. An overview. 1.5. Further reading -- 2. Gathering our tools: Basic concepts and notation. 2.1. Computability theory. 2.2. Computability theoretic reductions. 2.3. Forcing -- 3. Finding our path: Konig's lemma and computability. 3.1. II[symbol] classes, basis theorems, and PA degrees. 3.2. Versions of Konig's lemma -- 4. Gauging our strength: Reverse mathematics. 4.1. RCA[symbol]. 4.2. Working in RCA[symbol]. 4.3. ACA[symbol]. 4.4. WKL[symbol]. 4.5. [symbol]-models. (...) 4.6. First order axioms. 4.7. Further remarks -- 5. In defense of disarray -- 6. Achieving consensus: Ramsey's theorem. 6.1. Three proofs of Ramsey's theorem. 6.2. Ramsey's theorem and the arithmetic hierarchy. 6.3. RT, ACA[symbol], and the Paris-Harrington theorem. 6.4. Stability and cohesiveness. 6.5. Mathias forcing and cohesive sets. 6.6. Mathias forcing and stable colorings. 6.7. Seetapun's theorem and its extensions. 6.8. Ramsey's theorem and first order axioms. 6.9. Uniformity -- 7. Preserving our power: Conservativity. 7.1. Conservativity over first order systems. 7.2. WKL[symbol] and II[symbol]-conservativity. 7.3. COH and r-II[symbol]-conservativity -- 8. Drawing a map: Five diagrams -- 9. Exploring our surroundings: The world below RT[symbol]. 9.1. Ascending and descending sequences. 9.2. Other combinatorial principles provable from RT[symbol]. 9.3. Atomic models and omitting types -- 10. Charging ahead: Further topics. 10.1. The Dushnik-Miller theorem. 10.2. Linearizing well-founded partial orders. 10.3. The world above ACA[symbol]. 10.4. Still further topics, and a final exercise. (shrink)

We show that there is a minimal pair in the nonuniform generic degrees, and hence also in the uniform generic degrees. This fact contrasts with Igusa’s result that there are no minimal pairs for relative generic computability and answers a basic structural question mentioned in several papers in the area.

Goncharov and Peretyat'kin independently gave necessary and sufficient conditions for when a set of types of a complete theory is the type spectrum of some homogeneous model of. Their result can be stated as a principle of second order arithmetic, which is called the Homogeneous Model Theorem (HMT), and analyzed from the points of view of computability theory and reverse mathematics. Previous computability theoretic results by Lange suggested a close connection between HMT and the Atomic Model Theorem (AMT), which states (...) that every complete atomic theory has an atomic model. The authors show that HMT and AMT are indeed equivalent in the sense of reverse mathematics, as well as in a strong computability theoretic sense and do the same for an analogous result of Peretyat'kin giving necessary and sufficient conditions for when a set of types is the type spectrum of some model. (shrink)

We examine the reverse mathematical strength of a variation of Hindman’s Theorem (HT) constructed by essentially combining HT with the Thin Set Theorem to obtain a principle that we call thin-HT. This principle states that every coloring c:N→N has an infinite set S⊆N whose finite sums are thin for c, meaning that there is an i with c(s)≠i for all nonempty sums s of finitely many distinct elements of S. We show that there is a computable instance of thin-HT such (...) that every solution computes ∅′, as is the case with HT, as shown by Blass, Hirst, and Simpson (1987). In analyzing this proof, we deduce that thin-HT implies ACA0 over RCA0+IΣ20. On the other hand, using Rumyantsev and Shen’s computable version of the Lovász Local Lemma, we show that there is a computable instance of the restriction of thin-HT to sums of exactly 2 elements such that any solution has diagonally noncomputable degree relative to ∅′. Hence there is a computable instance of this restriction of thin-HT with no Σ20 solution. (shrink)

We report on some recent work centered on attempts to understand when one set is more random than another. We look at various methods of calibration by initial segment complexity, such as those introduced by Solovay [125], Downey, Hirschfeldt, and Nies [39], Downey, Hirschfeldt, and LaForte [36], and Downey [31]; as well as other methods such as lowness notions of Kučera and Terwijn [71], Terwijn and Zambella [133], Nies [101, 100], and Downey, Griffiths, and Reid [34]; higher level (...) randomness notions going back to the work of Kurtz [73], Kautz [61], and Solovay [125]; and other calibrations of randomness based on definitions along the lines of Schnorr [117].These notions have complex interrelationships, and connections to classical notions from computability theory such as relative computability and enumerability. Computability figures in obvious ways in definitions of effective randomness, but there are also applications of notions related to randomness in computability theory. For instance, an exciting by-product of the program we describe is a more-or-less naturalrequirement-freesolution to Post's Problem, much along the lines of the Dekker deficiency set. (shrink)

As a natural example of a 1-random real, Chaitin proposed the halting probability Ω of a universal prefix-free machine. We can relativize this example by considering a universal prefix-free oracle machine U. Let [Formula: see text] be the halting probability of UA; this gives a natural uniform way of producing an A-random real for every A ∈ 2ω. It is this operator which is our primary object of study. We can draw an analogy between the jump operator from computability theory (...) and this Omega operator. But unlike the jump, which is invariant under the choice of an effective enumeration of the partial computable functions, [Formula: see text] can be vastly different for different choices of U. Even for a fixed U, there are oracles A =* B such that [Formula: see text] and [Formula: see text] are 1-random relative to each other. We prove this and many other interesting properties of Omega operators. We investigate these operators from the perspective of analysis, computability theory, and of course, algorithmic randomness. (shrink)

We construct the set of the title, answering a question of Cholak, Jockusch, and Slaman [1], and discuss its connections with the study of the proof-theoretic strength and effective content of versions of Ramsey's Theorem. In particular, our result implies that every ω-model of RCA 0 + SRT 2 2 must contain a nonlow set.

This collection seeks to excavate the tradition of radical liberal class analysis, which predated and inspired Marx's reflections on class. Liberal class theory is distinctive because it regards relationship with the state as constitutive rather than just indicative of social class membership. Along with an introduction that frames the discussion historically and conceptually, Social Class and State Power provides readers with easy access to provocative texts from the early modern period to the present.

We show that the theory of the partial ordering of the computably enumerable degrees in any given nontrivial interval is undecidable and has uncountably many 1-types.

A set X is prime bounding if for every complete atomic decidable (CAD) theory T there is a prime model U of T decidable in X. It is easy to see that $X = 0\prime$ is prime bounding. Denisov claimed that every $X <_{T} 0\prime$ is not prime bounding, but we discovered this to be incorrect. Here we give the correct characterization that the prime bounding sets $X \leq_{T} 0\prime$ are exactly the sets which are not $low_2$ . Recall that (...) X is $low_2$ if $X\prime\prime$ $\leq_{T} 0\prime$ . To prove that a $low_2$ set X is not prime bounding we use a $0\prime$ -computable listing of the array of sets { Y : Y $\leq_{T}$ X } to build a CAD theory T which diagonalizes against all potential X-decidable prime models U of T. To prove that any $non-low_{2}$ ; X is indeed prime bounding, we fix a function f $\leq_T$ X that is not dominated by a certain $0\prime$ -computable function that picks out generators of principal types. Given a CAD theory T. we use f to eventually find, for every formula $\varphi (\bar{x})$ consistent with T, a principal type which contains it, and hence to build an X-decidable prime model of T. We prove the prime bounding property equivalent to several other combinatorial properties, including some related to the limitwise monotonic functions which have been introduced elsewhere in computable model theory. (shrink)

It is human nature to wonder how things might have turned out differently--either for the better or for the worse. For the past two decades psychologists have been intrigued by this phenomenon, which they call counterfactual thinking. Specifically, researchers have sought to answer the "big" questions: Why do people have such a strong propensity to generate counterfactuals, and what functions does counterfactual thinking serve? What are the determinants of counterfactual thinking, and what are its adaptive and psychological consequences? This important (...) work brings together a collection of thought-provoking papers by social and cognitive psychologists who have made important theoretical and empirical contributions to our understanding of this topic. The essays in this volume contain novel theoretical insights, and, in many cases descriptions of previously unpublished empirical studies. The Psychology of Counterfactual Thinking provides an excellent overview of this fascinating topic for researchers, as well as advanced undergraduates and graduates in psychology--particularly those with an interest in social cognition, social judgment, decision judgment, decision making, thinking and reasoning. (shrink)

Journal of Mathematical Logic, Volume 22, Issue 03, December 2022. Hirschfeldt and Jockusch (2016) introduced a two-player game in which winning strategies for one or the other player precisely correspond to implications and non-implications between [math] principles over [math]-models of [math]. They also introduced a version of this game that similarly captures provability over [math]. We generalize and extend this game-theoretic framework to other formal systems, and establish a certain compactness result that shows that if an implication [math] between (...) two principles holds, then there exists a winning strategy that achieves victory in a number of moves bounded by a number independent of the specific run of the game. This compactness result generalizes an old proof-theoretic fact noted by H. Wang (1981), and has applications to the reverse mathematics of combinatorial principles. We also demonstrate how this framework leads to a new kind of analysis of the logical strength of mathematical problems that refines both that of reverse mathematics and that of computability-theoretic notions such as Weihrauch reducibility, allowing for a kind of fine-structural comparison between [math] principles that has both computability-theoretic and proof-theoretic aspects, and can help us distinguish between these, for example by showing that a certain use of a principle in a proof is “purely proof-theoretic”, as opposed to relying on its computability-theoretic strength. We give examples of this analysis to a number of principles at the level of [math], uncovering new differences between their logical strengths. (shrink)