For a supersimple SU-rank 1 theory T we introduce the notion of a generic elementary pair of models of T . We show that the theory T* of all generic T-pairs is complete and supersimple. In the strongly minimal case, T* coincides with the theory of infinite dimensional pairs, which was used in 1184–1194) to study the geometric properties of T. In our SU-rank 1 setting, we use T* for the same purpose. In particular, we obtain a (...) characterization of linearity for SU-rank 1 structures by giving several equivalent conditions on T*, find a “weak” version of local modularity which is equivalent to linearity, show that linearity coincides with 1-basedness, and use the generic pairs to “recover” projective geometries over division rings from non-trivial linear SU-rank 1 structures. (shrink)

We continue the study of the connection between the “geometric” properties of SU -rank 1 structures and the properties of “generic” pairs of such structures, started in [8]. In particular, we show that the SU-rank of the theory of generic pairs of models of an SU -rank 1 theory T can only take values 1 , 2 or ω, generalizing the corresponding results for a strongly minimal T in [3]. We also use pairs to derive the implication from (...) pseudolinearity to linearity for ω -categorical SU -rank 1 structures, established in [7], from the conjecture that an ω -categorical supersimple theory has finite SU -rank, and find a condition on generic pairs, equivalent to pseudolinearity in the general case. (shrink)

A generic computation of a subset $A$ of $\mathbb{N}$ consists of a computation that correctly computes most of the bits of $A$, and never incorrectly computes any bits of $A$, but which does not necessarily give an answer for every input. The motivation for this concept comes from group theory and complexity theory, but the purely recursion theoretic analysis proves to be interesting, and often counterintuitive. The primary result of this paper is that there are no minimal pairs for (...) generic computability, answering a question of Jockusch and Schupp. (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.

We introduce the notion of a lovely pair of models of a simple theory T, generalizing Poizat's “belles paires” of models of a stable theory and the third author's “generic pairs” of models of an SU-rank 1 theory. We characterize when a saturated model of the theory TP of lovely pairs is a lovely pair, finding an analog of the nonfinite cover property for simple theories. We show that, under these hypotheses, TP is also simple, and we (...) study forking and canonical bases in TP. We also prove that assuming only that T is low, the existentially universal models of the universal part of a natural expansion TP+ of TP, are lovely pairs, and “simple Robinson universal domains”. (shrink)

In support of a recent conjecture by Nielsen (1999), we prove that the phenomena of ‘incomparable entanglement’— whereby, neither member of a pair of pure entangled states can be transformed into the other via local operations and classical communication (LOCC)—is a generic feature when the states at issue live in an infinite-dimensional Hilbert space.  2002 Elsevier Science B.V. All rights reserved.

In support of a recent conjecture by Nielsen (1999), we prove that the phenomena of ‘incomparable entanglement’— whereby, neither member of a pair of pure entangled states can be transformed into the other via local operations and classical communication (LOCC)—is a generic feature when the states at issue live in an infinite-dimensional Hilbert space.  2002 Elsevier Science B.V. All rights reserved.

We ask when, for a pair of structures $\mathcal {A}_1,\mathcal {A}_2$, there is a uniform effective procedure that, given copies of the two structures, unlabeled, always produces a copy of $\mathcal {A}_1$. We give some conditions guaranteeing that there is such a procedure. The conditions might suggest that for the pair of orderings $\mathcal {A}_1$ of type $\omega _1^{CK}$ and $\mathcal {A}_2$ of Harrison type, there should not be any such procedure, but, in fact, there is one. We (...) construct an example for which there is no such procedure. The construction involves forcing. On the way to constructing our example, we prove a general result on modifying Cohen generics. (shrink)

We show that every two-dimensional class of topological similarity, and hence every diagonal conjugacy class of pairs, is meager in the group of order preserving bijections of the rationale and in the group of automorphisms of the ordered rational Urysohn space.

A generic extension of the constructible universe by reals is defined, in which the union of ‐classes of x and y is a lightface set, but neither of these two ‐classes is separately ordinal‐definable.

The PBR theorem is widely seen as one of the most important no-go theorems in the foundations of quantum mechanics. Recently, in Cabbolet (Found Phys 53(3):64, 2023), it has been argued that there is no reality to the PBR theorem using a pair of bolts as a counterexample. In this addendum we expand on the argument: we disprove the PBR theorem by a generic counterexample, and we put the finger on the precise spot where Pusey, Barrett, and Rudolph (...) have made a tacit assumption that is false. (shrink)

How should AI-generated speech balance epistemic aims, such as precision and accuracy, with ethical and social considerations? This paper examines a subtle yet consequential aspect of LLM-driven communication: the use of generic generalizations that convey information about social groups (e.g., “immigrants work low-wage jobs”). While central to human epistemic and pedagogical practices, generics are theorized to reinforce stereotypes, essentialism, and injustice. Using ChatGPT-3.5 as a case study, I uncover tendencies for AI chatbots to inconsistently hedge and refuse generics, including (...) those that reflect well-documented social structural patterns, such as “women are more likely to get attacked while walking alone at night.” These tendencies are not only ethically dubious but also epistemically troubling, since they may reduce the accuracy and informativeness of outputs and impede the communication of important social truths. In response, I highlight the complexity of improving chatbots’ social generic use by evaluating four approaches and show that each approach carries significant tradeoffs. I suggest chatbots should be permitted to use social generics when such generics are paired with clarifications regarding their scope and fungibility. I then offer three complementary strategies to operationalize an Interdisciplinary Expert-Driven Counterfactual Dialogue proposal to improve the accuracy and social responsibility of descriptions of groups and patterns. (shrink)

Children and adults commonly produce more generic noun phrases (e.g., birds fly) about animals than artifacts. This may reflect differences in participants’ generic knowledge about specific animals/artifacts (e.g., dogs/chairs), or it may reflect a more general distinction. To test this, the current experiments asked adults and preschoolers to generate properties about novel animals and artifacts (Experiment 1: real animals/artifacts; Experiments 2 and 3: matched pairs of maximally similar, novel animals/artifacts). Data demonstrate that even without prior knowledge about these (...) items, the likelihood of producing a generic is significantly greater for animals than artifacts. These results leave open the question of whether this pattern is the product of experience and learned associations or instead a set of early-developing theories about animals and artifacts. (shrink)

Let be a countable first order structure and endow the universe of with the discrete topology. Then the automorphism group of becomes a topological group. A tuple of automorphisms is defined to be weakly generic iff its diagonal conjugacy class (in the algebraic sense) is dense (in the topological sense) and the ‐orbit of each is finite. Existence of tuples of weakly generic automorphisms are interesting from the point of view of model theory as well as from the (...) point of view of finite combinatorics.The main results of the present work are as follows. In Theorem 2.6 we characterize the existence of tuples of weakly generic automorphisms with the aid of the profinite topology of free groups. In Corollary 2.12 we will show that if has finite topological rank r (and satisfies a further, mild technical condition) then the existence of a weakly generic tuple in implies the existence of weakly generic tuples in for all natural number. Finally, in Theorem 3.2 we show that if is a countable model of an ℵ0‐categorical, simple theory in which all types over the empty set are stationary and has a pair of weakly generic automorphisms then it has tuples of weakly generic automorphisms of arbitrary finite length.At the technical level we will combine elementary investigations about the profinite topology of free groups with the results of [11] about topological ranks of the automorphism groups of some structures. (shrink)

In this paper, we study the power and limitations of computing effectively generic sequences using effectively random oracles. Previously, it was known that every 2-random sequence computes a 1-generic sequence and every 2-random sequence forms a minimal pair in the Turing degrees with every 2-generic sequence. We strengthen these results by showing that every Demuth random sequence computes a 1-generic sequence and that every Demuth random sequence forms a minimal pair with every pb-generic (...) sequence. Moreover, we prove that for every comeager${\cal G} \subseteq {2^\omega }$, there is some weakly 2-random sequenceXthat computes some$Y \in {\cal G}$, a result that allows us to provide a fairly complete classification as to how various notions of effective randomness interact in the Turing degrees with various notions of effective genericity. (shrink)

This paper examines the way Charles Darwin applied his domestic breeding analogy to the practical workings of species evolution: that application, it is argued, centered on Darwin's distinction between methodical and unconscious selection. Methodical selection, which entailed pairing particular individuals for mating purposes, represented conditions of strict geographic isolation, obviously useful for species multiplication (speciation). By contrast, unconscious selection represented an open landmass with a large breeding population. Yet Darwin held that this latter scenario, which often would include multiple ecological (...) subdistricts and thus partial isolation, was better suited for speciation than were isolated conditions. At the same time, many passages in Darwin's writings that apparently portrayed phyletic evolution exclusively (these including references to unconscious selection), actually applied to speciation as well, for phyletic change in a single district could constitute a local manifestation of a larger common-descent pattern. This generic use of "phyletic" change was reflected in Darwin's deployment of the unconscious selection analogy in his published writings as well as in his dispute with Moritz Wagner over the necessity of geographic isolation for speciation. We can thus understand Darwin's otherwise puzzling declaration in The Origin of Species that unconscious selection was 'more important' than the methodical approach. (shrink)

The aim of this work is an analysis of distal and non‐distal behavior in dense pairs of o‐minimal structures. A characterization of distal types is given through orthogonality to a generic type in, non‐distality is geometrically analyzed through Keisler measures, and a distal expansion for the case of pairs of ordered vector spaces is computed.

Given a countable model of set theory, we study the structure of its generic multiverse, the collection of its forcing extensions and ground models, ordered by inclusion. Mostowski showed that any finite poset embeds into the generic multiverse while preserving the nonexistence of upper bounds. We obtain several improvements of his result, using what we call the blockchain construction to build generic objects with varying degrees of mutual genericity. The method accommodates certain infinite posets, and we can (...) realize these embeddings via a wide variety of forcing notions, while providing control over lower bounds as well. We also give a generalization to class forcing in the context of second-order set theory, and exhibit some further structure in the generic multiverse, such as the existence of exact pairs. (shrink)

We show that the countable universal homogeneous meet-tree has a generic automorphism, but does not have a generic pair of automorphisms.

We study the class of weakly locally modular geometric theories introduced in [4], a common generalization of the classes of linear SU-rank 1 and linear o-minimal theories. We find new conditions equivalent to weak local modularity: "weak one-basedness", absence of type definable "almost quasidesigns", and "generic linearity". Among other things, we show that weak one-basedness is closed under reducts. We also show that the lovely pair expansion of a non-trivial weakly one-based ω-categorical geometric theory interprets an infinite vector (...) space over a finite field. (shrink)

In a pair of recent articles, William Hasker has attempted to defend Robert Adams’s new anti-Molinist argument. But I argue that the sense of explanatory priority operative in the argument is either equivocal or, if a univocal sense can be given to it, it is either so generic that we should have to deny its transitivity or so weak that it would not be incompatible with human freedom.

We show: There are pairs of universes V1V2 and there is a notion of forcing PV1 such that the change mentioned in the title occurs when going from V1[G] to V2[G] for a P-generic filter G over V2. We use forcing iterations with partial memories. Moreover, we implement highly transitive automorphism groups into the forcing orders.

A definable pair of disjoint non-OD sets of reals exists in the Sacks and ????0-large generic extensions of the constructible universe L. More specifically, if a∈2ω is eith...

This work establishes that the gauge dynamics of the Standard Model arise as structural necessities of the algebra M₃(ℂ) under Axioms 1–4 of the Cognitional Mechanics (CM) framework. Rather than assuming gauge groups, couplings, or renormalisation behaviour, the paper derives them from the minimal non‑commutative algebra capable of supporting internal distinction. The spectral space of the Dirac operator D_F is shown to possess a smooth structure without invoking Morse genericity, using only the isolated eigenvalue property and the geometry of the (...) pure‑state space P(A_F) ≅ CP² ⊔ {pt}. The dynamical weight κ is obtained non‑circularly as the unique eigenfunction of the dilation operator D = λ d/dλ, giving κ = c₀ λ² with λ² ≡ ξ² at Tier‑1. -/- A five‑functor chain F₄ ∘ H ∘ G ∘ F₂ ∘ F₁ maps the algebraic structure of M₃(ℂ) to the full gauge/sheaf structure of the Standard Model, with naturality verified at each stage. The Faddeev–Popov operator is proven elliptic independently of D_F, enabling BRST quantisation on the induced spectral manifold. The one‑loop β‑function coefficient b₀ = 11/3 is derived independently in three layers—algebraic (C_A = 3), geometric (scale invariance of a₄), and BRST (ghost pairing)—with n_f = 0 following from the Tier‑1 dissolution of particle ontology. -/- The Standard Model gauge sector is identified as the colimit of a sheaf over spectral patches, and three structural predictions follow without free parameters, including α_s/α_em = 13/7 from the cyclotomic structure Φ₃(3) and Φ₆(3). All results arise internally from the CM axioms, with no Tier‑2 assumptions. This establishes that the Standard Model is not an externally assembled structure but a forced projection of M₃(ℂ) under axiomatic closure. -/- Published on April 28, 2026 doi: 10.5281/zenodo.19833881. (shrink)

The expressive nature of musical improvisation is dissected, navigatingbetween two predominant theses: The Transparency Thesis (1) which proposes thatexpressiveness in improvisation transparently reflects the musician’s subjectiveaffectivity, and the Objective-Generic Expressiveness Thesis (2) asserting that ithinges only on the musicpractice’s objective components. This article challenges boththeses, arguing against (1) by emphasizing that musical expressivity transcends a merenatural outburst, and counteracting (2) by highlighting that it is not merely anenactment of objective expressive topoi. Introducing a novel perspective through theconceptual pair (...) of (social) Habitusand (individual) habit, a third stance is defended:the specific expressiveness in improvised music is significantly impacted by itsimprovisational quality, whereby the expressiveness emerges from the manipulationand transformation of an expressive Habitus within the musical performance’scontingent situation. (shrink)

There are known characterisations of several fragments of hybrid logic by means of invariance under bisimulations of some kind. The fragments include $\{\mathord {\downarrow }, \mathord {@}\}$ with or without nominals (Areces, Blackburn, Marx), $\mathord {@}$ with or without nominals (ten Cate), and $\mathord {\downarrow }$ without nominals (Hodkinson, Tahiri). Some pairs of these characterisations, however, are incompatible with one another. For other fragments of hybrid logic no such characterisations were known so far. We prove a generic bisimulation characterisation (...) theorem for all standard fragments of hybrid logic, in particular for the case with $\mathord {\downarrow }$ and nominals, left open by Hodkinson and Tahiri. Our characterisation is built on a common base and for each feature extension adds a specific condition, so it is modular in an engineering sense. (shrink)

A complete theory T has the Schröder–Bernstein property or simply the SB-property if any pair of elementarily bi-embeddable models are isomorphic. This property has been studied in the discrete first-order setting and can be seen as a first step towards classification theory. This paper deals with the SB-property on continuous theories. Examples of complete continuous theories that have this property include Hilbert spaces and any completion of the theory of probability algebras. We also study a weaker notion, the SB-property (...) up to perturbations. This property holds if any two elementarily bi-embeddable models are isomorphic up to perturbations. We prove that the theory of Hilbert spaces expanded with a bounded self-adjoint operator has the SB-property up to perturbations of the operator and that the theory of atomless probability algebras with a generic automorphism have the SB-property up to perturbations of the automorphism. We also study how the SB-property behaves with respect to randomizations. Finally we prove, in the continuous setting, that if T is a strictly stable theory then T does not have the SB-property. (shrink)

Journal of Mathematical Logic, Volume 22, Issue 02, August 2022. We generically construct a model in which the [math]-separation property is true, i.e. every pair of disjoint [math]-sets can be separated by a [math]-definable set. This answers an old question from the problem list “Surrealist landscape with figures” by A. Mathias from 1968. We also construct a model in which the (lightface) [math]-separation property is true.

We prove the preservation theorems for NATP; many of them extend the previously established preservation results for other model-theoretic tree properties. Using them, we also furnish proper examples of NATP theories which are simultaneously TP2 and SOP. First, we show that NATP is preserved by the parametrization and sum of the theories of Fraïssé limits of Fraïssé classes satisfying strong amalgamation property. Second, the preservation of NATP for two kinds of dense/co-dense expansions, i.e. the theories of lovely pairs and of (...) H-structures for geometric theories, and dense/co-dense expansion on vector spaces is proved. Next, we prove the preservation of NATP for the generic predicate expansion and the pair of an algebraically closed field and its distinguished subfield; for the latter, not only NATP, but also the preservation of NTP1 and NTP2 is considered. Finally, we present some proper examples of NATP theories using the results proved in this paper, including the parametrization of DLO and the expansion of an algebraically closed field by adding a generic linear order. In particular, we show that the model companion of the theory of algebraically closed fields with circular orders (ACFO) is NATP. (shrink)

Let A[ω]ω be a maximal almost disjoint family and assume P is a forcing notion. Say A is P-indestructible if A is still maximal in any P-generic extension. We investigate P-indestructibility for several classical forcing notions P. In particular, we provide a combinatorial characterization of P-indestructibility and, assuming a fragment of MA, we construct maximal almost disjoint families which are P-indestructible yet Q-destructible for several pairs of forcing notions. We close with a detailed investigation of iterated Sacks indestructibility.

Douglas Hofstadter coined the acronym ‘Jootsing’ to describe our endless ability to Jump Out Of The System (of rules, axioms, concepts, styles, etc.). Hofstadter viewed this as something possibly unique to human consciousness, which AI has not yet duplicated. Our minds can transcend computational and Godelian limitations and contemplate totalities. In this paper I argue that there is in fact a hard limit to jootsing (the “no-jootsing theorem”), but it is a limit that holds potentially important secrets about the nature (...) and role of consciousness. This appears to be a generic result of theories that aspire to totality (theories that include observers/agents in addition to what they observe/interact with). I show how we are faced with a pair of potential stances that have a distinctly Godelian flavour and that both include some kind of limitation on theories of totality: eternal inflation of knowledge versus closed loops. Using ideas from a recent version of dual-aspect monism based on the decomposition of a totality, I consider whether the two apparently conflicting stances can be reconciled, arguing that they can indeed and this reconciliation reveals what is special about consciousness. (shrink)

I postulate that the extension of a degree adjective is fixed by implicitly accepted non-analytic reference-fixing principles (“preconceptions”) that combine appeals to paradigmatic cases with generic principles designed to expand the extension of the adjective beyond the paradigmatic range. In regular occasions of use, the paradigm and generic preconceptions are jointly satisfied and determine the existence of an extension/anti-extension pair dividing the adjective’s comparison class into two mutually exclusive and jointly exhaustive subclasses. Sorites paradoxical occasions of use (...) are irregular occasions of use in which the paradigm and generic preconceptions are not jointly satisfied. In them, the relevant degree adjective lacks an extension, and utterances of sentences containing it appearing in sorites arguments do not have truth conditions. I also postulate a probable psychology of paradigm intuitions, used in a psychological explanation of the preference for solutions of the sorites paradox on which paradigm preconceptions retain their intuitive truth-values. (shrink)

Evolution and geometry generate complexity in similar ways. Evolution drives natural selection while geometry may capture the logic of this selection and express it visually, in terms of specific generic properties representing some kind of advantage. Geometry is ideally suited for expressing the logic of evolutionary selection for symmetry, which is found in the shape curves of vein systems and other natural objects such as leaves, cell membranes, or tunnel systems built by ants. The topology and geometry of symmetry (...) is controlled by numerical parameters, which act in analogy with a biological organism’s DNA. The introductory part of this paper reviews findings from experiments illustrating the critical role of two-dimensional (2D) design parameters, affine geometry and shape symmetry for visual or tactile shape sensation and perception-based decision making in populations of experts and non-experts. It will be shown that 2D fractal symmetry, referred to herein as the “symmetry of things in a thing”, results from principles very similar to those of affine projection. Results from experiments on aesthetic and visual preference judgments in response to 2D fractal trees with varying degrees of asymmetry are presented. In a first experiment (psychophysical scaling procedure), non-expert observers had to rate (on a scale from 0 to 10) the perceived beauty of a random series of 2D fractal trees with varying degrees of fractal symmetry. In a second experiment (two-alternative forced choice procedure), they had to express their preference for one of two shapes from the series. The shape pairs were presented successively in random order. Results show that the smallest possible fractal deviation from “symmetry of things in a thing” significantly reduces the perceived attractiveness of such shapes. The potential of future studies where different levels of complexity of fractal patterns are weighed against different degrees of symmetry is pointed out in the conclusion. (shrink)

Posner [6] has shown, by a nonuniform proof, that every ▵ 0 2 degree has a complement below 0'. We show that a 1-generic complement for each ▵ 0 2 set of degree between 0 and 0' can be found uniformly. Moreover, the methods just as easily can be used to produce a complement whose jump has the degree of any real recursively enumerable in and above $\varnothing'$ . In the second half of the paper, we show that the (...) complementation of the degrees below 0' does not extend to all recursively enumerable degrees. Namely, there is a pair of recursively enumerable degrees a above b such that no degree strictly below a joins b above a. (This result is independently due to S. B. Cooper.) We end with some open problems. (shrink)

This paper reports a literature review on the topic of ethical issues in in-depth interviews. The review returned three types of article: general discussion, issues in particular studies, and studies of interview-based research ethics. Whilst many of the issues discussed in these articles are generic to research ethics, such as confidentiality, they often had particular manifestations in this type of research. For example, privacy was a significant problem as interviews sometimes probe unexpected areas. For similar reasons, it is difficult (...) to give full information of the nature of a particular interview at the outset, hence informed consent is problematic. Where a pair is interviewed (such as carer and cared-for) there are major difficulties in maintaining confidentiality and protecting privacy. The potential for interviews to harm participants emotionally is noted in some papers, although this is often set against potential therapeutic benefit. As well as these generic issues, there are some ethical issues fairly specific to in-depth interviews. The problem of dual role is noted in many papers. It can take many forms: an interviewer might be nurse and researcher, scientist and counsellor, or reporter and evangelist. There are other specific issues such as taking sides in an interview, and protecting vulnerable groups. Little specific study of the ethics of in-depth interviews has taken place. However, that which has shows some important findings. For example, one study shows participants are not averse to discussing painful issues provided they feel the study is worthwhile. Some papers make recommendations for researchers. One such is that they should consider using a model of continuous (or process) consent rather than viewing consent as occurring once, at signature, prior to the interview. However, there is a need for further study of this area, both philosophical and empirical. (shrink)

We say that a computable structure $\mathcal {A}$ is computably categorical if for every computable copy $\mathcal {B}$, there exists a computable isomorphism $f:\mathcal {A}\to \mathcal {B}$. This notion can be relativized to a degree $\mathbf {d}$ by saying that a computable structure $\mathcal {A}$ is computably categorical relative to $\mathbf {d}$ if for every $\mathbf {d}$ -computable copy $\mathcal {B}$ of $\mathcal {A}$, there exists a $\mathbf {d}$ -computable isomorphism $f:\mathcal {A}\to \mathcal {B}$. A key part of this thesis (...) is to study the behavior of this notion of categoricity in the computably enumerable degrees.The main theorem in Chapter $1$ states that given any computable partially ordered set P and any computable partition $P=P_0\sqcup P_1$, there exists an embedding h of P into the c.e. degrees and a computable graph $\mathcal {G}$ which is computably categorical, computably categorical relative to all degrees in $h(P_0)$, and is not computably categorical relative to any degree in $h(P_1)$. We also show that by using largely the same techniques alongside a standard construction of minimal pairs, we can embed a four-element diamond lattice into the c.e. degrees in the style of the main result of Chapter $1$.We then apply some of the techniques used in this thesis to study the behavior of this notion in the context of generic degrees in Chapter $2$. Additionally, we show that several classes of structures admit a computable example that witnesses the pathological behavior of categoricity relative to a degree as seen in Chapter $1$ ’s main theorem.Lastly, in the context of reverse mathematics, we investigate the reverse mathematical strength of a topological principle named $\mathsf {wGS}^{\operatorname {cl}}$, a weakened version of the Ginsburg–Sands theorem which states that every infinite topological space contains one of the following five topologies as a subspace, with $\mathbb {N}$ as the underlying set: discrete, indiscrete, cofinite, initial segment, or final segment.Abstract prepared by Java Darleen VillanoE-mail: [email protected]: https://javavillano.github.io/thesisfinal.pdf. (shrink)

Let I be a countable jump ideal in $\mathscr{D} = \langle \text{The Turing degrees}, \leq\rangle$ . The central theorem of this paper is: a is a uniform upper bound on I iff a computes the join of an I-exact pair whose double jump a (1) computes. We may replace "the join of an I-exact pair" in the above theorem by "a weak uniform upper bound on I". We also answer two minimality questions: the class of uniform upper bounds (...) on I never has a minimal member; if ∪ I = L α [ A] ∩ ω ω for α admissible or a limit of admissibles, the same holds for nice uniform upper bounds. The central technique used in proving these theorems consists in this: by trial and error construct a generic sequence approximating the desired object; simultaneously settle definitely on finite pieces of that object; make sure that the guessing settles down to the object determined by the limit of these finite pieces. (shrink)

According to several prominent philosophers, pleasure and pain come in measurable quantities. This thesis is controversial, however, and many philosophers have presented or felt compelled to respond to arguments for the conclusion that it is false. One important class of these arguments concerns the problem of aggregation, which says that if pleasure and pain were measurable quantities, then, by definition, it would be possible to perform various mathematical and statistical operations on numbers representing amounts of them. It is sometimes argued (...) that such operations cannot be sensibly applied to pleasure and pain, and that sentences expressing such operations must be false or meaningless. The purpose of this paper is to present, explain, and rebut several versions of this argument. In the first section, I present a generic version of the argument. In the second section, I present a defense of its key premise based on a case involving comparisons of relief from pain, and explain why I think it fails. In the third section, I present and rebut another defense, based on a pair of analogies with temperature. In the final section, I present a third defense, based on an analogy with spatial distances. I then present my reasons for rejecting it. Along the way, I explain my reasons for thinking that pleasure and pain are amenable to interval measurement. (shrink)

We use Bub's (2016) correlation arrays and Pitowksy's (1989b) correlation polytopes to analyze an experimental setup due to Mermin (1981) for measurements on the singlet state of a pair of spin-12 particles. The class of correlations allowed by quantum mechanics in this setup is represented by an elliptope inscribed in a non-signaling cube. The class of correlations allowed by local hidden-variable theories is represented by a tetrahedron inscribed in this elliptope. We extend this analysis to pairs of particles of (...) arbitrary spin. The class of correlations allowed by quantum mechanics is still represented by the elliptope; the subclass of those allowed by local hidden-variable theories by polyhedra with increasing numbers of vertices and facets that get closer and closer to the elliptope. We use these results to advocate for an interpretation of quantum mechanics like Bub's. Probabilities and expectation values are primary in this interpretation. They are determined by inner products of vectors in Hilbert space. Such vectors do not themselves represent what is real in the quantum world. They encode families of probability distributions over values of different sets of observables. As in classical theory, these values ultimately represent what is real in the quantum world. Hilbert space puts constraints on possible combinations of such values, just as Minkowski space-time puts constraints on possible spatio-temporal constellations of events. Illustrating how generic such constraints are, the equation for the elliptope derived in this paper is a general constraint on correlation coefficients that can be found in older literature on statistics and probability theory. Yule (1896) already stated the constraint. De Finetti (1937) already gave it a geometrical interpretation. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:A Treatise on Efficacy: Between Western and Chinese ThinkingJeremy E. HenkelA Treatise on Efficacy: Between Western and Chinese Thinking. By François Jullien, translated by Janet Lloyd. Honolulu: University of Hawai'i Press, 2004. Pp. x + 202. $22.00.In A Treatise on Efficacy: Between Western and Chinese Thinking François Jullien argues that the different ways Chinese and Western thinkers have dealt with warfare and diplomacy reflect important differences in how (...) the two cultures understand human action in the world. When Chinese texts such as the Sunzi and the Guiguzi are set up as counterpoints to the Europeans Clausewitz and Machiavelli, Jullien argues that the Aristotelian inheritance of the latter pair is shown to be the cause of their inability to provide clean theories with the consistent predictive power they desired. Chinese strategists, by contrast, meet with no such limitations in the realm of human interaction. The basic difference between Western and Chinese thought that Jullien seeks to demonstrate is that "one constructs a model that is then projected onto the situation, which implies that the situation is momentarily 'frozen.' The other relies on the situation as on a disposition that is known to be constantly evolving" (p. 189; italics in original).Jullien's regular use of the generic terms "Western" and "Chinese" in constructing his comparison will likely give many comparative philosophers pause. But this [End Page 347] need not be a source of worry, as he is careful not to essentialize the two world-views. While he is pointing out patterns and tendencies in the thinking of Europe and China, he is not positing an essential incommensurable difference between them. Rather, he is pointing out that "what remains underdeveloped in the one context is more developed in the other" (p. 145). He goes on to say, "The purpose of our foray into China is not to imagine-let alone fabricate-other 'mentalities'.... It is simply to make use of other possible sources of intelligibility" (ibid.).Jullien begins by showing that Western thought regards human action as something that imposes itself on the world in order to bring about a predetermined goal. This is a natural consequence of the belief that the world is created: belief in creation implies something existing outside that creation, thus an intention behind that creation, and thus a transcendent norm toward which things within that creation aim. This two-leveled ontology of creator and creation yields a distinction between theory and practice, wherein practice always aims-and ultimately fails-at implementing an ideal that is established by theory. This worldview has led to demonstrable success in the sciences but meets with frequent frustration in the arena of human interaction. This limitation has been recognized by strategists in the West but has never been successfully resolved.Aristotle was the first to identify and try to resolve this problem. He posited phronesis as the skill that enables an individual to apply a model to the world successfully. For Aristotle such successful action consists of first identifying one's desired end and then reasoning back to one's present situation in order to establish the best means to achieve that end. Jullien's analysis shows that Clausewitz was working with the same model more than two thousand years later. Clausewitz believed that earlier attempts at creating a general theory of warfare had failed to achieve their goal-namely, to provide a model of warfare that a general could use to ensure victory-because the opponent's actions are unpredictable. The system that the military strategist is dealing with, then, is not mechanistic, and consequently plans are easily frustrated by ever-changing circumstances. But Clausewitz was unable to conceive of strategy as based on anything other than plans.For Clausewitz, warfare is composed of distinct engagements, separate moments in time where an army seeks to impose a plan on the world. Seeing warfare as composed of distinct moments means that time is an opponent, for time brings with it change and uncertainty and thus a greater chance that one's plans will be frustrated. Opportunity, then, is something unexpected-a chance happening at a specific moment that can be taken advantage of only... (shrink)

We give a necessary and sufficient condition in order that a type-shifting automorphism be constructed on a model of the Theory of Simple Types (TST) by forcing. Namely it is proved that, if for every n ≥ 1 there is a model of TST in the ground model M of ZFC that contains an n-extendible coherent pair, then there is a generic extension M[G] of M that contains a model of TST with a type-shifting automorphism, and hence M[G] (...) contains a model of NF. The converse holds trivially. It is also proved that there exist models of TST containing l-extendible coherent pairs. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:American Journal of Philology 1.1 (2000) 93-119 [Access article in PDF] Horace and His Fathers: Satires 1.4 and 1.6 Catherine Schlegel * Fere nulli alii sunt homines qui talem in filios suos habent potestatem qualem nos habemus. --Gaius Institutiones Iuris Civilis 1.55 No other ancient poet offers the sense of affectionate intimacy which Horace's "autobiography" in the Satires grants to his readers. 1 It is consequently with some initial (...) regret that readers recognize that Horace tells us very little about his life, 2 and that furthermore the "information" he supplies is motivated by its poetic context, rather than by the impulse which Horace beguilingly alludes to, of confessing his life to his books. Satires 1.4 and 1.6 are the well-known loci of Horace's upbringing by his father, told in the context of Horace's relation to Lucilius, his satiric forebear, and to Maecenas, the man conventionally known as Horace's patron. All four figures--father, son, satiric predecessor, and patron--are artifacts of the poet's generic construction, dramatis personae structured to provide a definition to Horace's satiric art. The freedman father who so famously raised his son, by hand as it were, serves to organize the relation between Horace and the two figures Horace makes to loom in his poetic life, Lucilius and Maecenas. Paired in their respective poems with Horace's father, Lucilius and Maecenas are given a fatherly relationship to Horace only to be displaced by the biological parent. More remarkably, Horace's biological father emerges from the poems as Horace's poetic father too, and this leaves Lucilius and Maecenas deprived of the poetically crucial role which they seemed bound to assume in the satire and life of the poet. These paternal maneuvers [End Page 93] in Satires 1.4 and 1.6 make the persona of Horace the poetic cause of his art, make the constructed "self" of Horace the unshakable source of his poetry, and secure a particular disposition for his satire. Although Horace appears to subordinate art to life, extracting the causes of his persona and his poetry from his father's training and social status, the eventual outcome works in reverse, and it is the poet's life which is subordinated to his art. Satire 1.4: Who is the Father of this Genre? It is especially important to recognize the artful selectivity of Horace's self-portrait in the Satires, because the satirist's persona emerges as a crucially defining element of the genre of satire for Horace. In Satire 1.4 he uses his own persona to explain, justify, and limit the satiric poetry he writes. Although he begins the poem by distinguishing himself from Lucilius stylistically, what evolves in the course of the poem is a contemplation of human character in which poetic style is only one outcome of that character. Horace's defense of his satire in 1.4 rests on a self-description couched in ethical, not poetic, terms. The merging of poetic style and personal character produces a picture of the satiric genre which is identified with the poet himself; the poetry is the inevitable outcome of the man. When Horace asks whether his poetry is justifiably suspectum (1.4.65), he answers by telling us who he is; the poet is the answer to the question about the genre. Style and ethos are thus made indistinguishable.That art can be wholly identified with its human source is in some sense a radical view, but it is nevertheless congenial to satire, a genre peculiarly fixed in the ordinariness of life, whose muse is, as Horace says later, pedestris. 3 Satire constantly finds its wisdom, parody, or bite in ordinary material reality, so Horace's strategy of equating the poem with its material cause, the poet, is perfectly consistent with the genre's [End Page 94] orientation. It is a genre, after all, whose name is derived from a food, the stuff (so to speak) of life. 4If the portrait of his father that Horace gives to us... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Perceiving Particulars-as-such Is Incoherent—A Reply to Mark SideritsMonima ChadhaI am honored by Mark Siderits' response to my article1 and thankful to him for the opportunity it affords me to clarify the arguments and develop the theses presented therein further. My discussion focuses primarily on a pair of epistemological theses drawing attention to what we can and cannot perceive.2 The negative thesis is that we cannot perceive particulars, and, (...) indeed, the very idea of "perceiving a particular-as-such" (which represents the position of some Buddhist philosophers, specifically Yogācāra-Sautrāntikas) is incoherent. The positive thesis, which draws its inspiration from the Navya-Naiyāyikas, is that we can perceive only universal features. Siderits focuses his response on the negative thesis. He does not explicitly complain about the positive thesis, although, being an enthusiast for the Buddhist approach, he cannot help but be suspicious about it. He complains that the argument for the negative thesis is wanting, for I seem to have "misconstrued those realist intuitions that stand behind the Yogācāra-Sautrāntika." Not only that, in section 2 of his response, he claims that my argument rests on a controversial thesis that presupposes a mistaken internalism about conscious content. I will argue that the difficulties Siderits raises for my argument are based on a misinterpretation of the central claim that underlies the argument for the negative thesis. This, in turn, will help to develop further the argument for my positive thesis.Before we turn to the arguments, I will attend to some terminological matters that Siderits draws attention to in the first section of his response. He complains that some distinctions, which are important in the context of this discussion, are conflated in my article. One of his suggestions—the distinction between particulars and individuals—is very helpful. The notion of "non-particular individuals" can be wheeled in here to clarify my theses and arguments.3 Henceforth, I will use "non-particular individuals" to refer to universal features.Siderits also underscores the differences between Nyāya universals and Kantian concepts by emphasizing the distinct metaphysical positions held by these philosophers. I am not convinced that he has shown that Nyāya universals are essentially incompatible with Kantian concepts. But, in any event, highlighting the differences [End Page 382] distracts us from the epistemological concerns raised in my earlier paper, so I will not examine those differences here.4Siderits rightly emphasizes the mind-independence of Nyāya universals: the thesis is indispensable to their metaphysics. Nyāya holds that some universals, namely those that pick out generic properties, are objectively real. Real universals are not mere mental constructions (figments of our imagination, as the Yogācāra-Sautrāntika puts it), or mere abstractions formed by putting together the common features of real particulars. Using the typical example of "cowness," the Naiyāyika explains that when we observe cows we not only grasp "that particular," we also grasp "cowness" as the common character manifested in distinct particular cows. For the Naiyāyikas, cowness is an example of a real universal: it presents a simple, indivisible property present in every cow and is directly grasped whenever we perceive a particular cow. Real universals are simple unanalyzable properties that exist "out there" and are cognized in perception. They are not complex ideas put together by the mind, and so in this sense they are mind-independent.However, to say that universals are mind-independent does not amount to saying that the mind cannot be implicated in their perception. Siderits' sympathies lie with the Buddhists, and, like them, he holds that since perception is our link with mind-independent reality, the perceptual link must be independent of any intervention by the mind. Nyāya universals are not "mind-independent" in this latter sense. The Naiyāyikas hold that the perception of universals requires abstraction in singling out one feature or property from the given. Furthermore, they would agree that a subject's perceptual grasp of universals is revealed in acquiring dispositional abilities to recognize other instances exemplifying the same property. The Naiyāyikas would readily admit a mental component for the perception... (shrink)

We generalize a result on True Arithmetic by Lachlan and Soare to certain other completions of Peano Arithmetic. If $\mathscr{T}$ is a completion of $\mathscr{PA}$, then Rep denotes the family of sets $X \subseteq \omega$ for which there exists a formula $\varphi$ such that for all $n \in \omega$, if $n \in X$, then $\mathscr{T} \vdash \varphi})$ ) and if $n \not\in X$, then $\mathscr{T} \vdash \neg\varphi})$. We show that if $\mathscr{S,J} \subseteq \mathscr{P}$ such that $\mathscr{S}$ is a Scott set, (...) $\mathscr{J}$ is a jump ideal, $\mathscr{S} \subset \mathscr{J}$ and for all $X \in \mathscr{J}$, there exists $C \in \mathscr{S}$ such that C is a "coding" set for the family of subtrees of 2$^{<\omega}$ computable in X, and if $\mathscr{T}$ is a completion of $\mathscr{PA}$ such that Rep = $\mathscr{S}$, then there exists a model $\mathscr{A}$ of $\mathscr{T}$ such that $\mathscr{J}$ is the Scott set of $\mathscr{A}$ and no enumeration of Rep is computable in $\mathscr{A}$. The model $\mathscr{A}$ of $\mathscr{T}$ is obtained via a new notion of forcing. Before proving our main result, we demonstrate the existence of uncountably many different pairs satisfying the conditions of our theorem. This involves a new characterization of 1-generic sets as coding sets for the computable subtrees of 2$^{<\omega}$. In particular, $C \subseteq \omega$ is a coding set for the family of subtrees of 2$^{<\omega}$ computable in X if and only if for all trees T $\subseteq 2^{<\omega}$ computable in X, if $\chi_C$ is a path through T, then there exists $\sigma \in T$ such that $\sigma \subset \chi_C$ and every extension of $\sigma$ is in T. Jockusch noted a connection between 1-generic sets and coding sets for computable subtrees of 2$^{<\omega}$. We show they are identical. (shrink)

This dissertation reconsiders some traditional issues in the foundations of quantum mechanics in the context of relativistic quantum field theory (RQFT); and it considers some novel foundational issues that arise first in the context of RQFT. The first part of the dissertation considers quantum nonlocality in RQFT. Here I show that the generic state of RQFT displays Bell correlations relative to measurements performed in any pair of spacelike separated regions, no matter how distant. I also show that local (...) systems in RQFT are "open" to influence from their environment, in the sense that it is generally impossible to perform local operations that would remove the entanglement between a local system and any other spacelike separated system. The second part of the dissertation argues that RQFT does not support a particle ontology -- at least if particles are understood to be localizable objects. In particular, while RQFT permits us to describe situations in which a determinate number of particles are present, it does not permit us to speak of the location of any individual particle, nor of the number of particles in any particular region of space. Nonetheless, the absence of localizable particles in RQFT does not threaten the integrity of our commonsense concept of a localized object. Indeed, RQFT itself predicts that descriptions in terms of localized objects can be quite accurate on the macroscopic level. The third part of the dissertation examines the so-called observer-dependence of the particle concept in RQFT -- that is, whether there are any particles present must be relativized to an observer's state of motion. Now, it is not uncommon for modern physical theories to subsume observer-dependent descriptions under a more general observer-independent description of some underlying state of affairs. However, I show that the conflicting accounts concerning the particle content of the field cannot be reconciled in this way. In fact, I argue that these conflicting accounts should be thought of as "complementary" in the same sense that position and momentum descriptions are complementary in elementary quantum mechanics. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Brief Mention: Shameless Interests: The Decent Scholarship of Indecency*Kenneth J. ReckfordGood intentions go astray. I had meant simply to celebrate the ease and naturalness with which classical scholars treat obscene subject-matter nowadays, but there were difficulties, which may prove instructive.I had felt oddly grateful, after reading and reviewing Dover’s 1993 Frogs, for how he explained (and of course, printed) the old scatological jokes that Merry (1905) had omitted, and (...) which Adam Parry copied onto the blackboard for our fascinated Greek 106a class in fall, 1951. How strange they seem now, all those expurgated editions, those unhelpful commentaries, those mystery-keeping professors. Like Byron, we all had comic encounters with the “grosser parts”:They only add them all in an appendix,Which saves, in fact, the trouble of an index...Perhaps we worked harder than Byron. I remember, early on, struggling with Cat. 56 (which even Housman misinterpreted in his notorious 1931 “Praefanda,” for Catullus does not look on, masturbating, but plunges in, making a threesome); and I struggled with Martial. I may even, at sixteen, have been the last of the Victorians, learning about sexual by-paths from Latin poetic grammar and syntax. (Loebs were forbidden under pain of death: but had I consulted Ker’s Martial, I would have found the offending passages translated into Italian, not English: “On the theory,” Rolfe Humphries once said, “that God doesn’t understand Italian.”) We laughed; it was good fun. Yet our half-prurient inquisitiveness was mixed with genuine philological research; we resented roadblocks to learning; and years later, when Fordyce’s Catullus appeared with twenty-two “dirty poems” left out, we felt absolutely betrayed.So I give thanks for unexpurgated texts and helpful commentaries (such as Dover’s Clouds and Frogs, and Henderson’s Lysistrata, and Ferguson’s Juvenal); for good translations, too, like Charles Martin’s The Poems of Catullus (Johns Hopkins, 1990), which combines plain speaking with a certain subtle elegance. I honor the pioneers: Arrowsmith and Parker, and Sullivan; Henderson’s Maculate Muse (1975) and Dover’s Greek Homosexuality (1978; earlier writings too); Adams’ Latin Sexual [End Page 311] Vocabulary (1982) and Richlin’s Garden of Priapus (1983).* Here were basic materials for further study and interpretation. Definitions and distinctions came first: between primary obscenities and euphemisms; between different generic conventions and expected attitudes (as in oratory, comedy, epigram). In art, too, the blacking-out of erect penises on Greek vases has been removed; wall-paintings from Pompeian brothels have been made accessible, even without a bribe; budding scholars of both sexes discourse learnedly on such subjects as “Male Discharge Imagery on Greek Vase-Painting” (at a 1995 colloquium); and now Martin Kilmer’s Greek Erotica (1994) has brought new precision to the study of, e.g., intercrural intercourse between “courting” men and youths, and heterosexual intercourse a tergo (where an oil flask may suggest anal penetration, as against vaginal). The details, the illustrations and descriptions, are increasingly accurate. Gradually, too, old-time scholarly taboos have been breached, so we can handle “dirty” words and pictures without fear of degradation. We are most grateful that obscenities are no longer kept in quarantine, or treated as the exclusive hunting preserve of a “gentleman’s club” (writing in Latin, of course), so that, taking our cue from Catullus and Martial, whose “poems are dirty but our lives were pure,” we may pursue the decent scholarship of indecency.Let me, though, introduce two caveats for philologists. First, that the poetic indecencies of Aristophanes, Catullus, Martial, and Juvenal remain “untranslatable”: not from embarrassment now, but from linguistic impotence. Take Catullus’ mentula, that ugly, dangling noun, so much more evocative than “prick” or “cock”; or take irrumabo and irrumator, those powerful, almost onomatopoeic four-syllable words. “Fuck in the [End Page 312] mouth” (early Richlin) explains but loses the tone. Martin does better with Cat. 16:I’ll fuck the pair of you as you prefer it,oral Aurelius, anal Furius...a touch of wit, accessible to non-Latinists. (Will Shackleton Bailey’s new Loeb of Martial send American students scurrying to their dictionaries to look up “sodomized”?)My second caveat is more nostalgic. In our innocent (or... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:A Treatise on Efficacy: Between Western and Chinese ThinkingJeremy E. HenkelA Treatise on Efficacy: Between Western and Chinese Thinking. By François Jullien, translated by Janet Lloyd. Honolulu: University of Hawai'i Press, 2004. Pp. x + 202. $22.00.In A Treatise on Efficacy: Between Western and Chinese Thinking François Jullien argues that the different ways Chinese and Western thinkers have dealt with warfare and diplomacy reflect important differences in how (...) the two cultures understand human action in the world. When Chinese texts such as the Sunzi and the Guiguzi are set up as counterpoints to the Europeans Clausewitz and Machiavelli, Jullien argues that the Aristotelian inheritance of the latter pair is shown to be the cause of their inability to provide clean theories with the consistent predictive power they desired. Chinese strategists, by contrast, meet with no such limitations in the realm of human interaction. The basic difference between Western and Chinese thought that Jullien seeks to demonstrate is that "one constructs a model that is then projected onto the situation, which implies that the situation is momentarily 'frozen.' The other relies on the situation as on a disposition that is known to be constantly evolving" (p. 189; italics in original).Jullien's regular use of the generic terms "Western" and "Chinese" in constructing his comparison will likely give many comparative philosophers pause. But this [End Page 347] need not be a source of worry, as he is careful not to essentialize the two world-views. While he is pointing out patterns and tendencies in the thinking of Europe and China, he is not positing an essential incommensurable difference between them. Rather, he is pointing out that "what remains underdeveloped in the one context is more developed in the other" (p. 145). He goes on to say, "The purpose of our foray into China is not to imagine-let alone fabricate-other 'mentalities'.... It is simply to make use of other possible sources of intelligibility" (ibid.).Jullien begins by showing that Western thought regards human action as something that imposes itself on the world in order to bring about a predetermined goal. This is a natural consequence of the belief that the world is created: belief in creation implies something existing outside that creation, thus an intention behind that creation, and thus a transcendent norm toward which things within that creation aim. This two-leveled ontology of creator and creation yields a distinction between theory and practice, wherein practice always aims-and ultimately fails-at implementing an ideal that is established by theory. This worldview has led to demonstrable success in the sciences but meets with frequent frustration in the arena of human interaction. This limitation has been recognized by strategists in the West but has never been successfully resolved.Aristotle was the first to identify and try to resolve this problem. He posited phronesis as the skill that enables an individual to apply a model to the world successfully. For Aristotle such successful action consists of first identifying one's desired end and then reasoning back to one's present situation in order to establish the best means to achieve that end. Jullien's analysis shows that Clausewitz was working with the same model more than two thousand years later. Clausewitz believed that earlier attempts at creating a general theory of warfare had failed to achieve their goal-namely, to provide a model of warfare that a general could use to ensure victory-because the opponent's actions are unpredictable. The system that the military strategist is dealing with, then, is not mechanistic, and consequently plans are easily frustrated by ever-changing circumstances. But Clausewitz was unable to conceive of strategy as based on anything other than plans.For Clausewitz, warfare is composed of distinct engagements, separate moments in time where an army seeks to impose a plan on the world. Seeing warfare as composed of distinct moments means that time is an opponent, for time brings with it change and uncertainty and thus a greater chance that one's plans will be frustrated. Opportunity, then, is something unexpected-a chance happening at a specific moment that can be taken advantage of only... (shrink)

In this chapter, we present results on dynamic modal operators that can change the accessibility relation of a model during the evaluation of a formula. In particular, we extend the basic modal language with modalities that are able to delete, add or swap an edge between pairs of elements in the domain of a model. We define a generic framework to characterize this kind of operations. First, we investigate relation-changing modal logics as fragments of other, better investigated, logics, and (...) in particular provide equivalence preserving translations into first order logic and hybrid logic. To investigate their expressive power, we define suitable notions of bisimulation for the logics introduced. We then turn to the complexity of different reasoning problems for these kind of logics. Finally, we discuss existing Hilbert-style axiomatizations, and the special techniques needed to establish completeness. (shrink)

In this chapter, we present a generic class of mean-field-type games in which the payoffs and the state dynamics depend not only on the state-action profile of the decision-makers but also on a measure of the state-action pair. The state dynamics is a measure-dependent process with jump-diffusion and regime switching. We derive novel equilibrium systems to be solved. Two solution approaches are presented: (i) dynamic programming principle and (ii) stochastic maximum principle. The relationship between dual function and adjoint (...) processes is provided. It is shown that the extension to the risk-sensitive case generates a nonlinearity to the adjoint process, and it involves three other processes associated with the diffusion, jump, and regime switching, respectively. (shrink)