We give a general framework for the treatment of perturbations of types and structures in continuous logic, allowing to specify which parts of the logic may be perturbed. We prove that separable, elementarily equivalent structures which are approximately $aleph_0$-saturated up to arbitrarily small perturbations are isomorphic up to arbitrarily small perturbations. As a corollary, we obtain a Ryll-Nardzewski style characterisation of complete theories all of whose separable models are isomorphic up to arbitrarily small perturbations.

We continue [2], developing simplicity in the framework of compact abstract theories. Due to the generality of the context we need to introduce definitions which differ somewhat from the ones use in first order theories. With these modified tools we obtain more or less classical behaviour: simplicity is characterized by the existence of a certain notion of independence, stability is characterized by simplicity and bounded multiplicity, and hyperimaginary canonical bases exist.

Accumulating data suggest that the eukaryotic cell originated from a merger of two prokaryotes, an archaeal host and a bacterial endosymbiont. However, since prokaryotes are unable to perform phagocytosis, the means by which the endosymbiont entered its host is an enigma. We suggest that a predatory or parasitic interaction between prokaryotes provides a reasonable explanation for this conundrum. According to the model presented here, the host in this interaction was an anaerobic archaeon with a periplasm‐like space. The predator was a (...) small (facultative) aerobic α‐proteobacterium, which penetrated and replicated within the host periplasm, and later became the mitochondria. Plausible conditions under which this interaction took place and circumstances that may have led to the contemporary complex eukaryotic cell are discussed. (shrink)

We develop positive model theory, which is a non first order analogue of classical model theory where compactness is kept at the expense of negation. The analogue of a first order theory in this framework is a compact abstract theory: several equivalent yet conceptually different presentations of this notion are given. We prove in particular that Banach and Hilbert spaces are compact abstract theories, and in fact very well-behaved as such.

We study in detail the blow-up procedure described in [BTW01]. We obtain a structure theorem for coreless polygroups as a double quotient space G//H, and a polygroup chunk theorem. Seeking to remove the arbitrary parameter needed for the blow-up, we find canonical Ø-invariant groupoids.

We develop the general theory of topometric spaces, i.e., topological spaces equipped with a well-behaved lower semi-continuous metric. Spaces of global and local types in continuous logic are the motivating examples for the study of such spaces. In particular, we develop Cantor-Bendixson analysis of topometric spaces, which can serve as a basis for the study of local stability (extending the ad hoc development in Ben Yaacov I and Usvyatsov A, Continuous first order logic and local stability. Trans Am Math (...) Soc, in press), as well as of global ${\aleph_0}$ -stability. We conclude with a study of perturbation systems (see Ben Yaacov I, On perturbations of continuous structures, submitted) in the formalism of topometric spaces. In particular, we show how the abstract development applies to ${\aleph_0}$ -stability up to perturbation. (shrink)

We present a framework for model theoretic forcing in a non first order context, and present some applications of this framework to Banach space theory.

In recent work, the authors have established the group configuration theorem for simple theories, as well as some of its main applications from geometric stability theory, such as the binding group theorem, or in the $\omega$-categorical case, the characterization of the forking geometry of a finitely based non-trivial locally modular regular type as projective geometry over a finite field and the equivalence of pseudolinearity and local modularity. The proof necessitated an extension of the model-theoretic framework to include almost hyperimaginaries, and (...) the study of polygroups. (shrink)

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)

We characterise imaginaries (up to interdefinability) in Hilbert spaces using a Galois theory for compact unitary groups.

We revisit the foundations of positive model theory, introducing h-inductive sentences. These allow a considerably simplified presentation of positive model theory, as well as a characterisation of Hausdorff cats by an amalgamation property of their h-inductive theory.

We prove that under reasonable assumptions, every cat (compact abstract theory) is metric, and develop some of the theory of metric cats. We generalise Morley's theorem: if a countable Hausdorff cat T has a unique complete model of density character Λ ≥ ω₁, then it has a unique complete model of density character Λ for every Λ ≥ ω₁.

Functional Neurological Symptom Disorder or Conversion Disorder, is a fairly common diagnosis among mental health patients in Pakistan. Despite its prevalence there's a dearth of research on the phenomenon, particularly on the experience of FNSD. The study was conducted with the aim to ascertain the lived experiences of individuals with Functional Neurological Symptom Disorder around stressful situations in their families in Pakistan. For this purpose, a total sample of 10 participants were recruited from the psychiatry department of a tertiary care (...) hospital in Lahore, Pakistan. Semi-structured interviews were conducted and analyzed through Interpretative Phenomenological Analysis. The two main themes revealed in the analyses were quarrels and unexpressed emotions. The sub-themes of quarrels included quarrels with family members, quarrels within family, parental/marital discord, and quarrels with extended family members. The subthemes for unexpressed emotions were hurt, anger, sadness, and jealousy. In conclusion, this study revealed that in Pakistan, stressors related to family serve as significant contributing factors in the development of FNSD. (shrink)

-/- Continuous first-order logic has found interest among model theorists who wish to extend the classical analysis of “algebraic” structures (such as fields, group, and graphs) to various natural classes of complete metric structures (such as probability algebras, Hilbert spaces, and Banach spaces). With research in continuous first-order logic preoccupied with studying the model theory of this framework, we find a natural question calls for attention. Is there an interesting set of axioms yielding a completeness result? -/- The primary purpose (...) of this article is to show that a certain, interesting set of axioms does indeed yield a completeness result for continuous first-order logic. In particular, we show that in continuous first-order logic a set of formulae is (completely) satisfiable if (and only if) it is consistent. From this result it follows that continuous first-order logic also satisfies an approximated form of strong completeness, whereby Σ⊧φ (if and) only if Σ⊢φ ∸2−n for all n < ω. This approximated form of strong completeness asserts that if Σ⊧φ, then proofs from Σ, being finite, can provide arbitrarily better approximations of the truth of φ. -/- Additionally, we consider a different kind of question traditionally arising in model theory—that of decidability. When is the set of all consequences of a theory (in a countable, recursive language) recursive? Say that a complete theory T is decidable if for every sentence φ, the value φ T is a recursive real, and moreover, uniformly computable from φ. If T is incomplete, we say it is decidable if for every sentence φ the real number φ T o is uniformly recursive from φ, where φ T o is the maximal value of φ consistent with T. As in classical first-order logic, it follows from the completeness theorem of continuous first-order logic that if a complete theory admits a recursive (or even recursively enumerable) axiomatization then it is decidable. (shrink)

We present an adaptation of continuous first order logic to unbounded metric structures. This has the advantage of being closer in spirit to C. Ward Henson's logic for Banach space structures than the unit ball approach, as well as of applying in situations where the unit ball approach does not apply. We also introduce the process of single point emph{emboundment}, allowing to bring unbounded structures back into the setting of bounded continuous first order logic. Together with results from cite{BenYaacov:Perturbations} regarding (...) perturbations of bounded metric structures, we prove a Ryll-Nardzewski style characterisation of theories of Banach spaces which are separably categorical up to small perturbation of the norm. This last result is motivated by an unpublished result of Henson. (shrink)

We prove that in a continuous ℵ₀-stable theory every type-definable group is definable. The two main ingredients in the proof are: 1. Results concerning Morley ranks (i.e., Cantor-Bendixson ranks) from [Ben08], allowing us to prove the theorem in case the metric is invariant under the group action; and 2. Results concerning the existence of translation-invariant definable metrics on type-definable groups and the extension of partial definable metrics to total ones.

This paper completes the proof of the group configuration theorem for simple theories started in [1]. We introduce the notion of an almost hyperdefinable structure, and show that it has a reasonable model theory. We then construct an almost hyperdefinable group from a polygroup chunk.

We prove that for every simple theory $T$ (or even simple thick compact abstract theory) there is a (unique) compact abstract theory $T^fP$ whose saturated models are the lovely pairs of $T$. Independence-theoretic results that were proved in [Ben Yaacov, Pillay, Vassiliev - Lovely pairs of models] when $T^fP$ is a first order theory are proved for the general case: in particular $T^fP$ is simple and we characterise independence.

We point out how the "Fundamental Theorem of Stability Theory", namely the equivalence between the "non order property" and definability of types, proved by Shelah in the 1970s, is in fact an immediate consequence of Grothendieck's "Criteres de compacite" from 1952. The familiar forms for the defining formulae then follow using Mazur's Lemma regarding weak convergence in Banach spaces.

We prove that the definition of supersimplicity in metric structures from [Ben Yaacov, Uncountable dense categoricity in cats] is equivalent to an textit{a priori} stronger variant. This stronger variant is then used to prove that if $T$ is a supersimple Hausdorff cat then so is its theory of lovely pairs.

In recent years, scientists have increasingly taken to investigate the predictive nature of cognition. We argue that prediction relies on abstraction, and thus theories of predictive cognition need an explicit theory of abstract representation. We propose such a theory of the abstract representational capacities that allow humans to transcend the “here-and-now.” Consistent with the predictive cognition literature, we suggest that the representational substrates of the mind are built as ahierarchy, ranging from the concrete to the abstract; however, we argue that (...) there are qualitative differences between elements along this hierarchy, generating meaningful, often unacknowledged,diversity.Echoing views from philosophy, we suggest that the representational hierarchy can be parsed into:modality-specificrepresentations, instantiated on perceptual similarity;multimodalrepresentations, instantiated primarily on the discovery of spatiotemporal contiguity; andcategoricalrepresentations, instantiated primarily on social interaction. These elements serve as the building blocks ofcomplex structures discussed in cognitive psychology (e.g., episodes, scripts) and are the inputs for mental representations that behave like functions, typically discussed in linguistics (i.e.,predicators). We support our argument for representational diversity by explaining how the elements in our ontology are all required to account for humans’ predictive cognition (e.g., in subserving logic-based prediction; in optimizing the trade-off between accurate and detailed predictions) and by examining how the neuroscientific evidence coheres with our account. In doing so, we provide a testable model of the neural bases of conceptual cognition and highlight several important implications to research on self-projection, reinforcement learning, and predictive-processing models of psychopathology. (shrink)

How compelling is radical normative pluralism, i.e. the view that contrary moral positions (deontological, consequentialist and so on) are all morally acceptable even in one given case? In ‘A Hostage Situation’ (2019), Saul Smilansky presents a thought experiment about moral decisions in life-and-death situations. According to Smilansky, the Hostage Situation (HS) reveals a rather puzzling and radical normative pluralistic picture, according to which even in life-and-death decisions, many moral choices that sometimes contradict each other are more or less equitable or (...) at least morally acceptable simultaneously. He argues that there is a paradigmatic difference between HS and Trolley-Problem-type cases (TP); according to him, the decision in TP is always single-valued, so one cannot be pluralistic in a plausible way—that is, one cannot rationally accept both options at the same time. TP, he claims, has been a misleading paradigmatic example; the radically pluralistic result of HS may be a better paradigm for much of morality. I argue that Smilansky should have gone further, and that a proper understanding of HS and TP reveals that, far from being in conflict, both thought experiments support a radical normative pluralism. Thus, building upon Smilansky’s work but offering a very different position, I present in this paper a radical, novel interpretation of Trolley-Problem-type cases, which, if convincing, should significantly affect the way we think about normative moral theory. (shrink)

We develop several aspects of local and global stability in continuous first order logic. In particular, we study type-definable groups and genericity.

We develop the theory of germs of generic functions in simple theories. Starting with an algebraic quadrangle (or other similar hypotheses), we obtain an "almost" generic group chunk, where the product is denned up to a bounded number of possible values. This is the first step towards the proof of the group configuration theorem for simple theories, which is completed in [3].

1. We show that if p is a real type which is internal in a set $\sigma$ of partial types in a simple theory, then there is a type p' interbounded with p, which is finitely generated over $\sigma$ , and possesses a fundamental system of solutions relative to $\sigma$ . 2. If p is a possibly hyperimaginary Lascar strong type, almost \sigma-internal$ , but almost orthogonal to $\sigma^{\omega}$ , then there is a canonical non-trivial almost hyperdefinable polygroup which multi-acts (...) on p while fixing $\sigma$ generically. In case p is $\sigma-internal$ and T is stable, this is the binding group of p over \sigma$. (shrink)

We lay down the groundwork for the treatment of almost hyperdefinable groups: notions from [5] are put into a natural hierarchy, and new notions, essential to the study to such groups, fit elegantly into this hierarchy. We show that "classical" properties of definable and hyperdefinable groups in simple theories can be generalised to this context. In particular, we prove the existence of stabilisers of Lascar strong types and of the connected and locally connected components of subgroups, and that in a (...) simple one-based theory an almost hyperdefinable group is bounded-by-abelian-by-bounded. (shrink)

We prove that the definition of supersimplicity in metric structures from [7] is equivalent to an a priori stronger variant. This stronger variant is then used to prove that if T is a supersimple Hausdorff cat then so is its theory of lovely pairs.

If ${\mathfrak {F}}$ is a type-definable family of commensurable subsets, subgroups or subvector spaces in a metric structure, then there is an invariant subset, subgroup or subvector space commensurable with ${\mathfrak {F}}$. This in particular applies to type-definable or hyper-definable objects in a classical first-order structure.

Given a continuous and isometric action of a Polish group G on an adequate Polish topometric space $(X,\tau,\rho )$ and $x \in X$, we find a necessary and sufficient condition for $\overline {Gx}^{\rho }$ to be co-meagre; we also obtain a criterion that characterizes when such a point exists. This work completes a criterion established in earlier work of the authors.

For a locally compact group G, we show that it is possible to present the class of continuous unitary representations of G as an elementary class of metric structures, in the sense of continuous logic. More precisely, we show how nondegenerate ∗-representations of a general ∗-algebra A (with some mild assumptions) can be viewed as an elementary class, in a many-sorted language, and use the correspondence between continuous unitary representations of G and nondegenerate ∗-representations of L1(G). We relate the notion (...) of ultraproduct of logical structures, under this presentation, with other notions of ultraproduct of representations appearing in the literature, and we characterize property (T) for G in terms of the definability of the sets of fixed points of L1 functions on G. (shrink)

ExcerptIntroduction One of the most profound changes to have engulfed the interdisciplinary study of religion, society, culture, and history has been what is now commonly termed the “post-secular” turn. This intellectual development, which has been gaining a growing momentum during the last two decades, offers a “revisionist” reconsideration of the “orthodox” paradigm of secularization. This revisionism involves practically all of the paradigm components, questioning not only the descriptive-analytical merit of the secularization thesis but also its political and philosophical implications. Critics (...) have pointed to, among other things, the thesis's misevaluation of the role of religion—as an…. (shrink)

Noting the prevalence of a misguided suspicion towards tradition, as well as an overt misunderstanding of the very notion of tradition in certain academic circles, this essay seeks to outline some of the basic tenets of an alternative understanding of tradition, based on a ‘sociological’ reading of several major philosophical works. It does so by revisiting and synthesizing some well-known, highly influential conceptual arguments that, taken together, offer a compelling, comprehensive interpretation and understanding of tradition, which manages to avoid and (...) overcome the false dichotomies that have dominated social-scientific thought. The article offers three corresponding analogies that capture the complex nature of tradition: tradition as language, tradition as narrative, and tradition as horizon. It then goes on to discuss the main implications these analogies carry to our understanding of tradition. (shrink)