Embedding locales and formal topologies into positive topologies

A positive topology is a set equipped with two particular relations between elements and subsets of that set: a convergent cover relation and a positivity relation. A set equipped with a convergent cover relation is a predicative counterpart of a locale, where the given set plays the role of a set of generators, typically a base, and the cover encodes the relations between generators. A positivity relation enriches the structure of a locale; among other things, it is a tool to study some particular subobjects, namely the overt weakly closed sublocales. We relate the category of locales to that of positive topologies and we show that the former is a re\ufb02ective subcategory of the latter. We then generalize such a result to the (opposite of the) category of suplattices, which we present by means of (not necessarily convergent) cover relations. Finally, we show that the category of positive topologies also generalizes that of formal topologies, that is, overt locales

The principle of pointfree continuity

In the setting of constructive pointfree topology, we introduce a notion of continuous operation between pointfree topologies and the corresponding principle of pointfree continuity. An operation between points of pointfree topologies is continuous if it is induced by a relation between the bases of the topologies; this gives a rigorous condition for Brouwer's continuity principle to hold. The principle of pointfree continuity for pointfree topologies $\mathcal{S}$ and $\mathcal{T}$ says that any relation which induces a continuous operation between points is a morphism from $\mathcal{S}$ to $\mathcal{T}$. The principle holds under the assumption of bi-spatiality of $\mathcal{S}$. When $\mathcal{S}$ is the formal Baire space or the formal unit interval and $\mathcal{T}$ is the formal topology of natural numbers, the principle is equivalent to spatiality of the formal Baire space and formal unit interval, respectively. Some of the well-known connections between spatiality, bar induction, and compactness of the unit interval are recast in terms of our principle of continuity. We adopt the Minimalist Foundation as our constructive foundation, and positive topology as the notion of pointfree topology. This allows us to distinguish ideal objects from constructive ones, and in particular, to interpret choice sequences as points of the formal Baire space

A constructive Galois connection between closure and interior

We construct a Galois connection between closure and interior operators on a given set. All arguments are intuitionistically valid. Our construction is an intuitionistic version of the classical correspondence between closure and interior operators via complement.Comment: This is a revised version. Content is reorganized so to separate clearly what requires an impredicative proof from what can be proven also predicatively. Moreover, some results are given in a more general form and some counterexamples are adde

Reducibility, a constructive dual of spatiality

An intuitionistic analysis of the relationship between pointfree and pointwise topology brings new notions to light that are hidden from a classical viewpoint. In this paper, we study one of these, namely the notion of reducibility for a pointfree topology, which is classically equivalent to spatiality. We study its basic properties and we relate it to spatiality and to other concepts in constructive topology. We also analyse some notable examples. For instance, reducibility for the pointfree Cantor space amounts to a strong version of Weak K\uf6nig\u2019s Lemma

The words of the body: psychophysiological patterns in dissociative narratives

Trauma has severe consequences on both psychological and somatic levels, even affecting the genetic expression and the cell\u2019s DNA repair ability. A key mechanism in the understanding of clinical disorders deriving from trauma is identified in dissociation, as a primitive defense against the fragmentation of the self originated by overwhelming experiences. The dysregulation of the interpersonal patterns due to the traumatic experience and its detrimental effects on the body are supported by influent neuroscientific models such as Damasio\u2019s somatic markers and Porges\u2019 polyvagal theory. On the basis of these premises, and supported by our previous empirical observations on 40 simulated clinical sessions, we will discuss the longitudinal process of a brief psychodynamic psychotherapy (16 sessions, weekly frequency) with a patient who suffered a relational trauma. The research design consists of the collection of self-report and projective tests, pre-post therapy and after each clinical session, in order to assess personality, empathy, clinical alliance and clinical progress, along with the verbatim analysis of the transcripts trough the Psychotherapy Process Q-Set and the Collaborative Interactions Scale. Furthermore, we collected simultaneous psychophysiological measures of the therapeutic dyad: skin conductance and hearth rate. Lastly, we employed a computerized analysis of non-verbal behaviors to assess synchrony in posture and gestures. These automated measures are able to highlight moments of affective concordance and discordance, allowing for a deep understanding of the mutual regulations between the patient and the therapist. Preliminary results showed that psychophysiological changes in dyadic synchrony, observed in body movements, skin conductance and hearth rate, occurred within sessions during the discussion of traumatic experiences, with levels of attunement that changed in both therapist and the patient depending on the quality of the emotional representation of the experience. These results go in the direction of understanding the relational process in trauma therapy, using an integrative language in which both clinical and neurophysiological knowledge may take advantage of each other

Characterization of quantum states in predicative logic

We develop a characterization of quantum states by means of first order variables and random variables, within a predicative logic with equality, in the framework of basic logic and its definitory equations. We introduce the notion of random first order domain and find a characterization of pure states in predicative logic and mixed states in propositional logic, due to a focusing condition. We discuss the role of first order variables and the related contextuality, in terms of sequents.Comment: 14 pages, Boston, IQSA10, to appea

The generalised type-theoretic interpretation of constructive set theory

We present a generalisation of the type-theoretic interpretation of constructive set theory into Martin-Löf type theory. The original interpretation treated logic in Martin-Löf type theory via the propositions-as-types interpretation. The generalisation involves replacing Martin-Löf type theory with a new type theory in which logic is treated as primitive. The primitive treatment of logic in type theories allows us to study reinterpretations of logic, such as the double-negation translation

Metric complements of overt closed sets

We show that the set of points of an overt closed subspace of a metric completion of a Bishop-locally compact metric space is located. Consequently, if the subspace is, moreover, compact, then its collection of points is Bishop compact.Comment: 9 pages, 1 figur

Abnormal default system functioning in depression: Implications for emotion regulation

open5noDepression is widely seen as the result of difficulties in regulating emotions. Based on neuroimaging studies on voluntary emotion regulation, neurobiological models have focused on the concept of cognitive control, considering emotion regulation as a shift toward involving controlled processes associated with activation of the prefrontal and parietal executive areas, instead of responding automatically to emotional stimuli. According to such models, the weaker executive area activation observed in depressed patients is attributable to a lack of cognitive control over negative emotions. Going beyond the concept of cognitive control, psychodynamic models describe the development of individuals’ capacity to regulate their emotional states in mother-infant interactions during childhood, through the construction of the representation of the self, others, and relationships. In this mini-review, we link these psychodynamic models with recent findings regarding the abnormal functioning of the default system in depression. Consistently with psychodynamic models, psychological functions associated with the default system include self-related processing, semantic processes, and implicit forms of emotion regulation. The abnormal activation of the default system observed in depression may explain the dysfunctional aspects of emotion regulation typical of the condition, such as an exaggerated negative self-focus and rumination on self-esteem issues. We also discuss the clinical implications of these findings with reference to the therapeutic relationship as a key tool for revisiting impaired or distorted representations of the self and relational objects.openMessina, Irene; Francesca, Bianco; Cusinato, Maria; Calvo, Vincenzo; Sambin, MarcoMessina, Irene; Bianco, Francesca; Cusinato, Maria; Calvo, Vincenzo; Sambin, Marc

Heyting-valued interpretations for Constructive Set Theory

We define and investigate Heyting-valued interpretations for Constructive Zermelo–Frankel set theory (CZF). These interpretations provide models for CZF that are analogous to Boolean-valued models for ZF and to Heyting-valued models for IZF. Heyting-valued interpretations are defined here using set-generated frames and formal topologies. As applications of Heyting-valued interpretations, we present a relative consistency result and an independence proof