Naming and Diagonalization, from Cantor to Godel to Kleene

We trace self-reference phenomena to the possibility of namingfunctions by names that belong to the domain over which thefunctions are defined. A naming system is a structure of theform (D, type( ),{ }), where D is a non-empty set; for everya D, which is a name of a k-ary function, {a}: D^k D is thefunction named by a, and type(a) is the type of a, which tellsus if a is a name and, if it is, the arity of the named function.Under quite general conditions we get a fixed point theorem,whose special cases include the fixed point theorem underlyingGödel's proof, Kleene's recursion theorem and many othertheorems of this nature, including the solution to simultaneousfixed point equations. Partial functions are accommodated byincluding "undefined" values; we investigate different systemsarising out of different ways of dealing with them. Many-sortednaming systems are suggested as a natural approach to generalcomputatability with many data types over arbitrary structures.The first part of the paper is a historical reconstruction ofthe way Gödel probably derived his proof from Cantor'sdiagonalization, through the semantic version of Richard. Theincompleteness proof–including the fixed point construction–resultfrom a natural line of thought, thereby dispelling the appearanceof a "magic trick". The analysis goes on to show how Kleene'srecursion theorem is obtained along the same lines.     

Undecidability without Arithmetization

In the present paper the well-known Gödels – Churchs argument concerning the undecidability of logic (of the first order functional calculus) is exhibited in a way which seems to be philosophically interestingfi The natural numbers are not used. (Neither Chinese Theorem nor other specifically mathematical tricks are applied.) Only elementary logic and very simple set-theoretical constructions are put into the proof. Instead of the arithmetization I use the theory of concatenation (formalized by Alfred Tarski). This theory proves to be an appropriate tool. The decidability is defined directly as the property of graphical discernibility of formulas.     

Labels and the Self: Identity Labels as Scaffold

This paper considers how labels may be used: “Neurodiverse,” “genderfluid,” “sex-positive,” “ADHD,” and “highly-sensitive” are just some of the labels that may be offered by patients in introducing themselves. Such labels can be thought of as shortcuts, a way to define identity and sum up a feeling state, attitude, or behaviour. While they may sometimes be “given” in the sense of a diagnosis, they are also “found” and self-adopted. Using scaffolding as a metaphor for allowing growth or development to take place (or compensate for its absence), the phenomenon of self-labelling is presented as fulfilling different functions, namely: Label as mirrored reflection; Label as creative defence; Label as something with which to play; Label as container for that which cannot yet be known; Label as calling something into being; and Label as collective dream image. The article starts with three brief composite clinical sketches, and goes on to explore some of the ways that labels may be used with reference to the presented clinical material.     

Randomized controlled trial of a prenatal focused coparenting consultation for unmarried black fathers and mothers: One-year infant and family outcomes

In the infant mental health field, scant conceptual attention has been given to coparenting and family adaptations of non-white family systems, with no evidence-based, community-informed coparenting interventions responsive to unmarried Black mothers’ and fathers’ life circumstances. This study examined 1-year post-partum child and family outcomes of a novel, modest dosage (six sessions) prenatal focused coparenting consultation (FCC) using randomized controlled trial methodology. One-hundred-thirty-eight expectant families (one or both parents identified as Black/African American) were randomized to an intervention (N = 70) or treatment-as-usual (TAU; control) condition (N = 68). TAU families received navigational support in accessing existing community services for pregnant families. Intervention families received TAU plus 6 dyadic FCC sessions led by a Black male-female Community Mentor team. When infants were three and 12 months old, parents reported on coparenting, father engagement, interparental aggression, depressive symptoms, and infant social and emotional adjustment. Intent-to-treat analyses focusing on 12-month post-partum data indicated significant intervention effects on coparenting, interparental psychological aggression, and infants’ emotional adjustment. Improvement was also seen in depression and father engagement, with gains for both groups. Results suggest FCC delivered by same-race Community Mentors to unmarried Black coparents transitioning to parenthood supports infant and family adaptation during the first year of life.     

Parenting stress and competence among mothers of young children with substance use disorders: The roles of trauma and reflective functioning

Posttraumatic stress symptoms are prominent in the lives of parents of young children with substance use disorders (SUD). Parenting experiences, particularly stress and competence, impact parenting behaviors and concomitant child growth and development. Factors that promote positive experiences of parenting, such as parental reflective functioning (PRF), and protect the mother and child from negative outcomes are crucial to understand to develop effective therapeutic interventions. The current US study analyzed baseline data from a parenting intervention evaluation to examine how length of substance misuse, PRF, and trauma symptoms were associated with parenting stress and parenting sense of competence among mothers in treatment for SUDs. Measures included the Addiction Severity Index, PTSD Symptom Scale-Self Report, Parental Reflective Functioning Questionnaire, Parenting Stress Index/Short Form, and Parenting Sense of Competence Scale. The sample included 54 predominantly White mothers with SUDs who had young children. Two multivariate regression analyses found that (1) lower parental reflective functioning and higher posttraumatic stress symptoms were associated with higher parenting stress, and (2) only higher posttraumatic stress symptoms were associated with lower levels of parenting sense of competence. Findings underscore the importance of addressing trauma symptoms and PRF when aiming to improve parenting experiences for women with an SUD.     

Type development in childhood and beyond

This article defines and explains type development across the lifespan and summarizes what other researchers have presented as a reference. A timeline for development is identified, and ways to differentiate development of a function from the development of other personality characteristics are described. Type development is about the dispersion of energy from an unconscious level to a managed conscious level. This development occurs in a spiral pattern, so development of each function occurs repeatedly but at higher levels of use and understanding each time. Although the dominant function develops most because of its innate drive to be the lead, each function can develop throughout the lifespan. The relative energy associated with each function always exists. When the energy is in the unconscious, it drives behavioural responses, but these are not within the individual’s control. The ideal is to develop the conscious awareness of each of the functions so each can be consciously used as needed to match a situation. Because development of the functions is essential, this article examines potential interferences with development and then offers some suggestions for ways to promote developmental opportunities.     

Voices of Dialogue and Directivity in Family Therapy With Refugees: Evolving Ideas About Dialogical Refugee Care

In this article, we reflect on our evolving ideas regarding a dialogical approach to refugee care. Broadening the predominant phased trauma care model and its engaging of directive expertise in symptom reduction, meaning making, and rebuilding connectedness, these developing dialogical notions involve the negotiation of silencing and disclosure, meaning and absurdity, hope and hopelessness in a therapeutic dialogue that accepts its encounter of cultural and social difference. In locating therapeutic practice within these divergent approaches, we argue an orientation on collaborative dialogue may operate together with notions from the phased trauma care model as heuristic background in engaging a polyphonic understanding of coping with individual and family sequelae of forced displacement. This locating of therapeutic practice, as informed by each perspective, invites us to remain present to fragments of therapeutic positioning that resonate power imbalance or appropriation in a therapeutic encounter imbued with a social context that silences refugees’ suffering. In a clinical case analysis, we further explore these relational complexities of negotiating directive expertise and collaborative dialogue in the therapeutic encounter with refugee clients.     

The Partners for Life Program: A Couples Approach to Cardiac Risk Reduction

Morbidity and mortality are reliably lower for the married compared with the unmarried across a variety of illnesses. What is less well understood is how a couple uses their relationship for recommended lifestyle changes associated with decreased risk for illness. Partners for Life compared a patient and partner approach to behavior change with a patient only approach on such factors as exercise, nutrition, and medication adherence. Ninety‐three patients and their spouses/partners consented to participate (26% of those eligible) and were randomized into either the individual or couples condition. However, only 80 couples, distributed across conditions, contributed data to the analyses, due to missing data and missing data points. For exercise, there was a significant effect of couples treatment on the increase in activity and a significant effect of couples treatment on the acceleration of treatment over time. In addition, there was an interaction between marital satisfaction and treatment condition such that patients who reported higher levels of marital distress in the individuals condition did not maintain their physical activity gains by the end of treatment, while both distressed and nondistressed patients in the couples treatment exhibited accelerating gains throughout treatment. In terms of medication adherence, patients in the couples treatment exhibited virtually no change in medication adherence over time, while patients in the individuals treatment showed a 9% relative decrease across time. There were no condition or time effects for nutritional outcomes. Finally, there was an interaction between baseline marital satisfaction and treatment condition such that patients in the individuals condition who reported lower levels of initial marital satisfaction showed deterioration in marital satisfaction, while non satisfied patients in the couples treatment showed improvement over time.