The Ackermann approach for modal logic,correspondence theory and second-order reduction

The problem of eliminating second-order quantification over predicate symbols is in general undecidable. Since an application of second-order quantifier elimination is correspondence theory in modal logic, understanding when second-order quantifier elimination methods succeed is an important problem that sheds light on the kinds of axioms that are equivalent to first-order correspondence properties and can be used to obtain complete axiomatizations for modal logics. This paper introduces a substitution-rewrite approach based on Ackermann?s Lemma to second-order quantifier elimination in modal logic. Compared to related approaches, the approach includes a number of enhancements: The quantified symbols that need to be eliminated can be flexibly specified. The inference rules are restricted by orderings compatible with the elimination order, which provides more control and reduces non-determinism in derivations thereby increasing the efficiency and success rate. The approach is equipped with a powerful notion of redundancy, allowing for the flexible definition of practical simplification and optimization techniques. We present correctness, termination and canonicity results, and consider two applications: (i) computing first-order frame correspondence properties for modal axioms and rules, and (ii) rewriting second-order modal problems to equivalent simpler forms. The approach allows us to define and characterize two new classes of formulae, which are elementary and canonical, and subsume the class of Sahlqvist formulae and the class of monadic inductive formulae.     

Those Easily Forgotten: The Impact of Emigration on Those Left Behind

Much has been written about the experiences and stresses of those who emigrate. By contrast, little attention has been paid to the experiences of those who stay behind—family members and friends who for various reasons do not to join their loved ones in the destination country. In this article, I describe the experiences of some South Africans whose families and friends have emigrated. This study forms part of a larger research project focusing on the impact of emigration on South African family life. Twenty‐one participants were interviewed by means of a semistructured interview at least 6 months after one or more family member(s) and/or friend(s) left South Africa, to explore participants’ experiences around their loved ones’ emigration. A thematic analysis of the data reveals that those left behind experience various emotions, ranging from emotional ambivalence to anger and distress. Emigration is mostly experienced as a vast loss, almost akin to a “death,” bringing about significant changes in social networks and relationships. The therapeutic significance of the findings for those working with emigrant families is also explored.     

The Development of Family Alliance From Pregnancy to Toddlerhood and Child Outcomes at 5 Years

This article presents a longitudinal study of the development of “family alliance” from pregnancy to toddlerhood in a community sample, as well as its links with the emotional and cognitive development of the child at age 5 years. Family alliance is defined as the quality of the interactive coordination between family members. We consider that the alliance constitutes a context for the child to learn emotion regulation and to develop an understanding of inner states. Family interactions (N = 38) were observed at the 5th month of pregnancy and at 3, 9, and 18 months after birth in a standardized situation of observation (Lausanne Trilogue Play). Marital satisfaction and child temperament were assessed through self‐reported questionnaires. Several outcomes of the child at age 5 years were measured: theory of mind performances, predominant emotional themes in pretend play, internalized and externalized symptoms. Results show that (a) three patterns of evolution of family alliance occur: “high stable” (n = 19), “high to low” (n = 10), and “low stable” (n = 9); (b) a high stable alliance is predictive of better outcomes in children at age 5 years, especially regarding theory of mind; (c) the temperament of the child is predictive of child outcomes; and (d) an interaction effect occurs between family alliance and temperament. These results highlight the importance of both family‐level and individual‐level variables for understanding individual differences in the social and cognitive development of children.