Focusing in linear meta-logic: extended report

It is well known how to use an intuitionistic meta-logic to specify natural deduction systems. It is also possible to use linear logic as a meta-logic for the specification of a variety of sequent calculus proof systems. Here, we show that if we adopt different {\em focusing} annotations for such linear logic specifications, a range of other proof systems can also be specified. In particular, we show that natural deduction (normal and non-normal), sequent proofs (with and without cut), tableaux, and proof systems using general elimination and general introduction rules can all be derived from essentially the same linear logic specification by altering focusing annotations. By using elementary linear logic equivalences and the completeness of focused proofs, we are able to derive new and modular proofs of the soundness and completeness of these various proofs systems for intuitionistic and classical logics

Young children's research: children aged 4-8 years finding solutions at home and at school

Children's research capacities have become increasingly recognised by adults, yet children remain excluded from the academy, with reports of their research participation generally located in adults' agenda. Such practice restricts children's freedom to make choices in matters affecting them, underestimates children’s capabilities and denies children particular rights. The present paper reports on one aspect of a small-scale critical ethnographic study adopting a constructivist grounded approach to conceptualise ways in which children's naturalistic behaviours may be perceived as research. The study builds on multi-disciplinary theoretical perspectives, embracing 'new' sociology, psychology, economics, philosophy and early childhood education and care (ECEC). Research questions include: 'What is the nature of ECEC research?' and 'Do children’s enquiries count as research?' Initially, data were collected from the academy: professional researchers (n=14) confirmed 'finding solutions' as a research behaviour and indicated children aged 4-8 years, their practitioners and primary carers as 'theoretical sampling'. Consequently, multi-modal case studies were constructed with children (n=138) and their practitioners (n=17) in three ‘good’ schools, with selected children and their primary carers also participating at home. This paper reports on data emerging from children aged 4-8 years at school (n=17) and at home (n=5). Outcomes indicate that participating children found diverse solutions to diverse problems, some of which they set themselves. Some solutions engaged children in high order thinking, whilst others did not; selecting resources and trialing activities engaged children in 'finding solutions'. Conversely, when children's time, provocations and activities were directed by adults, the quality of their solutions was limited, they focused on pleasing adults and their motivation to propose solutions decreased. In this study, professional researchers recognised 'finding solutions' as research behaviour and children aged 4-8 years naturalistically presented with capacities for finding solutions; however, the children's encounters with adults affected the solutions they found