Nominal Abstraction

Recursive relational specifications are commonly used to describe the computational structure of formal systems. Recent research in proof theory has identified two features that facilitate direct, logic-based reasoning about such descriptions: the interpretation of atomic judgments through recursive definitions and an encoding of binding constructs via generic judgments. However, logics encompassing these two features do not currently allow for the definition of relations that embody dynamic aspects related to binding, a capability needed in many reasoning tasks. We propose a new relation between terms called nominal abstraction as a means for overcoming this deficiency. We incorporate nominal abstraction into a rich logic also including definitions, generic quantification, induction, and co-induction that we then prove to be consistent. We present examples to show that this logic can provide elegant treatments of binding contexts that appear in many proofs, such as those establishing properties of typing calculi and of arbitrarily cascading substitutions that play a role in reducibility arguments.Comment: To appear in the Journal of Information and Computatio

Junior High School Students’ Abstraction In Learning Geometry

Abstraction is a fundamental process in learning mathematics. Although it is a fundamental process but it is still an unfamiliar issue in mathematics education. On the other side, geometry, one of the fields in mathematics, consists of abstracts ideas having big portion in Junior High School. It is known that in this stage most students’ still thinks in concrete orientation. That is why it is necessary to know how the abstraction process in learning geometry. The aims of this research are capturing the students’ abstraction process during geometry instruction process and capturing students’ abstraction process during solving geometry problems. It is a qualitative research study. This research was conducted at Public Junior High School I Cimahi in RSBI classes, which subjects are students in grade VII. The data were collected by observation, test, and interview. Further the data were analyzed using analytical induction and constant comparative techniques. The results of this research are (1) the type of students’ abstraction process when learning geometry is a theoretical abstraction process and (2) the students’ abstraction process in solving geometry problems in that class is a type of abstraction, namely empirical abstraction process. However, the student’s abstraction has emphasis in terms of aspects of abstraction. The aspect of identifying objects’ characteristics through field experiences is more dominant than others. Key Words: Abstraction, Geometry, Empirical abstraction, Theoretical Abstraction

Abstraction of Observables

Making use of the laws of physical transactions, we study symmetrical many-points systems. Relation of group-theory to physical transactions in such symmetrical systems is dealt with. Studying perturbations in the stability states in the attractor-maps for transactions, approximate values of the observables are to be predicted for such systems. Further, Abstraction Theory is typified with respect to studying the properties of irreducible representations, if any, inside a given such group

Two kinds of abstraction in schizophrenia : a thesis presented in partial fulfilment of the requirements for the degree of Master of Arts in Psychology at Massey University

An impairment in abstracting ability has frequently been proposed as a reason for schizophrenic thought disorder. The performance of hospitalized chronic paranoid schizophrenics and non-paranoid schizophrenics were compared to a normal control group on two types of abstraction; a traditional conceptual abstraction task (similarities, Trunnell, 1964) and an inferential abstraction task (relational abstraction, Bransford, Barclay & Franks, 1972). These two measures allowed a differential interpretation of the nature of the abstraction impairment in schizophrenia. The two clinical groups did not significantly differ on the traditional hierarchical measure of abstraction. Performance of both schizophrenic groups, however, differed significantly from that of controls in that schizophrenic subjects employed less abstract concepts to classify items in this task. On the second measure of abstraction no significant differences were found between schizophrenic subjects and the control group. Differences between paranoid and non-paranoid subjects did not reach significance on this task but there was some indication that each of these schizophrenic sub-groups used different cognitive strategies on this measure. Paranoid schizophrenics appeared not to elaborate information beyond its original form. The non-paranoids, on the other hand, appeared to elaborate stimulus material but were confused between inferential and original information. The present results indicate that chronic paranoid schizophrenics have a different type of abstraction impairment to chronic non-paranoid schizophrenics on the inferential conceptual abstraction task. These findings indicate the utility of using two indices of abstraction and the importance of not treating schizophrenics as a homogeneous group