Mathematical Problems from Applied Logic I (eBook)

Authors 21
Main Topics 9
Editors 10
Dov M Gabbay 11
Sergei S Goncharov 12
Michael Zakharyaschev 13
Authors 14
Franz Baader 17
Lev Beklemishev 17
Johan van Benthem 18
S Barry Cooper 19
John N Crossley 19
Wilfrid A Hodges 20
Ralf Küsters 20
Lawrence S Moss 21
Albert Visser 21
Content 23
Nonstandard Inferences in Description Logics: The Story So Far 29
1. Introduction 30
2. Description Logics and Standard Inferences 34
3. Nonstandard Inferences—Motivation and Definitions 39
4. A Structural Characterization of Subsumption 51
5. The Least Common Subsumer 60
6. The Most Specific Concept 64
7. Rewriting 72
8. Matching 81
9. Conclusion and Future Perspectives 92
References 94
Problems in the Logic of Provability 104
1. Introduction 105
2. Informal Concepts of Proof 108
3. Basics of Provability Logic 118
4. Provability Logic for Intuitionistic Arithmetic 120
5. Provability Logic and Bounded Arithmetic 129
6. Classification of Bimodal Provability Logics 133
7. Magari Algebras 136
8. Interpretability Logic 141
9. Graded Provability Algebras 147
10. List of Problems 152
References 156
Open Problems in Logical Dynamics 164
1. Logical Dynamics 164
2. Standard Epistemic Logic 166
3. Public Announcement: Epistemic Logic Dynamfied 171
4. Dynamic Epistemic Logic 184
5. Background in Standard Logics 191
6. From Information Update to Belief Revision 197
7. Temporal Epistemic Logic 205
8. Game Logics and Game Theory 211
9. Conclusion 212
References 213
Computability and Emergence 220
1. An Emergent World around Us 221
2. Descriptions, Algorithms, and the Breakdown of Inductive Structure 222
3. Ontology and Mathematical Structure 229
4. Where Does It All Start? 231
5. Towards a Model Based on Algorithmic Content 236
6. Levels of Reality 241
7. Algorithmic Content Revisited 248
8. What Is to Be Done? 251
References 255
Samsara† 259
1. Introduction 260
2. An Example of a Process 261
3. What Logics Do We Need? 262
4. What Are Logical Systems 284
and What Should They Be? 284
5. The Nature of Proof 288
6. Final Remarks 295
References 296
Two Doors to Open 303
1. Logic and Cognitive Science 305
2. Medieval Arabic Semantics 326
References 339
Applied Logic: A Manifesto 343
1. What is Applied Logic? 343
2. Mathematics and Logic, but Different from Mathematical Logic 345
3. Applied Philosophical Logic 352
4. What Does Computer Science Have to Do with It? 354
5. Other Case Studies 358
6. Being as catholic as Possible 367
References 369
Index 370

CHAPTER 15
Interaction in Small Groups (p. 363-364)

PETER J. BURKE

INTRODUCTION

In the early 1970s, the question "whatever happened to research on the group in social psychology" was raised (Steiner, 1974). A year earlier, small group research was declared dead in a chapter on small groups subtitled "the light that failed" (Mullins &, Mullins, 1973). In the early 1980s, Rosenberg and Turner's coverage of the field of social psychology included a chapter on small groups by Kurt Back (1981), but almost all of the research cited was done before the mid to late 1950s. The more recent coverage of the field of social psychology did not include a chapter on group processes (Cook, Fine, &, House, 1995). However, it did include a section under the rubric of social relationships and group processes in which seven chapters were placed. Small group research has not disappeared, rather, it has become ubiquitous, spread among a number of research issues (e.g., networks, exchange, bargaining, justice, group decision making, intergroup relations, jury studies, expectation states, minority influence, leadership, cohesion, therapy and self-analytic processes, and power and status) and disciplines (e.g., sociology, psychology, communications, organizational research) (Davis, 1996). In fact, research in all of these areas is active, though the outlets for such research are varied, and it is likely that no one is completely aware of the full range of activity.

On the other hand, research on groups has diminished in sociology as a result of the way in which much research on group processes is conducted. Following the insights of Zelditch (1969), laboratory practice in sociology has shifted from the earlier study of freely interacting persons in a group context to the study of particular processes, perceptions, and reactions that can often be studied on individuals within real or simulated social settings. This approach was often used in psychology from the early studies of Sherif on norms and the autokinetic effect (Sherif, 1936) and the Asch studies of conformity to group pressures (Asch, 1960), as well as the work on "groups" by Thibaut and Kelley (1959). As sociologists began to focus experimentally more on particular processes such as status or exchange, studies of the group qua group declined, but did not disappear.

In the present volume, two of the most active areas of group research have been elevated to theoretical orientations (expectation states and social exchange theory), and two other areas have their own chapters (intergroup relationships and interaction in networks). Still, the area of small group interaction contains a wide and rich history and set of empirical works that I attempt to summarize in this chapter.

This chapter is broken down into three sections. In the first, I review some of the historical foundations of small group research. I then cover selected research on three issues, again examining some important historical landmarks as well as more current theory and research. These three issue areas are status, power, and leadership, group integration and cohesion, and interaction.*

EARLY BEGINNINGS
Among the earliest writings on the small group is the work of Georg Simmel who, in the late 1800s and early 1900s, was concerned with general principles of groups and group formation (Wolff, 1950). At one end of the size continuum, he focused on how two person groups (dyads) differed from individuals in isolation and how groups of three (triads) differed from dyads (Wolff, 1950). At a more general level, he analyzed how people affiliate into groups of all sizes and how those multiple group affiliations influence the individual (Simmel, 1955). He also analyzed small groups, large groups, issues of divisions in groups, of authority and prestige as well as of superordination and subordination (Wolff, 1950)—all matters that still concern researchers in small groups.

Another writer in the early 1900s was Charles H. Cooley with interests in the nature of the social order. His work on conceptualizing primary groups reflected a general concern about changes in society, and how what are now called primary relationships (person to person) were giving way to more impersonal role to role relationships, what are now called secondary relationships (Cooley, 1909).

Thrasher's (1927) study of gangs in Chicago in the early 1920s focused on groups and group processes in a natural habitat. With discussions of status and leadership, the structure of and roles in the gang, social control of members. Thrasher examined many of the same group processes that continue to occupy researchers (cf. Short &, Strodtbeck, 1965). The rise of group therapy in the military during World War Two to handle the large numbers of battle stressed soldiers, who could not be accommodated in traditional individual therapy, gave rise to the study of what came to be known as T-groups (for therapy groups and [leadership] training groups). The study of therapy groups produced a plethora of research on group processes and the relationship between group processes and therapeutic processes (Bion, 1961, Scheflen, 1974, Whitaker &, Lieberman, 1967).