# Qualitative Models

and

### Alex Durig

###### Pp. 1582-1586 in E. Borgatta and R. J. V. Montgomery (Eds.), Encyclopedia of Sociology, Vol. IV. New York: Macmillan.

Qualitative models describe structure and metamorphoses among things or events or among properties of things or events. Sociologists have several ways of formulating qualitative models.

Qualitative modeling based on logic involves the following ideas. Propositions are simple sentences such as: “all humans are mortal,” and “a dictator is a human.” Propositions can be true or false, and negation of a proposition transforms truth into falsity, or falsity into truth. Compound statements are formed when two or more propositions are placed in disjunction or conjunction, signified in English by the words or (or nor) and and (or but). Compound statements are true if all their component propositions are true, and compound statements are false if all their component propositions are false. Disjunction of true and false propositions yields a compound statement that is true, whereas conjunction of true and false propositions yields a compound statement that is false. These definitions are sufficient for logical analyses, but a supplementary definition is useful: the conditional, "P implies Q" or "if P then Q", means that whenever proposition P is true, proposition Q is true also, but when P is false, Q may be either true or false.

Set theory corresponds closely with logic, to the point that logic formulations can be interpreted in terms of sets, and information about the existence of elements in sets and subsets can be interpreted in terms of logic. Logic also can be translated to Boolean algebra (which operates as does ordinary algebra except that there are only two numbers, 0 and 1, and 1+1=1), so any formulation in terms of logic can be transformed to an algebraic problem and processed mathematically.

Logic models have been used to define sociological constructs. Balzer (1990), for example, employed logic plus some additional mathematical ideas in order to construct a comprehensive definition of social institution. Logic models also can be used to compare competing sociological theories. Hannan (1998), for example, formalized different “stories” about how organizational age relates to organizational demise, and he used a computer program for automated deduction to prove that various empirical observations can be derived from different theoretical assumptions.

Znaniecki (1934) systematized analytic induction as a method for deriving logic models from statements known to be true as a result of sociological research. For example (alluding to a study by Becker, 1953, that applied the method), field research might have disclosed a set of fourteen males who are marijuana users, all of whom were taught to enjoy the drug; a set of three females who use marijuana though they never were taught to enjoy it; a set of six males who were taught how to enjoy marijuana, but who do not use it; and implicitly it is understood that other people never were taught to enjoy marijuana and do not use it. From this information one might conclude that for males like the ones who were studied, using marijuana implies being taught to enjoy the drug. Robinson's (1951) critique of analytic induction led to a hiatus in the development of logic models in sociology until modeling difficulties were understood better.

Ragin (1988) developed a method for constructing logic models from cross-sectional data. Empirically-valid propositions about all cases in a population are conjoined into a complex compound statement, transformed into Boolean-algebra format, and processed by a computer program. The result is a reduced compound statement that is empirically true for the cases and the propositions studied. The approach differs from statistical analysis of multi-fold tables in ignoring count information (other than whether a cell in a table has zero cases or more than zero cases), and in describing data patterns in terms of logic statements rather than in terms of the effects of variables and their interactions.

Abell (1987) and Heise (1989) developed a logic model approach for event sequence analyses. Logic models for sequences do not predict what will happen next but instead offer developmental accounts indicating what events must have preceded a focal event. A narrative of events is elicited from a culturally-competent consultant who also defines prerequisites of the events in terms of other events within the happening. Since prerequisites define implication relations, a logic model is obtained that accounts for sequencing of events within the happening and that can be tested as a possible explanation of event sequencing in other happenings. Routines that appear to have little surface similarity may be accountable by abstract events in a logic model; e.g., Corsaro and Heise (1990) showed that an abstract model account­ed for observed play routines among children in two different cultures. Abell (1987) suggested that ab­straction involves homomorphic reduction: That is, abstract events categorize concrete events that have identical logical relations with respect to events outside the category. Abbott (1995) reviewed logic models and other approaches to sequence analysis.

Careers are sequences in which the events are status transformations. Heise's (1990) logic model analysis of careers emphasized that individuals' sequences of status transformations are generated in limited patterns from institutional taxonomies of roles. Guttman scaling can be employed as a means of analyzing individual experiences in order to infer logic models that generate career sequences (e.g., see Wanderer 1984). Abbott and Hrycak (1990) applied optimal matching techniques to the problem of comparing career sequences, with the similarity of two sequences being measured as the minimum number of transformations required to change one sequence into the other; clusters of similar sequences discovered from the similarity measures are identified as genres of career patterns.

A formal grammar defines sequences of symbols that are acceptable in a language, being "essentially a deductive system of axioms and rules of inference, which generates the sentences of a language as its theorems" (Partee, ter Meulen, and Wall 1990). A grammar, like a logic model, is explanatory rather than predictive, interpreting why a sequence was constructed as it was or why a sequence is deviant in the sense of being unprincipled. Grammars have been applied for modeling episodes of social interaction, viewing sequences of social events as symbolic strings that are, or are not, legitimate within a language of action provided by a social institution (Skvoretz and Fararo 1980; Skvoretz 1984). The grammatical perspective on institutionalized action can be reformulated as a production system model in which a hierarchy of if-then rules defines how particular conditions instigate particular actions (Axten and Fararo 1977; Fararo and Skvoretz 1984).

Case frame grammar (Dirven and Radden 1987) deals with how syntactic position within a set of symbols designates function. For example, syntactic positioning in a sentence can designate an event's agent, action, object, instrument, product, beneficiary, and location (e.g., "the locksmith cut the blank with a grinder into a key for the customer in his shop"). Heise and Durig (1997) adapted case frame grammar to define an event frame for theoretical and empirical studies of social routines. The case-grammar perspective also informed Heise's (1979) symbolic interactionist modeling of social interaction by providing an agent-action-object-location framework for analyzing social events. Guttman's facet mapping sentences (see Shye 1978) implicitly employ a case-grammar framework for analyzing a conceptual domain in terms of sets of concepts that fit into different syntactic slots and thereby generate a large number of propositions related to the domain. For example, Grimshaw (1989) developed a complex mapping sentence that suggested how different kinds of ambiguities arise in conversation and are resolved as a function of a variety of factors.

The mathematics of abstract groups provide a means for modeling some deterministic systems. Suppose a few different situations exist, and combining any two situations establishes another one of the situations; the result of a string of combinations can be computed by combining adjacent situations two at a time in any order. Also suppose that any situation can be reproduced by combining it with one particular situation, and this identity situation can be obtained from any other situation through a single combination. Then the set of situations and the scheme for combining them together constitute a group, and the group describes a completely deterministic system of transformations. Kosaka (1989) suggested a possible application of abstract groups by modeling the aesthetic theory of a Japanese philosopher in which there are 64 defined transformations, such as "yabo" (rusticity) combines with "hade" (flamboyance) to produce "iki" (chic urbanity).

A classic sociological application of groups involved kinship. Classificatory kinship systems (which are common in aboriginal cultures) put every pair of people in a society into a kinship relationship that may have little relation to genetic closeness, and each person implicitly is in a society-wide kinship class that determines relationships with others. White (1963) showed through mathematical analysis that classificatory rules regarding marriage and parentage generate clans of people who are in the same kinship situation and that the resulting classificatory kinship system operates as an abstract group; then he demonstrated that existing kinship systems accord with analytic results.

Models of social networks sometimes employ the notion of semi-group—a set of situations and a scheme for combining them (i.e., a group without an identity situation). For example, Breiger and Pattison (1986) examined economic and marriage relations among elite families in fifteenth-century Florence and showed that each family's relations to other families constituted a semi-group that was part of the overall semi-group of family relations in the city; they were able to identify the allies and enemies of the famous Medici family from the structure of family relationships. Social network research, a sophisticated area of qualitative modeling in sociology, employs other algebraic and graph-theoretic notions as well (Marsden and Laumann 1984; Wasserman and Faust 1994).

In general, qualitative models describe systematic structures and processes, and developing qualitative models aids in interpretating nebulous phenomena. Creating and manipulating qualitative models confronts researchers with technical challenges, but software providing computer assistance is lessening the difficulties.

REFERENCES

Abbott, Andrew 1995 “Sequence Analysis: New Methods for Old Ideas.” Annual Review of Sociology 21:93-113.

— — and Alexandra Hrycak 1990 “Measuring Resemblance in Sequence Data: An Optimal Matching Analysis of Musicians' Careers.” American Journal of Sociology 96:144-185.

Abell, Peter 1987 The Syntax of Social Life: The Theory and Method of Comparative Narratives. New York: Oxford University Press.

Axten, N., and Thomas J. Fararo 1977 “The Information Processing Representation of Institutionalized Social Action.” In P. Krishnan, ed., Mathematical Models of Sociology. Sociological Review Monograph 24. Keele, UK: University of Keele. Reprinted 1979, Totowa, NJ: Rowan & Littlefield.

Balzer, Wolfgang 1990 “A Basic Model for Social Institutions.” Journal of Mathematical Sociology 16:1-29.

Becker, Howard S. 1953 “Becoming a Marihuana User.” American Journal of Sociology 59:235-243.

Breiger, Ronald L., and Philippa E. Pattison 1986. “Cumulated Social Roles: The Duality of Persons and Their Algebras.” Social Networks 8:215-256.

Corsaro, William and D. Heise 1990. “Event Structure Models From Ethnographic Data.” In C. Clogg, ed., Sociological Methodology: 1990. Cambridge, Mass.: Basil Blackwell.

Dirven, René, and Günter Radden, (eds.) 1987. Fillmore's Case Grammar: A Reader. Heidelberg, Germany: Julius Groos Verlag.

Fararo, Thomas J., and John Skvoretz 1984 “Institutions As Production Systems.” Journal of Mathematical Sociology 10:117-182.

Grimshaw, Allen D. 1989 Collegial Discourse: Professional Conversation Among Peers. Norwood NJ: Ablex.

Hannan, Michael T. 1998 “Rethinking Age Dependence in Organizational Mortality: Logical Formalizations.” American Journal of Sociology 104:126-164.

Heise, D. R. 1979 Understanding Events: Affect and the Construction of Social Action. New York: Cambridge University Press.

— — 1989 “Modeling Event Structures.” Journal of Mathematical Sociology 14:139-169.

— — 1990 “Careers, Career Trajectories, and the Self.” In J. Rodin, C. Schooler, and K. W. Schaie, eds., Self-Directedness: Cause and Effects Throughout the Life Course. New York: Lawrence Erlbaum.

— — and Alex Durig 1997 “A Frame for Organizational Actions and Macroactions.” Journal of Mathematical Sociology, 22:95-123.

Kosaka, Kenji 1989 “An Algebraic Reinterpretation of IKI NO KOZO (Structure of IKI).” Journal of Mathematical Sociology 14:293-304.

Marsden, Peter V., and Edward O. Laumann 1984 “Mathematical Ideas in Social Structure Analysis.” Journal of Mathematical Sociology 10:271-294.

Partee, B. H., A. ter Meulen, and R. E. Wall 1990 Mathematical Methods in Linguistics. Boston: Kluwer Academic Publishers.

Ragin, Charles C. 1988. Between Complexity and Generality: The Logic of Qualitative Comparison. Berkeley: University of California Press.

Robinson, W. S. 1951 “The Logical Structure of Analytic Induction. American Sociological Review 16:812-818.

Shye, S. (ed.) 1978. Theory Construction and Data Analysis in the Behavioral Sciences: A Volume in Honor of Louis Guttman. San Francisco: Jossey-Bass.

Skvoretz, John 1984 “Languages and Grammars of Action and Interaction: Some Further Results.” Behavioral Science 29:81-97.

— — and Thomas J. Fararo 1980 “Languages and Grammars of Action and Interaction: A Contribution to the Formal Theory of Action.” Behavioral Science 25:9-22.

Wanderer, J. J. 1984 “Scaling Delinquent Careers Over Time.” Criminology 22:83-95.

Wasserman, Stanley, and Katherine Faust 1994 Social Network Analysis: Methods and Applications. New York: Cambridge University Press.

White, Harrison C. 1963 An Anatomy of Kinship: Mathematical Models for Structures of Cumulated Roles. Englewood Cliffs NJ: Prentice-Hall.

Znaniecki, Florian 1934 The Method of Sociology. New York: Farrar & Rinehart.

David R. Heise
Alex Durig

[Word count: 2143 including heading and references.]

[28 February, 1999]