Department of Sociology

Indiana University

Bloomington, IN 47405

and

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*
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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

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[28 February, 1999]