Journal of Mathematical Sociology, 29: 33–71, 2005
Copyright # Taylor & Francis Inc.
ISSN: 0022-250X print/1545-5874 online
DOI: 10.1080/00222500590889749
THE LOGIC OF ROLE THEORY: ROLE CONFLICT
AND STABILITY OF THE SELF-CONCEPT
James D. Montgomery
Department of Sociology, University of Wisconsin – Madison,
Madison, Wisconsin, USA
Non-monotonic logic offers a useful framework for modeling human reasoning
in social settings where role conflict arises from contradictions among roles,
norms, and constraints. This paper uses non-monotonic logic to specify the
processes by which an individual chooses actions (based on self-concept and
norms) and observers then make attributions about the individual (based on
actions and norms). By linking the choice and attribution processes together
with the assumption that the individual gradually internalizes attributions,
we obtain a feedback loop governing change in the self-concept. Analysis of this
feedback loop reveals that the self-concept may reach a stable long-run state–an
‘‘absorbing self’’ – only if the normative system is logically consistent.
Keywords: role theory, role conflict, self-concept, moral dilemmas, nonmonotonic logic
I. INTRODUCTION
Sociologists have often viewed action as a consequence of an actor’s
reasoning about roles and norms, and socialization as a process by which
roles are merged into the actor’s self-concept.1 Surprisingly, despite much
informal elaboration, sociologists have made few sustained efforts toward
mathematical formalization of role-theoretic concepts and processes.2
I am grateful to Tom Fararo for helpful comments and encouragement.
Author can be reached at [email protected]
1
I refer especially to ‘‘interactionist’’ perspectives (J. Turner, 1991, Chaps. 18–23) which
include structural role theory (Parsons, 1951; Merton, 1957), process role theory (R. Turner,
1978), dramaturgical theory (Goffman, 1959), symbolic interactionism (Kuhn, 1964; Blumer,
1969), and identity theory (Stryker and Burke, 2000). See also Stryker and Statham (1985)
for an extended comparison of symbolic interactionism with role theory and an attempt to
unify these perspectives through ‘‘role’’ concept.
2
Notable exceptions include affect control theory (Heise, 1979) and production-system
models (Fararo and Skvoretz 1984). Some interesting earlier efforts to employ formal logic
(Anderson and Moore, 1957; Anderson, 1962) seem mostly to have been forgotten.
33
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J. D. Montgomery
Perhaps this lack of progress may be partly explained by the inadequacy
of conventional mathematical formalisms. Given the sociological emphasis
on the ablility of humans to create and use symbols (most explicit in
symbolic interactionism but implicit in other role-theoretic perspectives),
formal logic would seem the most obvious mathematical framework for
representation of decision-making processes. However, because it cannot
admit logical contradictions, standard logic is an inappropriate model of
human reasoning in real-world social systems where individuals experience role conflict as a consequence of inconsistent roles, norms, and
constraints.
Recent developments in formal logic thus warrant close attention from
mathematical sociologists. Standard logic is ‘‘monotonic’’ in the sense that
formulas, once derived, are never retracted. In contrast, recently
developed non-monotonic logics allow tentative derivations that may be
retracted in light of new information (Brewka, Dix, and Konolige,
1997). Researchers in computer science and artificial intelligence have
thus used non-monotonic logics to model common-sense reasoning in settings where information is either incomplete or contradictory. Perhaps
most relevant for sociological purposes, Horty (1993, 1994) has shown
how non-monotonic logic may be used to reason about ‘‘moral dilemmas’’
induced by normative contradictions. Thus, in contrast to standard logic,
non-monotonic logic seems well-suited to role-theoretic applications and
might provide a useful framework for formalizing sociological conceptions
of agency.3
This paper demonstrates how non-monotonic logic might be used to
represent key role-theoretic processes. Inspired partly by Ralph Turner’s
conception of the ‘‘role-person merger’’ (R. Turner, 1978), I focus on
the socialization process through which an individual’s self-concept is
modified over time. In the model, an individual (given her initial self
and norms) chooses actions, observers (given actions and norms) make
attributions, and the individual gradually internalizes these attributions
(modifying her self). Thus, the self is governed by a feedback loop that
runs from self to choices to attributions to self. I model the reasoning processes of the individual and observers using a simple non-monotonic logic
known as a level default theory. The feedback loop is specified as a
Markov chain, where the state of the chain in each period is the individual’s current self.
3
Of course, standard (first-order) logic may remain useful at the metatheoretical level as
researchers attempt to verify the logical consistency of sociological theories (Péli, et al., 1994;
Hannan, 1998; Kamps and P
olos, 1999). See also the metatheoretical use of non-monotonic
logic by P
olos and Hannan (2002) to reconcile some apparent theoretical inconsistencies in
organizational ecology.
The Logic of Role Theory
35
Although my model would also permit analysis of short-run dynamics,
the present paper focuses exclusively on the long-run state of the individual’s self. In particular, I examine whether the social system generates
one or more absorbing states—‘‘absorbing selves’’—from which the
self-concept would never exit. The general framework is developed
through an extended example based loosely on Tally’s Corner (Liebow,
1967). Variations on this example illustrate cases where the social system
generates no absorbing self, multiple absorbing selves, and a unique
absorbing self. I then offer a more general analysis which reveals how
the structure of norms affects the number of absorbing selves. The central
result is that an absorbing self exists only if norms are logically consistent.
Intuitively, logical inconsistency implies that observers can always give
multiple interpretations of the individual’s behavior, thus undermining
the stability of the individual’s self. Given my graph-theoretic representation of the attribution process, the analysis simultaneously uncovers
an interesting application of balance theory in the study of normative
systems.
In previous work (Montgomery, 2000), I developed a conceptually
similar model of the socialization process using fuzzy logic (rather than
non-monotonic logic) to derive the choices made by the individual and
the attributions made by observers. Arguably, fuzzy logic may have
some advantages over the (crisp) non-monotonic logic employed in
the present paper, allowing formal representation of ambiguity in settings where individuals may feel that roles or norms apply only partially.
However, as will become apparent in the analysis below, non-monotonic
logic has other advantages. Highlighting the logical contradictions faced
by actors, level default theories offer an elegant, tractable framework
for deriving the multiple courses of action that are ‘‘as consistent as
possible.’’ Similarly, highlighting the logical contractions faced by observers, level default theories can be used to derive multiple ‘‘interpretations’’ or ‘‘accounts’’ of the individual’s action. Further, unlike fuzzy
logic, the crisp logic employed in the present paper leads naturally to
a graph-theoretic treatment which suggests a connection to balance
theory.
The paper is organized as follows. Section II describes how the roleperson merger may be viewed as a dialectic process and operationalized
using non-monotonic logic. Section III develops the model through an
extended example. Section IV considers variations on this example (with
different sets of norms and constraints) and derives the implications for
the focal individual’s long-run self. Section V contains a more general analysis of the relation between the structure of norms and the number of
absorbing selves. I conclude in Section VI with a discussion of the model
and directions for future research.
36
J. D. Montgomery
II. CHANGE IN THE SELF-CONCEPT AS
A DIALECTIC PROCESS
Beginning with the notion of the ‘‘looking-glass self’’ (Cooley, 1983 [1902]),
social psychologists have often suggested that the self-concept conforms to
the views of others. Here, I draw especially on Ralph Turner’s (1978)
specification of the process by which roles are ‘‘merged’’ into the self.
Reduced to its essence, the ‘‘role-person merger’’ becomes the simple feedback loop depicted in Figure 1. The focal individual (‘‘ego’’) has a selfconcept (or simply ‘‘self’’) that is a set of roles. Norms specify the actions
that should be taken by roles. Thus, given an initial self, ego may (at least
attempt to) choose those actions required by norms. In turn, observers
(‘‘alters’’) observe ego’s actions, attributing her behaviour to roles in her
self-concept. A key substantive claim in Turner (1978) is that attributions
are systematically biased. In particular, alters tend to ‘‘overattribute’’ roles
to ego’s self. That is, alters presume that ego’s actions are generated by
roles in ego’s self, even when alternative explanations for ego’s actions
could be given. Finally, closing the feedback loop, these (biased) attributions are gradually internalized by ego. In the long run, ego’s self-concept
contains just those roles that alters attribute to it.
Building on this preliminary specification of the role-person merger, we
might suppose that the self evolves through a dialectic process. Having
already specified the self as a set of roles, we might further specify norms
as logical (material) implications from roles to actions. The first two links in
the feedback loop would then correspond to logical syllogisms. Ego might
derive appropriate actions through the syllogism
self ; norms
‘
actions
ð1Þ
which reflects derivation through modus ponens (P and P ! Q therefore
Q), while alters infer ego’s self through the syllogism
actions; norms
FIGURE 1 The Role-Person Merger.
‘
self
ð2Þ
The Logic of Role Theory
37
which might reflect derivation through modus tollens (:Q and P ! Q
therefore :P). To capture alters’ tendency to make stronger (logically more
problematic) inferences, we might revise the second syllogism so that
alters make inferences through abduction (Q and P ! Q therefore P) as
well as modus tollens.4 In particular, we might replace the implication
(P ! Q) in the second syllogism with the corresponding equivalence
(P $ Q).
To illustrate, consider an example (to be expanded in Sections III and
IV) where ego is a father (i.e., ego’s self contains the role f ). A social norm
requires that a father should provide for his family (an action p). Represented as an implication, this norm becomes f ! p. Thus, ego derives his
choice through the syllogism
f ; f ! p ‘ p:
ð3Þ
Alters observing this action would attribute it to a role in ego’s self through
the syllogism
p; f $ p ‘ f :
ð4Þ
Note that the attribution f is not a valid derivation from the action p and norm
f ! p (because ego might have chosen p even if he was not an f ), but follows
from alters’ tendency always to attribute behavior to roles (reflected in alters’
use of the equivalence f$p rather than the implication f ! p).
A Logical Framework to Model Reasoning in
Complex Social Systems
Reflection on this simple example suggests crucial modeling issues that will
arise in more complex (and realistic) social systems containing multiple
roles, actions, norms, and constraints. For instance, my specification of
norms as material implications might seem to ignore an important distinction between ‘‘structural’’ constraints (that cannnot be violated) and ‘‘normative’’ constraints (that are, in practice, often violated). To illustrate,
assume that ego can provide for his family only if he holds a good job
(an action g). Represented as a material implication, this constraint
becomes p ! g. Presuming that ego has no other source of (non-labor)
income, he cannot violate this structural constraint if he provides for his
family. In contrast, an individual who is a father should provide for his
family but may choose (or perhaps be forced) not to obey this norm.
4
Although Lemmon (1965, p. 17) simply labels it a logical fallacy (‘‘affirming the consequent’’), abduction does resemble (valid) probabilistic reasoning (through Bayes’ Rule) and
is used to generate tentative conclusions in various non-monotonic logics (Brewka, et al.,
1997, Chap. 5).
38
J. D. Montgomery
Clearly, normative constraints (such as f!p) are ‘‘optional’’ in a sense that
structural constraints (such as p!g) are not.
To distinguish formally between constraints and norms, perhaps the
most obvious modeling strategy is to introduce modal operators into
the logical framework, following standard deontic logic (Hilpinen, 1971;
Chellas, 1980; Hughes and Cresswell, 1996). In this way, while continuing
to represent the constraint in our example as the material implication
p!g, we might represent the norm as f!p or perhaps (f!p) where
is a modal operator denoting ‘‘ought.’’ However, because standard
deontic logic presumes that the set of all norms is logically consistent, it
is unable to capture the role conflicts (‘‘moral dilemmas’’) that arise in
real-world settings.5 Returning to our example, an individual who is a father
might choose not to provide for his family in light of other conflicting
norms (perhaps related to other roles in his self-concept) and constraints
(perhaps preventing him from holding a good job).
The need for a contradiction-tolerating logic to address choice in normative systems has led researchers to consider various alternatives to standard deontic logic. Most relevent for my present purposes is Horty’s
(1993, 1994) attempt to base deontic logic upon non-monotonic logic, in
particular Reiter’s (1980) default logic. In default logic, conclusions are
derived from a default theory D ¼ (W; D) where W is set of logical
formulas (representing ‘‘reliable’’ knowledge) and D is a set of default rules
(that generate more ‘‘tentative’’ inferences such as those made through
abduction). While the formulas in W must be logically consistent, the
default rules in D may lead to inconsistent inferences. Loosely, an extension of a default theory is a set of conclusions derived from the (entire)
set W and some (consistent) subset of D; inconsistencies among the
default rules lead to multiple extensions.6 Adopting this framework, Horty
(1994, p. 45) makes the key suggestion that social norms might be represented as default rules. Founded on default logic, deontic logic would thus
permit moral dilemmas, and inconsistent norms would lead to multiple
extensions of the default theory.
While Horty’s (1993, 1994) efforts remain a source of inspiration for the
present paper, I do not use Reiter’s (1980) default logic to model the
5
Formally, for any formula Q, standard deontic logic does not permit both Q and :Q to
be true. This restriction might be understood as a consequence of the possible-world semantics. In standard deontic logic, each world is associated with a non-empty set of ‘‘ideal’’ worlds.
If Q is true at a given world, then Q must be true at every associated ideal world. Because no
world (ideal or otherwise) can contain the contradiction Q and :Q, it is impossible for both
Q and :Q to be true at any world. See Horty (1993, 1994) for further discussion.
6
Extensions of default logic are defined formally through a more subtle fixed-point construction. See Horty (1993, 1994) or Brewka, et al. (1997, Chap. 4) for a more precise description of default logic and some standard examples.
The Logic of Role Theory
39
reasoning process of ego and alters, but instead adopt a simpler alternative
known as a level default theory (Brewka, et al., 1997, p. 56 ff).7 A level
default theory (LDT) is a set of logical formulas T partitioned into subsets
(T1, . . ., Tn) ranked by their ‘‘priority’’ or ‘‘reliability.’’ The formulas in T1
are considered more reliable than those in T2, which are more reliable than
those in T3, etc. Given this ranking, E ¼ E1 [ . . . [ En is an extension of the
level default theory T if and only if, for all m n, E1[ . . . [ Em is a maximal
consistent subset of T1 [ . . . [ Tm.8 More intuitively, to derive an extension
of the LDT, we begin with a maximal consistent subset of T1, extend this
subset by adding as many formulas from T2 as possible while still maintaining logical consistency, extend this subset by adding as many formulas from
T3 as consistently possible, and so on.9 Note that logical inconsistencies in
the set T may give rise to multiple extensions of the LDT.
Thus, adopting this framework to model the reasoning of ego and alters,
an LDT (like default logic) would allow us to consider the role conflicts
generated by contradictions in a social system. Moreover, we may distinguish the ‘‘must’’ implicit in structural constraints from the ‘‘should’’
implicit in normative constraints by placing structural constraints in the
first level of the LDT while placing normative constraints in some subsequent level. Multiple extensions generated by ego’s reasoning (the first
syllogism above) would reflect alternative choices that are ‘‘as consistent
as possible’’ given the moral dilemmas facing the individual. Multiple extensions generated by alters’ reasoning (the second syllogism above) would
reflect alternative ‘‘interpretations’’ or ‘‘accounts’’ of ego’s behavior.
Generic Theoretical Issues
Before moving in the next section to a full specification of the dynamic
process governing the self-concept, we might pause here to consider the
7
Both default logic and level default theories belong to the same family of consistencybased logics. See Brewka, et al. (1997, Chap. 4) for a detailed specification and comparison
of members of this family.
8
Brewka, et al. (1997) refer to the set E as a preferred subtheory of T. I use the term
extension to highlight the close analogy between preferred subtheories of LDTs and extensions of default theories.
9
A set of formulas F is logically consistent if and only if there is an assignment of truth
values to the (atomic) propositions in F such that all formulas in F are true. Thus, the set
F ¼ {x!y, x!:y} is logically consistent because both formulas in F are true if x is false. In
contrast, the set F ¼ {x, x!y, x!:y} is inconsistent (i.e., contradictory) because there is
no assignment of truth values to x and y such that all three propositions in F are true. To assess
logical consistency for a set of formulas, one mechanical procedure is to construct a truth
table, with each row corresponding to a different combination of truth values for the atomic
propositions (see, e.g., Lemmon, 1965).
40
J. D. Montgomery
generic issues that might be addressed within this theoretical framework.10
First, given the particular norms of the social system and the particular
constraints faced by ego, we might consider how ego’s self will evolve. That
is, given ego’s initial self S0, we might consider the sequence of selves (S0,
S1, S2, . . .) that ego will possess through time.11 Second, still holding fixed
norms and constraints, we might consider ego’s long-run self. In particular,
we might consider whether the social system generates one or more
‘‘absorbing’’ selves, or whether ego’s self will perpetually transition through
some subset of possible selves. Third, we might consider how changes in
the set of norms or constraints would affect either the short-run dynamics
of ego’s self (i.e., the sequence (S0, S1, S2, . . .)) or the long-run state of
ego’s self (i.e., the existence and number and location of absorbing selves).
Finally, we might consider the process through which norms themselves
change.
In the present paper, I pass over the first issue (short-run dynamics) in
order to focus on the second issue (long-run outcomes) and third issue
(how long-run outcomes depend on the set of norms). In particular, given
the extended example developed in Section III and IV, the analysis focuses
on the long-run absorbing selves generated by the social system. Then, Section V presents some more general results linking the logical (in)consistency of the set of norms to the (non)existence of absorbing selves. While
I hope that the theoretical framework will ultimately provide a solid foundation for addressing the evolution of norms, the fourth issue is beyond my
present scope. Throughout the present paper, I take social norms as
exogenously given.
III. THE THEORETICAL FRAMEWORK
This section develops the theoretical framework through an extended
example based loosely on the classic ethnography Tally’s Corner (Liebow,
1967). In the model, ego is a representative streetcorner man. Section IV
will consider some variations on this example incorporating different norms
and constraints.
Overview
In the present framework, the social system is characterized by a set of
social roles (R), a set of actions (X), and a set of social norms (N). Ego
faces a set of constraints (C). Given norms, constraints, and his current
10
Cf. Fararo’s (1989, p. 87) typology of theorems for dynamical system models.
This sequence was described as ego’s trajectory through ‘‘role space’’ in Montgomery
(2000).
11
The Logic of Role Theory
41
self-concept (S), ego uses a level default theory (T) to make a set of
choices (A 2 A). Alters observe these choices and, using a level default
theory (T0 ), make a collective attribution (S0 2 S) about ego’s self-concept. Ego gradually internalizes this attribution, revising S toward S0 before
making his next set of choices. The analysis will focus on ego’s long-run
self, examining whether ego’s self will eventually reach an absorbing state
(a self S that induces choices A that generate the attributions S0 ¼ S).
The Social System
Consider a simple social system where the set of roles is given by
R ¼ fm; f ; wg
ð5Þ
where m denotes ‘‘man,’’ f denotes ‘‘father,’’ and w denotes ‘‘worker.’’
Further let
R ¼ R [ f:r j r 2 Rg ¼ fm; f ; w; :m; :f ; :wg
ð6Þ
denote the set of all roles and their negations. In each period, for each role
r 2 R, ego is either certain that his self S contains this role (so that r 2 S),
certain that his self does not contain this role (so that :r 2 S), or is uncertain whether his self contains this role (so that neither r nor :r is contained
in S). Thus, ego’s self is a logically consistent subset of R. Formally, ego’s
self S is an element of
Sx ¼fS R j ðr 62 SÞ _ ð:r 62 SÞ for all r 2 Rg
¼ff:m; :f ; :wg; f:m; :f g; f:m; :f ; wg; . . . ; fm; f ; :wg;
fm; f g; fm; f ; wgg
ð7Þ
which contains the 3jRj ¼ 27 self-concepts possible in the present
example.12
Suppose that the set of actions is given by
X ¼ fp; g; tg
ð8Þ
where p denotes ‘‘provide for family,’’ g denotes ‘‘hold good job,’’ and t
denotes ‘‘spend time with streetcorner peer group.’’13 Further let
X ¼ X [ f:a j a 2 Xg ¼ fp; g; t; :p; :g; :tg
ð9Þ
denote the set of all actions and their negations. Each period, ego must
choose either a or :a for every a 2 X. Thus, ego’s choices A will be an
12
See the rows of Table 1 for a complete enumeration of these possible selves.
Attempting to rationalize my notation, I am using upper-case letters for sets, lower-case
italics for particular roles and lower-case regular type for particular actions.
13
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J. D. Montgomery
element of the Cartesian product
Ax ¼ f:p; pg f:g; gg f:t; tg
¼ ff:p; :g; :tg; f:p; :g; tg; f:p; g; :tg; f:p; g; tg;
fp; :g; :tg; fp; :g; tg; fp; g; :tg; fp; g; tgg
ð10Þ
which contains the 2jXj ¼ 8 choices possible in the present example.
Actions may be interrelated through structural constraints. Some of
these constraints may be common to all individuals while other constraints
may be specific to ego. A constraint might take the form of a material implication between actions (e.g., p ! g). More generally, we might permit constraints of any form, including simple requirements (e.g., p) or restrictions
(e.g., :p) or more complex formulas involving elements of X. Together,
these constraints comprise a set C. Given my conception of structural constraints, I require the set of formulas C to be logically consistent. The set of
choices A 2 Ax is feasible if the union A [ C is consistent. Conversely, the
set of choices A is infeasible if A [ C is contradictory. To begin, suppose
that ego faces the constraints.
C ¼ fp ! g; g ! :tg:
ð11Þ
In words, the first constraint ðp ! gÞ states that if ego provides for family, he must hold a good job. The second constraint ðg ! :tÞ states that if
ego holds a good job, then he cannot spend time with the street-corner peer
group.
A norm is an implication from a role to a formula involving roles and
actions. That is, a norm takes the form r!Q where r 2 R and Q is a formula
involving elements of R [ X. While I have assumed that the set C (of structural constraints) must be logically consistent, I do not assume that the set
N (of ‘‘normative constraints’’) must be logically consistent. To begin,
suppose that
N ¼ fm ! f ; m ! w; m ! t; f ! p; w ! g; f ! : tg:
ð12Þ
In words, these norms state that men should be fathers (m ! f), men
should be workers (m ! w), men should spend time with streetcorner
peers (m ! t), fathers should provide for their families (f ! p), workers
should hold good jobs (w ! g), and fathers should not spend time with
streetcorner peers (f !: t). Although N is not logically inconsistent, it does
imply :m given that m both requires t (directly through the norm m ! t)
and forbids t (indirectly through the norms m ! f and f !: t).14
14
Moreover, considering both norms and constraints, note that m forbids t indirectly
through the sets of formulas {m!f, f!p, p!g, g!:t} and {m!w, w!g, g!:t}.
The Logic of Role Theory
43
It is important to notice the distinction between my verbal renderings of
constraints (which specify what ego must do) and norms (which specify
what ego should do). This distinction is not yet evident in my formalization—both constraints and norms have been specified as material implications—but is embedded in the reasoning process (the level default theory)
specified below because ego gives higher priority to constraints than norms.
Ego’s Choices
In each period, ego derives his choices given his constraints (C), selfconcept (S), and social norms (N). Ideally, ego would simply derive his
choices from the union of these sets of formulas (C [ S [ N). But more
generally, ego’s choices will be either underdetermined or overdetermined:
either N contains too few norms to permit derivation of either a or :a for
some a 2 X, or else contradictions in N seem to obligate ego to choose both
a and :a for some a 2 X. Both underdetermination and overdetermination
lead to competing sets of choices – multiple extensions derived from the
union of formulas – from among which ego must decide.
As discussed in Section II, I assume that ego derives his choices using a
level default theory (LDT) in which subsets of formulas are ranked from
highest to lowest priority. In the present framework, I assume that the
set C is granted highest priority because ego cannot possibly violate these
(structural) constraints. I assume that ego grants the next highest
priority to formulas in his self-concept. The set of norms, which may be
inconsistent either internally or in union with C and S, is granted the next
highest priority. Note that, by ranking formulas contained in S above those
contained in N, I have assumed that ego’s beliefs about himself are more
reliable than his beliefs about norms – he will (within a period) retain his
self-concept while choosing among contradictory norms, rather than allowing contradictory norms to undermine his self-concept. Finally, to ensure
that ego’s choices are complete (i.e., that every extension of ego’s LDT
contains either a or :a for every a 2 X), the set X is included as the final
level in ego’s LDT. Ego’s LDT may be written generically as T ¼ (C, S, N, X).
In the present example, this becomes
T ¼ ðfp ! g; g ! :tg; S; fm ! f ; m ! w; m ! t; f ! p; w ! g; f ! :tg;
fp; g; t; :p; :g; :tgÞ
ð13Þ
given ego’s self S 2 Sx.
Having specified ego’s level default theory, we may now compute the
(possibly multiple) extensions of this LDT. Both underspecification and
overspecification of choices will generate multiple extensions. To illustrate,
I will derive ego’s choices given three alternative assumptions on his selfconcept: S ¼ 1, S ¼ { f }, and S ¼ {m, f }. These three cases will illustrate
44
J. D. Montgomery
how norms may underdetermine, (precisely) determine, or overdetermine
actions.
Given the empty self S ¼ 1, every extension of the level default theory
T ¼ (C, 1, N, X) must contain C [ 1 [ N because this union is logically
consistent. Each of the eight members of the Cartesian product Ax is a
maximal consistent subset of X. However, only four of these choices are
feasible, namely {p, g, :t}, {:p, g, :t}, {:p, :g, t}, and {:p, :g, :t}. Each
of these four sets is logically consistent with C [ 1 [ N. Thus, ego’s LDT
generates four extensions. Letting E denote the set of extensions of T,
E ¼ fC [ 1 [ N [ fp; g; :tg; C [ 1 [ N [ f:p; g; :tg;
C [ 1 [ N [ f:p; :g; :tg; C [ 1 [ N [ f:p; :g; tgg;
ð14Þ
The set of choices A implied by each extension E 2 E is given by E \ X .
Thus, ego’s choices A are selected from the choice set
A ¼ fE \ X jE 2 Eg
¼ ffp; g; :tg; f:p; g; :tg; f:p; :g; tg; f:p; :g; :tgg:
ð15Þ
Thus, in the case where ego’s self is empty, every feasible choice is contained
in ego’s choice set A.15 In the present paper, I do not attempt to place any
(preference) ordering over the elements of this set, but rather view each
A2A as ‘‘defensible’’ given the existing norms. Nevertheless, ego must make
a particular choice (i.e., choose some particular A 2 A) in each period. The
analysis below assumes merely that ego will choose each A 2 A with some
positive probability.
Given S ¼ { f }, any extension must again contain C [ S [ N because this
union remains logically consistent. But now, given the formulas { f, f ! p,
p!g, g!:t} contained in this union, there is a unique extension
E ¼ C [ f f g [ N [ fp; g; :tg
ð16Þ
and thus the choice set A contains the unique choice
A ¼ E \ X ¼ fp; g; :tg:
ð17Þ
In this case, ego’s choice is (precisely) determined.
Finally, consider the self S ¼ {m, f}. While C [ S remains consistent,
C [ S [ N is now inconsistent. Thus, each extension begins with C [ S and
is then extended by as many formulas from N as consistently possible.
15
It is interesting to note that the extensions hold different implications for ego’s self: the
extension C [ 1 [ N [ {p, g, :t} implies :m; the extension C [ 1 [ N [ {:p, g, :t} implies {:m,
:f}; the final two extensions imply {:m, :f, :w}. However, in the present framework, ego
makes no direct use of these implications; his self is altered only through feedback from alters’
attributions (described below).
45
The Logic of Role Theory
To derive these extensions, note that, because S contains both m and f,
every extension must contain the formula m!f. Moreover, if an extension
contained f!:t or f!p, then it could not contain m!t, but could contain
every other formula in N. Conversely, if an extension contained m!t then
it could contain neither f!p nor f!:t. While such an extension could not
contain both m!w and w!g, one of those formulas could remain. Thus,
we obtain three extensions,
E ¼ fC [ fm; f g [ fm ! f ; m ! w; f ! p; w ! g; f ! :tg [ fp; g; :tg;
C [ fm; f g [ fm ! f ; m ! w; m ! tg [ f:p; :g; tg;
C [ fm; f g [ fm ! f ; m ! t; w ! gg [ f:p; :g; tgg;
ð18Þ
which generate two distinct choices,
A ¼ ffp; g; :tg; f:p; :g; tgg:
ð19Þ
Thus, this third case illustrates how overspecification of norms – logical
inconsistencies in the union C [ S [ N – forces ego to abandon some norms
in order to follow others. In particular, in the present example, ego must
ignore either the norm m!t or else the norms f!p, f!:t and either
m!w or w!g.
Having examined closely these three cases, we may now undertake a
more general analysis of ego’s choices as a function of his self-concept.
To begin, recall that ego may possess 3jRj ¼ 27 different self-concepts,
and that ego has 2jXj ¼ 8 choices available. Thus, we might construct a
27 8 matrix characterizing ego’s choice set A Ax for every self-concept
S 2 Sx. Formally, letting A denote this self-by-choice-set matrix,
AðS; AÞ ¼ 1 if A 2 A ðSÞ
¼ 0 otherwise
ð21Þ
where A(S) denotes the set of choices generated by extensions of the
level default theory T ¼ (C, S, N, X). The non-zero columns of the A matrix
are given in Table 1.16 For each of the first 18 self-concepts (where m 62 S),
every extension must include C [ S [ N because this union is consistent.
Multiple extensions (when they occur) are due to underspecification of
the set of norms. For each of the last 9 self-concepts (where m 2 S), every
extension includes only some of the formulas in N because C [ S [ N is
inconsistent. One of these selves (S ¼ {m, :f, :w}) generates a unique
extension (E ¼ C [ S [ {m!t, f!p, w!g, f!:t} [ {:p, :g, t}) and thus a
unique choice (A ¼ {:p, :g, t}), while the other 8 selves generate multiple
extensions and choices due to overspecification of the set of norms.
16
Given the constraints C, only 4 of the 8 choices in Ax are feasible. The (all-zero) columns
corresponding to non-feasible actions have been omitted from Table 1.
46
J. D. Montgomery
TABLE 1 The Matrix A Characterizing Choices A(S) for all Selves S
f:m; :f ; :wg
f:m; :f g
f:m; :f ; wg
f:m; :wg
f:mg
f:m; wg
f:m; f ; :wg
f:m; f g
f:m; f ; wg
f:f ; :wg
f:f g
f:f ; wg
f:wg
1
fwg
f f ; :wg
ffg
f f ; wg
fm; :f ; :wg
fm; :f g
fm; :f ; wg
fm; :wg
fmg
fm; wg
fm; f ; :wg
fm; f g
fm; f ; wg
fp; g; :tg
f:p; g; :tg
f:p; :g; tg
f:p; :g; :tg
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
1
1
1
1
1
1
0
0
0
0
1
1
0
0
0
0
0
0
1
1
0
1
1
0
0
0
0
1
1
0
1
1
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
0
1
1
0
0
0
0
1
1
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
Note: In the present example, the A matrix is 27 8. This table reports only the (non-zero)
columns corresponding to the 4 feasible actions.
Alters’ Attributions
Given the choices made by ego in the current period (some A 2 A(S)),
alters make a single, collective attribution about ego’s self. (Thus, ‘‘alters’’
might be interpreted as a representative or ‘‘generalized’’ alter.) The
present framework assumes that, like ego, alters derive inferences from a
level default theory. Alters place the highest priority on ego’s choices and
give second priority to norms.17 However, while ego draws only logically
17
This ranking of actions above norms constitutes a substantive assumption that alters are
firmly convinced they correctly observe ego’s choices. The opposite ranking would imply that
alters ignore ego’s observed behavior when it conflicts with norms (perhaps rationalizing
‘‘could ego really have done that?’’).
47
The Logic of Role Theory
valid inferences from the material implications that comprise N, alters
make stronger (logically more problematic) inferences. To capture alters’
tendency to overattribute roles to ego’s self, I assume that, for each norm
r!Q contained in N, alters reason using the corresponding identity r$Q.
Together, these identities comprise a set of formulas
N0 ¼ fr $ Qjr ! Q 2 Ng
¼ fm $ f ; m $ w; m $ t; f $ p; w $ g; f $ :tg:
Alters’ LDT may written generically as T0 ¼ (A, N0 ).
this becomes
18
ð22Þ
In the present model,
T0 ¼ ðA; fm $ f ; m $ w; m $ t; f $ p; w $ g; f $ :tgÞ
ð23Þ
given ego’s current choice A 2 Ax :
Given this specification of alters0 level default theory, we may compute
the (possibly multiple) extensions. The attribution S0 corresponding to
extension E0 is given by the intersection
S0 ¼ ConðE0 Þ \ R
ð24Þ
where ConðE0 Þ denotes the consequence set of (all formulas that can be
derived from) E0 . The set of possible attributions made by alters is given by
S ¼ fConðE0 Þ \ RjE0 2 E 0 g
0
0
ð25Þ
where E denotes the set of all extensions of T . Note that S may be empty–
alters may draw no conclusions about ego’s self. Indeed, in the simplest
case where N ¼ 1 (and thus N0 ¼ 1), alters level default theory
T0 ¼ ðA; 1Þ generates the unique extension E0 ¼ A and thus S0 ¼ 1. On
the other hand, inconsistencies in the union A [ N0 may give rise to multiple extensions and thus multiple attributions – alters might give different
‘‘interpretations’’ or ‘‘accounts’’ of ego’s choices. Just as I did not attempt to
order ego’s choices (when A contained more than one element), I do not
here attempt to order the set of attributions (when S contains more than
one element). The analysis below merely assumes that, in each period,
each attribution S0 2 S is made with positive probability.
To illustrate how alters make attributions, consider the extensions of
alters’ LDT in the present example. Because N0 is inconsistent, T0 will
always (regardless of ego’s choices A) generate multiple extensions. But
while no extension can contain the entire subset {m$f, m$t, f$:t}, any
18
Given that alters may be unaware of (or simply ignore) the constraints faced by ego, I do
not assume that alters’ LDT contains the set C. However, given that any choices made by
ego must be feasible, and that alters observe every action, alters’ inferences would not be
changed if C was incorporated into T0.
48
J. D. Montgomery
two of these identities could be contained in an extension. Thus, for any A,
we will obtain at least three extensions, each resulting in a different attribution. If we further suppose that ego makes the choices A ¼ fp; g; :tg, we
obtain five extensions,
E 0 ¼ fA [ fm $ f ; m $ w; m $ tg; A [ fm $ f ; m $ t; w $ gg;
A [ fm $ w; m $ t; f $ p; f $ :tg; A [ fm $ t; f $ p; w $ g; f $ :tg;
A [ fm $ f ; m $ w; f $ p; w $ g; f $ :tgg;
ð26Þ
which generate five distinct attributions,
S ¼ ff:m; :f ; :wg; f:m; :f ; wg; f:m; f ; :wg; f:m; f ; wg; fm; f ; wgg:
ð27Þ
Moving to a more general analysis of alters’ attributions as a function of
ego’s choices, recall that there are 2jXj possible choices and 3jRj possible
attributions. Thus, for the present example we may form an 8 27 matrix
to characterize the set of possible attributions S Sx made by alters for
each set of choices A 2 Ax . Letting B denote this choice-by-attribution
matrix,
BðA; S0 Þ ¼ 1 if S0 2 S ðAÞ
¼ 0 otherwise
ð28Þ
where S ðAÞ denotes the set of attributions generated by alters’ level
default theory T0 ¼ ðA; N0 Þ. Table 2 gives the non-zero columns and rows
of the B matrix.19 Note that, as already discussed, alters always make multiple attributions regardless of ego’s choices in the present example.
Dynamics and Absorbing Selves
Ego gradually internalizes the attributions made by alters. This feedback
loop governs the evolution of ego’s self through time. Given an initial self
St (where the subscript t indexes time), ego chooses some At 2 A ðSt Þ:
Given ego’s choices At, alters make some attribution S0t 2 S ðAt Þ. Given this
attribution, ego forms a new self-concept Stþ1 that is ‘‘between’’ his old self
St and St0 . This new self Stþ1 provides a basis for ego’s next choices, and the
iteration continues. This dynamical system constitutes a Markov chain,
with each possible self ðS 2 Sx Þ as a state of the system. In the present
19
In the present example, every attribution must be a ‘‘full’’ self containing either r or :r for
every r 2 R: Thus, Figure 2 gives the 4 non-zero rows (corresponding to the feasible actions)
and 8 non-zero columns (corresponding to the full selves) of the B matrix.
49
1
1
1
1
1
1
0
0
f:m; :f ; wg
1
1
0
1
f:m; f ; :wg
1
1
0
0
f:m; f ; wg
0
0
1
0
fm; :f ; :wg
0
1
1
0
fm; :f ; wg
0
0
1
1
fm; f ; :wg
1
1
1
1
fm; f ; wg
Note: In the present example, the B matrix is 8 27. This table reports only the rows corresponding to the 4 feasible actions and the non-zero
columns corresponding to the 8 full selves.
fp; g; :tg
f:p; g; :tg
f:p; :g; tg
f:p; :g; :tg
f:m; :f ; :wg
TABLE 2 The Matrix B Characterizing Attributions S(A) for all Feasible Actions A
50
J. D. Montgomery
example, transitions from ego’s initial self (St) to new self ðStþ1 Þ would be
characterized by a 27 27 transition matrix.
A complete analysis of the short-run dynamics of ego’s self would require
further specification of the present model.20 But long-run outcomes are
already implicit in our preceding computations. In particular, we can determine whether the social system generates any absorbing states—‘‘absorbing selves’’—that ego will never exit after entry. Loosely, an absorbing
self may be understood as a fixed point of the equation.
S ¼ S0 ðAðSÞÞ:
ð29Þ
That is, we are looking for some self S that generates choices A(S) that
generate an attribution S0 ðAðSÞÞ that is equal to S. However, recognizing
that A is chosen from the set A ðSÞ which may contain multiple members,
and that S0 is chosen from the set S ðAÞ which may contain multiple members, the following definition is less ambiguous. An absorbing self is a self
S 2 Sx such that S is the unique member of S ðAÞ for all A 2 A ðSÞ: Given
this definition, an absorbing self S may generate multiple choices (i.e.,
A ðSÞ contains more than one member). But for any choice A 2 A ðSÞ,
alters would always make the unique attribution S0 ðAÞ equal to S.
Using the A and B matrices defined above, we may identify absorbing
selves by computing the Boolean product A B. This self-by-attribution
matrix is 27 27 in the present example. A self S is absorbing when S is
the unique attribution given the choice(s) generated by self S. Thus, S is
an absorbing self when row S of the A B matrix has a positive entry only
on the main diagonal.21 To illustrate, Table 3 gives the rows and columns of
the A B matrix corresponding to the 8 ‘‘full’’ selves.22 Inspection of this
matrix reveals that, while every diagonal element is positive (so that every
full self generates itself as one possible attribution), every row of the
matrix contains multiple positive elements (so that every full self also generates other attributions). Thus, there are no absorbing selves in the
present example. That is, ego’s self can change perpetually through time,
20
In particular, to determine the transition matrix, we would need to specify the probability
that ego makes each choice A 2 A ðSÞ for each self S, the probability that alters make each
attribution S0 2 S ðAÞ for each set of choices A, and the probability that ego would transition
to each new self for each pair ðS; S0 Þ characterizing his initial self and attributed self.
21
Note that this matrix test for the existence of an absorbing self is equivalent to the settheoretic definition given above. If the element on the main diagonal of row S is positive (i.e.,
(A B)(S,S) ¼ 1), then there must be some choice A such that A(S,A) ¼ 1 and B(A,S) ¼ 1
(i.e., A 2 A ðSÞ and S 2 S ðAÞÞ: If all non-diagonal elements of row S are zero (i.e.,
(A B)ðS; S0 Þ ¼ 0 for all S0 6¼ SÞ; then there cannot be a choice A and self S0 6¼ S such that
A(S,A) ¼ 1 and BðA; S0 Þ ¼ 1 (i.e., there can be no A 2 A ðSÞ such that S0 2 S ðAÞÞ:
22
In the present example, because every attribution will be a full self, only full selves are
potential absorbing selves.
51
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
1
f:m; :f ; wg
1
1
1
1
0
1
1
1
f:m; f ; :wg
1
1
1
1
0
1
1
1
f:m; f ; wg
1
0
0
0
1
1
1
1
fm; :f ; :wg
1
1
0
0
1
1
1
1
fm; :f ; wg
1
0
0
0
1
1
1
1
fm; f ; :wg
1
1
1
1
1
1
1
1
fm; f ; wg
Note: In the present example, the A B matrix is 27 27. This table reports only the submatrix of non-zero rows and columns corresponding to
the 8 full selves. Elements along the main diagonal are denoted in boldface.
f:m; :f ; :wg
f:m; :f ; wg
f:m; f ; :wg
f:m; f ; wg
fm; :f ; :wg
fm; :f ; wg
fm; f ; :wg
fm; f ; wg
f:m; :f ; :wg
TABLE 3 The Matrix A B Characterizing Attributions S(A) given Choices A 2 A(S) Generated by Self S
52
J. D. Montgomery
never reaching an absorbing state. Indeed, further inspection reveals that,
because every full self generates the first attribution f:m; :f ; :wg which in
turn generates every full self, ego’s self can perpetually transition through
every state.
IV. ALTERNATIVE SPECIFICATIONS OF NORMS
AND CONSTRAINTS
Having offered an extended example to illustrate the general theoretical
framework, we have seen one case where the social system generates no
absorbing selves. Modifying the sets of norms and constraints, we obtain
other cases where the social system generates multiple absorbing selves
or a unique absorbing self.23
A Disjunctive Specification of Norms
In the preceding example, every norm r ! q in N involved a simple consequent q that was an element of either R or X. More generally, a social
system might possess more complicated norms so that q is a formula
involving multiple elements of R and X . For instance, in place of the three
separate formulas fm ! f ; m ! w; m ! tg we might substitute the logically equivalent compound formula fm ! ðf ^ w ^ tÞg:24 Alternatively, we
might consider a disjunctive specification of this norm, m ! ð f _ w _ tÞ;
which states that a man should be either a father or a worker or spend time
with his peer group. Of course, we might also consider various hybrid constructions where a norm is partly conjunctive and partly disjunctive. One
reading of Tally’s Corner might suggest two competing conceptions of
manhood: a ‘‘mainstream’’ conception in which men are fathers and workers, and a ‘‘streetcorner’’ conception in which men receive validation
merely from spending time with the peer group. Formally, we might respecify the set of norms as
N ¼ fm ! ðð f ^ wÞ _ tÞ; f ! p; f ! :t; w ! gg
23
ð30Þ
While these modifications raise interesting substantive issues, my narrow goal in the
present section is to further illustrate the types of long-run outcomes that can be generated
by the model. See Montgomery (2000) for further discussion of Liebow’s (1967) original
account and the larger culture-of-poverty debate.
24
Although the formula ðm ! f Þ ^ ðm ! wÞ ^ ðm ! tÞ is logically equivalent to
m ! ðf ^ w ^ tÞ; an LDT which replaced the three separate formulas with the compound formula would not, in general, yield the same set of extensions. Note that an extension might
retain some of the separate formulas even when it could not retain the combination. Thus,
extensions may not be very robust to subtle restatements of the formulas contained in the
theorem set of the LDT.
53
The Logic of Role Theory
In this way, the man role implies that ego should be either a father and a
worker or else spend time on the streetcorner.
Retaining our initial set of constraints from Equation (11), we may use
ego’s level default theory T ¼ (C, S, N, X ) to derive ego’s choice set A ðSÞ
for every S 2 Sx : Table 4 gives the non-zero columns of the A matrix. In all
but two cases, the union C [ S [ N is consistent and multiple elements in
A ðSÞ arise when norms underdetermine behavior. Only for the selves
fm; :f ; wg and fm; f ; :wg does the union C [ S [ N become inconsistent,
so that multiple elements of A ðSÞ arise when norms overdetermine behavior. Comparing Table 4 to Table 1 (which gives choices under the previous
specification of norms), note that selves containing m more often generate
unique choices because behavior in (6 of 9 of) those cases is now precisely
determined by norms.
TABLE 4 The Matrix A given Norm m ! ððf ^ wÞ _ tÞ
f:m; :f ; :wg
f:m; :f g
f:m; :f ; wg
f:m; :wg
f:mg
f:m; wg
f:m; f ; :wg
f:m; f g
f:m; f ; wg
f:f ; :wg
f:f g
f:f ; wg
f:wg
1
fwg
ff ; :wg
ff g
ff ; wg
fm; :f ; :wg
fm; :f g
fm; :f ; wg
fm; :wg
fmg
fm; wg
fm; f ; :wg
fm; f g
fm; f ; wg
fp; g; :tg
f:p; g; :tg
f:p; :g; tg
f:p; :g; :tg
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
1
0
1
1
1
1
1
1
1
1
1
1
1
0
0
0
1
1
1
1
1
1
0
0
0
0
0
1
0
0
0
0
0
0
1
1
0
1
1
0
0
0
0
1
1
0
1
1
0
0
0
0
1
1
1
1
1
0
1
0
0
1
1
0
1
1
0
0
0
0
1
1
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
Note: In the present example, the A matrix is 27 8. This table reports only the non-zero
columns corresponding to the 4 feasible actions.
54
J. D. Montgomery
We may further use alters’ level default theory
T0 ¼ ðA; N0 Þ ¼ ðA; fm $ ððf ^ wÞ _ tÞ; f $ p; f $ :t; w $ ggÞ
ð31Þ
to derive alters’ attributions S ðAÞ for every A 2 Ax : Table 5 gives the relevant rows and columns of the B matrix. Comparing Table 5 to Table 2
(which characterized attributions under the previous specifications of
norms), note that the present disjunctive specification of norms has greatly
reduced the number of attributions possible for each set of choices. Indeed,
given the choices A ¼ fp; g; :tg or A ¼ f:p; :g; tg, the union A [ N0 is consistent and generates a unique attribution.
Finally, we might once again check for absorbing selves by computing the
Boolean product A B. Table 5 reveals the existence of two absorbing
selves: fm; :f ; :wg and fm; f ; wg. These absorbing selves are also apparent from the graph in Figure 2. In this figure, arrows pointing rightward
(from selves to actions) represent the choices A ðSÞ characterized by the
A matrix; arrows pointing leftward (from actions to selves) represent the
attributions S ðAÞ characterized by the B matrix. (To simplify, the graph
includes only those nodes with positive indegree and outdegree.) As the figure reveals, the self fm; f ; wg is absorbing because it generates the unique
set of choices fp; g; :tg which, in turn, generates the unique attribution
fm; f ; wg. Similarly, the self fm; :f ; :wg is absorbing because it generates
the unique set of choices f:p; :g; tg which, in turn, generates the unique
attribution fm; :f ; :wg. In contrast, the other selves given in Figure 2 do
not always generate themselves as unique attributions. For example, the
self f:m; :f ; wg generates two possible choices: fp; g; :tg and
f:p; g; :tg. While the latter choice would generate f:m; :f ; wg as an
attribution, the former choice would lead to the attribution fm; f ; wg
which is an absorbing self. Thus, given the disjunctive respecification of
the norms of manhood, the present example generates multiple absorbing
selves. Given two possible long-run outcomes, the particular absorbing self
that ego eventually reaches will depend (perhaps stochastically) on ego’s
initial self (i.e., the initial state of the Markov chain).
570
575
580
585
590
595
Disjunctive Norms with an Employment Constraint
A closer reading of Liebow’s (1967) account might further suggest that 600
streetcorner men were unable to find good jobs due to limited labormarket opportunities. Formally, we might extend the constraint set to include
the (structural) constraint :g. Thus, the revised set of constraints is given by
C ¼ fp ! g; g ! :t; :gg:
ð32Þ
Retaining the disjunctive specification of norms from Equation (30), ego’s 605
choice set A ðSÞ for every self S 2 Sx is given in Table 7. Alters’ attributions
55
0
0
0
1
0
1
0
0
f:m; :f ; wg
0
0
0
1
f:m; f ; :wg
0
0
0
0
f:m; f ; wg
0
0
1
0
fm; :f ; :wg
0
0
0
0
fm; :f ; wg
0
0
0
0
fm; f ; :wg
1
1
0
0
fm; f ; wg
Note: In the present example, the B matrix is 8 27. This table reports only the rows corresponding to the 4 feasible actions and the columns
corresponding to the 8 full selves.
fp; g; :tg
f:p; g; :tg
f:p; :g; tg
f:p; :g; :tg
f:m; :f ; :wg
TABLE 5 The Matrix B given Norm m ! ðð f ^ wÞ _ tÞ
56
J. D. Montgomery
FIGURE 2 Graph of A and B given Norm m ! ððf ^ wÞ _ tÞ Note: Arrows pointing
rightward represent choices given selves (positive elements of the A matrix);
arrows pointing leftward represent attributions given choices (positive elements
of the B matrix). To simplify, the graph includes only those nodes with positive
indegree and outdegree.
S ðAÞ for every feasible action A are given by the last 2 rows of Table 5. Table
8 reveals that there is now a unique absorbing self S ¼ fm; :f ; :wg. If, following Liebow (1967), we assume that ego’s initial self is fm; f ; wg, Figure 3
suggests the (possibly stochastic) path by which ego would reach this absorbing state.
V. SOME GENERAL RESULTS ON NORMS AND
ABSORBING SELVES
My various specifications of the Tally’s Corner example have illustrated
cases where there is no absorbing self, where there are multiple absorbing
57
1
0
0
0
0
0
0
0
1
1
0
0
0
1
0
0
f:m; :f ; wg
1
0
0
0
0
0
0
0
f:m; f ; :wg
0
0
0
0
0
0
0
0
f:m; f ; wg
1
0
0
0
1
1
1
0
fm; :f ; :wg
0
0
0
0
0
0
0
0
fm; :f ; wg
0
0
0
0
0
0
0
0
fm; f ; :wg
1
1
1
1
0
1
1
1
fm; f ; wg
Note: In the present example, the A B matrix is 27 27. This table reports only the submatrix corresponding to the 8 full selves. Elements along
the main diagonal are denoted in boldface.
f:m; :f ; :wg
f:m; :f ; wg
f:m; f ; :wg
f:m; f ; wg
fm; :f ; :wg
fm; :f ; wg
fm; f ; :wg
fm; f ; wg
f:m; :f ; :wg
TABLE 6 The Matrix A B given Norm m ! ððf ^ wÞ _ tÞ
58
J. D. Montgomery
TABLE 7 The Matrix A Given the Norm m ! ððf ^ wÞ _ tÞ and Constraint :g
f:m; :f ; :wg
f:m; :f g
f:m; :f ; wg
f:m; :wg
f:mg
f:m; wg
f:m; f ; :wg
f:m; f g
f:m; f ; wg
f:f ; :wg
f:f g
f:f ; wg
f:wg
1
fwg
ff ; :wg
ff g
ff ; wg
fm; :f ; :wg
fm; :f g
fm; :f ; wg
fm; :wg
fmg
fm; wg
fm; f ; :wg
fm; f g
fm; f ; wg
f:p; :g; tg
f:p; :g; :tg
1
1
1
1
1
1
0
0
0
1
1
1
1
1
1
0
0
0
1
1
1
1
1
1
1
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
1
1
1
1
1
Note: In the present example, the A matrix is 27 8. This table reports only the non-zero
columns corresponding to the 2 feasible actions.
selves, and where there is a unique absorbing self. In this section, I consider
more generally how properties of the set of norms influence the number of
absorbing selves. While I do not restrict attention to any particular example
(i.e., any particular set of roles, actions, or norms), the present analysis
does assume that norms take the simple form r ! q where r 2 R and
q 2 R [ X . Given this restriction, it is possible to develop an intuitive
graph-theoretic representation of the process by which alters make attributions, and the analysis uncovers an interesting application of balance
theory in the study of normative systems.
We begin by representing the set N0 (i.e., the set of equivalences used in
alters’ level default theory) as a signed graph. The nodes of this graph are
given by the elements of R and X that appear in any of the formulas in N0 .
59
1
1
1
1
0
0
1
1
0
0
0
0
0
0
0
0
{:m, :f, w}
1
1
1
1
0
0
1
1
{:m, f, :w}
0
0
0
0
0
0
0
0
{:m, f, w}
1
1
0
0
1
1
1
0
{m, :f, :w}
0
0
0
0
0
0
0
0
{m, :f, w}
0
0
0
0
0
0
0
0
{m, f, :w}
0
0
0
0
0
0
0
0
{m, f, w}
Note: In the present example, the A B matrix is 27 27. This table reports only the submatrix corresponding to the 8 full selves. Elements along
the main diagonal are denoted in boldface.
{:m, :f, :w}
{:m, :f, w}
{:m, f, :w}
{:m, f, w}
{m, :f, :w}
{m, :f, w}
{m, f, :w}
{m, f, w}
{:m, :f, :w}
TABLE 8 The Matrix A B Given Norm m ! (( f ^ w)_ t) and Constraint :g
60
J. D. Montgomery
FIGURE 3 Graph of A and B given Norm m ! ððf ^ wÞ _ tÞ and Constraint :g.
Note: Arrows pointing rightward represent choices given selves (positive elements
of the A matrix); arrows pointing leftward represent attributions given choices
(positive elements of the B matrix). To simplify, the graph includes only
fm; f ; wg and those nodes with positive indegree and outdegree.
Formally, the set of nodes is given by
fr 2 Rjr $ q 2 N0 g [ fq 2 R [ Xjðr $ q 2 N0 Þ _ ðr $ :q 2 N0 Þg:
ð33Þ
The edges of the graph correspond to the formulas in N0 and may be positive or negative. In particular, the graph contains a positive edge between r
and q whenever N0 contains r $ q, and a negative edge between r and q
whenever N0 contains r $ :q. Formally,
signðrqÞ ¼ 1 if r $ q 2 N0
¼ 1 if r $ :q 2 N0
ð34Þ
where rq denotes the edge connecting nodes r and q. To illustrate, the set
of equivalences from the Section III example (see Equation 22) is represented by the graph given in Figure 4. (Adopting the usual convention
for signed graphs, positive edges are denoted by solid lines while negative
edges are denoted by dotted lines.)
While the graph in Figure 4 has a single connected component (i.e., all
nodes are connected either directly or indirectly), other normative systems
may have multiple components. To illustrate, consider another (abstract)
The Logic of Role Theory
61
FIGURE 4 Graph of N0 ¼ fm $ f ; m $ w; m $ t; f $ p; w $ g; f $ :tg. Note:
Solid lines denote positive edges; dotted lines denote negative edges.
example where R ¼ fa; b; c; d; eg; X ¼ fy; zg; and N ¼ fa ! y; b ! z; b ! :y;
c ! d; c ! e; d ! :eg so that
N0 ¼ fa $ y; b $ z; b $ :y; c $ d; c $ e; d $ :eg
ð35Þ
which is represented by the graph in Figure 5. Note that this graph has two
connected components. Using this example, we may also introduce a distinction between role-action and role-only components. Role-action components of the graph contain both nodes that are elements of R and nodes
that are elements of X, while role-only components contain only nodes that
are elements of R. (There cannot be an action-only component because
every norm r ! q must have an antecedent r 2 R.) Thus, the component
of the graph depicted at the top of Figure 5 is a role-action component,
while the component depicted at the bottom of the figure is a role-only
component. Given the partition of the graph into connected components,
we may say that a particular role r 2 R is associated with a role-action
or role-only component whenever it is a node in that type of component.
Thus, given the set N0 represented by the graph in Figure 5, role a is associated with a role-action component, while role c is associated with a roleonly component.
The graph of N0 permits a (hopefully intuitive) graph-theoretic derivation of alters’ attributions. To derive an attribution using alters’ level
default theory (using the procedure described in previous sections), we
would begin with the particular choices made by ego (the first level of
alters’ LDT) and then add elements of N0 (from the second level of alters’
LDT) to generate an extension E0 that implies an attribution S0 . Analogously, using the graph of N0 , we first sign each of the action nodes as
either positive or negative, and then ‘‘activate’’ edges in order to sign the
62
J. D. Montgomery
FIGURE 5 Graph of N0 ¼ fa $ y; b $ z; b $ :y; c $ d; c $ e; d $ :eg. Note:
Solid lines denote positive edges; dotted lines denote negative edges.
role nodes. To be more precise, given the set of choices A made by ego, we
first assign
signðaÞ ¼ 1 if a 2 A
¼ 1 if :a 2 A
ð36Þ
for every action node a. Then, given the signs of the edges and action
nodes, it becomes possible to sign the role nodes in role-action components
by sequentially activating some of the edges. For instance, suppose the
graph contains the (signed) edge ra connecting the (signed) action node
a to the (not yet signed) role node r. Activating the edge ra, the sign of
node r is determined by the condition sign(r) ¼ sign(a) sign(ra). If the
graph further contains the (signed) edge rq connecting the (now signed)
role node r to the (not yet signed) role node q, we might next activate edge
rq to determine sign(q) ¼ sign(r) sign(rq). By sequentially activating
edges in this manner, it is possible to sign each role node within every
role-action component. Of course, depending on the graph of N0 , there
may be many possible sequences of edge activation, and it may be impossible to activate every edge without violating the sign condition. Moreover,
just as alters’ level default theory may generate multiple extensions, there
may be more than one possible set of activated edges. Formally, each set of
activated edges must satisfy the following conditions: edges within
The Logic of Role Theory
63
role-only components are never activated; for each role node in a roleaction component, there must be at least one path of activated edges to
some action node; and
signðrÞ signðqÞ ¼ signðrqÞ for all activated edges rq:
ð37Þ
Given a set of activated edges, the attribution S0 is determined by
r 2 S0 if signðrÞ ¼ 1
:r 2 S0 if signðrÞ ¼ 1
ð38Þ
for all nodes r in role-action components, and the attribution S0 will contain
neither r nor :r if role r is associated with a role-only component.25
To illustrate, we may use this graph-theoretic procedure to re-derive the
attributions given in Equation (27), which were generated from alters’ level
default theory T0 ¼ (A, N0 ) given ego’s choices A ¼ fp; g; :tg and the set of
equivalences N0 from Equation (22). Using the graph of N0 in Figure 4, we
begin by signing the action nodes so that sign(p) ¼ sign(g) ¼ 1 and
sign(t) ¼ 1. Then, starting at the (negative) node t, we may activate
(positive) edge mt, which implies that that node m is negative. Then, from
(negative) node m, we may activate (positive) edge mf, which implies that
node f is negative. Returning to (negative) node m, we may now activate
(positive) edge mw, which implies that node w is negative. In this
way, all of the role nodes have been negatively signed: sign(m)
¼ sign(f) ¼ sign(w) ¼ 1. Further inspection of Figure 4 reveals that
we cannot activate any of the remaining edges without violating the sign
condition. Through this graph-theoretic procedure, we have thus identified
one set of activated edges fmf ; mw; mtg which generates the attribution S0 ¼ f:m; :f ; :wg. Similar analysis would produce four other sets
of activated edges ðfmf ; mt; wgg; fmw; mt; f p; f tg; fmt; f p; wg; f tg;
fmf ; mw; f p; wg; f tgÞ generating the other four attributions given in
Equation (27).26
25
Because alters cannot assign a truth value to any of the roles associated with role-only
components, these roles never appear in any attribution, and role-only components can simply
be ignored in the graph-theoretic derivation of attributions. Because extensions of alters’ LDT
T0 ¼ (A, N0 ) are required to be maximal consistent subsets of T0, they do contain (consistent)
subsets of equivalences in N0 corresponding to edges in role-only components. However, these
equivalences are irrelevant for derivation of attributions, and could be removed from extensions of alters’ LDT without affecting alters’ attributions.
26
Note the correspondence between the sets of activated edges and the subsets of N0 found
in the extensions of alters’ LDT given in Equation (26). However, because I have not required
sets of activated edges to be maximal, a set of activated edges might contain fewer elements
than the subset of N0 found in the corresponding extension. For instance, the set of activated
edges fmw; mt; f p; f tg might be replaced by fmw; mt; f pg or fmw; mt; f tg given that all
three sets would generate the attribution f:m; f ; :wg.
64
J. D. Montgomery
This graph-theoretic analysis of the attribution process suggests several
propositions.
Proposition 1. The empty self (s ¼ 1) is an absorbing self if and only if
there are no role-action connected components.
Proof of Proposition 1. If there are no role-action components, none of
the edges of N0 will be activated, and hence none of the role nodes can be
signed. Because alters will always attribute the empty self to ego, the
empty self is the unique absorbing self. If there are role-action components,
every role in these components will be signed positive or negative. Because
alters will never attribute an empty self to ego, the empty self cannot be
absorbing.
QED
Social network analysts have often used balance theory to predict the
stability of social networks involving positive and negative ties. Formally,
a signed graph is balanced if it has no negative cycles or (equivalently)
if the nodes can be partitioned into two sets such that positive edges connect only nodes within the same set, and negative edges connect only
nodes in different sets (Chartrand, 1977, Chap. 8). A signed graph is
imbalanced when it does not satisfy these conditions. Balance theory
has generally been applied in settings where the nodes of the graph represent individuals and the edges represent social ties. However, given that
balance of the graph of N0 is equivalent to logical consistency of N0 , the following proposition suggests that balance theory is also useful in the study
of normative systems, where the nodes represent roles or actions and the
edges represent norms.
Proposition 2. If any role-action connected component is imbalanced,
there is no absorbing self.
Proof of Proposition 2. A signed graph is balanced if and only if, for
every pair of nodes, all paths joining the pair have the same sign (see proof
in Chartrand, 1977, p. 175). Thus, if some component of the graph of N0 is
imbalanced, this component must include some role node r and some
action node a connected by both a positive path and a negative path.
Because either path could be activated, sign(r) could be either positive
or negative regardless of sign(a). That is, no matter what choice ego
makes (a or :a), alters could always attribute either r or :r to ego’s self.
Because alters can always make multiple attributions, there is no absorbing
self.
QED
To illustrate Proposition 2, consider again the graph in Figure 4. Note
the negative cycle (mt, mf, ft), corresponding to the inconsistent set of
The Logic of Role Theory
65
equivalences fm $ t; m $ f ; f $ :tg. If ego chooses t, it is possible for
alters to conclude that ego is an m (by activating mt) or to conclude that
ego is not an m (by activating ft and mf ). But if ego chooses :t, it is still
possible for alters to conclude that ego is an m (by activating ft and mf ) or
to conclude that ego is not an m (by activating mt). Thus, regardless of
ego’s choice, alters may always form multiple attributions. Consequently,
imbalance of the graph implies that there is no absorbing self.27
On the other hand, Proposition 2 does not guarantee that an absorbing
self exists if every role-action component is balanced. That is, balance of
role-action components is a necessary but not sufficient condition for the
existence of an absorbing self. To see this, suppose that we eliminated
the norm f ! :t from the Section III example, so that the set of norms
becomes
N ¼ fm ! f ; m ! w; m ! t; f ! p; w ! gg;
ð40Þ
the corresponding set of equivalences becomes
N0 ¼ fm $ f ; m $ w; m $ t; f $ p; w $ gg;
ð41Þ
and thus the negative tie ft is eliminated from the graph in Figure 4.
Although the graph is now balanced (containing no negative cycles), one
can show that there is still no absorbing self when ego faces the set of constraints C ¼ fp ! g; g ! :tg. Three of the feasible actions (fp; g; :tg,
f:p; g; :tg, and f:p; :g; tg) generate multiple attributions, while the
fourth feasible action (f:p; :g; :tg) generates a unique attribution that
is not an absorbing self. The general point is that the existence of an
absorbing self depends upon both norms and constraints.
However, reflection on this example suggests a further restriction on the
set of absorbing selves. In our graph-theoretic analysis of the attribution
process, we started by signing action nodes and then activating a subset
of edges. Now suppose that we reverse this procedure. That is, given a
27
One reviewer offered a potential counterexample to Proposition 2, arguing that an
absorbing self would exist given the (unbalanced) graph of N0 with three roles nodes
fr; s; tg, one action node fag, one positive edge frag, and three negative edges frs; st; rtg.
(Note this graph is imbalanced because the three negative edges form a negative cycle.)
But, consistent with my result, there is no absorbing self in this case. Following the proof,
imbalance of this graph implies that there exists an action node (a) and some role node (s)
connected by both positive and negative paths. Hence, regardless of ego’s choice (a or :a),
alters could attribute either s or :s to ego. Further analysis reveals three possible sets of activated edges: fra; rs; rtg; fra; rs; stg; and fra; rt; stg. Assuming sign(a) ¼ 1, these three sets of
edges generate the attributions fr; :s; :tg; fr; :s; tg; and fr; s; :tg. Although this example fails
to invalidate Proposition 2, perhaps it does usefully highlight that some element of ego’s self
might be stable over time (in this example, the role r) even if there is no absorbing self S
(which requires stability of every role in S).
66
J. D. Montgomery
balanced (logically consistent) component, suppose that we activate every
edge. In this way, we have determined whether each pair of nodes will have
the same sign or the opposite signs, but have not yet determined whether
any particular node is positive or negative. Arbitrarily selecting some role
node r, suppose that we set sign(r) ¼ 1. This will determine the sign of
every other node of the component. The signs on the role nodes characterize one possible ‘‘subself’’ while the signs on the action nodes characterize
one possible set of ‘‘subchoices.’’ If we instead assume sign(r) ¼ 1, we
then obtain the opposite sign on each node, characterizing a second possible subself and subchoices.
Proposition 3. For each balanced role-action component, there are
exactly two subselves that could be contained in an absorbing self.
Proof of Proposition 3. If ego makes the subchoices determined by
sign(r) ¼ 1, alters’ unique attribution is the subself determined by
sign(r) ¼ 1. Similarly, if ego makes the subchoices determined by sign(r) ¼
1, alters’ unique attribution is the subself determined by sign(r) ¼
1. Any other set of subchoices will generate multiple attributions. Thus,
for the component, there are only two subselves that could possibly be
contained in an absorbing self.
QED
To illustrate, consider again the abstract example depicted in Figure 5.
Note that the role-action component is balanced (containing no negative
cycles).28 Activating all of the edges of this component, we obtain
signðaÞ ¼ signðyÞ ¼ signðbÞ ¼ signðzÞ:
ð42Þ
Setting sign(a) ¼ 1 determines the subself fa; :bg and subchoices
fy; :zg, while sign(a) ¼ 1 determines the subself f:a; bg and
subchoices f:y; zg. Each of these subchoices generates a unique
attribution,
S0 ðfy; :zgÞ ¼ fa; :bg;
S0 ðf:y; zgÞ ¼ f:a; bg;
ð43Þ
while the other possible subchoices generate multiple attributions
S0 ðfy; zgÞ ¼ ffa; :bg; fa; bgg;
S0 ðf:y; :zgÞ ¼ ff:a; bg; f:a; :bgg:
ð44Þ
28
Note that the imbalance of the role-only component of the graph (which contains a negative cycle) will have no bearing on the existence of an absorbing self. Logical inconsistency
undermines the existence of an absorbing self only when norms involve actions.
The Logic of Role Theory
67
Thus, if an absorbing self exists, it must contain either the subself
fa; :bg or else the subself f:a; bg.29
Note that Proposition 3 has placed an upper bound on the number of
absorbing selves.
Corollary. Given k > 0 role-action components, there are at most 2k
absorbing selves. In particular, given a single component (k ¼ 1), there
are at most 2 absorbing selves.
Thus, a society where all spheres of life are connected through norms (i.e.,
where the graph of N0 has a single component) generates at most two
absorbing selves. In cases where both absorbing selves exist, it may be
natural to label one self as ‘‘mainstream’’ and the other as ‘‘deviant.’’ Only
in a society where various spheres of life are normatively compartmentalized (i.e., where the graph of N0 has multiple connected components) is
it possible to have a richer array of absorbing selves.30
VI. DISCUSSION AND DIRECTIONS FOR
FUTURE RESEARCH
The present paper makes two important contributions toward the formalization of role theory. First, it offers a dialectic model of the process by
which individuals make choices and attributions. While it seems natural
to specify norms (and hence the process by which people reason about
norms) using formal logic, standard logic cannot cope with the contradictions inherent in real-world normative systems. Drawing upon recent innovations in formal logic, I have shown how non-monotonic logic can be used
by role theorists to model human reasoning in social settings where norms
are contradictory (and thus overdetermine behavior). Second, linking
these logical models of choice and attribution together with the assumption
that individuals gradually internalize attributions, the paper offers a model
of the process by which the self-concept changes over time. My analysis has
focused on the long-run ‘‘absorbing selves’’ generated by the social system.
The central result is that absorbing selves exist only when norms are logically consistent. Intuitively, logical inconsistencies in the normative system
permit multiple attributions, undermining the stability of the self.
29
While Proposition 3 establishes an upper bound, the number of subselves actually contained in an absorbing self will also depend on the set of constraints faced by ego. For instance,
in the present example, one can show that C ¼ 1 implies that f:a; bg is the unique absorbing
self, while C ¼ fy $ zg implies that there is no absorbing self.
30
Of course, societies with multiple connected components might have no absorbing self;
the Corollary merely gives an upper bound on the number of absorbing selves. Thus, as an
empirical matter, societies with more connected components could have fewer absorbing
selves than societies with fewer connected components.
68
J. D. Montgomery
The present paper runs parallel to Montgomery (2000), which modeled
the processes of choice and attribution using fuzzy logic, and formalized
the role-person merger as a fuzzy (dynamical) system. But my present
specification has some important advantages that arise because nonmonotonic is more ‘‘logical’’ and consequently ‘‘cleaner’’ than fuzzy logic.
Remaining closer to standard logic, consistency-based non-monotonic
logics (e.g., default logic and level default theories) highlight low logical
contradiction gives rise to multiple extensions (and hence multiple choices
or attributions). These logics should thus be of great interest to sociologists
studying identity who have emphasized that individuals often juggle multiple conflicting ‘‘accounts’’ or ‘‘interpretations’’ of their own behavior as
well as the behavior of others.31 Moreover, because these logics are more
transparent than fuzzy logic (which generates conclusions through various
forms of matrix composition), I was able in the present paper to derive
some general results on the relation between the structure of norms and
the existence of absorbing selves. Indeed, because my specification of
alters’ level default theory readily suggested a graph-theoretic treatment,
the analysis uncovered an interesting application of balance theory in the
study of normative systems. In this way, the paper contributes to the growing literature applying methods from social network analysis to the study of
culture (Mohr, 1998, 2000).
Future research might attempt to extend the present framework in several directions. While I have assumed that alters make a single, collective
attribution in each period (so that ‘‘alters’’ are essentially a ‘‘generalized
other’’), the model might be extended to allow multiple distinct alters
who might make different attributions (perhaps after observing different
subsets of ego’s choices). Relatedly, we might revise the process by which
ego internalizes attributions, allowing ego’s self to be differentially sensitive
to the attributions of various alters. Indeed, recognizing that ego’s social
ties might themselves be viewed as elements of ego’s self-concept (cf.
Montgomery 2000), future research might attempt to model endogenous
change in ego’s social network.32 And while the present paper focuses
solely on ego’s actions, future work might consider interaction systems in
which multiple individuals are making interdependent choices (and attributions), developing a role-theoretic alternative to game theory.
31
Consider, for example, the conception of ‘‘multivocality’’ in Padgett and Ansell (1993).
Their analysis of Cosimo de’ Medici not only suggests that various alters made different
attributions about Cosimo, but also hints that Cosimo himself must have maintained multiple,
conflicting interpretations of his own behavior.
32
While Turner’s (1978, p. 13) ‘‘consensual frames of reference’’ principle suggests that
ego’s self will be especially sensitive to the attributions of significant others, note that the significance of ego’s tie to alter might itself might itself depend on the content of the attributions
communicated by alter.
The Logic of Role Theory
69
Future research might also consider supplementing the present ‘‘rulefollowing’’ specification of ego’s choice-making process with some more
‘‘rational’’ features. While I have assumed that ego is myopic, making
choices without concern for the attributions that will follow, future versions
of the model might instead assume that ego acts in order to induce certain
attributions. In particular, one might assume that ego chooses actions in
order to generate attributions that will be as close as possible to his current
self.33 Alternatively, the logic itself might be expanded to encompass more
calculative modes of decision-making.34 While the simple social ontology in
the present paper includes only roles and actions (and norms as implications from roles to actions), future extensions might also include goals
(and means as implications from actions to goals). One could then endow
some roles (and perhaps also ego as the biological home of the self) with
preferences over goals.35 In this way, the present framework could be
applied in settings where role conflict emerges between ‘‘rule-following’’
and ‘‘calculative’’ roles (cf. Montgomery, 1998).
Finally, while this paper has addressed the micro-level process by which
the self evolves within an environment of fixed norms, future research
might consider the macro-level process by which norms themselves evolve.
Indeed, my application of balance theory immediately suggests one specification of this process. Just as network analysts have argued that social
networks will tend to become more balanced over time (see, e.g., Flament,
1963), role theorists might hypothesize that systems of norms will tend to
move from imbalance (contradiction) to balance (consistency). However,
reflection suggests that this specification is at best incomplete. Presumably,
any tendency towards balance might be offset by forces that make the normative system more complex and thus create new contradictions. Moreover, rather than assume that change in norms depends solely on the
structure of norms, we might suppose that pressure for normative change
will occur only if a sufficient number (or particular types) of individuals
experience role conflict. In this way, norms would co-evolve with the distribution of individuals across different selves (i.e., the distribution of individuals in ‘‘role space’’ [Montgomery, 2000]).
33
In this way, future research would adopt the control-theory perspective of Heise (1979).
Of course, future research might also consider less rational modes of behavior. For
instance, we might attempt to model habit formation by assuming that, after following the
norm r ! a for some time, ego’s LDT will contain the (non-contingent) norm a so that ego
would continue to try to choose the action a even if ego’s self no longer contained the role r.
35
In this context, it may useful to consider a natural generalization of level default theories
where, instead of ranking the formulas in T using levels, we specify an arbitrary partial order
on these formulas (see Brewka, et al., 1997, p. 57).
34
70
J. D. Montgomery
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