THE LOGIC OF ROLE THEORY: ROLE CONFLICT AND STABILITY OF THE SELF-CONCEPT

Abstract

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.

Key Words:

• role theory
• role conflict
• self-concept
• moral dilemmas
• non-monotonic logic

Acknowledgments

I am grateful to Tom Fararo for helpful comments and encouragement.

Notes

¹I 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.

²Notable 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.

³Of 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 Pólos, 1999). See also the metatheoretical use of non-monotonic logic by Pólos and Hannan (2002) to reconcile some apparent theoretical inconsistencies in organizational ecology.

⁴Although 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).

⁵Formally, 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.

⁶Extensions 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.

⁷Both default logic and level default theories belong to the same family of consistency-based logics. See Brewka, et al. (1997, Chap. 4) for a detailed specification and comparison of members of this family.

⁸Brewka, 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.

⁹A 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 = {x → y, x → ¬y} is logically consistent because both formulas in F are true if x is false. In contrast, the set F = {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).

¹⁰Cf. Fararo's (1989, p. 87) typology of theorems for dynamical system models.

¹¹This sequence was described as ego's trajectory through “role space” in Montgomery (2000).

¹²See the rows of Table 1 for a complete enumeration of these possible selves.

¹³Attempting 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.

¹⁴Moreover, 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}.

¹⁵It is interesting to note that the extensions hold different implications for ego's self: the extension C ∪ ∅ ∪ N ∪ {p, g, ¬t} implies ¬m; the extension C ∪ ∅ ∪ 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).

¹⁶Given the constraints C, only 4 of the 8 choices in A_x are feasible. The (all-zero) columns corresponding to non-feasible actions have been omitted from Table 1

¹⁷This 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?”).

¹⁸Given 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 T′.

¹⁹In the present example, every attribution must be a “full” self containing either r or for every r ∈ 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.

²⁰In particular, to determine the transition matrix, we would need to specify the probability that ego makes each choice A ∈ A*(S) for each self S, the probability that alters make each attribution for each set of choices A, and the probability that ego would transition to each new self for each pair (S, S′) characterizing his initial self and attributed self.

²¹Note that this matrix test for the existence of an absorbing self is equivalent to the set-theoretic definition given above. If the element on the main diagonal of row S is positive (i.e., (A ⊗ B)(S,S) = 1), then there must be some choice A such that A(S,A) = 1 and B(A,S) = 1 (i.e., A ∈ A*(S) and S ∈ S*(A)). If all non-diagonal elements of row S are zero (i.e., (A ⊗ B)(S, S′) = 0 for all then there cannot be a choice A and self such that A(S,A) = 1 and B(A, S′) = 1 (i.e., there can be no A ∈ A*(S) such that

²²In the present example, because every attribution will be a full self, only full selves are potential absorbing selves.

²³While 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.

²⁴Although the formula (m → f) ∧ (m → w) ∧ (m → t) is logically equivalent to m → (f ∧ w ∧ t), 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.

²⁵Because 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 T′ = (A, N′) are required to be maximal consistent subsets of T′, they do contain (consistent) subsets of equivalences in N′ 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.

²⁶Note the correspondence between the sets of activated edges and the subsets of N′ 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 N′ found in the corresponding extension. For instance, the set of activated edges {mw, mt, fp, ft} might be replaced by {mw, mt, fp} or {mw, mt, ft} given that all three sets would generate the attribution .

²⁷One reviewer offered a potential counterexample to Proposition 2, arguing that an absorbing self would exist given the (unbalanced) graph of N′ with three roles nodes {r, s, t}, one action node {a}, one positive edge {ra}, and three negative edges {rs, st, rt}. (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: {ra, rs, rt}, {ra, rs, st}, and{ra, rt, st}. Assuming sign(a) = 1, these three sets of edges generate the attributions . 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).

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.

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 = ∅ implies that is the unique absorbing self, while C = {y ↔ z} 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.

³¹Consider, 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.

³²While 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.

³³In this way, future research would adopt the control-theory perspective of Heise (1979).

³⁴Of 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.

³⁵In 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).