a formal model

©1997 by Richard Cullen Rath
[Introduction] [Discussion] [Formal Model] [Relations in General] [Conclusions]

N.B.: This is a work-in-progress. Please consult the most recent revision (at this URL) before quoting. Comments are welcome at [CONTACT PAGE].


In his system of force-dynamics (1988), Talmy's primary notion, as formalized and extended by Jackendoff (1990), is that the concept of causation can be decomposed. The parameters of causation's further constituents can be adjusted to result in the conceptual structures of a plexus of cause-related verbs. Jackendoff labels the compositional constituents AFF, REACT, and CS, roughly akin to affect, react and cause, respectively. The parameters for each of these constituents can be set to +,-, or u, for successful, unsuccessful, or unspecified, respectively. Furthermore, CS+ may have an additional parameter setting that specifies whether a CS+ action entrains (i.e., is temporally coextensive with) or launches an event. Jackendoff illustrates the difference with the verbs drag and throw, respectively.

The fact that the entrainment/launching parameter only comes into play in situations where cause has been successfully instantiated makes it inapplicable to situations where the outcomes have not yet been specified: "In indeterminate and failed causation, the Effect is asserted not to have taken place, so there cannot be a temporal relation between the Cause and the Effect." (Jackendoff 1990, 139) This has important implications for a class of CSu concepts involving power relations. Power can be conceived as latent force, force which could be, but in fact has not yet been, realized. Realization, of course, must take place in whatever its domain is. As such, power is irreal-but-possible force which would be constrained by its domain were the force realized. Thus power itself does not have outcomes until expended as a force. This event or process of realizing power as force changes all parties in the relationship to some extent, which in turn changes the nature of the power relations. In effect, a dynamic, changing system of reciprocal relations is maintained in any power relation.

Talmy's model, while dynamic, is also uni-directional. He posits two types of actors in a force-dynamic relationship, an agonist, with a particular set of natural tendencies, and an antagonist that somehow opposes this state. The extent that an antagonist's application of force can be considered successful is a function of the amount that the antagonist is able to make the agonist deviate from its natural tendencies. Counter-force on the part of the agonist is captured in the notion of resistance, with success being guaged as a function of the agonist's ability to maintain its natural tendency in light of the antagonist's exertion of force upon it. Here the direction of the force-dynamic interaction is reversed, extending from the agonist rather than from the antagonist.

However, force exerted from one source changes not only its target (if that), but the source as well, via the process of expenditure. The "forcer" is changed by its expending, the beneficiary or patient by its reaction, whether volitional or not. In the latter case, accommodation and resistance both involve a change in the expenditure of effort, even if that effort is expended in the service of 'keeping things the same'--i.e, fulfilling their "natural tendencies." These features render the agonist as a body at rest in its state of nature until such time as force is exerted upon it, and the antagonist as seemingly impervious--it is a source, not a recipient. These properties of the force dynamics model hinder the analysis of reciprocal relations and make the model best-suited to capture intended effects, not unintended ones.

This paper is devoted to extending the model, via Jackendoff's formalization of it, to account for reciprocal relations, particularly relations of power, in a fashion that retains the dynamics while overcoming the limitations of unidirectionality built into the antagonist/agonist distinction. The proposed analysis of power relations depends heavily on two tools, functional set notation(2) and a set of conceptual distinctions and relations among the categories substance, individual, group, and aggregate.(3) The paper is organized in the following manner: First, the line of thought driving the proposed model of power relations will be discussed informally; that explication will then be formally justified in a step-by-step fashion. This model of power relations, when formally justified, discloses some interesting properties of relations in general, which will be discussed briefly along with some conclusions.


The force-dynamics model of causation is concerned with "Who does what to whom." An analysis of power relations requires an additional parameter, marking whether a force is real or not. Irreal could be further distinguished as being impossible or possible. Power relations concern forces that are irreal but possible--i.e, latent forces. A working definition of power relations that distinguishes them from causation without losing their similarities is that

  1. Power relations are what enable who to do what to whom.

rather than "who does what to whom." A more structurally explicit and useful definition is:

  1. Power relations are the matrix of possible actors and their possible interactions.

To apply the definition, this matrix would be expressed as all the ways (an infinite set) that the following template could be completed in any one moment (an infinitely small interval), employing for each blank in order: a substantive property, a possible actor, a possible act, and a possible actor (can be reflexive). It needs to be kept in mind that the realization of an infinite set in an infinitely small time interval can only be approached, never obtained, a fact that would appear to disable the the perfect (though perhaps not the practical) application of the template to any actual situation.

  1. (what) enables (who) to (do what to) (whom).
    (wealth) enables (the west) to (exploit) (the rest)
    (labor) enables (the rest) to (sell to) (the west)

and so on, ad infinitum but with better poetics. The power relations that hold at a given moment between the putative "West" and the putative "rest" are defined by the "West's" "wealth" in the first instance, by the "rest's" "labor" in the second. In the interaction of the two, each changes the other as well. An interpretive problem with this system is that its dynamics are implied rather than explicit. The implication resides in the assertion that (3) is a template, the filling in of which is an infinite set in which the spelling out of one variable has repercussions on all others. When the template is begun to be filled, thereby applying the model, no completed definition of a particular set of power relations can ever be had. A specific description of particular power relations can be approached, but not arrived at. The distribution of latent force--power--cannot be conceived as being discretely contained by agonists or antagonists then. Jackendoff deals with this by placing volitional aspects of causation in the argument structure of the verb, effectively deconstructing the idea of agency, but leaving the concept of actors intact, the latter as, among other roles, a place where agonists and antagonists can reside (Jackendoff 1990 258). This effectively moves causation from actors to the act itself. This relocation of the source of power has some interesting implications, decentering the agent as the governor of power relations. Some other natural objects must be had to explain power relations. A place to start the search for would-be the candidates is in the constituents of the initial definition, (1).

How are the definitions of power relations derived? A derivative of the starting (who-gets-to-do-what-to-whom) definition is that:

  1. Power relations are the bounded portion of power that gives agents the ability to act on patients.

However, this tends to hide (by making implicit) the multilateral nature of relations. A way of highlighting this possibility for reciprocity (by making it explicit) would be to say:

  1. Power relations are the bounded portion of power that gives entities the ability to interact in a particular way.

The bounded portion of power, a latent domain, both generates and constrains the interaction in addition to affecting the participant entities. To clarify:

  1. Power relations are the domain of latent power(4) that makes it possible for entities to interact in a way that temporally manifests that power among them.

That is, relations of power are that which allows entities to interact effectively (in the sense of "with effect"). If the matrix of all the actions and actors possible at any one instant completely constitute the domain, then the definition may be stated more simply as (2) with no loss:

  1. Power relations are the matrix of possible actors and possible actions.

There are no outcomes, because power deployed unidirectionally instantaneously alters the relationship. This is the working definition. Its repercussions are important. Perhaps a more evocative way of stating what upon reflection amounts to the same thing is to say that

  1. Power relations are ever-shifting threats and promises.

Formal Model

Assuming the starting definition, can these definitions be formally justified? The following section is an explication of the definitions by means of function set notation. This is the means by which the definitions offered above were developed.

The statement of power relations as "what enable who to do what to whom" can be encoded as such:

  1. Relations (R) of (TYPE) power (P) are (=, ARE eq, EQUIVALENT TO) what (W) enable (ABILITY give) who (X) to (POSSIBLE/IRREAL) do (DO) what (Y) to whom(Z).

Yielding the functional set notation:

  1. {ARE eq [{Rtype[P]},

{ABILITY give [W, {FOR-TO possible [{Y do[X,Z]}]}]}

To paraphrase,

  1. Rtype[P] are the W that enable X-s to do Y to Z-s


P = unbounded, global properties--latent, and akin to substance; power.

R = a reciprocal path or channel, a relation

R(p) = Rtype[P]

R(p) = P is the content of R.

W = a bounded portion of P, a domain of P

Y = an action, either bounded (perfect) or continuous.

X = An agent: a doer capable of intentionally causing

Z = a patient/beneficiary: An object of the agent's doings

Paraphrased again, this yields the first derivative definition, (5):

  1. Power relations are the bounded portion of power that enables agents to interact in a particular way with patients/beneficiaries


  1. BE [R (p), (ENABLE [W (to [X (DO [Y (TO [Y, (Z)])])])])]

or it could be represented in binary branching trees:

     	   /  \	
     	R(p)  ENABLE
    	/  \	  / \
       R    p    W   TO
                    /  \
                   X	DO
                       /  \
                      X	   TO
                           / \
                          Y   Z

This definition works well as long as it is remembered that "X does Y to Z" is a possible, but as yet unrealized (irreal) and unspecified, act and that X and Z are also as yet unspecified. This leaves the reciprocity of the relations implicit but intact.

The implicit reciprocity in relational models is often overlooked, so it would be advisable to make it explicit. To do this, two definitions can be clarified. The only requirement of X and Z is that they each be able to participate effectively in the relationship. Each must possess an externally bounded locus; i.e., each entity has to be an aggregate or an individual, it cannot be a substance or a group. Unbounded matter, i.e. substance, has no effective potential--there is no way to discern whether power has acted upon or emanated from it. Groups are excluded from participating because an act upon them or from them instantaneously changes their constitution to include the other entity, else it is not a power relationship.

This participatory requirement is not unique to irrealis, being a necessary, perhaps even tautological, requirement for any entities in a relationship. However, the reduction to a synchronic analysis changes these properties in ways with global significance within any relational model. The difference between groups and aggregates is necessarily lost in synchronic analysis, and the difference between individuals and aggregates often follows it.

For example, if:

  1. Y = a temporal manifestation of P between X and Z reciprocally; interaction, either bounded (perfect) or not (i.e, continuous).


  1. (X,Z) = entities capable of participating in a relationship, each possessing an externally bounded locus; i.e., each entity is an aggregate or an individual. However, they do not have explicit agent and patient roles because the situation is both reciprocal and irreal.

then the following functions can then be applied to the definition of power relations, making reciprocity explicit:

  1. A = (X, Y, Z)

S(a) = {Y do reciprocally [X,Z]}

S(a) = {Or [(XYZ), (ZYX)]}

S(a) = {RESPECTIVELY [{Y do reciprocally [{or [X,Z]}, {or [Z,X]}]}]}, where RESPECTIVELY means "use each member once and only once."

N(s) = (POSSIBLEirreal[S(a)])

N(s) = An ordered relationship possible among participating entities (X,Z) and temporal manifestations of power (Y).

W(n) = {ABILITY give [W, N(s)]}

W(n) = W enable N(s), W is globally represented in N(s), W is a necessary condition of any N(s). No truth claims are involved as long as P remains latent/irreal; a possibility rather than history. If realized, it would be a force rather than power, changing the constitution of the attendant power relations.

W(n) = W, a domain of latent power, enables an ordered relationship to be possible among participating entities and temporal manifestations of power (i.e., forces).

Substitution yields:

  1. R(p) = {W(n)} = W([N([S(a)])])

W(n) = {ABILITY give [W, N(s)]} = {ENABLE [W, N(s)]}

N(s) = {POSSIBLEirreal[{Y do reciprocally [X,Z]}]}

Further substitution yields:

  1. {ARE eq [{R type [P]}, {ENABLE [W, {POSSIBLEirreal[{Ydo-reciprocally-to[X,Z]}]}]}]}


  1. R(p) are the W that make it possible for X and Z to interact in way Y

So the working definition is now (6):

  1. Power relations are a domain of latent power that makes it possible for entities to interact in a way that temporally manifests that power between them

Factoring out power gives the following definition of relations which is at the core of this discussion:

  1. Relations are a domain that makes it possible for entities to interact temporally.

Compare with the a priori definition given above:

  1. R = a reciprocal channel; a path.

This more general issue of relations will be returned to below.

An important issue is what happens to the definition when some portion of it is reduced to make it more manageable. A common way of doing this is to bound the time frame the model is working within as in a synchronic model. This approach to power relations would yield the following definition:

  1. As a result of X having power, X DOES Y to Z;

which can be glossed as:

  1. X has power (W), so X does Y to Z

where W can be assumed to be the remaining W(x) after W(y) and W(z) have been subtracted. The patient, Z, now passively and changelessly receives an invariant Y from the agent X with an unbounded supply of W (the unbounding of which makes it equivalent to homogenous P). In a synchronic analysis, the entity with the most power becomes the agent, all other power is factored out.

However, note that the alternative offered here, a dynamic definition, leaves no way of knowing how W is allotted to each entity (X,Y) or action (Y). This is because the the union (U) of the set of powers (W) cannot be disaggregated into portions unique to any member without constructing arbitrary divisions:

  1. W = Union[W(x), W(y), W(z)]

W = Union[W(x,y,z)]


  1. R(p) = Union[W(x,y,z)] ENABLE N(s)

It is only in the past tense, only when power has already been expressed through the relationship, that W can be accurately assigned to particular members of the set A. But expressing power (i.e., exerting force) alters the power relations in the expression instantaneously, making the answer to the question "What are power relations?" descriptively approachable, but ultimately unobtainable in any but a functional form.

In contrast with the synchronic reduction (and in agreement with the proffered definition), reconsider definition (6):

  1. Power relations are a domain of latent power that makes it possible for entities to interact in a way that temporally manifests that power between them

This means power relations allow entities to interact effectively. With that in place, consider the final working definition, (2):

  1. Power relations are the matrix of possible actors and possible actions

The advantage of (2) is that it explicates the fact that because power deployed unidirectionally alters the relationship instantly there are no outcomes.

Relations in General

In the course of defining relations of power, several insights into relations in general have been uncovered. First, factoring out power from the definitions yields an interesting general definition of relations. Second, the domain of relations is ineffable. Finally, a perfect definition of relations, while functionally approachable, is ultimately unobtainable.

The general definition for relations derived above, in example (19) was that:

  1. Relations are the domain that makes it possible for entities to interact temporally.

This can be stated more succinctly, in line with the final working definition of power relations, (10):

  1. Relations are the matrix of the possible.

It follows, that like power relations, the domain of relations in general is ineffable. None of the definitions (unless synchronic) of relations in general, nor of power relations in particular, provide any means of knowing how the contents a domain are allocated to any specific entity or action. The domain is ineffable as long as it remains irrealis. As such, it will be recalled, it cannot be disaggregated into portions unique to any member without first artificially imposing arbitrary divisions. For the same reasons as those given above for relations of power, a perfect definition of relations in general, though ultimately unobtainable, is infinitely and productively approachable.


The conceptual structure of power differs from that of force in its setting of one crucial parameter, namely realized/latent, with force marked as realized and power marked as latent. The latency and diffusion of power requires it to be treated as residing in a relation, rather than in individual entities. In contrast, force, while it perhaps could be treated in the domain of relations, is more often considered as emanating from some individual entity and affecting some other individual entity, as in Talmy's model.

The ability to locate force as emanating from individual entities stems directly from the fact that it is realized, and thus not diffuse or open-ended like power. Relations are by definition reciprocal to some extent, not unidirectional. This poses a problem for those seeking an absolute definition of power located in some entity or another and unambiguously realizable as a unidirectional force. The combination of Talmy's dynamic approach and the use of relations rather than entities as the object of inquiry is perhaps decentering to traditional dominance/submission models of power relations, but this approach closer approximates the intuitions of how such relations actually work than do approaches based on individual entities rather than relations. The reason for this is that the latter, in presumptively allocating power to units coextant with individual entities, basically functions as a blaming strategy rather than an attempt to understand the forces--and powers--at work in a particular situation.


1. This text is largely excerpted and edited from Rath (unpublished)

2. Functional set notation takes roughly the following form:

(AAA bbb [(CCC) (DDD)]

Which can be glossed as

CCC performs the bbb function of form AAA on DDD

Where AAA is a form of function bbb that takes two arguments, CCC and DDD. So, A one-argument clause,

She left


(LEFT do [She]).

A two argument clause,

John hits the ball


(HIT do [John, ball]).

And a three argument clause,

It gives her the creeps


(GIVE do [It, her, the creeps])

This is my reduced interpretation of the set notation employed by Jackendoff (1983, 1990).

3. The working model of conceptual distinctions employed here uses four conceptual categories:

    GROUP        AGGREGATE   

And a reciprocal movement transformation:

<== ( ==>un- ) specification

arranged along two axes:

     IDENTITY ....... specified <==> unspecified

    ENVIRONMENT ..........||

such that:

SUBSTANCE is something possessed of no specified identity (it is unbounded -- i.e. it may be possessed of a non-specific identity) operating in an unspecified environment;

INDIVIDUAL is something possessed of a specified identity (singular and countable) operating in an unspecified environment;

GROUP is a specified environment comprising a (plural and countable) number of specified identities; and

AGGREGATE is a specified environment possessed of a plural number of unspecified identities.

examples of each:


"a" dog meat

"an" event process


six dogs and a person observing them "The" pack of dogs

five related events "the" trend.

In addition to "horizontal" and "vertical" movement, diagonal movement is common. The path between INDIVIDUAL and AGGREGATE, is traversed by asserting identity to be environment or vice versa. For example, using upper case to represent specified, lower to represent unspecified and I/i to represent identity axis and E/e to represent position on the environment axis:


Ie ie



the move from AGGREGATE to INDIVIDUAL would be achieved in the following way:




a shift as simple as that from the pack of dogs to a pack of dogs.

Along with the above, the following issues, among others, are being developed in a separate paper. (1)The diagonal move from GROUP to SUBSTANCE is often commonly made under another interesting set of assumptions. (2)There may be another axis, something along the lines of "presence," "immediacy," or "reality." (3)With the third axis in place, this model directly addresses some of the issues ofBickerton's proposal for explaining a universal article-related system of specification in a new way (also addressed from a different perspective in Rath, "The Abduction of Form and Meaning"). (4)Issues of abduction (especially at the site of diagonal movement), emic-etic distinctions, Verbs, domains, and boundaries are addressed.

Much of this section is derived from another work by this author, in progress, on the conceptual structure of English articles. The four-cell distinction and the fact that there are boundary issues of some sort were presented by Jackendoff in his course in Semantics at Brandeis University in Spring, 1993. The impetus for my ideas comes from there. I do not know whether or not my consequent development of them is in agreement with Jackendoff's conception of these issues, though my understanding is that they are congruent (see also Semantic Structures, 29-30, 100-106).

4. Latent is redundant in this context, but I will occassionally employ it before power for the sake of emphasizing this feature in contexts where it could be overlooked.

Works Cited

Bickerton, D. (1981). Roots of Language. Karoma Press, Ann Arbor, MI.

Jackendoff, R. (1983). Semantics and Cognition. MIT Press, Cambridge, MA.

Jackendoff, R. (1990). Semantic Structures. MIT Press, Cambridge, MA.

Rath, R.C. (unpublished). The Abduction of Form and Meaning.

Rath, R.C. (unpublished). "Language, Culture and History Hypertext Project."

Talmy, L. (1988). "Force Dynamics in Language and Thought." Cognitive Science 12, 49-100.

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Copyright 1997 by Richard Cullen Rath. Readers may redistribute this article to other individuals for noncommercial use, provided that the text, all html codes, and this notice remain intact and unaltered in any way. This article may not be resold, reprinted, or redistributed for compensation of any kind without prior written permission from the author. If you have any questions about permissions, please contact <[CONTACT PAGE]> Preferred Citation: Richard Cullen Rath, "Relations of Power: A Formal Model," DISSONANCE (May 1, 1997 [http://way.net/dissonance/power.html]).