To do this will require using an unfamiliar mathematical and methodological approach. Vol. 4: War, Power, Peace will be devoted to conceptually unpacking this approach in detail, but for my purposes here I will try to be just precise enough for those familiar with the mathematics and methods to fill in the details, and conceptual and intuitive enough so that others will have a sense for the equation being developed.

Just to briefly review the psychological field, four major elements make intelligible our common behavior. The first is perception, the outcome of the dialectical struggle between reality and ourselves-a balance between the inward directed powers of reality and our powers projected outward as our perspective. Within our psychological field our perception forms a situation, which we are conscious of and which is interrelated with the forces of our personality. The second element of the field is our personality, the temperaments, abilities, motivations, moods, and states that are our relatively constant psychological traits and changing feelings and emotions. We relate to others in terms of our perceived distances between them and ourselves on our common personality components. Moreover, we tend to behave towards others as a product of situation and distances. How we are inclined to act towards another self depends in part on our personality differences and similarities, and on how these are weighted by the perceived situation.

But this is one side of the psychological field; the other consists of the remaining two elements: behavioral dispositions and expectations. How we are disposed to act depends on what behavioral possibilities are opened to us by our culture and learning. Moreover, which disposition we are inclined to manifest depends also on the expected outcome of our behavior. Manifest behavior is also a product of behavioral dispositions and expectations.

That is, for individuals i and j and their common behavior B_ij,h,

Equation 33.1:

B_ij,h = _kW_k,ij,

where

= the summation sign,

W_k,ij = the k^th behavior disposition of ij,

and _k = i's expectations of the outcome of behavioral disposition
W_k,ij.

In other terms, B is the inner product of a behavioral disposition vector and an expectation vector in the space of behavioral potentialities. All social behavior occurs within a matrix of dispositions and expectations, whether the result of antecedent causes or of free will. Accordingly, we can take equation (33.1) as fully descriptive of the behavior of one individual to another.

Moreover, behavior is partly a consequence of the weighting of distances i-j by the (perceived) situation. The "partly" reflects the omission of two factors which would complete this side of the equation. First, while we share certain communalities with others, each of us is also unique. And our uniqueness also can influence our behavior. For example, a boy who was attacked by a drunk at the age of twelve may subsequently be hostile towards those who drink. His antagonistic behavior may share many features with the actions of others but nonetheless flows from his unique experiences and traits. In the same way, individuals in a downtown crowd may act quite commonly in shopping, flagging cabs, and seeking shelter from the rain, even though each will have different reasons for being there.

Second, social behavior does not happen simply as a result of antecedent causes, conditions, and random movements. Social behavior is wined, directed to an end. It constitutes a choice of act and associated actions. Of course, the will may choose to act in line with causes or conditions, and thus for an observer the outcome often will be indistinguishable from a pure cause-effect relationship. However, the will has the power to choose spontaneously to act: to choose behavior independently of causes and conditions.

Any complete equation of behavior, therefore, should include the will and specific influences as independent powers along with distances and situations. Thus,

Equation 33.2:

B_ij,h = _imd_m,i-j + Q_{i j,h}.

where

d_m,i-j = the perceived distance vector between i and j on the m^th component of personality in i's psychological space,

_im = i's perception of the situation relevant to the m^th component of personality,

Q_{i j,h} =the specific effects on i's behavior h plus i's independent will.

We again have the inner product of two vectors: the vector of distances in i's psychological space and the vector of situation.²

Putting equations (33.1) and (33.2) together, we get

Equation 33.3:

_kW_k,ij = B_ij,h = _imd_m,i-j + Q_{i j,h}.

In words, how we behave towards another is a product of dispositions and expectations. However, the dispositions are themselves an outcome of distance-forces in the psychological field and their weighting by the perceived situation. Moreover, which expectations are salient and how much weight will be given them is partly a consequence of influences unique to the actor and to hi$ willful determination.

Equation (33.3) is the central equation of our psychological field. My aim, now, is to apply it to the conflict process.

First, the components of social potentialities must be inferred from manifest meanings, values, norms, status, and class-from manifest variance among social phenomena. To do this assumes that such phenomena represent a latent region of potentiality or probability. The more phenomena cluster in their social variance, the greater the underlying probability density. Moreover, if a variety of these clusters form independent patterns of covariance, then the underlying probability regions are independent. In other words, the multifold covariation of social phenomena provides insight into the underlying potentiality they manifest.

However, to move to the second assumption, the underlying potentiality cannot be measured precisely. We can only assess its approximate nature from its traces on phenomena. Sociocultural space is therefore indeterminate, a haze. Nonetheless, from its manifestations, from the variation among social phenomena, we can delineate the probable regions of sociocultural potentiality. Mathematically, this means assuming that social phenomena manifest an underlying multidimensional probability density function, a hyperellipsoid, whose orthogonal axes are the most likely components of the sociocultural space we wish to define.

Third, the social phenomena of interest are the covariances among social manifestations. We discern differences and similarities, and, indeed, discriminate social objects, in terms of their unity of action, or covariation. Thus the role of medical doctor has meaning reflected in a cluster of covarying social manifestation (nurse, diagnoses, medicine, and so on); thus, threats have social meaning manifested in the subsequent covarying actions and responses. Covariance will be my definition of relationship and my door to sociocultural space.

Fourth, only a certain kind of covariance is assumed above. Consider a particular social phenomenon, such as handshaking. We can discriminate three kinds of variance it embodies. The first covaries commonly with other social actions, as for example meeting a person and saying "Hi, glad to meet you" while shaking his hand. Handshakes also share variance with events, such as parties, business meetings, and appointments, and certain characteristics, such as sex, age (children do not shake hands), and nationality.

This common variance is of major interest, since it reflects the common sociocultural space underlying the conflict process and its structures and situations general to us all. Moreover, common variance reflects shared forces, expectations, perception, and choices. Even though our will be free, as reasoning creatures we may make similar decisions in similar circumstances, and the covariance among our behaviors mirrors this.

However, handshakes also manifest unique variance, an idiosyncratic manner or style which is unrelated to other social manifestations. There are two sources of this unique variance. One is the specific variance due to an individual's particular experience, perception, personality, heredity, and so on. Each shares communalities with others, but each also is his own self, and the specific variance manifests this. Second, there is the variance due to the will's independent action. People choose to act. By force of will we may intentionally project a specific feeling or image as by adding extra strength to our handshake, presenting a limp hand, pumping vigorously, or ignoring the other's outstretched hand.

Will is always present in social behavior, which is behavior directed towards an end or goal involving another self. However, willful behavior may comprise common variance (as when we stop at a service station for gasoline), specific behavior (as when we put maple syrup on bacon because our father did), or spontaneous behavior (as when we suddenly decide to quit our job). Will therefore enters into all social behavior, but I am discriminating here that behavior due to the independent, spontaneous decisions of the will.

In sum, then, social manifestations have common and unique variance, and the latter is divisible into specific and spontaneous (willful) variance.³ These are independent of each other: the total variance of some social manifestation is a sum of its common, specific and spontaneous variance.

The final assumption is that time is part of the sociocultural space--that time is subjective. Elsewhere (Rummel, 1974) I have considered this question for states, and will summarize my conclusions here. The time relevant to social affairs is that correlated with, but independent (linearly) of, calendar time. The passage and significance of time depends on our meanings, values, and norms; and on our activities and interests-in short, on our culture, and on what we are doing. Social time is subjective.

Clearly time is a component of sociocultural space. But the location of individuals on this component is a function not wholly of astronomical time, but partly of the other components of the space, which are in turn partly a function of the passage of time. This means that there is a common sociocultural space-time with space-time components.

This relationship of time to sociocultural space could have been denoted in the previous equations by subscripting the variables by t, standing for a specific time unit. This would localize jointly in space and social time the behavior, expectations, distances, and perception of an individual. I have not done this so far in order to avoid additional complexity. But as I move closer to empirical operations and assessment, I must introduce time-social time.

To combine all these assumptions in defining the sociocultural or conflict space, I am assuming: (1) that its components must be inferred from manifest variance; (2) the underlying potentialities are indeterminate and we must rely on multidimensional probability densities for their definition; (3) the probabilities of special concern are those associated with the manifest covariances among social phenomena; (4) this variance is divisible into common, specific, and spontaneous; (5) that its components are space-time components. In defining the components of sociocultural space,⁴ we are interested in the common covariance among manifestations, for if the sociocultural space is to have generality across individuals, if it is to aid our understanding of social behavior and conflict, then it is the common space-time components we must focus upon.

Next we must operationally define the components of sociocultural or conflict space. To begin, consider that the potentialities inherent in this space are manifest in social attributes, i.e., any social quality, characteristic, feature, or property of an individual. To say that Mary is a very religious Catholic teacher, a mild-mannered Democrat who is addicted to crossword puzzles, middle-aged and middle class, and a women who was born and lives in Los Angeles, is to specify a variety of her attributes and to approximately locate her in sociocultural space. Intuitively and instinctively we constantly fix strangers in sociocultural space as we perceive their dress, grooming, manners, speech, and so on. Note here the role of common variance. We read another's field in terms of experience with others and our cultural endowments. In other words, we match another's field of expression with the attributes we have come to associate with certain meanings, values, norms, status, and class. The unique aspect of another's field provides no such social understanding.

Given the common covariance among attributes, how can we then define sociocultural space and its space-time components? There are two theoretical considerations here that lead to the same mathematical result. First, as assumed. above, we can only deal with the underlying multidimensional probability densities. Second, the probabilities, dispositions, and powers that form sociocultural space are a complex of overlapping, intersecting, multifold, and entwined indeterminables. Tracking any one of these relationships mathematically is akin to reproducing a Persian rug's patterns with straight edge and compass. Therefore we must resort to representing the space through orthogonal functions.

From the probability perspective, the common covariances among attributes define a hyperellipsoid (a correlation-ellipsoid); the axes of this ellipsoid are then the common space-time component--the latent functions--defining sociocultural space. From the empirical perspective, the eigenvectors of the common covariance among social attributes will define the orthogonal functions spanning the space and subsuming manifest interrelationships. In both cases, we can define the connection between manifest attribute and sociocultural space as

Equation 33.4:

A_igt = a_gmS_imt + U_igt,

 _igt = a_gmS_imt,

where

A_igt = a manifest social attribute g at time t for individual i;

S_imt = i's value on the m^th common space-time component of sociocultural space at time t;

a_gm = the weight the m^th component has in producing manifestation A_igt;

U_igt = the unique contribution of i towards A_igt;

Â_igt = a manifest common social attribute g at time t for individual i.

Now, S_m represents the independent lines of space-time probabilities in sociocultural space underlying social manifestations, such as A_g. The parameter a_m measures the degree to which this particular probability underlies the specific manifestation. Each of these lines of probabilities, or components, are statistically independent. Therefore, the actual common manifestation A_g is a linear composite of these separate probabilities.⁵ In formal terms, the S_m are the principal axes of the space comprising the A_g.⁶

Variants of (33.4) as interpreted here (but without time) have been employed quite extensively in the social sciences, as by' Osgood, Suci and Tannenbaum (1957) to determine the underlying components of meaning, by Neal and Rettig (1963) to uncover the components of alienation, by Hart and colleagues (1943) to delineate the components of delinquency, by Borgatta (1965) to delimit the components of social desirability. Hundreds of such use of (33.4) are available,⁷ and show that science has had much experience with the mathematical and operational aspects of the theoretical equation presented.

So much should follow from the previous section. There is, however, a significant interpretation we can give the common space-time component of this space. To recapitulate, the common component is a probabilistic measure of the underlying potentialities inherent in the common sociocultural space, which are assumed to produce traces (as potentiality becomes disposition that is transformed to some specific observable) in social time on the phenomenal plane, especially in the covariance among observed attributes.

We can determine common components that lie along an independent pattern or cluster of covariance among the attributes across states and time. A cluster of interdependent attributes will define then a specific space-time component, a region in sociocultural space of high probability that some underlying factor common to state-societies exists--a region where potentialities are likely to become manifest. And this underlying factor may be a structure of expectations common to states.⁹

For example, common among all states is a pattern of economic activities, communications, and consumption. Attributes involved in this pattern cohere; they covary. If a state, for example, is low on GNP per capita, it usually has few automobiles, telephones, newspapers, manufacturing industries, energy consumption per capita, and so on. Those high on some of these attributes are high on the others. This coherence is no accident. These attributes are fundamentally part of the factor we call economic development or wealth.¹⁰ They form a causal-functional unity.

Wealth is a common space-time structure of expectations among states. Some states are poor, some very wealthy, but in each case wealth as a structure of expectations involves the same objective ingredients, although the extent of expectations and interactions will vary depending on the wealth. In poor states, norms, laws, and interactions are bound up with expectations, low income, few manufacturing jobs, few consumer goods, few telephones and so on. In wealthy states, expectations and interactions are quite different, but still regarding the same goods or opportunities.

A common structure of expectations is a particular space-time path of change for a state. Some development and slight increase in wealth create a new set of expectations. Additional change may upset this structure, and a higher level of development can again ensue. Development to higher levels of wealth is then as a succession of structures of expectations, punctuated by transforming conflict establishing new balances among interests, capabilities, and wills. Development is then like the individual's psychological growth from childhood,¹¹ which is a succession of surges marked by periods of personal and interpersonal conflict and crises (as with the adolescent).

A structure of expectations commonly varying among states, then, is a common path followed by a specific structure of expectations as it changes. It is a common covariance among--a clustering of--attributes on which states differ as to the level of their associated expectations, norms, and laws.

Operationally, a single solution solves two problems. The conflict or sociocultural space of states and its common space-time components are defined by probability densities underlying the covariance among observations-state attributes. And these components also define¹² the space-time structures of expectations commonly varying across states.