Steve Watson

For x ∈ |G|,
Let B_stat(x) ⊂ B(x) be all actions that constitute remaining in the group
Let B_exit(x) ⊂ B(x) be all actions that constitute leaving the group

(∀x ∈ |G|)[ E[ TS(C(B_stat(x), x), I(x), x) ] ≥ E[ TS(C(B_exit(x), x), I(x), x) ] → A(x)∈B_stat(x) ]

Where [ E[ TS(C(B_stat(x), x), I(x), x) ]
= ∑_{b∈E[ Bstat(x) ]} E[ P(b) ] (∑_j∈E[I(x)]w_j(x) ∑_{c∈E[C(b, x)]} E[ P(c) ]E[ S(c, j, x) ])
and [ E[ TS(C(B_exit(x), x), I(x), x) ]
= ∑_{b∈E[ Bexit(x) ]} E[ P(b) ] (∑_j∈E[I(x)]w_j(x) ∑_{c∈E[C(b, x)]} E[ P(c) ]E[ S(c, j, x) ])

We may similarly define a Recruitment Condition that needs to be satisfied in order that x should become a member of G:

For x ∈ |G|,
Let B_stat(x) ⊂ B(x) be all actions that constitute remaining outside the group
Let B_intrat(x) ⊂ B(x) be all actions that constitute joining the group

(∀x ∈ |G|)[ E[ TS(C(B_intrat(x), x), I(x), x) ] > E[ TS(C(B_stat(x), x), I(x), x) ] → A(x)∈B_intrat(x) ]

Where [ E[ TS(C(B_stat(x), x), I(x), x) ]
= ∑_{b∈E[ Bstat(x) ]} E[ P(b) ] (∑_j∈E[I(x)]w_j(x) ∑_{c∈E[C(b, x)]} E[ P(c) ]E[ S(c, j, x) ])
and [ E[ TS(C(B_intrat(x), x), I(x), x) ]
= ∑_{b∈E[ Bintrat(x) ]} E[ P(b) ] (∑_j∈E[I(x)]w_j(x) ∑_{c∈E[C(b, x)]} E[ P(c) ]E[ S(c, j, x) ])

Just for the sake of completeness, we’ll note two other conditions, which are perfectly obvious given the preceding. First, the Retirement Condition:

(∀x ∈ |G|)[ E[ TS(C(B_stat(x), x), I(x), x) ] < E[ TS(C(B_exit(x), x), I(x), x) ] → A(x)∈B_exit(x) ]

And, second, the Resistance Condition:

(∀x ∈ |G|)[ E[ TS(C(B_intrat(x), x), I(x), x) ] ≤ E[ TS(C(B_stat(x), x), I(x), x) ] → A(x)∈B_stat(x) ]

• The condition of equality between the two estimated total satisfactions will be taken to favour continuation of the status quo. It is assumed that inertia is preferred where we find ourselves in the position of Buridan’s Ass.
• It is not here assumed that (∀x ∈ |G|) [B_stat(x) = B_intrat(x)^c] or that (∀x ∈ |G|) [B_stat(x) = B_exit(x)^c], but that does seem like a reasonable assumption to make.

O(G) = {r₁, …, r_n} ∈ 2^|G| is the Organization of G, where r_i are the roles in G, satisfying

1. (∀x ∈ |G|)(∃r ∈ O(G)) [x ∈ r]
2. (∀x ∈ |G|)(∀r_i, r_j ∈ O(G)) [x ∈ r_i & x ∈ r_j → r_i = r_j]

Of course, merely defining a subset of subsets does not suffice to characterize an organization or the structure of a group.

The point of a group, we noted before, was to assist in the achievement of the interests of its members. Socially relevant groups, which are actors in the social realm, will have a structure that coordinates the actions of its members towards some end. Such coordination is achieved through the imposition of a system of dominance relations. The point of the roles is in large part that the dominance relations that are relevant to the function of the group are defined wrt the roles. We wish, therefore, to note the existence of a characteristic set of dominance relations amongst the roles in the organization, which can be done by referencing the dominance relationship defined between agents. First define a dominance relationship amongst sets of agents as follows:

Let X, Y be disjoint sets of agents. Y dominates X iff (∀x ∈ X) (∃y ∈ Y) [y/x].

We now apply this relation to the roles of a group, and claim that all the roles are involved in dominance relationships. We also need to specify some roles as being special in the sense that though they dominate others they are not themselves necessarily dominated by any other roles in the group. These are the Leadership Roles in an organization. In fact, we shall assume there is just one leadership role. This is certainly the case in most forms of organization, and certainly in the standard hierarchical organization. Let the leadership role be denoted r_L. Then we say that

3. (∀r_i ∈ O(G)-r_L) (∃r_j ∈ O(G)) [r_j / r_i]

• We should also distinguish leadership roles from Strict Leadership Roles. Strict leadership roles are undominated leadership roles. Not all LR are SLR. Every group has a LR but not every group has an SLR – in fact it’s likely that there are very few of the latter. The existence of an SLR is described thus:
  

  (∃r_SL ∈ O(G)) (∀r ∈ O(G)-r_SL) [~(r/r_SL)]

Setting that possibility to one side for now, there are many different structures possible under this minimal definition. It’s possible, for example, to have the leadership role entirely detached from the rest of the group roles. This is unacceptable – meaning just that it is not characteristic of significant social formations. We need to expand the definition of the leadership role to eliminate this possibility.

We say that there is a Domination Sequence (or a chain of command) in O(G) subordinating role p to role q iff (∃ < r_i: i=1, …,n > ∈ O(G)ⁿ) [r₁/p & r₂/r₁ & … & q/r_n], and we write q//p and read that as q Superdominates p

A reasonable suggestion, then, would be that for any other role in the group, the leadership role superdominates that role; thus:

4. (∀r ∈ O(G)-r_L) [r_L//r_n]

As an example of the kind of definition that is proposed to cover the latter conception they offer the following: “A has power over B when A can change B’s actions, or sanction B, and so forth.” This definition is only offered as an indicator by those authors, and it is, of course, unsatisfactory as it stands. For example, it does not distinguish between power and mere influence. To distinguish ‘power’ in the sense that it is usually used – and in which sense it has a special significance in informal sociological explanations – from ‘power’ as the mere existence of the effect of one agent on another, we need to concentrate on the implied coercive capabilities of A over B. Thus we would need to include the fact that B changes his behaviours because he believes that A has the ability to punish him if he fails to comply. Moreover, he believes that A actually will punish him if he fails to comply.

We can attempt a definition using the established formalism as follows:

x has power over y iff (∃b_response ∈ E(y))[ B(x) ] which satisfies the conditions

1. E(y)[ C(b_response, x) ] = c_response
2. E(y)[ TS(c_response, I(y), y) ] << E(y)[ TS(c_now, I(y), y) ]
3. (∃b ∈ E(y)[ B(y) ]) [(E(y)[ A(y)=b → A(x)=b_response ] & (E(y)[ (A(y)≠b → A(x)≠b_response) ]]

• Whenever x and y are in a relationship such as this we say that x Dominates y and write x/y
• As presented the relationship of dominance appears to have no essential structure. It is not necessarily transitive, and it is not necessarily non-symmetric.
• Degrees of power can be traced back to the degrees by which the total satisfaction functions are altered in 2.
• Note that E(y)[ statement ] in condition 3 is to be read as the subjective estimation of the fuzzy truth value of ‘statement,’ or the subjective degree of belief of y in ‘statement.’
• Complexity can be added by noting that in most cases b_response is merely one of a class of behaviours that are available to x which can punish y in various degrees, that b in condition 3 is merely one of a class of behaviours that are available to y which can prompt the response from x.
• c_now in condition 2 is the context at the time of consideration, but it should really refer to the context that exists when x does not respond punitively to the b in condition 3.
• Note that this is power through punishment, but there is a similar definition available for power through reward: simply replace condition 2 with the condition
  

  E(y)[ TS(c_response, I(y), y) ] >> E(y)[ TS(c_now, I(y), y) ]

In the light of the notes above, refine the original definition thus:

x dominates y iff
(∃B_response(x) ∈ E(y)[ B(x) ])
(∃B_neutral(x) ∈ E(y)[ B(x) ]-B_response(x))
(∃B_stimulus(y) ∈ E(y)[ B(y) ])
the following conditions are satisfied:

1. (∀b_stimulus ∈ B_stimulus(y)) (∀b_response ∈ B_response(x))
   E(y)[ P(B_neutral, x) | A(y)=b_stimulus & A(x)≠b_response ] ~ 1
2. ∀b_stimulus ∈ B_stimulus(y)) (∃b_response ∈ B_response(x))
   [(E(y)[ A(y)=b_stimulus → A(x)=b_response ]
3. ∀b_response ∈ B_response(x)) (∃b_stimulus ∈ B_stimulus(y))
   [(E(y)[ A(x)=b_response → A(y)=b_stimulus ]
4. ∀c_response ∈ E(y)[ C(B_response, x) ]) (∀c_neutral ∈ E(y)[ C(B_neutral, x) ])
   [E(y)[ TS(c_response, I(y), y) ] << E(y)[ TS(c_neutral, I(y), y) ]

There are a number of actions in the social world that can be associated with groups as actors.

The following are actions that are associated with single groups.

1. Formation: A group is formed when several individuals come together.
• A book club is formed to pursue a common interest in literature.
2. Expansion: More members attach themselves to the original group.
• Others who share the literary interest join the club when they observe that the club has been successful for the existing members.
3. Contraction: Membership declines
• Members leave and others do not join when it is seen that the organization of the club is poor, and does not function well.
4. Persistence: Individuals join and leave
• Some move away and others join, but the function of the club is unchanged.
5. Transformation: The nature of the group alters
• As membership changes the function of the club also drifts, becoming a political lobby group.
6. Dissolution: The group ceases to operate or to exist as a group.
• All the once-members join other clubs which suit them better and no others join, or the book club simply stops meeting through lack of interest.
7. Division: The group divides into separate groups.
• Those who are interested in genre fiction form a separate book club more in keeping with their interests. The two clubs have little to do with each other.
8. Subdivision: Subgroups form within the group.
• Those who prefer one literary style hold their own meetings, but without resigning or ceasing to participate in the general club activities.

The following are interactions that may be observed between groups

1. Absorption: A group attracts the membership of another group. (Set union.)
• The book club joins a larger literary society
2. Insertion: A group provides members to another group. (Set intersection.)
• Members of the university Trotskyist club infiltrate the book club (with a view to eventually taking it over and altering its function!)
3. Disruption: A group interrupts the functioning of another group
• The book club divides along political lines and it’s difficult to get any reading done with all the infighting happening. The Trotskyists insist that all book discussions are political meetings.
4. Concentration:A group causes the intensification of the membership relationships in another group
• The film club begins a campaign against the ‘old-fashioned’ book club, possibly threatening its unhampered continued operation. The book club members are motivated to defend their interests.

Functions and variables, V say, that are estimated by x or are otherwise subjective wrt x shall be denoted E(x)[ V ].

• The point of including x in that notation is that later we will want to be able to account for subjective judgements by x of subjective judgements by y, etc.
• Until that complexity is introduced we shall simply write E[ V ]

The partial satisfaction functions may be imperfectly known or imperfectly applied. Instead of the function S(c, j, x) we need to apply the function E[ S(c, j, x) ] which is a function that returns the Expected Partial Satisfaction of interest j for agent x in context c.

• We note that the expected partial satisfaction function may have little to no relationship to the partial satisfaction function.

We do assume that the Expected Total Satisfaction function (E[ TS ]) is unchanged in form (modulo the partial satisfactions) from TS.

• The weight functions are operationally determined, so they are not imperfectly known: they are essentially subjective. (I expect controversy on that point from champions of false consciousness arguments.)
• It is unlikely that the agent considers all the interests that he may have. Let the salient interests be denoted E[ I(x) ]
• We can restrict the E[ TS ] sum to just the psychologically salient interests.
  

  E[ TS(c, I(x), x) ] = ∑_j∈E[I(x)]w_j(x)E[ S(C(b, x) , j, x) ]

For each behaviour the agent will need to consider a range of Subjectively Possible Outcomes, rather than the single outcome that is actually determined by the laws of nature: let this be noted as E[ C(b, x) ] = {c₁, …, c_m}

• Each subjectively possible outcome, c, has an associated Subjective Estimate of Probability, E[ P(c) ], where ∑_{c∈E[C(b, x)]} E[ P(c) ] = 1
• To determine the satisfaction potential of an action b the agent x will consider the likely satisfactions to be had from each subjectively possible outcome of the action and weight it by the subjective estimate of probability of that outcome. Thus, for b ∈ B(x), j ∈ I(x),
  

  E[ S(C(b, x), j, x) ] = ∑_{c∈E[C(b,x)]} E[ P(c) ]E[ S(c, j, x) ]

• We thus require the further modification of the expected total satisfaction function for the action b of x:
  

  E[ TS(C(b, x), I(x), x) ] = ∑_j∈E[I(x)]w_j(x)E[ S(C(b, x) , j, x) ]
  = ∑_j∈E[I(x)]w_j(x)∑_{c∈E[C(b, x)]} E[ P(c) ]E[ S(c, j, x) ]

It is certain that the agent will not consider all possible behaviours. The set of Subjectively Possible Behaviours that the agent considers live options will be denoted E[ B(x) ]

Granted these forms of limited rationality, the behaviour produced by the agent will maximize the expected total satisfaction of all subjectively possible behaviours, yielding an output behaviour b_out, such that:

E[ TS(C(bout, x), I(x), x) ] = max{E[ TS(C(b, x), I (x), x) ]: b ∈ E[ B(x) ]}
= max{∑_j∈E[I(x)]w_j(x)∑_{c∈E[C(b, x)]} E[ P(c) ]E[ S(c, j, x) ]: b ∈ E[ B(x) ]}

Preliminaries

We have already defined I(x) = {i₁, …, i_n} to be the Interests of the individual x.
We have noted that there are scalars to account for the Weight of Interest for each of x’s interests:
W(x) = {W_j(x): j ∈ I(x)}
At any time there are a range of possible behaviours of x:
B(x = {b₁, …, b_m}
Each possible behavior b of x would result in a different social situation for x, which we call the Outcome of behaviour b of x. We write this as

C(b, x) = c,

• c is the Context of x. It is the set of relevant parameters describing the social context of x. The relevant parameters are only those that feature in the satisfaction function S, and this will be a very restricted set of all possible parameters.
• The same notation can be used unambiguously to refer to the outcomes of a range of x’s possible behaviours; thus:
  

  C(B(x), x) = {C(b₁, x), …, C(b_m, x)} = {c₁, …, c_m}

The different possible outcomes of x’s behaviour are responsible for differential advancement of x’s interests.

• They indicate/consist of changes to the relevant parameters of the satisfaction function described above.
• To record the fact that the satisfaction function is dependent upon the outcomes of particular behaviours of x, we include those outcomes as one of the arguments.

The outcomes of the behaviour b of x, yield the following satisfactions:

{S(C(b, x), j, x): b ∈ B(x), j ∈ I(x)}, or
{S(c, j, x): c ∈ C(B(x), x), j ∈ I(x)}

The perfectly informed actor will attempt to maximize the total satisfaction function by producing the specific behaviour b_out, such that:

TS(C(b_out, x), I(x), x) = max{TS(C(b, x), I(x), x): b ∈ B(x)}
= max{∑_j∈I(x)w_j(x)S(C(b, x), j, x): b ∈ B(x)}

A group is essentially an organization of individuals in pursuit of their common interests.

1. Each individual has an interest
2. That interest is advanced by group action
3. An organization exists to coordinate the actions of members in pursuit of that interest

Interests and Satisfaction

Let I(x) = {i₁, …, i_n} be the Interests of the individual x. These may be financial, cultural, social, psychological, etc.

Let I(G) = ∪_x∈|G| I(x)

• Interests may include ‘needs’ such as food, sex, etc. (Refer to Maslow’s hierarchy for one of many analyses that make the distinction negligible.)
• It is unclear as yet what level of specification is appropriate for sociological analysis. (It is not obvious, for example, whether or not a very specific interest of an individual should be listed as such or subsumed as a part of a more general interest.)

Let S_j(c, j, x) be a Partial Satisfaction Function measuring the satisfaction of the interest j∈I(x), where c is the set of relevant parameters describing the social context of x.

• Each partial satisfaction function, S_j, will be specific to the particular relevant interest, j. We may without ambiguity write the partial function S_j(c, j, x) as S(c, j, x) (and we may also omit both c and x when these are understood.)
• c is the Context. It is possible to see this as the total social context independent of x, but the relevant parameters are only those that feature in the function S, and this will be a very restricted set of all possible parameters.
• The input to a satisfaction function for a financial interest is easily quantified wrt monetary recompense, but other interests have less obvious inputs. What, for example is the proper measure of social interests such as having healthy friendships? And in either case, how do we determine a degree of satisfaction with the inputs. I suspect this will remain a purely notional function. In that case, the problem of specification is only significant if there is in principle no way to make sense of it.
• As a psychological factor of the agents involved, it’s reasonable to assume that there is a maximum degree of satisfaction that can be achieved for any interest. Therefore we can make the range of each S identical to the interval (0, 1).

Let TS(c, I(x), x) = w_i1(x)S(c, i₁, x) + … + w_in(x)S(c, i_n, x) be a measure of the Total Satisfaction of x, where w_ij(x) are weights representing the relative significance to x of the interests involved.

• TS(c, I(x), x) = ∑_j∈I(x)w_j(x)S(c, j, x) is a briefer way to write the satisfaction function.
• Each w_j(x) is the Weight of Interest for the interest j for x.
• For reasons as before, let the range of TS be (0, 1).

Organizations and Roles

Let O(G) = {r₁, …, r_n} ⊂ 2^|G| stand for the Organization of G.

• We call the elements of O(G) the Roles of the organization of G.
• (∀x ∈ |G|)(&exists;r ∈ O(G)) [x ∈ r]. Every member has a role – or, perhaps, if x has a role in the group organization, then x is a member of the group (though this would require a modification of the definition of O(G).)
• (∀x ∈ |G|)(∀r_i, r_j ∈ O(G)) [x ∈ r_i & x ∈ r_j → r_i = r_j]. Each member has just one role.
• There may be a use for a subset of 2^2|G| too. Consider the case where we wish to speak of the management of a company being the upper echelons of the financial, operational, etc. sectors of the company; or the heads of departments being a special organizational set within the company. For the purposes of simplicity, let us for now disregard this possibility.

The effect of group membership is to modify the satisfaction function. Suppose that the group is generally believed to particularly advance the interest j for its members; then j will be considered an Intentional Interest of the group. The assumption will be that:

(∀xx) [S(c|_xx∈|G|, j, x) > S(c|_xx∉|G|, j, x)]

If membership in fact tends to raise the satisfaction function wrt the interest j of group members then j may be considered an Extensional Interest of the group.

• Note that a group may have several intentional and extensional interests.
• Note that these interests have little to do with the supposed function for which the group exists. A corporation manufacturing widgets, considered as a group, does not primarily satisfy the interest in widgets of its members. Their interests lie elsewhere. Similarly for employees of the Department of Social Welfare, or for Members of Parliament, or for the armed forces, or for any number of other defined function groups.

Interests will be differentially advanced for group members depending on their organizational roles.

• We ought not to assume that (∀x,x ∈ r ∈ O(G)) (∀j ∈ I(G)) [S(c, j, x) = S(c, j, y)], which is the claim that all role members have the same satisfaction function, because x,x ∈ r ∈ O(G) are also likely to be in different roles of another group, or are in different other groups entirely. We need to consider the satisfaction due to G or r – the contribution that membership in a particular role in G makes to the satisfaction of x. To claim that for elements in the same role in G the satisfaction due to G is identical we can say:
  (∀x,x ∈ r ∈ O(G)) (∀j ∈ I(G)) [(S(c|_xx∈r, j, x) – S(c|_xx∉r, j, x)) = (S(c|_yx∈r, j, y) – S(c|_xy∉r, j, y))]
• As a matter of convenience, let S(c|_xx∈r, j, x) for r ∈ O(G) be written S(c, j, r)
• In order to minimize complexity, we will assume membership in just one group until it becomes necessary to do otherwise.

Within a group, therefore, we can define different intentional and extensional interests for the roles in the organization.

• The generally held expectation (of whatever reference group) that interest j will be advanced by membership of role r makes j an Intentional Interest of that role.
• If membership in fact tends to raise the satisfaction function wrt the interest j of members of role r, then j may be considered an Intentional Interest of that role.

Seeming public apathy over climate change is often attributed to a deficit in comprehension. The public knows too little science, it is claimed, to understand the evidence or avoid being misled1. Widespread limits on technical reasoning aggravate the problem by forcing citizens to use unreliable cognitive heuristics to assess risk2. We conducted a study to test this account and found no support for it. Members of the public with the highest degrees of science literacy and technical reasoning capacity were not the most concerned about climate change. Rather, they [the most scientifically and technically competent] were the ones among whom cultural polarization was greatest. This result suggests that public divisions over climate change stem not from the public’s incomprehension of science but from a distinctive conflict of interest: between the personal interest individuals have in forming beliefs in line with those held by others with whom they share close ties and the collective one they all share in making use of the best available science to promote common welfare.

Their conclusion is, of course, that those who are more scientifically literate are using their ability to deal with scientific material ‘dishonestly’ to justify positions that are culturally convenient for them to hold.

A long-established body of work examining motivated cognition supports this conjecture. Both to avoid dissonance and to secure their group standing, individuals unconsciously seek out and credit information supportive of “self-defining… values [and] attitudes”, such as the shared world-views featured in the study of cultural cognition. The predictive power of cultural world-views implies that the average member of the public performs these tasks quite proficiently.

Our data, consistent with that observed in other settings, suggest that those with the highest degree of science literacy and numeracy perform such tasks even more discerningly. Fitting information to identity-defining commitments makes demands on all manner of cognition—including both system 1 and system 2 reasoning. For ordinary citizens, the reward for acquiring greater scientific knowledge and more reliable technical-reasoning capacities is a greater facility to discover and use—or explain away—evidence relating to their groups’ positions.

And their recommendation is that climate science needs to be presented to the more technically literate in such a way that it does not culturally alienate them. This needs to be done because

Even if cultural cognition serves the personal interests of individuals, this form of reasoning can have a highly negative impact on collective decision making. What guides individual risk perception, on this account, is not the truth of those beliefs but rather their congruence with individuals’ cultural commitments. As a result, if beliefs about a societal risk such as climate change come to bear meanings congenial to some cultural outlooks but hostile to others, individuals motivated to adopt culturally congruent risk perceptions will fail to converge, or at least fail to converge as rapidly as they should, on scientific information essential to their common interests in health and prosperity. Although it is effectively costless for any individual to form a perception of climate-change risk that is wrong but culturally congenial, it is very harmful to collective welfare for individuals in aggregate to form beliefs this way.

There are a couple of alarming aspects to this:

1. Firstly, the idea that you can simply dismiss the arguments of those who disagree with you as being motivated by bad faith is extremely dangerous to the conduct of rational argument. It may or may not be the case that people tend to reject a certain position when that is against their interests, and to present arguments against it (and vice versa, of course) but it is still necessary to respond to their actual arguments.
2. The authors claim that one party is arguing dishonestly. That party happens to be the party that is rejecting a claim to which they are very much attached. (That is an assumption on my part, but I’d be very surprised if it wasn’t true; and it doesn’t matter for the argumentative point anyway.) They claim to know that this is possible because there is evidence that people will believe things that will support their interests, and notwithstanding the truth of things. But isn’t it equally open to their enemy party to claim that the only reason they believe the way that they do is because it’s in their interests to do so? Since we now both deny an interest in the truth to the other party, we must find other non-rationally persuasive means of making them do what we want. This doesn’t bode well, does it?
3. Have they considered the possibility that climate scientists are in exactly the same position as the rest of us wrt the effect of interests on the things that they will come to believe? This is one of the standard claims of CAGW resisters: all the non-truth-related incentives for climate scientists are on the side of support for one position whatever the truth of the matter might be. If there were in fact no CAGW, how could we trust them to resist those incentives and discover the truth? And if we can’t – because they are subject to the same psychological pressures that are supposed to be distorting our cognition – then what grounds do the authors have for accepting their claims?

The typical definition of a social group includes some combination of the following characteristics or characteristics that are equivalent to these:

1. Two or more people
2. Direct interaction amongst the members
3. Members are aware of their membership
4. Members share an interest or a cause
5. There is an organizational structure

Insofar as membership of the group is voluntary, the motivation for membership is that membership has the result that the interests of the member are advanced. (Note that these interests may be subjective, immaterial, etc. It is the evaluation of the member that counts in this case.)

Define a Group as a related set of individuals: G = <M, R>

1. The Membership of G is denoted |G|. In this case |G| = M = {m₁, …, m_n}.
2. The nature of a group is determined primarily by the nature of the relationship, R.
1. R stands for various parameters – to be clarified later – that determine the relationships that the group may enter into with other groups, and how the group may act in those relationships, how members interact within and without the group, and so on.
2. R is what distinguishes a group from just a collection of interacting individuals.

Group membership

The group relationship R has to be able to account for at least these characteristics of group membership

1. An individual may be a member of several groups.
• A person may be a male, Christian, labourer, homosexual, …
2. Some groups are mutually exclusive wrt membership. A collection of such groups D = {G₁, …, G_n}, that minimally covers another group, H, is a Division of that group and we write D|H
3. Thus, {G₁, …, G_n} is a division of H iff:
1. |G₁|, …, |G_n| are relatively disjoint, and
2. |H| ⊂ ∪_{i=1, …n} |G_i|, and
3. For i = 1, …, n, |H| ∩ |G_i| ≠ ∅.
• A group defined by religious affiliation might be mutually exclusive, since if one is a Christian one can’t be a Buddhist or a Hindu, and if one is a Hindu then one is not B or C, and, of course, if B then not C or H. Similarly for socio-economically defined groups, or educational, or residential groups, etc.
• An example of a division might be the confessional allegiances in a nation. If the citizens are one only of Buddhist or Hindu – as in Sri Lanka – then {Buddhists, Hindus} is a division of Sri Lanka. Since there are no Christians (we’ll pretend), {Buddhists, Hindus, Christians} is not a division of Sri Lanka, though it contains one.

The group memberships of an individual define its Identity wrt society. Thus for the individual x who is a member of groups G₁, …, G_n, the identity of x is that collection, and we write:

• ID(x) = {G₁, …, G_n} iff x ∈ |G₁| ∩ … ∩ |G_n|.

• The identity wrt society of the person mentioned above is male, Christian, labourer, homosexual, …

Where D is a division of H and x ∈ |H|, ID(x|D) is the member of D to which x belongs. Read it as the identity of x wrt D. Thus: For D = {G₁, …, G_n}, D|H, ID(x|D) = G_n iff x ∈ |G_n|

• Suppose we have H = Sri Lanka, D = {Buddhists, Hindus}, then it might be that ID(Bob|D) = Buddhist (if Bob is a Buddhist.)

Where D₁, …, D_n are divisions of H, we can call A = {D₁, …, D_n} an Analysis of H.

• An analysis of Sri Lanka might include divisions according to religious and racial and linguistic criteria. Thus we might have H = Sri Lanka, D₁ = {Buddhists, Hindus}, D₂ = {monolingual Tamil, monolingual Singhala, bilingual}, D₃ = {Tamil, Singhala}. Then A = {D₁, D₂, D₃} would be an analysis of Sri Lanka.

Let A = {D₁, …, D_n} be an analysis of H. Then the identity of x ∈ |H| wrt that analysis is:

• ID(x|A) = {ID(x|D₁), …, ID(x|D_n)}

In the example above we might have ID(Bob|A) = {Hindu, monolingual Tamil, Tamil}