By Leon James (c)19xx

                                                       Notes on the Geometry of Understanding  

    Suppose we wished to formalize a psychotherapeutic register along lines similar to geometry. How would we proceed and would its yield be more than an entertaining excercise? Consider the following attempt modeled after a college geometry text for teachers-in-training (Ringerberg, 1967, Chapter 7ff.): In MODERN ETHNOSEMANTICS, CULTURE is the global reference to the field of study, equivalent to the term SPACE in geometry (Euclidean). CULTURE is a conceptual space., In ethnosemantics, CULTURE is a set of' reciprocally ratifiable recognitions ("the new three R's"), every episode is a sequence of exchanges involving reciprocaly ratifiable recognitions, every group is a subset of reciprocally ratifiable recognitions, every setting is a set of exchanges. We make no attempt here to define CULTURE, GROUP, RECIPROCAL RATIFIABLE RECOGNITIONS, EXCHANGES, EPISODES, and SETTINGS., We officially declare them (and their, derivatives) as undefined terms.' We identify those properties of these, terms which we wish to accept officially without proof. We express these properties in statements called postulates., These properties, and their derivatives, are the properties we use in proving theorems. Theorems are empirical claims about how things actually are in culture, viz. laws of social interaction.

Definition: Culture is the set of all reciprocally ratifiable recognitions, called moves.

Definition: Reciprocally ratifiable recognitions (RRR) is the set of all meaningful transactions or interactions that can-take place in' cultural settings.

Postulate 1. A meaningful transaction contains at least two reciprocally ratifiable recognitions, called the initiating move and the ratification move, respectively.

Postulate 2. Every episode consists of a sequence of transactions and contains at least two moves.

Postulate 3. If (a) and (b) are two distinct moves, such that (b) follows (a) in an alternating sequence from one participant to another, then there is a particular functional relationship implied to the effect that (b) is a reply to (a) or a remedy to it.
     Space = setting culture
     Point = move
     Line = transaction

Plane = episode/relationship

                                                                 Postulates

1. Culture is the set of all moves.

2. Cultural event contains at least two distinct moves.

3. Every transaction is a set of moves and contains at least two distinct moves.

4. If M and N are two distinct moves, they can be linked to form one and only one transactional pair.

5. Two moves form a transactional pair if and only if they are reconstituted as the elements of a transaction, namely as move and reply move, respectively.

6. Alternating sequences of moves and transactions form an episode if they have-been marked ritually as a time-bound set (e.g. beginning, early, later, ending)

7. Episodes form relationship if they are cumulated as a joint record.

8. No transaction contains all moves possible in a culture.

9. Every episode is a set of moves and contains at least two pairs, one ritual which is obligatory and marks the beginning and ending, the other informational, which is obligatory as either "null" or "positive" (silence vs. mentioning something; not-displaying vs. displaying)

10. If AB represents the joint cumulative record of episodes between A and B, there exists a relationship between them.

11. No episode contains all moves or transactions.

11a. No relationship contains all possible transactions.

12. If two moves form a transactional pair, then all sub-moves within - each also belongs to the same transaction.

13.

Theorem 1

All relationships require an exchange of moves.

Proof:   (i) Relationship (implies)=====> joint record (P10)

             (ii) joint record ======> a history of episodes (self-evident) i.e. at least one episode.

             (iii) episode = set of moves (P9)

             Relationship requires at least one exchange of moves.

Theorem 2

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