structure

Korzybski proposed the term 'structure' as an undefined multiordinal term.

The common use of the term is to describe something that has form or parts which can be distinguished from one another.  It is also used to refer to something with solid substance.

I offer a formal definition for the term which includes both its multiordinal and undefined character.  I do this by distinguishing between the two common usages so as to designate one primary or primitive and the other as general or complex.

Backus-Naur form:

<complex structure> ::= <structure> <structure>

<structure> ::= <simple structure> | <structure>

where "simple structure" is undefined.

Recursive

A structure is (1) a simple primitive undifferentiated (undefined) structure (part, aspect, characteristic, etc.), or (2) it is composed of subordinate structures related and or ordered in some way.

Conditionally, we do not consider an "empty" structure to be a proper substructure, and, by definition of 'proper', neither is the entire structure.

A relation on a set is a subset of the set of all subsets of the set.

If the set is {a,b,c}, then {{a},{a,b}, {a,c} {a,b,c}} is an example relation.  This one could be called all parts that include a.  If abc represented a triangle, the aforementioned set might represent the vertex a, side ab, side ac, and the triangle abc itself..  {{a,b},{a,c}} might represent the sides containing vertex a.  The definition is VERY GENERAL, and these examples are extremely simple.  We are interested in when the elements of the set are "structures".  A relation on a structure is a subset of  the set of all sub-structures.

Korzybski proposed the term 'relation' also to be considered undefined.  The definition proposed is a formal definition that leaves the interpretation of the term completely determined by context, so it is, in this way, "undefined", it has no "proper" interpretation.

Summary:

A structure is either simple or complex.

• A simple structure has no subordinate parts, aspects, characteristics, etc.
• A complex structure has at least two distinct subordinate parts, aspects, characteristics, etc.
• Any differentiation of a complex structure is into subordinate structures.
• There may be multiple ways of differentiating a structure into subordinate structures.
• Differentiation of a structure (or not) may be a function of the level of observation.
• A differentiation of a structure is always at a lower level than an undifferentiation.