### CATEGORICAL RELATIONSHIPS

Given any two categories, there exists at least three and no more than four possible categorical relationships between them:

separate |
intersecting |
hierarchical |
---|---|---|

No members of one are members of the other. | Some members of each are contained, and not contained in the other. | One is contained in the other. |

If you want to count two identical categories as two categories, then "identical" is a fourth possible categorical relationship. You can also treat identical categories as one category. It usually depends on whether they are definitionally identical or coincidentally identical.

**definitionally identical:**(all bachelors) (all unmarried men)

These categories are always identical.

**coincidentally identical:**(all girls in the room) (all blonds in the room)

These categories are identical relative to a particular place at a particular time, but not necessarily always identical.

There are at least three kinds of hierarchical relationships. The contained set is either a subset, a part, or a subordinate of the container set. e.g.

Engines and tires are parts of cars.

Lieutenants and sergeants are subordinates of captains.

Sets may be categorized in at least three ways: e.g.

**number**of concepts in them

A

**null**set contains no concepts.

**finite**set contains at least one concept, and may include any number of concepts short of infinity.

**unitary**set is a subset of finite sets containing only one concept.

**infinite**set contains an infinite number of concepts.

**criteria**their members have in common

a. by the

**number of criteria**their members have in common

Composite sets have more than one criterion in common

**type of criteria**their members have in common

Simple sets have one type of criteria in common

Complex sets have more than one type of criteria in common

**clarity**of their boundaries

**Distinct**sets have clear boundaries

**Vague**sets have at least one unclear boundary.

Concepts may be categorized by boundary.

**Distinct** concepts have clear boundaries.

**Vague** concepts have at least one unclear boundary.

**Boundless**concepts have neither boundaries nor places for boundaries to be.

e.g. zero, emptiness, vacuum, nothing, absence, point

**Bounded**concepts have at least one boundary, and are therefore either totally finite, partly finite, or infinite.

e.g. Space is infinite in three dimensions (assuming either an infinite universe, or an infinite number of universes in one space continuum)

A plane is infinite in only two dimensions.

A line is infinite in only one dimension, but two directions.

A direction is finite on one end and infinite on the other.

e.g. everything, anything, existence

Paradoxically however, infinity becomes a limit in itself.