Limitation

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One Principle to Rule Them All

Defining Limitation

Limitation is a relation between a portion and an extent.

The portion and extent can be either:

an occasion, or
the absence of an occasion.

Visualizing Limitation

We may visualize limitation as one shape being bounded by another.

If a jigsaw puzzle piece is a portion and the remainder of the puzzle is an extent then the piece's presence in the puzzle is a limitation.

The portion cannot exceed the extent, but the extent need not be any more than the portion (.ie a puzzle of one piece)

Symbolizing Limitation

We use the symbol "⊇" for "superset of or equal to" to represent limitation.
The extent comes before this symbol and portion comes after it.

We are not speaking of sets, but we need a symbol for limitation, and this one comes close enough to our intuition of limitation as an act of inclusion or enclosure.

Formalizing Limitation

An Algebra of Limitation

An ADEPT LION occasion is a synthesis of its inputs. These inputs are the occasions at the other end of its channels. The synthesis is expressed as limitations. These limitations are interpreted in a process much like an algebraic calculation in mathematics. But rather than numbers, there are occasions of pattern. And rather than having multiple operators (addition, subtraction, division and multiplication), there is only limitation. And just as there are laws in mathematics to govern the use of operations (.ie associative, commutative), there are laws in pattern language to govern the operations of limitation.

Legend of Symbols

A Limitation: ⊇

An Occasion: a Greek letter like "α"

Absence of an occasion: ∅

The Limitation Laws

I. Self-Limitation

An occasion that limits itself entails itself

α ⊇ α ⊢ α

"alpha"

II. Non-Limitation

I. Self-Limitation

No occasion, being both limited by an initial occasion, and limiting another subsequent occasion, entails that the complement of the initial occasion limits a selection of the second occasion.

α ⊇ ∅ ⊇ β ⊢ ¬α ⊇ [β]

"all that is not alpha limits this beta"

III. Complement

I. Self-Limitation

III. Complement

An occasion that limits no occasion entails its complement.

α ⊇ ∅ ⊢ ¬α

"all that is not alpha"

IV. Selection

I. Self-Limitation

III. Complement

An occasion that is limited by no occasion entails the selection of itself.

∅ ⊇ α ⊢ [α]

"this alpha"

V. Approximate Mediation

VIII. Non-Exhaustive Division

V. Approximate Mediation

An occasion, which is limited by one occasion and limits another entails that the central occasion is an approximation of the limitation that exists between the other two occasions.

α ⊇ β ⊇ γ ⊢ β ≈ α ⊇ γ

"beta approximates alpha limiting gamma"

VI. Approximate Union

VIII. Non-Exhaustive Division

V. Approximate Mediation

An occasion, simultaneously limiting two occasions entails that the central occasion is an approximation of the union of the other two occasions.

α ⊆ β ⊇ γ ⊢ β ≈ α ∪ γ

⊢ β ≈ γ ∪ α

"beta approximates the union of alpha and gamma"; "beta approximates the union of gamma and alpha"

VII. Approximate Overlap

VIII. Non-Exhaustive Division

An occasion, simultaneously limited by two occasions entails that the central occasion is an approximation of the overlap of the other two occasions.

α ⊇ β ⊆ γ ⊢ β ≈ α ∩ γ

⊢ β ≈ γ ∩ α

"beta approximates the overlap of alpha and gamma"; "beta approximates the overlap of gamma and alpha"

VIII. Non-Exhaustive Division

No occasion, being simultaneously limited by two occasions entails the complement of either of these occasions properly limiting the other.

α ⊇ ∅ ⊆ β ⊢ α ⊂ ¬β

⊢ ¬α ⊃ β

"alpha is properly limited by all that is not beta"; "all that is not alpha properly limits beta"

IX. Exhaustive Division

No occasion, simultaneously limiting two occasions entails the complement of either of these occasions approximates the other.

α ⊆ ∅ ⊇ β ⊢ α ≈ ¬β

⊢ ¬α ≈ β

"alpha approximates all that is not beta" ; "all that is not alpha approximates beta"

X. Counter-Limitation

IX. Exhaustive Division

An occasion, that limits some other occasion entails that the other occasion is limited by the initial occasion.

α ⊇ β ⊢ α ⊇ β ⊆ α

⊢ β ⊆ α ⊇ β

⊢ β ⊆ α

"alpha limits beta limited by alpha"; "beta limited by alpha limits beta"; "beta limited by alpha"

Theory of Pattern

Limitation

One Principle to Rule Them All

Defining Limitation

Visualizing Limitation

Symbolizing Limitation

Formalizing Limitation

An Algebra of Limitation

Legend of Symbols

The Limitation Laws

I. Self-Limitation

I. Self-Limitation

I. Self-Limitation

II. Non-Limitation

I. Self-Limitation

I. Self-Limitation

III. Complement

I. Self-Limitation

III. Complement

IV. Selection

I. Self-Limitation

III. Complement

V. Approximate Mediation

VIII. Non-Exhaustive Division

V. Approximate Mediation

VI. Approximate Union

VIII. Non-Exhaustive Division

V. Approximate Mediation

VII. Approximate Overlap

VIII. Non-Exhaustive Division

VIII. Non-Exhaustive Division

VIII. Non-Exhaustive Division

VIII. Non-Exhaustive Division

VIII. Non-Exhaustive Division

IX. Exhaustive Division

IX. Exhaustive Division

IX. Exhaustive Division

X. Counter-Limitation

IX. Exhaustive Division

IX. Exhaustive Division

The Laws of Limitation and Pattern Interpretation

Sequences

Considerations

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