Algebraic Model

Pattern Interpretation in the First Consideration

Predication

Predication is curiously, the only concept upon which all of pattern logic is built. This is a surprising assertion given that pattern logic aspires to express not only logical systems and a new number theory but also the grammatical operations of natural languages. The aim of the following material is merely to establish the foundation for a demonstration of how a fixed system of patterns composed solely from the concept of predication can deliver on such a claim. We will explain predication and the laws which it is subject to.

An act of Predication can typically be identified through some form of the verb ‘to be’. To say ‘today is Tuesday’ is to predicate ‘Tuesday’ of ‘today’. To say ‘the sky is blue’ is to predicate the color blue of the sky. So, predication is found literally everywhere in language – it is the axis upon which sentence diagraming rests when we identify the subject of a sentence and model its relation to the object of that sentence through the verb. This is true even if the verb is not ‘to be’ because it can be made to be. For instance, ‘the cow jumped over the moon’ can be restated as the subject ‘cow’ of which we predicate ‘is’ the object ‘that which jumped over the moon’. These two ways of stating the core content of the sentence is why the designation of ‘the predicate’ is often given to the compound notion of the verb and object of a sentence.

In logic, the verb and object are typically teased apart to establish a clear delineation between the subject, the verb, and the object. Specifically, the predication verb is some form of the linking verb to be, which is also sometimes called a copulant.

Some Preliminary Points on the Laws of Pattern Logic

Constrained Coherence

Interpretation of any pattern

The rules of ADEPT LION provide enforceable constraints for what comprises a coherent pattern.

Interpretation of any pattern

Any pattern that follows the rules of ADEPT LION will be amenable to the deriving of an interpretation of its meaning through the laws of pattern logic.

Accumulation of Meaning

Interpretation of any pattern

Accumulation of Meaning

The interpreted meaning of a given pattern is the accumulation of the meanings of its inputs, including when an input is self-referencing.

Predication Alone

Composition before Equality

Accumulation of Meaning

While new laws will build cumulatively upon preceding laws, every law will have an expression that can be grounded in the meaning of predication alone.

Infinite Patterns

Composition before Equality

The count of laws in pattern logic may be fixed but if this is so, the laws will support a theoretically infinite number of patterns and interpretations.

Composition before Equality

Laws are presented as a statement of a pattern’s composition with a possible restatement as an equality.

Predication as the basis for the Laws of Pattern Logic

Predication & Counter-Predication

As a convenient convention, Occasions are represented by lowercase Greek letters.

Predication is represented by the mathematical symbol of ‘⊇’, which has the Unicode assigned definition of “SUPERSET OF OR EQUAL TO”. In pattern logic it can be rendered as "IS-SAID-OF", “predicates” or “predicating”. The “predicating” form will imply the grouping of two terms as in the use of parentheses in a mathematical formula and the “predicates” form will either be used in a simple binary relation or in higher order groupings such as the relationship between two parenthetical groupings in a mathematical formula (see the rendering of [x] below for an example of this usage).

For ease of conceptual or semantic rendering, a predication occasion can always be alternatively stated as an occasion of counter-predication. Counter-predication is given the Unicode symbol ‘⊆’ with definition “SUBSET OF OR EQUAL TO”. Counter-predication is simply an alternative way of expressing the same occasion in the opposite direction semantically. Counter-predication can be rendered "IS-PRESENT-IN", “counter-predicates” or “counter-predicating”.

An example of predication would be “mammal is (said of) dog”. An example of counter-predication, in which we say the same thing but in the opposite direction would be “dog is (present in) mammal”. Through the lens of set theory, predication would be rendered as “mammal is a superset of or equal to dog” and counter-predication would be rendered “dog is a subset of or equal to mammal”.

If we were to simply state “mammal is dog and dog is mammal”, then a listener would intuitively assign the roles of set and member to these terms. In the predication and counter-predication of pattern logic we have an explicit assignment of set and member to two different, but related senses of the word “is”.

Note: Counter-predication has been chosen as the default means of expressing a logical premise in the pattern logic syllogistic. So when we express a quantified premise like “all dogs are mammals”, the “are” is actually a counter-predication syntax for “all dogs IS-PRESENT-IN (an existent) mammal”. This can be succinctly expressed with quantifier and counter-predication symbols as “ꓯ dog ⊆ ꓱ mammal”.

The following laws will mostly be stated using the predication (⊇) symbol, but we introduce counter-predication (⊆) here simply to point out that the laws might also have been composed using counter-predication instead of predication. Pattern logic could equally have supported either starting point.

Pattern Logic and Set Theory

The ⊇ symbol and its corresponding definition from set theory above suffices for the purpose of pattern logic to represent predication, but should not be construed as a claim of complete correspondence between pattern logic and set theory. In many cases, this would give the wrong impression because ADEPT occasions are not limited to the concepts of sets and their membership although pattern logic will become expressive of these concepts as certain patterns become narrower in their interpretations.

Visualizations of pattern logic such as Frame-Region Diagrams leverage the idea of set membership much like how Venn Diagrams and Euler Diagrams can be used to represent set relations. The idea of a superset that includes or is equal to its expressed membership is still a useful mental construct when it comes to understanding predication conceptually. The including set would correspond to the extrinsic aspect of the use channel and the included member of a set to the intrinsic aspect of the thing channel. In this case, each act of set membership could be understood as an act of predication or counter-predication. The necessary caveat is that the expressive capabilities of predication will not be limited to this explicit means of concept construction as found in set theory.

Composition, Restatement and Equality

On Composition and Pattern

The composition of an occasion of predication is what follows the colon ‘:’ and can be rendered as ‘MEANS’. For instance the occasion beta may be composed of the predication between alpha (thing channel input) and gamma (use channel input). Beta's pattern would have a rivulet diagram as shown to the right. This can be symbolically stated as the following composition which is taken directly from the pattern:

α : γ ⊇ β

“alpha MEANS gamma predicates beta”

A restatement of a pattern’s composition is symbolized by “≡“ and can be rendered as “MAKES THE SAME STATEMENT AS”. For instance, the occasion alpha above can be restated as an equality like this:

α : γ ⊇ β ≡ α = γ ⊇ β

“alpha means gamma predicates beta” MAKES THE SAME STATEMENT AS “alpha equals gamma predicating beta”

This example is incidentally, the first law of pattern logic and is called “Interpretation”. Laws are assigned a number and typically written in square brackets such as “Law of Interpretation [i]”. In fact, all of the laws take the form of a restatement and the serial application of these laws is the act of interpreting pattern.

Given these preliminary points and introduction to the symbols, we are ready to introduce the following initial set of 16 laws for pattern logic.

Laws For Pattern Logic

Law of Interpretation

α : β ⊇ γ ≡ α = β ⊇ γ

“Alpha means beta predicates gamma” makes the same statement as "alpha equals beta predicating gamma”

Law of Commutation

α : β ⊇ γ ≡ β = α ⊇ γ

“Alpha means beta predicates gamma” makes the same statement as “beta equals alpha predicating gamma”

Law of Subsumption

iii

α : α ⊇ β ≡ α ⊇ β

“Alpha means alpha predicates beta" makes the same statement as "alpha predicates beta"

Law of Equivalence

α : β ⊇ β ≡ α = β

"Alpha means beta predicates beta” makes the same statement as “alpha equals beta”

Law of Non-Subsumption

α : _ ⊇ β ≡ α ⊉ β

"Alpha means no occasion predicates beta" makes the same statement as "alpha non-predicates beta"

Law of Disjoint

α : β ⊇ _ ≡ α ⊈ β ≡ β ⊈ α ≡ α φ β

"Alpha means Beta predicates no occasion" makes the same statement as "alpha non-counter-predicates beta" makes the same statement as “beta non-counter-predicates alpha” makes the same statement as "alpha and beta are disjoint"

Law of Intersection

vii

α : β ⊇ γ ≡ γ = α ∩ β ≡ γ = β ∩ α

“Alpha means beta predicates gamma” makes the same statement as “gamma equals alpha intersecting beta” makes the same statement as "gamma equals beta intersecting alpha"

Law of Extraction

viii

α : β ⊇ γ ≡ α = β ∩ (β ⊇ γ)

"Alpha means beta intersects gamma" makes the same statement as "alpha equals beta intersects beta predicating gamma"

Law of Injection

α : β ⊇ γ ≡ α = (β ∩ γ) ⊇ γ

"Alpha means beta predicates gamma" makes the same statement as "alpha equals beta intersecting gamma predicates gamma”

Law of Underlap

α : β ⊇ (γ ⊇ δ) ≡ α = (β ∩ γ) ⊇ δ

"Alpha means beta predicates gamma predicating delta" makes the same statement as "alpha equals beta intersecting gamma predicates delta". Therefore, the intersect of beta and gamma underlaps delta.

Law of Overlap

α : (β ⊇ γ) ⊇ δ ≡ α = γ ⊇ (β ∩ δ)

"Alpha means beta predicating gamma predicates delta" makes the same statement as "alpha equals gamma predicates beta intersecting delta". Therefore, the intersect of beta and delta overlaps gamma.

Law of Compound Predication

xii

α : (β ⊇ γ) ⊇ (δ ⊇ ε) ≡ α = (γ ∩ δ) ⊇ (β ∩ ε)

"Alpha means beta predicating gamma predicates delta predicating epsilon" makes the same statement as "alpha equals gamma intersecting delta predicates beta intersecting epsilon"

Law of Reflexivity of Predication

xiii

α ≡ α ⊇ α

"Alpha" makes the same statement as "alpha predicates alpha".

Law of Reflexivity of Intersection

xiv

α ≡ α ∩ α

"Alpha" makes the same statement as "alpha intersects alpha".

Law of Expansion

α : β ⊇ γ ≡ α = β ⊇ (β ⊇ γ)

"Alpha means beta predicates gamma" makes the same statement as "alpha equals beta predicates beta predicating gamma".

Law of Contraction

xvi

α : β ⊇ γ ≡ α = (β ⊇ γ) ⊇ γ

"Alpha means beta predicates gamma" makes the same statement as "alpha equals beta predicating gamma predicates gamma"

These laws have been presented entirely within a novel algebra but it will be useful and reassuring to many to know that the laws follow a syntax that corresponds very closely to the mathematical operations of division and multiplication. For instance, the Law of Compound Predication [xii] expressed like this: α : (β ⊇ γ) ⊇ (δ ⊇ ε) ≡ α = (γ ∩ δ) ⊇ (β ∩ ε) …will hopefully seem less foreign when compared to the more familiar rules of division and multiplication found in a comparable bit of arithmetic like this: α = (ε/ δ) / (γ/ β) = (ε ∙ β) / (δ ∙ γ)

Expressing Logic as Pattern

Algebraic Model For Pattern

Pattern Interpretation in the First Consideration

Predication

Some Preliminary Points on the Laws of Pattern Logic

Constrained Coherence

Interpretation of any pattern

Interpretation of any pattern

Interpretation of any pattern

Interpretation of any pattern

Interpretation of any pattern

Accumulation of Meaning

Interpretation of any pattern

Accumulation of Meaning

Predication Alone

Composition before Equality

Accumulation of Meaning

Infinite Patterns

Composition before Equality

Composition before Equality

Composition before Equality

Composition before Equality

Composition before Equality

Predication as the basis for the Laws of Pattern Logic

Predication & Counter-Predication

Pattern Logic and Set Theory

Composition, Restatement and Equality

On Composition and Pattern

Laws For Pattern Logic

Law of Interpretation

Law of Commutation

Law of Subsumption

Law of Equivalence

Law of Non-Subsumption

Law of Disjoint

Law of Intersection

Law of Extraction

Law of Injection

Law of Underlap

Law of Overlap

Law of Compound Predication

Law of Reflexivity of Predication

Law of Reflexivity of Intersection

Law of Expansion

Law of Contraction

Pattern Interpretations

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