Elements of Diamond Set Theory

Some more parts of the mosaic towards semiotics, logic, arithmetic and category theory

Rudolf Kaehr Dr.@

ThinkArt Lab Glasgow

 

Abstract

Further elements are sketched towards an interplay of polycontextural logic, semiotics, arithmetic and set theory. Basics for junctional and transjunctional quantification in polycontextural logic are presented. Hints to metamorphic changes between sets, classes and conglomerates in pluri-verses are given.

1.  Diamond set theory

1.1.  Sets, universes, conglomerations

It is said that category theory is a departure from set theory, other are more radical and insists that category theory has nothing to do with set theory at all.
From a foundational point of view, Herrlich makes it clear that a proper mathematical formalization of categories needs different sorts of collections of different generality. He distinguishes sets, classes and conglomerates as the collections of a universe appropriate to deal with categories.
Nevertheless, there is no special conflict necessary between set theory and category theory. Both are based on different, even complementary, thematizations of formal thinking. And as such, both are using mutually methods from each other. And both are, logically and semiotically, if blindness is not dominating, based on common grounds.

Collections of the universe U = [sets, classes, conglomerates].

The objects of category theory belong to these collections. Obviously, categorical objects are not simply sets but, e.g., categories of categories, hence surpassing all reasonable, i.e., contradiction-free notions of set theory. Hence, "One universe as a foundation of category theory", (Mac Lane, 1969)

Diamond theory is in no way less general than category theory, but the objects of diamonds are not only collections of different degrees of abstractions, but are bi-objects from their very beginning. Bi-objects are complementary objects constructed as an interplay between acceptional and rejectional aspects of diamond theory.

Hence the objects of diamonds are not simply belonging to the universe U of conglomerates with its classes and sets, but to the 2-verse (di-verse) as a complementarity of the universe of acceptional and the "universe" of rejectional objects.

Category theory happens in a universe, polycontexturality in a pluri-verse and diamond theory in a di-verse 2-U of complementarity.
Thus, 2-U = [collections || collections].
Hence, 2-U = [(set||set), (class||class), (conglomerate||conglomerate)].

A di-verse conception of collections opens up the possibility of metamorphic chiasms between their constituents [set, class, conglomerate].

This happens in a similar way like in polycontexturally disseminated categories. That is, a set in one contexture can be seen as a class in another contexture, etc. This happens on the base of as-abstractions. In category theory “a set is a set, a class is a class and a conglomerate is a conglomerate”; and nothing else happens. The hierarchy is strict and well defined. The notions, set, class, conglomerate, are defined by the is-abstraction.

This is different for polycontextural systems but also for diamond theory. For both, collections are still well defined and placed in their hierarchy. But because of the multitude of universes, interactions are possible between different kinds of collections. These interactions are strictly defined, too. They are ruled by the mechanism of chiastic metamorphosis.

Obviously, to describe the rules of sets, classes and conglomerates in di-verses we need some knowledge from diamond theory, which is based then just on such rules. That is, the whole idea of a di-verse is based on conceptions of diamond theory.
In diamond theory, conglomerates are not covering the situations of bi-objects. Bi-objects are polycontextural, thus they are members of disseminated conglomerates.

Contexture(Conglomerate(Class(Set)))

On the base of other conceptualizations of the diamond way of thematization, a transition from 2-verses to n-verses is not excluded. This should not be confused with the general multi-verses of polycontextural systems.

1.2.  Diamond strategies for bi-objects

Bi-objects are strictly divided into a saltatorical and a categorical part. With the interplay and interactivity between categories and saltatories, ruled by the bridging conditions and operations, a new type of object emerges: bi-objects with mixed parts. Hence, diamonds are involved not simply in bi-objects but in bridges, too.
Bridges are composed by difference operation into a combination of categorical and saltatorical parts. In this sense, they are the both-at-once aspect of diamond bi-objects. A change of perspective in favor to the bridging operation as such, abstracting from its bi-objects, the neither-nor structure of bi-objects might be constructed.
Hence, we have to distinguish 4 aspects of diamonds: categorical, saltatorical, interplay (bridging as a mix) and interactionality (bridging as such).

1.3.  Elements of a diamond theory of conglomerates

Both approaches, the polycontextural approach to logic and diamond theory as well the approach of mathematical semiotics, is first and mainly considered of abstract cognitive and volitive structures and transformations. Propositions about elements of the semiotic, polycontextural and diamond theoretical domains are not yet proposed in a formal and formalized way. Like classical propositional logic is enhanced by a theory of quantification, which allows to state statements about elements, properties and quantifications with all (∀), some (∃), exactly one (∃!), the same shall be introduced for transclassical approaches to formal thinking.
In classical logic, the logic of predicates, i.e. first-order logic, is defined on the base of a single, uniform or structured, universe of individual elements or objects, transclassical logic has to be defined in concert with a plurality of domains, called pluri-verse. Diamond theory of pluriverses is reflecting the otherness of any thematized universe of a pluriversal “set” or “domain” theory.
Classical logic and set theory is restricted to structure its single universe into sorts, sub-domains, layers, levels etc. without touching the strict hierarchy between the single universe and its parts or subsystems.
Today, set theory is enlarged, for the reasons of category theoretical aims, to a theory of universes, conglomerates, large and small sets, i.e classes and sets.

Laws for sets
Laws for classes
Laws for conglomerates
Laws for universes
Universes are founded in uniqueness
Laws for chiasms between universes.

Metamorphic interchanges between universes, conglomerates, classes and sets in a polycontextural framework are the fundamental mechanisms of change.
Changes might be iterative or accretive.
Iterative changes happens in a stable framework, accretive changes are augmenting the complexity of the framework.

This is not the place to enter into the intriguing world of mathematical foundations, its strategies of avoiding paradoxes and extending the fields of mathematical reasoning.

One small hint should be recalled. There is no primary need to avoid paradoxes in polycontextural theories because they are accessible to a paradox-free implementation based on the chiasm between elements and predicates, or sets and elements. It was sketched in (Kaehr 1978), that for each contexture, a contexture specific local paradox might be constructed and that the system as such is not involved, globally, into the well known unavoidable  paradoxes of self-referentiality.

 <br />    Chiasm  _ iter(Pluriverse^(m)) =  Pluriverse^(m)  ... sm  _ acc(Pluriverse^(m)) =  Pluriverse^(m + n)    <br /> <br />

[Graphics:HTMLFiles/Elements_2.gif]

Universe = Conglomerates [Classes [Sets [Elements]]] after Herrlich . <br />

Universal Category Theory = [               ]                                           Univer ...       Class                                 [Set      ]                                  {Element}

<br /> [                                                     ] <br /> [                        ... Element}                                     {Element}                                   {Element}

[                                                                   ]                          ... ass       [Set      ]                   [Set      ]        {Element}                     {Element}

<br /> Chiasm (Set, Element) = (           1                                         1) <br /> ... A0;                                    -                                               Set

1.3.1.  Ontology

General Set Theory (Boolos)
"
GST features a single primitive ontological notion, that of set, and a single ontological assumption, namely that all individuals in the universe of discourse (i.e., all mathematical objects) are sets. There is a single primitive binary relation, set membership; that set a is a member of set b is written a∈b (usually read "a is an element of b")." WiKi


1. Axion of Extensionality:
The sets x and y are the same set if they have the same members.
   ∀x ∀y [∀ z [z∈x <--> z∈ y] --> x = y].

2. Axiom Schema of Specification
Separation or Restricted Comprehension): If z is a set and ϕ is any property which may be satisfied by all, some, or no elements of z, then there exists a subset y of z containing just those elements x in z which satisfy the property ϕ.
  ∀z ∃y ∀x [x ∈ y <--> ( x ∈z typeset structure ϕ(x))].

3. Axiom of Adjunction
If x and y are sets, then there exists a set w, the adjunction of x and y, whose members are just y and the members of x.[1]
    ∀x ∀y ∃w ∀z [ z ∈ w<-> [z∈ x typeset structure z=y]].

General Diamond Set Theory
GDSTtypeset structure features for each contexture a single primitive contextural notion, that of bi-set, and for each a single contextural assumption, namely that all bi-individuals in the pluri-verse of thematizations (i.e., all mathematical objects) are bi-sets of complexity m.

There is locally for each contexture of a polycontexturality a single primitive binary relation, bi-set membership; that set (A,a) is a member of set (B,b) is written (A,a)∈∈(B,b) (usually read "a is an element of b").

There is globally for each constellation of contextures and uni-verses a chiastic exchange relation between contextures and uni-verses of the pluri-verse.

The logic of GDSTtypeset structure is the polycontextural diamond logic PolyLogictypeset structure

GST is derived from GDST by reducing pluri-verses to uni-verse, bi-sets to sets and by omitting chiasm.

(GDST               GST             )   pluri - verse      universe   bi - sets          sets  ... membership   diamond relation   binary relation   polylogic          logic   chiasm             --

Between the concept of "Urelement" and the concept of "Contexture" a duality holds.
A Urelement is an element which might be a member of a set but it doesn't contain itself any members.
A contexture isn’t a member of a set but contains all sets of itself.
In a chiastic scenario, a contexture might change its functionality into the functionality of an element of another contexture.  

2.  Quantification in polycontextural logics

First-order logic quantification is distributed over different contextural domains of polycontextural logic.

As for FOL, quantification in polylogics requires quantifiers which are applied to predicates and functions and their variables. And substitution is required too.

Quantification in polylogics is naturally realized by universal and existential quantification introduced in analogy to FOL for each contexture. Because of polycontexturality, additional to the separated actions of contextural quantifiers, quantifier for interactions between different contextures have to be introduced, i.e. transjunctional quantifiers.
For a 3-contextural logic Logtypeset structure  universal and existential quantifiers are distributed over 3 places. In general, for m there are typeset structureplaces to consider.
Hence, the patterns are in correspondence to the distribution of propositional conjunctions and disjunctions.

Typical polycontextural quantifiers are the transjunctional quantifiers Q and G.
Similar to the duality of logical quantifiers, ∀ and ∃, the transjunctional quantifiers Q and G are interchangeable.

Quantification is in analogy to propositional versions . That is, <br /> non  _ 3 ( QQ ... r /> non _ 3 (p ≀ ≀ ≀ non _ 1 q) = non _ 2 (p O O O q) .

non _ 3 (non _ 3  p ≀ ≀ ≀ non _ 3 p) = p ≀ ≀ ≀ q non _ 3 (non _ 3  p O O O non _ 3 p) = p O O O q .

<br /> Junctional quantifiers <br /> ∀ ∀ ∀, ∀ ∀ ∃, ∀ ... ∃, G ∃ G, <br /> ∃ GG, G ∃ ∃, ∃ G ∃, ∃ ∃ G .


Models

Q∃∀ xtypeset structurextypeset structuretypeset structuretypeset structuretypeset structuretypeset structuretypeset structuretypeset structure

Om typeset structuretypeset structure

Wording for first-Order Logic:

"The domain D is a set of "objects" of some kind. Intuitively, a first-order formula is a statement about objects; for example, ∃x.P(x) states the existence of an object x such that the predicate P is true where referred to it. The domain is the set of considered objects. As an example, one can take D to be the set of integer numbers.” (WiKi)

Because first-order logic objects are obviously characterized by a single domain (universe, contexture), the domain of polycontextural quantification is not a single universal set of objects but, metaphorically, a poly-set of relations.

An interesting example for “poly-sets of relations” is given by the system of triadic-trichotomic sign relations. An adequate thematization, modeling and logification of triadic-trichotomic domains, like Peircean semiotics, needs a structurally adequate logical apparatus. First-order logic is reducing such complex structures to mono-contextural predicative objects.

                                                                                     ...                                                                      1                           1

2.1.  Tableau rules for polycontextural quantifiers

2.1.1.  Syntactic schemes

<br />  <br /> Underscript[ (J^1 J^2 J^3) x ^(3) P^(3) ... ;    (Q^3(x _ 1^3, x _ 2^3)) ((P _ 1^3, P _ 2^3) (x _ 1^3, x _ 2^3)) <br />

 Relation R = (P _ 1^1, P _ 2^1) is a dual - predicate over the polycontextural tuple (x _ 1^1, x _ 2^1)

2.1.2.  Tableaux rules for typeset structure

<br /> <br /> Underscript[t  _ 1 Q ∃ ∀ x ^(3) P^(3)  x &#x ...                                                                                                  1

For all junctional quantificators there exists a quantificational representation with single objects a.
For all transjunctional quantificators there exists a quantificational representation with transcontextural tupels (a, b).

Objectivity and Objectionality
The objectivity of an object in First-Order Logic and set theory is characterized by its ontology of individual and predicate, i.e. substance and attribute. There is a strict hierarchic order between individuals and predicates. Objectivity is excluding, logically and ontologically, self-referential statements between individuals and predicates, i.e. a predicate can not change into an individual and vice versa without producing in its logical framework a contradiction.
Objectionality of a polycontextural object depends on the complexity of the interplay between entities and characteristics  of different contextures. (Kaehr, Materialien 1976, Siemens-Studie, 1985)

Linguistic example
A 3-contextural object Otypeset structure like a (contexturally) complementary object waveparcel (Heisenberg) has the property Ptypeset structure with locally P1 for the wave and for the parcel O2 the property P2 and globally the property P3 for the composed notion of wave and parcel, Otypeset structure waveparcel.
Quantification happens intra-contexturally for all contextures by the junctional quantifiers ∀ and ∃.
Quantification between different contextures, focusing different contextures at once, happens with transjunctional quantifiers, e.g. Q and G.

Famous   application   to   R ussell ' s   devilish   construction ,   but escaping paradox <b ...                       0     0                                                              0     0

2.1.3.  Mimicking General Set Theory (GST)

<br /> GST : <br /> 1. ∀ x ∀ y[∀ z[z ∈ x <--> z ∈ y] --> ...  3. ∀ x ∀ y ∃ w ∀ z [ z ∈ w <-> [z ∈ x ∨ z = y]] .

<br /> Axioms of Extensionality for 3 - contextural set theory GST^(3) <br /> <br /> H ... 712; ^(3) y^(3)] --> ^(3) x^(3) = ^(3) y^(3)] .

[                                                                                              ...       [x   =   y]                                                 [x   =   y]

FormBox[RowBox[{<br />, RowBox[{Cell[TextData[Cell[BoxData[[                                   ...        [x   =   y]                                                 [x   =   y]

More brackets at:
ConTeXtures. Programming Dynamic Complexity.
http://www.thinkartlab.com/pkl/lola/ConTeXtures.pdf

2.2.  Polylogical quantification rules

2.2.1.  DeMorgan rules


typeset structure

First - order logic duality ¬ ∀ ((x) (¬ P (x)) = (∃ x) P (x) <br /> ¬  ...   _ 2 P (x)) = (∀  _ 3 ∃  _ 2 ∀  _ 1) (x) P (x)

<br /> Polylogical self - duality for transjunctional quantifiers <br /> dual  _ 1 :   ... P (x) <br /> <br /> The quantifiers Q and G are self - dual, i . e dual (Q) = Q and dual (G) = G .

2.2.2.  Distribution rule for transjunctional quantifiers

    Propositional distribution rule         <br />      (&# ...                                                                                                 _

Underscript[             &nbs ...                                                                        3                         3

2.3.  Quantification for diamond theory

2.3.1.  3-contextural diamond

Scheme for diamond - quantification   (reduced) <br /> (S _ 1 S _ 2 S  ... #xF3A0;^(4)) > > <br /> <br /> Duality   for   diamond - quantificational   formulas <br />

                            (3) dual (dual (dual (D   ))) =     1     2     1                  ... #xF3A0;^(4) : dual _ 1(dual _ 2 (dual _ 1(D^(3)))) = dual _ 2(dual _ 1(dual _ 2(D^(3)))) . <br />

Example : Tableaux   rules   for   (Q _ 1 ∃  _ 2 ∀  _ 3 &# ...                                                                                                  3

2.3.2.  4-contextural diamond

[      id   ] : [      S ] Overscript[- ...                                                                                                  6

2.4.  Smullyan unification for diamond quantification

2.4.1.  Smullyan’s unification rules for  “propositional” constellations

PolyLogics
Towards a formalization of polycontextural Logics.
http://www.thinkartlab.com/pkl/lola/PolyLogics.pdf

From Dialogues to Polylogues
http://www.thinkartlab.com/pkl/lola/Games-short.pdf

Place-valued logics around Cybernetic Ontology, the BCL and AFOSR
http://www.thinkartlab.com/pkl/lola/AFOSR-Place-Valued-Logic.pdf

2.4.2.  Smullyan’s unification rules for  “quantificational” constellations

3.  Interplay of semiotics, logics, set theory and arithmetic

3.1.  Strategy

A study of polycontextural semiotics, focused on semiotics alone, is not yet guaranteeing its polycontexturality. The logical, arithmetical and set theoretical status of semiotics, mono- and polycontextural, remains undetermined if its corresponding logics, arithmetic and set theory (incl. category theory) are not determined and explicitly developed as polycontextural systems.

On the other hand, what value would have a semiotic system without any chances to proof statements, studying its arithmetical, set and category theoretical properties? Until now, arithmetic, e.g., in semiotics, is not recognizing semiotical complexity but is calculating some combinatorial properties which are independent of the genuine, say triadic-trichotomous structure. Similar mismatches happens with well known inadequate combinatorial studies of morpho- and kenogrammatics.

The same situation has to be recognized for other formal systems. A formalization of polycontextural logic is easily reduced  to monocontexturality by arithmetization (Gödelization) if there is not at the same time a polycontextural arithmetic at hand to defend the strategies of polycontextural logic. And obviously, because there is no initial origin, the carousel has to go through all stations of logic, arithmetic, semiotic, category and set theory, thematization, meta- and protolanguage, etc. to deliver and interplaying foundation for each other.
Proto- and meta-languagues of formal systems, as normed natural languages, are importand to rule the relation between natural and formal languages, especially in the case of the interpretation of formal terms for philosophical or applicative aims. If proto-language-based considerations are limiting the formal possibilities of formal constructions, the reasons for the restrictional decision should be made as explicite as possible. Also should formal possibilities be accepted which haven’t yet found an interpretation.

Earlier on, there was a big philosophical topic to fight against the advent of traditional many-valued logic with the argument that the natural meta-language used to motivate and to develop many-valuedness is a priori two-valued. Hence, there is no escape from the two-valuedness of human thinking with the help of many-valued logic. Today, not even the question is recognized.

3.2.  Sketch

For the purpose of recent introductory sketches of a descriptive characterization of the idea of poly-semiotics, it might be sufficient to hint to the decision to use 3-contextural subsystems of 4-contextural logics and arithmetics. Instead of the usual decomposition into elementary contextures.

Hence, from a 4-contextural logic, Logtypeset structurewith its six 2-contextures, Logtypeset structure its four 3-contextures, Logtypeset structure  only the four 3-contxtural subsystems are in direct correspondence to the 4-contextural (poly)semiotics, decomposed into its 3-contextural semiotic parts. The same holds in general for the interplay between arithmetic, set theory and semiotics.

<br />    Graphematics^(4, 3, 2) = (Sem^(4, 2), Log^(4.2), Arith༺ ... #xF3A0;^(4, 2)      = (Set^(3, 1), Set^(3, 2), Set^(3, 3), Set^(3, 4)) <br />


typeset structuretypeset structurerealizing the paradigmatic and conceptual transformations of the 4-contextural logics Logtypeset structureand arithmetic Arithtypeset structure

typeset structuretypeset structurerealizing the structural and deductional transformations of the 4-contextural semiotics Semtypeset structure

typeset structuretypeset structurerealizing the structural and computational transformations of the 4-contextural semiotics Semtypeset structure

typeset structuretypeset structurerealizing the objectional and predicational transformations of the 4-contextural semiotics Semtypeset structure

typeset structuretypeset structure