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PolyLogics.

Towards A Formalization of Polycontextural Logics

by Rudolf Kaehr

ThinkArt Lab Glasgow Hallowe’en 2005

"Interactivity is all there is to write about:

it is the paradox and

the horizon of realization."

*Sponsored, partly, by the German Software ThinkTank

ALGoLL AG

, Munich, Germany .


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PolyLogics.

Towards A Formalization of Polycontextural Logics

A. Rhizomatics for PolyLogics

1 Problems with logic and the logics of problem soving

1.1 A motivational scenario 6

2 From Propositions to Glyphs

2.1 Abstracting propositions 102.2 Generating contextures out of scriptures 112.3 Sentence vs. textures 122.4 Signs vs. Patterns 132.5 Contextures vs. Names 13

3 More Metaphorics

3.1 Tree farming of colored logics 143.2 Preliminary Comments 16

4 Architectonics

4.1 Proemial relationship of mediated systems 18

B. General Framework of PolyLogics

1 General Tableaux Scheme for G(3)

1.1 Proemiality of logical concepts 291.2 Interactional and reflectional logics 30

2 Syntactical structures for PolyLogics

2.1 Generel remarks 312.2 Balanced syntax 32

C. Tableaux rules for PolyLogics

1 Tableaux rules for junctions

1.1 Pattern [ id, id, id } 341.2 Pattern [ id, red, id 351.3 Tableaux rules for transjunctions 391.4 Rules and Definitions in PCL

(3) 51

1.5 Tableaux proofs in PolyLogics 531.6 Tableaux proofs for transjunctional constellations 601.7 Term representation of formula development H1 641.8 General Strategy 681.9 General term rules 71


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D. Unification for PolyLogics

1 Generalized Smullyan Unification for PolyLogics

1.1 General unification rules 741.2 Classification of unificators 761.3 General rules and tableaux rules 841.4 Patterns of unification 851.5 Contradiction and incompatibility 86


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PolyLogics*

Towards A Formalization of Aspects of Polycontextural Logics

*Sponsored, partly, by the German Software ThinkTank

ALGoLL AG

, Munich, Germany.


DRAFT


Rhizomatics for PolyLogics

PolyLogics

are a special thematization and formalization of the general idea of poly-contextural logics. There is a nice little book

"Logic with Trees"

from Colin Howson.PolyLogics in addition, could be called "Logics with Forrests" or

"Logics in Rhizomes"

to connect to the tradition of tree-based methods.

General Strategies

Polycontextural complexions are composed by components, contextures. Complex-ions are compounds of components. Compounds can be de-composed into compo-nents. Compositions are

super-additive

in respect to their components.In contrast to logocentric systems polycontexturality is involving complexity at the

very beginning, not as a static decision but as a dynamic exchange between its fun-damental architectonics and its metamorphic futures.

Metaphors and Motivations

PolyLogics are not based on propositions and their simple alphabet (atomicity, lin-earity), reducible to a one sign and a nil sign alphabet, based linguistically on speechbut on written forms, hieroglyphic, like Chinese writing with its tabularity, sign-complex-ity, contextuality, interpretability and dynamic multitude from the very beginning.

PolyLogics, in this picture, are the logics of Chinese writing. Dialectics.

How many characters?

The Chinese writing system an open-ended one, meaning that there is no upper limit to thenumber of characters. The largest Chinese dictionaries include about 56,000 characters,but most of them are archaic, obscure or rare variant forms.

Strokes

Chinese characters are written with the following twelve basic strokes:Basic strokes which are combined to make up all Chinese characters.

The strokes themselves are not characters, thus, they don’t have a meaning. They arethe conditions of the possibility of meaning at all. In polycontextural terms they are notsigns but kenograms inscribing morphograms which can be thematized, interpreted,transformed into logical meanings. First are the written characters, then the phoneticinterpretations. But this alone wouldn’t reflect the chiasm between writing and speechproperly. The written characters can contain some phonetic elements; and the meaningof the written forms has to be negotiated.

http://www.omniglot.com/writing/chinese.htmhttp://home.vicnet.net.au/~ozideas/writchin.htmhttp://cjvlang.com/Writing/writsys/writlinks.html

The Textual Dance: Allusion in the Oldest and Newest Poetry

A single Sumerian sign may have five, ten, twenty or more values. But the traditional andhistorical method of reading those signs ignores that multi-valent quality and instead of plas-ticizing and expanding meaning, traditional translation hardens meaning. Traditional read-ings do this by building a transliteration, a version of the text in which specific and singularmeanings are assigned to each of the signs, establishing a single definitive or "Ur" text.http://resonantconcept.typepad.com/experimental/2003/11/the_textual_dan.html

PolyLogics are fundamentally non-fundamentalist. There is no Ur-origin, only multi-tudes of beginnings and ends.

http://www.omniglot.com/writing/chinese.htm

http://home.vicnet.net.au/~ozideas/writchin.htm

http://cjvlang.com/Writing/writsys/writlinks.html

http://resonantconcept.typepad.com/experimental/2003/11/the_textual_dan.html

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1 Problems with logic and the logics of problem soving

1.1 A motivational scenario

Why not simply asking the experts from the MIT?

The Panalogy Principle:

If you 'understand' something in only one way then you scarcelyunderstand it at all—because when something goes wrong, you'll have no place to go. Butif you represent something in several ways, then when one of them fails you can switch toanother. That way, you can turn things around in your mind to see them from different pointsof view —until you find one that works well for you now. And that's one of the things that‘thinking” means!

We cannot expect much resourcefulness from a program that uses one single technique—because if that program works in only one way, then it will get stuck when that method fails.However, a program with multiple ‘ways to think’ could behave more like a person does:whenever you get frustrated enough, then you can switch to a different approach—perhapsthrough a change in emotional state.

Marvin Minsky

Implementing Panalogy

I will use the term Panalogy to refer to a family of techniques for synchronizing and sharinginformation between different ways of thinking concerned with the same or similar prob-lems. The term derives from ‘parallel analogy’. By maintaining panalogies between waysof thinking, we can rapidly switch from one way of thinking to another.We can also make more partial changes like the representation language they are using,the types of assumptions they are making, the methods that are available to them for solu-tion, and so forth. The key idea is to support representing multiple problem solving contextssimultaneously and the links between them. A graphical depiction of panalogy at work isshown below.

But is this exactly what I am looking for? Obviously not. To have the same wordingand to have the same diagram doesn’t yet mean that we are thematizing the same sit-uation in the same way of thinking and implementation.

The main difference between panalogy and PolyLogics is this. Panalogy is mono-con-textural, always only one method is running, not several at once and there is no inter-activity and reflectionality between successively different methods. They are appliedonly one after the other. If one method doesn’t work, take another.

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PolyLogics with its proemiality is ruling the interplay of different methods running andcooperating together at once.

Marvin Minsky offered the Six Level Model from his forthcoming book

The Emotion Machine

as an initial proposal for such an architecture. This architecture is being developed jointlyby himself and Aaron Sloman, and is based on several key ideas:

1. Use several approaches, at once, to each problem.

2. Have many ways to recognize and respond to internal and external problems.

3. Support many different “ways of thinking”.

Marvin Minsky, Push Singh, MIT, May 13, 2002

The wording

“Switching between parallel methods of thinking”

sounds quite promis-ing, but it doesn’t gives us a hint

how

the switch is working, what is the mechanism ofthe switch, and, how do we know that we are dealing with the same problem after theswitch to another domain. How much is the problem itself transformed by the switch ofcontext? And what is the notion of sameness involved in this switch? What do we meanby "parallel" in this context?

I will use the term Panalogy to refer to a family of techniques for synchronizing and sharinginformation between different ways of thinking concerned with the same or similar prob-lems.

The common term between the different domains of panalogy is obviously

informa-tion

. But how can we know that all the domains are ruled by the very same concept ofinformation? Why is the term information not in itself panalogical?

By maintaining panalogies between ways of thinking, we can rapidly switch from one wayof thinking to another.

This sounds really good! But, again,

how

does it work and who is operating thesedeliberating switches?

Minsky´s question is “

What

could cause the change?” and not “

How

does it hap-pen?” or "What is the

mechanism

of change?"

The key idea is to support representing multiple problem solving contexts simultaneously andthe links between them.

Different thematization, different strategies

The complementary aspect of Minsky´s approach to the polycontextural approach isexpressed by the statement

“We'll try to

design

(as opposed to

define

) machines that can do all those 'differentthings'."

Minsky

The question of definition is a logical one, the process of design belongs to the do-mains of modeling, simulation, implementation and not to formalization.

It seems not to be easy to escape the challenge of logics. All the tools and methodsof design, programming languages, LISP obviously too, are based on logic. The sameis the case for the machines.

Why should the process of design be restricted by the structure of its classical tools?

http://web.media.mit.edu/~push/Push.Phd.Proposal.pdfhttp://www.thinkartlab.com/pkl/media/DERRIDA/Panalogy.html

http://web.media.mit.edu/~push/Push.Phd.Proposal.pdf

http://www.thinkartlab.com/pkl/media/DERRIDA/Panalogy.html

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Peter Wegner on logics

Because of my focus on foundational studies my realization of the category of expla-nation is worked out in a more philosophical sense, and the category of implementa-tion too, is more foundational than empirical, that is, it is implementing new formalismsand programming languages with the help of today´s monolitical methods (existingprogramming languages) on monolitical machines.

"I first presented the idea that Turing machines cannot model interaction at the 1992 closingconference of the Japanese 5th generation computing project, showing that the project´sfailure to reduce computation to logic was due not to lack of cleverness on the part of logicprogramming researchers, but to theoretical impossibility of such a reduction. The key argu-ment is the inherent trade-off between logical completeness and commitment. Commitmentchoice to a course of action is inherently incomplete because commitment cuts off branchesof the proof tree that might contain the solution, and commitment is therefore incompatiblewith complete exploration.” Wegner, ECOOP ´99, p.1/2 http://www.cs.brown.edu/people/pw/

As mentioned above, Wegners strategy to surpass this limiting situation is not to de-liberate the paradigm of formality which is defining the very concept of logic and allthe concrete logical systems, but some form of regression to empiricism.

“Logic can in principle be extended to interaction by allowing nonlogical symbols to be in-teractively modified (reinterpreted) during the process of inference, for example by updatinga database of facts during the execution of logic programs. However, interactive discoveryof facts negates the monotonic property that true facts always remains true.”

Wegner, p. 25

This strategy of extending logical systems by non-logical symbols for modeling inter-action introduces into logic some non-logical elements of empiricism. For practical rea-sons this approach has its merits. Nevertheless, from a structural point of view ofoperativity and formality nothing has changed. Still the old logic is ruling the situationand dictating the possibilities of design.

http://www.cse.uconn.edu/~dqg/papers/#interaction

The polycontextural approach

Thus, we can distinguish 3 levels of reflection or even 3 different paradigm to com-putation and logic:

1. The

classic

paradigm of logic, arithmetics, computation, etc.2. The

interactive

paradigm (Peter Wegner, Marvin Minsky, Rolf Pfeifer, et al). Withthemes of interaction, morphologic computation based on a new interpretation of com-puting reality in modeling and design, accepting the first paradigm of formality, butcorrecting it with new more realistic models based in empirical approaches. This par-adigm is more heterogeneous and interactive than the first but doesn’t have any for-malism to mediate the interacting heterarchic parts of the complex system in anoperativ way. This approach is still descriptive and not operative.

3. The

polycontextural

paradigm which accepts both previous paradigms in theircontext, but is trying to surpass the very limits of operativity and formality posed by thefirst paradigm. Polycontexturality tries to keep the level of operativity of the classic par-adigm but concerns with needs of a operative theory of mediation of interacting andreflecting parts which has no place in the 2 previous paradigms.

Short: Simultaneity on a strict logico-mathematical level. Paradox approach to theidea of an extension of logics: there is no extension of logic, in principle, but a dissem-ination of the very same logic and its logical systems over a grid of contextural loci.

http://www.ifi.unizh.ch/ailab/people/iida/research/pfeifer_iida_JSM05.pdfhttp://www.ifi.unizh.ch/ailab/people/llicht/morphcomp/slides/MorphologicalComputa-tion.htm

http://www.cs.brown.edu/people/pw/

http://www.cse.uconn.edu/~dqg/papers/#interaction

http://www.ifi.unizh.ch/ailab/people/iida/research/pfeifer_iida_JSM05.pdf

http://www.ifi.unizh.ch/ailab/people/llicht/morphcomp/slides/MorphologicalComputa-tion

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DRAFT


2 From Propositions to Glyphs

"The idea of

naming

something is a process of

abstraction

."

O-LUDICS.pdf, Jean-Yves Girard, 2000

Time is changing quickly, now, we are in 2005, and it seems that category theoryhas lost its leading function to

polymathematics

with its m-categories.

"I shall take heart from this dream and extend here a scheme I outlined in Chapter 10 ofmy book, an amalgamation of a scheme of Sir Michael Atiyah with one of Baez and Dolan,which derives in part from another giant of the twentieth century, Alexandre Grothendieck:

19th century

The study of functions of one (complex) variable The codification of 0-category theory (set theory).

20th century

The study of functions of many variables The codification of 1-category theory

21st century

Infinite-dimensional mathematics The codification of n-category theory, and infinite dimensional-category theory." David Corfieldhttp://www-users.york.ac.uk/~dc23/phorem.htm

To connect motivations and metaphors for an introduction of PolyLogics to the 21stcentury additional to grammatological speculations about Chinese writing, the eventof a revolution in category theory could play a significant role. It will still be an analogyand its interpretation full of risks but easier to handle. The nice symmetry between log-ic, computation and 1-category theory is in a process of displacement by the newmovement of n-category theory, challenges of interactivity in computing (Peter Wegner)and approaches in polycontexturality to transform logic; and more.

The Tale of n-Categories:

http://math.ucr.edu/home/baez/week78.html#tale

n-Categories: Foundations and Applications:

http://www.ima.umn.edu/categories/

http://www-users.york.ac.uk/~dc23/phorem.htm

http://math.ucr.edu/home/baez/week78.html#tale

http://www.ima.umn.edu/categories/

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DRAFT DERRIDA‘S MACHINES 12

Even if not well studied, we know well that 1-categories are based on triadic con-cepts. We know well dyadic concepts and their logics. But we still try to understandtriadic concepts (semiotics, categories) in the framework of dyadics. Maybe of com-bined dyadics. But do we have an idea about n-categories? Are they iterations, evenindefinite iterations ("infinite dimensionality") of triadic concepts of 1-category, stillbased on dyadic logics? Is the term "infinite" not understood as a dyadic and not asa genuine n-category theoretical term? What is the difference in the meaning of thenotion "indefinite" in the 3 different conceptualisations; the dyadic, the triadic 1-cate-gorial and the magic n-categorial?

Is the notion of the infinite chain of 1-categorial concepts constituting n-categoriesitself a 1-categorial concept?

PolyLogics are both: combinations of dyadic logics and genuinely m-categorial. Be-cause mediation (combination) in PolyLogics is super-additive, decomposition into sin-gle dyadic systems is not working without reduction, that is, denying the interactionaland reflectional parts of the whole. PolyLogics are based on morphograms. Morpho-grammatics: The calculus of kenomic loci.

Thus, category theory is philosophically relevant in many ways and which will undoubtedlyhave to be taken into account in the years to come.Introduction to CT: http://plato.stanford.edu/entries/category-theory/Manifesto for CT: http://www-cse.ucsd.edu/users/goguen/pps/manif.psCT and Computer Science: http://www.cwru.edu/artsci/math/wells/pub/ctcs.html

Locus SolumOnly locations matters. Jean-Yves Girard

What’s about the location of different n-categorial systems, where are they placed,do they occupy a locus? And what’s the calculus of these loci? Locations in the senseof Girard are intra-contextural loci of a system, they are not thematizing the genuinelocus of the system itself. Does n-category theory reflect any loci?

Logic and category theory

Classic category theory is well founded in classic logic. On the other hand, logicalsystems can well be modeled in category theory.

William S. Hatcher: http://www.rbjones.com/rbjpub/philos/bibliog/hatch82.htm

Now, n-category theory claims to be a kind of a revolution transforming the old con-cepts of 1-category to new concepts of n-categories. My question remains, what arethe logics of n-category theory? The plural of logics means the different roles logic canplay in the construction of n-categories. What is the use of logic in developing n-cate-gories, what is the deduction system for n-categories and what is the foundational roleof logic for the new category theory?

Ultimative presentation: Tom Leinster: http://www.maths.gla.ac.uk/~tl/#book

1-category n-categoryextension

logic ???

foundation

(PolyLogics)

http://plato.stanford.edu/entries/category-theory/

http://www-cse.ucsd.edu/users/goguen/pps/manif.ps

http://www.cwru.edu/artsci/math/wells/pub/ctcs.html

http://www.rbjones.com/rbjpub/philos/bibliog/hatch82.htm

http://www.maths.gla.ac.uk/~tl/#book

2.1 Abstracting propositions

"Abstraktion schwächt, Reflexion stärkt". Novalis

2.1.1 Abstracting sentences as propositionsWriting in the phono-logocentric conception of rationality is a secondary event.

It is the inscription of thoughts and spoken language into a linear scripture basedon atomic elements. The process of abstracting propositions is modeled along theconcept of naming, giving something a name, producing the whole machinery ofidentity. Abstractionn as giving something a name is emphazing the act of classi-fication in contrast to creation.

Abstraction

Out of the dynamic complexity of language sentences are abstracted which aredecontextualized enough to play the role of propositions1. Logical propositionsare propositions which can be true or false independent of all kind of modalitiesand parameters. That is, independent of who, where, how, when, why, etc. a prop-osition is stated. Propositions are formal statements which can be true or false.

Formalization

Formalization is connecting the abstract concept of propositions with operation-al methods, mainly mathematical, like geometric, algebraic, functional, categoricetc. to develop formal logics, which is also called symbolic or mathematical (prop-ositional) logic. In other words, formalization is producing what we know todayas logic and logical systems, universal logic and combined logics.

Codification

Intensional logics are based on propositional logic. But enriched, step by step,by what had to be excluded from language to develop abstract propositional log-ic. That is, all sorts of intensional operators recognized by linguists are transformedfrom their vagueness in ordinary language via mathematical linguistics to mathe-matization, that is concretization, in intensional logics.

Concretization

Concretizations happens by parametrization of aspects of formal logics and byadopting further formal features of natural language, like temporal, modal, spa-cial, etc. aspects leading to modal logics, temporal logics, multi-valued logics, etc.

Formal semantics, operational hermeneutics, common sense logics, etc. are oth-er labels for intensional logics.

1. Usually I don’t give references for obvious things. But there seems to be some de-mand to justify my thoughts by referencing it to the well known (?) academic literature.Thus: The procedure of "abstracting" logical propositions out of ordinary language isdeveloped with high accuracy by: Wilhelm Kamlah/Paul Lorenzen, Logische Propä-deutik, Vorschule des Vernünftigen Redens, B.I. 227-227a, Mannheim, 1967, 242pp.

language propositionabstraction

intensional logics formal logics

codification

concretisation

formalization

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Rudolf Kaehr Dezember 26, 2005 9/16/05 DRAFT DERRIDA‘S MACHINES 14

Again, alphabetism and signatures

To speak about alphabetism in formal systems, with its atomicity, linearity, iterability,and ideality is not forgetting the conceptual move from alphabets as sign repertoiresto the more abstract concept of signatures of institutions introduced by Goguen. Thismove is connected with the move from set to category theoretic conceptualizations.

Institutions accomplish this formalization by passing from "vocabularies" to signatures,which are abstract objects, and from "translations among vocabularies" to abstract map-pings between objects, called signature morphisms;

then the parameterization of sentences by signatures is given by as assignment of a setSen(S) of sentences to each signature S, and a translation Sen(f) from Sen(S) to Sen(S') foreach signature morphism f: S -> S', while the parameterization of models by signatures isgiven by an assignment of a class Mod(S) of models for each signature S, and a translationMod(S') -> Mod(S) for each f: S -> S' (please note the contravariance here).

More technically, an institution consists of an abstract category Sign, the objects of whichare signatures, a functor Sen: Sign -> Set, and a contravariant functor Mod: Sign -> Setop(more technically, we might uses classes instead of sets here). Satisfaction is then a parameterized relation |=S between Mod(S) and Sen(S), such that thefollowing satisfaction condition holds, for any signature morphism f: S -> S', any S-modelM, and any S'-sentence e:

M |=S f(e) iff f(M) |=S' e

This condition expresses the invariance of truth under change of notation.

http://www.cs.ucsd.edu/users/goguen/projs/inst.htmlIdeality: Abstractness of the change of notation

Signatures are even better realizing alphabetism than sign repertoires because theyare empathizing the abstractness of alphabetical signs, that is, the ideality of signs,and sugn systems, in contrast to concrete occurrence of signs, independent of the con-tent of the sign repertoire, i.e., the concrete notational material. That is, sign systemsare not only characterized by atomicity, linearity, iterability, but also by ideality. Ideal-ity is the medium of realization of signs. Sign systems are not concrete systems but idealsystems. Notational systems of sign systems are, to some degree, the concrete realiza-tions, that is, the representations of abstract sign systems. And signatures as they aredefined in the theory of institutions are the themes of thematizations.

Goguen’s "This condition expresses the invariance of truth under change of nota-tion."

and Makowski’s „Computing does not deal with the creation of notational systems.“ Makowsky, in: Herken, p. 457,was quite motivational for my studies in "Strukturationen der Interaktivität" (SKIZZE).

More about semiotic, grammmatological and graphematic reflexions in/of formallanguagues (systems, logics, computation) can be found in my German texts:

http://www.thinkartlab/.com/pkl/media/SKIZZE-0.9.5-Prop.pdfhttp://www.thinkartlab.com/pkl/media/DISSEM-final.pdf

http://www.cs.ucsd.edu/users/goguen/projs/inst.html

http://www.thinkartlab/.com/pkl/media/SKIZZE-0.9.5-Prop.pdf

http://www.thinkartlab.com/pkl/media/DISSEM-final.pdf

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2.2 Generating contextures out of scripturesThematization is not only classification, like abstraction, but creation and evocation

of new horizons of thinking.

Thematization

An abstraction is delivering an abstract proposition excluding its context of abstrac-tion. Thematizations are "abstractions" of the difference "proposition/context" deliver-ing propositions as propositions of a context, that is, a contexture. As a consequenceof this as-abstraction there are always a plurality of contextures involved. Therefore, aproposition as belonging to one contexture is conceptually different to a propositionas belonging to another contexture.

Mediation

Contextures and their propositions are not isolated but mediated to complexions ofcontextures and propositions. The mechanisms of such textual mediations have to beanalyzed and to be formalized towards a realization of polycontextural logics, espe-cially to PolyLogics.

Codification

General patterns of writing have to be thematized. Not along the lines of phono-log-ical linguistics but specific to the practice of writing. Such grammatological andgraphematic rules are delivering the basis for operational realizations.

Realization

A realization of graphematics is a further abstraction froom the rules of polycontex-tural logics connected with the insights into the rules of scriptural codifications. Poly-contexturality and graphematics are the two new domains of scriptural research.

scriptures contexturesthematization

graphematics polylogics

codification

realization

mediation

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2.3 Sentence vs. texturesThe name giving process is identifying its object and installing the laws of identity,

thus these name givers are also called "identifiers". Like the lambda abstraction, whichis defined or introduced intra-contexturally, the new samba abstraction is a trans-con-textural operation. To distinguish it from the lambda abstraction, it should be calledsamba thematization.

Philosophically, to give a name is a special linguistic operation of the general modeof thematizing. Thematizing is more textual, to give a name is propositional or senten-tional. It is connected with the concept of a sentence as a statement. The philosophy ofthe lambda calculus is even stressing this further to the point, that the definition of asentence is build by naming. To give a name is fundamental for the lambda calculusand radicalized, brought to the point by A++.

Hermeneutics and in radicalizing it, deconstructivism, tried to surpass this restrictionand to focus more on texts, intertextuality, interpretability, iterability and ambiguity incontrast to well-formed single isolated sentences and propositions. In this sense poly-A++ can be considered as a further extension of the lambda calculus not from the in-side but by distribution of the very idea and apparatus of the lambda calculus overdifferent loci, empty places. Surely not in changing at all anything of the lambda cal-culus itself, but in disseminating it over the loci of the graphematic matrix.

Every programming language must somehow provide a `name giving' mechanism.Thus, every polycontextural programming language-system must somehow provide ageneral ’thematizing’ mechanism as a general feature allowing disseminated ’namegiving’ mechanisms which each of them allows to call procedures or functions andhave the possibility to refer to variables inside the ’name giving’ systems and betweendifferent ’name giving’ systems.

A ’name giving’ procedure is also an identifier. To be able to identify something ithas to be separated from its environment, but something can be separated from othersonly if it can be identified. We don’t want to go into this paradoxical situation whichis nevertheless the beginning of all formalism at all. But it should be mentioned that toidentify something is including also a semiotic-ontological principle of identity: thenamed has not to be changed in the process of its naming. To name is to identify andnot to change. But this is true only for the very special class of identical beings. Itdoesn’t apply for living systems and even quantum physics is running in some troubleswith this identity principle.

Said all that, it seems to be obvious, that the "references" of ConTeXtures are notsymbols, variables and the data like for the intra-contextural ARS systems but the pro-cesses of interaction and reflection between ARS systems itself as it is realized in thetexturality/textuality of texts, that is contextures. In this sense, ConTeXtures are abstract-ing from the process of abstraction as it is realized in the Lambda Calculus. The refer-ence is the processuality of the abstraction and not its topics.

Abstraction, again:"The idea of naming something is a process of abstraction.When we calculate 2+1, 3+1, 5+1, 16+1 we detect a pattern and feel that it might beuseful to calculate x+1 for any x – or at any rate for a numeral x. This concept is of coursecentral to mathematics and to computing where it is of the essence that we should try to de-velop programs not just to do one job but to be as general as possible. The replacing of awhole class of objects by a name representative of an element of the class is roughly whatwe mean by abstraction and it allows us to approach functions naturally." A J T Davie, An Introduction to Functional Programming Systems using HASKELL, pp. 79/80

To name a function by name, that is to name a name, as it happens in the defi-nition for recursive function, can be done in two ways: intra-contextural, using thecircularity of the Y-operator or distributed over different contextures, that is trans-contextural.

2.4 Signs vs. PatternsConTeXtures is not based on statements, but on intertextu(r)ality. It is not starting

with signs but with complex graphemes. ConTextures, thus, are not primarily guid-ed by philosophical concept of logos but by graphein. The logos, and all its secu-lar derivations, like the lambda calculus statements, have to be listened. You haveto listen the command sentences. ConTeXtures have to be read. Their elements arenot semiotic atoms but scriptural patterns which have to be described and deci-phered. Signs are based on atomic unities, patterns are complex, antagonistic, dy-namic events. If there is a cultural change involved with ConTeXtures then it is thetransition from the Greek alphabetism to the Chinese emblematics. ConTeXturesare not belonging to a new logical paradigm but to the grammatological subver-sion of Graphematics.

The polycontextural matrix can not be read like a sentence and to be listened toits meaning. The matrix has to be read, involving different viewpoints and memory.It can not be memorized internally as a subjective truth. There is no truth in Con-TeXtures, but a multitude of truths space-ing multiple ways of practice. The play ofdifferences has to be written in a mundane game of traces and marks. The poly-contextural matrix is a first step/jump to Graphematics transforming idealistic logicto materialistic dialectics.

A further step is introduced with the "abstraction" (subversion) of morphizingcontextures. This is realized by Morphogrammatics and Kenogrammatics, thegrammar of kenos, the game of emptiness, studying the invariance of dynamicscriptural patterns.

2.5 Contextures vs. NamesThe textuality of ConTeXtures is involving the game of the as-abstraction based

on the proemial relation. Therefore there are no new stable beginnings of the cal-culus. ConTeXtures are not simply changing from a name dominated to a text dom-inated paradigm of disseminated contextures. In the same sense as names canbecome contextures, e.g. in a process of de-nominalization, contextures can benominalized and can become names. We can call contextures by name. Contex-tures can be named and names can be contexturalized. This is not a return hometo the name dominated logocentrism of the Lambda Calculus but part of the gameof proemial metamorphosis.

There is no fixed hierarchy for the operators samba and lambda. It is allowed touse these operators in a "circular" and "parallel" way which also shows, that thereis no ultimate beginning and origin of the game: (...(lambda(samba(lamb-da))...).This too, may give a hint to understand that ConTeXtures are combining al-gebraic and co-algebraic concepts and methods of formalizing.

ARS allows a very general use of the lambda abstraction. Everything named canbe called everywhere and every time by name. It seems, that the conceptualizationof ARS is more "self-referential" than other Lambda-based programming approach-es. In general, ARS could even call itself by name. But this makes not much sense,because there is no conceptual space to use this way of naming and to put itsnamed object (ARS) somewhere. In a polycontextural environment things are verydifferent and it is not only possible but also necessary to call a whole system, likeARS, by name and to use it in other contextures.

p


3 More Metaphorics

3.1 Tree farming of colored logicsTo put all these more philosophical descriptions and ideas together in a more strict

formal terminology and operative apparatus we connect these ideas of colored logicswith the metaphor of the tree. In classical logic trees figure as metaphor and as math-ematical concepts on a very basic level. We postulated that every colored logic is lo-cally classical, e.g. each colored logic has the structure of a classic tree logic in itssyntax, semantics and proof theory. It is helpful to represent the structure of the bodyor the tectonics of a logical system by its conceptual graph.3.1.1 Notation of an institution

Diagramm 1 Conceptual Graph of an Institution

„The arrows in this diagram represents conceptual dependencies. The notation model ––> signaturefor example, means that:the concept of model varies as signature varies.In particular, it means that the concept of model, the one that we have in mind, cannot beindependent of the concept of signature and neither can a particular model be independentof its particular signature.In a conceptual diagram, 1 represents the absolute. The notion institution––>1expresses that the institution notion is absolute, for it tells us that the institution notion variesas the absolute varies – which is not at all.“ p. 488

absolute

The absolute 1 expresses that there is only one logic as such. There are many differ-ent logical systems but from a more philosophical and logical and not only mathemat-ical point of view all these logical systems are isomorphic to one and only one logic.This is a (not provable Hypo) thesis.

If we do not like this absolutism we should remember that the wording holds also inthe more relativistic case. Considering a logical system working in it and with it meansthat we are working with this system and not at once or at the same time with anotherone. We can speak therefore of a relative absolute of the logic under consideration.

satisfaction

model sentence

signature

institution

1

p


Even for mixed logics as in the project of Combining Logics there is a (relative) notionof the absolute of the system.

As we will see the situation will be totally different for polycontextural logics wherea plurality of absolutes ordered in a heterarchical manner exists and the desire to havea mega-absolute for the whole complex system would turn out to be (simply) a new ab-solute within a plurality of other neighbored absolutes.

From the point of view of PCL the absolute means that the whole system is definedunder the operation of identity ID. The system viewed as an object and as a morphismcoincide.

Diagramm 2 Monoforme Mediation of two Institutions

satisfaction

model sentence

signature

institution

1

satisfaction

model sentence

signature

satisfaction

model sentence

signature

institution

satisfaction

model sentence

signature

institution

1 121

p


3.2 Preliminary Comments3.2.1 Syntax

For each tree of the colored logical systems the ancestral property of its formulaeholds. In each tree there is a immediate predecessor relation which decompose the for-mula in its subformulas. All that is ruled by the principle of induction over the formulasfor each logical system. The induction principle is distributed over all logical systems.

Additionally to the concept of a predecessor for each system we have to introducein the trans-classical context the operation or relation of the immediate neighbor. Thesemakes possible some kind of permutation of logical systems over the range of the com-plex of distributed systems.

Further on we have to introduce the very special concept of the immediate bifurca-tion of a formula of a logical system into subformulas distributed over other logical sys-tems and one part of the formula remains in its original system3.2.2 Unification

On a meta-logical level, total functions are supporting symmetrical classificationsand categorisations of logical particles.

This nice property of a logical symmetry is lost in trans-classical logics because oftheir transjunctions which are composed of partial functions. But with the help of theconcept of partial functions we can introduce a new idea of a slightly more complexsymmetry composed by partiality. The classical case of symmetry is then introduced asa regular composition of partial functions. This idea of a complex symmetry composedof asymmetrical functions needs additionally to the classical operator of conjugation anew operator of composition of partial functions.3.2.3 Semantics

Truth-values in classical systems are connotated with the formal logical explication oftruth and false. Formal truth and formal false do not involve ontological questions abouttruth and falsehood of sentence. This belongs to the level of examples for formal logicalsentences.

In PCL systems truth values, if we are choosing a truth value semantics, have to real-ize two jobs, the first is more or less the same as in classical systems, they representthe formal logical concept of true and false of propositions of their logic. The secondjob is very different, they have to mark in which logical system the difference of true/false holds. Therefore they have an index of the system they belong or origin: {Ti, Fi}.As the splitting function of transjunction shows these truth values can occur in differentsystems at once. As a result, the whole semantics of propositions, sentences, phrasesand truth-values has to be deconstructed.

As explained metaphorically earlier these logics are not isolated from each other butcombined to a complex logical, or ultra-logical, system. Otherwise they would behavetotally in parallel and it would be at least at first only an application of one classicallogic at different epistemological places without any interaction or mediation.

A first, quite natural and elementary, connection is given by a (special) linear order-ing of the systems and their truth-values.

To not to confuse this kind of order with other ordering systems I call it a chiastic lin-ear order of truth-values. A chiasm is defined as a tupel of order, exchange, coinci-dence and positioning relations.

Therefore the semantics of PCL is not defined over a set of truth-values but over anorder, a chiastically ordered structure of truth-values.

The difference becomes obvious for the semantics of ternary and general n-ary log-ical functions or logical compositions. This difference between set based and order

based semantics is hidden for the typical binary case.As a natural consequence the notions of sentence, model and satisfaction have

to be distributed over the indices of their semantics.3.2.4 Consequence relations and proof theory

For each single logical system of the PCL complexion there exist a consequencerelation and a proof theory for this logic. The consequence relation for the wholesystem of logic is composed of the distributed single consequence relations of eachlogic.

We will choose the analytical tableaux method as our proof procedure.


Proemiality of PolyLogics

4 Architectonics

4.1 Proemial relationship of mediated systemsThe architectonics of a polycontextural systems defines the kind and complexity of

distribution and mediation of the contextures involved, short, the complexity of dissem-ination.

For introductory reasons the order of mediation of basic contextures will be restrictedto linearity in excluding other combinatorial patterns like stars, etc. as architectonic ba-sic structures. That doesn’t mean, that polycontexturality is subsumed under the generallogocentric principle of linearity. Simply because this logocentric principle of linearityis an intra-contextural and not a trans-contextural principle governing the polycontex-turality of a multitude of contextures, all inhibited locally by a classical principle of lin-earity.

The modus of mediation will be realized by the proemial relation, its structure is oftencalled chiasm. This too, is supporting tabularity and distribution of different principlesof linearity, in contrast to classic linearity, even for the case of "linearily" mediated sys-tems.

On this level the presentation and notation of architectonics and proemiality is re-stricted to conceptual graphs.

Diagramm 3 Basic proemial relationship

Terminology

The cascading order of the two order relation, mediated by the exchange and coin-cidence relations, an elementary case of "linearily" ordered mediation of subystemsrepresented by order relations. Not included in the diagram is the additional sub-sys-tem at position3, produced super-additively by the difference of position1 and 2.

Diagramm 4 Basic proemial relation table

order relation

order relation

exchange relation

coincidence relation

position1

position2

coinc ( x y ) exch ( x y ) ord ( x y ) x1 coinc x2 x1 exch y2 x1 ord y1 y1 coinc y2 y1 exch x2 x2 ord y2


The mediator is defined by the proemial relationship of (coinc, exch, ord, pos).The operator "coinc" is representing the binary relation of coincidence, in the sense

of the sameness, categorial analogy, of two operands or two operators.The operator "ord" is representing the binary relation of order, in the sense of the

asymmetry of an ordered pair of operator, operand and uniqueness.The operator "exch" is representing the binary relation of symmetrical exchange, in

the sense of a symmetric difference between operator and operand.The object 1 is the initial object defining the relative uniqueness of the relationship.

relator

relatum

relation

1

relator

relatum

relation

1


4.1.1 Objectionality of mediated proemial Objects Atomic objects are not simple and identical in ConTeXtures but complexions of

"parts", "aspects" belonging to different contextures. Contextures are mediated by theproemial relationship. Additional to this proemial or chiastic pattern we introduce aspecial kind of mediating systems, realizing the super-additivity of mediated or com-bined systems. I call their relationality towards the other systems "siml" for similarity.

samba ( samba, 3, (ord, coinc, exch, siml, opp), (x, y) )

Diagramm 5 Diagram of mediated proemiality of objects

Diagramm 6 Relational table of mediated proemial object

Diagramm 7 Symmetric mediation table of mediated proemial object

c-ob(3) x1 y1 x2 y2 x3 y3

x1 id ord coinc exch siml exch

y1 ord id exch coinc opp siml

x2 coinc exch id ord siml opp

y2 exch coinc ord id opp siml

x3 siml opp siml opp id ord

y3 exch coinc opp siml ord id

x1 x1

x2 x2

x3 x3

Obj(3) : Obj (3) ––> Obj(3)X

X

X

X

( )

.

.

.

3

1 3

1 2

2 3

( )

coinc ( x y ) exch ( x y ) siml ( x y ) ord ( x y ) opp ( x y ) x1 coinc x2 x1 exch y2 x1 siml x3 x1 ord y1 x2 opp y3 y1 coinc y2 y1 exch x2 y2 siml y3 x2 ord y2 x3 opp y2 y1 coinc y3 x1 exch y3 x3 ord y3 x3 opp y1

x2 coinc x3


Proemial object

x(3) == ( x1 coinc x2 coinc x3 , x1 siml x3 )

y(3) == ( y1 coinc y2 coinc y3 , y1 siml y3 )x(3) exch y(3) ;; x(3) ord y(3)

An atomic 3-contextural object is therefore introduced as a mapping in itself whichmakes sense only if the object is not an atomic object ob (formal object, Curry) but acomplex c-ob(m) (proemial object). Obviously, an atomic object ob is a reduction of ac-ob to its mono-contextural status.

In formal systems atomic terms are distinguished from composed terms. Composedterms are linear compositions of concatenations. Polycontextural c-obs are complex,not in the linear but in the tabular sense. Tabular complex terms are composed not byconcatenation but by mediation.

4.1.2 Super-additivity of mediated systems

Super-additivity, as well as super-subtractivity, appears naturally on all levels of poly-contextural systems and is "based" in the super-additivity of the proemial relationship.Proem (A) + Proem (B) < Proem (A + B)

Obj(5) : Obj (5) ––> Obj(5)

x1 y1

x2

y4

y2

x4

x3

y3x5

y5

x6

y6

x7y7

x8

y8

x9

y9

x10

y10

super-additive systems mediated systems


4.1.3 Numbering of sub-systems

Diagramm 8 Enumeration of sub-systems

Directly mediated systems like S1, S2, S4,S7 are mediated by indirect mediated su-per-additive systems which becomes medi-ating systems relative to their positions. ((S1 med S2) add S3),((S2 med S4) add S5), ((S4 med S7) add S8), ((S3 med S5) add S6), ((S5 med S8) add SS9), ((S6 med S9) add S10).

1 ––> 2 ––> 3 ––> 4 ––> 5

1 ––––––––> 3 2 ––––––––> 4

1––––––––––––––> 4

3 ––––––––> 5

2 ––––––––––––––> 5

1 –––––––––––––––––––> 5

mediated systems

super-additive systems

- short version -

|

|

|

| |

S1 S2 S4 S7

S3 S5 S8

S6 S9

S10

mediated

super-

additive

sub-systems

p p y


5 Notational problems with complexity

5.1 Linear vs. tabular notationsFor traditional and historical reasons formulas have been written in linear form. But

this comes to an end if we introduce more complex situations. The following examplesare considering only the case of balanced matrices and OiMj, i=j, that is, the diagonalsystems only.

To write down and to deal with, say a 47-contextural logic, would simply fall out ofthe scope of perception. Linearity, even for the representations of the operators of aformula, not mentioning composed formulas, has failed to work.5.1.1 Operational patterns

A kind of a tabular numeration of the operators may help a step further in dealingwith notational complexity.

On the half way to combinatory logic operational patterns are abstracting, as far aspossible, from the variables of the formulas and inscribing only the operational pat-terns and their transformations.

Numeration

Diagramm 9 Enumeration of sub-systems, O-matrix

Binomial numbers onto operator patterns.

X Y

X Y

X

4 4

5 5

6

( ) ( )

( ) ( )

( )

∨∧∨∧∨∧

∨∧∨∧∨∧∨∧∨∧

∨∧∨

� �

�

� ∧∧∨∧∨∧∨∧∧∨∧∨∧

( )�

� � � :�

Y

and so on binomialm

6

2

1 3 6 10 15 21�� ...

O

O

O O

O O O

OO O

num:��

�

�

�

�

1 2 3 4 5

10

6 9

3 5 8

1 2

[ ] ⇒

44 7O �

p p y


5.1.2 Examples: Dualizations

Diagramm 10 Dualization

Duality of sub-system 6 and transpositions of affected neighbor-systemsGreen: not affected, red: dualized, black: transposed sub-systems.Interpretation: sub-systems 1, 9, 10 are isolated in respect to the dualization of sub-

system 4. Thus, sub-system 4 can be logically manipulated without disturbing the basicsub-system 1 and the mediating sub-systems 9 and 10.

The dualizations D1 and D4 are commutative: D1 ( D4 ) = D4 ( D1 ).

5.1.3 Dualization cycles

5.1.4 N, K and D calculus

D4

5 6

3 5 8

2 7

10

9

1 4

( )

∨

∨

∨

∧

∨ ∧ ∨

∧ ∧

∨

:��

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⇒∧

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∧ ∨

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10

9

1

3

6 2 7

5 84

Dualization cycles

D D D D

D D D

�

��1 3 3 1

1 2 1

( ) = ( )( )( )��

��

:�

= ( )( )( )( ) = ( )( )

D D D

D D D D D D

DC D

2 1 2

2 3 2 3 2 3

111 2 1 2 1 2

1 3 1 3

1 2 1 2 1 3 1

2

3

. . . . .

. . .

. . . . . .

:�

:

DC D

DC D.. . . . . . . . . . . . . . . .2 1 2 3 1 2 3 2 1 2 1 2 3 2 1 3

K N X Y

K X N Y

D

i i

i i

i

1

2

�: � � � �

�: � � � �

�:

= ( )= ( )

=�� ,� ,N K K ii i i

1 2 1 2( )( ) =

5.1.5 Proemiality in ConTeXturesDespite the intricate complex-ity of proemial objects Con-TeXtures will be based mainlyon the basic relations of pro-emiality, order, exchangeand for the mediated systemssiml. Without producing con-fusion siml will also be calledthe coincidence relation.

5.1.6 Comparability of proemial objects5.1.6.1 Unary objects

x(3) = (x1, x2, x3), more explicit because it is not a tuple but a mediation andagain visualized by the diagram below.

x(3) = (x1 § x2 § x3)

Signatures of x(3)

T1,3 x §F1,2 x (or F1, T2 x)§F2,3 x (or F2,3 x )

There are 5 unary morphogrammatic objects for a 3x1-matrix with max 3 differ-ent occupations, giving base for all possible interpretations, mapping of obs ontoit.5.1.6.2 Binary Objects

There are 3281 morphograms for a 3x3-matrix with max 3 different occupations(kenograms).

x1 x1

x2 x2

x3 x3

Obj(3) : Obj (3) ––> Obj(3)X

X

X

X

( )

.

.

.

3

1 3

1 2

2 3

( )

T1 F1

T2 F2

F3 T3

X

X

X

X

( )

.

.

.

3

1 3

1 2

2 3

( )

X

X

X

1 1

1 1

1 1

.

.

.

T T

F T

F F

1 3

1 2

2 3

,�,��

,� ,�

,� ,� �

∅

∅

∅

term

comparable incomparable

equal non-equal

p p y

DRAFT 1

6��.


General Framework of PolyLogics

PolyLogics

sketch logic horizon

build arc

m− −

−

( )

hhitectures

interactionalreflectionaldeductioonal

thematize scenar

− iios

choose logic types

classicintuitionistic

− −

ddialogicalmultiple valuedcontextual

−

−decide prooff style

HilbertSmullyanFitchGentzen

−

− −select logic topics

Boooleanclassrelationalreflectionalactional�

identtify frameworks

define operations

abstract

−

−

−

ffunction

propose statements�� −{ }

sketch horizon

build architectures

themat

m−

−

( )

iize scenarios

identify frameworks

define o

−

−

−

pperations

abstract function

propose sta

−

−�� ttements{ }

Tableaux rules for PolyLogics

1 Tableaux rules for junctions

1.1 Pattern [ id, id, id ]

t X Y

t X

t Y

f X Y

f X f Y1

1

1

1

1 1

� � � �

�

�

��

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∧ ∧ ∧ ∧ ∧ ∧

tt X Yt X

t Y

f X Y

f X f2

2

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� � ��

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∧ ∧ ∧ ∧ ∧ ∧

YY

t X Yt X

t Y

f X Y

f

��

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3

3

3

3

∧ ∧ ∧ ∧ ∧ ∧

XX f Y�� 3

t X Y

t X t Y

f X Yf X

f Y

1

1 1

1

1

1

� � � �

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�

∨ ∨ ∨ ∨ ∨ ∨

tt X Y

t X t Y

f X Yf X

f

2

2 2

2

2

2

� � �

� ��

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∨ ∨ ∨ ∨ ∨ ∨

YY

t X Y

t X t Y

f X Yf

��

� � �

� � ��

� � �3

3 3

3∨ ∨ ∨ ∨ ∨ ∨

33

3

�

��

X

f Y

PM O O OM logM logM log

PM O O O1 2 31 12 23 3

1 2∅ ∅

∅ ∅∅ ∅

��

33123

M andM andM and

∅ ∅∅ ∅∅ ∅

JJJ L L L L L L( ) →

( ) ( ) ( ): �*�� :� ,� ,�3 3 31 2 3

LLog L L L

Log L Ljunct J1 1 1 1

2 2 2

: �*� ��

: �*��

→

��

: �*� ��

�

junct J

junct J

L

Log L L

→

→2

3 3 3

�L

3

j


1.2 Pattern [ id, id, red ]

t X Y

t X t Y

f X Yf X

f Y

1

1 1

1

1

1

� � � �

� ��

� � ��

�

∨ ∧ ∨ ∨ ∧ ∨

tt X Y

t X

t Y

f X Y

f X f Y2

2

2

2

2 2

� � �

�

�

��

� ��

∨ ∧ ∨ ∨ ∧ ∨

��

� � �

� � ��

� � �t X Y

t X t Y

f X Yf

3

3 3

3

3

∨ ∧ ∨ ∨ ∧ ∨��

��

X

f Y3

t X Y

t X

t Y

f X Y

f X f Y1

1

1

1

1 1

� � � �

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tt X Y

t X t Y

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f

2

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∧ ∨ ∧ ∧ ∨ ∧

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t X Yt X

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f X Y

f

��

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3

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∧ ∨ ∧ ∧ ∨ ∧

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t X Y

f X f Y f X f Y

f X1

1 1 3 3

1� � � �

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��

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f Y

f X

f Y

t X Y

f X t Y

1

1

3

3

2

2 2

∨→ ∨��

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��

f X Yt X

f Y

2

2

2

∨→ ∨


PM O O O1 2 31 12 23 3

1 2∅ ∅

∅ ∅∅ ∅

��

33123

M orM implM or

∅ ∅∅ ∅

∅ ∅

∨→ ∨( ) → ∅( ) ( ) ( ): �*�� :�( | ),� ,�L L L L L L3 3 31 3 2

→∨

Log L L Ldisjunction1 1 1 1

: �*� ��

��|� �

: �*� ��

L

Log L Limlication

3

2 2 2 → →��

: �*� ��

L

Log L L Ldisjunction

2

3 3 3 1∨ →

1.3 Pattern [ id, id, red ], part II

t X Y

f X t Y f X t Y

f X1

1 1 3 3

1� � � �

� � � ��

� ��→→→ →→→→

→→→

��

��

�

�

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Yt X

f Y

t X

f Y

t X Y

f X t Y

1

1

3

3

3

2 2 ��

� ��

��

�

f X Yt X

f Y

3

2

2

→→→

t X Y

f X t Y

f X Yt X

f

i

i i

i

i

i

� � � �

� � � ��

� ��

�

→⇒→ →⇒→

YY

i

��,� �,�,��= 1 2 3

t X Y

f X t Y f X t Y f X t Y1

1 1 2 2 3 3

� � � �

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→⇒→��

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��

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f X Y

t X

f Y

t X

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t1

1

1

2

2

3

→⇒→

XX

f Y3 �

t X Y

t X f Y t X f Y

f X3

2 2 3 3

3� � �

� � � � ��

� ��←←← ←←←←��

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t Y

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t Y2

2

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t X Y

t X f Y

f X Yf X

t

i

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i

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←⇐← ←⇐←

YY

i

��,� �,�,��= 1 2 3

t X Y

f X t Y f X t Y

f X1

1 1 3 3

1� � � �

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� ��→←→ →←→→

→←→

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t X Y

t X f

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3

3

2 2 ��

� ��

��

�Y

f X Yf X

t Y

3

2

2

→←→

j


1.4 Pattern [id, red, red]

t X Y

f X t Y t X f Y f X t Y1

1 1 2 2 3 3

� � � �

� � � ��

→⇐→��

� ��

�

��

� �

��

�

f X Y

t X

f Y

f X

t Y

t1

1

1

2

2

3

→⇐→

XX

f Y3 �


PM O O O1 2 31 12 23 3

1 2∅ ∅∅ ∅∅ ∅

��

33123

M implM replM impl

∅ ∅∅ ∅∅ ∅

→⇐→( ) →( ) ( ) ( ): �* ��

��:� ,�

L L L

L

3 3 3

1LL L

Log L L

Log Limpl

1 1

1 1 1 1

2

,�

: � � �

:��

→

22 1 1

3 3 1 1

� � �

: � ��

�

repl

impl

L

Log L L

→

→

g g


2 Tableaux rules for negations Neg(3)

2.1 Permutational patterns

t X

f X

t X

t X1 1

3

13

2 13

33

� �

��

� �

�

( )

( )

( )

( )

¬( ) ¬( )��

� �

�

� �

�

( )

( )

( )

( )

t X

t X

f X

t X

3 13

23

1 13

13

¬( )

¬( )��

� �

��

� �( )

( )

( )f X

f X

f X

f2 1

3

33

3 13

2

¬( ) ¬( )�� ( )X 3

t X

t X

t X

f X1 2

3

33

2 23

23

� �

��

� �

�

( )

( )

( )

( )

¬( ) ¬( )��

� �

�

� �

�

( )

( )

( )

( )

t X

t X

f X

f X

3 23

13

1 23

33

¬( )

¬( )��

� �

��

� �( )

( )

( )f X

t X

f X

f2 2

3

23

3 23

1

¬( ) ¬( )�� ( )X 3

t t t X

f t t X

f f f X1 2 3 1

3

1 3 23

1 2 3 1� �

��

� �( )

( )

¬( ) ¬ (( )

( )

( )

(

�

� �

�

3

1 3 23

1 2 3 23

3 2 13

( )

¬( )

t f f X

t t t X

t f t X ))

( )

( )��

� �

�

f f f X

f t f X1 2 3 2

3

3 2 13

¬( )

non L L L L L

Log L1

3 31 3 2

1

( ) →

( ) ( ): �� :� ,� ,�

:11 1 1

2 2 1 3

3

� � �

: � � �

:

��neg

perm

L

Log L L

Log

→

→

LL Lperm3 1 2

� � →

non L L L L L

Log L pe

23 3

3 2 1

1 1

( ) → [ ]( ) ( ): �� :� ,� ,�

: � rrm

neg

L

Log L L

Log

2 3

2 2 2 2

3

��

��

� �

: � � �

:

→

→

LL Lperm3 2 1� �� →

g g


2.2 Bifurcational patterns: Tableaux rules for transjunctionsTransjunctions are the primary interactional operations in polylogical systems.

Diagramm 11 Tableaux rules for (X trans and and Y)

t X Y

t X

t Y

f X Y

f X

f Y

1

1

1

1

1

1

� � ��

�

�

��

�

�

<>∧∧ <>∧∧

tt X Y

t X

t Y

f X

f Y

f X2

2

2

1

1

2� � �

��

��

�

�

�� <>∧∧ <>∧∧∧

<>

�

� ��

��

�

�

��

� �

Y

f X f Yf X

t Y

t X

f Y

t X

2 21

1

1

1

3∧∧∧ <>∧∧�

��

��

�

�

�� Y

t X

t Y

t X

t Y

f X Y

f3

3

1

1

3

3��

�

��

�

�X f Y

f X

t Y

t X

f Y31

1

1

1

PM O O OM log log logM logM log

P1 2 31 1 1 12 23 3

∅ ∅∅ ∅

��

MM O O OM trans trans transM andM and

1 2 3123

∅ ∅∅ ∅

<>∧∧( ) →( ) ( ) ( ): �*�� :� ,�( || ),L L L L L L3 3 31 2 1

LL L

Log L Ltransjunct

3 1

1 1 1

||

: �*� ��

( )

<>→→

→

→� :�

�*�� ,�

�*�� ,�L

f t f f

t f t t11 1 2 3

1 1 1 3

→Log L Lconjunction2 2 2

: �*� ��

LL L

Log L Lconjunction

2 1

3 3 3

�||�

: �*� ��

→ �� ||L L3 1

g g


T1,3 : truth value true for systems 1 an3

f1: value false for system 1 (= f1)f2: value true for system 2 (= t2)

F2,3: values false for systems 2, 3

t: transjunction o: disjunctiona: conjunction(this terminology (o, a, t, i, j ) holdsfor the ML implementation)

Diagramm 12 Tableaux rules for ( or trans or )

Matrix representationof oto

�� (� �)��oof taa

T T T

T f

�(� �)

��

��,1 3 1 3

1 11 2 1 3

3 1 3 2 3

, ,

, ,

��

��

��

��

T

T T F

TT F F

F f F

F F F

1 3 2 3 3

2 3 1 2 2

3 2 2

, ,

, ,

,

��

��

�� 33

t X Y

t X t Yf X

t Y

t X

f Y

1

1 12

2

2

2

� � ��

� ��

��

�

�

��∨<>∨

��

��

��

�

�

� �

f X Y

f X

f Y

t X

t Y

t X

1

1

1

2

2

2

∨<>∨

∨<>∨∨ ∨<>∨

∨

�

�

�

��

�

�

��

� �

Y

t X

t Y

f X Y

f X

f Y

t X

2

2

2

2

2

3<<>∨�

��

��

�

�

��Y

t X t Yf X

t Y

t X

f Y

f

3 32

2

2

2

33

3

3

2

2

� � �

�

��

�

��

X Y

f X

f Y

f X

f Y

∨<>∨

∨<>∨( ) →( ) ( ) ( ): �*�� :�( || ), ,�L L L L L L3 3 31 2 2

LL L

Log L Ldisjunction

3 2

1 1 1

||

: �*� ��

( )

→

→<>

� ||

: �*� ��

L L

Log L Ltransjunct

1 2

2 2 2�� :

�*�� ,�

�*�� ,�L

f t t t

t f t t22 2 1 3

2 2 1 3

→

→

→

Log L L L Ldisjunction3 3 3 3 2

: �*� �� ||

g g


For total transjunctions and junctions.J is representing junctional operators. The compposition of (J<>J) has, as for all other

cases too, to fuliil the conditions of mediation (CM).

t X J J Y

t Jf X

t Y

t X

f Y

f1

12

2

2

2

1� � ��

��

��

�

�

��<> ��

��

�

� � �

�

X J J Y

f Jt X

t Y

t X J J Y

t X

t

<>

<>

12

2

2

2

22

2

2

2

3

3

�

��

�

�

��

� � �

�

Y

f X J J Y

f X

f Y

t X J J Y

t

<>

<>

��

��

�

�

��

�Jf X

t Y

t X

f Y

f X J J Y

f J2

2

2

2

3

3

<>

��

��

f X

f Y2

2

t X JJ Y

t X

t Y

f X JJ Y

f X

f Y

1

1

1

1

1

1

� � ��

�

�

��

�

�

<> <>

tt X JJ Y

Jf X

f Y

f X JJ Y

J

2

21

1

2

2

� � �

��

��

�

�� <> <>

��

��

�

�

��

� � �

� ��

f X

t Y

t X

f Y

t X JJ Y

Jt

1

1

1

1

3

31

<>

��

�

��

��

��

�

�

X

t Y

f X JJ Y

Jf X

t Y

t X

f1

3

31

1

1

1

<>

YY

J J L L L L L L<>( ) →( ) ( ) ( ): �*�� :�( || ), ,�3 3 31 2 2

LL L

Log L L Ljunct J

3 2

1 1 1

||

: �*� ��

( )

→11 2

2 2 2 2

||

: �*� ��

L

Log L L Ltransjunct <>

→ ::�*�� ,�

�*�� ,�

f t t t

t f t t

Lo

2 2 1 3

2 2 1 3

→

→

gg L L L Ljunct J3 3 3 3 2

: �*� �� ||�

→

<>( ) →( ) ( ) ( )JJ L L L L L: �*�� :� ,��

3 3 31 22 1 3 1

1 1 1

|| ,� ||

: �*� ��

L L L

Log L Ltrans

( ) ( )

jjunctL

f t f f

t f�� :�

�*�� ,�

�*�<> →

→1

1 1 2 3

1 1�� ,�

: �*� ��

→

f f

Log L Ltransjunct

2 3

2 2 2 ��

��

� ||

: �*� ��

→ L L

Log L Ljunct J

2 1

3 3 3 �� || →

L L

3 1

g g


2.2.1 A Classification of transjunctions

We can distinguish between total and partial transjunctions.

Total transjunctions can be full total transjunctions (<>) or reduced (<>’, ’<>) totaltransjunctions.

Full total transjunctions can be separated into equivalential or into kontra-valentialfull transjunctions.

Partial transjunction, also called conditional transjunctions, can be classified as rep-licative (<) and implicative (>) partial transjunctions.

Implicative and replicative partial transjunctions can be devided into conjunctive anddisjunctive parts.

For systems with more than 3 contexture a new type of transjunctions and its differ-entiations occurs: the global transjunction. In contrast to the above transjunctions, novalue is repeated in global transjunctions. Each place of the morphogram is occupiedby a different value from different systems.

This terminology is modeled, obviously, on the names of the classical junctions, otherclassifications are in use, too.

g g


2.2.2 Conditional transjunctivity

Conditional transjunction, also called implicative (>) and replicative (<) transjunc-tions, only one part of the two alternatives is chosen.

t X J J Y

t Jt X

f Y

f X J J Y

f

1

12

2

1

1

� � ��

��

�

��

�

> >

��

��

��

� � �

�

Jt X

t Y

f X

t X

t X J J Y

t X

2

2

2

2

2

2

>

��

��

��

�

��

��

t Y

f X

t Y

f X J J Y

f X

f Y

t

2

2

2

2

2

2

>

33

32

2

3

3

� � �

��

�

��

�

X J J Y

t Jt X

f Y

f X J J Y

f

> >

JJf X

f Y��

�

��2

2

t X J J Y

t Jt X

f Y

f X J J Y

f

1

12

2

1

1

� � ��

��

�

��

��

�

��

� � �

�

� ��

��

Jt X

t Y

t X J J Y

t X

t Y

2

2

2

2

2

��

� ��

� � �

��

f X J J Y

f X

t X J J Y

t Jt X

f

2

2

3

32

22

3

3 2

�

��

��

Y

f X J J Y

f J f X

J J L L L L L L L>( ) →( ) ( ) ( ): �*�� :�( || ), ,� |3 3 31 3 2 3

||

: �*� �� ||�

L

Log L L Ljunct J

2

1 1 1 1

( )

→ LL

Log L L Lf

transjunct

2

2 2 2 22

�

: �*� �� :�>

→��*�� ,�

�*�� ,�

:

t t t

t f f t

Log

2 1 3

2 2 1 2

3

→

→

LL L L L

J J

junct J3 3 3 2�*� �� ||

� →

<( )) →( ) ( ) ( ): �*�� :�( || ), ,� ||L L L L L L L L3 3 31 3 2 3 2(( )

→Log L L L L

L

junct J1 1 1 1 2: �*� �� ||

�

oog L L Lf t

transjunct2 2 2 22: �*� �� :�*�

�< → 22 1 2

2 2 1 3

3 3

� �,�

�*�� ,�

: �*

→

→

f t

t f t f

Log L �� ||�

L L Ljunct J3 3 1

→

t X J J Y

t Jf X

t Y

f X J J Y

f

1

12

2

1

1

� � ��

��

�

��

�

< <

��

��

�

� � �

�

� ��

Jt X

t Y

t X

f X

t X J J Y

t X

t Y

2

2

2

2

2

2

2

<

tt X

f Y

f X J J Y

f X

f Y

t X J

2

2

2

2

2

3

�

��

��

�

��

� �

<

< JJ Y

t Jf X

t Y

f X J J Y

f Jf

�

��

�

��

��

32

2

3

32

<

XX

f Y2�

�

�

�

��

�

��

f X

f Y

t X

f Yf X2

2

2

22

=

g g


2.2.3 implicative and replicative transjunctivity

both wrong !!!

Systematics and semantics

t X J J Y

t Jt X

f Y

f X J J Y

f

1

12

2

1

1

� � ��

��

�

��

��

�

��

� � �

�

� ��

��

Jt X

t Y

t X J J Y

t X

t Y

2

2

2

2

2

��

�

��

�

��

��

� � �

f X J J Y

f X

f Y

f X

f Y

t X J J Y

2

2

2

2

2

3

��

�

��

��

t Jt X

f Y

f X J J Y

f Jf X

f32

2

3

32

2��

��

��

Y

f X

f Y2

2

t X J J Y

t Jt X

t Y

f X

t Y

f1

11

1

2

2

1� � ��

��

��

�

�

��XX J J Y

f Jf X

f Y

t X

f Y

t X J J Y

� �

��

��

�

� � �

11

1

2

2

2

tt X

t Y

f X J J Y

f X

f Y

t X J

2

2

2

2

2

3

�

� ��

��

�

��

� � J Y

t Jf X

t Y

f X J J Y

f Jf

�

��

�

��

��3

2

2

3

32��

��

X

f Y2

J J L L L L L L L( ) →( ) ( ) ( ): �*�� :�( || ), ,� |3 3 31 3 2 3

||

: �*� �� ||�

L

Log L L Ljunct J

2

1 1 1 1

( )

→ LL

Log L L Lf

transjunct

2

2 2 2 22

�

: �*� �� :�

→��*�� ,�

�*�� ,�

:

t f f

t f t t

Log

2 2 3

2 2 1 3

3

→

→

LL L L L

J J

junct J3 3 3 2�*� �� ||

� →

( )) →( ) ( ) ( ): �*�� :�( || ), ,� ||L L L L L L L L3 3 31 3 2 3 2(( )

→Log L L L L

L

junct J1 1 1 1 2: �*� �� ||

�

oog L L Lf t

transjunct2 2 2 22: �*� �� :�*�

� → 22 1 3

2 2 2 1

3 3

� �,�

�*�� ,�

: �*

→

→

t t

t f t f

Log L �� ||�

L L Ljunct J3 3 1

→

g g


2.2.4 Reductional transjunctive constellations

Total transjunctions, but reduced to only one sub-system interaction.

Reduced transjunctions occurs for all types of transjunctions. It has to be applied gen-erally for all 8 transjunctional types.

>’

t X J J Y

t Jt X

f Y

f X

t Y

f1

12

2

2

2

1� � ��

��

��

�

��

��> ��

��

�

� � �

�

�

X J J Y

f Jt X

t Y

t X J J Y

t X

t

>

>

12

2

2

2

2YY

f X J J Y

f X

f Y

t X J J Y

t J

��

�

�

��

� � �

��

2

2

2

3

3

>

>��

� � �

��

f X J J Y

f J3

3

>

t X J J Y

t J

f X J J Y

f J

t X J

1

1

1

1

2

� � ��

��

� � �

��

� �

< <

<< <J Y

t X

t Y

f X J J Y

f X

f Y

t X J

�

�

�

��

�

�

��

� �

2

2

2

2

2

3<< <J Y

t Jt X

f Y

f X

t Y

f X J J�

��

��

�

�

��

32

2

2

2

3YY

f Jf X

f Y32

2

��

��

g g


2.2.5 Composed transjunctional constellationsLogical operators with more than one transjunction can be composed out of the gen-

eral tableaux rules for single transjunctions. Like for all mediated compositions theyhave to fulfil the conditions of mediation (CM).

Systematic semantics

Archtectonics

Composition of transjunctions

trans J J

� �

, , ��( ) ⊕ JJ trans J trans trans JExample

J

, , �� , ,:

, ,

( ) = ( )

<> JJ J J J

J J CM and

( ) ⊕ <>( ) = <> <>( )<>( )∈

�� , , �� , ,

, , � �JJ J CM J CM, , � � , , �<>( )∈ ⇒ <> <>( )∈

<><>( ) →( ) ( )J L Ltranstransjunct

: �*�� 3 3 LL L L L L L L L31 2 2 1 3 2 1

( ) ( ) ( ):�( || ),� ||� ,� || ||�

→<>

Log L L Ltransjunct1 1 1 1

: �*� �� ||�LLf t f f

t f f f21 1 2 3

1 1 2 3

��:��*�� ,�

�*�� ,��

→

→

→<>

Log L L Ltransjunct2 2 2 2

: �*� ��

|||� �:�*�� ,�

�*�� ,�L

f t t t

t f t t12 2 1 3

2 2 1 3

→

→

→Log L L L L Ljunct J3 3 3 3 2

: �*� �� || �||�� 11

�

PM O O OM log log logM logM log

P1 2 31 1 1 12 23 3

∅ ∅∅ ∅

��

MM O O OM logM log log logM log

1 2 31 12 2 2 23 3

∅ ∅

∅ ∅

��

PM O O OM log log logM log log logM

1 2 31 1 1 12 2 2 233 3∅ ∅

< >

log

Op trans J J Op

��

,�,� :�� << > <J trans J Op trans trans,� ,� :�� ,� ,�JJPM O O OM tran trans transM JM J

P>

∅ ∅∅ ∅

:

��

1 2 3123

MM O O OM JM trans tran transM J

PM O O1 2 3123

1 2∅ ∅

∅ ∅

��

OOM tran trans transM trans tran transM J

3123 ∅ ∅

g g


Diagramm 13 Composition of a double transjunction tableaux

Op trans J J Op J trans J Op tra< > ⊕ < > = <,�,� �� ,� ,� �� nns trans Jt X JJ Y

t X

t Y

t,� ,� :

� � ��

�

�

��

><>

1

1

1

11

12

2

2

2

� � ��

��

��

�

�

��X J J Y

t Jf X

t Y

t X

f Y

<>��

� � ��

�

��

�

��

�

�

t X J Y

t X

t Y

f X

t Y

t X

f Y

1

1

1

2

2

2

2

<><>

ff X JJ Y

f X

f Y

f X J J Y

f J

1

1

1

1

1

� � �

�

�

��

��

<> <>

tt X

t Y

f X J Y

f X

f Y2

2

1

1

1

�

�

��

��

�

<><>

��

�

� � �

��

�

��

t X

t Y

t X JJ Y

t Jf X

f Y

t

2

2

2

21

1

<>22

2

2

2� � �

�

�

�� X J J Y

t X

t Y

t X J<> <><> YY

t X

t Y

f X

f Y

f X J J Y

f X

f Y

��

��

�

�

� � �

�

�

��

2

2

1

1

2

2

2

<>��

� � �

��

��

�

�

��f X JJ Y

f Jf X

t Y

t X

f Y

2

21

1

1

1

<>��

� � �

�

��

�

��

f X J Y

f X

f Y

f X

t Y

t2

2

2

1

1

1

<><>

��

�

��

� � �

��

�

��

X

f Y

t X JJ Y

t Jt X

t Y

t X

1

3

31

1

3<> JJ J Y

t Jf X

t Y

t X

f Y

t<> �

��

��

�

�

��

32

2

2

2

3��

��

��

�

��

�

X J Y

t Jt X

t Y

f X

t Y

t X

<><>

31

1

2

2

2

ff Y

f X JJ Y

f Jf X

t Y

t X

f

2

3

31

1

1

�

��

� � �

��

��

�

<>

11

3

32

2�

��

��

��

��

Y

f X J J Y

f Jf X

f Y

<>��

� � �

��

��

�

��

f X J Y

f Jf X

f Y

f X

t Y

3

32

2

1

1

<><>

tt X

f Y1

1

�

�

g g


Composition as definition

The composition operation Op+ is not simply a conjunction between two formulasbecause it overwrites the J-part.

But what does a conjunction of the parts mean?

Part 1: Tabl1

H X J J Y X Y

t X J J Y

� � � � � � ��

� � � �

= <>( ) ∧∧∧ <>∧∧( )<>( )

1∧∧∧∧ <>∧∧( )

<>( )

<>∧∧(

� � ��

� � � �

� � ��

X Y

t X J J Y

t X Y1

1))

t X

t Y

t Jf X

t Y

t X

f Y

1

1

12

2

2

2

�

�

��

��

�

�

f X J J Y X Y X J1� � � � � � �� <>( ) ∧∧∧ <>∧∧( )( ) →→→ <><> ��

� � � � � � ��

� � �

Y

t X J J Y X Y

f X J1

1

<>( ) ∧∧∧ <>∧∧( )

<><> YY

t X J J Y

t X Y

t X

t Y

t J

1

1

1

1

1

� � � �

� � ��

�

�

��

<>( )

<>∧∧( )

��

��

�

�

� �

��

�

�

f X

t Y

t X

f Y

f X

f Y

xx

t X

t Y

xx

2

2

2

2

1

1

2

2

g g


Part 2: Tabl2

f X J J Y X Y X J1� � � � � � �� <>( ) ∧∧∧ <>∧∧( )( ) →→→ <><> ��

� � � � � � ��

� � �

Y

t X J J Y X Y

f X J3

3

<>( ) ∧∧∧ <>∧∧( )

<><> YY

t X J J Y

t X JJ Y

t Jt X

3

3

31

� � � �

� � ��

��

�

<>( )

<>( )

tt Y

t Jf X

t Y

t X

f Y

1

31

1

1

1

�

��

��

�

�

�?��xxx xx

f Jf X

t Y

t X

f Y

f X

f

��

��

��

�

��

�3

1

1

1

1

2

22�

�� ??� � �

Y

xx xx xx closed w⇒ iith Tabl�1

g g


2.3 Rules and Definitions in PCL(3)

2.3.1 Unary and binary constellations

2.3.2 Dualities for junctions

2.3.3 Dualities for transjunctions

¬ ¬( ) ⇔⇔⇔ =

¬ ¬ ¬

i i X X i

X

� � � � ,� ,�

� � ��

( ) ( )

(

3 3

1 2 13

1 2

)) ( )� � �� ( )( ) ⇔⇔⇔ ¬ ¬ ¬( )( )2 1 23X

¬ = ¬ ¬( )¬ = ¬ ¬

33

1 23

43

2 1

� �: � � �

� �: � �(�

( )�

( )

( )�

X X

X XX

X X

( )

( )�

( )

�)

� �: � � �

��

3

53

1 2 13¬ = ¬ ¬ ¬( )( )

��: � � ��( )= ¬ ¬ ¬( )( )2 1 23X

¬ ∨ = → ⊕⊕

¬ ∨1

3 3 3 3 3

23

� � � � � � � �

� �

( ) ( ) ( ) ( ) ( )

( )

X Y X Y

X (( ) ( ) ( ) ( )� � � � �3 3 3 3Y X Y= ∨→ ∨

¬ ¬ ∧ ∨ ∧ ¬( ) ⇔⇔⇔ ∨∧∨5 53

53 3 3� � �� ( ) ( ) ( ) ( )X Y X Y

¬ ¬ ∧ ∨ ∧ ¬( ) ⇔⇔⇔ ∨∧∨5 53

53 3 3� � �� ( ) ( ) ( ) ( )X Y X Y

¬ ¬ ⊕⊕⊕ ¬( ) ⇔⇔⇔ ⊗⊗⊗5 53

53 3 3� � � � � � � � �

�

( ) ( ) ( ) ( )X Y X Y

�� ,� � � ,�⊕ ⊗ = ∧ ∨{ }

g g


Full total transjunction defined as a conjunction of 2 reductional transjunctions. (???)

Ternary and n-ary constellations

X J J Y X J J Y X J J Y� � �: � � � � � � �<>( ) === >( ) ∧∧∧ <( )

g g


2.4 Tableaux proofs in PolyLogics2.4.1 Junctional and negational constellations

At a first glance it seems that our job can easily be realized by combining logics andto combine logics to combined logics, PolyLogics, seems to be an equally easy job. Asusual, the devil is in the detail, and because PolyLogics are introduced on a very basiclevel, we are confronted, step by step, by such concrete problems.

polylog id id red

thematize Boole

( ) ( , ,� �)

�(�

3

aan definition

identify frames

lambda

,� �)

�

(

( )3

�� )

(� �)

(� �)

( ) ( )

( )

X Y

lambda neg

lambda or

3 3

2

3

iidentify frame

define or

lambda X Y

�

( )

.1 1

1

( � � � )perm X or Y21

�

� .identify frame

define impl

lam

2 2

2

bbda X Y

neg X or Y

( )

�( � � � )22

identify frame

defi

� .3 1

nne or

lambda X Y

perm X or Y

3

23

( )

( � � � )

¬ ∨ = ∨→ ∨23 3 3 3 3� � � � � � �( ) ( ) ( ) ( ) ( )X Y X Y


PM O O O1 2 31 12 23 3

1 2∅ ∅

∅ ∅∅ ∅

��

33123

M orM implM or

∅ ∅∅ ∅

∅ ∅

g g


2.4.2 Junctional constellations without negations

0 3 113 113 113.� � � � �� X Y Z X Y∧( ) → ≡ → →( ) ( ) ( ) ( ) ZZ

f X Y Z X

( )

∧( ) → → →( ) ( ) ( )11

3 113 113 1

.�� . .� �113 113

0

21

3

( ) ( )

→( )( ) ( )

∧( ) →( )

� � � ��

.��

Y Z

t X Y 33

13 3

1

3 1

4

( )

( ) ( )

( )

→ →( ) ( )��

.��

Z

f X Y Z

..��

.��

t X Y

f Z1

3

1

2

5

∧( ) ( )( )

��

.��

2

61

( )

t X ��

.��

.�

4

7 4

8

1

( )

( )t Y

ff X

t Y Z

f Y t Z1

13

1 1

3

8� �

��

��

→( ) ( )( )

(( )

�

�� xx xx xx

g g


2.4.3 Junctional constellations with negations

Diagramm 14 Junctional formula

Incomparability occursfor the signatures f1and t3, also for f3 andt1. The tableau is decom-posing the mediatedformula into its sub-sys-tems. Therefore the in-dices of the signatureshave to be taken intoaccount. There is noglobal signature frothem, say t and f, with-out indices.In the tree example,the notational frame of

different sub-systems OiMi is not used because the indices are doing the job. Otherwise it would be evenmore clear that the signaturesand terms are not comparableas the diagram shows.That is, the branches to com-pare are localized in differentsub-systems OiMj.

Our tableau development ofthe formula shows us that weare running into incompatibili-ties disproving the formula.

But, interestingly, the definition of the local implication impl2 can be verified.

f X Y X Y1 23 3 3 3 3� � � � � � � �( ) ( ) ( ) ( ) ( )¬ ∨( ) →→→ ∨→ ∨( )( ))

¬ ∨( )∨→ ∨( )

t X Y

f X Y

1 23 3 3

13 3

� � � �

� � �

�

( ) ( ) ( )

( ) ( )

��

��

�

( )

( )

f X

f Y

t

13

13

1 ¬¬

¬ ∨

23

33 1

3

3 23

� �

�

??

��

��

� � �

( )

( )( )

( ) (

X

t Xt Y

xx

t X 33 3

33 3

3

) ( )

( ) ( )

�

� � �

��

Y

f X Y

f X

( )∨→ ∨( )

(( )

( )

( )

( )

��

� �

�

??

�

3

33

3 23

13

f Y

t X

t X

¬tt Y

xx3

3� ( )

M1 M2 M3 M1 M2 M3 M1 M2 M3

O1 O2

G103 G000 G103

# #

O3

# # #

g g


The environment ofimpl2 is not to satisfy.Asymmetric permuta-tions in binary logicalfunction, (neg X opnY), are often destroy-ing the harmony ofmed ia t ion o f t hewhole function. From a local point ofview, the definition ofimpl via neg plus disjis working.Obviously, the formulahas to be set into a

more complex environment to fulfill the conditions of mediation for the whole complex-ion. Another approach is to rethink the definitions of the involved implications.

Other methods of formalizing PCL may use additionally global values to tackle theproblem. But global values don’t tell us much about the local conditions. A mechanismof mappings could be introduced which is negotiating between local and global view-points and tries to resolve the local problem of disturbed harmony and incompatibility.But this would involve additional operators not yet accessible in the presented frame-works.

Comparability: Contradiction and Distance

Contradiction in PolyLogics is, until now, defined strictly local. The global aspectcomes into play only with the collection of local situations. Thus, a formula is globallytrue iff it is true for all local sub-logics. There is nothing wrong with that.

Until now, strictly global considerations had been reductional, denying the localsubs-system attributes. Thus reducing truth values to natural numbers as a set of truthvalues. Like in usual multi-valued logics, but surely with different use for functions.

A new approach can be introduced with the concept of distance between locally in-comparable truth values. If a term is truei for systemi and there is a term in systemj withvalue falsej and both occur in a common development, a new systemk can be intro-duced reflecting in systemk on the constellation (truei , falsej) as the constellation (truek,falsek) and observing a contradiction in systemk with a specific distance between sys-temi and systemj.

Thus, agent of systemi insists on true, agent of systemj insists on false.And for an agent of a systemk , which is on a higher or lower level than both, it can

be comparable because both are arguing still in the framework of semantics of truthand false. From the 3rd position it turns out to be a contradiction. But this new interpre-tation is not denying the incomparability between the local systemi and systemj.

It could be called a negotional mediation. And is a resolution of a global conflictivesituation. This negotiation as a re-interpretation happens between different local sys-tems and is therefore not itself local but global. But the result of the re-interpretation isrealized and localized in an own local system, thus, again local.

Thus, (truei, falsej)=incomparabel, becomes (truek, falsek)=contradiction.Obviously, the old constellation has to be enlarged and re-interpreted by this model-

ing procedure. For this reason too, it isn’t a reduction to global numeric truth values.

f X Y X Y3 23 3 3 3 3� � � � � � � �( ) ( ) ( ) ( ) ( )¬ ∨( ) →→→ ∨→ ∨( )( ))

¬ ∨( )∨→ ∨( )

t X Y

f X Y

2 23 3 3

23 3

� � � �

� � �

�

( ) ( ) ( )

( ) ( )

��

��

�

( )

( )

t X

f Y

t

23

23

2 ¬223

23 2

3�

� ��

( )

( )

( )X

f X

xx

t Y

xx

g g


2.4.4 Balanced junctional formulas

Diagramm 15 A DeMorgan formula

f X Y X Y1 2 23 3

23 3� � � � � � � � �( ) ( ) ( ) ( ) (¬ ¬ ∨ ¬( ) →→→ ∨∧∨ 33

1 2 23 3

23

)

( ) ( ) ( )� � � � �

��

( )( )¬ ¬ ∨ ¬( )t X Y

f113 3

13

� � �

��

( ) ( )

(

X Y

f X

∨ ∧ ∨( )))

( )

( ) (

��

� � �

f Y

t X

13

1 2 23¬ ¬ ∨ 33

23

3 23 3

23

3

) ( )

( ) ( ) ( )

� � �

� � � � �

�

¬( )¬ ∨ ¬( )¬

Y

t X Y

t 223

13

3 23

13

3

�

� ��

� �

�

�

�

��

( )

( )

( )

( )

X

t X

xx

t Y

t Y

xx

t

¬

��

� � �

( ) ( ) ( )

( ) ( )

¬ ¬ ∨ ¬( )∨ ∧ ∨(

2 23 3

23

33 3

X Y

f X Y ))

¬

��

��

�

( )

( )

f X

f Y

t

33

33

1 223 3

23

1 23

33

� � � �

� � �

� �

( ) ( ) ( )

( )

( )

X Y

t X

t X

xx

∨ ¬( )¬ tt Y

t Y

xx

1 23

33

� �

�

( )

( )

¬

f X Y X Y3 2 23 3

23 3 3� � � � � � � �( ) ( ) ( ) ( ) (¬ ¬ ∨ ¬( ) →→→ ∨∧∨ ))

( ) ( ) ( )

( )

�

� � � � �

�

( )( )¬ ¬ ∨ ¬( )¬

t X Y

f X

2 2 23 3

23

2 23 ��

� � �

� �

( ) ( )

( ) ( )

(

∨ ¬( )∨ ∧ ∨( )

¬

32

3

23 3

2 23

Y

f X Y

f X ))

( )

( )

( )

( ) (

� �

�

�

��

�

f Y

t X

t Y

f X

xx

f Y

2 23

23

23

23

23

¬

))

��

xx

g g


Notational convention

If we know in which polylogical constellation we are working, all sorts of indices,indicating complexity, type of logic, etc., can be omitted.

If a formula is properly decomposable in its sub-systems we obviously also can omitits neighbor operators and variables and deal with the separated sub-systems only.

f X Y X Y

t3 2 2 2

2 2 2

� � � � � � � �

�

�

¬ ¬ ∨∨∨ ¬( ) →→→ ∨∧∨( )( )¬ ¬ ��

� ��

� ��

� �

X Y

f X Y

f X Y

f

∨ ¬( )¬ ∨ ¬( )∧( )

¬

2

2 2 2

2

2 2XX

f Y

t X

t Y

f X

xx

f Y

xx

2 2

2

2

2 2

� �

�

�

��

�

��

¬

��

g g


2.4.5 Tautology propertyThe above formula H is closing for both signatures F1 and F3. The formula H is a

tautology iff its tableaux Tabl close in all branches for the signatures F1 and F3.A branch is closed iff it contains the signatures Ti, Fi for the same terms.A tableaux is a connected graph with a tree structure. That is, with an origin and a

succession of branches connected to the origin.

A polylogical complexion, a rhizome, is a ordered list of trees with an ordered setof origins. The order of the ordered set of origins is ruled by the proemial relation.

To use more familiar terminology we can say, a polylogical complexion is represent-ed by a list of trees. More exactly it is a polycontextural list of trees. Polycontextural,because the listed trees are not only listed but mediated.

Also the tautology test or other formula development can be done successively, es-pecially in the manual case of formula developments, it is ruled by definition that thesedifferent developments have to be realized simultaneously, say in parallel. It is thesame formula which is tested for the different sub-system related semantic attributes.This has consequences for the implementation not considered in the present presenta-tion. Formally the consequences are related to the fact of polycontexturally distributedsyntactical induction principle. The decomposition steps are ruled by induction andtherefore each sub-system has its own induction.

The definition of the tautology attribute thus can be condensed to the following for-mulation:

2.4.6 Transjunctional constellations without negations

Pattern id red S S S S� ,� :� � � �;� �;�{ } →123 113 133 1311

3 1 3 3

( )∈( ) ( )

:

�H taut iff ETF H simul ETF HH 3( )

H taut iff E T F F H 13 133 3 3 1 3 3( ) ( ) ( ) ( )∈�

g g


2.5 Reductional constellations with negationsIncompatible situations turned into reductions.

� ��incompatibility ��

�

,� ,�

reduction rules

p p3 1 3( ) ( )∧ ∧ ∧ ¬ ≡ ⊥ ∅ ∅[ ]pp p

p p

3 2 3

3 3 3

( ) ( )

( ) ( )

∧ ∧ ∧ ¬ ≡ ∅ ⊥ ∅[ ]∧ ∧ ∧ ¬ ≡ ∅ ∅

,� ,�

,� ,��

,� ,�

⊥[ ]∨ ∧ ∧ ¬ ≡ ∅ ∅[ ]∧ ∨ ∧ ¬ ≡ ∅

( ) ( )

( ) ( )

p p T

p p

3 1 3

3 2 3 ,,� ,�

,� ,�

�

��

T

p p T

p

∅[ ]∧ ∧ ∨ ¬ ≡ ∅ ∅[ ]

∧ ∧ ∧

( ) ( )

( )

3 3 3

3 ¬¬ ≡ ⊥

∨ ∧ ∧ ¬ ≡

( )

( ) ( )

1 3 2 2

3 1 3 3

p p p

p p T p p

,� ,�

,� ,� 33

3 2 3 1 1

3 2

∧ ∨ ∧ ¬ ≡ ⊥

∧ ∨ ∧ ¬

( ) ( )

( )

p p p p

p p

,� ,�

33 3 3

3 3 3 1 1 1

( )

( ) ( )

≡

∧ ∧ ∨ ¬ ≡ ¬

p T p

p p p p

,� ,�

� ,� ,�⊥⊥

∧ ∧ ∨ ¬ ≡ ¬ ( ) ( )� ,� ,�

�

p p p p T3 3 3 2 2 2

PM O O O

M

M p

M p

PM O O O

M T

M

1 2 3

1

2

3

1 2 3

1

2

⊥ ∅ ∅

∅ ∅

∅ ∅

∅ ∅

∅��

∅∅

∅ ∅

∅

∅ ⊥ ∅

∅ ∅ ∅

p

M p

PM O O O

M p p

M

M

PM

3

1 2 3

1

2

3

��

��

��

OO O O

M

M T

M p p

PM O O O

M p p

M

1 2 3

1

2

3

1 2 3

1

2

∅ ∅ ∅

∅ ∅

∅

¬ ∅

∅ ∅

��

∅∅

∅ ∅ ⊥

∅ ∅ ∅

¬ ∅

∅ ∅M

PM O O O

M

M p p

M T3

1 2 3

1

2

3

��

g g


2.6 Transjunctional constellations and tableaux proofsStep-wise concretization of the presentation. First, simple formulas can be written

without considering the OM-structures, using only the rules of the sub-system-indices ofthe formulas. Second, especially for transjunctional formulas, the sub-system structure,O (=S), is involved but omitting the M-structure. Third, for full interactional and reflec-tional formulas, the whole OM-structure has to be used. The following diagrams showthe development of a simple negational and transjunctional formula, using only the sub-system structure. All those notational forms are for manual use only and to guide nec-essary further implementations. A further step follows by the application of meta-rulesof the term calculus. Finally, a semi-automated proof by the LOLA-implementation is pre-sented. A machine oriented presentation can be set into a different scheme.

Diagramm 16 Tableau presentation

Nr S1

1

2

3

S2 S3

t3 X tr.et.et Y

f3 N5 ( N5 X vel.tr.vel N5 Y)

t3 N5 X vel.tr.vel N5 Y

t3 X

t3 Y

Nr

(0)

(0)

(2)

(1)

(1)

(3)

(6)

t3 N5 X t3 N5 Y

f3 X f3 Y

t2 N5 X

f2 N5 Y

f2 N5 X

t2 N5 Y

(3)

t1 X

t1 Y (1)

(1)

S3

f1 X t1 X

t1 Y f1 Y

(6)

(7)

(S3)

(S2)

(S3)

x x

x x

4

5

6

7

8

9

10

(0) f3 H1 = f3 ((X tr.et.et Y ) .iij. N5 ( N5 X vel.tr.vel N5 Y ))

(�,��,�� )(� ,�,�� )(�,�� ,� �)(

∅ ∅∅

∅

SS S

S S

31 3

2 3�� ,�,�� )

(�,��,��)

S S

x x

1 3∅

∅

S3

S3S1|

S3S2

S2S1

|

�,�,� �

� ,�,� �

�,� ,�

∅ ∅( )↓

∅( )↓

∅

S M

S M S M

S M

3 3

1 3 3 3

2 3 SS M

S M S M

3 3

1 3 3 3

�

� ,�,� �

( )↓

∅( )

Unification

��

(� � )

(� � )�

( )α

α γ

β δ

3

3 1

3 1

(3)

trans

trans

neg

rules:

tr: transjunctionet: conjunctionvel: disjunction

i,j: implicationS3

g g


Diagramm 17 Tableaux presentation

Sub-systems S1 and S2 are closing directly, S1 at step 8 and S2 at step 9. This wouldbe enough to close the tableaux for S1 and S2. But there is an additional part of theformula which is closing separately in S1, closing at step 13, encircled in red.

The different structural diagrams above should show clear enough how the formuladevelopment is working. The step-wise development of the formula guarantees the con-nectedness of the branches of the different trees, despite the jumps into other sub-sys-tems, which are necessary to produce a semantic result on the base of the signatures.

Nr S1

1

2

3

S2


t2 N5 X vel.tr.vel N5 Y (2)

Nr

(0)

(0)

(2)

(3)

(3)

(4)

(5)

S2

4

5

6

7

8

9

10

(0) f1 H1 = f1 ((X tr.et.et Y ) .iij. N5 ( N5 X vel.tr.vel N5 Y ))

t1 X tr.et.et Y t2 X tr.et.et Y


t1 X (1)

t1 Y (1) t2 N5 X (3)

t2 N5 Y (3)f1 X (4)

f1 Y (5)

x

t2 X (1)

t2 Y (1)

f1 X (1 )

f1 Y (1)

t1 N5 X | t1 N5 Y ( 8)

11

12

13

(10)

f2 X | f2 Y (9)

f2 N5 X t2 N5 Xt2 N5 Y f2 N5 Y

(8)(8) t1 X f1 X

f1 Y t1 Y ------------- x x

t1 N5 X vel.tr.vel N5 Y

--- --- x x

x

x

x

------------

M1 M2 M3 M1 M2 M3 M1 M2 M3

O1 O2

G110 G120 G000

#

O3

# # #

xxxx

x

x

#

g g


DRAFT


2.6.1 Reduction of complexity by unification and meta-rules

The concrete complexity of the tableaux tree of formula H1 can be reduced with thehelp of the unification method of Smullyan and, as a step further, with the applicationof some meta-rules over tableaux trees, that is, the term rules of polylogic. The diagramstructure is represented by the indices of the sub-systems only.

Diagramm 18

Unification tableaux tree development of f1H1

The example of unification preserves the tree structure induced by the formula. Thenext example, which applies additionally the term rule R1, is reducing and separatingjunctional and transjunctional parts, and therefore, offering a better economy of thesub-system parts which in the latter example are still distributed over the whole tree.

Diagramm 19

Unification of f1H1 with meta-rules

unv f H( � )�:

��

11

3( )

��

α3( )��

�� α31

��

��

α

α

32

33 1 3 2��

��

δ β γ( ) ( )��

�α δ3

1 11

��

�� α δ32 2

1 ββ βγ

13

23

1

��

��

( )

��

��

��γ

γ

γ

γ11

21

31

41

Term rule R realized as

simul et

� � � � :

��

1

α δ( ) ��

��

β γ

α β δ γ

simul

et simul et

( )( ) ( ))��

nv f H( � )�:��

11

3( )

��

α3( )��

�� α31

��

��

α

α

β

32

3

33

1

1

��

��

δ

γ��

��α

α

31

332

11

21

��

��

δ

δ

�� β β

γ

γ

γ

γ13

2

3 11

21

31

41

g g


DRAFT


The tableau tree for f3H1 is even more confusing in its full concrete manual develop-ment, which first has to be studied before it can be reduced with the application ofmeta-rules to the following unification.

Diagramm 20

Unification tableaux tree development of f3H1

Again, the meta-rules are applied, producing a simple house holding of the branchesand their sub-systems.

Diagramm 21

Unification of f3H1 with meta-rules

System changes, represented asa change in the index of the for-mula, say f rom gamma2 togamma1, are produced by logi-cal negators. Their rules are givenwith the tableaux rules for nega-tions of signed formulas. Signedformulas are very convenient forcomplex logics but the polylogi-cal rules are not depending onthe method of signed formulas.

Each method which is keeping the sub-system indices right is doing the job.

Comment:

dec: decomposition of the complexion into alpha1 and alpha2 parts,tabl: tableaux rules, producing intra-contextural sub-formulas,R1: term rule R1, collecting junctional and transjunctional parts separately.

unv f H( � )�:

��

31

3( )

��

α α1 2( )��

��

α α1 2( ) ( )��

�� α α α11

21

12 ��

��

α

α α

22

1 2( ) (( ) ( ) ( )��

��

α δ β2 1 1

�� α α α δ β11 1

12

11 2( ) ��

��

γ

α

2

21

( )αα α δ

β βγ1

122

21

12

22

1��

� ��

��

( )

��

α21

��

��

γ

γ

γ

γ11

21

31

41

�� α α1 2( )

�� dec α α1( ) 22( )

��

��tabl α11 2 1 2 2

11

( ) ( ) ( )

( )

�� ||� �� ||��

�� |�

α δ β γ

αR �� α

β

δ

γ

2

2

1

1

��

g g


DRAFT


2.7 Term representation of formula development H1

The formula development can be represented by a term calculus. The example in-cludes rule R

0

and R

1

of the term list below. This, again, is a linearizations of the nota-tional approach, but it is exactly what we need for machine readability of animplemented tableaux prover even if it has a tree representation to comfort the user.

Once, the rules of polylogical systems are clear, it is necessary to realize a (semi-)automated prove system. It makes not much sense to go one with manual work. Proofsystems like LOLA are supporting the experimental exploration of polylogics, their re-sults can be implemented to improve the theorem prover. Until now a strict mathemat-ical approach hasn’t produced much valuable insights beyond existing experimentalknowledge.

Diagramm 22

Term development for f

3

H1

The structure of step-wise develop-ment of the terms can be shown bybrackets only and using the LOLAnotation. Term developments can beproduced semi-automatically by LO-LA.

f H

t X et t Y sim t X et tY

3

3 3 1 11

� :

.� � � � � � �( ) ( )

�� et f X et fY sim f X et tY or t X e

3 3 1 1 1( ) ( ) tt fY

t X et t Y et f

�

.� � � � �

1

3 3 32

( )( )

( ) XX et fY sim t X et tY et f X e� � � � � � � � �3 1 1 1( )

( ) tt tY or t X et fY R� � � � � �:� / .

.�

1 1 1 11

3

( ) ( )( )

∅∅ ( ) ( )(3 1 1 1 1� � � � � � � �sim t X et tY et f X et tY )) ( ) ( )( )

� � � � � � � �or t X et tY et t X et fY1 1 1 1

∅ ∅

��:� / .

.� � �

R

sim

0

3 1

2

4 ∅

( )

∅ ∅

� �

.� � �

or

sim

1

35

11

f H3

1

� :

.� �|||� �& &� �|||� �||�( ) ( )

( ) ( ) ( )( ))

( ) ( )[ ] ( ) ( )2.� �& &� �|||� �& &� �||��

.� �|||� �& &� �||�

( )( )

∅

( ) ( )( )33

(( ) ( )( )

∅ ∅

�& &� �

.� �|||�43 1

∅

( )

∅ ∅

�||�

.� �|||�

1

35

11

Term rule R realized as

simul et

� � � � :

��

1

α δ( ) ��

��

β γ

α β δ γ

simul

et simul et

( )( ) ( ))��

g g


DRAFT


Diagramm 23

Semi-automated tree-proof of f3H in LOLA (1992)

f3: ((X taa Y) iij n5 (n5 X oto n5 Y))

|

+------ && ----------+

| |

t3: (X taa Y) f3: n5 (n5 X oto n5 Y)

|

+------------------- && ---------+

| |

| t3: (n5 X oto n5 Y)

+------- /// -------+

| |

| +--------------+

+-- && --+ | | |

| | | +-- && --+ |

t3: X t3: Y | | | |

|t1: X t1: Y|

+--------------+

|

+--------------------------------------------------------------------------- /// -------+

| |

| +--------------+

+---------- && -----------------------+ | | |

| | | +-- && --+ |

| | | | | |

+---- && ---+ +---------- /// ----------------------+ |t1: X t1: Y|

| | | | +--------------+

+------+ +------+ | +--------------------------------------------+

|[t3:X]| |[t3:Y]| +---- || ---+ | | |

+------+ +------+ | | | +---------- || ---------+ |

t3: n5 X t3: n5 Y | | | |

| | | |

| +---- && ---+ +---- && ---+ |

| | | | | |

|f2: n5 X t2: n5 Y t2: n5 X f2: n5 Y|

+--------------------------------------------+

|

+------------------------------------------------- /// -------+

| |

| +--------------+

+---------------- /// ----------------------+ | | |

| | | +-- && --+ |

| +--------------------------------------------+ | | | |

+------- && ------+ | | | |t1: X t1: Y|

| | | +---------- || ---------+ | +--------------+

+------------+ | | | | |

|[t3:X, t3:Y]| +-- || --+ | | | |

+------------+ | | | +---- && ---+ +---- && ---+ |

f3: X f3: Y | | | | | |

|f2: n5 X t2: n5 Y t2: n5 X f2: n5 Y|

+--------------------------------------------+

|

+---------------------- /// -------------------------------+

| |

| +--------------------------------------------------------------+

+------- && ---------+ | | |

| | | +------- && ---------------------+ |

+------------+ | | | | |

|[t3:X, t3:Y]| +---- || ---+ | | | |

+------------+ | | | +-- && --+ +---------- || ---------+ |

+------+ +------+ | | | | | |

|[f3:X]| |[f3:Y]| |t1: X t1: Y | | |

g g


DRAFT


+------+ +------+ | +---- && ---+ +---- && ---+ |

| | | | | |

| f2: n5 X t2: n5 Y t2: n5 X f2: n5 Y|

+--------------------------------------------------------------+

|

+---------- /// -------------------------------+

| |

| +--------------------------------------------------------------+

+------- && ---+ | | |

| | | +------- && ---------------------+ |

+------------+ +------+ | | | |

|[t3:X, t3:Y]| |[f3:Y]| | | | |

+------------+ |[f3:X]| | +-- && --+ +---------- || ---------+ |

+------+ | | | | | |

|t1: X t1: Y | | |

| +---- && ---+ +---- && ---+ |

| | | | | |

| f2: n5 X t2: n5 Y t2: n5 X f2: n5 Y|

+--------------------------------------------------------------+

|

+- /// -------------------------------+

| |

+-+ +--------------------------------------------------------------+

|$| | | |

+-+ | +------- && ---------------------+ |

| | | |

| | | |

| +-- && --+ +---------- || ---------+ |

| | | | | |

|t1: X t1: Y | | |

| +---- && ---+ +---- && ---+ |

| | | | | |

| f2: n5 X t2: n5 Y t2: n5 X f2: n5 Y|

+--------------------------------------------------------------+

|

+- /// ----------------------------+

| |

+-+ +--------------------------------------------------------+

|$| | | |

+-+ | +---------- && ---------------+ |

| | | |

| | | |

| +---- && ---+ +------- || ------+ |

| | | | | |

|+------+ +------+ | | |

||[t1:X]| |[t1:Y]| +-- && --+ +-- && --+ |

|+------+ +------+ | | | | |

| t1: X f1: Y f1: X t1: Y|

+--------------------------------------------------------+

|

+- /// -------------------------------+

| |

+-+ +--------------------------------------------------------------+

|$| | | |

+-+ | +------- && ---------------------+ |

| | | |

|+------------+ | |

||[t1:X, t1:Y]| +---------- || ---------+ |

|+------------+ | | |

| | | |

| +---- && ---+ +---- && ---+ |

| | | | | |

| +------+ +------+ +------+ +------+|

| |[t1:X]| |[f1:Y]| |[f1:X]| |[t1:Y]||

| +------+ +------+ +------+ +------+|

g g


DRAFT


+--------------------------------------------------------------+

|

+- /// -------------------------+

| |

+-+ +--------------------------------------------------+

|$| | | |

+-+ | +------- && ---------------+ |

| | | |

|+------------+ | |

||[t1:X, t1:Y]| +------- || ------+ |

|+------------+ | | |

| +------------+ +------------+|

| |[t1:X, f1:Y]| |[f1:X, t1:Y]||

| +------------+ +------------+|

+--------------------------------------------------+

|

+- /// ----------------+

| |

+-+ +--------------------------------+

|$| | | |

+-+ | +------- && ------+ |

| | | |

|+------------+ +------------+|

||[t1:X, t1:Y]| |[f1:X, t1:Y]||

|+------------+ |[t1:X, f1:Y]||

| +------------+|

+--------------------------------+

|

+- /// --+

| |

+-+ +---+

|$| |+-+|

+-+ ||$||

|+-+|

+---+

g g


DRAFT


2.8 General Strategy

Distribution of the sub-systems over different places according the OM-matrix.Using signed formulas to realize the sub-system management.Developing the sub-formulas of the distributed sub-systems according to their tab-

leaux rules.Applying first alpha and delta rules to beta and gamma rules.Applying the meta-rules of the term calculus to organize the distributed results collect-

ed in the trees Searching for closed branches.

Alternatively, a strict term development which requires its own strategy can be ap-plied.

g g


DRAFT


2.8.1 Interpretations for formula H1

Junctions vs. transjunctions

The difference between junctional and transjunctional logical function has to be ex-plained first.

Junctions are total functions accepting the values offered by their system.Transjunctions are composed of partial functions. They reject alternatives of their sys-

tem in favor to values of other systems. Therefore they are composed of acception andrejection values distributed over different sub-systems. They are intrinsically intra-con-textural. Transjunctions are obviously trans-contextural, involved in at least two differentcontextures (sub-systems).

Junctions, like conjunction and disjunction, but also implication and replication, aredirectly dual under DeMorgan operators.

Transjunctions

Between the term (tr.et.et) and the term (vel.tr.vel) a duality is established with helpof the negator N5.

D5 ( tr.et.et ) = ( vel.tr.vel)

This holds for junctions in general: D5 ( tr. J.J ) = ( DJ, tr, DJ )But this duality is, as all negational situations in polylogics, involved in permutations.

The duality of the transjunction tr is due to its symmetric definition as a total transjunc-tion.

In general, transjunction are classified in commutative and non-commutative, or sym-metric and non-symmetric transjunctions. Only commutative transjunctions are directlydual. They are also called total transjunctions.

Duality-D5 for ( tr. et.et ): ( S123, S2, S3 ) ––> ( S1, S213, S3 )

WordingsSystem S1 is modeling in itself situations of systems S2 and S3.System S2 is modeling in itself situations of systems S1 and S3.

2.8.2 General interpretation of term rules: Separability of interactionality

It was emphasized before that the operators of interactionality in polylogical sys-tems are realized by the logical operators of transjunctions while the intra-contex-tural logical situations are ruled by junctional operations.

The term rules for formulas with transjunctions are separating the junctional andthe transjunctional parts of the complex formula. These rules correspond to theDNF of classical logics which are also included in polylogics.

Additional to their proof-technical meaning it is possible to understand theserules as the rules of separating the intra-systemic part from their trans-systemic re-lations which are exactly the interactional patterns. That is, the term rules for tran-sjunctional formulas are the rules, or meta-rules, for interactionality in polylogicalsystems.

The set of term rules for transjunctional formulas are based on experiences, intu-ition and experiments with the implementation in ML of the theorem prover LOLA(1992) for polylogical systems. They are still waiting for a strict mathematical proofof correctnes!

The set above of meta-rules is not dealing with systems change produced by per-mutational operators, also not with reduction and other topics. But it seems thatthese topics are not producing any special obstacles.

These meta-rules are of great unificational importance because of their meta-log-ical status they help to reduce the enormous magnitude of concrete single tableauxrules to some understandable and accessible structures. The multitude of concretelogical operators shows the flexibility of polylogical systems to the challenge ofreal-world applications. Each concrete situation can be handled by its own con-crete set of operators. This is not excluding the search for minimal sets of operatorsbut these new kind of sets turn out to be flexible and adapted to concrete demandsof logical modeling.

The combination of Smullyan unification and the meta-rules makes this polylogi-cal complexity of concrete situations accessible to further studies.

Following the example of Gentzen and Moisil a calculus focussed mainly on themeta-rules can be developed. It would emphasize more clear the structure of inter-actionality between logical systems in addition to their intra-contextural deductibil-ity.

From a polycontextural point of view, meta- and meta-meta-rules can be under-stood as a reflectional activity towards the object-system. Therefore it is possible tomodel meta-concepts of all degrees in the framework of reflectionality of polylog-ics. With this, meta-considerations are free to chose between hierarchical and alsoheterarchical strategies.

The term rules, as listed below, are emphasizing the separation, or separability,of transjunctional terms from the junctional. Therefore, they offer a method of sep-arating from their actionality and of studying interactionality between systems assuch.


DRAFT


3 General term rules

3.1 Term rules for junction and transjunctions

The term calculus for poly-contextural logics was firstintroduced in the work"Tableaux Beweiser" byBashford/Kaehr 1992 as afirst attempt to deal with thequestion of meta-rules.

Term Rules

R t et t or t

�

�:��

��

0 1 2 3( )��

��

t et t or t et t1 2 1 3( ) ( )

��

��

t or t et t

t et t

1 2 3

1

( )33 2 3

1

( ) ( )

( )

� ��

:��

��

or t et t

R

t simul ta ��'� �' �

��'��

t simul t a

t t simul ta

( )

( ) ��' ��

:

�� '� �

t a

R

t et t simul

( )

2

tt a

t et t simul ta

' �

�� '��

( )

( )

��

�� '

��

t simul ta et t

t

( )

eet t simul ta

R

t simul

�'��

:

��

( )

{ }3

�� ' � � '�

��

ta or t simul ta

t or

( ) { }( )

��'�� ' ��

:

�

t simul ta or t a

R

( ) ( )

4

�� ' � �' �

��

t or t simul t a{ } { }( )

�� '�� ' ��

��

t or t simul t a( )

tt simul ta or t

t or t

{ }( ) { }

( )� � �� '

�� '��

:

��

simul ta

R

t simul ta

5( ))

( )� �'

�� ' ��

simul t a

t simul ta et t a ��

y y g


Unification for PolyLogics

1 Generalized Smullyan Unification for PolyLogics

Obviously, as easily visible, the full magnitude of combinatory possible functions isnot simple to handle, therefore more classifications and unifications has to be intro-duced. The best guide is given by Raymond Smullyan’s well known method of unifica-tion. We are studying the familiar case of linear-mediated polylogics in contrast totabular architectonics.

1.1 General unification rules

Diagramm 24 Unified junctions and total and partial transjunctions

Diagramm 25 Basic unification for total transjunctions

junctions and total transjunction scheme� � � ��α��

�

��

��

��

�

��

�α

α

β

β β

δδ

δ

γ

1

2

1 2 1

2

γγ

γ

γ

γ1

2

3

4

� �

�

�

� � �junctions and partial transjuncction schemes��

�

��

��

αα

α

β

β β1

2

1 2

��

� ��

��

�

��

�

��

� �

δ

δ

δ

γ

γ

γγ

γ

αα

α

1

2

3

4

1

2

1

2

ββ

β β

δ

δ

δ

γ

γ

γγ

γ

��

��

� ��

��

�� 1 2 1

2

1

2

3

44

��

�

��

��

��

�

��α

α

α

β

β β

δδ

δ1

2

1 2 1

2

γγ

γ

γ

γ

γ� �� 1

2

1

2

γ

γ

γ

γ1

2

3

4

� ��

=

1.1.1 Signed formulas

1.1.2 Junctional unification table

Diagramm 26 Unification for junctions

��

� �

��

��

� �

π απ α

π α

π β

π β π β1 1

2 2

1 1 2 2

��

��

�

��

��

��

��

π δπ δ

π δ

π γ

π γ

π γ

π γ

π1 1

2 2

1 1

2 2

3 3

44 4�

�γ

� | | ��

� ��

�

�

�

α α αi i i

i

i

i

i

t X Yf X Yf X Yf

1 2

∧∨→

��

�

�

�

�

�

�

�

��

� ��

�

X Y

t Xf Xt Xf X

t Yf Yf Yt Y

i

i

i

i

i

i

i

i←

�� | | ��

� ��

�

�

�

β β βi i i

i

i

i

f X Yt X Yt X Yt

1 2

∧∨→

ii

i

i

i

i

i

i

i

iX Y

f Xt Xf Xt X

f Yt Yt Yf�

�

�

�

�

�

�

�

��

� ��

�

← YY

y y g


1.2 Classification of unificators

1.2.1 (f,c)- and (klor)-analysis

The rules of mediation are restricting the combinatory of distributed operators.

The sets f and c alsocontains the negatedfunctions.

Junctional mediation

Smediation

� :

,� �/α β{ } ∈ ( )3

⇔⇔ { }∈= ( ) ( ) ( ) ( )

� ,� �

� ,� ,� ,�

α β CF

CF ccc cff fcf ffc ,,�

� ,� ��

� ,� ,�

fff

f

c

( ){ }= ∧ ∨{ }= → ← ↔{ }

Transjunctional mediation

medi

� :

,�,�,� /α β δ γ{ }aation S CF

CF ccc cff

∈ ⇔ { }∈= ( )

( )3

� � ,��,� �

� ,�

α β δ γ

(( ) ( ) ( ) ( ){ }= ∧ ∨ < > <>

,� ,� ,�

� ,�,�,�,�

fcf ffc fff

f ,,�,� ��

� ,� ,� ,�,�

{ }= → ← ↔ ⇑ ⇓{ }c

� ,�

��

α β

ααα

ααβ

αβα

αββ

βα

{ }∈( )

( )( )( )

( )S3

αα

βαβ

ββα

βββ

( )( )( )( )

∈ ( )S 3 ��

α α α

ααα

∈ ∈ ∈

( )∈ ( )

S S S

S

1 2 3

3

,� ,�

��

1.2.2 Mediation and unification

1.2.2.1 Identity patterns

pattern id id id S S

med tid

� ,�,� :� � →123 123

1�� ,� � ,� �X Y t X Y t X Y∧( )∈ ∧( )∈ ∧( )∈

α α α1 2 2 3 3 ∈∈ ( ) ⇒ ( )∈

∧( )∈

( )fff S

med f X Y f

� � �

� ,��

α α α

α

1 2 3 3

1 1 2�� ,� � � �X Y f X Y fff∧( )∈ ∧( )∈

∈ ( ) ⇒α α β β2 3 3 1 22 3 3

1 1 2 2

β

α α

( )∈∧( )∈ ∨( )∈

( )S

med t X Y t X Y

�

� ,� � ,�� tt X Y fff S

med

3 3 1 2 3 3� � � � �∧( )∈

∈ ( ) ⇒ ( )∈ ( )α α β α

tt X Y t X Y t X Y1 1 2 2 3 3� � �� ,� � ,� �∧( )∈ ∧( )∈ ∨( )∈ α α α ∈ ( ) ⇒ ( )∈

∧( )∈

( )fff S

med t X Y

� � �

� ,��

α α β

α

1 2 3 3

1 1 tt X Y t X Y fff2 2 3 3 1� �� ,� � � �∨( )∈ ∨( )∈

∈ ( ) ⇒α α α β22 3 3β( )∈ ( )S �

pattern id id id S S

Xid

� ,�,� :� �

�

→

∧∧∧123 123

YY fff t t t S

X Y

( )∈ ( ) ⇒ ( )∈∧∧∧(

( )� � �

�

� � �1 2 3 1 2 3 3α α α))∈ ( ) ⇒ ( )∈

∧∨∧( )∈

( )fff f f f S

X Y

� � �

�

� � �1 2 3 1 2 3 3β β β

ffff t t t S

X Y ff

( ) ⇒ ( )∈∧∧∨( )∈

( )� � �

�

� � �1 2 3 1 2 3 3α β α

ff t t t S

X Y fff

( ) ⇒ ( )∈∧∨∨( )∈ (

( )� � � �

�

� �1 2 3 1 2 3 3α α β

)) ⇒ ( )∈ ( )� � � �� t t t S1 2 3 1 2 3 3α β β

y y g


1.2.2.2 Reductional patterns

pattern id red red Sidredred

� ,� ,� :� →123

��

� ,� � ,� ��

S

med t X Y t X Y f X111

1 1 2 2 3∧( )∈ ∧( )∈ →α α YY ffc S( )∈

∈ ( ) ⇒ ( )∈ ( )α α α α3 1 1 1 3� � �

pattern id red id Sidredrid

� ,� ,� :� � →123

SS

med t X Y t X Y f X Y113

1 1 2 2 3� � �� ,� � ,� �∧( )∈ ∧( )∈ →α α (( )∈

∈ ( ) ⇒ ( )∈ ( )α α α α3 1 1 3 3ffc S� � �

y y g


1.2.3 Transjunctional unification tableThe general unificational schemes for transjunctions are not as simple as the junction-

al counterparts but there is a clear structure behind the transjunctional table (situation,configuration).

signature schemespartial transjunction

� :� �( )< ,,� , , �

��

�

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i mod

t

f

t

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i

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ii

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it

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1.2.4 Consistency properties

Diagramm 27 Propositional consistency properties for PolyLogic(3)

1.2.5 Conjugations

Symmetric conjugations

Between α-terms and β-terms we observe a nice and simple conjugation. Both, α-and β-terms, are dual.

Conjugation rules

Propositional consistency property for Poly� � � � LLogics

i j s m

F S T S

Zi

ii

i

∀ ∀ ∈ ( )

∉ ¬ ∉

¬¬ ∈

�:

.� �;�

.� �

1

2 SS S Z C

S S C

i i i

i i i

� �

.� � � ,

.��

⇒ ∪ { }∈∈ ⇒ ∪ { }∈3

41 2

α α α

β ∈∈ ⇒ ∪ { }∈ ∪ { }∈

S S C or S Ci i i i i� � � �

�β β

1 2

∈ ⇒ ∪ { }∈

�

.� � � ,

.��

junctions

S S Cj j j5

61 2

δ δ δ

γ ∈∈ ⇒ ∪ { }∈ ∪ { }∈

S S C or S Cj j j j j� � , � � ,� �

γ γ γ γ1 2 3 4

∈ ⇒ ∪

� �

'.�' � �

total transjunctions

S Sj j5 δ δ11 2 1 2

6� �, � � � ,

' .�' � �

δ γ γ

γ

{ }∈ ∪ { }∈∈ ⇒

C or S C

S S

j j j

j jj j

j j j

C

S S C

∪ { }∈∈ ⇒ ∪ { }∈

γ γ

δ δ δ3 4

1 25

�

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,

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S S

j j

j j

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∪ { }∈∈ ⇒ ∪ { }∈

γ γ

γ γ γ3 4

1 26 CC

partial transjunctions

j

� �

ρ α β

ρ α ρ α ρ α

ρ β

� ��

� � � � � ��

��

( ) = ( )( ) ⇔ ( ) ( )( )1 2

and

(( ) ⇔ ( ) ( )( )( ) ⇔

� � ��

� � ��

ρ β ρ β

ρ α β β1 2

1 2

or

or(( )

y y g


Asymmetric conjugations

This elegant symmetric duality is excluding logical function like equivalence and con-tra-valence. This two functions are not symmetric in respect of their possible α-terms andβ-terms. Thus, they are treated as derived functions, composed of implication and con-njunction. But nevertheless, the tableaux rules for the equivalence shows a clear asym-metry between the two parts, the truth-part and the false-part.

This solution is perfect for classical logic because there are only a few asymmetriccases but in polylogical constellations they are quantitatively and from their signifi-cance in the vast majority. Morphogrammatically it looks more balanced, from the 15basic morphograms for polylogical functions, 8 are junctional and 7 are transjunction-al. But this balance disappears quickly because from the 8 junctional morphogramsonly 4 are relevant, producing the 8 classical logical functions. On the other hand, the7 transjunctional morphograms are basic and all full in the game additionally, the de-liver the structure for different logical interpretations, mirrored as different realizationsof semantic mappings into the morphograms.

To restore a reasonable conjugation theory for asymmetric situations too, we haveto involve the fact that transjunctional functions are better understood not as total butas combinations of partial functions. Thus, transjunctions can be introduced by apply-ing the methods used by defining derived functions. Transjunctions are not only com-positions of partial functions but are the functions which are reflecting the environmentof the logical place they are set. This two observations, partiality and environment,gives a clue how to construct a general setting for asymmetric conjugations.

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t X Y f X Y

t X

t Y

f X

f

i i

i

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i

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↔ ↔

YY

t X

f Y

f X

t Yi

i

i

i

��

�

�

�

Composed conjugation for equivalence

X Y

� � � :

� ��↔ == →( ) ∧ ←( )

∨( ) ←� � � ��

� � ��

X Y X Y

If ti dualα α

1 2→→ ∧( )

← →

� � �� ,

� ,��

f

theni

dual du

β β

α β α1 2

1 1 2 aal← → β

2

y y g


1.3 General rules and tableaux rulesOnly for the junctional situation we have a one-to-one correspondence between the

general unification rules and the tableaux rules connected to their sub-systems.Even between the general rule for total transjunction and its tableaux their is some

degree of differentiation. In the case of reductional transjunctions the general rules forthe transjunction has not to be applied to all its sub-systems. In other words, we cancompose full transjunctions as conjunctions of conditional transjunctions. This appliesto total as well as to conditional transjunctions.

The general rules are defined for "isolated" sub-systems. Between the α−β-rules thereis a duality, and for the case of only one sub-system, that is, for i=1, the whole logicalsystem is defined. There is no ambiguity of differentiation in that. Polylogical systemsin contrast have to take their whole constellation into account and have therefore to becharacterized by the tableaux and the rules for the whole operators. They can not bedetermined by the local characteristics only. But this difference between the generalrules and the tableaux rules is, conceptionally, also at place for the junctional situationof classical logic. The only difference is that in the classical case there is a coincidencebetween the uniqueness of the logical system and the uniqueness of the rules. Never-theless, they have to fit together.

Even for the classical case, some conventions, which could be questioned, are nec-essary to adapt of the logical operators to their unificational rules. Examples for a needof some kind technical conventions are negation and logical equivalence.

The operator J represents junctions as α− or β-parts.

t X J J Y

t J

f X J J Y

f J

t X J

1

1

1

1

2

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< <

<< <

<

J Y

t

f X J J Y

f

t X J J Y

t

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2

2

2

3

3

δ δ

JJ

f X J J Y

f J��

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3

<

t X J J Y

t J

f X J J Y

f J1

1

1

1

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> >

γ δδ

δ δ

t X J J Y

t

f X J J Y

f

t X J

2

2

2

2

3

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> >

> JJ Y

t J

f X J J Y

f J

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3

3

>

y y g


1.4 Patterns of unification

Patterns of unification correspond the application of super-operators to standard (id,id, id)-patterns of logical operators.

1.4.1 General classisficationJunctional patterns: {a, b } –> {id, red, perm}

Transjunctional patterns: (a, b) –> {id, red, perm, bif}

1.4.2 Special classificationDepending on the distribution of the super-operators {id, perm, red, bif} different

classification systems can be established.

1.5 Contradiction and incompatibilityA branch δ of a tableau T is called closed by contradiction if both X and non-X oc-

curs on δ for some propositional formula X, or if F occurs on δ.

A branch δ of a tableau T is called closed by incompatibility iff X, Y and Z occurson δ for some propositional formula X with val(X) /= val(Y)/=val(Z). Such a branch canbe eliminated, cutted away, from the tree.

Branches can be eliminated if they contain at least 3 different variables with 3 dif-ferent values (for the case of m=3).

perm Z S Z S

red Z

i j j i

i j

:� � � � �

:��

. .

.

∈ ⇒ ( )∈∈

( ) ( )3 3

�� .S Z Si i3 3( ) ( )⇒ ( )∈

red α α α α α α α α α α α α1 2 3 1 1 3 1 3 1 1 3 3( ) ⇒ ( ) ( ) ( )� � ,� ,� ,��α α α1 1 1( ){ }

DEERRRRIIDDAA’’SS MMAACCHHIINNEESS PPAARRTT IIIIII...

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