| |
|
|
|
ThinkArt Lab Animation: A.T. Kelemen© November 12, 1998 Dr. Rudolf Kaehr
"When classical logic is applied to non-mathematical examples, the examples are first ´mathematicized´. In the real world we might argue about whether block B is behind block A or not - maybe it depends on one's point of view. But we can create an ideal world, a mathematical model, in which either B is behind A, or it isn't. Classical logic can be used to reason correctly about such a model. Whether the model accurately reflects the real world is a separate issue." Fitting, 1990, p. 1
Trans-classical logic is aimed to model the situation of rational reasoning between different agents where each agent has its own logic, that is, its own point of view in respect to his world. Therefore trans-classical logic reflects the world from a multitude of different logical points of view. Each locus has its own mathematicized" formal apparatus, its own mathematical formal logic. As a consequence, the monolitical or erratic concept of world disappears as a very special case of disambiguity in a dynamic multi-verse.
The mathematicization of a world including a multitude of different logical loci, points of view, is obviously different from the more abstract model of the classical mono-logical mathematicization of the world.
Maybe, the classical model reflects an ideal world or even investigates the principles of reasoning for perfect worlds" (Fitting) trans-classical logic reflects the rational principle of a conflicting, interacting, co-operating world, where the participants of these interactions creates together their own worlds in a co-creating manner. In this sense reasoning and modeling are not structures but actions.
The new concept of trans-classical logic as a complex logic of a multitude of points of view does not introduce some shades of grey" between the strictness of the classical concepts. There is now fuzziness here. It introduces something different, each (classical) logic gets an index that indicates the point of view of the rational agent, which indicates the separation between the different agents. Trans-classical logic is not a logic of white and black" nor a logic with shades of grey, it is a logic of colors, a colored logic. The logics of the living tissue are colored ones.
Each color has its own formal and operative strength.
There is no ambiguity and fuzziness in this notion of colored logics.
But these colors are not only simply identical with themselves. Each colored system is able to reflect the other systems simultaneously in its own domain. This new ambiguity is produced by the complexity of the polycontectural logic as a whole with its interaction and reflection.
Another interesting feature of colored logical systems is given by the mechanism of change of systems. A system may change from one colour to an other colour. Or some systems may permute their colours.
In the example, the blue colored logical system has a picture of the green system, the blue system is able to reflect, to mirror, to model the other colored systems in its own domain. It has replications of the other systems.
This means, green is not simply green. Green as green is green, but green as blue is the blue of the green. Green is green and not blue but green can have aspects of blue.
All this is possible only because the logical and ontological principle of identity is abandoned and transcended by the game of sameness.
As a result of the plurality of formal systems as differently colored logics, each logic and complexion of these logics is localized in a structural space. Every logic has its own locus. Each logical locus gives place for the replication of other logics which are located at other logical loci.
The theory of these ontological or pre-logical loci is called kenogrammatics.
In other words, we can say, that the mirroring of one contexture by another, is a belief function. One system beliefes something about another system. This more linguistic perspective opens up a connection to the work about beliefe systems, beliefe logics etc. in Artificial Intelligence (Konolige, van Harmelen,´).
But also to the Algebra of Reflection as it is proposed by Levebvre.
1 General poly-contextural diagram
As a result of the analysis of the structure of interaction I can introduce a general diagram or scheme of polycontextural logical and arithmetical reasoning and computation.
Here I have restricted the complexitiy of interaction to the special case of 3 actors.
At each locus Oi which gives place for the logical system of the place as such the locus Oi also offers place for the modeling of the neighbor logical systems. That is, for the modeling the logics of the logics of the other interacting agents.
In a classical setting, this situation is not modeled or as in computational reflection, the meta-level approach does not map on a structural level the complex logical constellations between different interacting agents.
(This can be shown for Smith, Maes, Sloman, Kennedy et al.)
O1: (M1, M2, ...., Mn) § O2: (M1, M2, ...., Mn) § ... § On: (M1, M2, ...., Mn)
The difference between contextures as objects and contextures as morphisms has to be considered. In the very special case of m = 3, both concepts coincide
Diagramm 1
Basic Modul of Metamorphosis
2 Dissemination of deductive systems
2.1 Deductive system
2.2 Combining deductive systems
2.3 Disseminating deductive systems
3 Dissemination of a framework of Tableaux Logics
3.1 ASM specification of tableaux logics
3.2 Dissemination of ASM specification
Specifications after Börger and the problem of chiastification of their basic terms.
On the first level of specification Tableaus are lists of Branches. These Branches are connected with the Universe of Formulas. On this level of abstraction there is no root and there are no nodes specified.
These two universes, Branches and Formulas, are the basic concepts of the definition of tableaux-systems on this level of specification.
The question arise, how are they interconnected classically and how can they be involved in a interlocking mechanism of a chiasm in a next step?
The answer to the first part of the question is given by construction.
Universe: Branch = Fml*, list of formulas
Universe: Tableau = Branch*, list of branches
The second part of the question has to be developed.
On the second level of specification Trees and their Nodes are used as the basis or place were formulas are located.
In a reflectional system the structure of the presupposed tableaux and later the trees of tableaux systems can be modelled as a mathemetical theory.
Maybe there are new insights in the theory and techniques of graph theoretical systems like tableau and tree systems, new and faster algorithms or more abstract construtions and the introduction of new concepts.
This knowledge can be given back to the first system which is based on graph theory.
As a consequence of the chiasm of Formulas and Branches in Tableau-Systems we can establish as an application an interlocking mechanism between hardware and software of a computer system evon on the first level of specification.
In the second system the structure of the hardware, e.g. the logic of the processor, is modelled. As an object of reflection it can be transformed and reimplemented in the former system.
3.3 Polycontextural tableaux logics
Despite of the true-value semantic approach in the first steps of introducing polycontextural logics, for instance, as a new interpretation for multi-valued logics by Gunther, recently by Meixl/Pfalzgraf, it should be insisted to the fact that the very idea of dissemination is independent from the system being disseminated. This was pointed out at least in my paper (1981).
This remark is not only of scientific relevance, it has also a political significance. The reason is simple. In the 70th when I applied for a research project I got a denial because from the point of view of the constructivist Dialog logic of Lorenzen, truth values are obsolete, at least secondary. And because the whole polycontextural, multi-negational, morpho- and kenogrammatic constructions of Gunther can be read as depending on truth-values, the whole proposal has to be rejecteted because it is simply nonsensical. This argument is based on a reference paper to a Gunther lecture at the Hegel Congress by Lorenzen (1962). In my 1981 paper I refuted this lack of thinking. Propaganda and thinking are obviosly two different activities. Nevertheless this attitude was not only influential but also killing. Also its history it can easily be repeated, now from computerscience oriented logicians.
Nevertheless, a lot can be learned, and is still to learn, by studying disseminated semantic based logics. (Semantics of combined logics is hard, Diss)
First it should be clear that the "mixing" of logics, neither in the project of combining logics nor in the polycontextural approach, is restricted at all to truth-value based logics. Mediated logics can be based on all known methods and the mix can "easily" be heterogeneous: semantic based logics mixed with constructivist systems with paraconsistent, etc.
3.4 Implementing PCL fragments in ML
4 Classical and polycontextural logics
4.1 Tree farming of colored logics
To put all these more philosophical descriptions and ideas together in a more strict formal terminology and operative apparatus we connect these ideas of colored logics with the metaphor of the tree.
In classical logic trees figure as metaphor and as mathematical concepts on a very basic level.
We postulated that every colored logic is locally classical, e.g. each colored logic has the structure of a classic tree logic in its syntax, semantics and proof theory.
It is helpful to represent the structure of the body or the tectonics of a logical system by its conceptual graph.
Diagramm 2 Conceptual Graph of an Institution
The arrows in this diagram represents conceptual dependencies. The notation
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 be independent of the concept of signature and neither can a particular model be independent of its particular signature.
In a conceptual diagram, 1 represents the absolute. The notion
expresses that the institution notion is absolute, for it tells us that the institution notion varies as the absolute varies - which is not at all." p. 488
The absolute 1 expresses that there is only one logic as such. There are many different logical systems but from a more philosophical and logical and not only mathematical 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 in the more relativistic case. Considering a logical system working in it and with it means that we are working with this system and not at once or at the same time with another one. We can speak therefore of a relative absolute of the logic under consideration.
Even for mixed logics as in the project of Combining Logics there is a (relative) notion of the absolute of the system.
As we will see the situation will be totally different for polycontextural logics where a plurality of absolutes ordered in a heterarchical manner exists and the desire to have a mega-absolute for the whole complex system would turn out to be (simply) a new absolute within a plurality of other neighbored absolutes.
From the point of view of PCL the absolute means that the whole system is defined under the operation of identity ID. The system viewed as an object and as a morphism coincide.
Diagramm 3 Monoforme Mediation of two Institutions
The institution represents the logical system under consideration as a whole, as a logic in its uniqueness.
The signatures provides the vocabularies for the sentences of the logical system.
4.2 Preliminary Comments
For each tree of the colored logical systems the ancestral property of its formulae holds.
In each tree there is a immediate predecessor relation which decompose the formula in its subformulas.
All that is ruled by the principle of induction over the formulas for 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 introduce in the trans-classical context the operation or relation of the immediate neighbor. These makes possible some kind of permutation of logical systems over the range of the complex of distributed systems.
Further on we have to introduce the very special concept of the immediate bifurcation of a formula of a logical system into subformulas distributed over other logical systems and one part of the formula remains in its original system
From the point of view of functions and relations the operation of identical distribution, permutation and reduction are covered by the concept of total functions and relations.
Remember, all classical logical functions, e.g. of propositional logic, semantics and syntactics, are total functions.
The concept of distribution by bifurcation is possible only with the help of partial functions and relations. We call such logical functions transjunctions because they surpass the boundary of their own system and hold simultaneously in their own and in other systems. On a metalanguage level this way of speaking leads to the distinction of global and local concepts. PCL systems studies the interlocking mechanism of logical globality and locality. This mechanism guarantees that there is no hierarchical suprematy of the global over the local. Local and global concepts interlock together in a heterarchical manner.
On a meta-logical level, total functions are supporting symmetrical classifications and categorisations of logical particles.
This nice property of a logical symmetry is lost in trans-classical logics because of their transjunctions which are composed of partial functions. But with the help of the concept of partial functions we can introduce a new idea of a slightly more complex symmetry composed by partiality. The classical case of symmetry is then introduced as a regular composition of partial functions. This idea of a complex symmetry composed of asymmetrical functions needs additionally to the classical operator of conjugation a new operator of composition of partial functions.
Truth-values in classical systems are connotated with the formal logical explication of truth and false. Formal truth and formal false do not involve ontological questions about truth and falsehood of sentence. This belongs to the level of examples for formal logical sentences.
In PCL systems truth values, if we are choosing a truth value semantics, have to realize two jobs, the first is more or less the same as in classical systems, they represent the formal logical concept of true and false of propositions of their logic. The second job 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 different systems at once. As a result, the whole semantics of propositions, sentences, phrases and truth-values has to be deconstructed.
As explained metaphorically earlier these logics are not isolated from each other but combined to a complex logical, or ultra-logical, system. Otherwise they would behave totally in parallel and it would be at least at first only an application of one classical logic at different epistemological places without any interaction or mediation.
A first, quite natural and elementary, connection is given by a (special) linear ordering of the systems and their truth-values.
To not to confuse this kind of order with other ordering systems I call it a chiastic linear order of truth-values. A chiasm is defined as a tupel of order, exchange, coincidence and positioning relations.
therefore the semantics of PCL is not defined over a set of truth-values but over an order, a chiastically ordered structure of truth-values.
The difference becomes obvious for the semantics of ternary and general n-ary logical 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.
For each single logical system of the PCL complex there exist a consequence relation and a proof theory for this logic.
The consequence relation for the whole system of logic is composed of the distributed single consequence relation of each logic.
We will choose the analytical tableaux method as our proof procedure.
4.3 Signatures and Vocabularies
The distribution of vocabularies in PCL is produced not only by the genuin vocabularies of the single logics but also by the interaction between the logics. Vocabularies in general are not only objects or sets but morphisms.
In this sense vocabularies are not the initial objects of their category. They can be initial but only from a local point of view,
with Id, Perm, Red, Bif as operators.
------------------------------------------------
Voc1|Voc2|Voc3||Voc1|Voc2|Voc3||Voc1|Voc2|Voc3
For the sake of a gentle introduction I choose a monoform setting of the vocabularies (alphabets) and the sentences. All vocabularies have the same, but not the identical, signs. All sentences are balanced, that is, they have the same syntactical structure. That means, that everything is in full parallelism or accordance with the classical setting.
For all logical systems we have locally:
Voc i = {p, q, r, ..., and, or, not, trans, ...) and the same type of brackets.
The sentences Seni are defined locally recursively over the Voci.
X, Y, Z, .. are metasymbols for the object symbols build up with p, q, r, ...
------------ o={and, or, impl, ...}
Decompositions of Transjunctions
---------------------- o={transj1, ....}
As in classical syntax where not all possible combinations of signs from the alphabet are defining sentences I have to introduce a new rule of combining signs, especially between logical connectives, negators, junctors and transjunctors from different logical systems. These rules are called in the literature of polycontextural logics the VB-Rules (Vermittlungs-Bedingungen = Conditions of mediation).
As in classical logic where the pure combinatorial possibilities are restricted by semantical considerations I have to do the same for the VB-rules. In relation to a specific choise of the concept of mediaton between logics or of logics which is reflected by the VB-rules the possibilities of combining logical connectives of different systems is ruled. A lot of combinations are ruled out. Not everything can be mediated whith everything.
To state it just at the very beginning of my exposition of PCL, if someone needs to collect these 3 different vocabularies under one new all 3 subsuming vocabulary, it would have been produced simply a new vocabulary as a basis for a new 4-contextural logic.#
Each system has its own vocabulary and is additionally able to reflect the terms of the neighbour systems in its own contexture. Therefore, the terms of the other systems can occur in its range. This doesn´t mean that all systems have the vocabularies of all other systems included in their own vocabulary. The representation of the vocabularies in other systems is the result of operations of interactions between the genuin vocabularies, like Bifurcation. The other vocabularies don´t occur in the vocabulary of a system but in the contexture or at the locus of the system. Therefore they are still disjunct or irreducible different by their belonging to another system, marked by their colour or their index.#
On a metatheoretical level we have to be aware about the following situation. If we speak about signatures as morphisms and as categories we are using category theoretical terms. And it sounds as if PCL turns out to be simply a special theory of category theory.
This is the case e.g. of fibered logics. "However, the concept of abstract fiberings is general enough to allow us to "mix logics" in the sense that different logics can occur as local fibres." Pfalzgraf
If we speak about a plurality of irreducibly different institutions and signatures or vocabularies we are forced to use also a multitude of irreducibly different category theories. We don´t have this polycontextural category theory at disposal. This very new theory would have to be developed on the basis of polycontextural logics in a similar sense as classical category theory is based on classical logic (many-sorted eqational logic). But I am just introducing PCL.
Therefore, we have to use terms and methods of classical theories as our first step tools of thematizing, modelling and formalizing the trans-classical domain of PCL.#
5 Modi of interactions: super-operators
It seems to be natural to accept that there are at least the following operations of interaction between the systems to observe.
ID: Identity. Mappings of a system onto itself
RED: Reduction. Mappings of systems into other systems (Acceptance)
PERM: Permutation. Transversions between systems, exchange of positions
BIF: Bifurcation. Mappings of systems onto themselves and simultaneously into others.
IDi: (G1G2...Gi...Gn) ===> (G1G2...Gi...Gn)
PERMij: (G1G2...Gi Gj...Gn) ===> (G1G2...GjGi...Gn)
REDij: (G1G2...Gi Gj...Gn) ===> (G1G2...GiGi...Gn)
BIFi : (G1G2...Gi...Gn) ===> (G1G2...(Gi1...Gin)...Gn)
I call these additional operators super-operators. In contrast to the first operators of the natural system, zero and suc, which are defined inside the system, e.g. locally, the superoperators are defined between the natural systems of a collection of cloned systems and are therefore of a global character.
5.1 Super-operators in disseminated natural systems
DISS(3) (object nat0) = object1nat0 § object2nat0 § object3nat0
(ID PERM2 PERM3) zero(3) : --> nat(3) with
(ID BIF1,3 ID)zero(3) : --> nat(3) with
(RED2 ID ID) zero(3) : --> nat(3) with
The same procedure is to apply to the successor function suc and later to other operators like the addition add.
In contrast to the purely parallel construction we have to introduce a more complex notation for the general case.
Untill now I have treated zero as an object and different types of zeros as objects belonging to different contextures. This is a quite conservative introduction. In correspondence to the idea of proemiality and polycontexturality, it is more appropriate to think of zero as an action. In this sense zero is the notation of the action of beginning. There are many beginnings but no single origin. Actions, and especially simultaneous actions, are not necessarily connected with the notion of identity. In contrast, objects are very close to the notion of identity. The classiscal concept of an object coincides more or less with this notion of identity.
The Operator (ID BIF1,3 ID)zero(3) suggests that there are additionally to the genuine objects of the systems objects from the neighbor systems too.
The first object of the application of the operator (ID BIF1,3 ID) to zero(3) is therefore:
(zero, zero, #), (#, zero, #) (#, zero, zero)
(ID, BIF1,3, ID) (OP1, OP2, OP3)
--------------------------------------
(OP1, OP2, #) (#, OP2, #) (#,OP2, OP3)
In this sense the Operator (ID, BIF1,3, ID) is only a perhaps misleading abbreviation of the above more explicit notation.
Diagramm 4
Transition-Diagramm of the Operation (ID BIF1,3 ID)
In contrast to the previous ebook SKIZZE" the idea of unrestricted cloning" also of non-transjunctional functions is new. This idea is realized by a more radical approach to the super-operator construction between formal systems as defined before and additionally by a more radical application of the category of sameness" to all objects.
The idea of reflectional modeling is more advanced and more explicit with the concept of reflectional architectures of formal systems.
The concept of arithmetical neighbors is more explicit and the status of the initial object zero in contrast to its neighbors is explained philosophically and formally.
The idea of categorial metamorphosis on the basis of the proemial relation is introduced.
The phenomenon of over-determination, that is of the possibility of a multitude of interpretations of arithmetically interpreted morphograms is linked to the metaphor of togetherness.
|
ThinkArt Lab http://www.ThinkArtLab.com sales@ThinkArtLab.com |
| |
|
|
|
