An Outline of Pragmatologic Model-Theory
(sec. Stachowiak). Semiotic Subjectivity II


Second Lecture by Dr. Gerhard GELBMANN, guest researcher at the WAB,
held at the HIT Centre, 3rd June 2002, 15:15-17:00







Herbert STACHOWIAK wrote in 1973 his book on General Model Theory (hence GMT) which starts from a specific and at the same time very general notion of "model"; his later works and also the contributions of others might lead to something which during the last two years I came to address as "Pragmatologic Model-Theory" or, probably better: "Pragmatologic Theory of Models".
 

1. the term "model"


The crucial term "model" is not to be read in the way as it is done in formal-semantic model-theory: there a model is an interpretation of a theory making all of the theory's statements true.

The pragmatologic conception of model means a "functional" (or "operational") mapping of attributes (of the same sort) of a so-called original onto attributes of a model. This shall serve as a first, but not ultimate or exhaustive definition. It is worth mentioning that we usually restrict ourselves in this transformation of attributes to the same sort of attributes, and that meas that an attribute of the original will be transformed into the same sort of attribute as an attribute of the model (but attributes of the original can also be identical with attributes of the model).

See the image for a first and rough "pictorial" understanding (via Venn-diagrams); a thorough explanation follows below.

While the formal-semantic approach belongs to a statement-view, this understanding here adheres to a non-statement-view. This Theory of modelling is "pragmatologic" in the sense of trying to present a frame for a formalization of pragmatics. This will become especially relevant when we come to talk about the term "Theory (in non-statement-view)" in contrast to "theory (in statement-view)".

Already Lasar Ossipowitsch RESNIKOW went 1968 in a direction of a pragmatologic critique of formal-semantic conceptions of model and semiotics, but from a clearly Marxist and materialist view-point in his account of epistemology. Although I am impressed by RESNIKOW's results, he did not develop any other attempt of a model-theory as STACHOWIAK did, and STACHOWIAK did not refer to RESNIKOW. And none of them referred much to WITTGENSTEIN, if WITTGENSTEIN himself ever had a clear idea about his conception of "model" or his application of the term. It is not that easy to state whether WITTGENSTEIN thought about the concept of "theory" on the lines of a statement-view or of a non-statement-view, and it is also not that clear which understanding of model he had. Yet some of what we find in his writings, can be understood within STACHOWIAK's GMT, and this is a further reason why I am dealing with it.

STACHOWIAK himself has undeniably a cybernetic background. His thinking about modelling is influenced from the Cybernetic Revolution of the Fifties and Sixties (of the 20th century). Although Norbert WIENER did not write much about the notion of model, WIENER's work can be looked upon as an anticipation of such an understanding of the term "model", just as the modern contributions to the Natural Sciences and Technology have a much stronger tendency and nearness to the pragmatologic conception of "model" than to the formal-semantic.
 

2. symbolic representation and definitions


I have to give a brief list of symbols and definitions in order to prepare an adequate understanding of STACHOWIAK's GMT and of the symbolism he applies. It is quite easy:

Let the symbols Ai, Bi, Ci, … refer to attributes; the index "i" shall signify any natural number (the number of attributes thus is infinite, yet countable). We discriminate between sorts of attributes, signified by different capital letters (in italics). So an attribute X100 is of the same sort as an attribute X18, whereas neither the attribute Y100 nor the attribute Z18 is.i

Attributes themselves can be colours, states, properties, qualities, relations, but also relations between colours, relations between states, relations between properties, relations between qualities, relations between relations, etc (cf. STACHOWIAK 1973: 134).- Just a side-remark: Attributes can themselves be modelled, like originals, and hence a model can contain modells as its attributes. [Due to the discussion with Ralph JEWELL, Alois PICHLER, and Simo SÄÄTELÄ, I have to admit that the conception of attributes being themselves modelled is "totally wrong", to use an expression of Ralph; G.G. evening of 3rd June 2002.]

The symbol O abbreviates the term original; O is (taken as) a class of attributes, e.g. O = {A1, B5, C9, D4}. The original can be any modellable object of our thinking, a physical or a mental, a concrete or an abstract entity.

Thy symbol M stands for model; M is (taken as) a class of attributes as well, e.g. M = {A1, C3, D2, E2}. That some object M models some object O, is more or less an aspective, conventional, and pragmatic matter (in the semiotic sense of the word). There is no modelling beyond a frame of a Theory (cf. inf.); modelling is an operation involving semiotic subjects and their operations.

We introduce now the sign OP. OP refers to a class of so-called non-preterated attributes of O, ergo the class of attributes of O which are not neglected by the modelling operation F (cf. inf.). For example: OP = {A1, C9, D4} if O = {A1, B5, C9, D4}. The class of neglected attributes of O would be O \ OP, i.e. in our case {B5}.ii In other words: OP is a sub-class of O: In mathematical symbolism we can express this as: OPO. In extreme cases O \ OP = {} or even OP = {}.iii

The complex sign MA refers to class of so-called abundant attributes of M; and ME means the class of essential (or non-abundant) attributes of M. Ergo: M \ ME = MA or MAME = M. In the case of our example we can express the sub-class of essential attributes of the model M as: ME = {A1, C3, D2}, whereas the sub-class of abundant attributes of M are: MA = {E2}. It holds, of course, to scribeiv in terms of sub-classes of attributes that MEM and MAM. In extreme cases MA = {} or even ME = {}.

Especially the second case can hardly be interpreted; it means that even if no attribute of the object is in common with the model or is of the same sort as at least one attribute of the model as a functional mapping of an attribute of the original, the model would model the original; in other words, two things which have not even sorts of attributes in common could be regarded as standing to each other in a modelling relation. But isn't just this the case of a pure semantic signification? Does the typographical signs "coffee" have anything in common with what we signify with it? What sort of attribute of "coffee" can we also find in the drink we call coffee? Probably none! (A further side remark: this is a hint to Karl BÜHLER's principle of abstractive relevance: Not all properties of a sign are relevant for its significative function; cf. BÜHLER 1982: 44.)-

The mapping of attributes of O onto attributes of M is accomplished through a functional operationv F such that F(OP) = ME, with a useful practical restriction, viz. that attributes of O are mapped into attributes of M of the same sort,vi as already has been said. (This functional operation F will in case of a pure semantical signification become a semantic function.)

Back to our example: In our example the operation F would render the following list:

F(A1OP) = A1ME
since O and M can have attributes in common, thus it could be that OPME~={};
F(C9OP) = C3ME;
F(D4OP) =  D2ME,
whereas the attribute B5O is neglected (preterated), hence we scribe: B5(O \ OP).

The attribute E2M is abundant, ergo E2MA with: ~x(x = F(B5)) and ~y(F(E2) = y).

A graphical image (Venn-diagram) shall illustrate the easiest (and idealized) case of modelling (with OM={}vii):
 
 



Stachowiak 1973: 157




The elements of OP and ME correspond to each other in one-to-one relations, but not those of O \ OP and MA!
 

3. the term "Theory"


As mentioned (cf. sup.), Pragmatology presents a non-statement-view. This does not merely concern the notion of "model" but as well the notion of "theoreticity" (theoreticity is the abstract concept used in the definitional predication of "... is a theory" by giving some conditions for the application of this term; especially in the usage and understanding of this notion the pragmatologic non-statement-view differs from the statement-view of neo-positivists, formalists, logicists, critical rationalists etc., and thus I venture to say that there are different conceptions of theoreticity). We henceforward prefer the spelling with capital later "Theory", abbreviated as "Th", in order to signify a conception of Theory in non-statement view.viii

The concept of "Theory" makes use of this concept of "model" as has just been developed above and can in symbolic representation be defined as follows:

We define: Th = <O, M, k, t, Z>. A Theory Th is a tupel of five parametersix of which it is remarkable that an object O and a model M occur as filling the first and the second of its places thus extensionally representing the functional operation F.

The symbol k is written for the Operator who perfors the functional operation F which models O in M. This operator usually can be conceived of as a semiotic subject.

With t we refer to a certain point or span of time as the that for the performance(s) of the operator.

And Z abbreviates the interests or aims, purposes, targets, calibrating values which are to be accounted for by the Theory Th in modelling O in M.

This does not mean that at a repetitive process of such modelling stages, arriving at several successive Theories Th1-n, M and Z have to be identical.x Z just says to which degree M is a satisfying model of O, i.e. which selection of essentially modelling attributes is relevant. It is my conviction that conceptually these interests have to be socially implemented.xi
 

4. higher order entities


Even if it sounds simple, it should be mentioned that the instrumentarium so far introduced already leads to higher order entities, especially in regard to questions like "Can models themselves be modelled?".

An alternate application of representational modelling and operative Theorizing produces such higher order entities, among which I also sense sentences, insofar as sentences (or statements/propositions or well-formed formulae, i.e. wffs, of a given calculus of a logical language) can be produced according to a functional syntax and semantics.xii In a rough sketch I can outline the production of "sentences" and hence the derivation of a statement-view of theory (viz. as calculus) as a consequence of a certain form of modelling and functional representation. The attributes thereby modelled will at the first step be composing the syntax' alphabet. At the second step a selection of the possible combinations of these syntactically applied signs will be taken as representing the wffs. Thirdly the interpretation of these wffs will have semantic value in taking them as (fulfilled) predicative attributes or functional predications (with referent signs/names or denotations as their arguments). So the first step gives us the foundation of a syntax, the second the rules of formation of formulae, whereas the third step provides the semantic interpretation of the signs so far generated. A fourth step will then deal with derivations (transformations of wffs within arguments), a fifth concerns truth-functions.xiii

A model M of a model M, in short MM, shall be called a second-order model or model of the logical type 2, in short M2. It is clear that two second order models of the same object need not be identical, since they might differ in their selection of attributes (in their preterition as well as in their abundance). Equally interesting is the predication of equifinality, viz. that several succeeding modellations can lead to the same higher-type result, which in fact states an important insight about (a certain, quite common feature of) dynamic systems.xiv

Furthermore, the term "original" is relativised, since everything which is modellable (cf. inf.) can serve as a (relative) original (for some other process of modellation). Thus a model of type n, i.e. Mn, models an entity of type n-1, i.e. Mn-1, or a model of type m-1, i.e. Mm-1, models a model of type m, i.e. Mm-1 (the superscribed indices "n" and "m" are hereby to be taken as natural numbers). An original can itself be already modelled!

We can also model Theories by treating Theories as objects; the attributes of Theories are then of special interest.xv Thus MTh signifies a model of a Theory Th, and since Th contains already a model of the (relative) type r, the model of the Theory will be typed r+1: Mr+1Th. To the Theory Th itself we attach the highest type which belongs to any parameter contained in it, and this is usually a model of type Mr itself, hence we can scribe: Thr, and we could complete the model of a Theory of type r to: Mr+1Thr.

It might be of interest to find out what kinds of attributes of Theories I am thinking about: First of all, one has to mention networks of Theories (what might even cover something like family-resemblances between representations of Theories). So in modelling a Theory (in some schematic form like MTh), one might especially be interested in the (family-)resemblances, connections, and relations this Theory has to other Theories, one representation of a certain Theory has to other representations of other Theories. It is not only the case that almost every scientific Theory makes use of other theoretical achievements, like e.g. Astronomy uses Optics, or that the current state of the Natural Sciences consists in entire networks of Theories; the operations of measurement are as well based on a Theory, and hence what is observable depends on a Theoryxvi and can be looked upon as an attribute of a Theory, represented in its modelling.

Another point about attributes of Theories are parameters. Within GMT (of STACHOWIAK) the parameters are restricted in their number and order. But would not be a more general formal-pragmatologic conception of theoreticity be conceiveable which takes such parameters as variable in the number and function?

A remark at the end of this section: The typing is always relative, hence questions like "Which object has type 0?" or "Is there any absolute original?" are quite senseless.
 

5. modellability:


Let me state some six conditions for an object being modellable (signified by "@").xvii

(1.) the presumed existence of that which is signified by @;

(2.) the givenness of at least one attribute of that which is signified by @;

(3.) existence is then not regarded as attribute;xviii

(4.) modellability is itself nothing being modelled within first-order-model-Theory (i.e. within GMT);

(5.) there is no actualisation of modellability without the frame of a Theory;

(6.) modellability presupposes the performance(s) of at least one semiotic subject.

I do not claim the sufficiency or completeness of this list, but it contains the core of my pragmatologic understanding of the (meta-)predicate "... is modellable" or "... can be modelled" as applied to some object of our thinking or perception.

It is remarkable that the same (metalinguistic) predicate gets a total different meaning when used in the context of formal semantic model theory, where "... is modellable" means something like "We can give an interpretation of this calculus/theory called ... such that all sentences of ... become true". There only theories in statement-view can be modelled, whereas in the pragmatologic approach all kind of objects can be modelled (theories/Theories included).

The terms "modellable" or "modellability" have a certain modal touch, namely in applying the modality of possibility. Nevertheless it is no term of modal logics.
 

6. some connections to Ludwig WITTGENSTEIN:


At the end of my lecture I would like to point out some connections of this Theory of models in a pragmatic point of view to the philosophy of Ludwig WITTGENSTEIN (sup. we mentioned already the concept of family-resemblance). As is widely known, WITTGENSTEIN was influenced by HERTZ' conception of "picture" and "(dynamic) system" (cf. TLP 4.04, TLP 6.361), as well as of MAXWELL's term of "mechanical model" and of BOLTZMANN.xix I prefer a reading of these terms as well as of WITTGENSTEIN's notion of "Bild", "Abbild", "Modell" which brings these terms in the context of the Theory of models outlined above.xx I want to emphasise the point that my interpretation of WITTGENSTEIN in this regard does not make use of a semantic conception of "model"!

Let me now quote some of WITTGENSTEIN's propositions and utterances which I see in connection to my considerations:

German: "Das Bild ist ein Modell der Wirklichkeit." (LPA 2.12; cf. NB entry from Oct. 27th 1914)

English Translation: "A picture is a model of reality." (TLP 2.12)
 

German: "Der Satz ist ein Bild der Wirklichkeit.
Der Satz ist ein Modell der Wirklichkeit, so wie wir sie uns denken." (LPA 4.01)

English Translation: "A proposition is a picture of reality.
A proposition is a model of reality as we imagine it." (TLP 4.01)


These quotations fit to taking the term "model" (sensu STACHOWIAK) wide enough to cover not only physical artefacts modelling some natural process or object, but also to regard drawings, graphs, images, pictures as models. In the case of picture, the correspondence with objects can easily be imagined (in TLP 2.13 WITTGENSTEIN uses the term "object" in a sense instead of which we would probably prefer the term "attribute" to be applied).

TLP 2.14 states something which makes pictures modellable (cf. inf.) by talking about its attributes:xxi

German: "Das Bild besteht darin, daß sich seine Elemente in bestimmter Art und Weise zueinander verhalten." (LPA  2.14)

English Translation: "What constitutes a picture is that its elements are related to one another in a determinate way." (TLP 2.14)
 

German: "Die abbildende Beziehung besteht aus den Zuordnungen der Elemente des Bildes und der Sachen" (LPA 2.1514)

English Translation: "The pictorial relationship consists of the correlations of the picture's elements with things." (TLP 2.1514)


The astonishing consequence, which WITTGENSTEIN draws, follows from the notion of modellability of picture:

German: "Das Bild ist eine Tatsache." (LPA 2.141)

English Translation: "A picture is a fact." (TLP 2.141)


From this I infer the allowance to scribe (cf. sup. my list of conditions about modellability):

(7.) Models are facts. (But not every fact is modelled.)


But even in a more psychological sense, WITTGENSTEIN applied the term "Modell". Look at the following passage:

German: "Unsere Erwartung antizipiert das Ereignis. Sie macht in diesem Sinne ein Modell des Ereignisses.
Wir können aber nur ein Modell von einer Tatsache in der Welt machen, in der wir leben. D.h., das Modell muß in seinem Wesen die Beziehung auf die Welt haben, in der wir leben, und zwar gleichgültig, ob es richtig oder falsch ist." (PB §34 p.71)

English Translation: "Our expectation anticipates the event. In this sense, it makes a model of the event. But we can only make a model of a fact in the world we live in, i.e. the model must be essentially related to the world we live in and what's more, independently of whether it's true or false." (PR §34 S.71)


These lines could probably be interpreted as linking modelling anticipation in expectation to interest-sensitivity. More liberally, but nonetheless within the frame of my WITTGENSTEIN interpretation,xxii I would say that for anticipating and expecting (as well as for other forms of modelling), the social-ontological background, especially the form of life, is decisive (cf. i.a. PU I §19, PU I §23, PU I §241). (This is a point I missed a lot in LUNTLEY's lectures, and in HINTIKKA's understanding of pragmatics, too; cf. my first lecture.)

There are certainly a lot of other quotations from WITTGENSTEIN's works which could be seen in (family-)resemblance to this model-theoretical approach, but let me be content with what so far has been presented.

Thank you for your attention.




 

Notes:

i We could also introduce sub-sorts of attributes (think of shades of colours like light-green, dark-green, or of the sorting in cold and warm colours), e.g. by fixing sub-sorts of Øi by an extensional counting of the attributes Ø1 to Øi-n as a sub-sort I of Øi, in brief: ØI, and sub-sort II of Øi as the extension of Øi-n+1 to Øi, abbreviated as ØII, and so on (until some definite number, of course). Thus it can be scribed (for the terminology cf. inf. 3): ØIØII ... ØN.N. = Øi, where the index "N.N." denotes the number of the last such scribed sub-sort.
Sub-sorts are not sub-classes of objects or models taken as classes of attributes; the creation of sub-sorts only makes our universe of attributes richer and more precisely modellable, but it has no bearing on our conception of a pragmatologic Theory of modelling: Any question whether a sort is divided into sub-sorts by modelling sub-classes of attributes can be resolved by a new distribution of the symbols denoting attributes in such a way that any ambiguity is avoided. (back)-


ii In this case a whole sort of attributes has been neglected.- It is a bad terminology to talk about OP as "essential attributes" of O, but one could call them the attributes corresponding to the essential attributes of M, i.e. to ME (cf. sup.). (back)-


iii The interpretation of these extreme cases begs the question. The case of OP = {}, if the operation of modelling is understood as signification, is then the case of a pure semantic denotation of some (material) object as a whole (i.e. with all its attributes) via some model M taken as its sign in mind (and ME would then also be empty, cf. sup.).
The purity of such a semantic denotation in my view is owed to the modelling sign M then being mental such there can be no attributes in common between the object and its sign. But if M signifies some mental object we can hardly speak of semantic purity, and I fear that it then will difficult to scribe anything like OP = {}. That something is pure and/or mental can not be attributed to objects, models, or signs but is an effect of a special form of modelling which goes beyond the borders of GMT. (back)-


iv This queer semiotic term "scribe" is Peircean. Cf. HOUSER & ROBERTS & EVRA 1997. (back)-


v Stachowiak talks about an "ikostrukturelle Funktion". (back)-


vi This practical restriction is ours, originally STACHOWIAK 1973 did not restrict himself to it. (back)-


vii This is, of course, an idealisation (of a not always desirable, even quite impractical character). (back)-


viii To this symbol "Th" there might be added subscribed or superscribed indexes indicating the modelling sort or logical type of what is symbolized thereby, just as it can be done in the case of models. (back)-


ix The role of parameters is a difficult one. We can address them as attributes of theoreticity in non-statement-view, and they are attributes of something we talk about in pragmatology by applying them as the frame of our such talking.
Curiously enough, if one conceives of these parametrical attributes as variable in their number, one arrives at seeing a statement-view of theoreticity as a reduced and simplified case of pragmatologic theoreticity insofar as in a pure syntactico-semantic approach a theory is then nothing as an interpreted calculus who "models" the predicative attributes in so-called well-formed formulae or sentences/statements (and the verb "model" is then to be taken in the sense of "forming", "transforming", and "denoting").
This is in my opinion the way of recognizing that the formal-semantic understanding of "model" can be looked upon as a derivation of the far more complex and more embracing pragmatologic conception of a Theory of models. (back)-


x This would be a very special feature of a particular Theory of Interests becoming automorph and self-referential. (back)-


xi When I say that the social implementation of these interests is a conceptual matter, I do not mean that it cannot be an empirical question (this is a clear reference to my first lecture); yet in my understanding a semiotic subject k is in its interests and representational operations always already potentially social since it cannot prevent or foresee whether and how the results of its representational operations will influence other semiotic subjects, and furthermore it can not know qua operator to what goals its operations shall come up to. (We regard the universe of semiotic subjects as not empty, i.e. the operator is never the only semiotic subject.) Thus the targets of these operations can not be immanent in themselves.
There is a certain moment of incalculability involved which makes these interests and especially their bearers non-trivial; and the quality of this operator lies exactly therein that a semiotic subject is a social subject (cf. GELBMANN 2002c and see my third lecture). Thus a second order cybernetics (in the sense of FOERSTER 1979a and FOERSTER 1990a) is the frame for describing and dealing with these interests which are not barely attributes of the semiotic subject as attributes of its operative functioning, but show that the capability of the semiotic subject is not exhausted by its modelling operations.
The interests are externally put on the modelling process, just as the rate of speed allowed for a speeding car on a city road are external to the engine moving the car, yet the limits one sets to using the car's machine in actu are implemented in the person in charge of the car's machine. So if the machine in its operations is a matter of physics and mechanics, the question how it is operated and what limits externally are superimposed on this machine is clearly a social one, transcending the realm of the pure operation itself. (back)-


xii In the case of a pure involvement of predicative attributes in the constitution of wffs I prefer to speak of "syntax" instead of "syntactics", cf. my first lecture. (back)-


xiii In this respect, the property (or even quality) of undecideability of certain algorithmical systems (think of GÖDEL's proof and TURING's incomputable numbers) is a feature which can be addressed as the necessarily inexhaustible abundance of its representation. (back)-


xivCf. BERTALANFFY 1995: 40 und BERTALANFFY 1995: 132. (back)-


xv In winter 2002 I proposed a project to the University of Vienna, investigating a modelling of attributes of Theories. Unfortunately this proposal was rejected due to lack of funds. A decision about a proposal for a related project, delivered to the Theodor Körner Fund in Vienna, is still pending. (back)-


xviCf. the famous exclamation of Albert EINSTEIN, in turning away from positivism:
"Aber vom prinzipiellen Standpunkt aus ist es ganz falsch, eine Theorie nur auf beobachtbare Größen gründen zu wollen. Denn es ist ja in Wirklichkeit genau umgekehrt. Erst die Theorie entscheidet darüber, was man beobachten kann." (EINSTEIN according to HEISENBERG 1985: 80)
I think, this renders a comprehension of theoreticity which in its kernel has already shades of a non-statement-view. It is one of the most distinctive properties of a Theory in its pragmatologic understanding that it is an operation which decides what can be represented within its frame. (back)-


xvii These considerations I have copied from an unpublished book in the field of this lecture, tentatively titled "The Pragmatologic Conception of »Model«. An Essay into Theoreticity, Objectivism and an Algebra of Mind". (back)-


xviii Condition (3.) shall guarantee that existence is independent from modellability. This presupposition safeguards us against the danger of a subjective idealism. (back)-


xixCf. the famous "Principles of Mechanics" in HERTZ 1894 and A. JANIK 1994/95. (back)-


xx My contention of the close relation, if not synonymy, between the terms "Bild" (but not or not always "Abbild") and "Modell" is backed by i.a. MS 102: 32r, MS 104: 4, MS 104: 41, MS 110: 261, MS 113: 123r, MS 136: 112a, TS 210: 65, TS 211: 284, TS 213: 168, TS 232: 698. In MS 114: 34 WITTGENSTEIN even brings the terms "Vorstellung" (representation, idea) and "Modell" close to each other, what anticipates my personal conviction that "Vorstellung" as "mental image" or "idea" and "picture" are just instances or forms of what is signified by "model" (cf. PG I §18 S.56). Often WITTGENSTEIN means with the word "Modell" a physical object and with "Bild" only a picture in its literal sense, but the pragmatologic term "model" is abstract enough to cover both meanings.
For my semiotic considerations of taking the pragmatologic concept "model" as a term for "sign" confer WITTGENSTEIN in MS 116: 110 and TS 211: 511, TS 212: 518, TS 229: 290, and in PG I §18 S.56, where he explicitly talks of the sign-role of a model (see also BPP I §593). (back)-


xxi It is an exciting, striking question whether modelling of pictures must make use of predicative attributes. But even if a proposition/sentence (Satz) is a picture (which shows its sense, cf. TLP 4.022), it is a model. (back)-


xxii The subject of "Social Ontology" was brought up by me the first time in the "Philosophy Round Table" at the University of Bergen (a discussion forum set up by me and a graduate student, Deirdre SMITH); before that, I saw some connections between WITTGENSTEIN and radical (social-psychological) constructivists like Paul WATZLAWICK (cf. GELBMANN 2000 and GELBMANN 2001b). (back)-



 

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