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

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 20^th 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 A_i, B_i, C_i, … 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 O_P. O_P 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: O_P = {A₁, C₉, D₄} if O = {A₁, B₅, C₉, D₄}. The class of neglected attributes of O would be O \ O_P, i.e. in our case {B₅}.ⁱⁱ In other words: O_P is a sub-class of O: In mathematical symbolism we can express this as: O_PO. In extreme cases O \ O_P = {} or even O_P = {}.ⁱⁱⁱ

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

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 operation^v F such that F(O_P) = M_E, 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(A₁O_P) = A₁M_E
since O and M can have attributes in common, thus it could be that O_PM_E~={};
F(C₉O_P) = C₃M_E;
F(D₄O_P) =  D₂M_E,
whereas the attribute B₅O is neglected (preterated), hence we scribe: B₅(O \ O_P).

The attribute E₂M is abundant, ergo E₂M_A 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):
 
 

The elements of O_P and M_E correspond to each other in one-to-one relations, but not those of O \ O_P and M_A!
 

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 parameters^ix 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 Th_1-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. M^n-1, or a model of type m-1, i.e. M^m-1, models a model of type m, i.e. M^m-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: M^r+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 M^r itself, hence we can scribe: Th^r, and we could complete the model of a Theory of type r to: M^r+1Th^r.

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 Theory^xvi 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.

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. 27^th 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.

Ø_IØ_II

"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)

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