## Stachowiak on Preterition and Abundance in Modelling

In his 1973 book “Allgemeine Modelltheorie” (General Model Theory) Herbert Stachowiak describes the fundamental properties that make a Model.  Unfortunately this is still not available in english language, so I thought why not try a (quite free-style) translation of the essential bits (see here for all):

## Part I: Excerpt

2.1.4.5 Preterition, Abundance and Contrastation

Three fundamental concepts (out of many) from the model theoretical apparatus are presented here (their explication is done not here).

Preteritive (= omitted, ignored) attributes are not covered by the Original-Model mapping, since they are operationally irrelevant. The preteritive class of an Original (wrt one of its Models) is the class of the preterited Original attributes (wrt this O-M mapping).

Abundant (= superfluous, redundant) attributes in the Model correspond to no attributes in the Original. This can often be found by retrospective empirical analysis of concrete Models. They constitute the abundance class of the Model wrt its Original. Abundant attributes have no direct function in the O-M mapping. They are either just ‘technical vehicles’ of Model construction or ‘hypothetical bridge-over’ constructs. As technical vehicles they are chosen by economies of efficiency, construction, but also presentation and explanation. As hypothetical bridge-overs they close a gap in the mapping, i.e. they imply imaginary attributes of the Original from which they are mapped. This may be in close conjunction with the intention to gain new insights into ‘yet unknown’ properties of the Original.

Contrastation (not in the picture) refers to the feature of many concrete Models to accentuate certain attributes or attribute classes wrt the other original attributes, i.e. to contrast with each other. (e.g. image processing like sharpening or increasing contrast)

Stachowiak, Herbert (1973) (in german (DE)). Allgemeine Modelltheorie [General Model Theory]. Springer. ISBN 3-211-81106-0.

## Part II: Interpretation

From a logical point of view, I would put it like this:

### Interpretation I

An Original-Model mapping, can be seen as a homomorphism. In this case, the homomorphism preservation theorem holds, stating that just Original attributes of a certain limited expressiveness are always preserved in the Model. Others may or may not hold in the Model.
In detail: Sentences equivalent to a positive first-order existential expression are always preserved by the mapping (this also holds when restricted to finite models only). E.g. ‘my relation has a non-reflexive element’ might hold in the Original, whereas ‘all my elements are reflexive’ might hold in the model.
Thus, for non-preserved expressions from the Original, their negation might hold in the Model. The earlier could be interpreted as Preterition, the latter as Abundance. However, I don’t think, this was what Stachowiak had in mind.

### Interpretation II

Beyond these unpreserved attributes, there are cases like in figure (unpreserved colouring). The Model could be interpreted s.t. now all nodes and edges are coloured black. Alternatively the Model could be seen as not covering the concept of colour at all anymore, i.e. colour has been abstracted away. However, even in the case of a non-coloured model the nodes and edges physical have to be drawn in some colour. Thus, here black means neutral. Imho this was the kind of Abundance Stachowiak had in mind.

So long
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PS
The book at Google books

Modeling Theory and Abstraction Awareness in strive for scientific rigour and relevance to information systems engineering.
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### 10 Responses to Stachowiak on Preterition and Abundance in Modelling

1. TY says:

Well, let me try to understand some your thoughts 🙂
Have a question: why is it *have to* drawn in some color but not in “unknown” (as the NULL in relational model)?

• Hi TY
Thx, good point. I mean this solely physical, since there physical is no ‘non-coloured’ colour one has to declare a colour to have the meaning of ‘unknown’ explicitly. (just re-edited it a bit to make it clearer)

This way, one has to take the abundant attributes (here: being black) out of the Model explicitly, s.t. they cannot be taken for actual properties.

Think, this is especially important from your point of view, when you say that everything can be a model of everything, since if you take e.g. a coffee mug as a model for a building one could say “but its open on the top, so it can rain in”, but of course this is not an actual property of the building you wanted to address by using the mug.

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• TY says:

agree! and what is interesting is that how to determine that an attribute is valid (i.e., an actual property of the thing being modeled) when use a model. My opinion is, it is necessary to some modeling knowledge (metamodels) about the model which is (are) used by the device who uses the model, in which they are formed as model-driven mechanism.

• think, from a practical point of view, this hits the point:
“In some sense, the semantics of a language takes into account more knowledge about the domain than what is expressed in the language: the language only allows users to express things that are particular to the specific system/application/instance they describe with the model. The semantics, however, also takes into account the knowledge about all the stuff in the domain that is identical for every system/application/instance in that domain.”

from Markus Völter’s “DSL best practices”
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• TY says:

That’s a sharp point of view, and it’s to be a rather interesting comparison with my opinion. Although I have my reservations about DSL approach, relatively speaking, I prefer to DSLs than UML.

2. How would Abundance be defined in terms of Model-driven mechanism?
perhaps sth like this: “makes all or part [but definitely not more] of functions and behaviors (or the structure and form) of a system to be in control of or mastered by model.”

• TY says:

hummm…the statement you cited is about model-driven system, what MDM presented is that, in a certain case using a model, the Abundance (Preterition as well) of the model is indeed determined by the modeling knowledge (metamodels), and I prefer to say that the issues about Abundance / Preterition is removed in this situation.