Beyond the colloquial fundamental model properties (mapping, reduction, pragmatics) Stachowiak gives a more formal description of model. The definitions of the terms used in the explication of the concept of model take a whole chapter in the book, however I want to give a rough idea what I think it all means:
k is an intelligent modeller, that is able to recognize the signs relevant to it (1), that express the original O and the model M as appropriate. It is assumed that the original model can be expressed by propositions P (2, 3). The O to M mapping takes place between the corresponding propositions (4) where not all of them need to be mapped. (5) seems actually redundant, saying mainly the same as (4) on the level of signs allocated to propositions. Finally, in the eye of the organism k, M replaces O.
So far it is just some allocation of any M to an O performed by k, satisfying the mapping and reduction properties. However in order to make ‘sense’ the pragmatic aspect is required, that is explicated by the remaining points (6 – 12). Therefore the M*, P* and finally O* are introduced, where O* represents an “improved” original (by inverse mapping from M* via P*), in the sense that it provides some improved understanding of the original O to k. In other words, modeling created something new, a new insight to the modeller! (replace model by abstraction if you like)
Stachowiak’s original “explication” looks like this:
Explication of the Concept of Model
Thus, [by the tuple (M, O, k, t, Z)] the concept of model can be presented in the following more rigorous way:
Let M and O be some (semiotic or non-semiotic) objects. Then for a K-organism k the object M is said to be a model of the object O in the time interval t wrt. the operation objective (the aim, the intention of the modelling) Z, if k in the time interval t
- is L-rational,
- realises a description Po of O,
- realises a description Pm of M,
- realises an icostructural mapping F from Po to Pm,
- realises a transcoding T from Po to Pm in the transcoding class TK,
- realises the replacement of O by M,
- realises certain operations Op(i, Z), with i=1, 2, …, n for the (partial or complete) attainment of Z for M, s.t. M is transformed to an object M*,
- realises a description Pm* of M*,
- realises a revers of the icostructural mapping F from P m to Po with the result Pm*,
- accepts the predicate class Po* as description of O*,
- accepts the replacement of O* by M*, and
- realises the re-coding of the transcoding T of Po in Pm wrt. Po* and Pm*
(Notice that the reduction property is satisfied, since 1. the domain and the codomain of F, UO and UM contain the same number of of elements, 2. this number is less or equal the number of elements of PO, 3. the number of elements of PM is less or equal of that of PO. The boundary cases mentioned in 2 and 3, usually never be reached in real modelling.)
Stachowiak, Herbert (1973) (in german (DE)). Allgemeine Modelltheorie [General Model Theory]. Springer. ISBN 3-211-81106-0.
For a discussion on the interpretation of the K-system see also:
- Stachowiak’s (M, O, K, t, Z): a case for model as use by TY.
- An Outline of Pragmatologic Model-Theory by Gerhard Gelbmann.