Theory - Model

The usage of the concepts 'theory' and 'model' in the literature is very broad and does not allow a unique definition. In this text we will follow the tradition of modern theory of science (cf. e.g. the books of F.Suppe (1979) [372], Bourbaki (1970) [29], Balzer (1982)[14], Balzer et. (1987)[15], Ludwig (1978)[238]).

From this tradition we derive the following general assumptions:

Figure 17.1: Paradim of an Empirical Theory

• OBSERVER, Primary Language: Every kind of description is finally rooted in a 'community of observers' [OBS] which are using a 'primary language' $L_{1}$ (usually a natural language which has to be presupposed 'as it is'). This primary language $L_{1}$ is 'pre-scientific' in the sense that it is not yet a 'theory'. By using the primary language $L_{1}$ the observers can agree (if possible) in the 'light' of their 'presupposed knowledge' $K_{1}$ a 'point of view' $V_{i}$ what they want to 'observe' in the 'shared real world'.

• MEASUREMENT, Data: If the community of observers [OBS] can define a 'common measurement procedure' $M_{i}$ they can start science by making reproducible invariant measurements generating isolated 'data' $DAT$ to be written in an 'observer language' $L_{2}$ (in many cases the measured events $E$ are not perceivable by human persons!). The data expressions from $L_{2}$ as such have no further meaning then to 'refer' to some 'event' in the 'space of events' $E$ called 'real world' $e_{i} \in E$.

• DATA, Domains: Depending from the 'view' of the observers one can investigate a manifold of different 'domains' of events. Here, in this text, we will look to (i) 'observable behavior' located in a specified 'environment'; we will correlate this with findings in the area of the 'brain' of animals and humans; (iii) we will correlate this with 'phenomenological knowledge' rooted in the human 'consciousness'. The priority has the observable behavior. We will define a list of experimental environments, which are documented in the literature and which we will use as our 'points of reference'.

• RELATIONS, Expressions To 'explain' the data with regard to some 'forecast' one has to 'construct' a relationship between the data which allows to 'infer' a statement which can be used as a forecast. This can be done by explicitly constructing a 'theory' [TH]. This deserves an additional 'theory language' $L_{3}$ which allows the generation of a set of possible theoretical expressions $TH$ which represent the 'true' sentences of the theory $TH \subseteq 2^{L3}$.

• INTERPRETATION: To relate a theory expression $e_{t} \in TH$ to a measurement expression $e_{m} \in L_{2}$ one has to establish a mapping from $L_{2}$ onto a subset $L_{3}*$ of $L_{3}$ as $L_{2} \leftrightarrow L_{3}*$. This mapping works as an 'interpretation' establishing a 'meaning' relation from $L_{3}*$ to $L_{2}$.

• LOGIC: To make 'inferences' from expressions of $L_{3}$ one needs additionally a 'logic' given as a set of 'rules of inference'. It has to be proven that these rules of inference generate only 'true' statements if applied to a set of theoretical expressions which are 'true' (e.g. the axioms of the theory).

• THEORY While a 'theory' $TH$ can be understood as a set of expressions of some language $L_{3}$ it is common usage in the 'structural approach' to assume as 'basic categories' of the world 'sets' $M_{i}$ of objects as well as 'relations'$R_{i}$ between these. Thus the 'kernel' of a theory is assumed to have the format $\langle M_{1}, M_{2}, ..., R_{1}, R_{2}, ...\rangle$ enumerating all assumed basic sets and relations. The relations $R_{i}$ are 'typefied' by applying set operators onto the sets $M_{i}$ (cf. Bourbaki [29]:chapt.7). Example: having a structure $T1 = \langle M, R\rangle$ one could typefy the relational term $R$ as $R\subseteq M \times 2^{M}$ saying that $R$ is a relation between the set $M$ and the power-set of $M$. Including functions as special cases of relations one could construct typefications like $f: M \times M \longmapsto M$ assuming a structure like $T1b = \langle M, R, f\rangle$.

• COMPLEXITY: The usage of structures allows the introduction of the term 'complexity'. 'Complexity' understood as depending on (i) the number of elements in a structure, (ii) the degree of typification of the relations, as well as (iii) the internal structure of the elements.

• MODEL: A formal theory as a structure describes no concrete 'object in the world'. To use such a structural theory with our investigation we have to construct 'instances' $th$ of such structures by mapping the abstract sets $M_{i}$ into sets of expressions of $L_{2}$ and to map the abstract relations and functions $R_{i}, f_{j}$ into 'real computing functions' of a 'working theory' $th$. This can be done e.g. by using a 'programming language' or the language of 'automata' which can further be translated into a program written with a programming language. There can be different models $th_{i}$ derived from a theory structure $TH$ $TH \Vdash \{th_{1}, ..., th_{n}\}$.

• VALIDATION: To 'test' the 'validity' of a model $th$ we have to test the possible models against those domains of possible behavior, which are explicitly given in the experimental environments. But, even if a model $th$ is in 'agreement' with the selected set of an experimental environment, this does not tell us, whether this model $th$ will 'work' in any experimental environment. To validate a model $th$ one has to 'expose' the model in the environment and then one has to observe the performance of the model there. This observation has to be specified by defined methods of measurement, which have to be realized as 'automatic protocols' (log-files). The outcome of the measurements should allow at least a comparison between different models in the same environment with regard to the differing properties of the models, giving 'hints' to the implicit properties of the model. This would allow the definition of a relative taxonomy of models. A different kind of evaluation would use experiments which are related to those kinds of experiments, which are used to measure the IQ of children. This would allow an IQ-based taxonomy of models. Other kinds of taxonomy are conceivable.