In addition to former excerpts of Herbert Stachowiak’s 1973 book “Allgemeine Modelltheorie” (General Model Theory) here is a part on empirical-theoretical models wrt. empirical science (chapter 22.214.171.124 part 1 (formal semantic approach)).
Although I’m a software requirements analyst and not an empirical scientist, this describes pretty close the very basics of my kind of modelling.
First he recapitulates the formal (scientific) models (iow: basic maths) from an earlier chapter:
Starting point of the subsequently following considerations is the concept of the formal-scientific evaluation model in the closer sense, as introduced in the last chapter. It was said to be a (concrete or abstract) relational structure (M, a), that satisfies a given (sound) axiom system Σ. Latter consists of conditions in formal language, equipped with variables (argument places). The relational structure was a (n + 1)-tuple (M; Ru …, Rn), defined on a domain set M and n relations defined on M, satisfying the condition system Σ. With Σ also the theory induced by Σ (in addition of deduction rules) holds, since the satisfying Σ-evaluation entails a satisfying T(Σ) evaluation.
Such formal models can be applied to turn data (more or less well) into understanding of the world by abstraction:
These characterisations and in particular the differentiation between a theory and its possible satisfying evaluations are transferable to certain empirical scientific subject areas. Therefore, such a subject area has to be either available in a completely axiomatised way or its axiomatisation has to be progressed so far that linguistic description models and their associated allocation models can be sensibly distinguished. In other words, the conceptual apparatus of mathematical structural theory and the logical-semantic model theory can be applied to empirical scientific modelling in such cases. Here, the formal linguistic axiomatic constraint system Σ is abstracted from empirical data (with actually a broad subjective variety in the theory construction process) or it is at least constructed wrt. empirical data. This is due to the systematic summarising, explaining and prediction-enabling description of parts of the empirically accessible world it provides, especially the space-timewise-energetic entity and event world.
And here is how to do it, namely by interpreting the formal domain and relations by empirical entities and relationships:
The axiom system of an axiomatised empirical scientific theory is at best a system of logically fully formalised predicative statements, employing mathematical means of representation and descriptive, i.e. non-logical and non-mathematical, base terms. Let such an axiom system be interpreted by an empirical relational construct satisfying the axiom system of the associated empirical scientific theory T(Σ). Then this construct is said to be an evaluation model (in the closer sense) of Σ, or an (empirical) realisation of the axiom system Σ. The domain M of the construct consists of individuals of a part of the empirically accessible world of entities and events. Relations on M are empirically deducible respectively empirically verifiable properties of these entities as well as empirically deducible respectively empirically verifiable relationships amongst these entities.
For example, consider the axioms of classical particle mechanics:
(A1) P is a finite nonempty set.
(A2) T is an interval of real numbers.
(A3) Let p ∈ P and t ∈ T. Then s(p, t) is a twice t-differentiable vector-valued function.
Above, the set symbols P and T as well as the relation symbols s, m, and f are interpreted descriptive base terms, only implicitly defined by the formal axiomatic conditions. Their empirical interpretation takes place by the evaluation by terms of the used observation resp. measurement language. Let it be characterized on the level of the observation language as follows:
On (A1) P is a set of physical objects, e.g. the set of planet objects of the solar system, or more abstract, the set of centers of masses of the physical objects. The finiteness condition shell ensure the definability of the masses and the kinetic energy of the whole particle system.
On (A2) The elements of T represent periods of time. Concerning time measurement only, rational numbers had been sufficient. Reals are required to allow differentiation and integration wrt T.
On (A3) s = s (p, t) is a 2-ary location function, that describes the spacial changes of the physical objects (planets) p over time t. (A3) requires the existence of the 2nd derivative of s(p, t) wrt. t, s.t. (A7) can be satisfied.
In order to leap back to requirements modelling, as mentioned above, we take a look at the famous example by Jackson/ Zave 1995 of the requirements of a turnstile. They call an empirical realisation a designation set:
Each designation of the set gives a careful informal description by which certain phenomena may be recognised in the environment; it also gives a term by which the phenomena may be denoted in requirement and specification descriptions:
a) in event e a visitor pushes the barrier to its intermediate position ≈ Push(e)
b) in event e a visitor pushes the barrier fully home and so gains entry to the zoo ≈ Enter(e)
c) in event e a valid coin is inserted into the coin slot ≈ Coin(e)
In terms of Stachowiak: The unary relations Push(e), Enter(e), Coin(e) that are part of the formal requirements modelling language are defined by an empirical realisation in an informal observation language.
Thus, the theory of empirical-theoretical models by Stachowiak, provides a perfectly suitable foundation of semantic requirements modelling.
1. Please excuse my rough and incomplete translation. Professional improvements welcome.
2. Ref: Michael Jackson, Pamela Zave (1995) Deriving specifications from Requirements: An Example
3. See also: Wilfrid Hodges (2009) Functional Modelling and mathematical Models: A semantic Analysis
4. More by Herbert Stachowiak: Fundamental Model Properties, Preterition and Abundance, …