# The Notion of “Model” in Philosophy of Science

## Abstract

One intuitive idea of a model is a possible interpretation in which a theory is satisfied in the Tarskian sense, that is, according to Tarski (1956) a model of a sentence in some appropriate language is a possible interpretation of the language in which the sentence of the language are satisfied. This is the basis of model-theoretic analyses of scientific theories, i.e. a model of a theory is a possible interpretation in which all sentences of the theory are satisfied (i.e., in which the sentences are “true”). Model theory was initially developed for explicitly constructed formal languages with the purpose of studying certain mathematical issues, until Evert Beth and others initiated the application of model theory to semantic analyses of so-called “empirical” scientific theories.^{1}

## Keywords

Scientific Theory Empirical Adequacy Rational Reconstruction Intended Model Gedanken Experiment## Preview

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## Notes: Chapter 1

- 1.For a good introduction into model theory for first-order languages, read
*Model theory*(1990, 3r^{d}. edition) by C.C. Chang and H.J. Keisler.Google Scholar - 2.Suppes (1960, 290, 291) writes: “To define formally a model as a set-theoretical entity which is a certain kind of ordered tuple consisting of a set of objects and relations and operations on these objects is not to rule out the physical model of the kind which is appealing to physicists, for the physical model may be simply taken to define the set of objects in the set-theoretical model”.Google Scholar
- 3.I shall from now on sometimes speak of Newton’s “theory” when referring to his laws of motion and his law of gravitation.Google Scholar
- 4.See also Hesse, M. 1963.
*Models and analogies in science*. As Redhead (1980, 149) remarks, Hesse does however point to the role of mathematical models in the development of theories, although she pays much more attention to the role of models in science in terms of analogies.Google Scholar - 5.I am not implying here of course that these (Tarskian) models are “necessarily true” depictions of systems in reality, but simply wish to point out that here at least the possibility of such a turn of events is possible, albeit sometimes by rather complex means. I shall come back often to this point.Google Scholar
- 6.Another example that illustrates the relation between Hesse’s analogous models and so-called iconic models is given in Da Costa French (1990, p.250). They (ibid.) write: “In a nucleus… there are too few particles for a statistical treatment, and there is no overriding centre of force which would enable us to treat the forces between nucleus as small perturbations. For this reason, physicists have fallen back on the ‘as if methods of attack, also known… as the method of nuclear models. This method consists of looking around for a physical system, the ‘model’, with which we are familiar and which in some of its properties resembles the nucleus. The physics of the models are then investigated and it is hoped that any properties discovered will also be properties of the nucleus.… In this way the nucleus has been treated ‘as if it were a gas, a liquid drop, an atom, and several other things”.Google Scholar
- 7.The arguments that Da Costa and French (1990, p.260) offer in support of their claim that models can only be false do not have anything to do with mathematical models, but only with the use of models as iconic models. Therefore they can only be allowed to conclude that
*iconic*models are false, which after all is rather obvious, given the analogous “as if’ role of these models.Google Scholar - 8.Redhead (1980, p.147) remarks that models are used as “impoverished theories” if a theory is so complicated that it is very difficult to draw any kind of empirical conclusion from it, since comparisons between the theory and experimental results prove to be too complex. He also shows clearly that the role of these kind of models is not to be confused with the role a Tarskian model plays in the process of science: — “… the important ingredient … [is] that [the model] and [the theory] logically contradict each other, so that we believe [the model] to be false insofar we believe [the theory] to be true” (Redhead, 1980, p.147).Google Scholar
- 9.Note that also theories which are proved somehow empirically inadequate through experimental or some other type of empirical investigation, turn into “impoverishments of theories” — for example (Redhead, 1980, p.147) Maxwell’s kinetic theory of gases is now known as the billiard ball model of gases.Google Scholar
- 10.See Ruttkamp (1997a), as well as Chapter 2 of this text.Google Scholar
- 11.In my scheme of things, only the “intended” model of some theory has the potential to develop into a “full-fledged theory”. See my explanation and discussion of these notions in Chapter 2.Google Scholar
- 12.See Suppes (1960, p.296) for his reference to Mach in this context.Google Scholar
- 13.Tuomela (1972b, 1974) goes to great lengths to point out the problems involved in Przelecki’s assumptions concerning the fixing of the universe of the language L in advance — which Przelecki claims is necessary to do in order to “explain the fact of empirical interpretations” (Przelecki, 1974, p.404).Google Scholar
- 14.See Van Fraassen (1980, pp.45ff.), and Chapter 5 of this text.Google Scholar
- 15.Suppes and Giere and Wójcicki all seem to think that scale models and — even more physical perhaps — models of aeroplanes and cars are at least part of the notion of a “physical” model. In my terms part of the conceptualising that culminates in the intended model may well be directed towards such a type of model — or not, depending on the particular line of research in question. (See Chapter 2.)Google Scholar
- 16.Note that “physical” here does not necessarily mean concretely physical, but merely serves to show the more “direct” link with the real system of reality being examined. Any activity ending in the construction of a model is conceptual in the sense that various activities of abstraction and even idealisation are performed. As far as the very few times that an actual concrete model is built go — think for example of the model that Watson and Crick (see Giere, 1991) built of the DNA molecular structure — I would say that, usually, even then, at the same time some kind of (interpretative)conceptual model is also created.Google Scholar
- 17.See Hausman (1992, p.77) for a table of differences between models and theories.Google Scholar
- 18.See also Morgan (1990) and Hoover (1994).Google Scholar
- 19.Chiang (ibid.) also points out that economic models do not necessarily have to be mathematical, but if they are, they consist of a set of mathematical equations describing the structure of the model, that gives mathematical form to the set of analytical assumptions adopted by the model, via the relation of variables to each other in certain specific ways. Then, via some mathematical operation to these equations, a definition of a set of conclusions which logically follows from the assumptions of the model, can be formulated.Google Scholar
- 20.Model“ in
*The Fonatana dictionary of modern thought*(1977).Google Scholar