# A Model-Theoretic Account of Science

Chapter

## Abstract

What is it that philosophers can or should say about science? Should we explain the actions of scientists? Should we explain the methodology of science? Should we explain the process and progress of science? Should we study the knowledge claims offered by scientists and decide which of these are so-called “scientific” knowledge claims? Should we study the history of science and its influence on current scientific practice? What — if anything — should we say about reality and the links — again, if any — that exist between science and reality?

## Keywords

Empirical Model Scientific Theory Default Rule Empirical Adequacy Interpretative Model
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## Notes: Chapter 2

- 2.Although not as common-sensical as Fine’s (1986b) “natural ontological attitude” perhaps.Google Scholar
- 3.One may ask how — and even if — it is possible to distinguish between conceptual and linguistic levels without giving a clear and valid answer to the question of whether it is possible to think without language. I am however not making rigid distinctions here. What I am doing, in fact, is to depict the development of scientific research by emphasising one by one the real, conceptual, and linguistic aspects of this evolutionary process. And, moreover, I am claiming that there always is interplay between these aspects.Google Scholar
- 4.Think, for example, of students able to cite all the rules (or laws) of a specific area of their subject matter, who are still unable to
*apply*this knowledge in any concrete way.Google Scholar - 5.See for instance Balzer’s articles “A logical reconstruction of pure exchange economics” (1982), and “The proper reconstruction of exchange economics” (1985); Hands’s article “The structuralist view of economic theories: A review essay” (1985); and Janssen’s article entitled “Structuralist reconstructions of classical and Keynesian macroeconomics” (1989); as well as Balzer and Hamminga’s
*Philosophy of economics*(1989), and many others.Google Scholar - 6.In a language such as L, we usually have the following eight categories of basic symbols available:Google Scholar
- 7.A mathematical structure U = < A, R,> consists of a set A, which is the domain of U, and a set of relations R (one for each a from some index set) defined on domain A. The sets A and both may be infinite. A relation R
_{a}on domain A is defined as a set of ordered µ(a)-tuples of elements from domain A, where µ(a) is a unique non-negative integer associated with the relation.Google Scholar - 8.In other words the mathematical structure U will count as an interpretation of the language L if and only if the arity of the relations R
_{a}correspond to the arity of the predicate letters**P**_{a}. (That is, if 8(a) = µ(a).) In this case U is called a realisation of the language L (and we can say that L is appropriate for the structure U). We call the relation R_{•}the value of P_{•}in the realisation U of language L. (If L has constant and function symbols, they are interpreted as elements of A and functions — of the proper arities — on A.)Google Scholar - 9.E.g., consider the formula Pxy. If P is interpreted as the relation < and if x and y are given the values of 3 and 5 respectively, then we say that Pxy is
*true under that interpretation*and we say that formula Pxy in language L is*satisfied*by the valuation in the domain of interpretation U, ascribing the given values to variables x and y. (Because 3 is indeed smaller than 5.)Google Scholar - 10.Note that a realisation of language L is in principle a realisation of all the sentences in L, and this implies that every sentence in L is either true or false in that particular realisation.Google Scholar
- 11.Einstein referred to these convictions as “free conventions” (Holton, 1995, p.464). “These themata, to which [Einstein] was obstinately devoted, explain why he would continue his work in a given direction even when tests against experience were difficult or unavailable (as in General Theory of Relativity), or, conversely, why he refused to accept theories well supported by phenomena, but, as in the case of Bohr’s quantum mechanics, based on presuppositions opposite to his own,…” (ibid., p.457).Google Scholar
- 12.Chalmers (1993, p.202) gives another example of these events: “We may abstract the falling of [a]… leaf from other aspects of its motion… We then apply the appropriate fundamental laws [axioms of mostly ”background“ theories] to [this model] that [is] the result of our abstraction. We apply Newton’s laws to the leaf as a mass subject to the gravitational attraction of the earth only, and derive the law of fall from it. Of course, since we have abstracted from winds, air resistance and the like, our model will not in general serve to describe the fall of any particular leaf. After all, the model is an abstraction. Nevertheless, provided we understand the leaf to have a capacity to fall, governed by Newton’s laws, the
*theoretical treatment via the abstract model*does explain the falling of the leaf, as distinct from its fluttering in the breeze”. This is a point about which Nancy Cartwright has realist reservations, but which I still interpret as “realist” within a model-theoretic context. See Chapters 4 and 5 for more on Cartwright.Google Scholar - 13.Johannes Heidema (University of South Africa) introduced me to thinking in terms of this hierarchy.Google Scholar
- 14.Einstein referred to the movement from conceptual structures or models to theories as a “creative leap” and in this sense referred to theories as “free creations of the human mind”Google Scholar
- 15.Kepler’s laws:Google Scholar
- 16.See the examples of the discoveries of Neptune and Pluto, as well as other applications of Newton’s theory in the following section.Google Scholar
- 17.Another type of approach to the interpretation of language terms is offered, for example, by Hans Lenk’s methodological or schema interpretationism. See for instance Lenk (1993), (1995).Google Scholar
- 18.Whenever I speak of models of theories, I am referring to the notion of model in the Tarskian sense that a model of a theory is an interpretation of the theory under which the set of sentences comprising the theory is true. As mentioned in the previous section, at the start of theory formulation the intended “model” scientists work with is not (initially) such a mathematical model, although at the stage of theory interpretation it becomes obvious that such (intended) models can be easily adapted such that they also are elements of the set of all (mathematical) models of the theory in question.Google Scholar
- 19.These terms are the terms traditionally referred to as “theoretical” terms. Note that therefore I do not follow in the footsteps of advocates of the traditional version of the statement approach, in the sense that I do not need the kind of (too simple) distinction they make between theoretical and observational terms in the language of the theory. Rather than this forced division, I propose an approach in which theoretical and observational terms, as well as the difficult “correspondence rules” or “bridge principles” supposedly acting between these kinds of terms, all have natural interrelated non-unique and co-dependent roles to play at various levels of the scientific process. See Chapter 3.Google Scholar
- 20.I claim that Giere’s theoretical models, Wójcicki’s theoretical and semantic models, and Suppes’s physical and set-theoretic models are all mathematical models in this sense. Some of these authors make a similar kind of distinction that I make between these models as “intended” models — Wójcicki’s theoretical models and Suppes’s physical models — and these models as “interpretative models” interpreting the theory — Giere’s theoretical models, Wójcicki’s semantic models, and Suppes’s set-theoretic models. See Chapter 4.Google Scholar
- 21.This notion of changing the original set of assumptions made by the original scientist, naturally may be connected to Popper’s theory of falsification.Google Scholar
- 22.Note that in this case, the embedding function simply is the identitiy function, mapping elements of E
_{e}onto elements of E_{w}. See*Figure 2*below.Google Scholar - 23.Note that this distinction between so-called “theoretical” and “empirical” predicates is model-specific rather than unique or absolute. See Chapter 3.Google Scholar
- 24.it is always legitimate for scientists to ask and sometimes possible for them to answer, questions about whether gasses are really composed of molecules or whether the earth really moves. Such questions cannot
*be*rephrased as questions about the plausibility of our conceptions“ (Bhaskar, 1978, pp.155). Well, model-theoretically, the verification of our interpretative models depends on being able to show how experiments concerning the data in question may be linked to these models (via certain empirical models). However, what Bhaskar means, I think, is rather that science does not determine the structure of reality, but rather discovers it.Google Scholar - 25.Formally, in Lipsey’s (1983) terms, we have the demand function as cl
_{o}^{d}= D(p_{o}, p„…,p„ Y, E), and the supply function as qos = S(p_{n}, F_{1},…,F_{n-1}).Google Scholar - 26.Hausman’s ( 1991, Chapter 3) term, in the sense of Giere’s (1983), (1991) definition of “theoretical models”.Google Scholar
- 27.Now we can write the functions in endnote 25 as q
^{d}= a - bp, a,b > 0, and q’ = dp - c, c,d > O.Google Scholar - 28.Other examples: Models of neo-classical growth theory for instance are models of monetary growth, the one-sector model with exogenous population growth and technical progress, two-sector models, models with endogenous technical progress, and so on.Google Scholar
- 29.See Torr (1999).Google Scholar
- 30.Keuzenkamp and Magnus (1995) actually went so far as to challenge readers of the
*Journal of econometrics*to “name a paper that contains significance tests which significantly changed the way economists think about some economic proposition” (ibid., p.21).Google Scholar - 31.Whether or not econometrics has an explanatory role — which is more important from a realist point of view — is a very difficult question, which can only be answered contextually.Google Scholar
- 32.In the context of the Wahasian theory of tatonnement Patinkin (1965) asks who actually solves the equations. He (ibid., p.38) writes: “The fact that the number of independent excess-demand equations is equal to the number of unknown money prices and that the system can be formally solved, might some day interest a Central Planning Bureau duly equipped with… computers and charged with setting equilibrium prices by decree. But what is the relevance of this fact for a free market functioning under conditions of perfect competition?”.Google Scholar
- 33.Think of caricatures exaggerating certain features of their subjects. If it is a good caricature, it is however possible to recognise the subject in question at a glance. (Chris Torr, Department of Economics, at the University of south Africa)Google Scholar
- 34.Parts of this section forthcoming in the Poznan Studies’s volume on Theo Kuipers (Amsterdam: Rodopi).Google Scholar
- 35.More precisely, traditionally the nature of under-determination has been understood in terms of two kinds of relation between the “real world” and scientific theories. The first kind is taken to exist between phenomena (or whole systems) in reality and the observation terms of theories, while the second kind of relation is said to exist between sets of protocol sentences (formed from the observation terms and expressing data) and possible theories incorporating or explaining such a set of protocol sentences — that is, the existence of incompatible but empirically equivalent theories.Google Scholar
- 36.Heidema and Burger (Forthcoming, p.1) note Paul’s (1993) remark that abduction is often related to conjecture; diagnosis, induction, inference to the best explanation, hypothesis formulation, disambiguation, and pattern recognition.Google Scholar
- 37.For instance: Clark’s (1978) predicate completion, Reiter’s (1980) default logic, McDermott and McDoyle’s (1980) non-monotonic logic, McCarthy’s (1980) circumscription, or McDermott’s (1982) non-monotonic logic II. See also Ginsberg (1987), Kraus, Lehmann and Magidor (1990), and Shoham (1987).Google Scholar
- 38.Where `interpretation’ means truth assignment for [propositional calculus], a first-order interpretation for [first-order predicate calculus], and a <Kripke interpretation, world>-pair for modal logic. (Ibid.)Google Scholar
- 39.This example is borrowed from discussions with Willem Labuschagne from the Department of Computer Science at Otago University, Dunedin, New Zealand.Google Scholar
- 40.There are two approaches to ordering possible worlds: by using numbers, or without using numbers. The best known numerical ways are those using
*fuzzy sets*or using*probabilities*. Neither of these would give us the kind of formal mechanism I am looking for in the current context, and therefore I choose the mechanism of non-numerical default rules, applied by defining certain total pre-order relations to induce such orderings.Google Scholar - 41.It is natural to wonder whether a default rule can be expressed by a sentence of the formal language in the same way as our knowledge that, say, the light is on. The answer is no. A default rule, as we have construed it, says something about the ordering of states according to decreasing normality or typicality. Such default rules cannot be expressed as sentences of the logic language whose valuations or interpretations are intended to correspond to states of the system. That is why we cannot simply add in a default rule as an axiom and resort to familiar ‘classical’ logic. To see that default rules cannot be expressed as axioms, take a transparent propositional language for talking about the light-fan system. The predicate symbols are about the components, expressing such notions as ’is on’ or ’is malfunctioning’. The predicate symbols are not about the system as a whole. And if we add in constants denoting states and predicate symbols denoting relative normality, then interpretations of the new language would correspond to a new, more complex, system instead of to the old system.Google Scholar
- 42.See Einstein, (1956, pp.l lff., 65ff.).Google Scholar
- 43.Where i equals i-1.Google Scholar

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© Springer Science+Business Media Dordrecht 2002