The semantic approach developed by van Fraassen was also motivated by Tarski’s work in model theory but indirectly through the work of Beth (1948, 1949, 1961), himself drawing upon the work of von Neumann, Birkhoff, Destouches and Weyl. This approach is a state-space approach, treating a theory fundamentally in terms of the space of possible states of a system and the measurable physical magnitudes that characterize the state of a system. Extending the work of Beth, van Fraassen
(1970) developed a general framework intended to characterize the formal structure of nonrelativistic physical theories. Unlike Suppes’s approach, which works informally with the notions of set theory in order to specify the set of models of a theory, van Fraassen explicitly uses a model-theoretic approach to characterize the semantics of theories. We will see that van Fraassen’s formal semantic approach bears the closest similarity to the basic framework we will use in this study.
A theory is characterized, on van Fraassen’s formal approach, in terms of three fundamental components. One is the state-space of a given kind of physical system.
7This argument will be important for our purposes, since the constraint approach that I use in
this study will make similar use of well-defined mathematical concepts in the interests of modeling theory application.
This includes the system (position, configuration, phase) spaces described in the pre- vious chapter, as well as the Hilbert spaces used in quantum theory. The second component is a certain set of measurable physical quantities provided by the theory, which are capable of characterizing a physical system, e.g., position and momentum in classical mechanics. In van Fraassen’s picture, this then determines the set of
elementary statements of the theory. These are statementsU that express the propo- sition that a given physical magnitude m has a certain value r at a certain time t, schematically written, e.g., U(m, r, t). The third component is a function s(U) that specifies a region of the state-space over which U is satisfied. This function is called the satisfaction function. To illustrate, in the case of the 6-dimensional Euclidean phase space for a classical point particle, the statement U ⇔ “The x-component px of momentum is r” is satisfied over the 5-dimensional hyperplane defined by px =r. So in this case, s(U) is that 5-dimensional hyperplane. A given statement U about a physical system is then true provided that the system’s actual state is represented by an element of s(U).
Thus, on van Fraassen’s formal approach a theory is understood to specify a set
E of elementary statements, a state-space S and a satisfaction function s, which maps the set E of elementary statements to some set of subsets of S.8 This triple L = ⟨E, S, s⟩ is then called a semi-interpreted language, since the language E of L
is provided with a certain partial interpretation in terms of the satisfaction function
s(U). It is partial, because which statements are actually true depends on the actual state of a physical system. In this way, van Fraassen’s formal approach specifies a definite theoretical semantics in terms of the state-space S, which provides via s(U)
a semi-interpretation for the language E that specifies the possible statements U
about a given physical system. Thus, van Fraassen’s formal approach provides a limited representation of the symbolic constructions of a physical theory in terms of the elementary sentences E and relates them explicitly to a phase-space semantics via the satisfaction function.
A semi-interpreted language is also given a formal empirical semantics. A model
for a semi-interpreted language L is a couple ⟨loc, X⟩, where X is a system of the type specified by L and loc is a function that assigns a location to X inS. What X
is intended to be is not made clear, but it plays a role quite analogous to that of a data model in Suppes’s framework, viz., an abstraction from a real physical system that correlates measurement data from experiment with what can be interpreted via the function loc as a state or trajectory in S. This formal empirical semantics is given a truth definition by defining an elementary statement U of L to be true in M = ⟨loc, X⟩ iff loc(X) ∈s(U). A set X of statements is true in a model M just in case each member is true in M. This then allows the following:
Definition 2.2.1 (Semantic Entailment) A set X⊆E semantically entails U in L, writ- ten X ⊩U, iff U is true in every modelM of L where X is true.
The (semantically) valid sentences then come out as a special degenerate case where
U is true in every model of L. The logic of a semi-interpreted language is then “essentially a syntactic description of the set of valid sentences and the semantic entailment relation in that language” (van Fraassen, 1970, 335).
The picture of a theory, then, for van Fraassen, at least in the context of physics, is that of Beth, viz., that the postulates of a theory serve to provide “the description of a state-space together with a mapping correlating the state-space with elementary statements about measurable physical magnitudes” (van Fraassen, 1970, 337). The formal description above provides a general formulation of this idea for non-relativistic physical theories. This serves to provide both a theoretical semantics, in terms of the state-space, and an empirical semantics, in terms of models construed as mappings from physical systems (or some abstract representation of them), to locations in the state-space. This faithfully represents the basic theoretical framework of a dynamical system and a specific mathematical model within it that describes the behaviour of a physical system, specified by the model ⟨loc, X⟩ of the semi-interpreted language
L = ⟨E, S, s⟩. This approach also preserves the natural description of dynamical systems in terms of state spaces and the manner in which trajectories represent the dynamical behaviour of physical systems in the world.
This provides a technical formulation underlying van Fraassen’s semantic ap- proach. In his description of his constructive empiricism,van Fraassen(1980) provides an intuitive version of this approach. This is specified as follows (for the correspond- ing version in terms of semi-interpreted languages see table 2.2). For van Fraassen, just as for Suppes, to present a theory is to specify a set ofstructures. Certain parts
scientific terminology constructive empiricism semi-interpreted languages
theory set ofstructures set of languages
L= ⟨E, S, s⟩
model structure semi-interpreted
languageL theoretical description of physical system empirical substructure of a structure trajectory or region of state-space S instance of theory-data relation isomorphism of appearance to empirical substructure of a structure model M = ⟨loc, X⟩ experiential
phenomena appearances descriptionsX
Table 2.2: Roughly equivalent interpretations of scientific concepts as construed in van Fraassen’s constructive empiricism (van Fraassen, 1980) and semi-interpreted languages (van Fraassen,1970) semantic approaches to the theory-world relation.
of these structures, called the empirical substructures, are to be specified as candi- dates for the direct representation of observable phenomena, which would include trajectories or regions of a system space. Structures that can be described in exper- imental and measurement reports he calls appearances. The appearances, then, are the observed “phenomena” that the theory is supposed to describe, explain or pre- dict. Observational adequacy (not to be confused with empirical adequacy) is then defined in terms of a theory T having a structure S ∈ T such that every member of the set At(T) of appearances, the correlate of the observation sentences Ot(T) of the received view, is embeddable via isomorphism to empirical substructures of
S. Confirmation then grows as the set of appearances At(T)grows while preserving isomorphic embeddability in some structure S, and T is falsified if At(T) comes to have a member that cannot be isomorphically embedded.9
Van Fraassen’s concept of empirical adequacy is stronger than the concept of observational adequacy. A theory T is empirically adequate if the theory has some model M ∈ T such that all appearances (within the domain of applicability of the
9This view is described by van Fraassen as a ‘picture’,viz., “something to guide the imagination
as we go along” (van Fraassen, 1980, 64), which he recognizes is a very limited description of the relationship between theory and data. The assumption, then, is that, as far as it goes, any given theory can (in principle) be adequately formalized in this way by appropriately specifying the appearances it is meant to capture and their relation to some model of the theory.
theory) are isomorphic to empirical substructures of that model. Empirical adequacy requires not only that the theory captures all appearances that have actually been observed, but also all appearances there have been and any possible appearances in the future. We could denote this set of appearances A∞(T), in which case a theory is empirically adequate provided the theory has some model such that each member of A∞(T) is isomorphic to an empirical substructure of that model.
Assessing van Fraassen’s approach as a methodology avoidance strategy helps to bring into focus some of the main issues in this chapter. Recall the adequacy con- ditions (A) and (B) on a reconstruction of a theory and its relation to the world considered in the previous section, which pertained, respectively, to preserving essen- tial content and eliminating only inessential content in an abstraction from actual science. The semi-interpreted languages approach is seen to preserve a great deal of the structure of dynamical system models, which are common in theoretical physics, including the state space, dynamics on that space and that part of the symbolic lan- guage that specifies states. There is clearly a great deal that is not represented here, however, including the vector field on the state-space, the equations that specify the dynamics, any consideration of solution of equations, and any methods and models used to construct the given model. This is certainly not epistemologically insignificant content but neither is it clearly essential for the purposes of the 1970 paper, which were,inter alia, to provide a picture of a physical theory that was in line with work in foundations of physics. The interest is thus a limited task in classical epistemology,
viz., the clarification of the foundational structure of physical theories. The structure eliminated is not clearly relevant for this task. Given this limited aim, van Fraassen’s approach meets the adequacy conditions.
Another aim of van Fraassen in the 1970 paper, however, is to capture is the relation between such a mathematical model and a physical system, which is done in terms of the “model”⟨loc, X⟩. This essentially provides an embedding of a description of the dynamics of a physical phenomenon into the mathematical framework of the language L. So, this captures in some way the relation between the measurable quantities of a physical system and the mathematical framework, but it does so without any consideration of how the description X of the system is specified. This serves the classical purposes of picking out a relation between the phenomenon and
the mathematical model that must obtain, in a manner similar to Giere’s notions of “interpretation” and “indentification”, but it obscures the detailed nature of that relation and the limitations on when such a relation actually obtains. It is here where methodology avoidance turns into epistemology avoidance, i.e., eliminating complex details about what, and how, knowledge is gained or provided by a scientific theory. Of course van Fraassen’s approach is not intended to characterize precisely the knowledge state-space models provide about physical phenomena, but it is intended to clarify the nature of the knowledge about phenomena provided by physical theories. Whether van Fraassen’s approach is adequate for this purpose, even from the point of view of a classical general epistemology of scientific theories is less clear than is its adequacy as a methodology avoidance strategy.
It is useful to note here that included in the structure eliminated in the interests of classical epistemology is structure that is essential to understanding the feasible epistemology of physical theories. This includes the relationship to measurement de- vices, any consideration of experimental questions and conditions, statistical analysis and error analysis, but also all of the mathematical methods that are essential to ac- count for theprocess of theory application. The semi-interpreted languages approach provides a snapshot of a particular mathematical model and a schematic connection to the world; as a result, it does not represent the process of constructing such a model, the relationships between the models used in the construction process, includ- ing those from other theories, or the complex relationship between these models and actual experimental situations. This serves both to highlight the manner in which the interests of classical and feasible epistemology are complementary and just how much methodologically and epistemologically significant structure classical accounts leave out.