Review of Van Fraassen's Semantic Conception of theories (mainly Laws and Symmetry, 1989)

The next stage of our journey is van Fraassen’s presentations of the semantic conception of theories. I will focus here mainly on parts of chapter 8 and chapter 9 of his “Laws and Symmetry” (1989), as well as the appendix of chapter 1 in his “Scientific Representation” (2008), and to a less extent his 1970 article “On the Extension of Beth’s Semantics of Physical Theories”. I will also mention in passing a presentation by Thompson “Formalisations of Evolutionary Biology” that is cited favourably in Scientific Representation.

Summary of the material

Laws and Symmetries is largely concerned with giving an account of theoretical laws that does not take them to represent real aspects of nature (actual laws), but to be mere structural features of our representation of nature, characterised in particular by symmetries. Laws are laws of models only, and it is ultimately the content of these models, not their overarching structure, that is supposed to be adequate. As we can see, the whole project is tangential to our concerns, but it still rests heavily on a semantic conception of theories.

This conception, already developed in other work, is reintroduced in chapter 8 mainly on a polemic mode, claiming that the received view that theories are axiomatic systems of first-order logic is responsible for a “tragedy” that badly affects the philosophy of science. I identified two main lines of criticism.

The first criticism is the following: first order logic is too poor to capture meaningful mathematical distinctions, and so, too poor a language to really identify the structures that are intended by theoreticians. Of course these structures are described using a language, but this is the language of mathematics, not first-order logic, and philosophers should do the same as scientists: use mathematics directly to present theories. So theories are perhaps axiomatised in a weak sense of being expressed by statements, but not necessarily in the stronger sense of being an “elementary class” of a system of axioms of first-order logic (I will explain this technical point in my comments). Typically, even in mathematics, we would describe kinds of structures (such as Euclidean spaces) instead of giving a set of axioms. And in physics, the laws found in textbooks cannot really be taken to correspond to axioms anyway, because they do not have full generality: Schrödinger’s equation, for example, only applies to conservative systems, not to dissipative ones. So, this is the first gist: scientific theories describe structures using math, they are not first-order logic axiomatic systems.

The second criticism concerns language. Interestingly, the move away from the syntactic view is motivated by a move away from ideal language semantics (p. 212), which is the idea that we could characterise language-world relations in a very generic way, but not to deny any role for language. Van Fraassen thinks that we should be content with capturing fragments of scientific language, in particular the attribution of values to physical magnitudes that is involved when interpreting models. This corresponds to what he calls a partial interpretation of models in his 1970 article “On the Extension of Beth’s Semantics of Physical Theories”. We could extend this a bit (attribution of states, of causal relations, modalities), but we should be careful: maybe these are descriptions of non-representational parts of the models, just like idealisations are not supposed to represent actual aspects. This is the case in particular, according to van Fraassen, of everything that has to do with modalities (and of laws and symmetries): they are not really representational. So, we should be careful not to attempt to fit everything in the mould of a linguistic construal that would interpret everything in terms of worldly aspects, and this is why models and structures must play the central role in our account of science instead of linguistic formulations. Models are structural entities that mediate between the linguistic formulation of the theory and the world and, from what I understand, allow for a more fine-grained interpretation of theories.

All this excessive focus on linguistic formulations led, according to van Fraassen, to a disconnection between philosophy and actual science: this is the “tragedy”. Van Fraassen cites, as examples of disconnected notions, Ramsey sentences, Craig reduction, but also projectible predicates (!) reduction sentences, disposition terms (!!)…

Chapter 9 is dedicated to a more positive presentation of the semantic conception, illustrating it with toy theories and models (a figure satisfying some axioms). The general picture that van Fraassen promotes is that models are defined on state spaces, associating a trajectory in state space and abstract objects in a domain, and theories are defined by various related clusters of models, putting constraints on them (laws of succession etc,), plus a claim that real systems stand in some relationship with models. This is very similar to Suppe's account detailed in the previous post.

The claim that is part of the definition of a theory is important: it is what makes the theory either true or false (because a theory is indeed something that can be believed, doubted, etc.). The claim is that real systems belong to the same class of structure as models, or, equivalently, that one of the possible worlds assumed by the theory is the actual world. Now of course van Fraassen is an empiricist, so he is not claiming that theories are true here, only that this is what theoretical truth amounts to. He also notes that there is leeway in metaphysical interpretation, but that empirical content at least must stay the same if we’re talking about the same theory.

I was interested in knowing whether van Fraassen’s semantic conception had evolved in time, so I looked up his more recent book Scientific Representation. There is a full appendix dedicated to the topic, responding to more recent criticism that came from the pragmatic side, in particular, from Cartwright, Suárez, Frigg and Morrison. These authors basically think that models are more autonomous from theories than is usually acknowledged by the semantic view, and that it makes sense to identify theories with sets of laws or statements, and models as something connected but distinct that is used to represent real entities. For example, one can take theories to be tools for model construction, as Cartwright and Suárez propose (I will review this more recent literature on this blog soon).

Van Fraassen’s defense of the semantic conception is roughly that this is mainly a difference in focus. Models must still respect the laws of the theory (at least in approximation I guess), otherwise they would be counter-instances refuting it. So much is accepted by Cartwright and Suárez. He argues that when given laws, even taken as tools, potential constructible models are somehow “already there” in the sense that there is a set of structures satisfying these laws that could be constructed, just like the solutions of an equation are somehow “already there” even if we haven’t found them. So it still makes sense to understand the theory as a family of models. But of course, we can also focus on the process of model construction and emphasise other aspects, including aspects external to the theory. In sum, van Fraassen wants the semantic conception to be compatible with these recent pragmatic developments.

This appendix mentions favourably Thompson as a presentation of the semantic conception applied to biology. I mention it here because it is indeed a good resource. It explains clearly what is at stake in the contrast between the semantic and syntactic views: according to the semantic view, there are two stages in representation (1) the theory presentation describes (potential) models, as we could describe pictures, and (2) the models have some relation to real systems, as pictures can be taken to represent real objects. The first stage is usually formalised using Tarski's model theory and the second in terms of morphisms between structures. This is the main thesis that is common to all defenders of the semantic conception: that there are two stages involved, and as already mentioned in my previous post, this is very similar in spirit to what has been called indirect fictionalism about models (notably Frigg’s version).

Comments on first-order logic versus mathematical language

Some years ago I adhered more or less to the semantic conception, mostly because it was in the air. But the more I read about it, the more silly it sounds to me. All authors insist that a theory can be true or false, believed or doubted, etc. So of course a theory is a linguistic entity! They also all claim that the formulation in a particular language is not what matters. But as I said in my previous reviews, of course, just as the same thought can be expressed in English or Spanish, the particular language does not really matter, and logical empiricists never pretended otherwise! And there’s not much difference between presenting axioms and presenting the set of structures satisfying them. The structure must be characterised after all, in a language, so in both cases, what is given to us is exactly the same, a set of statements, and what is described by these statements is the same as well. The semantic conception is weird, making distinctions that don't make a difference as if it changed everything. But I must grant that at least in the case of van Fraassen there are real, explicit attempts to address these worries of mine (shared by others in a more recent literature), in particular with the remarks on first order logic.

Having said that, I wasn’t convinced in the end. First a point of logic. Van Fraassen rightly says that to be axiomatisable is not so trivial, and that not all mathematically definable sets of structures are elementary classes of a first-order axiomatic system. But this is because the ones that are not, in so far as they can still be defined, are actually restrictions of an elementary class, suppressing part of its vocabulary. These are called pseudo-elementary. In other words, the way of defining them is by using a richer language, and then “hiding” the parts that make use of this richer language: what results is not strictly axiomatisable without the rich language. If this were the case of the structures of a scientific theory, this would mean that theories are intentionally restricted to substructures of a richer axiomatisable theory, but I doubt that this is the case. Ironically, the point of rejecting instrumentalism (which van Fraassen does) is precisely to incorporate all the theoretical vocabulary that is essential in identifying the right class of models, all the "purely theoretical" vocabulary that serves axiomatisation/unification purposes rather than direct empirical purposes. So, I would dare say that rejecting instrumentalism implies looking for fully axiomatisable theories instead of their non-axiomatisable restrictions to a sub-language.

Perhaps I’m being unfair here, because I’m still considering first-order logic, whereas van Fraassen’s main point is that mathematical language is more expressive. One can show, using higher-order logics, that there are structures that are neither elementary nor pseudo-elementary classes of first-order axiomatic systems (although I don't know if they can be constructed or presented as a theory can!) But as other critics have noted (I’ll find the reference for a future post), the logical empiricists didn’t see any problem in expressing theories using Ramsey sentences, which are sentences in second-order logic, and second-order logic is nearly as expressive as set theory, and quite natural as a language for mathematics with that, except for applications that I doubt are really relevant to empirical sciences. So there’s no real obstacle to the idea that theories are axiomatic systems of second-order logic.

In the end, perhaps it's fair to note that first-order logic is too limited a tool for philosophers of science, but what language to use to characterise theories and whether we should identify them with statements or models are very distinct issues. It's a mistake to conflate the two. Even if the language of mathematics is more appropriate than first-order logic for philosophers of science, this does nothing to show that theories are models instead of statements, because they could be mathematical statements (or models of first-order logic for that matter). The limitations of first-order logic is a red herring.

The most serious objection against the idea that theories are axiomatic systems might be found in van Fraassen’s claim that Schrödinger’s equation, for instance, is only valid for a class of system, and so, doesn't count as an axiom of quantum theory. But this move does not play as well as one could hope for the semantic conception. If this is really the case, and if no more general statements can count as sufficient axioms (a definition of Hilbert space say), then the set of models of the theory is not well-defined in textbooks, so there's a problem with identifying theories and models. And van Fraassen himself insisted, when responding to pragmatists, that models must not constitute counter-instances to the theoretical laws! On the other hand, if the models of the theory are well-defined by the Schrödinger equation or by a more general statement, then the theory is axiomatisable after all. If it's Schrödinger's equation, then the models constructed by scientists have some autonomy with regards to theories, they can slightly depart from it in some occasions if the context justifies it (when the system is dissipative), and we cannot strictly identify the theory and the models. In this case, we reach a conception that is more pragmatic than semantic. But in any case there's no reason so far to resist the natural idea that theories are charactersiable as sets of axioms. And I think it has proved fruitful to think of them in this way (think of Einstein developing relativity from first principles, of Newton's formulation of mechanics in terms of laws, of the various theorems that have been proved in foundations of quantum mechanics...).

Comments about language

In the end, the idea that theories are structures rather than statements has more to do with the idea that the linguistic analyses proposed by logical empiricists were irrelevant to understand science (but note, it's only correspondence rules that are ever attacked!). In this respect, I was happy to read from van Fraassen that the problem with the syntactic view is not so much linguistic analysis per se, but its reliance on ideal language semantics. I am completely on board with this, and I think we could use many tools from pragmatics (a departure from ideal language semantics that addresses its limitations by taking into account contexts and locutors, not only word-world relations) to shed light on science. Unfortunately, this is not what is proposed. What is proposed is to focus on structures instead of language: this is still something a-contextual, how is it going to help? The more I think about it, the less I understand this move, and this might be a bit redundant with comments from the previous posts, but let us explain why again.

To be sure, van Fraassen accepts that "fragments of language" are relevant in science, in particular partial interpretations of models in terms of attribution of properties to systems. This notion of partial interpretation is something actually very similar to the logical empiricists’ partial interpretation of theories in terms of observations, to the point that I'm not sure what the difference is.

Side note on more general interpretative issues... It's quite weird, in this respect, that what van Fraassen wants to retain from the logical empiricists, the ideas that necessity is analytic and corresponds to nothing in the world (in particular, rejecting the notion of disposition) and that a distinction between what is observable or not would have an epistemological role to play in science, these aspects are responsible, for most authors, of the failure of the doctrine. The rejection of objective modalities because they are essential to account for scientific practice and discourse, and the notion of observable because on the contrary it fails to match with anything relevant to science. But what he and defenders of the semantic conception want to amend, this idea that theories are statements about the world expressed with theoretical terms that are presumed to be projectible, and must have some general linguistic interpretation that connects with experimental practice, all this is not particularly problematic after all and much in line with the practice and discourses of science (see Chang for a defence of operational definitions of theoretical terms). What he does is very much like throwing the baby and keeping the bath water to me! We should do the exact contrary: retain this idea that theories are statements to be interpreted and connected to experience using linguistic tools, but incorporate modalities in our interpretive toolbox and forget about this observable/unobservable distinction in favour of something more pragmatic.

Back to the main thread. So van Fraassen has this notion of partial interpretation that gives a small role to language for interpreting models, which makes sense. However, if the state-space of a model can be so interpreted, this is only because it is not a pure structure.

Most models in science are linguistic entities, dare I say. Theoretical models are: they are expressed in a theoretical language. This is why they are intepretable: their state-space is just a possibility space linked to theoretical terms, from which property attribution follows directly. Phenomenological models, data models and models of the experiment are also linguistic entities attributing properties to systems expressed in a theoretical vocabulary. The fact that a dimension in a model is time, and not space, matters: this is a matter of linguistic interpretation of the model. Furthermore, scientists do not describe a model with statements, as claimed by semanticists. They express the model with statements, as we could express an idea or a belief. Models are not systems or structures, either abstract, concrete or fictional, as they claim. Rather they are used to describe or characterise actual systems and their structures, treating them as such and such, representing them as such and such, ascribing properties to the system they represent. They make (or incorporate) assumptions about these targets. They are not compared, in terms of similarity or morphism, to the systems they represent. At best, they are compared to other representations of the same systems in order to assess their accuracy. This is how to properly talk about modelling activities in science. Take any scientific article, and this is what you will read.

The confusion of taking models to be abstract or fictional objects described by theoretical statements is a confusion between the source and target of representation that likely stems from an illegitimate application of model theory in logic, where models are not representations of anything, but rather a part of the world to be represented by propositions. Thinking that this has anything to do with models in science was a mistake. In effect, in model theory, we describe logical models (parts of the world) using set theory, and we also describe a language, and we characterise the relations between the two, still using set theory. This could possibly be used to understand world–theory relations. But this is not what scientists do when they model a system: they represent the world itself, not any relation between language and the world. So it does not make any sense to identify logical models and scientific models.

But, one will object, the main point is to make a helpful distinction between two stages in representation: the construction of a model first, using the theory, and then its comparison with an actual phenomena. This is how we account for idealisations, etc. Right. Like the two stages involved in logical empiricism: the theoretical statements on the one hand and the correspondence rules on the other. Oh no, not these two stages? What then? Of course the logical empiricists were wrong to assume that applying a theory was a matter of applying semantic rule, or maybe more accurately, they were overly optimistic about the idea that this could be reconstructed as such. But this focus on “structures instead of statements” does nothing to address the issue. Models play a mediation role: I'm fully on board with this, and I accept that the focus on models was an important move in the end, but models are linguistic entities expressed using mathematical statements and a theoretical vocabulary (equations and the like), they are not objects described by these statements, for these statements, and so, the model that is expressed by them, are about worldly entities or kinds. Models are complex, highly structured, ok, but the point is still to describe the structure of real entities such as atoms or molecules, not a platonic realm, and this is done with sophisticated mathematical statements and a scientific vocabulary. If sometimes a fictional entity is invoked in classroom instead of an actual atom, well, this is because we don't have a full laboratory at disposition and still want to present the model, and even if the assumptions made by the model are known to be false of the actual entities it represent, the model is still about the world.

In sum, my view is that models are just applications of theories to particular cases, and that both are linguistic. What really does all the mediation job in modelling activities is taking into account some contextual aspects that are not expressed by the general theory. It’s not focusing on “structure instead of statements”, on something extra-linguistic, or whatever. So, really, we should put these ideas at rest, and assume, as we should always have, that scientific theories and theoretical models are linguistic entities.

And this the end of my rant.