Here are our conclusions so far: what the semantic conception of theories claims is that a theory provides us with a certain family of representational vehicles that are non-linguistic in nature, its models; however, the idea that representation in science is non-linguistic is unwarranted: for sure, the informal features of experimentation and the contextual nature of idealisations are hard to address using the correspondence rules put forth by logical empiricists, but this seems to point to a more pragmatic conception of linguistic interpretation, not to the idea that representation would be structural instead of linguistic.
Having said that, I think that there is still a grain of truth in the semantic view, which is the following:
Theories do not represent or describe nature directly as a whole. They are used to build models that represent it piecemeal.
This, I take it, is an assumption that is shared by semantic and pragmatist conceptions of theories, and that is rather at odds with syntactic conceptions. However, I think that the semantic view gets things wrong because beyond this opportune emphasis on models, which, dare I say, is responsible for all the fruitfulness of the view, it takes the wrong focus by contrasting languages and structures. What the emphasis on model is useful for is rather that it allows to bring the pragmatic aspects of representation into our picture of science. I mean pragmatic, as in philosophy of language, in reference to the users of theories and contexts of use, and I take this insight to be the main defining characteristic of a pragmatic conception of theories, and also its main difference with the semantic view:
Epistemic agents and communities and associated contexts must be taken into account to understand how scientific models represent specific parts of nature, and these aspects are not encoded at the level of the abstract theories that are used when building these models.
According to pragmatists, what makes of models an unavoidable middle-person between theories and phenomena is not their structural, as opposed to linguistic nature. What makes them an unavoidable middle-person is the way they incorporate features that are not present at the theory level. This is the main theme developed by pragmatically oriented philosophers of science, which is summarised by Cartwright’s contention that a theory is not a vending machine, or by the slogan that models are mediators between theory and experience (see my (non-)Review of Models as Mediators).
In what follows, I will sketch the main elements of a pragmatist conception as I conceive it, and how it differs from a semantic or syntactic view.
What is a model?
I take a theoretical model (not any model, but the kind that can be used to interpret a theory) to be a description schema couched in the vocabulary of the theory. This means a set of statements or propositions that describe a situation. However, these statements contain free variables and parameters. This include, for example, a system S, its mass M, the centre of coordinate O, the position of the system x along an axis, and so on. The fact that these variables are free means that they do not refer, not outside of a context of application at least. The same model can be used to represent many situations of a given kind in many different contexts. However, the model also comes with implicit rules of interpretation telling us how to interpret the free variables in particular contexts: M is the mass, O the centre of coordinate, and so on. In this respect, models are indexical: they are functions from context to content. These rules of interpretation are associated with norms of use at play in the epistemic community where the model is deployed. They sometimes directly refer to experimental techniques available to this community.
This is the kind of view that I have developed in my past research (initially in (Ruyant 2021); I applied it to interpret theoretical symmetries (2021), quantum mechanics (2023) and cosmological arguments (2024) in various papers). I explained as well in (Ruyant 2025) how it can account for some of the motivations behind fictionalist accounts of models: outside of a concrete application context, free variables cannot take value, so we need a fictional target to present the model and reason about its content. This is why model presentation in science often looks very much like fictional story telling. However, it would be a mistake to think that this fictional target plays a mediation role between model descriptions and concrete target systems: it only plays a role outside of concrete applications for presenting the model. This is a mistake to think that models are fictions in general.
I think that the semantic view roughly makes the same kind of mistake when it views models qua abstract structures as an intermediary level between statements and concrete applications. What is really going on is that a fictional target, which looks very much like an abstract structure (because it isn’t concretely instantiated and need not have perfectly determined properties), is used outside of concrete applications to present the model. But in concrete applications, free variables acquire a reference or a value and the model can be used to directly describe its target by means of mathematical statements and make inferences about it. The only model of these statements in the sense of truth-maker (satisfying structure) of model theory is the target system itself. All this becomes clear once we realise that a model, abstractly conceived, is schematic or indexical rather than directly descriptive.
Having said all this, I'm not really against van Fraassen's conception of models in terms of state space: I think this is just a canonical way of analysing the content of a set of descriptions (like a possible world semantics, but limited to some aspects). The state space will typically be defined in terms of the free dynamical variables of the model.
What is a theory?
Now that we're clear on what models are, let's turn to theories, which, remember, are only interpreted through model building. In what sense are theories used to build models?
If both are linguistic entities, I would say that the way they are used is simply that theoretical laws are used to make deductions from model descriptions. So, the model can be seen as a deductively closed set of indexical statements, all the statements that can be deduced from a an initial schematic description of a situation, where the deduction rules are provided by the laws of the theory (or alternatively, theoretical laws are just part of the initial model descriptions). This is the sense in which models “make true” the theory: they are not represented by the theory, but compatible with it. And contrarily to the theory, which only provides general statements, they provide existential statements describing a particular object or configuration as well, which are ultimately associated with its free variables.
The model uses the vocabulary of the theory, which means that it can make full use of its mathematical formalism (defining a particular force function or Hamiltonian operator for the system for instance). However, it also typically uses an external vocabulary, and it is this external vocabulary that is typically involved in the interpretation of the model (where free variables occur, that is).
For example, a model could use Newtonian mechanics to describe particles in interaction, with the additional claim that the average kinetic energy in a region is the temperature of the target system in this region. Here, "temperature" isn't part of the vocabulary of the theory that is used to build the model. Furthermore, the microscopic description in terms of kinetic energy won't be directly interpreted in terms of the target situation, but temperature will. It is associated with a free variable T that is taken to denote temperature in any context of application.
To take another example notably developed by Suárez and Cartwright (see review of models as mediators linked above), a model can claim that a system associated with a given configuration and dynamics is a superconductor. The term "superconductor" isn't part of the vocabulary of the fundamental theory, but it will be interpreted in application contexts in terms of specific experimental configurations, whereas many aspects of the microscopic configuration associated with it won't be directly interpreted (they aren't associated with free variables).
As Cartwright explains, the way a model of superconduction is built from quantum mechanics is only loosely constrained by the theory. The theoretical framework only provides stock models with standard Hamiltonians, but they need to be adapted to the case using various assumptions (that ions shield the electromagnetic field, etc.: see Cartwright’s The Dappled World ch. 8 for a detailed analysis of this case). These are aspects that the model adds on top of the theory. And this is how a model works as a bridge between the theory and some phenomena.
Free variables and their values play a very special role here, since they are more directly associated with elements or aspects of target systems, whereas other descriptions in the model might be interpreted less literally, in terms of how they contribute to constrain the free variables. We can also assume that there are several layers of models involved in some applications (abstract models being used in the same way theories are, and combined together) until something directly associated with observations and manipulation appears.
If this is the right picture, then the idea that a theory is a family (or class or kind) of models that the semantic view puts forth is wrong, or only true in a very loose sense. This is because the models of the theory, in the sense of the models that can be built using it, do not constitute a well-defined set. Indeed, even assuming that all models of the theory are strictly speaking compatible with its laws, the collection of models that are compatible with theoretical laws is too big and too open ended to be of scientific relevance, so claiming that a theory is a collection of models is completely uninformative. This collection actually becomes larger for every new application of the theory, when some new vocabulary, associated for example with new experimental techniques, is introduced: if the theory were characterised by its models, it would indeed change every time new techniques are developed, which, remember, was precisely the ground for rejecting the correspondence rules of the syntactic view (see my review of Suppe (1989)). It makes more sense to claim that a theory is just a set of laws, or general statements.
Here the semanticist could respond by restricting the models of the theory to those structures that only use the vocabulary of the theory, so as to limit this set to something well-defined and fixed rather than open ended and evolving (but still very big if, say, any kind of force function is acceptable). However, even the models of this set become less relevant to account for actual modelling practice in science, where models must be enriched to get empirical significance in most applications.
Another question is whether the fact that some idealised theoretical models do not strictly satisfy theoretical laws is essential to their function, or only a practical matter (simplifying calculus etc). As discussed in the previous article, the semanticist needs to assume the latter, that in the end, all models respect theoretical laws, and van Fraassen does indeed assume something along these lines (see my review of van Fraassen’s semantic conception of theories). A pragmatist can be more open here, since the link between theories and models is looser than in the semantic view. However, assuming that in principle, the laws of the theory could be respected everywhere could be an advantage if one wants to maintain that theories are capable of being true or false, and generally aim at being true. I think the two options remain on the table under a pragmatic view.
This doesn't prevent the pragmatic view from providing an original account of the idealisations that caricature target systems without violating theoretical laws. For example, some authors have argued that idealisations such as infinite gases are indispensable to represent some phenomena such as phase transitions. Idealised parts of models, in so far as they are not directly associated with free variables that refer in context, could receive a non-literal interpretation in a pragmatic view.
How is a pragmatic conception different?
You could think that this conception of models and theories is compatible with a syntactic view. Theories are general statements couched in a theoretical vocabulary. Then models, if they are functions from context to content, are a bit like correspondence rules associating any context constituted of observation reports, with a richer vocabulary than the theoretical one, to some theoretical statements. This can include notably models of experiments and models of instruments, which are indispensable for interpreting the theoretical vocabulary. Then we are back to the problems of the syntactic view: if these correspondence postulates aren’t part of the theory, then it is a pure calculus without any empirical interpretation, and if they are, then the theory seems to change every time a new technology is developed.
I cannot present here a fully fleshed out pragmatic conception of theories that would respond satisfactorily to these worries, but we can presume that the main difference between a syntactic view and a pragmatic view will lie in a reference to model users and epistemic communities, their perspectives, aims and focuses, their abilities and standards, in particular when it comes to understanding what concrete models add to more abstract models and to theories. It must lie in the assumption that a theory isn’t pure calculus, that the empirical interpretation of its vocabulary is sufficiently constrained, but at the same time, sufficiently unspecific to leave room for contextual applications.
If this is the right approach, then again, viewing theories as well-defined families of models is wrong, because only specific models that acquire meaning in specific contexts are scientifically relevant, and the theory itself doesn’t tell us which: we need to refer to epistemic agents and communities, and associated experimental abilities, to account for this.
Now semanticists such as Giere and van Fraassen also accept that users are involved in representation. Doesn’t it mean that there version of the semantic view is subtle enough to accommodate these aspects?
I think not. It is perfectly possible to involve users in scientific representation while also accepting that theories are very well-defined families of model with a systematic interpretation. This can be the idea that theories provide strict indexical rules of interpretation for their models. And I think that this is more or less what van Fraassen (2008) is after, with his emphasis on indexicality in his book about representation. This, indeed, is quite compatible with a semantic view of theories, while incorporating part of what I’ve been advocating for here: the idea of piecemeal, indexical representation. But only part of it, since models never really add anything on top of theories in this view: building a model for a new application is just figuring out which of the models already available actually fits the situation, or locating this situation in the modal structure of the theory. And I think that this is idealistic and not entirely faithful to scientific practice.
As for Giere, he was much closer to a pragmatist conception of theories: he thought of models as linguistic entities, he thought that model theory is quite irrelevant for the philosophy of science, he assumed a hierarchy of models that is open ended, he thought that models, theories and instruments are perspectival, he conceived of representation as involving not only agents but also aims and the like… This, to me, is a pragmatic conception of theories. Part of my contention is terminological: we have to put the divide somewhere, but I don’t see much in common between Giere’s view and other semanticists.
What about theoretical truth?
One could think that an advantage of van Fraassen’s view is that it allows to have a well-defined notion of theoretical truth. It’s quite common to say that a theory is true or false. If we want to do justice to this way of speaking, we need theories to make substantial claims.
Presumably, under a semantic view, the claim is that the models of the theory accurately represent relevant real systems in relevant respects (semanticists generally talk about hypotheses linking models and real systems), and all this can take the form of statements specifying, at the theory level, what kind of model represents what kind of situation in what respect, and what it means for them to be accurate. Roughly speaking, we can take theories to be very complex functions from context to truth-conditions, and then claim that the theory is true if these truth-conditions are satisfied whatever the context (we can quantify as we want on contexts to get different notions of truth). This conception of theoretical truth is available in a semantic view, but not in a pragmatic view, precisely because models add something to theories.
Right… But the other side of the coin is that if really theories have such systematic interpretations in the semantic view, we are, again, defending something that is very much like a syntactic view: these functions from context to content that tell us, in any given context, what model applies, and what are its truth or accuracy conditions, are pretty much like correspondence rules. In particular, they share with them the problem already discussed: that they must be part of the theory if we want the theory to have empirical content, but at the same time, that they capture aspects, such as experimental techniques or auxiliary hypotheses, that are better thought of as external to the theory, since we don’t want to say that the theory has changed every time a new experimental technique is used.
In this respect, a pragmatic conception might be better apt to take its distance from logical empiricism and the syntactic view, by viewing things in a more modular way. However, this opens a challenge: how exactly shall we account for theoretical truth or acceptability in this context?
Good question. Thanks for asking. Answering this kind of question is more or less what fleshing out a pragmatic conception amounts to I think (instead of just alluding to such a conception). I’m not aware of anyone having done so. I personally have some ideas, but they are very much work in progress. Maybe next year I will publish them here.
Conclusion
In this article, I have examined what I take to be the main gist of the semantic conception: this idea that theories are interpreted piecemeal through model building rather than globally. What I take to be the main difference between a semantic view and a pragmatic view is that the former assumes systematic interpretations for models that are fixed at the theory level, even if agents are ultimately involved, whereas the latter assumes that models add something contextual that is not specified by the theory. The former makes it impossible, or at least mostly irrelevant, to think of theories as families of models, since the collection of models compatible with a theory is open ended. At the same time, it purports to be more faithful to scientific practice. The latter ultimately threatens to collapse to something equivalent to a syntactic view sharing the same problems. Having said that, a full fledged pragmatic conception of theories that would uphold all these challenges largely remains to be developed.
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