1. Introduction

Hans Halvorson (Reference Halvorson2012, Reference Halvorson2013) takes the semantic view of theories to be that a theory is something language-independent, namely, a set of models, with models understood to be mathematical structures with no relation to language. He supports this understanding of what the semantic view is with quotable quotes that appear to say precisely that. Halvorson then provides examples to show that theories may be distinct even though the relevant sets of models are the same, and theories may be identical or relevantly equivalent although the sets of models are distinct. Therefore, that semantic view of theories, thus understood, stands refuted. This critique then opens the way to a truly interesting program for what we might think of as a neo-syntactic view of theories in the sciences.

An initial reaction to Halvorson’s critique may properly focus on what it does for the relation between the semantic view and the philosophy of the empirical sciences where it has been applied, as Halvorson generously describes at the outset. Then I will also voice a reaction to Halvorson’s and Glymour’s subsequent interaction, which illuminates the more technical logical questions that Halvorson brought to the fore, quite independently of any relevance to philosophy of the empirical sciences.

Third, I would like to comment on what Halvorson took to be the semantic view of theories. It would indeed be astonishing if anything of this sort could have been of any use whatever to philosophers of biology, psychology, or physics, or practically anyone else.

2. Initial Reaction: What Is a Scientific Model or a Scientific Theory?

An empirical scientist is concerned with a certain class of phenomena, and when she presents a theory, she provides a class of models for the representation of those phenomena. So a scientific model is a representation, and we properly speak of a model of combustion or of the San Francisco Bay in the way we speak of a painting of fire or of the Giaconda.

Is it also true that a scientific model is a mathematical structure? Certainly. A model is a mathematical structure in the same sense that the Mona Lisa is a painted piece of wood.

In a philosophical discussion of art we can distinguish between the painted piece of wood and its representational content, though of course these are both involved inseparably in what the Mona Lisa is (cf. the sustained argument in Suárez [Reference Suárez, Magnani, Nersessian and Thagard1999], with examples from Velázquez and Picasso). In the same way we can distinguish between the mathematician’s Hilbert space adorned with a special family of operators and a quantum mechanical model of the Stern-Gerlach apparatus, of a neutron interference experiment, or of a Fabry-Perot laser. That model is a Hilbert space adorned with a special family of operators, but to say no more is to ignore its representational content.

Will much light be shed on modeling if we restrict our study to model theory in logic and meta-mathematics? These disciplines are branches of mathematics, and remain in pure mathematics; in that respect they are no different from the study of Hilbert space in pure mathematics. In both disciplines the term “interpretation” plays a role. In logic or meta-mathematics a model is typically defined as a couple consisting of, first of all, a structure (typically, a set with some relations on it) and, second, a mapping of expressions of a language to elements of that structure: an interpretation (e.g., Robinson Reference Robinson1965, 9–10; Chang and Keisler Reference Chang and Keisler1973, 20). The sentences true under this interpretation form a description of that structure. It is natural thus to say that a model is a structure: an interpreted structure or a structure with an interpretation.Footnote ¹

Although models in the sciences are not exactly what are called models in meta-mathematics, similar terminology has been adopted in some places in philosophy of science. And there is similar terminology to be found in philosophy of art and literature. There are interesting connections between those different uses of the term “interpretation” worthy of exploration. But it is very doubtful that assimilating all to the use in logical model theory would prove illuminating. A rational reconstruction (were there ever Carnapian philosophers of art?) could perhaps reify the meaning of the Mona Lisa as a mapping of natural language expressions to aspects of that painted piece of wood, such that certain sentences in catalogs or art books would come out true under that interpretation. Even upon such a philistine effort it would still be natural to point to the painted piece of wood, saying “that is the Mona Lisa” and unnatural to identify the Mona Lisa as a mapping defined on natural language.

The main point, terminology aside, is that a scientific model is a representation. So even if a scientific theory is a set of scientific models, and those literally are mathematical structures, it does not follow that the identity of a theory can be defined in terms of the corresponding set of mathematical structures without reference to their representational function.

None of this seems in accord with what Halvorson takes to be the semantic view, but in a footnote he acknowledges that there is a difference between his target and the semantic view as presented elsewhere: “obviously, semanticists do not reduce theories to a mere class of models. As explicated by Giere, Suppe, and van Fraassen, a theory is a class of models plus a theoretical hypothesis. But my attack has nothing to do with this second component of the semantic view of theories. I mean only to show that the first component is a mistake, i.e. a class of models is not the correct mathematical component of a theory” (Halvorson Reference Halvorson2012, 189).

Well, for his arguments about identity, difference, and equivalence of theories he cannot have it both ways. Either

• a) theories are classes of such structures, and the identity criteria must be understood to be definable on those, or else

• b) a class of structures is only a component of a theory, and the identity criteria must take the other components into account as well.

For example, taking for granted what Halvorson appears to think in this footnote, theories would be identical only if both the theoretical definition and the theoretical hypothesis were relevantly equivalent.Footnote ² But in that case his contention, that different theories could have the same mathematical component and that essentially identical theories could have different mathematical components, cuts no ice.

There were familiar examples of that already! If the same diffusion equation is presented to describe gas diffusion and, elsewhere, temperature distribution over time, would anyone think that one and only one theory was being presented? Or if, in two axiomatizations of classical mechanics, mass is left undefined under certain conditions, in the one, and under those conditions assigned an arbitrary value, in the other, would we think that it was not classical mechanics that was being axiomatized after all? These sorts of examples had ample discussion in familiar literature; no need to see them in elegant formal abstraction as well.

But Halvorson’s text aside, there is both an instructive mistake and a valuable suggestion to be found in his ideas. To the valuable suggestion I will come back below. The instructive mistake is to take the truth that a scientific model is a mathematical structure and let it lead one up the garden path. (The Mona Lisa = a certain painted piece of wood. … But the general theory of paint on wood does not throw much light on it.) Model theory is part of meta-mathematics, and meta-mathematics is mathematics; it does not go beyond the study of mathematical structures (of a special sort). A scientific model is a mathematical structure offered as representation for certain phenomena. A representation has content. A representation of gas diffusion is not the same thing as a representation of temperature distribution, even if the math is the same.

3. Second Reaction: What Is That Claim of Language Independence?

Clark Glymour’s response to Halvorson devotes considerable space to the demonstration that when a theory is presented by displaying a family of models, those models are being described, and that description is something done in language. So there is no getting away from language, “the ‘yoke’ to language … remains” (Glymour Reference Cohen2013, 288).

Perhaps Glymour too feels sometimes the twinge of embarrassment I have on rereading my own truisms, like “the semantic view of theories makes language largely irrelevant. … Of course, to present a theory, we must present it in and by language. That is a trivial point” (van Fraassen Reference Colodny1989, 222). But there must be a serious point to Glymour’s iteration of this trivial point. Either he wants to strongly endorse Halvorson’s take on what the semantic view really is or else he spends those several impassioned paragraphs to warn Halvorson away from tilting at windmills. I think of Glymour as a person of serious intent, so the latter seems more likely.

With that out of the way, and language allowed its proper role, Glymour can make his main point. Even within the rough confines of what Halvorson addresses, theory formulations in first-order language, a good equivalence relation can be defined in standard model theory, along standard lines, begun by De Bouvere in 1965. Halvorson must have been familiar with this (even I have been familiar with it, since reading Glymour [Reference Glymour, Buck and Cohen1970], 43 years ago). Glymour’s point must be that Halvorson, by ignoring that trivial point about language, had focused on a trivially absurd version of the semantic view, which precluded any feasible equivalence relation between theories.

Halvorson’s reply is that Glymour’s version of a tenable semantic view should be called “semantic+L,” whereas he had focused on the ‘semantic−L’ view, which holds that theories consist in sets of mathematical structures with no relation to language.Footnote ³ And for reasons much more technical than those Glymour adduced, Halvorson had also established that the semantic−L view was trivially absurd. Indeed, he had.

But did Halvorson not have textual support for his take on the semantic view? Were not all of those passages in which the received (syntactic) view was disparaged precisely an official formulation of the semantic view construable as semantic−L?

I would suggest not—that context makes a difference. Thus, Halvorson quoted, “if the theory as such, is to be identified with anything at all—if theories are to be reified—then a theory should be identified with its class of models” (van Fraassen Reference Colodny1989, 222). That sentence was part of a brief exposition of Patrick Suppes’s approach in the 1950s, a precursor to the semantic view. Here “models” has the sense it has in Suppes’s presentation: “When a theory is axiomatized by defining a set-theoretical predicate, by a model for the theory we mean simply an entity which satisfies the predicate” (Suppes Reference Suppes1957, 253). No one could surely infer from this that Suppes is unaware that the set-theoretical predicate is defined in set-theory language. The sentence Halvorson then quotes, to support his contention that any relation to language altogether was to be thrown away, was a footnote to that same exposition of Suppes. It emphasizes that “models” refers there to structures, consisting of sets with relations on them, and not including interpretation functions among their elements. This was not an idiosyncratic change of meaning of “model”; it continued a use that is richly illustrated in Suppes’s writings (Reference Suppes1957, chap. 12 and elsewhere).

I must confess that the surrounding text in Laws and Symmetry does not make clear how the semantic view differs from Suppes’s, to which it was indebted. An earlier, longer exposition indicated that the aim was to find a middle way between Montague’s ‘first order language formalization’ and Suppes’s ‘no language’ approaches. This included a critique of Suppes’s approach as unable to deal with certain language-related issues (van Fraassen Reference van Fraassen and Colodny1972, 310–11), but that part was omitted in Laws and Symmetry. That is no excuse. If to construe certain passages as Halvorson does is to be misled, then it follows that they can be so construed, and the fault lies with the author (mea, mea culpa).

But what about the historical context? I submit that nothing like Halvorson’s take on the semantic view can be reconciled with the more substantive presentations that brought the semantic view onto the philosophical stage.

Tellingly, even Halvorson’s take on what the syntactic view was is not in accord with what the semantic view offered to replace. In his postscript to the 1968 Illinois conference, Stephen Toulmin summed it up properly and concisely:

This revival of Leibniz’s mathesis universalis is what Fred Suppe took on in his dissertation (Suppe Reference Suppe1967; cf. Suppe Reference Suppe1974, 221–30). It was equally what I saw as being replaceable due to the lesson that Evert Beth drew from Birkhoff and von Neumann.

The idea was not to banish language from scientific theorizing—a trivially absurd idea—but to put it in its proper place and to direct philosophers’ attention away from vocabulary and grammar to models and modeling. Thus “On the Extension of Beth’s Semantics of Physical Theories” (van Fraassen Reference van Fraassen1970) was in good part an essay on the description of scientific models in semi-interpreted languages.

Beth’s semantic analysis, my paradigm, does not resemble either the semantic−L or the semantic+L view. Following Beth, I proposed “an approach to the formal structure of theories that does not place much emphasis on syntax or proof theory, but yet makes it possible to focus on certain linguistic structures associated with given theories” (van Fraassen Reference van Fraassen and Colodny1972, 311). For each theory there is a primary language, whose statements (‘elementary statements’) attribute values to its theoretical and measurable quantities. The equations that typically present the theory establish the consistency and implication relations among those statements, for those equations are statements about how the values of different quantities are related to each other. The important structures are those defined in terms of quantities, states, and trajectories, and the elementary statements are assigned semantic content in terms of these (van Fraassen Reference van Fraassen1970; see further the elaboration in van Fraassen Reference van Fraassen and Colodny1972). The parallels to the semantic analysis of modal logic (at that time exciting news in philosophy) were clear, emphatically highlighted here, and motivating the term “semantic analysis.”

4. Ending on a Positive Note

Halvorson refers to various studies of interpretability relations among formalized theories. As I read his definitions and suggestions for further work, I discern the fine hand of Albert Visser of the University of Utrecht, where Halvorson recently spent a year as visiting researcher. Visser has developed a grand scheme that, in moments of true exhilaration, I could very well see as the replacement for the poverty-stricken syntactic approach of Carnap and his cohort.Footnote ⁴ Could this grand scheme be developed as a new formal approach in philosophy of science? What if it were to be developed while attending to the recent literature on models as mediating between phenomena and theory in scientific practice? What if the problem of defining empirical adequacy were newly broached in this new framework? If it were applied constructively to philosophical issues, the impact could be as great as Carnap’s conveyance of Russell’s logic to philosophy! Will that development perhaps happen at Halvorson’s hands? Let a hundred flowers blossom!