Tuesday, 19 July 2011

On the Model-Theoretic Conception of Scientific Theories

Ordinarily, in mathematical and scientific practice, the notion of a “theory” is understood as follows:
(SCT) Standard Conception of Theories:
A theory $T$ is a collection of statements, propositions, conjectures, etc.
A theory claims that things are thus and so. The theory may be true, and may be false. A theory $T$ is true if things are as $T$ says they are, and $T$ is false if things are not as T says they are.

One can make this Aristotelian explanation more precise, as Tarski showed, in the cases where we understand how to give precise logical analyses of theories, by identifying an interpreted language $(\mathcal{L}, \mathcal{I})$ in which $T$ may be formulated. Here $\mathcal{L}$ is some formalized language and $\mathcal{I}$ is an $\mathcal{L}$-interpretation. (The language can be higher-order, infinitary, contain infinitely many predicates, or uncountably many constants, etc.) One can define the satisfaction relation $\models$, holding between $\mathcal{L}$-sentences $\phi$ and $\mathcal{L}$-interpretations, and then define the notion "$\mathcal{L}$-sentence $\phi$ is true in $(\mathcal{L}, \mathcal{I})$" as "$\mathcal{I} \models \phi$". What is essential about this is that theories are $\textit{truth bearers}$. They are bearers of semantic properties.

The standard conception of theories thus takes theories to be truth bearers. In particular, given the history of science, we need to be able to make sense of saying of a theory $T$ that it is $\textit{false}$. Furthermore, it seems clear that any account of theories according to which a theory is not a truth bearer – i.e., an account which rejects semantics – is surely not acceptable. We are obliged by the facts concerning scientific practice to provide an analysis of the notion of a theory in such a way that electromagnetic theory, special relativity, evolutionary theory, and all the various other theories that may interest us, are capable of being either true or false.

Over the last forty years, a contrasting view has appeared:
(MCT) The Model-Theoretic Conception of Theories:
A theory $T$ is a collection $\Sigma$ of structures.
This view has been advocated by Suppe, van Fraassen, French, Ladyman and others. E.g.,
Van Fraassen elaborated and generalized Beth's approach, arguing that theories and models are essentially mathematical structures ... (Ladyman & Ross 2007, p. 116).
The semantic view encourages us to think about the relation between theories and the world in terms of mathematical and formal structures. (Ladyman & Ross 2007, p. 118.)
In the simplest case, a structure $\mathcal{M}$ is a package of the form $(D, \{R_i\}_{i \in I})$, where $D$ is some non-empty set, and the $R_i$ are relations on $D$. (This can be generalized in various ways.) The Model-Theoretic conception thus rejects the standard conception of theories described above. For it is meaningless to say of a structure $\mathcal{M}$ that it is true. Therefore,
According to the Model-Theoretic View, theories are not truth-bearers.
This consequence of MCT is a refutation of it. It is a minimal constraint on what a theory is that it be a truth bearer. If something isn’t a truth bearer, then it isn’t a theory.

Having mentioned this objection on various occasions, the reply one hears is that:
(R) A structure $\mathcal{M}$ is true iff $\mathcal{M}$ “represents the world”.
However, there is no such notion as that of a structure $\mathcal{M}$ “representing the world”! So, one is led to the question:
(Q) What does it mean to say of a structure $\mathcal{M}$ that it represents the world?
Advocates of MCT sometimes say that a structure $\mathcal{M}$ represents the world by "being isomorphic to it". However, prima facie, it doesn’t make any sense whatsoever to say of a structure $\mathcal{M}$ that it is "isomorphic to the world", because isomorphism is a relation that holds $\textit{between structures}$. Is the world a structure? (We return to this in a moment.)

The only answer I can think of, at least consistent with the intentions of advocates of this view, is the following:
(D) A structure $\mathcal{M} = (D, R_1, \dots, R_n)$ $\textit{represents the world}$ iff there is a subset $W$ of things in the world, and there are relations $S_1, \dots, S_n$ on $W$ such that $(D, R_1, \dots, R_n) \cong (W, S_1, \dots, S_n)$.
(Where "$\cong$" stands for “is isomorphic to”.)

However, according to definition (D), a claim of the form "$\mathcal{M}$ represents the world" is a Ramsey sentence. And then it is not difficult to prove the following:
(N) $(D, R_1, \dots, R_n)$ represents the world iff, for some subset $W$ of things in the world, $|D| = |W|$.
(This is a version of Newman’s Objection to structuralism. The left-to-right direction is trivial. The right-to-left direction is proved by assuming that $|D| = |W|$, and considering an bijection $f : D \rightarrow W$. Take the images $f(R_i)$ under $f$ of the relations $R_1, \dots, R_n$. The result is the structure $(W, f(R_1), \dots, f(R_n))$ isomorphic to $(D, R_1, \dots, R_n)$ by construction.)

This tells us that $\textit{any}$ structure $\mathcal{M}$ "represents the world" so long as the world has enough things in it.

This is surely unacceptable. The only way around this problem is to say something like the following:
(S) The world $\mathbf{is}$ a structure $\mathbb{W} = (W, S_1, \dots, S_n)$.
(One cannot replace "is" by "can be represented by". Go back and re-read the definition (D) again. For unless one accepts (S), then, as (N) tells us, the world "can be represented" by any structure, cardinality permitting.)

Then, if one accepts (S), one can say that a structure $\mathcal{M}$ "represents the world" iff it is isomorphic to $\mathbb{W}$.

However, if we say this, then we have returned to a (rather strong) formulation of the original, standard, conception of theories. For $\mathbb{W}$ is simply the intended interpretation of some language $\mathcal{L}$ with which we may formulate the original theory $T$ of which $\mathcal{M}$ may be a model (in the usual sense). Indeed, the correct theory $Th(\mathbb{W})$ is simply the set of truths in the structure $\mathbb{W}$, and a theory $T$ in this interpreted language is then true when it is subset of $Th(\mathbb{W})$.

In summary, this post gives two major objections to the Model-Theoretic View. The first is the Truth-Bearer Objection. The second is the Newman Objection. The standard conception of theories takes theories to be truth bearers. These may be taken to be propositional or may be take to be statements in an interpreted language. In contrast, the model-theoretic conception of theories, associated with van Fraassen, French, Ladyman and others, identifies theories with collections of structures. Hence, the model-theoretic conception denies that theories are truth bearers. Unless we are prepared to accept the idea that theories are uninterpreted calculi for making predictions (i.e., radical instrumentalism about the semantics of theories), then this is unacceptable.

If one tries to remedy this problem, by defining "$\mathcal{M}$ represents the world", the definition yields a Ramsey sentence, and then one is faced with a version of the Newman objection. The only way around this problem is to $\textit{identify}$ the world itself with a structure $\mathbb{W}$. However, this merely plays the role of the intended interpretation of a language $\mathcal{L}$ which might be used to formulate the theory $T$ in question. One is then back to the standard conception, along with the the very strong metaphysical assumptions that the world is a structure $\mathbb{W}$.

Update:
I should include at least some background references for anyone curious about the literature.

A defence of the model-theoretic conception can be found in James Ladyman & Don Ross, 2007, Every Thing Must Go, Chapter 2, pp. 111-118. Here, Ladyman & Ross implicitly reject the notion that theories have semantic properties (e.g., reference, truth):
In the context of the syntactic approach, within which a theory is taken to be a set of sentences, realism amounts to the commitment to standard (correspondence) referential semantics, and to truth, for the whole theory. (Ladyman & Ross 2007, p. 117.)
Anjan Chakravartty raised the truth-bearer objection in his 2001 article, "The Semantic or Model-Theoretic Conception of Scientific Theories", Synthese (online here).

A version of the Newman objection is mentioned in footnote 10 of Jeffrey Ketland, 2004, "Empirical Adequacy and Ramsification", British Journal for the Philosophy of Science.

Roman Frigg raised a version of the Newman objection in his 2006 article, "Scientific Representation and the Semantic View of Theories", Theoria (online here), Sections 5 and 6.

6 comments:

  1. Will Bausman19 July 2011 18:40

    Hello Jeff,

    I am curious and interested in your statement, "We are obliged by the facts concerning scientific practice to provide an analysis of the notion of a theory in such a way that electromagnetic theory, special relativity, evolutionary theory, and all the various other theories that may interest us, are capable of being either true or false."

    What sort of facts about practice do you have in mind here? Don't you think that maybe there is a difference between what scientists say about what they are doing (looking for true theories) and what they are doing/practicing (alternative account)?

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  2. Hello Will,

    Einstein made a distinction between looking at what scientists do and what they say they are doing. But I would be loathe to ignore what scientists say they are doing, and if a conflict arises between that and what a philosophical position claims, I think the philosophical claim loses.

    Here's my argument that, in scientific practice, scientists treat theories as truth bearers.

    The scientist studies certain kinds of phenomena (not necessarily directly observable), and is trying to explain those phenomena. In general, a scientist may talk about evidence that shows that a theory is false, or that perhaps confirms it. For example,

    (*) Newtonian mechanics is (strictly speaking) false.

    because,

    (**) According to Newton's theory, the kinetic energy of particle is 1/2 mv^2; but this statement becomes inaccurate as v gets nears to c.

    The statements (*) and (**) would be agreed by any knowledgeable physicist. It seems clear that such examples could be multiplied endlessly, throughout physics, chemistry, biology, etc.

    Of course, the philosopher may simply adopt an instrumentalism about scientific theories, and insist that theories are not truth bearers. The empirical constraint is: "consistent with data". But this is an extreme view.

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  3. I am similarly interested in the general question of what it might mean for a formal structure to represent.

    In my paper 'On the Interpretation of the Propositional Calculus', section 4, I consider something I call The Modelling Approach.

    On this approach, one specifies a formal structure (or the state of the formal structure) *conditionally* on the state of the world. (For example, one might say "Let the formula 'p' be assigned the value 1 if snow is white, and let it be assigned 0 otherwise.")

    See here (section 4) for a fuller explanation: http://sites.google.com/site/tristanhaze/OnPC.pdf

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  4. I am humbled and honored to have found this blog.

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  5. Hi Jeff,

    I think you are completely right with what you´re saying.

    But couldn´t it be plausible to say that:

    1. I´m fine with not talking about truth when I talk about scientific theories. I don´t want to talk about truth, I am interested in studying intertheoretical relations, in analyzing the epistemological status of our scientific concepts (theoreticity/obersvability), I´m interested in reconstructing logically as most of our empirical theories as possible to get clearness about the connections between them.

    2. What is Philosophy of Science then? Is it analyzing whether our scientific theories are true or false? Could be, yes. But is Philosophy of Science not also studying intertheoretical relations and logical reconstruction of theories in order to get clear about the epistemological status of our theories? Does this last conception of Philosophy of Science make one an anti-realist? I don´t know. That´s why I am studying, obviously.

    Cheers and many thanks for posting!

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  6. Hi Thomas, the problem is the elimination of semantics from the analysis of scientific theories. So, semantic notions like "T is false", "T is approximately true", "T and T' make incompatible claims about some entity x", etc., are all removed from the philosophy of science. Suppose one does this: what has been achieved?

    Anti-realism (more exactly, idealist anti-realism) is the view that an empirically adequate scientific theory cannot be false. Do you think Maxwell's theory could be empirically adequate and yet false? If so, how do you define "empirically adequate" and "false"?

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