Thursday, February 5, 2009

The Semantic View of Theories Again

I was thinking more about Gabriele's post on the semantic view of theories. Consider this quick argument in favour of a semantic view of theories:

  1. Scientists believe (or at least accept) scientific theories;
  2. Attitudes like belief and acceptance are held to propositions (certainly belief and acceptance ascriptions involve embedded 'that' clauses which seem to denote propositions);
  3. So, scientific theories are propositions.

This is pretty clearly a semantic view of theories. But Gabriele, and I presume others, seem not to believe this is the semantic view of theories, as discussed in the literature. Could someone explain the difference, if there is one? For I do not at this stage see it, for reasons I'll now explain.

The conclusion of this argument is what I have always understood to be the semantic view of theories. If propositions are structured (i.e., not just sets of possible worlds), then the propositions which express scientific theories can easily be models in the model-theoretic sense; if theories have merely qualitative content, no one model will capture the proposition expressed (as qualitatively indistinguishable models will equally satisfy the theory), so the proposition expressed should be a set of models. (Things seem to be a little, but not much, trickier if propositions are unstructured.) In any case, there seems to be a clear correspondence between the propositions expressed by the sentences of some presentation of the theory and the models which satisfy those sentences, a close enough correspondence that reducing the one to the other doesn't seem unreasonable.


  1. Antony,

    As far as I know, all supporters of the semantic view seem to agree that models are not truth-apt and most philosophers seem to agree that, if there are propositions, they are truth-apt (in fact, they seem to be among main candidates for playing the role of truth-bearers). This seems to suggest that the supporters of the semantic view do not see models as (structured) propositions.
    It is much harder to understand what they take models to be (abstract structures? imaginary systems?), but I think that one can safely say that they are not usually understood as being propositions.
    I like your argument but at least from the point of view of ordinary language I don't know if how many people would be happy to say (as in your premise 1) that one believes a scientific theories (as opposed to one believing it to be (approximately) true/empirically adequate/empirically successful/instrumentally useful). In other words, ordinary language does not seem to take theories (as opposed to propositions about theories) to be the direct objects of beliefs. But this should not be that surprising--if theories are *sets* of propositions and propositions are the objects of belief, there is no reason to think that theories are the object of belief unless one assumes that sets of propositions are propositions.

  2. I guess I'm not hearing the contrast you're drawing between believing a theory and believing it to be true. There seems no difference in the propositional case between believing that p and believing p to be true. I certainly don't think it is problematic in natural language to say one believes a theory; googling "believe the theory that" (to rule out spurious cases like 'Three-quarters of Americans say they believe the theory of global warming is a proven fact'), I get 36500 hits of the right form, which is at least some evidence this is no idiosyncratic judgement on my part.

  3. Hey there Anthony,

    I think that your argument is interesting, but I'm inclined to agree with Gabriele in both of his points.

    I think there are cases where we can believe P to be true without believing P. For instance, maybe I believe the story Hamlet is true. But I don't believe Hamlet. (This doesn't even look well-formed since it doesn't seem like Hamlet can be an object of belief.)

    Maybe you meant to rule out these cases by saying "in the propositional case", but in that case, it seems like your point hinges on the truth-aimed-ness of belief, which sidesteps the issue: I'm inclined to think that sentences of the form "S believe T" where T is a scientific theory do not express propositions that are structurally similar to those expressed by "S believes that P" where P is a proposition. Rather, they express the proposition that S takes the theory to be (approximately) true/empirically adequate/empirically successful/instrumentally useful, etc.

  4. Antony,

    The point I was trying to make is that 'Isaac believes that Newtonian Mechanics' and 'Isaac believes Newtonian Mechanics' do not seem to be well-formed sentences (the point that also Daniel is making). At most, 'Isaac believes *in* NM' seems to be one but its more like 'Anselm believes in God'.
    Btw, let me add that if we take propositions to be truthbearers and sets of propositions not to be propositions themselves, we ahave to concede that theories can only be true derivatively--i.e. iff all the propositions in them are true.

    Hope this clarifies my point.

  5. Dear Anthony,

    you wrote that "If propositions are structured [...], then the propositions which express scientific theories can easily be made models in the model-theoretic sense", which I find difficult to follow. Maybe you could elaborate. Here are my concerns:

    a) Your conclusion surely can't follow just from the fact that propositions are structured. After all, snowflakes are also structured, but I can't see how they can easily be made models in the model-theoretic sense.

    b) Let's take for example the statement "Between any two points there is a line" as being part of my scientific theory. For a model, following Tarski, you need a set of things that are called points, another set that are called lines, and an incidence relation that relates elements from these two sets. How are you going to achieve this in terms of the proposition that is expressed by "Between any two points there is a line"?

  6. To Dirk: I didn't say anything about what the structured propositions were, but I was assuming that they were set theoretic constructions of some kind out of some 'constituents' which are intuitively the subject matter of the proposition. Quite how to handle quantification, as in your second question, is a subtle issue; but I see no in principle reason why the structured proposition expressed by a universally quantified sentence shouldn't be sufficient to either be or be directly reducible to the kind of model you mention; after all, they are what that sentence says.

    To Gabriele and Daniel: Gabriele said

    The point I was trying to make is that 'Isaac believes that Newtonian Mechanics' and 'Isaac believes Newtonian Mechanics' do not seem to be well-formed sentences (the point that also Daniel is making).

    I still don't see it. It's quite clear that the 'I believe that p' construction won't accept the name of a proposition substituted for p—compare 'I believe that the Liar', which is similarly defective. And it is perfectly natural to think that 'Newtonian Mechanics' is the name of a theory, and hence the name of a proposition, in my view, so the failures of substitutivity you point to are no argument against that view.

    The second sentence you report as being ill-formed seems perfectly fine to me, by the way, and to many other native speakers as the google counts I put forward seem to suggest.

    I agree with Gabriele's second point, his suggestion that the sentence 'I believe in Newtonian Mechanics' doesn't succeed in expressing my commitment to the content of the theory. For while I don't believe the phlogiston theory, I certainly believe in that theory, as I certainly believe that theory exists. Since one can believe in a theory without subscribing to it—or, as I would put it, believing it—I agree that response couldn't help here.

  7. Antony,

    You have a point--as my colleagues in linguistics like to remind me, as a non-native speaker I can only defer to native speakers when it comes to the grammaticality of English sentences. But if, unlike Heidegger, you don't believe that some specific natural language is the repository of all philosophical wisdom (did Heidegger really believe that of German was such a language or is it just a philosophical urban legend?), I have to report that "Isacco crede la Meccanica Newtoniana" is not a well-formed sentence in Italian (and this, I suspect, is because Italian has many more overt case indicators than English).

    In any case, I think that the grammaticality of the constriuction 'S believes P' (unlike that of 'S believes that P') hardly proves that 'P' is the name of a propositions. At most it proves that 'P' is a name. I suspect that most native English-speaking philosophers would accept that 'I believe John' is well-formed English sentence but John is not a proposition but a person. (Am I wrong? Again, I defer.)

  8. I agree with your first two paragraphs, Anthony, and have said this for years. In order to be consistent with scientific practice, a theory must be a set of propositions. Theories must be truth-bearers. They must bear other semantic relations to other propositions (implication, etc.). Furthermore, there is no obvious sense in which a structure - in the standard sense used in mathematics (e.g., a linear ordering or a group) - is a truth-bearer. The models of arithmetic, set theory, etc., or, if we knew how to formalize scientific theories, of Newtonian mechanics, special relativity, etc., are not truth-bearers. The truth-bearers are the relevant propositions involved, expressed by the statements of the theory.

    Also, theories are the content of propositional attitudes - the sorts of things that we can believe, accept, reject, deny, etc. (Propositions are usually defined as either the semantic content of interpreted sentences or as the content of propositional attitudes.) But it doesn't make sense to say that I believe (M, g), where M is a topological manifold and g is a (0,2) tensor on M. We might believe something about such a model - that is, we believe some proposition about such a model - but we don't believe it. If I describe a model M of Peano arithmetic, it doesn't make sense to say you believe M. It does make sense to say you believe that M is isomorphic to the intended model, (N, 0, S, +, x). And the object of your belief is a proposition. The same is true of the statement "F = ma", which expresses the proposition that the net force on a body is equal to its mass times its acceleration. In believing Newtonian mechanics, this is one of the propositions that I believe. The same is true of Maxwell's laws, "E = mc^2", the Dirac equation, etc.

    I don't see any particular need to provide a further analysis of what propositions are, unless one happens to be specifically interested in that topic from metaphysics and philosophy of language.

    But regarding your proposed analysis, you write "so the proposition expressed should be a set of models". I cannot see how this works as an analysis of propositions. Consider the English sentence "Derrida is Spanish". This expresses, in English, a certain proposition, namely that Derrida is Spanish.

    Then consider the models of this sentence. Namely, all structures of the form (D, a, X), where D is a non-empty set, X is a subset of D and a is an element of D, and a is an element X.

    I can't see any obvious relation between the set of such structures and the proposition that Derrida is Spanish. The proposition is specifically about Derrida and the property of being Spanish. The models are irrelevant.

    Note also that the models of "Derrida is Spanish" are the same as the models of "Foucault is Hungarian", but clearly these express (in English) different propositions.

    So, what is the relation between:
    1. the proposition expressed, in an interpreted language L, by a sentence S;
    2. and its class of models, Mod(S)?

  9. I am also very curious to hear an answer to Jeff's question. (Possibly, also addressing the situation where S is part of an inconsistent theory, which doesn't have any models.)

  10. It is probably important to distinguish between 'models' as commonly used in science (a ball and stick model of a molecule, for instance), and the sort of structures that the semantic view refers to (which are confusingly also called models, but I guess most people on here appreciate the difference).

    I agree that it is hard to see how the latter can relate to reality, or how it can be said that such a model is true in a 'direct' sense. What would be required is an account of how such structures represent reality.

    In the Sneed / Balzer framework the matter of truth tends to become one of a relationship between structures, for instance potential and partial potential models of a theory, the latter being (structural) representations of observable reality, which is one way to deal with the issue at the cost of having an infinite regress of models.

    In my view, one of the stronger supporting arguments that can be made in defense of a semantic view of theories is that it provides a much improved context in which to discuss the role of theoretical terms (for instance in the form of a Ramsey sentence, though Ramseyification would not be strictly required) and thus it does away with the problems associated with 'partial interpretation' of theoretical terms.

    While it is clear why partial interpretation is needed in a syntactic view of theories it is much harder to work out what it exactly is, and how partially interpreted terms (can) relate to reality.

    One serious difficulty that to my knowledge is still unresolved is that the syntactic and semantic views on theories are only equivalent for consistent and simple theories; it is somewhat doubtful that actual scientific theories can be axiomatised in such a way that the equivalence holds.

    For instance, actual scientific theories might have inconsistencies that prohibit the equivalence.

    On balance, though I believe that the semantic view of theories provides a useful framework in which to discuss such matters.

  11. Jeff:

    I agree with a lot of what you say, and I'm certainly sympathetic to the idea of giving propositions a serious look-in. I've argued that the Bohr model of the hydrogen atom, and the nuclear model of the cell are better thought of as sets of propositions ("propositional models") than mathematical structures; following on from that, I've suggested that we explore the idea that a theory might be best thought of as a collection of propositional models.

    Someone who maintains that theories are collections of mathematical models (of the state-spaces-with-trajectories variety, let's say) does have room for manoeuvre in the face of your arguments to the contrary, however, thanks to either of a couple of moves of an entirely standard sort. (Of course, the view that a theory is a collection of propositional models needs to find that room, too: collections of sets of propositions aren't the sorts of things we usually think of as being truth-bearers or objects of propositional attitudes, either, even though they're composed of things which are.) She – the naïve semantic view theorist – can acknowledge that we say things like 'Theory X is true' and 'I believe theory X', and even manage to say true things in the process, but then propose that the surface grammar is misleading: uttering 'Theory X is true' is just a shorthand way of saying that theory X contains a model which is isomorphic to (some relevant part of) the world; 'I believe theory X' is a shorthand way of saying that you believe that theory X contains a model which.... Alternatively, she could propose an error theory of such utterances: they make false claims – or perhaps non-truth-valued claims – because theories aren't the kinds of things that can be true, or be believed, but we treat them as true because they're acceptable stand-ins for various true claims.

    Of course, on the "paraphrase" view the semantics of the utterances in question is messier, and the semantics of our talk overall will be less unified, than on the view you prefer; and in the case of the error-theoretic approach, I suppose it's the pragmatics that will be messier in places and less unified overall. But then, of course, the naïve semantic view theorist moves on to try and make a case that these admitted costs are outweighed by various other theoretical benefits the view (allegedly) has. So there's no knock-down argument against the naïve semantic view here.

    The other point I wanted to make is that it's quite hard to find the unqualified claim that theories are collections of models in the work of either Suppes or van Fraassen (the two seminal figures, in my view). Indeed, Suppes says some remarkably syntactic-viewy things at some points in at least one of the papers that's standardly cited in connection with the semantic view; and see first two pages of the appendix to chapter 1 of van Fraassen's latest book (in which he refers back to work from 1970 and 1991) for a useful corrective to the idea that the semantic view as he understands it is the view that a theory is (just) a collection of mathematical structures. (Particularly notable in the context of the present discussion is this sentence, from pp. 309-310 of _Scientific Representation_: "While not settling on any official definition of what a theory is, I emphasized [in 1991] that it must be the sort of thing that can be believed, disbelieved, doubted, and so forth.")

    Closing disclaimer: I'm not ultimately out to defend the semantic view.