1. Introduction

The so-called Received View in the philosophy of science (rv), developed by the logical empiricists and especially by Carnap and Hempel, assumes the formalization of scientific theories in predicate logic and a bipartition of the nonlogical vocabulary into observational and theoretical terms. While it had never been generally accepted (Richardson Reference Richardson, Richardson and Uebel2007, 362–65), it suffered a complete fall from grace (and then some) in the second half of the twentieth century (see Lutz Reference Lutz2012, sec. 1). Only recently have there been defenses of Carnap’s general philosophical method (Carus Reference Carus2007; Maher Reference Maher2007; Lutz Reference Lutz2009; Justus Reference Justus2012) and of the use of predicate logic or more generally syntactic means for the analysis of scientific theories (Halvorson Reference Halvorson2012, Reference Halvorson2013; Lutz, Reference Lutzforthcoming). The rv itself, however, has so far mainly seen historical defenses (Mormann Reference Mormann, Richardson and Uebel2007; Lutz Reference Lutz2012). It is the aim of this article to provide a conceptual defense relative to one of its main competitors, Constructive Empiricism (ce; van Fraassen Reference van Fraassen1980). Specifically, I argue that the relation between theory and observations that is described in the ce-formalism can also be described in the rv-formalism, while the converse does not hold.

The only previous defense of the rv-formalism relative to ce (Friedman Reference Friedman and van Fraassen1982, 276–77) consists of a valuable but very short outline of an argument and furthermore assumes a restriction to first-order logic. Other relative defenses have focused on syntactic approaches in general (Worrall Reference Worrall and van Fraassen1984, sec. 2; Turney Reference Turney1990). Turney provides the technically most elaborate defense. After proving the equivalence of van Fraassen’s notion of embedding and his syntactic notion of implanting (under specific assumptions), Turney concludes: “I agree with van Fraassen, that the positivist account does not yield good definitions for empirical equivalence and empirical adequacy. However, I do not agree with his claim that this is because the positivist account is syntactic, while his is semantic. … The relevant distinction here … is between two ways of linking theory and observation: Correspondence rules versus embedding/implanting” (Reference Turney1990, 448–49). Turney provides a detailed argument for the equivalence of his syntactic and van Fraassen’s semantic account, but that van Fraassen’s embeddings are an improvement over correspondence rules remains a conjecture.

To disprove Turney’s (and van Fraassen’s) conjecture, I provide a definition of the rv-formalism and a definition of the formal core of ce, empirical adequacy. The latter is in conflict with Turney’s definition, which I show to be inadequate (sec. 2). Next, I provide a syntactic description of empirical adequacy (sec. 3), which forms the basis for a description of empirical adequacy on the basis of correspondence rules (sec. 4). This shows that whatever the relation between theory and observations is, if it can be described in the ce-formalism, then it can be described in the rv-formalism as well. The converse does not hold without significantly rewriting scientific theories, however, and I show that this puts ce at a disadvantage (sec. 5). Finally, I give a precise description of a problem first pointed out by Worrall (Reference Worrall and van Fraassen1984, sec. 3) and show how it can be solved in ce by taking a cue from the rv.

2. Two Accounts of the Relation between Theory and Observation

Like the ce-formalism, the rv-formalism provides a framework for analyzing scientific theories and their relation to observations. The most important reconstructions of theories and most important tools developed in the rv assume a formalization of theories in higher-order predicate logic with a vocabulary of distinct symbols, where each c_k is an object symbol and each P_i and F_j is a relation symbol or, respectively, function symbol of a specific arity and type. Formal model theory is used for the definition of logical entailment (Lutz Reference Lutz2012, 83). What distinguishes the rv from syntactic approaches in general is the assumed bipartition of the nonlogical vocabulary of a theory into a set of observation terms and a set of theoretical terms.Footnote ¹ This bipartition leads to a natural tripartition of the set of sentences into -sentences, -sentences, and correspondence rules, the latter of which are sentences containing both - and -terms (Carnap Reference Schilpp1966, chap. 24).

Sentences in the extended observation language can be of any order. This is, for example, assumed by Carnap’s explication of a theory’s synthetic component as its Ramsey sentence (Carnap Reference Carnap and Schilpp1963, sec. 24.D).

The moniker ‘observations’ is of course somewhat misleading, but it will be convenient in what follows. What is important is that sets of -sentences are, in a sense, completely interpreted (cf. Carnap Reference Carnap and Schilpp1963, sec. 24.A).

In contradistinction to the rv, the ce-formalism relies on model theory.Footnote ² Van Fraassen (Reference van Fraassen1980, 64) describes it as follows: “To present a theory is to specify a family of structures, its models; and second, to specify certain parts of those models (the empirical substructures) as candidates for the direct representation of observable phenomena.” Furthermore the models of the theory “are describable only up to structural isomorphism” (van Fraassen Reference Monton2008, 238; cf. Reference van Fraassen2002, 22). More formally, this can be phrased as follows:

Here and in what follows, . As a shorthand, define furthermore . Definition 3 assumes a standard formalization of the notion of a structure (Chang and Keisler Reference Chang and Keisler1990, 19–20; Hodges Reference Hodges1993, 2).Van Fraassen (Reference van Fraassen1989, 366 n. 4) objects to structures being “yolked to a particular syntax,” as in “many standard logic texts,” which suggests that he would prefer a definition phrased in the formalism of Bell and Machover (Reference Bell and Machover1977, 164), who use an index set instead of a vocabulary. This difference is little more than notational, however, since in their definitions of further model theoretic concepts, the index set plays the role of the vocabulary. For examples relevant in what follows, compare the definitions of reduct (or “reduction”), isomorphism, and substructure by Bell and Machover (Reference Bell and Machover1977, 164–65) and by Chang and Keisler (Reference Chang and Keisler1990, 20–23), respectively (cf. Lutz, Reference Lutzforthcoming, sec. 3). A reader preferring Bell and Machover’s notation will have no problem adapting the following discussion.

Strictly distinguishing between observable events (phenomena) and unobservable events, van Fraassen (Reference van Fraassen1980, 64) suggests representing phenomena by structures as well: “The structures which can be described in experimental and measurement reports we can call appearances.”Footnote ⁵ This suggests

In early works van Fraassen (Reference van Fraassen1970, sec. 3) relies on “elementary statements” and a “satisfaction function” to give the relation between theory and observations, but later van Fraassen (Reference van Fraassen1980, 64) defines a theory to be “empirically adequate if it has some model such that all appearances are isomorphic to empirical substructures of that model” (cf. van Fraassen Reference van Fraassen1991, 12).

Definition 5 defines the empirical adequacy of a theory relative to a set of appearances. In contradistinction, empirical adequacy simpliciter is defined as empirical adequacy for the set of all appearances (Monton and Mohler Reference Monton, Mohler and Zalta2008, sec. 1.5).

Definition 5 assumes the standard notion of isomorphism (e.g., Hodges Reference Hodges1993). Turney (Reference Turney1990, 430) notes that his definitions are “loosely based” on Bell and Machover (Reference Bell and Machover1977, 161–68), but he defines the isomorphism of two relational structures as follows (430–31; my terminology):

In contradistinction, Bell and Machover (Reference Bell and Machover1977, 165) effectively assume that the bijection g is determined to map P_i to for all (which leads, up to notational change, to the definition given by Hodges [Reference Hodges1993]). Clearly, then, more structures are idiosyncratically isomorphic than isomorphic to each other—too many more, as one can see from the following example loosely inspired by Halvorson (Reference Halvorson2012, 192). Let the ce-theory contain all _n and E_n that fulfill the following conditions: _n is a model of with a set of objects with a_i ≠ a_j for i ≠ j, and E_n contains the substructures of _n with domains . ce-theory makes the substantial claim about the appearances that everything with property P_k also has all properties P ₁, … , P _k−1. But in Turney’s definition, is empirically adequate for any class A of appearances with singleton domains and properties P ₁, … , P_k.

It is for this reason that Turney’s definition is inadequate.

3. Syntactic Empirical Adequacy

I now want to provide a syntactic description of the notion of a ce-theory and of empirical adequacy. In a ce-theory, each element is uniquely determined as the restriction _n | E of _n to ’s domain E. Thus for each _n, is determined by the set , and each ce-theory is completely determined by its corresponding family, defined as follows:

Thus, is the extension of a predicate S of higher order, and for any predicate P, SP is the claim that P has an empirical extension. If is a ce-theory, its corresponding family can be described up to isomorphism by a set Θ of higher-order sentences or a class thereof, where is the union of the models of each set Θ_m (see Lutz, Reference Lutzforthcoming, sec. 2).Footnote ⁶ For a class to be necessary, however, M or at least one Θ_m has to be infinite, for otherwise the corresponding family would be the class of models of the single sentence . I am not aware of any scientific theory that would require an infinite M or Θ_m. For instance, the typical mathematical structures needed in physics can be defined by a single higher-order sentence (Enderton Reference Enderton and Zalta2009, secs. 2, 4). Nonetheless, I will first discuss ce-theories whose corresponding family can be described by a single set of sentences and then discuss the general case.

Each appearance can be described up to isomorphism by a single set of sentences . Of course, each may have its own distinct domain A, so that the sentences in must be taken as relativized to A. Using the symbol ‘E ’ as name for A, this leads to a set of relativized sentences.Footnote ⁷ The set of all appearances can then be described as the set of sentences containing the relativized descriptions of each appearance and, for each appearance’s domain E , the higher-order sentence SE stating that E has an empirical extension. Empirical adequacy then becomes semantic consistency:

Claim 1.— Let be the family corresponding to ce-theory , and let Θ describe up to isomorphism. Let each appearance be described up to isomorphism by . Then is empirically adequate for A iff

is semantically consistent.

If a ce-theory can be described by a class of sets of sentences, claim 1 entails that is empirically adequate iff is semantically consistent for at least one m ∈ M. The use of semantic consistency is not forbidden for syntactic approaches since their core assumption is not that semantic concepts cannot be used at all but rather that the description of theories has to be syntactical (cf. Lutz, Reference Lutzforthcoming, sec. 5).

The syntactic description of empirical adequacy that is given in claim 1 differs from the one given by Turney (Reference Turney1990, 446–47) in a number of respects. For example, Turney assumes that a ce-theory is an elementary class that contains only relational structures. The former arguably puts him at odds with van Fraassen (Reference van Fraassen1989, 211), while the latter requires elaborate reformulations of most scientific theories, which typically contain constants and functions. Most important for what follows, Turney provides a syntactic reformulation of each concept of definition 5 but keeps its overall structure intact. Specifically, he does not provide a single set of sentences that captures both the family of models and the family of sets of empirical substructures of a ce-theory. For this reason, the possibility of capturing empirical adequacy in the rv is not obvious in his syntactic description.

4. Empirical Adequacy in a Bipartitioned Vocabulary

Given that the empirical adequacy of a theory can be phrased syntactically in a single vocabulary, capturing empirical adequacy in the bipartitioned vocabulary of the rv is very simple. One can simply treat the original vocabulary as theoretical () and introduce a second observational term for each theoretical one. Thus, for each P_i one can introduce an observational predicate symbol , and analogously for function and constant symbols, leading to an observational vocabulary . Let be the set of sentences that results from replacing each nonlogical constant in the set Φ of sentences by its observational counterpart. Then the empirical adequacy of a ce-theory can be expressed as the compatibility of an rv-theory with the observations:

Corollary 1.— Let be the family corresponding to ce-theory , and let Π describe up to isomorphism. Let each appearance be described up to isomorphism by . Then is empirically adequate for A iff

is semantically consistent, where

Corollary 1 establishes that empirical adequacy is nothing but the compatibility of an rv-theory consisting of theoretical postulates Π and correspondence rules Γ (definition 1) with observation sentences (definition 2). Like claim 1, corollary 1 applies only to ce-theories whose corresponding family can be described by a single set of higher-order sentences. Again with the caveat that few if any theories will require a description of their models by a class of sets of sentences, corollary 1 can be generalized to this case: the ce-theory is then empirically adequate iff the set is semantically consistent for at least one m ∈ M. The use of semantic consistency is not incompatible with the rv since it is essentially the same as the use of semantic entailment.

Corollary 1 is somewhat unsatisfying in that there is no difference between theoretical and observational concepts except their names. This is not a problem of the rv itself but rather inherited from ce—which will become important in section 5. Nonetheless, there are some ways to provide the correspondence rules with a somewhat more elaborate role, even within the confines of ce. In keeping with Carnap’s hope that all observational terms can be defined in theoretical terms and his almost universally accepted conjecture that the converse is not generally possible (Carnap Reference Schilpp1966, 234), one can restrict the identification of theoretical with observational terms to observable objects. For the relation symbols, the correspondence rules then have the form

where m_i is the arity of P_i and stands for a string of m_i distinct variables.Footnote ⁹ Things are not as straightforward for functions since functions have to be defined over the whole domain. One could thus substitute an n_j + 1-ary observational relation for each n_j-ary theoretical function F_j with

which would also mean that the descriptions of the appearances have to be changed accordingly. It is therefore more convenient to disappoint Carnap’s hope for explicit definitions and rely for each F_j on the conditional definition

However, since the value of the observational functions is irrelevant outside the domains of the empirical substructures, the conditional definitions could be completed by conditionally defining that outside of the empirical subdomains, the observational functions all have the same arbitrarily chosen value.

For reasons that will be discussed in section 5, observational constant symbols still have to be identified with their theoretical counterparts, which leads to

This, then, shows that van Fraassen’s and Turney’s conjecture about the impossibility of capturing empirical adequacy in the rv is incorrect. The use of embeddings is thus not superior to the use of correspondence rules when connecting theory and observations. In fact, the ce-formalism is but a special case of the rv-formalism, with specific restrictions on the form of observation sentences, correspondence rules Γ, and postulates Π. In the next section, I will argue that because of these restrictions, the ce-formalism is actually inferior to the rv-formalism.

5. Beyond Empirical Substructures

Definition 3 puts enormous constraints on the formalization of theories. For it follows from the definition of a substructure that every constant of a model _n of a ce-theory has to be in the domain E of each of its empirical substructures . Furthermore, every function of the structure must map all (tuples of) elements of E to elements of E (Hodges Reference Hodges1993, lemma 1.2.2). For a ce-theory to be possibly empirically adequate, it must be possible that every appearance is isomorphic to some empirical substructure of some _n. And since is closed under isomorphism, according to every empirical substructure of every _n could be an appearance (Hodges Reference Hodges1993, ex. 1.2.4b). Therefore, according to , every constant of _n represents something observable (since it represents an aspect of a phenomenon) and all functions of _n map (tuples of) elements of T_n that partake in the representation of phenomena only to other such elements of T_n. It is clear that this is not the case for actual theories about, say, elementary particles.

One may object to these criticisms (and definitions 3–5) on the grounds that the terms ‘embedding’, ‘substructure’, and ‘isomorphism’ are not meant literally. However, the terms are well defined within but not outside of model theory, where in general they also do not occur together. And the objection makes van Fraassen use these terms in different, undefined ways without pointing this out and subsequently helping himself to results from model theory without any justification. For example, van Fraassen (Reference van Fraassen1980, 43) discusses cases in which “every model of T ₁ can be embedded in (identified with a substructure of) a model of T ₂.” The parenthetical equivalence claim relies on the model theoretic definition of ‘embedding’ and ‘substructure’, the closure of the class of models of a ce-theory under isomorphism, and the equivalence of embedding and the substructure relation for isomorphically closed classes of structures (Hodges Reference Hodges1993, ex. 1.2.4b). If the terms were not meant in the model theoretic sense, there would be no justification for this equivalence claim.

The problems resulting from van Fraassen’s reliance on substructures can easily be solved in the rv because rv-theories can contain nonobservational terms, which themselves do not have to be connected directly to observations. Thus even without loosening the restrictions on the observation sentences inherited from ce, the rv allows for connections between arbitrarily complex theoretical concepts, named by -terms, and simple observational concepts, named by -terms: postulates Π can introduce -terms and correspondence rules Γ can go beyond simple identification or identification in a restricted domain. Hence in an rv-theory, there can be constant symbols in that name unobservable events and function symbols in that name functions from observable to unobservable events. The empirical adequacy of such a more general rv-theory can still be defined as the semantic consistency of the set (2) of sentences. This solution in the rv no doubt has a counterpart in model theory, but it would in all likelihood either require a significant reformulation of most scientific theories before they can be analyzed in the ce-formalism or involve a generalization of the ce-formalism.Footnote ¹⁰ In the latter case the formalism would not be the one criticized here and conjectured by van Fraassen to be superior to the rv-formalism.

6. Beyond Consistency

Corollary 1 allows a direct comparison of the rv and ce. Its relevance is demonstrated by a previous comparison given by Muller (Reference Muller2010, 90), who, in line with Carnap (Reference Schilpp1966, 240–41), states that one aim of scientific theories in the rv is observational adequacy (“ObsAdeq”), for which a theory must entail the observations made so far. He points out three differences to empirical adequacy (“EmpAdeq”): “(i) ObsAdeq relies on the distinction between theoretical and observational concepts whereas EmpAdeq does not … ; (ii) ObsAdeq depends on historical time whereas EmpAdeq is timeless (quantifies universally over historical times); (iii) EmpAdeq requires for its definition a conception of a scientific theory that … cannot be defined in the [rv]” (Muller Reference Muller2010, 90 n. 8). Corollary 1 shows that Muller’s claim iii, which is again but a conjecture, is false. It also shows that difference i is easily overcome by introducing an observational vocabulary and identity statements as correspondence rules. The discussion in section 5 furthermore shows that this difference speaks in favor of the rv. Difference ii is not one of principle but at best one of primacy of definition: one could define ‘preliminary empirical adequacy’ as ‘empirical adequacy given the appearances A observed so far’ and define ‘eternal observational adequacy’ as ‘observational adequacy at all times’.

Muller misses the real difference, which is obvious in the syntactic paraphrase: a theory is observationally adequate if it entails all observations, while it is empirically adequate if it is not incompatible with them. Compared to observational adequacy, then, empirical significance is cheap: a tautology is compatible with any set of sentences describing the appearances, and thus its set of models is empirically adequate. This is obvious given claim 1, and it can also be translated into the ce-formalism: the ce-theory corresponding to a tautology contains the class of all structures and for each structure the set of all its substructures as empirical substructures. This ce-theory is empirically adequate given any appearances A. Thus claim 1 throws doubt on van Fraassen’s claim that science “aims to give us theories that are empirically adequate” (van Fraassen Reference van Fraassen1980, 12), at least if empirical adequacy is meant to be the only aim of science, or even a particularly interesting one.

This criticism has already been put forth by Worrall (Reference Worrall and van Fraassen1984, sec. 3) and Percival (Reference Percival and Monton2007, 87), who suggests as one of three possible replacements for van Fraassen’s account of the aim of science that for each context c, science aims for theories that are empirically adequate and “fully empirically informative with respect to all the phenomena that are relevant at c.”Footnote ¹¹ Unfortunately, Percival does not provide an explication of the latter notion in the ce-formalism but rather stipulates that a theory “is ‘fully empirically informative’ with respect to certain phenomena iff it expresses all true propositions that are solely about those phenomena” (n. 44). In his reply, van Fraassen (Reference van Fraassen and Monton2007, sec. 1.1) acknowledges the need for informative theories but does not give an explication either.

To arrive at a precise notion of full empirical informativeness, one can look to the rv. An rv-theory Θ expresses all and only true propositions about some appearances A iff . For a ce-theory , then, every model of the corresponding family described by Θ is also a model of the observations. It is not very difficult to see that this is the case iff every empirical substructure of every structure in is isomorphic to an appearance and, since every is described up to isomorphism, vice versa. Since this condition already entails empirical adequacy, it formalizes the aim of science as suggested by Percival if context c is taken to determine the appearances A.

7. Conclusion

The preceding discussion has brought out a number of points. First, the rv can describe scientific theories and their relation to the observations if ce can, while the opposite is not obviously the case. Second, the ce-formalism can and should be extended in at least two respects. It should allow for concepts in theories that do not also occur in the appearances, and it should include the explication of ‘full empirical informativeness’ among its formal desiderata for a ce-theory. Finally, and as a suggestive corollary to this and previous discussions, the philosophy of science would be better off if syntactic and semantic methods in general and the rv and ce in particular were not seen as in opposition but, correctly, as means of mutual support and improvement.