On Theories

Logical Empiricism and the Methodology of Modern Physics

William Demopoulos, Michael Friedman

• Publisher: Harvard University Press
• Year: 2022
• ISBN:
  9780674269729
• Topic: Physical Sciences
• Subtopic: Quantum Theory

A renowned philosopher's final work, illuminating how the logical empiricist tradition has failed to appreciate the role of actual experiments in forming its philosophy of science. The logical empiricist treatment of physics dominated twentieth-century philosophy of science. But the logical empiricist tradition, for all it accomplished, does not do justice to the way in which empirical evidence functions in modern physics.In his final work, the late philosopher of science William Demopoulos contends that philosophers have failed to provide an adequate epistemology of science because they have failed to appreciate the tightly woven character of theory and evidence. As a consequence, theory comes apart from evidence. This trouble is nowhere more evident than in theorizing about particle and quantum physics. Arguing that we must consider actual experiments as they have unfolded across history, Demopoulos provides a new epistemology of theories and evidence, albeit one that stands on the shoulders of giants. On Theories finds clarity in Isaac Newton's suspicion of mere "hypotheses." Newton's methodology lies in the background of Jean Perrin's experimental investigations of molecular reality and of the subatomic investigations of J. J. Thomson and Robert Millikan. Demopoulos extends this account to offer novel insights into the distinctive nature of quantum reality, where a logico-mathematical reconstruction of Bohrian complementarity meets John Stewart Bell's empirical analysis of Einstein's "local realism." On Theories ultimately provides a new interpretation of quantum probabilities as themselves objectively representing empirical reality.

1 Logical Empiricist and Related Reconstructions of Theoretical Knowledge

1.1 The Partial Interpretation Account of Theories

The partial interpretation account of theories holds that the vocabulary of a theory consists of an observational and a theoretical component, where the application of the observation-theory distinction is based on whether a vocabulary item is held to apply to entities in the intended domain of the theory which are observable (in which case the item belongs to the observational vocabulary) or unobservable (in which case it belongs to the theoretical vocabulary).¹ In its classical formulation, the distinction partitions vocabulary items into just these two classes.² (The possibility of a third class of terms—mixed vocabulary items that are understood to apply to both observable and unobservable entities—will be addressed later in this chapter .) Given this distinction in vocabulary, the sentences of the language of a theory are divided into three classes: one consisting of sentences which are generated from just the observation vocabulary, another of sentences generated from just the theoretical vocabulary, and a third consisting of sentences generated from the combined observation and theoretical vocabularies. These are, respectively, the observation sentences, theoretical sentences, and correspondence rules of the language. A theory is the conjunction of a selection of theoretical sentences and correspondence rules; I will ignore various possible generalizations of the notion of a theory that are irrelevant to the conceptual issues on which I will focus. There are no special restrictions on the logic of a theory, and it may be either first order or higher order.

Historically, the partial interpretation account of theoretical knowledge derives from the idea that theoretical terms are introduced by sentences which, taken by themselves, are indistinguishable in their epistemic status from the statements of a pure mathematical theory. Of chief importance to the partial interpretation account is the notion that theoretical statements share with the statements of a mathematical theory the property that their interpretation is responsible only to the logical category of their constituent nonlogical constants. This view of theoretical statements is a consequence of the fact that the partial interpretation account is a continuation and extension to the theories of physics of the axiomatic tradition that Hilbert initiated in pure mathematics. Especially influential was Hilbert’s contention that the primitives of a mathematical theory are whatever satisfies its axioms. This contention—that the postulates of a theory “implicitly define” its primitive notions—swept away the subjective associations that characterized an older tradition’s understanding of a mathematical theory’s primitives, even in the case of geometry, where they were thought to have a familiar “intuitive” content.³ The partial interpretation account sought to extend Hilbert’s analysis of mathematical theories to physics by providing an account of the empirical content of the theoretical statements of physics based on the connections between theoretical terms and observation terms that are expressed by correspondence rules.

One can see in this brief sketch the two characteristic theses of the partial interpretation view: the first, its claim that only the observation vocabulary is completely understood; and the second, the correlative claim that the interpretation of the theoretical vocabulary is limited by constraints which depend only on the logical category of the theoretical terms and whatever restrictions the true observation sentences impose on the domain of unobservable entities over which the theoretical sentences and correspondence rules are evaluated. I will refer to this second claim as the structuralist thesis. We have yet to explain how the partial interpretation view conceives the relation between interpretations, true interpretations, and truth.

1.2 Carnap on Ramsey Sentences and the Explicit Definition of Theoretical Terms

Carnap’s mature reconstruction of the language of science⁴ builds on and extends the partial interpretation view of theories. The central notion of this account is the Ramsey sentence of a theory: the sentence formed by replacing theoretical terms by (new) variables of the appropriate logical category, then closing the resulting formula by adding an existential quantifier for each of the new variables. It is a very short step from the two characteristic theses of the partial interpretation account of theories to the notion that a partially interpreted theory’s Ramsey sentence captures its “factual content”: the Ramsey sentence is observationally equivalent to the theory in the sense that any argument from the partially interpreted theory to a sentence of the observation language can be recovered using the Ramsey sentence instead; and the Ramsey sentence’s use of variables in place of uninterpreted theoretical terms simply makes explicit the commitment of the partial interpretation account to the structuralist thesis. As Ramsey expressed it:

So far … as reasoning is concerned, that the [transforms of the theoretical sentences and correspondence rules in the matrix⁵ of the Ramsey sentence of the theory] are not complete propositions makes no difference, provided we interpret all logical combinations as taking place within the scope of a [single existential] prefix.… For we can reason about the characters in a story just as well as if they were really identified, provided we don’t take part of what we say as about one story, part about another.⁶

Carnap’s mature reconstruction refines the doctrine of partial interpretation in two principal respects. As we have already noted, Carnap explicates the factual content of a partially interpreted theory in terms of its Ramsey sentence. But Carnap took things a step further by combining his account of the factual content of a theory with an explication of theoretical analyticity—analyticity relative to a theory—in terms of what has come to be known as the Carnap sentence of a theory: the conditional whose antecedent is the theory’s Ramsey sentence and whose consequent is the partially interpreted theory. Before Carnap, the distinction between the factual and analytic (and hence, nonfactual) components of a theory followed the distinction between postulates and definitions. But since this distinction is inherently arbitrary, its utility for a dichotomy that is supposed to reveal our factual commitments may be doubted.

The Carnap sentence is justifiably regarded as analytic because it is a kind of “implicit definition” of the theoretical vocabulary, one that is provably nonfactual in the sense that the only observation sentences it logically implies are logical truths. And as John Winnie (1970) later showed, the Carnap sentence, like a proper definition, satisfies a special noncreativity condition similar to the noncreativity condition that is customary for proper explicit definitions.⁷

Carnap advanced the Ramsey sentence not just as a clarification of the partial interpretation view of theories but also as a correct representation of how scientists understand their theoretical claims. They intend, Carnap held, an “indeterminate” claim, one that may have many interpretations under which it comes out true. As scientists understand them, theoretical claims are indeterminate as to the interpretation of their theoretical vocabulary, and any representative class or relation which makes true the Ramsey sentence of the theory to which the claim belongs is as acceptable as any other. To narrow down the interpretation any further than is demanded by the truth of the Ramsey sentence would, for Carnap, violate the intentions of the scientist who constructed the theoretical system.

In one of his last papers on the subject,⁸ Carnap converts the implicit definition of theoretical terms by the Carnap sentence into a sequence of explicit definitions of them. But these explicit definitions do not eliminate—and were not intended by Carnap to eliminate—the indeterminateness of his earlier account. Indeed, Carnap formulates his explicit definitions in what he calls a “logically indeterminate” language. The language L_ε which he employs is a standard first- or higher-order language enriched with Hilbert’s epsilon operator and the extensional axioms which govern its use. There are just two such axioms. Given a formula Fx in one free variable, the first axiom tells us that if there is something satisfying Fx, then there is an “ε-representative” of F, denoted “εx(Fx),” that is selected by the choice function which interprets the epsilon operator. The second axiom tells us that if the formulas Fx and Gx are extensionally equivalent, their ε-representatives are the same.

That it should be possible to apply Hilbert’s epsilon operator to the Ramsey sentence reconstruction of theories is a consequence of Carnap’s observation that the Carnap sentence of a theory can be derived from a sentence that is in the same form as the first of the axioms governing the epsilon operator. This sentence can be understood as asserting that if there is a sequence of classes of the appropriate type which satisfies the matrix of the Ramsey sentence of a theory, then there is an ε-representative such sequence (i.e., a sequence consisting of n ε-representative classes, where n is the number of new variables that were introduced when the partially interpreted theory was replaced by its Ramsey sentence). Carnap observed that if the theoretical terms Ti(1 ≤ i ≤ n) are now explicitly defined as the ε-representatives of such a sequence, the Carnap sentence follows.

For Carnap, the principal virtue of his proposal is that it incorporates the convenience of having the use of a theoretical vocabulary while retaining all the characteristic indeterminateness of that vocabulary, which is the hallmark of the partial interpretation view and of his mature reconstruction in terms of Ramsey and Carnap sentences. Carnap writes that the theoretical postulates and correspondence rules “are intended by the scientist who constructs the system to specify the meaning of [a theoretical term] to just this extent: if there is an entity satisfying the postulates, then [the term] is to be understood as denoting one such entity. Therefore the definition [of a theoretical term by means of the epsilon operator] gives to the indeterminate [theoretical term] just the intended meaning with just the intended degree of indeterminacy” (Carnap 1961, p. 163; emphasis added).

1.3 A Proposal of David Lewis and Two Theorems of John Winnie

David Lewis’s (1970) article is sometimes credited with having refined Carnap’s and Ramsey’s reconstructions and to have improved on Carnap’s approach to the explicit definition of theoretical terms by showing how it might be possible to avoid multiple interpretations of the theoretical vocabulary under which the theory comes out true. Lewis maintained that allowing for what he calls “multiple realizations” concedes too much to instrumentalism. Lewis does not say why multiple realizability is a concession to instrumentalism, but let us for the moment grant the point and consider how he makes the case that there are many theories which, if realizable, are uniquely realized. Lewis is clear that he must provide an independent defense of this contention, since the possibility of there being just one realization appears to be excluded by two theorems of John Winnie (1967). Modulo the conceptually unimportant technical restriction that not all theoretical properties and not all theoretical relations are universal, Winnie shows that on the partial interpretation view of theories, if a theory has one realization, there is always another; and if a theory is realizable at all, it is arithmetically realizable.

Lewis’s response to Winnie rests on two features of his conception of the language in which theories are formulated. First of all, Lewis follows the partial interpretation account by dividing the vocabulary of a theory into two parts, which he calls the “O-vocabulary” and the “T-vocabulary” of the theory. However Lewis’s “O-T distinction” is not the distinction between observational and theoretical vocabulary of the partial interpretation account. Lewis’s distinction concerns old vocabulary, vocabulary which is understood prior to the formulation of the vocabulary-introducing theory; and the contrast Lewis’s distinction draws between old and T- or new vocabulary has nothing to do with observation or observability. In principle, Lewis’s O-T distinction could be completely orthogonal to the observational-theoretical distinction of Carnap and the doctrine of partial interpretation. A second, related, difference involves Lewis’s notion of an “O-mixed term.” This is a notion that does real work for Lewis, but before explaining it, some further background regarding the partial interpretation view and its relation to Lewis’s O-T distinction is necessary.

In his exposition of the partial interpretation reconstruction, Winnie includes in addition to the observational and theoretical predicates a separate category of mixed predicates, predicates that apply to both observable and unobservable entities. Lewis has the notion of a mixed term, and of special importance are those he calls “O-mixed” terms. These are terms which, like Winnie’s mixed predicates, can apply to both observable and unobservable entities. But their characterization—unlike Winnie’s characterization of observational and theoretical predicates—has nothing to do with observability. And while Winnie’s mixed predicates are distinguished from observation predicates, Lewis’s O-mixed terms count as O-terms, and as such, are assumed to be fully understood whether they apply to observable or unobservable entities; therefore the interpretation of O-terms—whether they are unmixed and refer only to observable entities, or are mixed and refer also to unobservable entities—must be preserved as we pass from one realization of a theory to another.

The situation is altogether different for Winnie and for the standard view. When Winnie defines the permutation map which establishes the existence of alternative realizations, it is only the entities in the observable part of the domain that cannot be permuted, and it is only the interpretation of the observation predicates—predicates which apply only to observable entities—that must be the same in any intended realization of a partially interpreted theory. No such requirement applies to the interpretation of theoretical predicates; nor does it apply to mixed predicates.

Since Winnie’s permutation map is the identity on observable entities, it is trivially true that observable relations are isomorphic to their images under his mapping. B...