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Logic and the Philosophy of Science

Logic and the Philosophy of Science

Bas C. van Fraassen Department of Philosophy San Francisco State University

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Penultimate draft published: Journal of the Indian Council of Philosophical Research 27 (2011), #2

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Abstract

While logic has sometimes tended to lead to oversimplification and abstraction, it has also made it possible to refine philosophical problems pertaining to science so as to give them rigor and precision, and in some cases, to solve them definitively. There are too many different cases to provide a helpful overview, so I will discuss several examples that I have found especially telling concerning the value of logic. I will take up two issues concerning definability and one issue in epistemology. They concern the problem of understanding theoretical terms in physics and what is known as the problem of old evidence.

1 The Problem of `Implicit' Definability

How theoretical terms are related to what we can observe and measure has been a recurrent problem in philosophy of science. When the theoretical scene changes, new terms appear and to understand what they mean seems to require learning the new theories, as if they can only be understood `from within' those new theories.

I will begin with a famous historical example. In the 17th century, Cartesians considered Newton's introduction of the new concepts of mass and force a return to the `occult qualities' of the medievals. In the 19th century, however, there were sustained efforts to provide reductive accounts of those concepts. Mach's work is the best known. His attempt and its difficulties presaged the wider ranging controversies about theoretical terms in our own time, which we will address in turn.

1.1 Attempted definitions of mass

In the context of classical physics, all measurements are reducible to series of measurements of time and position, so we may designate as basic

1. The Problem of `Implicit' Definability

1.1. Attempted definitions of mass a. Critique of the definition b. Bressan: Necessity rather than counterfactuals c. Alternative approaches to mass in mechanics

1.2 Attempted eliminations of theoretical terms

a. Hilbert's introduction of `implicit definition' b. Quine's `Implicit definition sustained' c. Keeping something fixed: Winnie's rejection d. Lewis on the definition of theoretical terms e. The relevance of Beth's theorem

2. The Problem of Old Evidence 2.1 Evidence, confirmation, and probability 2.2 Responses to the problem of old evidence 2.3 Postulating new `logical' evidence 2.4 Probabilist versions of implication and modus ponens 2.5 The Conditional Proof Requirement

3. Conclusion

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observables all quantities that are functions of time and position alone. Called the kinematic quantities, these include velocity and acceleration, relative distances and angles of separation. They do not include mass, force, momentum, kinetic energy (the dynamic quantities). To some extent the values of the latter can be calculated (within the theory) from the basic observables. That is precisely what inspired the many proposed `definitions' of force and mass in the nineteenth century, and the more recent axiomatic theories of mechanics in which mass is not a primitive quantity.

The seminal text is Mach's proposed `definition' of this concept within the theory, in a chapter called "Criticism Of The Principle Of Reaction And Of The Concept Of Mass" (Mach 1883: 264ff). He writes, in somewhat tentative fashion:

If, however, mechanical experiences clearly and indubitably point to the existence in bodies of a special and distinct property determinative of accelerations, nothing stands in the way of our arbitrarily establishing the following definition:

All those bodies are bodies of equal mass, which, mutually acting on each other, produce in each other equal and opposite accelerations.

[...] That these accelerations always have opposite signs, that there are therefore, by our definition, only positive masses, is a point that experience teaches, and experience alone can teach. In our concept of mass no theory is involved; "quantity of matter" is wholly unnecessary in it; all it contains is the exact establishment, designation, and denomination of a fact. (pp. 266-7)

While the `definition' is formulated in terms that are purely kinematic, Mach clearly realized that there is an empirical fact behind it, so to speak, in that at least the law of equality of action and reaction must be presupposed. But Mach's idea of a definition does not meet the standards developed since then.

1.1.1 Critique of the definition

As Patrick Suppes emphasized, if we postulate with Newton that every body has a mass, then mass is not definable in terms of the basic observables, not even if we take force for granted (cf. Suppes 1957: 298). For, consider, as simplest example, a model of mechanics in which a given body

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has constant velocity throughout its existence. We deduce, within the theory, that the total force on it equals zero throughout. But every value for its mass is compatible with this information.

Could Mach possibly have missed this obvious point? It seems unlikely. It appears rather that his purpose was to present what in his book on the theory of heat was called a coordination (Mach 1896: 52). The concept of mass is introduced within the theory by specifying precisely what will count as a measurement of mass, by procedures that presuppose the empirical correctness of Newton's third law of action and reaction. These procedures are explored in some detail in a previous section (Mach 1883: 247ff.). What Mach thinks of as the axiomatizations of the empirical core and the general theory of classical mechanics is not the same enterprise as what 20th century logicians consider to be definition and axiomatization. But Mach's work has the virtue of focusing on the relation between the mathematics and experimental practice.

Nevertheless, Suppes has a point, and it is clear that if Mach wants to have the theory imply that every body has a mass, then he is involved in a modal or counterfactual assertion about what would happen under suitable, possible but not always actual, conditions.

What are those conditions, and to what extent do they determine the masses of bodies, relative to the theory? Here the seminal work was by Pendse (1937, 1939, 1940) to determine how much information about a body (possibly concerning a number of distinct times) would allow one to calculate its mass (cf. Jammer1964: 92-95). It appeared that in almost all cases, the kinematic data would determine the mass. In response to Suppes, Herbert Simon (1954, 1959, 1966) discussed a measure on the class of models of Newtonian particle mechanics, and proved that by that measure mass is definable almost everywhere. However, although this measure was presented as `natural', we must acknowledge the infinity of exceptions this "almost everywhere" allows.1 Such results are scant comfort for someone who wishes to eliminate mass as a primitive concept.

To sum up: there are models of mechanics (that is, worlds allowed as possible by this theory) in which a complete specification of the basic observable quantities does not suffice to determine the values of all the other quantities. Thus the same observable phenomena equally fit more than one distinct model of the theory.

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1.1.2 Bressan: Necessity rather than counterfactuals

Use of counterfactual language can raise eyebrows even among the friends of modality. Is it possible at least in the present case to replace the counterfactuals with the alethic modalities, necessity, possibility, ...? To do so was the aim of Aldo Bressan (1973), who devised a subtle and rich account of the modalities and constructed formal proofs of adequacy for his axiomatization of mechanics. While I do not wish to discuss this work in detail, there is a problem concerning the specific formulation adopted by Bressan, and it concerns a quite general difficulty for the understanding of modalities in nature. In his formulation, Bressan asserts for each body U a conditional :

(If a certain experiment is performed on U then the outcome is real number )

which he ranks as necessary, and as satisfied by a unique number . Upon analysis, it then appears that this means that in all physically possible cases, this experiment upon U yields .

What we need to ask then, however, is: what are the physically possible cases? They cannot be those logically possible cases that are compatible with the laws of mechanics, for the laws, being general, will not entail information about characteristics of individual, specific bodies. If to those laws we add factual information about U, phrased in purely kinematic terms, there will still be in general many alternatives left open. That is just our initial problem returning: relative to all that, if U is always unaccelerated, it is as possible that U has one mass as that it has another.

So the necessity cannot be understood as `nomological' in the sense of `deriving from laws plus kinematic factors'. It would have to be a sort of necessity that is specific and different from body to body. In other words, this program needs very specific de re modalities or essences or the like. Suppose, however, that the mode of response of body U to a certain kind of experiment is introduced as an essential property of U, by postulate. Then we can hardly count the manner in which mass has been eliminated from the primitive concepts of mechanics as a gain over the reliance on counterfactuals.

1.1.3 Alternative approaches to mass in mechanics

In the axiomatic theories of mechanics developed in this century, we see many different treatments of mass. In the theory of McKinsey, Sugar, and Suppes (1953), as I think in Newton's own, each body has a mass. In

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Hermes's theory, the mass ratio is so defined that if a given body never collides with another one, there is no number which is the ratio of its mass to that of any other given body. In Simon's, if a body X is never accelerated, the term `the mass of X' is not defined. In Mackey's any two bodies which are never accelerated, are arbitrarily assigned the same mass.2

What explains this divergence, and the conviction of these authors that they have axiomatized classical mechanics? Well, the theories they developed are demonstrably empirically equivalent in exactly the sense that any phenomena which can be accommodated by a model of any one of them can be thus accommodated by all. Therefore, from the point of view of empirical adequacy, they are indeed equal. And this, an empiricist would wish to submit, is just the basic criterion of success in science, to which all other criteria are subordinate.

1.2 Attempted eliminations of theoretical terms

The dispute about mass is one specific example of the wider problem of how to understand theoretical terms, that is, terms newly introduced to formulate new theories that could apparently not be expressed in the language up till that moment. While the program of providing defnitions of the sort that Mach sought, or Bridgman's `operational definitions' for all such terms, or the early positivist attempt to understand the language of science through its relation to just `observation vocabulary' alone, have long since been definitively rejected, there is still a related idea. That is the persistently seductive philosophical notion that all theoretical terms have a meaning precisely determined by the roles they play in scientific discourse. That is the idea of implicit definition.

Hilbert is generally credited with making this idea precise. When it ran into heavy weather, many logical tours-de-force were tried to either defend or reformulate it in defensible form. After Hilbert, the idea journeyed through writings of Frank Ramsey, David Lewis, and Frank Jackson to morph recently into the `Canberra Plan' (Braddon-Mitchell and Nola 2009b). Here I will describe some of this history, and the argument that new developments in logic - most especially, Beth's theorem on definability ? should have spelled the end of these attempts to use the notion of `implicit definition' and rescued philosophy of science from the seduction of this mirage.

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1.2.1 Hilbert's introduction of `implicit definition'

Early in the century, Hilbert introduced the notion of "implicit definition" in connection with the meanings of terms in geometry. In commenting on his own axiomatization of Euclidean geometry, where such terms as "point" and "between" occur as primitive, Hilbert wrote "The axioms of [order] define the idea expressed by the word `between'." (Hilbert 1902: 5) And more generally, he took the axioms to be components of the definition of the terms that are primitives of the theory.

What are we to make of this? At first blush, Hilbert's proposal may sound very plausible. When we understand the axioms, and are able to deduce theorems, to solve problems posed concerning the system, how could we be said not to understand what we are doing? Yet the doing consists entirely in the systematic use of the terms introduced in the formulation of that theory ... . But the question started with the words "When we understand the axioms"; how could we be said to understand the axioms if we do not already understand the terms in them?

Hilbert's views about "implicit definition" were immediately subjected to criticism by Frege in correspondence. Hilbert did not agree to have the correspondence published, but Frege then presented his side in a review (Frege 1903), to express what must surely puzzle everyone about Hilbert's notion. Perhaps the word "between" will have its meaning fixed by some axioms, in which other terms occur that we already understand. And perhaps another of those terms could have its meaning fixed by those axioms if we take "between" as understood. But circularity seems to threaten if we suppose that all the terms occurring in the axioms have their meaning fixed in this manner.

I say "seems to threaten"; not everyone has seen this as a real threat. In fact, despite Frege's vigorous critique, the idea proved tremendously appealing and continued in a long life, though in various forms, often highly ingenious and always apparently responsive to philosophical puzzles.

1.2.2 Quine's `Implicit definition sustained'

In an article that we must mainly read as ironic, Quine [1964] purported to have rescued the idea of implicit definition, to his own dismay. His dismay with the idea is clearly expressed at the beginning:

What is exasperating about the doctrine is its facility, or cheapness, as a way of endowing statements with the security of analytic truths without ever having to show that they follow

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from definitions properly so called, definitions with eliminable definienda. (Quine 1964: 71)

So what is the argument that purports to support that doctrine? Assume that a certain empirical theory -for example, chemistry- is true, and can be formulated with predicates F1, ..., Fn and a single axiom A(F1, ..., Fn). Then it is satisfiable, and there will be a structurally similar statement in arithmetic A(K1, ..., Kn) that is an arithmetic truth. (Here Quine is drawing on the Loewenheim-Skolem theorem and some related results in metalogic; we'll return to those in a moment.) Now proceed as follows: have a language that includes arithmetic and also predicates G1, ..., Gn which are interpreted to mean the same as the original chemical predicates F1, ..., Fn. Do not introduce any axiom at all! As Quine shows, it is now possible to define the predicates F1, ..., Fn in terms of those new predicates Gi and defined predicates Ki so that

1. the statement A(F1, ..., Fn) will be true just because A(G1, ..., Gn) happens to be true (assumption!) and A(K1, ..., Kn) is an arithmetic truth, and

2. the statement A(F1, ..., Fn) is deducible from the arithmetic truth A(K1, ..., Kn)

How did Quine perform this leger-de-main? The assumption that the theory he is considering is true played a crucial part. On that assumption, the augmenting definitions do indeed introduce expressions co-extensive with what they purport to define; but only on that assumption (cf. Wilson 1965).3 So the theory is `mimicked' among the arithmetic truths; but the idea that the formula A(F1, ..., Fn) is sufficient to give meaning to the predicates F1, ..., Fn is spurious. Quine himself drops his ironic tone toward the end, and calls the manoever "farcical" and "hocus pocus".

1.2.3 Keeping something fixed: Winnie's rejection

While Quine kept something - the truth of the empirical theory in question ? fixed rather surreptitiously, Winnie [1967] proposes that we explicitly suppose the extension of `observational' terms to be fixed, and asks whether the remaining `theoretical' terms can then be said to be implicitly defined by the axioms of the theory. The conclusion he reaches is negative.

Winnie assumes that the domain of discourse is divided into two disjoint parts. The candidates for referents of these two sorts are thought of as the

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