Philosophy of Science, Problems of

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The scope of the philosophy of science is sufficiently broad to encompass, at one extreme, conceptual problems so intimately connected with science itself that their solution may as readily be regarded a contribution to science as to philosophy and, at the other extreme, problems of so general a philosophical bearing that their solution would as much be a contribution to metaphysics or epistemology as to philosophy of science proper. Similarly, the range of issues investigated by philosophers of science may be so narrow as to concern the explication of a single concept, considered of importance in a single branch of science, and so general as to be concerned with structural features invariant to all the branches of science, taken as a class. Accordingly, it is difficult to draw boundaries that neatly separate philosophy of science from philosophy, from science, or even from the history of science, broadly interpreted. But we can give some characterization of the main groups of problems if we think of science as concerned with providing descriptions of phenomena under which significant regularities emerge and with explaining these regularities. Problems thus arise in connection with terms, with laws, and with theories where a theory is understood as explaining a law and a law is understood as stating the regularities that appear in connection with descriptions of phenomena.


Ordinary language provides us the wherewithal to offer indefinitely rich descriptions of individual objects, and, as a matter of logical fact, no description, however rich, will exhaustively describe a given object, however simple. Science chooses a deliberately circumscribed vocabulary for describing objects, and scientists may be said to be concerned only with those objects described with the vocabulary of their science and with these only insofar as they are so describable. Historically, the terms first applied by scientists were continuous with their cognates in ordinary speech, just as science itself was continuous with common experience. But special usages quickly developed, and an important class of philosophical problems concerns the relation between scientific and ordinary language, as well as that between those terms selected for purposes of scientific description and other terms that, though applicable to all the same objects as the former, have no obvious scientific use. Scientists from Galileo Galilei to Arthur Eddington have sometimes tended to impugn as unreal those properties of things not covered by scientific description or at least have thought that the question of which are the real properties is an important one. Certainly, it would destroy the very concept of science to suppose it possible to account for all the distinctions between things under all the descriptions of them that are feasible, but there is no recipe for selecting the scientifically relevant predicates.

In practice, terms have been chosen when there seem to be interesting and systematic patterns of change in the properties picked out by these termsfor instance, between the distance a body travels and the time it takes to do so, between the temperature and the pressure of a gas, between the density of a fluid and the deviation from a norm of a light ray passing into it, and so forth. It has often been immensely difficult to set aside manifest and cherished differences among objects and the subtle language for expressing these in favor of the spare vocabulary of science under which such seemingly crucial distinctions are obliterated, as, for example, between celestial and terrestrial objects or between "noble" and base metals.

Not only do scientific terms cut across the distinctions of common sense, but they also permit distinctions not ordinarily made and allow comparisons more precise than ordinarily demandedfor example, between differential amounts and precisely determinable degrees. For the class of terms discussed here are those that may be said to apply or not to apply to a given object by means of an act of observation rendered precise through some device of mensurationfor example, that the distance traveled is n units along a scale, that the temperature of a gas is n degrees along another scale, that the density of a fluid is m grams per cubic centimeter. The last measurement, which involves reference to different scalesnamely, measures of mass and volumeis sometimes called a "derived" in contrast with a "fundamental" measurement, where only single scales are involved. But even when we speak of derived measurements, as with pressure (in terms of foot-pounds), velocity (in terms of feet per second), or stress (in terms of force per unit area), we remain within the domain of observation; the coincidence of a needle with a mark on a gauge, the angle of a balance, the appearance of a color, a bubble between lines, or a certain buzz, inform us that a given term is true or false with respect to whatever we are studying.

Philosophers may press for a further reduction of the observational language of a science to a favored idiomfor example, to a sense-datum languagebut within science observational vocabulary enjoys a certain ultimacy. There are many questions as to whether observational language, thus construed, is sufficient for the entire conduct of science, whether the whole language of science can be expressed in purely observational terms so that recourse need never be made to covert entities, hidden processes, or occult structures unamenable to direct observation and measurement. This issue cannot be fruitfully discussed until we come to the topic of theories, but it has been recognized that while observation has an essential role to play as the occasion for framing and the basis for testing scientific hypotheses, the no less important feature of measurement sets a limit on the program of thoroughgoing observationalism. For the algorithms, in connection with which it first makes scientific sense to assign numerical values and to apply scales, require use of the real number system, the class of whose values has the power of the continuum.

Hence, as Carl G. Hempel remarked, "A full definition of metrical terms by means of observables is not possible." Nevertheless, it has been through the efforts of reductionists to assimilate the entirety of scientific language to observation terms that other sorts of terms, having logically distinct roles within science, have been discovered, and a main task in philosophy of science has been to identify and determine the relation between terms occurring at different levels, and variously related to observation, within the idiom of developed scientific theories.


One cannot very readily treat the syntactical features of laws in isolation from their semantic properties or, for that matter, from pragmatic considerations. Syntax here concerns the formal conditions of "lawlikeness" for sentences, and semantics concerns the truth conditions for lawlike sentences, it being customary to define a law as a true lawlike sentence. But some philosophers will reject this definition since it might rule out any sentence as having the status of a law, inasmuch as laws are not, they feel, the sorts of sentences that it makes sense to regard as admitting truth-values in the normal way or even at all; for these a law would be a lawlike sentence which has a certain use.

It is commonly supposed that a universally quantified conditional sentence(x )(Fx Gx )is the simplest form with which a lawlike sentence may be expressed. The chief syntactical problems arise, however in connection with the nonlogical terms F and G. For an important class of cases these will be observational, so that it is in principle possible to determine whether a given instance is both F and G, and the law is generally based upon some known favorable instances, Yet there are cases in which the terms satisfy observational criteria, in which there are a large class of favorable instances and no known counterinstances, and still the appearance of these terms in a lawlike sentence L disqualifies L as a law even if it is true. Such terms are unduly restricted in scope, whereas it is thought that the terms suitable for laws should be unrestricted in scope. "All the hairs on my head are black" employs the restrictive term "the hairs on my head" and thus is disqualified as a law.

A criterion sometimes advanced for identifying restrictive terms as antecedents in possible laws is that if the requisite universal conditional supports a true counterfactual, it is a law, but if the counterfactual is false, as (with reference to a certain white hair) "If that hair were on my head, it would be black" is false, then the corresponding sentence is not a law, and the term is restricted. However, this criterion begs the question insofar as it seems that counterfactuals must be analyzed in terms of general laws; at any rate, the analysis of counterfactuals, as well as the basis for distinguishing true from false counterfactuals, remains to be given by philosophers. In what sense "the hairs on my head" is restrictive, whereas ravens in "All ravens are black" is not, is difficult to specify, though the former does refer to a specific object (my head) and it is believed that the terms in a law must not make such references. This restriction, however, makes Johannes Kepler's laws laws in name only and forestalls the possibility of any laws for the universe as a whole. And though Kepler's laws may be retained since they are derivable from laws that employ unrestricted and generally referential terms, the laws of the universe hardly could be thus derived; moreover, it could be argued that "All the hairs on my head are black" might be derivable from some general laws of hirsuteness, making use only of purely qualitative predicates. Thus, precise and rigorous criteria for lawlikeness are difficult to specify.

If the terms of a lawlike sentence L must be unrestricted, L cannot be known as true through induction by finite enumeration; since there must in principle always be uninspected instances under F, the law (x )(Fx Gx ) cannot be known true no matter how many known favorable instances there are. Of course, laws are not always (and perhaps not even often) inductive generalizations from large samplesGalileo's laws, for instance, were based upon few observations indeedand it has been maintained by anti-inductivists (chiefly Karl Popper and his followers) that observations function as tests rather than inductive bases for laws; in this view laws need not be generalizations from observation but only be in principle falsifiable on the basis of observation. Some lawlike sentences may be known false, at least to the extent that they admit of observational consequences, but often the antecedent of a lawlike sentence is sufficiently hedged with ceteris paribus riders, to which we may add indefinitely, that one need not surrender a law save as an act of will.

This suggests that the criteria for accepting a lawlike sentence as a law are more complex than either inductivists or their opponents have recognized, and an instrumentalist position may be taken, in accord with which laws are neither true nor false but serve as instruments in the facilitation of inference"inference-tickets," as Gilbert Ryle put it. In this view, as Stephen Toulmin pointed out, the question is not "'Is it true?' but 'When does it hold?'" Here laws are regarded not as sentences about the world but as rules for conducting ourselves in it, and semantic considerations thus yield to pragmatic ones in that there is surely some agreement that a criterion for accepting L as a law is that it should, in conjunction with information, furnish successful predictions. Whether, in addition, a successful law is true and, if so, in what sense it is true other than that it successfully enables predictions cannot be discussed independently of larger philosophical considerations.

Many laws in science are statistical in form, but the suggestion that a law may be truly scientific and yet affirm a merely probable connection among phenomena has been offensive to scientists and philosophers with antecedent commitments to determinism as a metaphysical fact or a scientific ideal. For these nothing less than deterministic (nonstatistical) laws are ultimately tolerable, so that statistical laws, while countenanced as interim makeshifts, are, ideally, to be replaced in every instance with deterministic ones. As a program, however, the projected reconstruction of statistical laws and the theories that contain them has encountered an impressive obstacle in the quantum theory of matter, upon which the whole of atomic physics is based, for the laws here are demonstrably irreducible to deterministic form.

To be sure, there is a logical possibility that quantum theory could be replaced in toto. But there is no wayfor instance through the discovery of hidden variablesin which its laws may be rendered deterministic, and since there is scant evidence for any alternative and the evidence for quantum theory is overwhelming, most members of the scientific community are reconciled to an obdurate indeterminism at the core of one of its most fundamental theories. If the quantum theory should be true, certain events are objectively probable, or indeterministic; that is, they are probable independently of the state of our knowledge or ignorance.

An epistemological sense of probability, connected with our concepts of induction and confirmation, is not incompatible with determinism; we may even speak of the probability of a deterministic law, meaning that relative to our evidence its degree of confirmation is equal to a number between 0 and 1. It is nonepistemological probability, according to which we could conceivably be certain that a given event were objectively probable, which is allegedly repugnant to determinism. It should be pointed out, however, that indeterministic laws may be deterministic in at least the sense that the values of certain probability variables are precisely determined by the values of other variables. At any rate, the extent of incompatibility between determinism and indeterministic laws and the precise explication of the two kinds of probability are topics of continuing philosophical investigation and controversy.

Laws are believed to play an important role in explanation as well as in prediction. It has been maintained that a necessary condition for explaining an event E consists in bringing E under the same general law with which it could have been predicted. Hempel regards the temporal position of the scientist vis-à-vis the event as the sole difference between explaining and predicting that event. This symmetry has been challenged (notably by Israel Scheffler), but we might still maintain Hempel's thesis by distinguishing among laws. Not every law used in prediction has explanatory force if we think of explanations as causal explanations, for causal laws do not exhaust the class of scientific laws, which also includes functional expressions of covariation among magnitudes, statistical laws, and so on, all of which are used in predicting. Even so, it has been questioned whether even causal explanation requires the use of causal laws, either in science or in history or the social sciences, where this controversy has been chiefly focused.

Be this as it may, the explanation of particular events has less importance in science proper than the explanation of regularities, and it is therefore the explanation of laws that characterizes scientific achievement in its most creative aspect. This brings us to theories, for it is commonly held that to explain a law L is to derive L from a theory T when T satisfies certain conditions.


Let us characterize a law all of whose nonlogical terms are observational as an empirical law. A theory may be regarded as a system of laws, some of which are empirical. Not every empirical law is part of a theory, nor are all the laws of a theory empirical, for some of a theory's laws employ theoretical terms, which are nonobservational. Theoretical terms, if they denote at all, refer to unobservable entities or processes, and it is with respect to changes at this covert level that one explains the observed regularities as covered by empirical laws. Thus one explains the regularities covered by the BoyleCharles law (all the terms of which are observational) in terms of the (unobservable) behavior of the gas molecules of which the gas is theoretically composed. The status of theoretical terms (and the theoretical entities they would designate if they designated anything) has been the subject of intense philosophical investigation. It is not mere unobservabilityJulius Caesar is at this point in time unobservable though his name is not a theoretical termbut unobservability in principle that characterizes these entities; it is unclear whether there would be any sense in speaking of observing, say, Psi-functions, electrons, fields, superegos, and the like. Moreover, the behavior of theoretical entities, supposing the theory to be true, is (as with certain fundamental particles) often so grossly disanalogous to the behavior of the entities they are invoked to explain that our ordinary framework of concepts fails to apply to them.

Yet theoretical terms seem deeply embedded in scientific language. Empiricist strategies of eliminating them by explicit definition in observational language or of tying them to observation by reduction sentences have failed, although there exist techniques by which they may be formally replaced with striking ease. William Craig demonstrated that any theory containing both theoretical and observational predicates may be replaced with another employing only observational ones but yielding, nevertheless, all the observational theorems (or empirical laws) of the original. Craig's result, however, has not been a victory for empiricism; the reasons for this are somewhat obscure, but it is due in part at least to the realization that theoretical terms play a role and have a meaning in terms of the total structure of the theory and therefore cannot be neatly extricated to leave anything to be called a "theory." Indeed, it often happens that rather than theoretical terms being defined in observational terms, observational terms are defined with reference to the theoretical vocabulary, so that one must, in effect, master the theory in order to make the relevant observations.

With the elaboration of a theory, however, the inferential route from observation to (predicted) observation becomes complex (there may be many intervening steps and intermediate computations) and far removed from the simple universal conditional used to represent a law. A theory, in Hempel's words, "may be likened to a complex spatial network [which] floats, as it were, above the plane of observation and is anchored to it by rules of interpretation." Theories, that is, impinge upon experience as wholes but not in all their parts, and the rules of interpretation, or correspondence, which permit them to be applied, are not part of the theory; indeed, the same formal theoretical network might, through different interpretations, have application to different domains of experience.

We may think of a theory as a formal system distinguishable, in principle, from its interpretation, regarding the former (in R. B. Braithwaite's terms) as a calculus and the latter as its model. In point of scientific history and practice, however, model and calculus emerge together. The distinction first began to be clear through the advent of non-Euclidean geometries and the consequent agitated question of which was physically descriptive, and geometry, perhaps because it has been almost paradigmatic of axiomatic systems, has served as a pattern, at least for analytical purposes, for the calculi of theories generally. Thus, philosophers think of theories as employing primitive and derived terms, primitive and derived sentences, satisfying explicit formation and transformation rules, and the like. But whether, apart from the purposes of philosophical representation, actual scientific theories exhibit axiomatized form and whether axiomatization is even a desideratum for scientific theory-formation are moot points.

At any rate, the framing of theories in the course of history has almost always involved some intuitive model on the scientist's part, the pattern of thought being (whether this is or is not the "logic of discovery" that N. R. Hanson suggested) this, that the regularities for which explanation is sought would hold as a matter of course if certain states of affairs (those postulated by the theory) held in fact. Whether the theoretical states do hold in fact is, of course, the immediate question, and it is through the obligation to provide an answer that the scientific imagination is disciplined. Without the formal means of deriving testable consequences from a theory, the theory would merely be ad hoc, and one wants more than the mere deduction of the laws that the theory was intended to explain. Indeed, it is by and large the ability of a theory to permit derivations far afield from its original domain that serves as a criterion for accepting a theory, for in addition to the obvious fruitfulness such a criterion emphasizes, such derivations permit an increasingly broad and diversified basis for testing the theory. The great theories in the development of scienceIsaac Newton's, Albert Einstein's, Paul Dirac'shave brought into a single comprehensive system great numbers of phenomena not previously known to have been connected.

It is impossible to say, of course, whether the whole of scientific knowledge might someday be embraced in a single unified theory, but piecemeal assimilation of one theory to another is constantly taking place, and the conceptual issues that arise through such reductions are of immense philosophical interest. The careful elucidation of the logic of scientific reductionof thermodynamics to mechanics, of wave and matrix mechanicsdraws attention to features that lie, far more obscurely, within the oldest philosophical problems and controversies: problems of emergence, of natural kinds, of free will and determinism, of body and mind, and so on. The treatment of these questions is often not so much philosophy of science proper as the philosophical interpretation of science, in which the philosophy of science serves as a technique of philosophical clarification, illuminating topics remote from the conceptual issues of science as such.

See also Braithwaite, Richard Bevan; Eddington, Arthur Stanley; Empiricism; Explanation; Force; Galileo Galilei; Hempel, Carl Gustav; Laws, Scientific; Matter; Popper, Karl Raimund; Quantum Mechanics; Ryle, Gilbert; Thought Experiments in Science.


The literature on the philosophy of science is immense and often technical. Many influential papers are anthologized in Philosophy of Science, edited by Arthur C. Danto and Sidney Morgenbesser (New York: Meridian, 1960), and in Readings in Philosophy of Science, edited by Herbert Feigl and May Brodbeck (New York: Appleton-Century-Crofts, 1953).

The following may be consulted as representative and excellent discussions. On preliminary definition of the field see R. B. Braithwaite, Scientific Explanation (Cambridge, U.K.: Cambridge University Press, 1953), pp. 19, and Israel Scheffler, The Anatomy of Inquiry (New York: Knopf, 1963), pp. 315. On the observational bases of science see Carl G. Hempel, Fundamentals of Concept Formation in Empirical Science (Chicago: University of Chicago Press, 1952), especially pp. 2050, and Scheffler, op. cit., pp. 127222. On measurement see Ernest Nagel "Measurement," in Danto and Morgenbesser, op. cit., pp. 121140. On laws see C. F. Presley, "Laws and Theories in the Physical Sciences," in Danto and Morgenbesser, op. cit., pp. 205215; Ernest Nagel, The Structure of Science (New York: Harcourt Brace, 1960), especially pp. 2978; and Stephen Toulmin, Philosophy of Science (London: Hutchinson, 1953), pp. 57104. On the nonlogical terms of laws see Carl G. Hempel and Paul Oppenheim, "The Logic of Explanation," in Feigl and Brodbeck, op. cit., pp. 319352. On laws and falsifiability see Karl Popper, The Logic of Scientific Discovery (New York: Hutchinson, 1959), pp. 2748. On explanation see Hempel and Oppenheim, op. cit. On the parity between explanation and prediction see Scheffler, op. cit., pp. 4388. On theories see Braithwaite, op. cit., pp. 50114; Nagel, The Structure of Science, op. cit., pp. 79152; Presley, op. cit., pp. 215225; Toulmin, op. cit., pp. 105139. On Craig's theorem see Carl G. Hempel, "The Theoretician's Dilemma," in Minnesota Studies in the Philosophy of Science, edited by Herbert Feigl et al., Vol. II (Minneapolis: University of Minnesota Press, 1958), and Scheffler, op. cit., pp. 193203. On the logic of discovery see N. R. Hanson, Patterns of Discovery (Cambridge, U.K.: Cambridge University Press, 1958), passim. On reduction of theories and related issues, see Nagel, The Structure of Science, pp. 336397.

Arthur C. Danto (1967)

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