The Received View

Unit 5

The Received View

1 Introduction

The received view is the conception of scientific theories developed by the logical positivists, which dominated the philosophy of science dur

ing the first half of the twentieth century. It was subsequently subjected to devastating critique, with the result that it is no longer viable. No one standpoint has risen as its successor; instead, a number of alternative approaches are now on

the table, distinguishable by what in the received view they reject.

The history sketched in the last paragraph is commonly accepted, and there is certainly much truth in it. But it does obscure two crucial points. First, the received view underwent considerable elaboration and alteration during its tenure, so that criticisms applicable to early

versions do not apply to later ones. Second, the title "the received view" is misleading in that it suggests one unified package of inseparable doc

trines. But at least many of those doctrines are separable and were, in fact, separated. Certain advocates of the view endorsed only some of them,

and not always the same ones. And a number of those doctrines survived long after the received

view's downfall; some, indeed, are accepted by many present-day philosophers of science.

The received view is represented primarily by the first reading by Rudolf Carnap, excerpted from his Philosophical Foundations of Physics. The second reading by Carl Hempel, excerpted

from his Aspects of Scientific Explanation, contains

Hempel' s classic presentation of the fundamentals of the received view of explanation including the deductive-nomological (D-N) and inductivestatistical (I-S) models. It also includes Hempel's discussion of the problem of ambiguity afflicting the I-S model and his requirement of maximal specificity that, he hoped, would solve that problem.

The third reading is Carnap's essay, "Empiricism, Semantics, and Ontology," which represents his attitude toward a variety of metaphysical dis

putes, including that between realists and antirealists in the philosophy of science (see Unit 9).

The views expressed in this essay are also the primary targets of one of the most famous critics of positivism, W. V. 0. Quine.1

The fourth and fifth readings contain two of the most famous responses to David Hume's problem

of induction.2 There is no standard positivistresponse to Hume's problem; they more typically just dismissed it as a pseudo-problem. But these two responses are arguably those most compat

ible with the overall positivist standpoint. The first is by Hans Reichenbach, who was himself a positivist (specifically, the founder of the Berlin Society; see §3 of the Part II commentary). The author of the second response, Peter Strawson, was

not a positivist. But he was one of the ordinarylanguage philosophers that shared the positivist's

inclination to characterize many philosophical disputes as resulting from confusions concerning language and resolvable by conceptual analysis.3

316 THE RECEIVED VIEW

His response to the problem of induction is indeed virtually identical to (although more detailed than) that of the English positivist, A. J. Ayer in Ayer's Language, Truth, and Logic.4

2 Empirical Laws and Confirmation

In keeping with their general logico-linguistic orientation (cf. §§4-8 of Part II commentary), the positivists understood a scientific theory to consist in a collection of sentences. These sentences exhibit the structure of an axiom system from which theorems can be derived, much as one might derive various theorems concerning, for example, the properties of a triangle from the axioms of Euclidean geometry. Unlike (pure) geometry, however, a scientific theory has empirical consequences; the scientist derives predictions from the axioms that can then be checked by observation.5

The view that a scientific theory is a linguistic entity, in particular a formalized axiom system, was a tenet of the received view that survived long after that view was abandoned. It is now called the syntactic account of scientific theories, and is contrasted with the semantic account, according to which a scientific theory is a set of models picked out by a linguistic formulation, rather than the linguistic formulation itself. The relative merits of these two approaches continue to be assessed.6

An axiom system requires a logical scaffolding in order to permit the derivation of theorems. The system employed by almost all versions of the received view7 is first-order predicate logic with identity, now standard among philosophers and logicians. In addition to the logical system, a vocabulary is required for the description of observations. The positivists assumed (without much argument) that there is an isolable observation-language that refers exclusively to objects and properties that can be "directly perceived by the senses."8 This vocabulary was essentially borrowed from everyday speech, and included such concepts as 'blue', 'hard', 'hot', and so on.9 At least some of the theorems derived from the axioms of a scientific theory must be couched exclusively in the observation vocabulary in order to be capable of empirical check, without which the system is not an empirical scientific theory at all.

As Carnap points out in reading 5.1, scientists do not usually employ such a restrictive conception of observation, applying the term also to the results of simple measurements, such as temperature, weight, length, time-duration, and so on. There is, Carnap insisted, no question which vocabulary is the right observation-vocabulary; there is a continuum here, and any observable/ unobservable line through it is bound to be "highly arbitrary."1° Carnap was happy to extend the observation vocabulary to terms that referred, not just to directly observable properties, but also to the quantitative results of simple measurements.

The theory does not, however, imply observation-sentences concerning specific dateable and locatable observations or measurements. The kinetic theory of gases, for example, does not on its own imply that the temperature, pressure, and volume of a particular gas measured at a specific time and place will have such-and-such values. Instead, the theory implies empirical laws, such as the combined gas law PY/T1 = P2V/T2

(where the variables stand for pressure, volume, and temperature, respectively). Although such laws are typically formulated as mathematical equations, most are taken by the positivists to be universal generalizations.1 1 A universal generalization is an expression of the form "All P are Q" (such as "all ravens are black" or "all copper expands when heated"). In predicate logic this is written as "(x)(Px :::i Qx)," which reads "Every x is such that, if x is a P, then x is a Q." ("Px" is called the antecedent and "Qx" the consequent.) Such a law will apply to a specific observation so long as the antecedent is satisfied; its being true will then require that its consequent is satisfied as well. If the gas law is correct, for example, then any particular isolated quantity of gas will satisfy that equation.

Empirical laws are tested by deriving predictions from them. We begin by observing cases where the antecedent is realized. If the consequent is realized as well, then the law is confirmed; if not, it is disconfirmed. Carnap suggests that there is an important difference between confirmation and disconfirmation. We can never decisively establish that a law is correct because the law concerns an infinite number of cases. The gas law, for example, concerns every quantity of gas, past, present, and future. It even concerns possible but not actual gases: although I am not boiling the water in my cup, if I had done so, thereby

THE RECEIVED VIEW 317

producing water vapor, the law would apply to that vapor as well. But the number of observations that we have made or ever will make is finite (and of only actual cases). So it is always possible that another observation that we do not in fact make would, if we had made it, disconfirm the law ( the next volume of gas would not satisfy the equation, the next raven would not be black, and so on). So empirical laws cannot be proven by observation to be correct, at least not if "proof " is meant in the mathematical sense that rules out the possibility of being incorrect. However, so long as the observation report itself is correct, it seems that an empirical law can be decisively refuted, since so much as one non-black raven will demonstrate that the law "all ravens are black" is incorrect. 12

This is called the hypothetico-deductive (H-D) account of confirmation: a prediction is deduced from an hypothesis (or theory or law) and the prediction is then checked; if the prediction is correct the hypothesis is confirmed, and if not it is disconfirmed.13

3 Explanation

Empirical laws are also used to explain observable phenomena. The positivists considered prediction and explanation to be structurally identical. In both cases, one presents an argument in which one logically deduces an observation statement from premises that include at least one law and a set of initial conditions. If the deduced observation statement has not already been empirically verified, then the conclusion of the argument is a prediction. (A prediction will typically concern an event that has not yet taken place; but it might concern a past or present event that is not known, at the time of the prediction, to have occurred.) But if the deduced observation statement has been empirically verified, then the argument provides an explanation of that observed phenomenon. For example, from the law "all copper expands when heated" we might predict that if we heat a particular copper bar it will expand. But, if the bar has already been heated and expanded, then that expansion can be explained by citing the law and the condition of its having been heated. The phenomenon to be explained (that the bar expands, in this case) is called the explanandum;

the law and the initial conditions that explain the phenomenon are called the explanans. This is called the deductive-nomological (D-N) model of explanation ('nomos' is Greek for 'law', and the inference is deductive). The structure of a D-N explanation is:

1) L1-L11

2) Ci-Ck

3) E

(the laws) ( the initial conditions) ( observation statement)

The D-N model was elaborated and defended primarily by Carl Hempel, 14 and constitutes the received-view account of explanation. The claim that an explanation-after-the-observed-fact could equally well have served as a prediction-beforethe-observed-fact is called the prediction/explanation symmetry thesis.

The laws we have considered so far have been universal, that is, they apply in all cases ( to all ravens, all quantities of gas, and so on). The positivists recognized, however, that not all laws are universal; some report only statistical regularities. Such laws will therefore be formulated, not as "All

P are Q," but "the probability of a P's being a Q is r." Statistical laws can still be used to generate predictions, so long as the value of r is fairly high. For example, if most weeds sprayed with a particular herbicide die, then I can reasonably expect that a particular weed that is sprayed will die. I cannot, however, be certain of it; given the premises, it is only likely that the conclusion will be true. The argument from which the prediction is derived is inductive, rather than deductive.

Hempel also thought that statistical laws can be used in explanations. Since such laws cannot guarantee the phenomenon's occurrence, the structure of such explanations will be inductive rather than deductive. He therefore called them inductive-statistical (I-S) explanations (because the argument structure is inductive and the law used is statistical). The structure of an 1-S explanation is:

1 p(QIP) = r. 2 i is aP. Therefore it is probable to degree q that: 3 i is a Q.

Here, "p(QIP)" reads "the probability of something's being a Q given that it is a P" (the


probability that a weed will die given that it is sprayed, for example). "i is a P" reads "this particular case is an instance of P" ("this is a sprayed weed"), and similarly with "i is a Q". In most cases, the probabilities r and q will be the same.

Hempel insisted that q - the probability of the explanandum given the explanans - must be high enough that the explanandum is expectable. As a result, the prediction/explanation symmetry thesis holds for I-S as well as D-N explanations: an I-S explanation would provide rational grounds for a prediction of the explanandum.

4 The Requirement of Maximal Specificity

Notwithstanding their similarities, there is a problem that afflicts I-S explanation from which D-N explanations are immune. Consider one ofHempel's examples: suppose that you are surprisedat the warm and sunny weather on November 27.You are offered the explanation that in fact theweather is very typically (95 percent of the time)warm and sunny in November. That explanationconforms to the I-S pattern as follows:

I The probability that it is warm and sunny, given that today is in November, is 0.95.

2 Today is in November. Therefore it is 0.95 probable that: 3 Today is warm and sunny.

But now suppose that it was cold and rainy yesterday, and that when the previous day was cold and rainy it is very rare ( only 20 percent of the time) that the next day is warm and sunny. This can also be presented as an I-S "explanation":

I The probability that it is not warm and sunny today, given that it was cold and rainy yesterday, is 0.8.

2 It was cold and rainy yesterday. Therefore, it is 0.8 probable that: 3 Today is not warm and sunny.

This of course could not really be an explanation, since it is in fact warm and sunny (so that the conclusion of this last argument is false). But Hempel nevertheless considered the fact that we can construct two I-S explanations with con-

tradictory explananda to be intolerable.15 For it means that, regardless of what the weather is like, we have available an I-S explanation that will explain it. Moreover, assuming the prediction/ explanation symmetry thesis (that every explanation is a potential prediction and vice versa) the result is incoherent: we could yesterday have predicted both that it would be warm and sunny today and that it would not be warm and sunny today. But to endorse that prediction would be to accept a contradiction. Hempel called this the problem of ambiguity in I-S explanation.

The problem is that the case we are interested in - today's weather, in the example - is, according to our background knowledge, a member of a great many classes: days that are in November, days that follow rainy and cold days, days that are in the last two months of the year, and so on. And the probability (again, according to our background knowledge) that a day that falls in one of these classes will be warm and sunny may well be different from the probability that a day that falls in another of these classes will be warm and sunny. So to which of these classes are we to refer in estimating the probability that it will be warm and sunny today?

Hempel's answer, in essence, is that we should pick the most specific reference class (according to our background knowledge). Return to the weather example. Should we pick days in November, or days that follow cold and rainy days? Answer: neither. We should pick days in November that also follow cold and rainy days. We then determine the probability that such days are warm and sunny. In general, if there is a more specific class whose probability of warm and sunny days is different from the more general class, then we must always pick that more specific class. Hempel called this requirement that we use the most specific class that generates a change in probability - the requirement of maximal specificity (RMS).

However, if the probability of the property ofinterest (being warm and sunny) is no different for the more specific reference class than it is for the more general class according to our background knowledge, then the more general class is acceptable for use in an I-S explanation. For example, suppose that the probability that a day that falls in November and that also follows a cold and rainy day is warm and sunny is 0.19. Suppose also that the probability that a day that


falls in November and that follows a cold and

rainy day and that is a Tuesday, and which is warm and sunny, is also 0.19. Then the more general class ( without specifying that the day is Tuesday) is acceptable for use as a premise in 1-S explanations.16

This problem does not arise for D-N explanations because it is impossible to have two deductively valid arguments, both with true premises, which imply contradictory conclusions. One of the contradictory conclusions must be false (P and -P cannot both be true). 17 So consider the argument

(whichever it is) that has the false conclusion. Since it is valid, it is impossible for the premises to be true and the conclusion false. So at least one of the premises of that argument is false. So it cannot be true that all the premises of both arguments are true. So there cannot be two D-N explanations, both with true explanans, one of which explains P and the other of which explains -P. (The same applies if the premises are drawn from our background knowledge, so long as that background knowledge is consistent.)

You might have noticed the frequent references to our background knowledge. Hempel

maintained that the RMS applies, not to the actual various classes of which the particular case

is a member, but to the classes of which our best scientific information at the time says the particular case is a member, and the probabilities that that information assigns to members of those classes having the relevant property. That is, we gather together all of the currently accepted scientific claims. Call the set of all those claims K. K will imply that the particular case in question - November 27, in our example - is a memberof a variety of different classes (is in November,follows a cold and rainy day, etc.). It will also(typically) assign a probability that members ofthose classes have the relevant property (being

warm and sunny). The RMS then applies tothose classes, with those probabilities.

That the RMS applies to K in this way is very significant. For it means that what counts as an acceptable 1-S explanation can change with changes in the body of accepted scientific claims. There is therefore no such thing as the right 1-S explanation, except relative to such a body;change in that body means a change in what theright explanation is. This is radically differentfrom Hempel's conception ofD-N explanations,

which are not relative in this way. 1-S explanations are therefore subjective in a way that D-N explanations are not, on Hempel's view.

5 Two Kinds of Probability

Notice that two probability statements occur in the 1-S explanation structure: first, within the statistical law that occurs in (at least) one of the premises; and second, the probability that the conclusion is true given that the premises are true. Although these are both formulated as statements of probability, the positivists (Carnap in particular) were careful to distinguish them. They understood the first to constitute a statistical probability: it indicates the proportion of Ps among the Qs ( e.g., the proportion of sprayed weeds that die). Sometimes this is the actual frequency

of the Ps among the Qs, as when the effectiveness of a particular drug is measured by the proportion of successful outcomes to all cases in which the drug is ingested. But, as we saw earlier, most scientific laws concern an infinite domain; the

probability is then taken to concern the statistical frequency "in the long run," or "in the limit," that is, as the proportion to which the observed proportions tend as observations continue to be made. A statistical-frequency interpretation of

the claim that the probability of heads on a coin toss is 0.5, for example, is not that any finite sequence will result in precisely one half heads (which is unlikely), but that, as the tosses increase, the proportion will more closely approximate that value.

The probability statement that concerns the support given to the conclusion by the premises was, however, taken to be a quasi-logical relation akin to that of deductive validity, indicating the degree of rational support that the premises provide for the conclusion. Carnap accordingly called it logical probability. Unlike statistical probability, logical probability is always a relation between sentences (premises and conclusion, or evidence and hypothesis). Just as it makes no sense to say that a conclusion of a deductive argu

ment is valid (since validity is a relation between the premises and the conclusion rather than just a property of the conclusion on its own), it makes no sense to say that the logical probability, for example, of a sentence is 0.8, since 0.8 is


a measure of the support that other sentences provide for that sentence rather than a property of that sentence on its own.

Statements of logical probability are statements in the meta-language since they report a relation between sentences, whereas statistical probabilities are object-language statements since they concern frequencies with which certain properties are realized. Statistical probabilities, as reported in statistical laws, are synthetic a posteriori claims about the world, but Carnap thought that statements of logical probability are analytic a priori claims (just as he, and the positivists generally, understood logical - and mathematical - statements to be analytic a priori claims).

Logical probabilities play three roles in the received view. First, they describe the relation between the explanans and explanandum in an I-S explanation. Second, they indicate the degreeto which a prediction from a statistical lawshould be expected to be successful given thatthe law is true (that a sprayed weed will die, forexample). Third, they measure the extent to whicha collection of successful predictions confirm a theory. Notice that the last applies even when the lawso confirmed is universal; many observations ofblack ravens confer a certain logical probabilityon the universal generalization that all ravensare black. Not surprisingly, Carnap spent muchof his time later in life attempting to formulate asystem of logical probability (or "inductive logic,"or "logic of confirmation").

6 Explaining Laws and the Theoretical Language

Scientific theories do not only explain particular empirical facts; sometimes they provide explanations for the laws themselves. For example, I might derive the law of expansion of copper when heated from a law of thermal expansion for metals in general. In accordance with the account of explanation sketched above, the laws are themselves deduced from other laws (which might, in turn, be themselves deduced from further laws). When the explained law is universal, then the explanation is counted as an instance of the D-N model; when it is a statistical law, the explanation is what Hempel called a "D-S"

explanation because it is a deductive derivation of the statistical law from other laws (which might themselves be statistical laws).

One law might explain another if the explaining law is more general (as in the derivation of the law of expansion of copper from a law of thermal expansion for metals). However, no matter how general the law is, it will always employ terms from the observation vocabulary. This is because the generalization will always lead to broader categories of observable properties (e.g., from copper to metal to solid body). But the positivists were well aware of the fact that scientific theories very often include laws that refer to putative unobservable, theoretical entities and processes ( e.g., electrons, electromagnetic waves, tectonic plates). They also recognized that these theoretical laws were routinely taken to be confirmed in scientific practice and that they played an integral role in the prediction and explanation of observable phenomena.

The positivists therefore understood the language of a scientific theory to contain, in addition to the observation language, a theoretical language in which such laws were couched. The received view, therefore, divides the scientific language into three parts: the language of the underlying system of logic (which is also taken to include any mathematics employed in the theory's expression); the observation language, used to report specific observations as well as to formulate empirical generalizations; and the theoretical language, used to formulate theoretical laws.

The scientific theory was understood by the positivists to be composed of three levels. The top level consists in the theoretical laws which are used to derive, and thereby both predict and explain, empirical generalizations, which generalizations constitute the second level. Those generalizations are then used to derive, and thereby both predict and explain, specific empirical facts, which constitute the third level. And in the other direction, those specific empirical facts confirm the empirical generalizations, and the (confirmed) generalizations in tum confirm the theoretical laws from which they derive.18

The theoretical entities and laws, and the theoretical vocabulary used to refer to them, however, presented the positivists with a dilemma. On the one hand, given their verificationist criterion


of cognitive significance, the apparent remoteness of such putatively unobservable entities from sensory experience might seem to require their banishment from science.19 But on the other hand, the positivists well knew that, for whatever reason, appeal to such entities and laws has proven to be utterly indispensable in the development of the tremendously empirically successful theories that so impressed them. The development of the different versions of the received view was primarily an attempt to solve the "problem of theoretical terms" that this dilemma generates.

7 Theoretical Terms and

Correspondence Rules

Theoretical laws contain only terms from the theoretical vocabulary, whereas empirical laws contain only those from the observation vocabulary. But it is not possible to deduce a sentence which employs one vocabulary from other sentences that exclusively employ another vocabulary. There must in addition be "mixed" sentences that contain terms from both. Such sentences, called correspondence rules by Carnap,20 had two roles. First, they provided the link between the theoretical and empirical strata of the theory, allowing prediction and explanation in the downward direction and confirmation in the upward. Second, they were to provide the solution to the problem of theoretical terms.

An early, straightforward attempt at characterizing correspondence rules was to simply define the theoretical terms directly by equating them with expressions in the observation vocabulary. Such definitions proceeded by identifying the theoretical property with the operation by which that property was measured; these were called operational definitions. For example, an operational definition of "has a mass of 3 grams" might be "if placed on a spring scale, the scale's pointer will coincide with the number 3."21

But operationism (or operationalism), as this approach is called, faces a number of serious problems. First, according to it, no one property can be measured by more than one operation, since the operation itself defines the property. For example, mass-as-measured-by-a-balancescale must be an entirely different property

than mass-as-measured-by-a-spring-scale, which is different than mass-as-measured-by-waterdisplacement. For the same reason, it is not possible to discover a new method of measuring a property ( the new method implies a new property, which is therefore distinct from the old). Such a proliferation of different concepts of mass is both unintuitive and contrary to scientific practice, in which these different operations are routinely taken to measure the same property. Second, while some theoretical properties ( e.g., mass, temperature, and length) might seem sufficiently close to the empirical ground, as it were, to be characterizable in terms of the methods by which they are measured, the prospects for definitions for 'electron', 'tectonic plate', and other concepts more deeply embedded within the theoretical system seem dim indeed.

Finally, a certain logical problem afflicts operational definitions. Such definitions treat a theoretical property as a disposition (viz. to produce such-and-such a result under such-andsuch measurement conditions). But consider an everyday disposition, such as the fragility of a pane of glass. An obvious definition of this disposition is "Something is fragile if and only if it breaks when it is struck." In predicate logic this is written (x)(Fx if and only if (if Sx then Bx)); that is, "for all x, x is fragile if and only if, if it is struck, then it breaks." The problem is that the right-hand side of the biconditional (viz. "if it is struck, then it breaks") was understood to be a material conditional, and a material conditional is true if its antecedent is false. 22 So it follows from this definition of fragility (and the definition of the material conditional) that so long as an object is never struck - even if it is made of lead - it will be fragile; obviously the wrong result.

One possible response is to employ subjunctive conditionals, like "were it to be struck, then it would break," rather than the material conditional "if it is struck, then it will break." The logic of such conditionals was, however, far too underdeveloped when the received view was formulated to rely on them. (In fact, controversy concerning their correct interpretation continues to this day.)

Instead, Carnap formulated an alternative, called reduction sentences. A reduction sentence for fragility is: "If x is struck, then it is fragile if and only if it breaks." When the antecedent is true


this tells us that it will break if and only if it is fragile, which is intuitively correct. But if the antecedent is false then the conditional will count as true regardless of whether the consequent is true or false ( again, a material conditional is true if the antecedent is false). So the fact that it did not break will not imply that it is fragile. But neither will it imply that it is not fragile; neither follows from the fact that the antecedent is false, since that leaves open whether the consequent is true or false.

The upshot is that a reduction sentence provides a definition for a dispositional term whenever the operation is performed or the test-condition realized (e.g., when the glass is struck). But it is silent concerning whether the term applies when the test-condition is not realized (e.g., when the glass is not struck). It is therefore what Carnap called a partial definition, since it provides necessa1y, but not sufficient, conditions for application of the concept. Other reduction sentences may be added to minimize this; we might add, for example, "If x is dropped onto a ceramic floor, then it is fragile if and only if it breaks." But so long as there are other conditions under which that disposition would be realized, no collection of such definitions will be complete.

In addition to avoiding the logical problem, reduction sentences avoid the problem of multiple methods of measurement facing operationism since there is no reason why we cannot think of the different reduction sentences as each specifying part (but never all) of the definition of the same property. They apply under different test-conditions, after all, and so do not compete with each other. But reduction sentences are still unlikely to be applicable to deeply theoretical terms like "electron", since it is hard to imagine what test-condition we could use in the formulation of either an operational definition or a reduction sentence.

Carnap's eventual solution was to suggest that most theoretical terms receive empirical content indirectly, through the network of theoretical laws in which they are embedded and, finally, by their relation to the empirical laws and facts derived from that network. To understand this approach, first consider the system of axioms-plusderived-theorems that constitutes a scientific theory, not as a collection of meaningful sentences,

but just as a formal, uninterpreted calculus or game of symbol-manipulations whose rules dictate only that certain sequences of symbols can be written down when other symbol-sequences are given. The question is then how to provide the expressions in the system with empirical significance so that the result is a scientific the01y rather than just a formal calculus.

The obvious way to do this is to first assign a meaning to the basic or "primitive" theoretical terms (that is, those that are not defined by other terms) that occur in the axioms or basic laws of the theory. Then the meaning of defined expressions (and of the theorems that contain those defined expressions) will flow from those initial assignments, much as "triangle" in the system of Euclidean geometry might earn its meaning by definition from the already-meaningful terms "line segment" and "point" (and "three"). Carnap referred to this as the "top-down" approach, since meaning is assigned to the primitive terms at the top and then flows downward to the defined expressions. Both operationism and the method of reduction sentences are top-down approaches since, according to them, primitive theoretical terms are directly assigned an empirical meaning by the operational definition or reduction sentence.

Carnap, however, proposed another "bottomup" method. We can set out the entire formal system, using the primitive theoretical terms ( and the axioms containing those primitive terms) to define other expressions (and derive theorems containing those defined expressions). So far, no empirical meaning has been assigned to any expression. Then we could assign a meaning, not to the primitive terms at the top of the system, but instead to the defined expressions at the bottom, by correspondence rules (which will typically be reduction sentences). For example, measurements made by a thermometer of the temperature of a gas would be used to assign a meaning, not to a primitive term lil<e "molecule", but rather to the expression "mean molecular kinetic energy" (which is obviously constructed using the primitive term "molecule").

In this bottom-up approach, no empirical significance is assigned directly to the primitive theoretical terms; "molecule", for example, is not directly provided with any empirical meaning


as it is in the top-down approach. These terms acquire the only meaning they have in virtue of their role in generating the defined expressions, which are directly provided with empirical meaning. Empirical significance is therefore drawn up from the defined expressions at the bottom to the primitive theoretical terms at the top by a kind of semantic analogue to osmosis.

A complete interpretation of an expression would provide necessary and sufficient conditions for the application of that expression. But in the bottom-up approach the theoretical terms only inherit their empirical meaning in the indirect fashion described above rather than directly by the specification of necessary and sufficient conditions. Their interpretation is therefore only partial. This partial interpretation solution to the problem of theoretical terms is now standardly associated with the received view.23

A consequence of this bottom-up approach is that the theoretical terms do not enjoy empirical significance in isolation from one another, but only in virtue of their place in the system. So the only legitimate answer to questions concerning, for example, the meaning of "electron" or "molecule" is to present the entire theoretical system itself to which those terms belong. Appeal to pictures, models, or analogies (such as picturing molecules impacting the walls of the container as like little billiard balls hitting the walls of a pool table) is therefore at best unnecessary for a complete understanding of the meaning of theoretical terms, and, at worst, misleading. This is because such surplus content would be independent of the theoretical term's role in generating empirical consequences. But such surplus content would then contribute no empirical significance; it would therefore introduce transcendent metaphysics into the heart of theoretical science.

Another consequence is that the only way

to distinguish a cognitively significant theoretical term from a cognitively insignificant metaphysical expression is to notice that removal of the theoretical term from the system in which it is embedded will result in an impoverishment of the empirical consequences that are derivable from that system, whereas removal of a metaphysical expression will have no such impact (the latter came to be called isolated sentences). The partial

interpretation approach therefore introduces a kind of semantic ho/ism into the received view, at least for the theoretical terms (although not for the observation terms).

8 Ancillary Doctrines

There are three other doctrines widely advocated among the positivists to consider. These are not obvious consequences of the central tenets of the received view as presented above. They are, however, strongly associated with it, and subsequent criticisms raised against them have inevitably been seen to attack the received view itself.

The first is the doctrine of the unity of science. This is the view that, notwithstanding the variety of scientific disciplines ranging from physics to biology to economics, there is only one scientific method employed (with varying success, perhaps) across all such disciplines. There is, moreover, ultimately one domain of objects and properties that are the concern of all of the sciences. That domain is most directly characterized within physics; the subject matter of the other sciences, if legitimate, must in the end be reducible to (translatable into) the language of physics. Call the first claim concerning scientific method the doctrine of synchronic methodological reductionism and the second concerning object-domains synchronic ontological reductionism ("synchronic" means "at the same time").

The positivists also typically advocated diachronic reductionism of both the ontological and methodological varieties ("diachronic" means "over time"). According to diachronic ontological reductionism, when successor theories replaced reasonably successful earlier ones, this did not amount to wholesale replacement of one theory, then considered false, by an entirely different theory. Instead, the positivists saw it in one of

two ways. First, the old theory might be extended into new domains to which it was not originally applied, as when classical particle mechanics was extended to rigid-body mechanics. Second, the older theory with its limited domain might be absorbed into another theory with a wider domain, as when Kepler's astronomical theory was derived from Newton's physics. In such a case, the original theory is understood to be true


of the objects that are its concern, but only as a part of the greater story told by the reducing theory.

Diachronic ontological reductionism implies that the sequence of theories over time is both progressive and cumulative. It is progressive because successor theories constitute improvements on their predecessors (either because the successors expand the domain of their predecessors' application or because they are accurate in domains in which their predecessors were not). And it is cumulative because the successor theories endorse the previous theories as true within their domains but contribute further truths (in the domains outside the scope of the predecessor theory); the truth-content of the succession of theories is therefore steadily increasing.

According to diachronic methodological reductionism, there is no temporal change in scientific methodology; the scientific method that is employed by all scientists at a time is the same as that employed by all scientists across time, and therefore across theory-change. Of course certain methods might be improved, new instruments developed, and so on. But the logic of science itself - which underlies the explanation, prediction, confirmation, and interpretation relations between theory and observation - remains the same.

Finally, notice that the relations between theory and observation just mentioned do not include a procedure for the discovery of scientific theories. A logic of discovery would provide - ideally in algorithmic or recipe form a procedure that would tal<e a scientist from observations to the correct ( or best) theory that those observations support. Having distinguished the projectof formulating a method of theory-development(the context of discovery) from the method forthe evaluation of theories ( the context of justification), the positivists decided that the former isat best a subject for psychology, and abandonedit. The context of justification, they claimed, isthe only context to which the philosopher ofscience is responsible.24

This ends our presentation of the various doctrines that make up the received view. Some, such as the partial interpretation account, were rejected fairly quickly; some, such as the D-N model of explanation, are no longer accepted, but survived much longer than positivism itself;

and some, such as the syntactic approach and the attempt to develop a formal confirmation theory, still constitute viable positions in the philosophy of science. The influence of the received view, therefore, continues to be felt long after its demise.

9 "Empiricism, Semantics, and Ontology''

In the essay by this title, Carnap introduces a distinction between internal and external questions of existence. They are internal and external with respect to what he called a linguistic framework. Linguistic frameworks introduce the resources needed for discourse concerning certain kinds of entities, such as numbers, material objects, properties, and events. This requires both introducing a general term for the class of entity (e.g., "number") and allowing a particular class of expressions (e.g., numerals) to be substituted by variables that can be bound by quantifiers. Start, for example, with the use of numerals as adjectives, as in "There are three cats." The objects with which this sentence is concerned are obviously only the cats. But now allow substitution of the number by a variable n that comes within the scope of the existential quantifier which asserts that there is at least one n. The result in predicate logic is "(3n)(There are n cats and n equals three)" which says that there is a number that is the number of cats, and that number is three. This sentence now speaks, not only of the cats, but of their number (which is identified with the number three). Numbers are thereby introduced as objects referred to by the numerals. Such frameworks also introduce rules that govern the assertion of the new sentences that they make available. To be proficient with the framework of numbers, for example, is not only to be able to form sentences that concern numbers; it is also to be able to determine the answers to such questions as what the value of two plus three is, or whether there is an even prime.

When a question of existence is answered in accordance with the rules of the framework, the question is internal to the framework. When doing your math homework, you know that there is an even prime, that three is the square root of nine, and so on. Notice that, since primes are numbers, if there is an even prime, then it trivially follows


that there are numbers. But when philosophers question the existence of numbers, they are not so easily satisfied. They intend to ask a question, not within mathematics and answerable according to its rules, but about mathematics. They want to know whether there really are numbers, whether mathematics accurately represents some aspect of reality.

These are what Carnap called external questions of existence. And he simply denied that such questions are meaningful.25 There is no way to answer them since the only procedures for their evaluation are specified by the framework, and the external questioner dismisses the answers provided by that evaluation as beside the point of the question.

However, that is not to say that there are no legitimate external questions whatsoever. We can consider the advantages and disadvantages of introducing a particular framework of entities, and propose that we do so (or not). But these are pragmatic questions concerning how the language is best engineered; they are not, Carnap insisted, questions of what to believe there is.

The distinction between internal and external questions also applies to questions of existence concerning theoretical entities within a system of science. The system provides rules for the evaluation of a particular existential question - for example, whether any molecules remain in an evacuated chamber - the answer to which is a matter of empirical investigation conducted by the scientist. But the philosopher wants to know whether there really are molecules and is not satisfied with the answer that the theory and its rules of application might deliver. After reflection (and argument) the scientific realist decides that indeed there are, whereas the anti-realist says that there are not ( or else that we cannot tell whether there are or not).

Carnap, again, denies that such external questions make sense. There is the internal answer provided by empirical investigation conducted according to the rules of the scientific framework, and that is it. To demand more than this is to assume that there is more to the meaning of the term "molecule" than is provided by that term's role within the theory and its empirical consequences. But, as we saw above, Carnap insisted that such a role exhausts the term's legitimate interpretation.

It is not easy to say what Carnap's position on the realism debate in the philosophy of science

amounts to. Traditionally, he is seen as an antirealist. This is sometimes because he (and the positivists generally) are taken to be reductionists,26

according to which assertions about theoretical entities can be completely translated into assertions about observable entities. This is not an accurate portrayal, however; Carnap explicitly rejected reductionism in this sense very early on. Certainly the method of partial interpretation is not a form of such reductionism. 27 So Carnap would have endorsed the claim that electrons exist, for example, since this follows from the application of the rules of the theory in conjunction with experimental results. And he would deny that the sentence "electrons exist" is translatable into a sentence about observables. This would suggest that he was a realist.

However, one might view the partial interpretation account as embodying anti-realism nevertheless. After all, the only significance that the theoretical terms receive is that which they absorb from their role in implying observation sentences; they, and the laws that embed them, are otherwise purely formal. This has struck many, including some positivists, to amount to a version of instrumentalism (which is a version of antirealism) according to which theoretical terms and laws are really just instruments, or tools, mediating the prediction of observable phenomena. Such tools are useful or not for their purpose, but they are no more true (or false) than is a hammer.

Whether Carnap's "plague on both your houses" stance collapses into a form of either realism or anti-realism remains the subject of debate. Interestingly Carnap's stance foreshadows a similarly dismissive response to the realism debate formulated by Arthur Fine, which is otherwise very far removed from the received view. Fine calls his position the Natural Ontological Attitude (reading 9.4). A parallel debate has ensued concerning whether the Natural Ontological Attitude amounts to a form of either realism or anti-realism. We will discuss Fine's view, and that debate, in § 11 of the Unit 9 commentary.

10 Reichenbach's Pragmatic Vindication

Hume presented a dilemma: induction can either be justified by "demonstrative reasoning" (what


might be now called deductive proof) or by "moral reasoning" (reasoning on the basis of experience). But demonstrative reasoning guarantees its conclusions on the assumption that the premises are true, whereas inductive arguments do not. And to appeal to past and present experience of successful inductions in order to warrant future uses of induction is itself an inductive argument and so begs the question. There is therefore no justification for our belief that induction is a reliable method for the generation of a reasonably high proportion of true beliefs.

Hans Reichenbach agreed. But he claimed that we can nevertheless justify (or "vindicate") our use of induction, because we can argue that if there is any reliable method of predicting the future on the basis of the past, induction is it. The goal of an inductive argument, he suggested, is that of determining the limit of the frequency of "favorable" cases to all cases ( e.g., of coin tosses that are heads, of ravens that are black, and so on; see §5 above). But if there is such a limit, then setting our prediction of that frequency to the frequency observed so far is guaranteed to eventually approach that limit. This is a consequence of the definition of a limit: to say that the limit of the frequency is f just is to say that, for any number £ (no matter how small), there is a number n of observed cases where the observed frequency will thereafter not deviate from f by more than £. So, if there is a limit of the frequency, it just is that limit to which the observed proportion tends as those observations continue to be made.

There is, however, no guarantee that there is a limit. There is also no guarantee even that the number of observed cases is of a sufficient size as to provide us with any reason to believe that we are within any value £ of the limit, even assuming that one exists. These are, Reichenbach conceded, consequences of Hume's dilemma that must be accepted. However, it remains the case, he claimed, that if there is any method that will work, the above argument demonstrates that induction will do so. So, Reichenbach claims, it remains rational to use it, even in the face of Humean skepticism.

Consider an analogy. Suppose you are lost in the woods, and you discover a path. You have no idea, no justification for believing, that the path leads out of the woods at all. But you know this

much: if there is any way out, then this path is it. In that case it remains rational for you to follow it, even in the face of no assurance of success. That is, Reichenbach suggests, our situation with respect to induction: we are not justified in believing that it will be successful, but we are nevertheless justified in using it to determine our expectations for the future.

There are various technical objections to Reichenbach's vindication, but the most forceful objection is simply that it provides far too little comfort. It might vindicate the act of our using induction, but it does not vindicate our belief in its reliability. It provides no support, for example, for the belief that the sun will very likely rise again tomorrow. One might reasonably consider Hume's critique to be directed, not at the act of performing an induction, but at the rationality of the belief that induction is reliable; and Reichenbach's vindication provides no protection against the critique so understood.

11 Strawson's "Ordinary Language" Response

Reichenbach took Hume's challenge seriously. We cannot, he conceded, provide such a justification as would meet that challenge, although we can nevertheless vindicate our use of induction. Peter Strawson, on the other hand, rejected the challenge itself, insisting that the question whether induction is rational is simply incoherent.

If someone routinely makes financial decisions on the basis of their horoscope, then we might challenge them, pointing out that the track record of those decisions has been disastrous. In so evaluating their behavior we implicitly (or explicitly) appeal to certain standards of rational belief, emphasizing accuracy in observation, adequacy of sample size, representativeness of the sample, avoidance of bias, and so on. To do all this is to appeal to the standards by which inductive reasoning is evaluated.

But if we are then asked to evaluate the inductive standard itself, Strawson asks, to what further standard are we to appeal? To demand that it meet the deductive standard, as Hume rightly noted, is to apply a standard applicable only to a distinct form of reasoning. But there is no other higher standard; induction just is the standard of


rationality for beliefs formulated in response to

experience. To ask whether such beliefs are

rational is to judge them by the standards im

posed by the inductive method. It is incoherent,

therefore, to demand a justification for induction, when induction determines what it means to call a belief that is based on experience justified, just

as it is incoherent to ask whether the laws of the land themselves are legal.

So attempting to respond to Hume's challenge is rather like attempting to answer the question

what is north of the North Pole. Properly understood, the problem lies, not in the unavailability

of satisfactory answers to a legitimate question,

but in the posing of an incoherent question as a coherent one that demands an answer.

A problem with Strawson's solution is ana

logous to the problem with Reichenbach's vindication discussed above. Strawson understands

Hume's challenge to concern the rationality of

induction. But one might instead understand it

to concern our belief in induction's reliability. We believe that the use of induction will generate a

reasonably high proportion of true beliefs, and we

want to know if that belief is correct. To be told that this belief is rational simply in virtue of the

definition of "rational" seems to miss the point of the question. Suppose that we concede to Strawson

that induction is rational. What we want to know

is whether our judgment of its rationality is res

ponsive to the reliability of the inductive method. And that still seems a coherent question to ask.

Notes

1 See reading 6.3 and §§4-6 of Unit 6 commentary. 2 See reading 3.7 and §5 of Unit 3 commentary. 3 See note 14 of Part II introduction. 4 Ayer 1946, Language, Truth and Logic, 2nd edn.,

London: Gollancz, p. 50. 5 The positivists distinguished pure geometry,

which is purely mathematical, from applied or physical geometry, which purports to characterize the structure of physical space. The former has no empirical consequences of its own, while the latter amounts to an empirical scientific theory.

6 An advantage of the semantic account is that it allows the same theory to be expressed in two different languages, and also permits two different axiomatizations to count as expressions of the

same theory.

7 The exception is Carnap's Ramsey-sentence approach, which employs second-order logic. See note 23 below.

8 Reading 5.1, p. 336. In a different work, Carnap characterized a property P as observable when a person "is able under suitable circumstances to come to a decision with the help of a few observations about a full sentence, say '-P(b)', i.e. to a confirmation of either 'P(b )' or 'P(b )' of such a high degree that he will either accept or reject 'P(b)' " (Carnap 1936-37, p. 455). We will discuss challenges to the observable/unobservable distinction in Unit 6.

9 The received view's observation vocabulary is often thought to be phenomenalistic, that is, as referring to sense-data concerning, for example, patches and colors in the visual field of the scientist's experience (rather than to features of the scientist's physical environment). But a phenomenalistic language was rejected early on as the observation basis in favor of a physicalistic language concerning ordinary physical objects, and was never the basis of the more sophisticated versions of the received view. See §8 of Part II introduction.

10 Reading 5.1, p. 337. 11 The rest are statistical generalizations; see below. 12 The claim that such decisive refutations of a the-

ory are possible has, we will see, been challenged. See readings 4.13 and 6.3, as well as Unit 7 commentary, especially §§2-3.

13 Notice that the law so confirmed need not be an empirical law; it could concern unobservable theoretical entities. See §6 below.

14 See §3 of Part II introduction. 15 "Explananda" is the plural of "explanandum." 16 It is significant that Hempel did not insist that we

use only the most general such class. In Unit 8 we will see Wesley Salmon suggest that Hempel's failing to do so is responsible for the counterexamples to Hempel's I-S model that are called irrelevance cases.

17 '-P' means "It is not the case that P." 18 As a result the positivist's account is sometimes

called the "layer-cake" model of scientific theories. 19 See §5 of Part II introduction. 20 They were also called "operational definitions,"

"coordinating definitions," "dictionaries," and "interpretation rules."

21 The philosopher most closely associated with operationism is Percy Bridgman. See Bridgman 1927.

22 Consider the sentence "if you get an A, you will pass." This of course rules out your getting an A and failing (antecedent true, consequent false). But it does not rule out your not getting an A and


passing, since you could get a B, also a passing grade (antecedent false, consequent true). Nor does it rule out your not getting an A and failing, since you could get an F (antecedent false, consequent false). So logicians count the sentence true under those cases, which are the two cases when the antecedent is false.

23 It was not Carnap's last proposal, however. Very soon after formulating the partial interpretation approach, he abandoned it in favor of the "Ramseysentence" method (in fact he discusses both in the book from which selection 5.1 is excerpted). But that method faces serious difficulties, and, at any rate is not now associated with the standard version of the received view.

24 The distinction between the contexts of discovery and justification originates in Reichenbach, Hans, 1951, The Rise of Scientific Philosophy, Berkeley/ Los Angeles: University of California Press.

25 It is not clear precisely how Carnap's charge of the meaninglessness of external questions was to be understood. Many assume it to constitute an expression of verificationism ( for example Stroud, B. 1984, Significance of Philosophical Scepticism,

Oxford: Oxford University Press). Others see in itan application of the analytic/synthetic distinction (Quine, "On Carnap's Views on Ontology" inPhilosophical Studies 2, repr. in Quine, 1966,Ways of Paradox, Cambridge, MA: Harvard University Press, 2nd enlarged edn. 1976, pp. 203-11).See Alspector-Kelly, M., 2001, "On Quine onCarnap on Ontology", Philosophical Studies,

Vol. 102 (January), pp. 93-122, and 2002,"Stroud's Carnap," Philosophy and Phenomeno

logical Research, Vol. 64, No. 2 (March), pp. 276-302, for discussion.

26 The sense of this multiply abused term here is not that of ontological or methodological reductionism (either synchronic or diachronic), nor that in "reduction sentence," and should not be confused with those uses.

27 Ironically, the reduction sentences discussed in §7 above also do not provide reductions in the present sense of the term, since reduction sentencesonly constitute partial definitions.

Suggestions for Further Reading

Carnap 1956 contains a detailed statement of the received view and the role of the empiricist criterion of significance within it. His 1936-7 presents his earlier development of reduction sentences. The classic statement of operationism is Percy Bridgman 1927. Nagel 1979 is another detailed presentation of the received view that was highly influential in the United States. The D-N model of explanation was originally presented in Hempel and Oppenheim 1948. A fuller treatment of the D-N model, along with introduction of the D-S and I-S models, is given in Hempel 1965b (from which reading 5.2 is excerpted). An excellent, detailed discussion of the origins, development, critique, and downfall of the received view is Suppe 1977.

Bridgman, P. W., 1927, Thelogic ofModernPhysics. New York: Macmillan.

Carnap, R., 1936-7, "Testability and Meaning," in Philosophy of Science 3( 4): 419-71 and Philosophy of

Science 4(1): 1-40. Carnap, R., 1956, "The Methodological Character of

Theoretical Concepts," in Herbert Feig! and Michael Scriven (eds.), The Foundations of Science and the

Concepts of Science and Psychology. Minnesota: University of Minneapolis Press, pp. 38-76.

Hempel, C., 1965a, Aspects of Scientific Explanation

and Other Essays in the Philosophy of Science. New York: Free Press.

Hempel, C., 1965b, "Aspects of Scientific Explanation," in Hempel 1965a, pp. 331-496.

Hempel, C. and Oppenheim, P., 1948, "Studies in the Logic of Explanation," Philosophy of Science 15: 135-75. Reprinted in Hempel 1965a, pp. 245-90.

Nagel, E., 1979, The Structure of Science: Problems in

the Logic of Scientific Explanation. Indianapolis: Hackett.

Suppe, F., 1977, "The Search for Philosophic Understanding of Scientific Theories," in Suppe, F., The

Structure of Scientific Theories. Chicago: University of Illinois Press, pp. 3-232.

5.1

Theory and Observation

Rudolf Carnap

Rudolf Carnap (1891-1970) was a member of the Vienna Circle and the leading figure in the positivist philosophical movement that dominated the first half of the twentieth century. He was also the primary author of the received view of scientific theories. The following selection is an excerpt from his textbook in the philosophy of science, and it provides an overview of his views on the role of laws in scientific theories, the distinction between two concepts of probability, and the empirical significance of theoretical terms.

The Value of Laws: Explanation and Prediction

The observations we make in everyday life as well as the more systematic observations of science reveal certain repetitions or regularities in the world. Day always follows night; the seasons repeat themselves in the same order; fire always feels hot; objects fall when we drop them; and so on. The laws of science are nothing more than statements expressing these regularities as precisely as possible.

If a certain regularity is observed at all times and all places, without exception, then the regularity is expressed in the form of a "universal law". An example from daily life is, "All ice is cold." This statement asserts that any piece of ice - at any place

in the universe, at any time, past, present, or future - is (was, or will be) cold. Not all laws of science are universal. Instead of asserting that a regularity occurs in all cases, some laws assert that it occurs in only a certain percentage of cases. If the percentage is specified or if in some other way a quantitative statement is made about the relation of one event to another, then the statement is called a "statistical law". For example: "Ripe apples are usually red", or "Approximately half the children born each year are boys." Both types of law - universal and statistical - are needed in science. The universal laws are logically simpler, and for this reason we shall consider them first. In the early part of this discussion "laws" will usually mean universal laws.

From Rudolf Carnap, Philosophical Foundations of Physics: An Introduction to the Philosophy of Science, ed. Martin Gardner (New York: Basic Books, 1966), pp. 3-8, 16-22, 32-9, 225-39.

330 RUDOLF CARNAP

Universal laws are expressed in the logical form of what is called in formal logic a "universal conditional statement". (In this book, we shall occasionally make use of symbolic logic, but only in a very elementary way.) For example, let us consider a law of the simplest possible type. It asserts that, whatever x may be, if x is P, then x is also Q. This is written symbolically as follows:

(x)(Px ::J Qx).

The expression "(x)" on the left is called a "universal quantifier." It tells us that the statement refers to all cases of x, rather than to just a certain percentage of cases. "Px" says that x is P, and "Qx" says that xis Q. The symbol "::J" is a connective. It links the term on its left to the term on its right. In English, it corresponds roughly to the assertion, "If . . . then . . . "

If "x" stands for any material body, then the law states that, for any material body x, if x has the property P, it also has the property Q. For instance, in physics we might say: "For every body x, if that body is heated, that body will expand." This is the law of thermal expansion in its simplest, nonquantitative form. In physics, of course, one tries to obtain quantitative laws and to qualify them so as to exclude exceptions; but, if we forget about such refinements, then this universal conditional statement is the basic logical form of all universal laws. Sometimes we may say that, not only does Qx hold whenever Px holds, but the reverse is also true; whenever Qx holds, Px holds also. Logicians call this a biconditional statement - a statement that is conditional in both directions. But of course this does not contradict the fact that in all universal laws we deal with universal conditionals, because a biconditional may be regarded as the conjunction of two conditionals.

Not all statements made by scientists have this logical form. A scientist may say: "Yesterday in Brazil, Professor Smith discovered a new species of butterfly." This is not the statement of a law. It speaks about a specified single time and place; it states that something happened at that time and place. Because statements such as this are about single facts, they are called "singular" statements. Of course, all our lmowledge has its origin in singular statements - the particular observations of particular individuals. One of the big, perplexing

questions in the philosophy of science is how we are able to go from such singular statements to the assertion of universal laws.

[ . . . ]

When we use the word "fact", we will mean it in the singular sense in order to distinguish it clearly from universal statements. Such universal statements will be called "laws" even when they are as elementary as the law of thermal expansion or, still more elementary, the statement, "All ravens are black." I do not know whether this statement is true, but, assuming its truth, we will call such a statement a law of zoology. Zoologists may speak informally of such "facts" as "the raven is black" or "the octopus has eight arms", but, in our more precise terminology, statements of this sort will be called "laws".

Later we shall distinguish between two kinds of law - empirical and theoretical. Laws of the simple kind that I have just mentioned are sometimes called "empirical generalizations" or "empirical laws". They are simple because they speak of properties, like the color black or the magnetic properties of a piece of iron, that can be directly observed. The law of thermal expansion, for example, is a generalization based on many direct observations of bodies that expand when heated. In contrast, theoretical, nonobservable concepts, such as elementary particles and electromagnetic fields, must be dealt with by theoretical laws. We will discuss all this later. I mention it here because otherwise you might think that the examples I have given do not cover the kind of laws you have perhaps learned in theoretical physics.

To summarize, science begins with direct observations of single facts. Nothing else is observable. Certainly a regularity is not directly observable. It is only when many observations are compared with one another that regularities are discovered. These regularities are expressed by statements called "laws".

What good are such laws? What purposes do they serve in science and everyday life? The answer is twofold: they are used to explain facts already !mown, and they are used to predict facts not yet known.

First, let us see how laws of science are used for explanation. No explanation - that is, nothing that

THEORY AND OBSERVATION 331

deserves the honorific title of "explanation " -can be given without referring to at least one law. (In simple cases, there is only one law, but in more complicated cases a set of many laws may be involved.) It is important to emphasize this point, because philosophers have often maintained that they could explain certain facts in history, nature, or human life in some other way. They usually do this by specifying some type of agent or force that is made responsible for the occurrence to be explained.

[ ... ]

[F]act explanations are really law explanationsin disguise. When we examine them more carefully, we find them to be abbreviated, incomplete statements that tacitly assume certain laws, but laws so familiar that it is unnecessary to express them.[ ... ]

Consider one ... example. We ask little Tommy why he is crying, and he answers with another fact: "Jimmy hit me on the nose." Why do we consider this a sufficient explanation? Because we know that a blow on the nose causes pain and that, when children feel pain, they cry. These are general psychological laws. They are so well known that they are assumed even by Tommy when he tells us why he is crying. If we were dealing with, say, a Martian child and knew very little about Martian psychological laws, then a simple statement of fact might not be considered an adequate explanation of the child's behavior. Unless facts can be connected with other facts by means of at least one law, explicitly stated or tacitly understood, they do not provide explanations.

The general schema involved in all explanation can be expressed symbolically as follows:

(x)(Px ::i Qx) 2 Pa 3 Qa

The first statement is the universal law that applies to any object x. The second statement asserts that a particular object a has the property P. These two statements taken together enable usto derive logically the third statement: object a hasthe property Q.

In science, as in everyday life, the universal law is not always explicitly stated. If you ask a

physicist: "Why is it that this iron rod, which a moment ago fitted exactly into the apparatus, is now a trifle too long to fit?", he may reply by saying: "While you were out of the room, I heated the rod." He assumes, of course, that you know the law of thermal expansion; otherwise, in order to be understood, he would have added, "and, whenever a body is heated, it expands ". The general law is essential to his explanation. If you know the law, however, and he knows that you know it, he may not feel it necessary to state the law. For this reason, explanations, especially in everyday life where common-sense laws are taken for granted, often seem quite different from the schema I have given.

At times, in giving an explanation, the only known laws that apply are statistical rather than universal. In such cases, we must be content with a statistical explanation. For example, we may know that a certain kind of mushroom is slightly poisonous and causes certain symptoms of illness in 90 per cent of those who eat it. If a doctor finds these symptoms when he examines a patient and the patient informs the doctor that yesterday he ate this particular kind of mushroom, the doctor will consider this an explanation of the symptoms even though the law involved is only a statistical one. And it is, indeed, an explanation.

[ ... ]

In addition to providing explanations for observed facts, the laws of science also provide a means for predicting new facts not yet observed. The logical schema involved here is exactly the same as the schema underlying explanation. This, you recall, was expressed symbolically:

(x)(Px ::i Qx) 2 Pa 3 Qa

First we have a universal law: for any object x, if it has the property P, then it also has the property Q. Second, we have a statement saying that object a has the property P. Third, we deduce by elementary logic that object a has the property Q. This schema underlies both explanation andprediction; only the knowledge situation is different. In explanation, the fact Qa is alreadyknown. We explain Qa by showing how it can be

332 RUDOLF CARNAP

deduced from statements 1 and 2. In prediction, Qa is a fact not yet known. We have a law, and we have the fact Pa. We conclude that Qa must also be a fact, even though it has not yet been observed. For example, I know the law of thermal expansion. I also know that I have heated a certain rod. By applying logic in the way shown in the schema, I infer that if I now measure the rod, I will find that it is longer than it was before.

In most cases, the unknown fact is actually a future event (for example, an astronomer predicts the time of the neJci eclipse of the sun); that is why I use the term "prediction" for this second use of laws. It need not, however, be prediction in the literal sense. In many cases the unknown fact is simultaneous with the known fact, as is the case in the example of the heated rod. The expansion of the rod occurs simultaneously with the heating. It is only our observation of the expansion that takes place after our observation of the heating.

In other cases, the unknown fact may even be in the past. On the basis of psychological laws, together with certain facts derived from historical documents, a historian infers certain unknown facts of history. An astronomer may infer that an eclipse of the moon must have taken place at a certain date in the past. A geologist may infer from striations on boulders that at one time in the past a region must have been covered by a glacier. I use the term "prediction" for all these examples because in every case we have the same logical schema and the same knowledge situation - a known fact and a known law from which anunknown fact is derived.

In many cases, the law involved may be statistical rather than universal. The prediction will then be only probable. A meteorologist, for instance, deals with a mixture of exact physical laws and various statistical laws. He cannot say that it will rain tomorrow; he can only say that rain is very likely.

This uncertainty is also characteristic of prediction about human behavior. On the basis of knowing certain psychological laws of a statistical nature and certain facts about a person, we can predict with varying degrees of probability how he will behave. Perhaps we ask a psychologist to tell us what effect a certain event will have on our child. He replies: "As I see the situation, your child will probably react in this way. Of course, the laws

of psychology are not very exact. It is a young science, and as yet we know very little about its laws. But on the basis of what is known, I think it advisable that you plan to ... ". And so he gives us advice based on the best prediction he can mal<e, with his probabilistic laws, about the future behavior of our child.

When the law is universal, then elementary deductive logic is involved in inferring unknown facts. If the law is statistical, we must use a different logic - the logic of probability. To give a simple example: a law states that 90 per cent of the residents of a certain region have black hair. I know that an individual is a resident of that region, but I do not know the color of his hair. I can infer, however, on the basis of the statistical law, that the probability his hair is black is 9/io.

Prediction is, of course, as essential to everyday life as it is to science. Even the most trivial acts we perform during the day are based on predictions. You turn a doorknob. You do so because past observations of facts, together with universal laws, lead you to believe that turning the knob will open the door. You may not be conscious of the logical schema involved - no doubt you are thinking about other things -but all such deliberate actions presuppose the schema. There is a knowledge of specific facts, a knowledge of certain observed regularities

that can be expressed as universal or statistical laws and provide a basis for the prediction of unknown facts. Prediction is involved in every act of human behavior that involves deliberate choice. Without it, both science and everyday life would be impossible.

Induction and Statistical Probability

[W) e [earlier) assumed that laws of science were available. We saw how such laws are used, in both science and everyday life, as explanations of known facts and as a means for predicting unknown facts. Let us now ask how we arrive at such laws. On what basis are we justified in believing that a law holds? We know, of course, that all laws are based on the observation of certain regularities. They constitute indirect knowledge, as opposed to direct knowledge of facts. What justifies us in going from the direct observation of facts to a law that expresses


certain regularities of nature? This is what in traditional terminology is called "the problem of induction".

Induction is often contrasted with deduction

by saying that deduction goes from the general to the specific or singular, whereas induction goes the other way, from the singular to the general. This is a misleading oversimplification. In deduction, there are kinds of inferences other than those from the general to the specific; in induction there are also many kinds of inference. The traditional distinction is also misleading because it suggests that deduction and induc

tion are simply two branches of a single kind of logic. John Stuart Mill's famous work, A System of Logic, contains a lengthy description of what he called "inductive logic" and states various canons of inductive procedure. Today we are more reluctant to use the term "inductive inference". If it is used at all, we must realize that it refers to a kind of inference that differs fundamentally from deduction.

In deductive logic, inference leads from a set of premisses to a conclusion just as certain as the premisses. If you have reason to believe the premisses, you have equally valid reason to believe the conclusion that follows logically from the premisses. If the premisses are true, the conclusion cannot be false. With respect to induction, the situation is entirely different. The truth of an inductive conclusion is never certain. I do not mean only that the conclusion cannot be certain because it rests on premisses that cannot be known with certainty. Even if the premisses are assumed to be true and the inference is a valid inductive inference, the conclusion may be false. The most we can say is that, with respect to given premisses, the conclusion has a certain degree of probability. Inductive logic tells us how to calculate the value of this probability.

We know that singular statements of fact, obtained by observation, are never absolutely certain because we may make errors in our observations; but, in respect to laws, there is still

greater uncertainty. A law about the world states that, in any particular case, at any place and any time, if one thing is true, another thing is true. Clearly, this speaks about an infinity of possible instances. The actual instances may not be infinite, but there is an infinity of possible instances.

A physiological law says that, if you stick a

dagger into the heart of any human being, he will die. Since no exception to this law has ever been observed, it is accepted as universal. It is true, of course, that the number of instances so far observed of daggers being thrust into human hearts is finite. It is possible that some day humanity may cease to exist; in that case, the number of human beings, both past and future, is finite. But we do not know that humanity will cease to exist. Therefore, we must say that there is an infinity of possible instances, all of which are covered by the law. And, if there is an infinity of instances, no number of finite observations, however large, can make the "universal" law certain.

Of course, we may go on and make more and more observations, making them in as careful and scientific a manner as we can, until eventually we may say: "This law has been tested so many times that we can have complete confidence in its truth. It is a well-established, well-founded law."

If we think about it, however, we see that even the best-founded laws of physics must rest on only a finite number of observations. It is always possible that tomorrow a counterinstance may be found. At no time is it possible to arrive at complete verification of a law. In fact, we should not speak of "verification" at all if by the word we mean a definitive establishment of truth - but only

of confirmation. Interestingly enough, although there is no way

in which a law can be verified (in the strict sense), there is a simple way it can be falsified. One need find only a single counterinstance. The knowledge of a counterinstance may, in itself, be uncertain. You may have made an error of observation or have been deceived in some way. But, if we assume that the counterinstance is a fact, then the negation of the law follows immediately. If a law says that every object that is P is also Q and we find an object that is P and not Q,

the law is refuted. A million positive instances are insufficient to verify the law; one counterinstance is sufficient to falsify it. The situation is strongly asymmetric. It is easy to refute a law; it is exceedingly difficult to find strong confirmation.

How do we find confirmation of a law? If we have observed a great many positive instances and no negative instance, we say that the confirmation is strong. How strong it is and whether the strength can be expressed numerically is still a controversial question in the philosophy of science.

334 RUDOLF CARNAP

We will return to this in a moment. Here we are concerned only with making clear that our first task in seeking confirmation of a law is to test instances to determine whether they are positive or negative. This is done by using our logical schema to make predictions. A law states that (x)(Px :::::i Qx); hence, for a given object a, Pa :::i Qa. We try to find as many objects as we can (here symbolized by "a") that have the property P. We then observe whether they also fulfill the condition Q. If we find a negative instance, the matter is settled. Otherwise, each positive instance is additional evidence adding to the strength of our confirmation.

There are, of course, various methodological rules for efficient testing. For example, instances should be diversified as much as possible. If you are testing the law of thermal expansion, you should not limit your tests to solid substances. If you are testing the law that all metals are good conductors of electricity, you should not confine your tests to specimens of copper. You should test as many metals as possible under various conditions-hot, cold, and so on. We will not go into the many methodological rules for testing; we will only point out that in all cases the law is tested by making predictions and then seeing whether those predictions hold. In some cases, we find in nature the objects that we wish to test. In other cases, we have to produce them. In testing the law of thermal expansion, for example, we do not look for objects that are hot; we take certain objects and heat them. Producing conditions for testing has the great advantage that we can more easily follow the methodological rule of diversification; but whether we create the situations to be tested or find them ready-made in nature, the underlying schema is the same.

A moment ago I raised the question of whether the degree of confirmation of a law ( or a singular statement that we are predicting by means of the law) can be expressed in quantitative form. Instead of saying that one law is "well founded" and that another law "rests on flimsy evidence", we might say that the first law has a .8 degree of confirmation, whereas the degree of confirmation for the second law is only .2. This question has long been debated. My own view is that such a procedure is legitimate and that what I have called "degree of confirmation" is identical with logical probability.

Such a statement does not mean much until we know what is meant by "logical probability". Why do I add the adjective "logical"? It is not customary practice; most books on probability do not make a distinction between various kinds of probability, one of which is called "logical". It is my belief, however, that there are two fundamentally different kinds of probability, and I distinguish between them by calling one "statistical probability", and the other "logical probability". It is unfortunate that the same word, "probability", has been used in two such widely differing senses. Failing to make the distinction is a source of enormous confusion in books on the philosophy of science as well as in the discourse of scientists themselves.

Instead of "logical probability", I sometimes use the term "inductive probability", because in my conception this is the kind of probability that is meant whenever we make an inductive inference. By "inductive inference" I mean, not only inference from facts to laws, but also any inference that is "nondemonstrative"; that is, an inference such that the conclusion does not follow with logical necessity when the truth of the premisses is granted. Such inferences must be expressed in degrees of what I call "logical probability" or "inductive probability".

[ . . . ]

In my conception, logical probability is a logical relation somewhat similar to logical implication; indeed, I think probability may be regarded as a partial implication. If the evidence is so strong that the hypothesis follows logically from it - is logically implied by it - we have one extreme case in which the probability is 1. (Probability 1 also occurs in other cases, but this is one special case where is occurs.) Similarly, if the negation of a hypothesis is logically implied by the evidence, the logical probability of the hypothesis is 0. In between, there is a continuum of cases about which deductive logic tells us nothing beyond the negative assertion that neither the hypothesis nor its negation can be deduced from the evidence. On this continuum inductive logic must tal<e over. But inductive logic is lil<e deductive logic in being concerned solely with the statements involved, not with the facts of nature. By a logical analysis of a stated hypothesis h and stated


evidence e, we conclude that h is not logically implied but is, so to speak, partially implied by e

to the degree of so-and-so much. At this point, we are justified, in my view, in

assigning numerical value to the probability. If possible, we should like to construct a system of inductive logic of such a kind that for any pair of sentences, one asserting evidence e and the other stating a hypothesis h, we can assign a number giving the logical probability of h with respect to e. (We do not consider the trivial case in whichthe sentence e is contradictory; in such instances,no probability value can be assigned to h.) [ ...]

When I say I think it is possible to apply an inductive logic to the language of science, I do not mean that it is possible to formulate a set of rules, fixed once and for all, that will lead automatically, in any field, from facts to theories. It seems doubtful, for example, that rules can be formulated to enable a scientist to survey a hundred thousand sentences giving various observational reports and then find, by a mechanical application of those rules, a general theory (system oflaws) that would explain the observed phenomena. This is usually not possible, because theories, especially the more abstract ones dealing with such nonobservables as particles and fields, use a conceptual framework that goes far beyond the framework used for the description of observation material. One cannot simply follow a mechanical procedure based on fixed rules to devise a new system of theoretical concepts, and with its help a theory. Creative ingenuity is required. This point is sometimes expressed by saying that there cannot be an inductive machine - a computer into which we can put all the relevant observational sentences and get, as an output, a neat system of laws that will explain the observed phenomena.

I agree that there cannot be an inductive machine if the purpose of the machine is to invent new theories. I believe, however, that there can be an inductive machine with a much more modest aim. Given certain observations e

and a hypothesis h (in the form, say, of a prediction or even of a set of laws), then I believe it is in many cases possible to determine, by mechanical procedures, the logical probability, or degree of confirmation, of h on the basis of e. For this concept of probability, I also use the term "inductive probability", because I am convinced that this is the basic concept involved in all inductive rea-

saning and that the chief task of inductive reasoning is the evaluation of this probability. [ ... ]

Statements giving values of statistical probability are not purely logical; they are factual statements in the language of science. When a medical man says that the probability is "very good" ( or perhaps he uses a numerical value and says .7) that a patient will react positively to a certain injection, he is making a statement in medical science. When a physicist says that the probability of a certain radioactive phenomenon is so-and-so much, he is making a statement in physics. Statistical probability is a scientific, empirical concept. Statements about statistical probability are "synthetic" statements, statements that cannot be decided by logic but which rest on empirical investigations ...

[ . . . ]

On the other hand, we also need the concept of logical probability. It is especially useful in metascientific statements, that is, statements about science. We say to a scientist: "You tell me that I can rely on this law in making a certain prediction. How well established is the law? How trustworthy is the prediction?" The scientist today may or may not be willing to answer a meta -scientific question of this kind in quantitative terms. But I believe that, once inductive logic is sufficiently developed, he could reply: "This hypothesis is confirmed to degree .8 on the basis of the available evidence." A scientist who answers in this way is making a statement about a logical relation between the evidence and the hypothesis in question. The sort of probability he has in mind is logical probability, which I also call "degree of confirmation". His statement that the value of this probability is .8 is, in this context, not a synthetic (empirical) statement, but an analytic one. It is analytic because no empirical investigation is demanded. It expresses a logical relation between a sentence that states the evidence and a sentence that states the hypothesis.

Note that, in making an analytic statement of probability, it is always necessary to specify the evidence explicitly. The scientist must not say: "The hypothesis has a probability of .8." He must add, "with respect to such and such evidence." If this is not added, his statement might be taken as a statement of statistical probability. If he intends

336 RUDOLF CARNAP

it to be a statement of logical probability, it is an elliptical statement in which an important component has been left out. In quantum theory, for instance, it is often difficult to know whether a physicist means statistical probability or logical probability. Physicists usually do not draw this distinction. They talk as though there were only one concept of probability with which they work. "We mean that kind of probability that fulfills the ordinary axioms of probability theory", they may say. But the ordinary axioms of probability theory are fulfilled by both concepts, so this remark does not clear up the question of exactly what type of probability they mean.

[ . . . ]

I will not go into greater detail here about my view of probability, because many technicalities are involved. But I will discuss the one inference in which the two concepts of probability may come together. This occurs when either the hypothesis or one of the premisses for the inductive inference contains a concept of statistical probability. We can see this easily by modifying the basic schema used in our discussion of universal laws. Instead of a universal law (1), we take as the first premiss a statistical law (I'), which says that the relative frequency (if) of Q with respect to Pis (say) .8. The second premiss (2) states, as before, that a certain individual a has the property P. The third statement (3) asserts that a has the property Q. This third statement, Qa, is the hypothesis wewish to consider on the basis of the two premisses.

In symbolic form:

(I') if( Q,P) = .8 (2) Pa

(3) Qa

What can we say about the logical relation of (3) to (l') and (2)? In the previous case - theschema for a universal law - we could make thefollowing logical statement:

(4) Statement (3) is logically implied by (1)and (2).

We cannot make such a statement about the schema given above because the new premiss (1') is weaker than the former premiss (1); it states

a relative frequency rather than a universal law. We can, however, make the following statement, which also asserts a logical relation, but in terms of logical probability or degree of confirmation, rather than in terms of implication:

(4') Statement (3), on the basis of (1') and (2), has a probability of .8.

Note that this statement, like statement ( 4), is not a logical inference from (l') and (2). Both ( 4) and ( 4') are statements in what is called ametalanguage; they are logical statements about

three assertions: (1) [or (1'), respectively], (2),and (3).

[ . . . ]

The main points that I wish to stress here are these: Both types of probability - statistical and logical - may occur together in the same chain of reasoning. Statistical probability is part of the object language of science. To statements about statistical probability we can apply logical probability, which is part of the metalanguage of science. It is my conviction that this point of view gives a much clearer picture of statistical inference than is commonly found in books on statistics and that it provides an essential groundwork for the construction of an adequate inductive logic of science.

Theories and Nonobservables

One of the most important distinctions between two types of laws in science is the distinction between what may be called (there is no generally accepted terminology for them) empirical laws and theoretical laws. Empirical laws are laws that can be confirmed directly by empirical observations. The term "observable" is often used for any phenomenon that can be directly observed, so it can be said that empirical laws are laws about observables.

Here, a warning must be issued. Philosophers and scientists have quite different ways of using the terms "observable" and "nonobservable". To a philosopher, "observable" has a very narrow meaning. It applies to such properties as "blue", "hard", "hot". These are properties directly per-


ceived by the senses. To the physicist, the word has a much broader meaning. It includes any quantitative magnitude that can be measured in a relatively simple, direct way. A philosopher would not consider a temperature of, perhaps, 80 degrees centigrade, or a weight of 931/2 pounds,an observable because there is no direct sensory perception of such magnitudes. To a physicist, both are observables because they can be measured in an extremely simple way. The object to be weighed is placed on a balance scale. The temperature is measured with a thermometer. The physicist would not say that the mass of a molecule, let alone the mass of an electron, is something observable, because here the procedures of measurement are much more complicated and indirect. But magnitudes that can be established by relatively simple procedures length with a ruler, time with a clock, or frequency oflight waves with a spectrometer are called observables.

A philosopher might object that the intensity of an electric current is not really observed. Only a pointer position was observed. An ammeter was attached to the circuit and it was noted that the pointer pointed to a mark labeled 5.3. Certainly the current's intensity was not observed. It was inferred from what was observed.

The physicist would reply that this was true enough, but the inference was not very complicated. The procedure of measurement is so simple, so well established, that it could not be doubted that the ammeter would give an accurate measurement of current intensity. Therefore, it is included among what are called observables.

There is no question here of who is using the term "observable" in a right or proper way. There is a continuum which starts with direct sensory observations and proceeds to enormously complex, indirect methods of observation. Obviously no sharp line can be drawn across this continuum; it is a matter of degree. A philosopher is sure that the sound of his wife's voice, coming from across the room, is an observable. But suppose he listens to her on the telephone. Is her voice an observable or isn't it? A physicist would certainly say that when he looks at something through an ordinary microscope, he is observing it directly. Is this also the case when he looks into an electron microscope? Does he observe the path of a particle when he sees the track it makes in a bubble chamber? In general, the physicist speaks

of observables in a very wide sense compared with the narrow sense of the philosopher, but, in both cases, the line separating observable from nonobservable is highly arbitrary. It is well to keep this in mind whenever these terms are encountered in a book by a philosopher or scientist. Individual authors will draw the line where it is most convenient, depending on their points of view, and there is no reason why they should not have this privilege.

Empirical laws, in my terminology, are laws containing terms either directly observable by the senses or measurable by relatively simple techniques. Sometimes such laws are called empirical generalizations, as a reminder that they have been obtained by generalizing results found by observations and measurements. They include not only simple qualitative laws (such as, "All ravens are black") but also quantitative laws that arise from simple measurements. The laws relating pressure, volume, and temperature of gases are of this type. Ohm's law, connecting the electric potential difference, resistance, and intensity of current, is another familiar example. The scientist makes repeated measurements, finds certain

regularities, and expresses them in a law. These are the empirical laws. As indicated in earlier chapters, they are used for explaining observed facts and for predicting future observable events.

There is no commonly accepted term for the second kind of laws, which I call theoretical laws. Sometimes they are called abstract or hypothetical laws. "Hypothetical" is perhaps not suitable because it suggests that the distinction between the two types of laws is based on the degree to which the laws are confirmed. But an empirical law, if it is a tentative hypothesis, confirmed only to a low degree, would still be an empirical law although it might be said that it was rather hypothetical. A theoretical law is not to be distinguished from an empirical law by the fact that it is not well established, but by the fact that it contains terms of a different kind. The terms of a theoretical law do not refer to observables even when the physicist's wide meaning for what can be observed is adopted. They are laws about such entities as molecules, atoms, electrons, protons, electromagnetic fields, and others that cannot be measured in simple, direct ways.

[ . . . ]

338 RUDOLF CARNAP

It is true, as shown earlier, that the concepts "observable" and "nonobservable" cannot be sharply defined because they lie on a continuum. In actual practice, however, the difference is usually great enough so there is not likely to be debate. All physicists would agree that the laws relating pressure, volume, and temperature of a gas, for example, are empirical laws. Here the amount of gas is large enough so that the magnitudes to be measured remain constant over a sufficiently large volume of space and period of time to permit direct, simple measurements which can then be generalized into laws. All physicists would agree that laws about the behavior of single molecules are theoretical. Such laws concern a microprocess about which generalizations cannot be based on simple, direct measurements.

Theoretical laws are, of course, more general than empirical laws. It is important to understand, however, that theoretical laws cannot be arrived at simply by taking the empirical laws, then generalizing a few steps further. How does a physicist arrive at an empirical law? He observes certain events in nature. He notices a certain regularity. He describes this regularity by mal<ing an inductive generalization. It might be supposed that he could now put together a group of empirical laws, observe some sort of pattern, make a wider inductive generalization, and arrive at a theoretical law. Such is not the case.

To mal<e this clear, suppose it has been observed that a certain iron bar expands when heated. After the experiment has been repeated many times, always with the same result, the regularity is generalized by saying that this bar expands when heated. An empirical law has been stated, even though it has a narrow range and applies only to one particular iron bar. Now further tests are made of other iron objects with the ensuing discovery that every time an iron object is heated it expands. This permits a more general law to be formulated, namely that all bodies of iron expand when heated. In similar fashion, the still more general laws "All metals ... ", then "All solid bodies ... ", are developed. These are all simple generalizations, each a bit more general than the previous one, but they are all empirical laws. Why? Because in each case, the objects dealt with are observable (iron, copper, metal, solid bodies); in each case the increases in temperature and length are measurable by simple, direct techniques.

In contrast, a theoretical law relating to this process would refer to the behavior of molecules in the iron bar. In what way is the behavior of the molecules connected with the expansion of the bar when heated? You see at once that we are now speal<ing of nonobservables. We must introduce a theory - the atomic theory of matter - and we are quickly plunged into atomic laws involving concepts radically different from those we had before. It is true that these theoretical concepts differ from concepts of length and temperature only in the degree to which they are directly or indirectly observable, but the difference is so great that there is no debate about the radically different nature of the laws that must be formulated.

Theoretical laws are related to empirical laws in a way somewhat analogous to the way empirical laws are related to single facts. An empirical law helps to explain a fact that has been observed and to predict a fact not yet observed. In similar fashion, the theoretical law helps to explain empirical laws already formulated, and to permit the derivation of new empirical laws. Just as the single, separate facts fall into place in an orderly pattern when they are generalized in an empirical law, the single and separate empirical laws fit into the orderly pattern of a theoretical law. This raises one of the main problems in the methodology of science. How can the !<ind of knowledge that will justify the assertion of a theoretical law be obtained? An empirical law may be justified by mal<ing observations of single facts. But to justify a theoretical law, comparable observations cannot be made because the entities referred to in theoretical laws are nonobservables.

Before tal<ing up this problem, some remarks made in an earlier chapter, about the use of the word "fact", should be repeated. It is important in the present context to be extremely careful in the use of this word because some authors, especially scientists, use "fact" or "empirical fact" for some propositions which I would call empirical laws. For example, many physicists will refer to the "fact" that the specific heat of copper is .090. I would call this a law because in its full formulation it is seen to be a universal conditional statement: "For any x, and any time t, if xis a solid body of copper, then the specific heat of x at tis

. 090." Some physicists may even speak of the law of thermal expansion, Ohm's law, and others, as


facts. Of course, they can then say that theoretical laws help explain such facts. This sounds like my statement that empirical laws explain facts, but the word "fact" is being used here in two different ways. I restrict the word to particular, concrete facts that can be spatiotemporally specified, not thermal expansion in general, but the

expansion of this iron bar observed this morning at ten o'clock when it was heated. It is important to bear in mind the restricted way in which I speak of facts. If the word "fact" is used in an ambiguous manner, the important difference between the ways in which empirical and theoretical laws serve for explanation will be entirely blurred.

How can theoretical laws be discovered? We cannot say: "Let's just collect more and more data, then generalize beyond the empirical laws until we reach theoretical ones." No theoretical law was ever found that way. We observe stones and trees and flowers, noting various regularities and describing them by empirical laws. But no matter how long or how carefully we observe such things, we never reach a point at which we observe a molecule. The term "molecule" never arises as a result of observations. For this reason, no amount of generalization from observations will ever produce a theory of molecular processes. Such a theory must arise in another way. It is stated not as a generalization of facts but as a hypothesis. The hypothesis is then tested in a manner analogous in certain ways to the testing of an empirical law. From the hypothesis, certain empirical laws are derived, and these empirical laws are tested in turn by observation of facts. Perhaps the empirical laws derived from the theory are already known and well confirmed. (Such laws may even have motivated the formulation of the theoretical law.) Regardless of whether the derived empirical laws are known and confirmed, or whether they are new laws confirmed by new observations, the confirmation of such derived laws provides indirect confirmation of the theoretical law.

The point to be made clear is this. A scientist does not start with one empirical law, perhaps Boyle's law for gases, and then seek a theory about molecules from which this law can be derived. The scientist tries to formulate a much more general theory from which a variety of empirical laws can be derived. The more such laws, the greater their variety and apparent lack of

connection with one another, the stronger will be the theory that explains them. Some of these derived laws may have been known before, but the theory may also make it possible to derive new empirical laws which can be confirmed by new tests. If this is the case, it can be said that the theory made it possible to predict new empirical laws. The prediction is understood in a hypothetical way. If the theory holds, certain empirical laws will also hold. The predicted empirical law speaks about relations between observables, so it is now possible to make experiments to see if the empirical law holds. If the empirical law is confirmed, it provides indirect confirmation of the theory. Every confirmation of a law, empirical or theoretical, is, of course, only partial, never complete and absolute. But in the case of empirical laws, it is a more direct confirmation. The confirmation of a theoretical law is indirect, because it takes place only through the confirmation of empirical laws derived from the theory.

The supreme value of a new theory is its power to predict new empirical laws. It is true that it also has value in explaining known empirical laws, but this is a minor value. If a scientist proposes a new theoretical system, from which no new laws can be derived, then it is logically equivalent to the set of all known empirical laws. The theory may have a certain elegance, and it may simplify to some degree the set of all known laws, although it is not likely that there would be an essential simplification. On the other hand, every new theory in physics that has led to a great leap forward has been a theory from which new empirical laws could be derived. If Einstein had done no more than propose his theory of relativity as an elegant new theory that would embrace certain known laws - perhaps also simplify them to a certain degree - then his theory would not have had such a revolutionary effect.

Of course it was quite otherwise. The theory of relativity led to new empirical laws which explained for the first time such phenomena as the movement of the perihelion of Mercury, and the bending of light rays in the neighborhood of the sun. These predictions showed that relativity theory was more than just a new way of expressing the old laws. Indeed, it was a theory of great predictive power. The consequences that can be derived from Einstein's theory are far

340 RUDOLF CARNAP

from being exhausted. These are consequences that could not have been derived from earlier theories. Usually a theory of such power does have an elegance, and a unifying effect on known laws.

It is simpler than the total collection of known laws. But the great value of the theory lies in its

power to suggest new laws that can be confirmed

by empirical means.

Correspondence Rules

An important qualification must now be added to the discussion of theoretical laws and terms given in the last chapter. The statement that empirical laws are derived from theoretical laws is an oversimplification. It is not possible to derive them directly because a theoretical law contains theoretical terms, whereas an empirical law contains only observable terms. This prevents any direct deduction of an empirical law from a theoretical one.

To understand this, imagine that we are back in the nineteenth century, preparing to state for the first time some theoretical laws about molecules in a gas. These laws are to describe the

number of molecules per unit volume of the gas, the molecular velocities, and so forth. To simplify matters, we assume that all the molecules have the same velocity. (This was indeed the original assumption; later it was abandoned in favor of a certain probability distribution of velocities.) Further assumptions must be made about what happens when molecules collide. We do not know the exact shape of molecules, so let us suppose that they are tiny spheres. How do spheres

collide? There are laws about colliding spheres, but they concern large bodies. Since we cannot directly observe molecules, we assume their collisions are analogous to those of large bodies;

perhaps they behave like perfect billiard balls on a frictionless table. These are, of course, only assumptions; guesses suggested by analogies with

known macrolaws. But now we come up against a difficult prob

lem. Our theoretical laws deal exclusively with the behavior of molecules, which cannot be seen. How, therefore, can we deduce from such laws a law about observable properties such as the pressure or temperature of a gas or properties

of sound waves that pass through the gas? The

theoretical laws contain only theoretical terms. What we seek are empirical laws containing

observable terms. Obviously, such laws cannot be derived without having something else given in addition to the theoretical laws.

The something else that must be given is this: a set of rules connecting the theoretical terms with the observable terms. Scientists and philosophers

of science have long recognized the need for such a set of rules, and their nature has been often discussed. An example of such a rule is: "If there is an electromagnetic oscillation of a specified frequency, then there is a visible greenish-blue color of a certain hue." Here something observable is connected with a nonobservable microprocess.

Another example is: "The temperature (measured by a thermometer and, therefore, an observable in the wider sense explained earlier) of a gas is proportional to the mean kinetic energy of its molecules." This rule connects a nonobservable in molecular theory, the kinetic energy of

molecules, with an observable, the temperature of

the gas. If statements of this kind did not exist, there would be no way of deriving empirical laws about observables from theoretical laws about nonobservables.

Different writers have different names for these rules. I call them "correspondence rules". P. W.

Bridgman calls them operational rules. Norman R. Campbell speal<s of them as the "Dictionary" . 1

Since the rules connect a term in one termino

logy with a term in another terminology, the useof the rules is analogous to the use of a FrenchEnglish dictionary. What does the French word"cheval" mean? You look it up in the dictionaryand find that it means "horse". It is not really thatsimple when a set of rules is used for connecting

nonobservables with observables; nevertheless,there is an analogy here that makes Campbell's"Dictionary" a suggestive name for the set of rules.

There is a temptation at times to think that the set of rules provides a means for defining theoretical terms, whereas just the opposite is

really true. A theoretical term can never be explicitly defined on the basis of observable terms, although sometimes an observable can be defined in theoretical terms. For example, "iron" can be defined as a substance consisting of small crystalline parts, each having a certain arrangement

of atoms and each atom being a configuration of particles of a certain type. In theoretical terms


then, it is possible to express what is meant by the observable term "iron", but the reverse is not true.

There is no answer to the question: "Exactly what is an electron?" Later we shall come back to this question, because it is the kind that philosophers are always asking scientists. They want

the physicist to tell them just what he means by "electricity", "magnetism", "gravity", "a molecule". If the physicist explains them in theoretical terms, the philosopher may be disappointed. "That is not what I meant at all", he will say. "I want you to tell me, in ordinary language, what those terms mean." Sometimes the philosopher writes a book in which he talks about the great mysteries of nature. "No one", he writes, "has been able so far, and perhaps no one ever will be able, to give us a straightforward answer to the question: 'What is electricity?' And so electricity remains forever one of the great, unfathomable mysteries of the universe."

There is no special mystery here. There is only an improperly phrased question. Definitions that cannot, in the nature of the case, be given, should not be demanded. If a child does not know what an elephant is, we can tell him it is a huge animal with big ears and a long trunk. We can show him a picture of an elephant. It serves admirably to define an elephant in observable terms that a child can understand. By analogy, there is a temptation to believe that, when a scientist introduces theoretical terms, he should also be able to define them in familiar terms. But this is not possible. There is no way a physicist can show us a picture of electricity in the way he can show his child a picture of an elephant. Even the cell of an organism, although it cannot be seen with the unaided eye, can be represented by a picture because the cell can be seen when it is viewed through a microscope. But we do not possess a picture of the electron. We cannot say how it looks or how it feels, because it cannot be seen or touched. The best we can do is to say that it is an extremely small body that behaves in a certain manner. This may seem to be analogous to our description of an elephant. We can describe an elephant as a large animal that behaves in a certain manner. Why not do the same with an electron?

The answer is that a physicist can describe the behavior of an electron only by stating theoretical laws, and these laws contain only theoretical

terms. They describe the field produced by an electron, the reaction of an electron to a field, and so on. If an electron is in an electrostatic field, its velocity will accelerate in a certain way. Unfortunately, the electron's acceleration is an unobservable. It is not like the acceleration of a billiard ball, which can be studied by direct observation. There is no way that a theoretical concept can be defined in terms of observables. We must, therefore, resign ourselves to the fact that definitions of the kind that can be supplied for observable terms cannot be formulated for theoretical terms.

It is true that some authors, including Bridgman, have spoken of the rules as "operational definitions". Bridgman had a certain justification, because he used his rules in a somewhat different way, I believe, than most physicists use them. He was a great physicist and was certainly aware of his departure from the usual use of rules, but he was willing to accept certain forms of speech that are not customary, and this explains his departure. It was pointed out in a previous chapter that Bridgman preferred to say that there is not just one concept of intensity of electric current, but a dozen concepts. Each procedure by which a magnitude can be measured provides an operational definition for that magnitude. Since there are different procedures for measuring current, there are different concepts. For the sake of convenience, the physicist speaks of just one concept of current. Strictly speaking, Bridgman believed, he should recognize many different concepts, each defined by a different operational procedure of measurement.

We are faced here with a choice between two different physical languages. If the customary procedure among physicists is followed, the various concepts of current will be replaced by one concept. This means, however, that you place the concept in your theoretical laws, because the operational rules are just correspondence rules, as I call them, which connect the theoretical terms with the empirical ones. Any claim to possessing a definition - that is, an operational definition - of the theoretical concept must be given up.Bridgman could speak of having operationaldefinitions for his theoretical terms only becausehe was not speaking of a general concept. He wasspeaking of partial concepts, each defined by adifferent empirical procedure.

342 RUDOLF CARNAP

Even in Bridgman's terminology, the question of whether his partial concepts can be adequately defined by operational rules is problematic. Reichenbach speaks often of what he calls "correlative definitions". (In his German publications, he calls them Zuordnungsdefinitionen, from zuordnen, which means to correlate.) Perhaps correlation is a better term than definition for what Bridgman's rules actually do. In geometry, for instance, Reichenbach points out that the axiom system of geometry, as developed by David Hilbert, for example, is an uninterpreted axiom system. The basic concepts of point, line, and plane could just as well be called "class alpha", "class beta", and "class gamma". We must not be seduced by the sound of familiar words, such as "point" and "line", into thinking they must be taken in their ordinary meaning. In the axiom system, they are uninterpreted terms. But when geometry is applied to physics, these terms must be connected with something in the physical world. We can say, for example, that the lines of the geometry are exemplified by rays of light in a vacuum or by stretched cords. In order to connect the uninterpreted terms with observable physical phenomena, we must have rules for establishing the connection.

What we call these rules is, of course, only a terminological question; we should be cautious and not speak of them as definitions. They are not definitions in any strict sense. We cannot give a really adequate definition of the geometrical concept of "line" by referring to anything in nature. Light rays, stretched strings, and so on are only approximately straight; moreover, they are not lines, but only segments of lines. In geometry, a line is infinite in length and absolutely straight. Neither property is exhibited by any phenomenon in nature. For that reason, it is not possible to give an operational definition, in the strict sense of the word, of concepts in theoretical geometry. The same is true of all the other theoretical concepts of physics. Strictly speaking, there are no "definitions" of such concepts. I prefer not to speak of "operational definitions" or even to use Reichenbach's term "correlative definitions". In my publications ( only in recent years have I written about this question), I have called them "rules of correspondence" or, more simply, "correspondence rules".

Campbell and other authors often speak of the entities in theoretical physics as mathemat-

ical entities. They mean by this that the entities are related to each other in ways that can be expressed by mathematical functions. But they are not mathematical entities of the sort that can be defined in pure mathematics. In pure mathematics, it is possible to define various kinds of numbers, the function of logarithm, the exponential function, and so forth. It is not possible, however, to define such terms as "electron" and "temperature" by pure mathematics. Physical terms can be introduced only with the help of nonlogical constants, based on observations of the actual world. Here we have an essential difference between an axiomatic system in mathematics and an axiomatic system in physics.

If we wish to give an interpretation to a term in a mathematical axiom system, we can do it by giving a definition in logic. Consider, for example, the term "number" as it is used in Peano's axiom system. We can define it in logical terms, by the Frege-Russell method, for example. In this way the concept of "number" acquires a complete, explicit definition on the basis of pure logic. There is no need to establish a connection between the number 5 and such observables as "blue" and "hot". The terms have only a logical interpretation; no connection with the actual world is needed. Sometimes an axiom system in mathematics is called a theory. Mathematicians speal< of set theory, group theory, matrix theory, probability theory. Here the word "theory" is used in a purely analytic way. It denotes a deductive system that makes no reference to the actual world. We must always bear in mind that such a use of the word "theory" is entirely different from its use in reference to empirical theories such as relativity theory, quantum theory, psychoanalytical theory, and Keynesian economic theory.

A postulate system in physics cannot have, as mathematical theories have, a splendid isolation from the world. Its axiomatic terms - "electron", "field", and so on - must be interpreted by correspondence rules that connect the terms with observable phenomena. This interpretation is necessarily incomplete. Because it is always incomplete, the system is left open to make it possible to add new rules of correspondence. Indeed, this is what continually happens in the history of physics. I am not thinking now of a revolution in physics, in which an entirely new theory is developed, but of less radical changes


that modify existing theories. Nineteenth-century physics provides a good example, because classical mechanics and electromagnetics had been established, and, for many decades, there was relatively little change in fundamental laws. The basic theories of physics remained unchanged. There was, however, a steady addition of new correspondence rules, because new procedures were continually being developed for measuring this or that magnitude.

Of course, physicists always face the danger that they may develop correspondence rules that will be incompatible with each other or with the theoretical laws. As long as such incompatibility does not occur, however, they are free to add new correspondence rules. The procedure is neverending. There is always the possibility of adding new rules, thereby increasing the amount of interpretation specified for the theoretical terms; but no matter how much this is increased, the interpretation is never final. In a mathematical system, it is otherwise. There a logical interpretation of an axiomatic term is complete. Here we find another reason for reluctance in speaking of theoretical terms as "defined" by correspondence rules. It tends to blur the important distinction between the nature of an axiom system in pure mathematics and one in theoretical physics.

Is it not possible to interpret a theoretical term by correspondence rules so completely that no further interpretation would be possible? Perhaps the actual world is limited in its structure and laws. Eventually a point may be reached beyond which there will be no room for strengthening the interpretation of a term by new correspondence rules. Would not the rules then provide a final, explicit definition for the term? Yes, but then the term would no longer be theoretical. It would become part of the observation language. The history of physics has not yet indicated that physics will become complete; there has been only a

steady addition of new correspondence rules and a continual modification in the interpretations of theoretical terms. There is no way of knowing whether this is an infinite process or whether it will eventually come to some sort of end.

It may be looked at this way. There is no prohibition in physics against making the correspondence rules for a term so strong that the term becomes explicitly defined and therefore ceases to be theoretical. Neither is there any basis for assuming that it will always be possible to add new correspondence rules. Because the history of physics has shown such a steady, unceasing modification of theoretical concepts, most physicists would advise against correspondence rules so strong that a theoretical term becomes explicitly defined. Moreover, it is a wholly unnecessary procedure. Nothing is gained by it. It may even have the adverse effect of blocking progress.

Of course, here again we must recognize that the distinction between observables and nonobservables is a matter of degree. We might give an explicit definition, by empirical procedures, to a concept such as length, because it is so easily and directly measured, and is unlikely to be modified by new observations. But it would be rash to seek such strong correspondence rules that "electron" would be explicitly defined. The concept "electron" is so far removed from simple, direct observations that it is best to keep it theoretical, open to modifications by new observations.

Note

See Percy W. Bridgman, The Logic of Modern

Physics (New York: Macmillan, 1927), and Norman R. Campbell, Physics: The Elements ( Cambridge:

Cambridge University Press, 1920). Rules of cor

respondence are discussed by Ernest Nagel, The

Structure of Science (New York: Harcourt, Brace &

World, 1961), pp. 97-105.

5.2

Scientific Explanation

Carl Hempel

Carl Hempel (1905-1997) was a member of the Berlin Society and the primary author of the received view on explanation. The following selection is an excerpt from his classic "Aspects of Scientific Explanation" and presents the deductive-nomological (D-N) and inductive-statistical (I-S) models of explanation. It also includes the requirement of maximal specificity (RMS) that constitutes his response to the problem of ambiguity of I-S explanations.

2 Deductive-Nomological Explanation

2.1 Fundamentals: D-N explanation and the concept of law

In his book, How We Think,1 John Dewey describes a phenomenon he observed one day while washing dishes. Having removed some glass tumblers from the hot suds and placed them upside down on a plate, he noticed that soap bubbles emerged from under the tumbler's rims, grew for a while, came to a standstill and finally receded into the tumblers. Why did this happen? Dewey outlines an explanation to this effect: Transferring the tumblers to the plate, he had trapped cool air in them; that air was gradually

warmed by the glass, which initially had the temperature of the hot suds. This led to an increase in the volume of the trapped air, and thus to an expansion of the soap film that had formed between the plate and the tumblers' rims. But gradually, the glass cooled off, and so did the air inside, and as a result, the soap bubbles receded.

The explanation here outlined may be regarded as an argument to the effect that the phenomenon to be explained, the explanandum phenomenon, was to be expected in virtue of certain explanatory facts. These fall into two groups: (i) particular facts and (ii) uniformities expressible by means of generallaws. The first group includes facts such as these:the tumblers had been immersed in soap suds ofa temperature considerably higher than that of thesurrounding air; they were put, upside down, on

From Carl Hempel, Aspects of Scientific Explanation and Other Essays in the Philosophy of Science (New York: Free Press,

1965), pp. 335-8, 380-4, 394-403. © 1965 by The Free Press. Copyright renewed© 1997 by Carl G. Hempel. Reprinted

with the permission of The Free Press, a Division of Simon & Schuster, Inc.

SCIENTIFIC EXPLANATION 345

a plate on which a puddle of soapy water had formed that provided a connecting soap film, and so on. The second group of explanatory facts would be expressed by the gas laws and by various other laws concerning the exchange of heat between bodies of different temperature, the elastic behavior of soap bubbles, and so on. While some of these laws are only hinted at by such phrasings as 'the warming of the trapped air led to an increase in its pressure', and others are not referred to even in this oblique fashion, they are clearly presupposed in the claim that certain stages in the process yielded others as their results. If we imagine the various explicit or tacit explanatory assumption to be fully stated, then the explanation may be conceived as a deductive argument of the form

(D-N) Explanans S

Explanandumsentence

Here, Cp C2, ••• , Ck are sentences describing the particular facts invoked; Lp L2, • • • , L, are the general laws on which the explanation rests.Jointly these sentences will be said to form theexplanans S, where S may be thought of alternatively as the set of the explanatory sentencesor as their conjunction. The conclusion E ofthe argument is a sentence describing theexplanandum-phenomenon; I will call E theexplanandum-sentence or explanandumstatement; the word 'explanandum' alone willbe used to refer either to the explanandumphenomenon or to the explanandum-sentence:the context will show which is meant.

The kind of explanation whose logical structure is suggested by the schema (D-N) will be called deductive-nomological explanation or D-N explanation for short; for it effects a deductive subsumption of the explanandum under principles that have the character of general laws. Thus a DN explanation answers the question 'Why did the explanandum-phenomenon occur?' by showing that the phenomenon resulted from certain particular circumstances, specified in Cp C2, • •• ,

Ck, in accordance with the laws Lp L2, • • • , L,. By pointing this out, the argument shows that,given the particular circumstances and the lawsin question, the occurrence of the phenomenon

was to be expected; and it is in this sense that the explanation enables us to understand why the phenomenon occurred.2

In a D-N explanation, then, the explanandum is a logical consequence of the explanans. Furthermore, reliance on general laws is essential to a D-N explanation; it is in virtue of such laws that the particular facts cited in the explanans possess explanatory relevance to the explanandum phenomenon. Thus, in the case of Dewey's soap bubbles, the gradual warming of the cool air trapped under the hot tumblers would constitute a mere accidental antecedent rather than an explanatory factor for the growth of the bubbles, if it were not for the gas laws, which connect the two events. But what if the explanandum sentence E in an argument of the form (D-N) is a logical consequence of the sentences Ci, C2, ••• ,

Ck alone? Then, surely, no empirical laws are required to deduce E from the explanans; and any laws included in the latter are gratuitous, dispensable premises. Quite so; but in this case, the argument would not count as an explanation. For example, the argument:

The soap bubbles first expanded and then receded

The soap bubbles first expanded

though deductively valid, clearly cannot qualify as an explanation of why the bubbles first expanded. The same remark applies to all other cases of this kind. A D-N explanation will have to contain, in its explanans, some general laws that are required for the deduction of the explanandum, i.e. whose deletion would make the argument invalid.

If the explanans of a given D-N explanation is true, i.e. if the conjunction of its constituent sentences is true, we will call the explanation true; a true explanation, of course, has a true explanandum as well. Next, let us call a D-N explanation more or less strongly supported or confirmed by a given body of evidence according as its explanans is more or less strongly confirmed by the given evidence. ( One factor to be considered in appraising the empirical soundness of a given explanation will be the extent to which its explanans is supported by the total relevant evidence available.) Finally, by a potential D-N explanation, let us understand any argument that has the

346 CARL HEMPEL

character of a D-N explanation except that the sentences constituting its explanans need not be true. In a potential D-N explanation, therefore, L i, L2, ••• , L,. will be what Goodman has called lawlike sentences, i.e. sentences that are like laws except for possibly being false. Sentences of this kind will also be referred to as nomic or nomological. We use the notion of a potential explanation, for example, when we ask whether a novel and as yet untested law or theory would provide an explanation for some empirical phenomenon; or when we say that the phlogiston theory, though now discarded, afforded an explanation for certain aspects of combustion.3

Strictly speaking, only true lawlike statements can count as laws - one would hardly want to speak of false laws of nature. But for convenience I will occasionally use the term 'law' without implying that the sentence in question is true, as in fact, I have done already in the preceding sentence.

[ . . . ]

3.2 Deductive-statistical explanation

It is an instance of the so-called gambler's fallacy to assume that when several successive tossings of a fair coin have yielded heads, the next toss will more probably yield tails than heads. Why this is not the case can be explained by means of two hypotheses that have the form of statistical laws. The first is that the random experiment of flipping a fair coin yields heads with a statistical probability of 1/2. The second hypothesis is that the outcomes of different tossings of the coin are statistically independent, so that the probability of any specified sequence of outcomes - such as heads twice, then tails, then heads, then tails three times - equals the product of the probabilities of the constituent single outcomes. These two hypothesis in terms of statistical probabilities imply deductively that the probability for heads to come up after a long sequence of heads is still 1/2.

Certain statistical explanations offered in science are of the same deductive character, though often quite complex mathematically. Consider, for example, the hypothesis that for the atoms of every radioactive substance there is a characteristic probability of disintegrating during a given unit time interval. This complex statistical

hypothesis explains, by deductive implication, various other statistical aspects of radioactive decay, among them, the following: Suppose that the decay of individual atoms of some radioactive substance is recorded by means of the scintillations produced upon a sensitive screen by the alpha particles emitted by the disintegrating atoms. Then the time intervals separating successive scintillations will vary considerably in length, but intervals of different lengths will occur with different statistical probabilities. Specifically, if the mean time interval between successive scintillations is s seconds, then the probability for two successive scintillations to be separated by more than n·s seconds is (lie)", where e is the base of the natural logarithms.4

Explanations of the kind here illustrated will be called deductive-statistical explanations, or DS explanations. They involve the deduction of a statement in the form of a statistical law from an explanans that contains indispensably at least one law or theoretical principle of statistical form. The deduction is effected by means of the mathematical theory of statistical probability, which makes it possible to calculate certain derivative probabilities (those referred to in the explanandum) on the basis of other probabilities (specified in the explanans) which have been empirically ascertained or hypothetically assumed. What a D-S explanation accounts for is thus always ageneral uniformity expressed by a presumptivelaw of statistical form.

Ultimately, however, statistical laws are meant to be applied to particular occurrences and to establish explanatory and predictive connections among them. In the next subsection, we will examine the statistical explanation of particu

lar events. Our discussion will be limited to the case where the explanatory statistical laws are of basic form: this will suffice to exhibit the basic logical differences between the statistical and the deductive-nomological explanation of individual occurrences.

3.3 Inductive-statistical explanation

As an explanation of why patient John Jones recovered from a streptococcus infection, we might be told that Jones had been given penicillin. But if we try to amplify this explanatory claim by indicating a general connection between penicillin


treatment and the subsiding of a streptococcus infection we cannot justifiably invoke a general law to the effect that in all cases of such infection, administration of penicillin will lead to recovery.

What can be asserted, and what surely is taken for granted here, is only that penicillin will effect

a cure in a high percentage of cases, or with a

high statistical probability. This statement has the general character of a law of statistical form,

and while the probability value is not specified,

the statement indicates that it is high. But in contrast to the cases of deductive-nomological and

deductive-statistical explanation, the explanans consisting of this statistical law together with the statement that the patient did receive peni

cillin obviously does not imply the explanandum

statement, 'the patient recovered', with deductive certainty, but only, as we might say, with high

likelihood, or near-certainty. Briefly, then, the explanation amounts to this argument:

(3a) The particular case of illness of John Jones - let us call it j - was an instance of severe

streptococcal infection (Sj) which was treated

with large doses of penicillin (Pj); and the statistical probability p(R, S·P) of recovery in cases where S and P are present is close to 1; hence,

the case was practically certain to end in recovery (Rj).

This argument might invite the following

schematization:

p(R, S·P) is close to 1

(3b) Sj · Pj -------------

(Therefore:) It is practically certain

(very likely) that Rj

In the literature on inductive inference, argu

ments thus based on statistical hypotheses have

often been construed as having this form or a similar one. On this construal, the conclusion

characteristically contains a modal qualifier such

as 'almost certainly', 'with high probability', 'very likely', etc. But the conception of arguments

having this character is untenable. For phrases of the form 'it is practically certain that p' or 'It is very likely that p', where the place of 'p' is taken by some statement, are not complete self-contained

sentences that can be qualified as either true or

false. The statement that takes the place of 'p' -for example, 'Rj' - is either true or false, quite inde

pendently of whatever relevant evidence may be

available, but it can be qualified as more or less

likely, probable, certain, or the like only relative to some body of evidence. One and the same statement, such as 'Rj', will be certain, very likely,

not very likely, highly unlikely, and so forth,

depending upon what evidence is considered. The phrase 'it is almost certain that Rj' taken

by itself is therefore neither true nor false; and it cannot be inferred from the premises specified in

(3b) nor from any other statements. The confusion underlying the schematization

(3b) might be further illuminated by considering

its analogue for the case of deductive arguments. The force of a deductive inference, such as that

from 'all Fare G' and 'a is F' to 'a is G', is sometimes indicated by saying that if the premises are true, then the conclusion is necessarily true or is

certain to be true - a phrasing that might suggest the schematization

All Fare G

a is F (Therefore:) It is necessary (certain) that

a is G.

But clearly the given premises - which might

be, for example, 'all men are mortal' and 'Socrates is a man' - do not establish the sentence

'a is G' ('Socrates is mortal') as a necessary or certain truth. The certainty referred to in the informal paraphrase of the argument is relational: the statement 'a is G' is certain, or necessary,

relative to the specified premises; i.e., their truth

will guarantee its truth - which means nothing more than that 'a is G' is a logical consequence

of those premises. Analogously, to present our statistical explana

tion in the manner of schema (3b) is to miscon

strue the function of the words 'almost certain' or 'very likely' as they occur in the formal word

ing of the explanation. Those words clearly must be taken to indicate that on the evidence provided by the explanans, or relative to that evidence, the

explanandum is practically certain or very likely, i.e., that

(3c) 'Rj' is practically certain (very likely)relative to the explanans containing the sentences

'p (R, S·P) is close to 1' and 'Sj • Pj'.5

The explanatory argument misrepresented by

(3b) might therefore suitably be schematized as

follows:

348 CARL HEMPEL

(3d)

p(R, S·P) is close to 1 Sj·Pj

Rj [makes practically certain ( very likely)]

In this schema, the double line separating the "premises" from the "conclusion" is to signify that the relation of the former to the latter is not that of deductive implication but that of inductive support, the strength of which is indicated in square brackets. 6'

7

[ . . .]

3.4 The ambiguity of inductive-statistical explanation and the requirement of maximal specificity

3.4.1 The problem of explanatory ambiguity

Consider once more the explanation (3d) of recovery in the particular case j of John Jones's illness. The statistical law there invoked claims recovery in response to penicillin only for a high percentage of streptococcal infections, but not for all of them; and in fact, certain streptococcus strains are resistant to penicillin. Let us say that an occurrence, e.g., a particular case of illness, has the property S* (or belongs to the class S*) if it is an instance of infection with a penicillinresistant streptococcus strain. Then the probability of recovery among randomly chosen instances of S* which are treated with penicillin will be quite small, i.e., p(R, S*·P) will be close to O and the probability of non-recovery, p(R, S*·P) will be close to 1. But suppose now that Jones's illness is in fact a streptococcal infection of the penicillin-resistant variety, and consider the following argument:

(3k) p(R, S*·P) is close to 1 S*j·Pj

Rj [makes practically certain]

This "rival" argument has the same form as (3d), and on our assumptions, its premises are true, just like those of (3d). Yet its conclusion is the contradictory of the conclusion of (3d).

Or suppose that Jones is an octogenarian with a weak heart, and that in this group, S**, the probability of recovery from a streptococcus infection in response to penicillin treatment, p(R, S**·P), is quite small. Then, there is the following rival argument to (3d), which presents Jones's nonrecovery as practically certain in the light of premises which are true:

(31)

p(R, S**·P) is close to 1 S**j·Pj ========= [makes practically R;·

certain]

The peculiar logical phenomenon here illustrated will be called the ambiguity of inductivestatistical explanation or, briefly, of statistical explanation. This ambiguity derives from the fact that a given individual event (e.g., Jones's illness) will often be obtainable by random selection from any one of several "reference classes" ( such as S·P, S*·P, S**·P), with respect to which the kind of occurrence (e.g., R) instantiated by the given event has very different statistical probabilities. Hence, for a proposed probabilistic explanation with true explanans which confers near-certainty upon a particular event, there will often exist a rival argument of the same probabilistic form and with equally true premises which confers near-certainty upon the nonoccurrence of the same event. And any statistical explanation for the occurrence of an event must seem suspect if there is the possibility of a logically and empirically equally sound probabilistic account for its nonoccurrence. This predicament has no analogue in the case of deductive explanation; for if the premises of a proposed deductive explanation are true then so is its conclusion; and its contradictory, being false, cannot be a logical consequence of a rival set of premises that are equally true.

Here is another example of the ambiguity of I-S explanation: Upon expressing surprise atfinding the weather in Stanford warm and sunnyon a date as autumnal as November 27, I mightbe told, by way of explanation, that this wasrather to be expected because the probability ofwarm and sunny weather (W) on a November dayin Stanford (N) is, say, .95. Schematically, thisaccount would take the following form, where 'n' stands for 'November 27':


(3m)

p(W, N) = .95 Nn

Wn [.95]

But suppose it happens to be the case that the day before, November 26, was cold and rainy, and that the probability for the immediate successors (S) of cold and rainy days in Stanford to bewarm and sunny is .2; then the account (3m) hasa rival in the following argument which, byreference to equally true premises, presents it asfairly certain that November 27 is not warm andsunny:

(3n)

p(W, S) = .8 Sn

Wn [.8]

In this form, the problem of ambiguity concerns I-S arguments whose premises are in fact true, no matter whether we are aware of this or not. But, as will now be shown, the problem has a variant that concerns explanations whose explanans statements, no matter whether in fact true or not, are asserted or accepted by empirical science at the time when the explanation is proffered or contemplated. This variant will be called the problem of the epistemic ambiguity of statistical explanation, since it refers to what is presumed to be known in science rather than to what, perhaps unknown to anyone, is in fact the case.

Let K, be the class of all statements asserted or accepted by empirical science at time t. This class then represents the total scientific information, or "scientific knowledge" at time t. The word 'knowledge' is here used in the sense in which we commonly speak of the scientific knowledge at a given time. It is not meant to convey the claim that the elements of K, are true, and hence neither that they are definitely known to be true. No such claim can justifiably be made for any of the statements established by empirical science; and the basic standards of scientific inquiry demand that an empirical statement, however well supported, be accepted and thus admitted to membership in K, only tentatively, i.e., with the understanding that the privilege may be withdrawn if unfavorable evidence should be discovered. The membership of K, therefore changes in the course of time; for as a result of continuing

research, new statements are admitted into that class; others may come to be discredited and dropped. Henceforth, the class of accepted statements will be referred to simply as K when specific reference to the time in question is not required. We will assume that K is logically consistent and that it is closed under logical implication, i.e., that it contains every statement that is logically implied by any of its subsets.

The epistemic ambiguity of I-S explanation can now be characterized as follows: The total set K of accepted scientific statements contains different subsets of statements which can be used as premises in arguments of the probabilistic form just considered, and which confer high probabilities on logically contradictory "conclusions." Our earlier examples (3k), (31) and (3m), (3n) illustrate this point if we assume that the premises of those arguments all belong to K rather than that they are all true. If one of two such rival arguments with premises in K is proposed as an explanation of an event considered, or acknowledged, in science to have occurred, then the conclusion of the argument, i.e., the explanandum statement, will accordingly belong to K as well. And since K is consistent, the conclusion of the rival argument will not belong to K. Nonetheless it is disquieting that we should be able to say: No matter whether we are informed that the event in question ( e.g., warm and sunny weather on November 27 in Stanford) did occur or that it did not occur, we can produce an explanation of the reported outcome in either case; and an explanation, moreover, whose premises are scientifically established statements that confer a high logical probability upon the reported outcome.

This epistemic ambiguity, again, has no analogue for deductive explanation; for since K is logically consistent, it cannot contain premise-sets that imply logically contradictory conclusions.

Epistemic ambiguity also bedevils the predictive use of statistical arguments. Here, it has the alarming aspect of presenting us with two rival arguments whose premises are scientifically well established, but one of which characterizes a contemplated future occurrence as practically certain, whereas the other characterizes it as practically impossible. Which of such conflicting arguments, if any, are rationally to be relied on for explanation or for prediction?

350 CARL HEMPEL

3.4.2 The requirement of maximal specificity and the epistemic relativity of inductive-statistical explanation

Our illustrations of explanatory ambiguity suggest that a decision on the acceptability of a proposed probabilistic explanation or prediction will have to be made in the light of all the relevant information at our disposal. This is indicated also by a general principle whose importance for inductive reasoning has been acknowledged, if not always very explicitly, by many writers, and which has recently been strongly emphasized by Carnap, who calls it the requirement of total evidence. Carnap formulates it as follows: "in the application of inductive logic to a given knowledge situation, the total evidence available must be taken as basis for determining the degree of confirmation."8 Using only a part of the total evidence is permissible if the balance of the evidence is irrelevant to the inductive "conclusion," i.e., if on the partial evidence alone, the conclusion has the same confirmation, or logical probability, as on the total evidence.9

The requirement of total evidence is not a postulate nor a theorem of inductive logic; it is not concerned with the formal validity of inductive arguments. Rather, as Carnap has stressed, it is a maxim for the application of inductive logic; we might say that it states a necessary condition of rationality of any such application in a given "knowledge situation," which we will think of as represented by the set K of all statements accepted in the situation.

But in what manner should the basic idea of this requirement be brought to bear upon probabilistic explanation? Surely we should not insist that the explanans must contain all and only the empirical information available at the time. Not all the available information, because otherwise all probabilistic explanations acceptable at time t would have to have the same explanans, K,; and not only the available information, because a proffered explanation may meet the intent of the requirement in not overlooking any relevant information available, and may nevertheless invoke some explanans statements which have not as yet been sufficiently tested to be included in K,.

The extent to which the requirement of total evidence should be imposed upon statistical explanations is suggested by considerations such as

the following. A proffered explanation ofJones's recovery based on the information that Jones had a streptococcal infection and was treated with penicillin, and that the statistical probability for recovery in such cases is very high is unacceptable if K includes the further information that Jones's streptococci were resistant to penicillin, or that Jones was an octogenarian with a wealc heart, and that in these reference classes the probability of recovery is small. Indeed, one would want an acceptable explanation to be based on a statistical probability statement pertaining to the narrowest reference class of which, according to our total information, the particular occurrence under consideration is a member. Thus, if K tells us not only that Jones had a streptococcus infection and was treated with penicillin, but also that he was an octogenarian with a wealc heart (and if K provides no information more specific than that) then we would require that an acceptable explanation of Jones's response to the treatment be based on a statistical law stating the probability of that response in the narrowest reference class to which our total information assigns Jones's illness, i.e., the class of streptococcal infections suffered by octogenarians with weak hearts.10

Let me amplify this suggestion by reference to our earlier example concerning the use of the law that the half-life of radon is 3.82 days in accounting for the fact that the residual amount of radon to which a sample of 10 milligrams was reduced in 7.64 days was within the range from 2.4 to 2.6 milligrams. According to present scientific knowledge, the rate of decay of a radioactive element depends solely upon its atomic structure as characterized by its atomic number and its mass number, and it is thus unaffected by the age of the sample and by such factors as temperature, pressure, magnetic and electric forces, and chemical interactions. Thus, by specifying the half-life of radon as well as the initial mass of the sample and the time interval in question, the explanans takes into account all the available information that is relevant to appraising the probability of the given outcome by means of statistical laws. To state the point somewhat differently: Under the circumstances here assumed, our total information K assigns the case under study first of all to the reference class say Pp

of cases where a 10 milligram sample of radon is


allowed to decay for 7.64 days; and the half-life law for radon assigns a very high probability, within F1 , to the "outcome," say G, consisting in the fact that the residual mass of radon lies between 2.4 and 2.6 milligrams. Suppose now that K also contains information about the temperature of the given sample, the pressure and relative humidity under which it is kept, the surrounding electric and magnetic conditions, and so forth, so that K assigns the given case to a reference class much narrower than F1, let us say, F1F2F3 ••• F,,. Now the theory of radioactive decay, which is equally included in K, tells us that the statistical probability of G within this narrower class is the same as within G. For this reason, it suffices in our explanation to rely on the probability p(G, F1 ).

Let us note, however, that "knowledge situations" are conceivable in which the same argument would not be an acceptable explanation. Suppose, for example, that in the case of the radon sample under study, the amount remaining one hour before the end of the 7.64 day period happens to have been measured and found to be 2.7 milligrams, and thus markedly in excess of 2.6 milligrams - an occurrence which, considering the decay law for radon, is highly improbable, but not impossible. That finding, which then forms part of the total evidence K, assigns the particular case at hand to a reference class, say F*, within which, according to the decay law for radon, the outcome G is highly improbable since it would require a quite unusual spurt in the decay of the given sample to reduce the 2.7 milligrams, within the one final hour of the test, to an amount falling between 2.4 and 2.6 milligrams. Hence, the additional information here considered may not be disregarded, and an explanation of the observed outcome will be acceptable only if it takes account of the probability of G in the narrower reference class, i.e., p( G, F

1 P). (The theory of radioactive decay implies that this probability equals p( G, P), so that as a consequence the membership of the given case in F

1 need not

be explicitly taken into account.) The requirement suggested by the preceding

considerations can now be stated more explicitly; we will call it the requirement of maximal specificity for inductive-statistical explanations. Consider a proposed explanation of the basic statistical form

(3o)

p(G, F) = r Pb

Gb [r]

Let s be the conjunction of the premises, and, if K is the set of all statements accepted at the given time, let k be a sentence that is logically equivalent to K (in the sense that k is implied by K and in turn implies every sentence in K). Then, to be rationally acceptable in the knowledge situation represented by K, the proposed explanation (3o) must meet the following condition (the requirement of maximal specificity): If s·k implies11 that b belongs to a class Fi, and that F1 is a subclass of F, then s•k must also imply a statement specifying the statistical probability of G in F

i, say

Here, r1 must equal r unless the probability

statement just cited is simply a theorem of mathematical probability theory.

The qualifying unless-clause here appended is quite proper, and its omission would result in undesirable consequences. It is proper because theorems of pure mathematical probability theory cannot provide an explanation of empirical subject matter. They may therefore be discounted when we inquire whether s·/c might not give us statistical laws specifying the probability of G in reference classes narrower than F. And the omission of the clause would prove troublesome, for if ( 3o) is proffered as an explanation, then it is presumably accepted as a fact that Gb; hence 'Gb' belongs to K. Thus K assigns b to the narrower class F·G, and concerning the probability of G in that class, s·k trivially implies the statement that p( G, F·G) = 1, which is simply a consequence of the measure-theoretical postulates for statistical probability. Since s·/c thus implies a more specific probability statement for G than that invoked in (3o), the requirement of maximal specificity would be violated by (3o) - and analogously by any proffered statistical explanation of an event that we take to have occurred -were it not for the unless-clause, which, in effect, disqualifies the notion that the statement 'p( G, F·G) = l' affords a more appropriate law to account for the presumed fact that Gb.

352 CARL HEMPEL

The requirement of maximal specificity, then, is here tentatively put forward as characterizing the extent to which the requirement of total evidence properly applies to inductive-statistical explanations. The general idea thus suggested comes to this: In formulating or appraising an IS explanation, we should take into account all that information provided by K which is of potential explanatory relevance to the explanandum event; i.e., all pertinent statistical laws, and such particular facts as might be connected, by the statistical laws, with the explanandum event.12

The requirement of maximal specificity disposes of the problem of epistemic ambiguity; for it is readily seen that of two rival statistical arguments with high associated probabilities and with premises that all belong to K, at least one violates the requirement of maximum specificity. Indeed, let

p(G, P) = r 1

Pb

Gb

p(G, H) = r2

Hb

Gb

be the arguments in question, with r 1 and r2 close to 1. Then, since K contains the premises of both arguments, it assigns b to both P and H and hence to PH. Hence if both arguments satisfy the requirement of maximal specificity, K must imply that

p(G, PH)= p(G, P) = r1

p(G, PH)= p(G, H) = r2

But p(G, PH)+ p(G, PH)= l Hence r

1 + r2 = 1

and this is an arithmetic falsehood, since 1\ and r

2 are both close to l; hence it cannot be implied

by the consistent class K. Thus, for I-S explanations that meet the

requirement of maximal specificity the problem of epistemic ambiguity no longer arises. We are never in a position to say: No matter whether this particular event did or did not occur, we can produce an acceptable explanation of either outcome; and an explanation, moreover, whose premises are scientifically accepted statements which confer a high logical probability upon the given outcome.

While the problem of epistemic ambiguity has thus been resolved, ambiguity in the first sense

discussed in this section remains unaffected by our requirement; i.e., it remains the case that for a given statistical argument with true premises and a high associated probability, there may exist a rival one with equally true premises and with a high associated probability, whose conclusion contradicts that of the first argument. And though the set K of statements accepted at any time never includes all statements that are in fact true (and no doubt many that are false), it is perfectly possible that K should contain the premises of two such conflicting arguments; but as we have seen, at least one of the latter will fail to be rationally acceptable because it violates the requirement of maximal specificity.

The preceding considerations show that the concept of statistical explanation for particular events is essentially relative to a given knowledge situation as represented by a class K of accepted statements. Indeed, the requirement of maximal specificity makes explicit and unavoidable reference to such a class, and it thus serves to characterize the concept of "I-S explanation relative to the knowledge situation represented by K." We will refer to this characteristic as the epistemic relativity of statistical explanation.

It might seem that the concept of deductive explanation possesses the same kind of relativity, since whether a proposed D-N or D-S account is acceptable will depend not only on whether it is deductively valid and makes essential use of the proper type of general law, but also on whether its premises are well supported by the relevant evidence at hand. Quite so; and this condition of empirical confirmation applies equally to statistical explanations that are to be acceptable in a given knowledge situation. But the epistemic relativity that the requirement of maximal specificity implies for I-S explanations is of quite a different kind and has no analogue for D-N explanations. For the specificity requirement is not concerned with the evidential support that the total evidence K affords for the explanans statements: it does not demand that the latter be included in K, nor even that K supply supporting evidence for them. It rather concerns what may be called the concept of a potential statistical explanation. For it stipulates that no matter how much evidential support there may be for the explanans, a proposed I-S explanation is not acceptable if its potential explanatory force with respect to the


specified explanandum is vitiated by statistical laws which are included in K but not in the explanans, and which might permit the production of rival statistical arguments. As we have seen, this danger never arises for deductive explanations. Hence, these are not subject to any such restrictive condition, and the notion of a potential deductive explanation (as contradistinguished from a deductive explanation with well-confirmed explanans) requires no relativization with respect to K.

As a consequence, we can significantly speak of true D-N and D-S explanations: they are those potential D-N and D-S explanations whose premises (and hence also conclusions) are true - no matter whether this happens to be known or believed, and thus no matter whether the premises are included in K. But this idea has no significant analogue for I-S explanation since, as we have seen, the concept of potential statistical explanation requires relativization with respect to K.

Notes

1 Dewey (1910), chap. VI. 2 A general conception of scientific explanation as

involving a deductive subsumption under general laws was espoused, though not always clearly stated, by various thinkers in the past, and has been advocated by several recent or contemporary writers, among them N. R. Campbell [(1920), (1921)], who developed the idea in considerable detail. In a textbook published in 1934, the conception was concisely stated as follows: "Scientific explanation consists in subsuming under some rule or law which expresses an invariant character of a group of events, the particular events it is said to explain. Laws themselves may be explained, and in the same manner, by showing that they are consequences of more comprehensive theories." (Cohen and Nagel 1934, p. 397.) Popper has set forth this construal of explanation in several of his publications; cf. the note at the end of section 3 in Hempel and Oppenheim (1948) His earliest statement appears in section 12 of his book (1935), of which his work (1959) is an expanded English version. His book (1962) contains further observations on scientific explanation. For some additional references to other proponents of the general idea, see Donagan (1957), footnote 2; Scriven (1959), footnote 3. However, as will be shown in section 3, deductive subsumption

under general laws does not constitute the only form of scientific explanation.

3 The explanatory role of the phlogiston theory is described in Conant (1951), pp. 164-71. The concept of potential explanation was introduced in Hempel and Oppenheim (1948), section 7. The concept of lawlike sentence, in the sense here indicated, is due to Goodman (1947).

4 Cf. Mises (1939), pp. 272-8, where both the empirical findings and the explanatory argument are presented. This book also contains many other illustrations of what is here called deductivestatistical explanation.

5 Phrases such as 'It is almost certain (very likely) that j recovers', even when given the relational construal here suggested, are ostensibly concerned with relations between propositions, such as those expressed by the sentences forming the conclusion and the premises of an argument. For the purpose of the present discussion, however, involvement with propositions can be avoided by construing the phrases in question as expressing logical relations between corresponding sentences, e.g., the conclusion-sentence and the premise-sentence of an argument. This construal, which underlies the formulation of (3c), will be adopted in this essay, though for the sake of convenience we may occasionally use a paraphrase.

6 In the familiar schematization of deductive arguments, with a single line separating the premises from the conclusion, no explicit distinction is made between a weaker and a stronger claim, either of which might be intended; namely (i) that the premises logically imply the conclusion and (ii) that, in addition, the premises are true. In the case of our probabilistic argument, (3c) expresses a weaker claim, analogous to (i), whereas (3d) may be taken to express a "proffered explanation" ( the term is borrowed from Scheffler (1957), section 1) in which, in addition, the explanatory premises are

however tentatively - asserted as true. 7 The considerations here outlined concerning the

use of terms like 'probably' and 'certainly' as modal qualifiers of individual statements seem to me to militate also against the notion of categorical probability statement that C. I. Lewis sets forth in the following passage (italics the author's):

Just as 'If D then (certainly) P, and D is the fact,' leads to the categorical consequence, 'Therefore (certainly) P'; so too, 'If D then probably P, and D is the fact', leads to a categorical consequence expressed by 'It is probable that P'. And this conclusion is not merely the statement over again of the probability relation between 'P' and 'D'; any more than 'Therefore (certainly) P' is the

354 CARL HEMPEL

statement over again of 'If D then (certainly) P'. 'If the barometer is high, tomorrow will probably be fair; and the barometer is high', categorically assures something expressed by 'Tomorrow will probably be fair'. This probability is still relative to the grounds of judgment; but if these grounds are actual, and contain all the available evidence which is pertinent, then it is not only categorical but may fairly be called the probability of the event in question. (1946, p. 319).

This position seems to me to be open to just those objections suggested in the main text. If 'P' is a statement, then the expressions 'certainly P'

and 'probably P' as envisaged in the quoted passage are not statements. If we ask how one would go about trying to ascertain whether they were true, we realize that we are entirely at a loss unless and until a reference set of statements or assumptions has been specified relative to which P may then be found to be certain, or to be highly probable, or neither. The expressions in question, then, are essentially incomplete; they are elliptic formulations of relational statements; neither of them can be the conclusion of an inference. However plausible Lewis's suggestion may seem, there is no analogue in inductive logic to modus ponens, or the "rule of detachment," of deductive logic, which, given the information that 'D', and also 'if D then P', are true statements, authorizes us to detach the

consequent 'P' in the conditional premise and to assert it as a self-contained statement which must then be true as well.

At the end of the quoted passage, Lewis suggests the important idea that 'probably P' might be tal<en to mean that the total relevant evidence avail

able at the time confers high probability upon P. But even this statement is relational in that it tacitly refers to some unspecified time, and, besides, his general notion of a categorical prob

ability statement as a conclusion of an argumentis not made dependent on the assumption that the premises of the argument include all the relevant evidence available.

It must be stressed, however, that elsewhere in his discussion, Lewis emphasizes the relativity of (logical) probability, and, thus, the very characteristic that rules out the conception of categorical probability statements.

Similar objections apply, I think, to Toulmin's construal of probabilistic arguments; cf. Toulmin (1958) and the discussion in Hempel (1960), sections 1-3.

8 Carnap (1950), p. 211. The requirement is suggested, for example, in

the passage from Lewis (1946) quoted in note 7

for this section. Similarly Williams speaks of "the most fundamental of all rules of probability logic, that 'the' probability of any proposition is its probability in relation to the known premises and them only." (Williams, 1947, p. 72).

I am greatly indebted to Professor Carnap for having pointed out to me in 1945, when I first noticed the ambiguity of probabilistic arguments, that this was but one of several apparent paradoxes of inductive logic that result from disregard of the requirement of total evidence.

Barker (1957), pp. 70-8, has given a lucid independent presentation of the basic ambiguity of probabilistic arguments, and a skeptical appraisal

of the requirement of total evidence as a means of dealing with the problem. However, I will presently suggest a way of remedying the ambiguity of probabilistic explanation with the help of a rather severely modified version of the requirement of total

evidence. It will be called the requirement of maximal specificity, and is not open to the same criticism.

9 Cf. Carnap (1950), p. 211 and p. 494. IO This idea is closely related to one used by Reichen

bach (cf. (1949), section 72) in an attempt to show that it is possible to assign probabilities to individual events within the framework of a strictly statistical conception of probability. Reichenbach proposed that the probability of a single

event, such as the safe completion of a particular scheduled flight of a given commercial plane, be construed as the statistical probability which the kind of event considered (safe completion of a flight) possesses within the narrowest reference class to which the given case (the specified flight

of the given plane) belongs, and for which reliable statistical information is available (for example, the class of scheduled flights undertaken so far by planes of the line to which the given plane belongs, and under weather conditions similar to those prevailing at the time of the flight in question).

11 Reference to s·k rather than to k is called for because, as was noted earlier, we do not construe the condition here under discussion as requiring that all the explanans statements invoked be scientifically accepted at the time in question, and thus be included in the corresponding class K.

12 By its reliance on this general idea, and specifically on the requirement of maximal specificity, the method here suggested for eliminating the epistemic ambiguity of statistical explanation differs substantially from the way in which I attempted in an earlier study (Hempel, 1962, especially section IO) to deal with the same problem. In that study,

which did not distinguish explicitly between the


two types of explanatory ambiguity characterized earlier in this section, I applied the requirement of total evidence to statistical explanations in a manner which presupposed that the explanans of any acceptable explanation belongs to the class K, and which then demanded that the probability which the explanans confers upon the explanandum be equal to that which the total evidence, K, imparts to the explanandum. The reasons why this approach seems unsatisfactory to me are suggested by the arguments set forth in the present section. Note in particular that, if strictly enforced, the requirement of total evidence would preclude the possibility of any significant statistical explanation for events whose occurrence is regarded as an established fact in science; for any sentence describing such an occurrence is logically implied by K and thus trivially has the logical probability 1 relative to K.

References

Barker, S. F., 1957, Induction and Hypothesis. Ithaca, NY: Cornell University Press.

Campbell, N. R., 1920, Physics: The Elements. Cambridge: Cambridge University Press.

Campbell, N. R., 1952[1921], What is Science? New York: Dover.

Carnap, R., 1950, Logical Foundations of Probability.

Chicago: University of Chicago Press. Cohen, M. R. and Nagel, E., 1934, An Introduction to

Logic and Scientific Method. New York: Harcourt, Brace and World.

Conant, James B., 1951, Science and Common Sense.

New Haven: Yale University Press. Dewey, John, 1910, How We Think. Boston: D. C.

Heath.

Donagan, A., 1957, "Explanation in History." Mind 66:

145-64.Goodman, N., 1947, "The Problem of Counterfac

tual Conditionals." The Journal of Philosophy 44: 113-28.

Hempel, C. G., 1960, "Inductive Inconsistencies." Synthese 12: 439-69.

Hempel, C. G., 1962, "Deductive-Nomological vs. Statistical Explanation," in H. Feig! and G. Maxwell (eds.), Minnesota Studies in the Philosophy of Science.

Vol. III, Minneapolis: University of Minnesota Press.

Hempel, C. G. and Oppenheim, P., 1948, "Studies in the Logic of Explanation." Philosophy of Science 15: 135-75.

Lewis, C. I., 1946, An Analysis of Knowledge and

Valuation. La Salle, Ill.: Open Court Publishing. Mises, R. van, 1939, Probability, Statistics and Truth.

London: William Hodge & Co. Popper, K. R., 1935, Logic der Forschung. Vienna:

Springer. Popper, K. R., 1959, The Logic of Scientific Discovery.

London: Hutchinson.

Popper, K. R., 1962, Conjectures and Refutations. New Yark: Basic Books.

Reichenbach, H., 1949, The Theory of Probability.

Berkeley and Los Angeles: The University of California Press.

Scheffler, I., 1957, "Explanation, Prediction, and Abstraction." The British Journal for the Philosophy

of Science 7: 293-309. Scriven, M., 1959, "Truisms as the Grounds for His

torical Explanations," in P. Gardiner (ed.), Theories

of History. New York: The Free Press. Toulmin, S., 1958, The Uses of Argument. Cambridge:

Cambridge University Press. Williams, D. C., 1947, The Ground of Induction.

Cambridge, MA: Harvard University Press.

5.3

Empiricism, Semantics, and Ontology

Rudolf Carnap

In this selection, Carnap draws a distinction between "internal" and "external" questions concerning the existence of various kinds of entity (e.g., numbers). Internal questions are those answered by appeal to the rules of the "linguistic framework" ( there is, for example, an even prime number according to the rules for mathematics). External questions are those posed by philosophers when they ask, for example, whether there really are numbers, notwithstanding what those rules imply. Carnap claims that only the internal questions of existence are meaningful; he therefore repudiates the external question of existence, and the philosophers' disputes concerning them, as incoherent.

I The Problem of Abstract Entities

Empiricists are in general rather suspicious with respect to any kind of abstract entities like properties, classes, relations, numbers, propositions, etc. They usually feel much more in sympathy with nominalists than with realists (in the medieval sense). As, far as possible they try to avoid any reference to abstract entities and to restrict themselves to what is sometimes called a nominalistic language, i.e., one not containing such references. However, within certain scientific contexts it seems hardly possible to avoid them. In the case of mathematics some empiricists try to find a way

out by treating the whole of mathematics as a mere calculus, a formal system for which no interpretation is given, or can be given. Accordingly, the mathematician is said to speak not about numbers, functions and infinite classes but merely about meaningless symbols and formulas manipulated according to given formal rules. In physics it is more difficult to shun the suspected entities because the language of physics serves for the communication of reports and predictions and hence cannot be talcen as a mere calculus. A physicist who is suspicious of abstract entities may perhaps try to declare a certain part of the language of physics as uninterpreted and uninterpretable,

From Revue Internationale de Philosophie 4 (1950): 20-40. As reprinted in the Supplement to Meaning and Necessity: A

Study in Semantics and Modal Logic, enlarged edn. (University of Chicago Press, 1956). © 1950. Reprinted by permis

sion of Revue Internationale de Philosophie.

EMPIRICISM, SEMANTICS, AND ONTOLOGY 357

that part which refers to real numbers as spacetime coordinates or as values of physical magnitudes, to functions, limits, etc. More probably he will just speak about all these things like anybody else but with an uneasy conscience, like a man who in his everyday life does with qualms many things which are not in accord with the high moral principles he professes on Sundays. Recently the problem of abstract entities has arisen again in connection with semantics, the theory of meaning and truth. Some semanticists say that certain expressions designate certain entities, and among these designated entities they include not only concrete material things but also abstract entities e.g., properties as designated by predicates and propositions as designated by sentences.1 Others object strongly to this procedure as violating the basic principles of empiricism and leading back to a metaphysical ontology of the Platonic kind.

It is the purpose of this article to clarify this controversial issue. The nature and implications of the acceptance of a language referring to abstract entities will first be discussed in general; it will be shown that using such a language does not imply embracing a Platonic ontology but is perfectly compatible with empiricism and strictly scientific thinking. Then the special question of the role of abstract entities in semantics will be discussed. It is hoped that the clarification of the issue will be useful to those who would like to accept abstract entities in their work in mathematics, physics, semantics, or any other field; it may help them to overcome nominalistic scruples.

2 Linguistic Frameworks

Are there properties classes, numbers, propositions? In order to understand more clearly the nature of these and related problems, it is above all necessary to recognize a fundamental distinction between two kinds of questions concerning the existence or reality of entities. If someone wishes to speak in his language about a new kind of entities, he has to introduce a system of new ways of speaking, subject to new rules; we shall call this procedure the construction of a linguistic fi-amework for the new entities in question. And now we must distinguish two kinds of questions of existence: first, questions of the existence of certain entities of the new kind within the framework;

we call them internal questions; and second, questions concerning the existence or reality of the system of entities as a whole, called external questions. Internal questions and possible answers to them are formulated with the help of the new forms of expressions. The answers may be found either by purely logical methods or by empirical methods, depending upon whether the framework is a logical or a factual one. An external question is of a problematic character which is in need of closer examination.

The world of things. Let us consider as an example the simplest kind of entities dealt with in the everyday language: the spatio-temporally ordered system of observable things and events. Once we have accepted the thing language with its framework for things, we can raise and answer internal questions, e.g., "Is there a white piece of paper on my desk?" "Did King Arthur actually live?", "Are unicorns and centaurs real or merely imaginary?" and the like. These questions are to be answered by empirical investigations. Results of observations are evaluated according to certain rules as confirming or disconfirming evidence for possible answers. (This evaluation is usually carried out, of course, as a matter of habit rather than a deliberate, rational procedure. But it is possible, in a rational reconstruction, to lay down explicit rules for the evaluation. This is one of the main tasks of a pure, as distinguished from a psychological, epistemology.) The concept of reality occurring in these internal questions is an empirical scientific non-metaphysical concept. To recognize something as a real thing or event means to succeed in incorporating it into the system of things at a particular space-time position so that it fits together with the other things as real, according to the rules of the framework.

From these questions we must distinguish the external question of the reality of the thing world itself. In contrast to the former questions, this question is raised neither by the man in the street nor by scientists, but only by philosophers. Realists give an affirmative answer, subjective idealists a negative one, and the controversy goes on for centuries without ever being solved. And it cannot be solved because it is framed in a wrong way. To be real in the scientific sense means to be an element of the system; hence this concept cannot be meaningfully applied to the system itself. Those who raise the question of the

358 RUDOLF CARNAP

reality of the thing world itself have perhaps in mind not a theoretical question as their formulation seems to suggest, but rather a practical question, a matter of a practical decision concerning the structure of our language. We have to make the choice whether or not to accept and use the forms of expression in the framework in question.

In the case of this particular example, there is usually no deliberate choice because we all have accepted the thing language early in our lives as a matter of course. Nevertheless, we may regard it as a matter of decision in this sense: we are free to choose to continue using the thing language or not; in the latter case we could restrict ourselves to a language of sense data and other "phenomenal" entities, or construct an alternative to the customary thing language with another structure, or, finally, we could refrain from speaking. If someone decides to accept the thing language, there is no objection against saying that he has accepted the world of things. But this must not be interpreted as if it meant his acceptance of a belief in the reality of the thing world; there is no such belief or assertion or assumption, because it is not a theoretical question. To accept the thing world means nothing more than to accept a certain form of language, in other words, to accept rules for forming statements and for testing accepting or rejecting them. The acceptance of the thing language leads on the basis of observations made, also to the acceptance, belief, and assertion of certain statements. But the thesis of the reality of the thing world cannot be among these statements, because it cannot be formulated in the thing language or, it seems, in any other theoretical language.

The decision of accepting the thing language, although itself not of a cognitive nature, will nevertheless usually be influenced by theoretical knowledge, just like any other deliberate decision concerning the acceptance of linguistic or other rules. The purposes for which the language is intended to be used, for instance, the purpose of communicating factual knowledge, will determine which factors are relevant for the decision. The efficiency, fruitfulness, and simplicity of the use of the thing language may be among the decisive factors. And the questions concerning these qualities are indeed of a theoretical nature. But these questions cannot be identified with the

question of realism. They are not yes-no questions but questions of degree. The thing language in the customary form works indeed with a high degree of efficiency for most purposes of everyday life. This is a matter of fact, based upon the content of our experiences. However, it would be wrong to describe this situation by saying: "The fact of the efficiency of the thing language is confirming evidence for the reality of the thing world; we should rather say instead: "This fact malces it advisable to accept the thing language."

The system of numbers. As an example of a system which is of a logical rather than a factual nature let us take the system of natural numbers. The framework for this system is constructed by introducing into the language new expressions with suitable rules: (I) numerals like "five" and sentence forms like "there are five books on the table"; (2) the general term "number" for the new entities, and sentence forms like "five is a number"; (3) expressions for properties of numbers (e.g. "odd," "prime"), relations (e.g., "greater than") and functions (e.g. "plus"), and sentence forms like "two plus three is five"; (4) numerical variables ("m," "n," etc.) and quantifiers for universal sentences ("for every n . . . ") and existential sentences ("there is an n such that . . . ") with the customary deductive rules.

Here again there are internal questions, e.g., "Is there a prime number greater than a hundred?" Here however the answers are found not by empirical investigation based on observations but by logical analysis based on the rules for the new expressions. Therefore the answers are here analytic, i.e., logically true.

What is now the nature of the philosophical question concerning the existence or reality of numbers? To begin with, there is the internal question which together with the affirmative answer, can be formulated in the new terms, say by "There are numbers" or, more explicitly, "There is an n such that n is a number." This statement follows from the analytic statement "five is a number" and is therefore itself analytic. Moreover, it is rather trivial (in contradistinction to a statement like "There is a prime number greater than a million which is likewise analytic but far from trivial), because it does not say more than that the new system is not empty; but this is immediately seen from the rule which states that words like "five" are substitutable for the new variables.


Therefore nobody who meant the question "Are there numbers?" in the internal sense would either assert or even seriously consider a negative answer. This makes it plausible to assume that those philosophers who treat the question of the existence of numbers as a serious philosophical problem and offer lengthy arguments on either side, do not have in mind the internal question. And indeed, if we were to ask them: "Do you mean the question as to whether the framework of numbers, if we were to accept it, would be found to be empty or not?" they would probably reply: "Not at all; we mean a question prior to the acceptance of the new framework." They might try to explain what they mean by saying that it is a question of the ontological status of numbers; the question whether or not numbers have a certain metaphysical characteristic called reality (but a kind of ideal reality, different from the material reality of the thing world) or subsistence or status of "independent entities." Unfortunately, these philosophers have so far not given a formulation of their question in terms of the common scientific language. Therefore our judgment must be that they have not succeeded in giving to the external question and to the possible answers any cognitive content. Unless and until they supply a clear cognitive interpretation, we are justified in our suspicion that their question is a pseudo-question, that is, one disguised in the form of a theoretical question while in fact it is a non-theoretical; in the present case it is the practical problem whether or not to incorporate into the language the new linguistic forms which constitute the framework of numbers.

The system of propositions. New variables, "p," "q," etc., are introduced with a role to the effect that any (declarative) sentence may be substituted for a variable of this kind; this includes, in addition to the sentences of the original thing language, also all general sentences with variables of any kind which may have been introduced into the language. Further, the general term "proposition" is introduced. "p is a proposition" may be defined by "p or not p" (or by any other sentence form yielding only analytic sentences). Therefore every sentence of the form " . . . is a proposition" (where any sentence may stand in the place of the dots) is analytic. This holds, for example, for the sentence:

(a) Chicago is large is a proposition.

(We disregard here the fact that the rules of English grammar require not a sentence but a that-clause as the subject of another sentence; accordingly instead of (a) we should have to say "That Chicago is large is a proposition.") Predicates may be admitted whose argument expressions are sentences; these predicates may be either extensional (e.g. the customary truth-functional connectives) or not (e.g. modal predicates like "possible," "necessaty," etc.). With the help of the new variables, general sentences may be formed, e.g.,

(b) "For every p, either p or not-p."

(c) "There is a p such that p is not necessary and not-p is not necessary."

( d) "There is a p such that p is a proposition."

(c) and (d) are internal assertions of existence.The statement "There are propositions" may be meant in the sense of (d); in this case it is analytic (since it follows from (a)) and even trivial. If, however, the statement is meant in an external sense, then it is non-cognitive.

It is important to notice that the system of rules for the linguistic expressions of the propositional framework ( of which only a few rules have here been briefly indicated) is sufficient for the introduction of the framework. Any further explanations as to the nature of the propositions (i.e., the elements of the system indicated, the values of the variables "p," "q," etc.) are theoretically unnecessary because, if correct, they follow from the rules. For example, are propositions mental events (as in Russell's theory)? A look at the rules shows us that they are not, because otherwise existential statements would be of the form: "If the mental state of the person in question fulfills such and such conditions, then there is a p such that . . . . " The fact that no references to mental conditions occur in existential statements (like (c), (d), etc.) shows that propositions are not mental entities. Further, a statement of the existence of linguistic entities ( e.g., expressions, classes of expressions, etc.) must contain a reference to a language. The fact that no such reference occurs in the existential statements here, shows that propositions are not linguistic entities. The fact that in these statements no reference to a subject

360 RUDOLF CARNAP

(an observer or knower) occurs (nothing like: "There is a p which is necessary for Mr X."), shows that the propositions (and their properties, like necessity, etc.) are not subjective. Although characterizations of these or similar kinds are, strictly speaking, unnecessary, they may nevertheless be practically useful. If they are given, they should be understood, not as ingredient parts of the system, but merely as marginal notes with the purpose of supplying to the reader helpful hints or convenient pictorial associations which may make his learning of the use of the expressions easier than the bare system of the rules would do. Such a characterization is analogous to an extra-systematic explanation which a physicist sometimes gives to the beginner. He might, for example, tell him to imagine the atoms of a gas as small balls rushing around with great speed, or the electromagnetic field and its oscillations as quasi-elastic tensions and vibrations in an ether. In fact, however, all that can accurately be said about atoms or the field is implicitly contained in the physical laws of the theories in question.2

The system of thing properties The thing language contains words like "red," "hard," "stone," "house," etc., which we used for describing what things are like. Now we may introduce new variables, say "f," "g," etc., for which those words are substitutable and furthermore the general term "property." New rules are laid down which admit sentences like "Red is a property," "Red is a color," "These two pieces of paper have at least one color in common" (i.e., "There is an f such that fis a color, and . . . "). The last sentence is an internal assertion. It is an empirical, factual nature. However, the external statement, the philosophical statement of the reality of properties - a special case of the thesis of the reality ofuniversals - is devoid of cognitive content.

The system of integers and rational numbers. Into a language containing the framework of natural numbers we may introduce first the (positive and negative) integers as relations among natural numbers and then the rational numbers as relations among integers. This involves introducing new types of variables, expressions substitutable for them, and the general terms "integer" and "rational number."

The system of real numbers. On the basis of the rational numbers, the real numbers may be introduced as classes of a special kind (segments) of

rational numbers (according to the method developed by Dedekind and Frege). Here again a new type of variables is introduced, expressions substitutable for them (e.g., "✓2"), and the general term "real number."

The spatio-temporal coordinate system for physics. The new entities are the space-time points. Each is an ordered quadruple of four real numbers, called its coordinates, consisting of three spatial and one temporal coordinates. The physical state of a spatio-temporal point or region is described either with the help of qualitative predicates (e.g., "hot") or by ascribing numbers as values of a physical magnitude ( e.g., mass, temperature, and the Iilce). The step from the system of things (which does not contain space-time points but only extended objects with spatial and temporal relations between them) to the physical coordinate system is again a matter of decision. Our choice of certain features, although itself not theoretical, is suggested by theoretical knowledge, either logical or factual. For example, the choice of real numbers rather than rational numbers or integers as coordinates is not much influenced by the facts of experience but mainly due to considerations of mathematical simplicity. The restriction to rational coordinates would not be in conflict with any experimental knowledge we have, because the result of any measurement is a rational number. However, it would prevent the use of ordinary geometry (which says, e.g., that the diagonal of a square with the side I has the irrational value ✓2) and thus lead to great complications. On the other hand, the decision to use three rather than two or four spatial coordinates is strongly suggested, but still not forced upon us, by the result of common observations. If certain events allegedly observed in spiritualistic seances, e.g., a ball moving out of a sealed box, were confirmed beyond any reasonable doubt, it might seem advisable to use four spatial coordinates. Internal questions are here, in general, empirical questions to be answered by empirical investigations. On the other hand, the external questions of the reality of physical space and physical time are pseudo-questions. A question like: "Are there (really) space-time points?" is ambiguous. It may be meant as an internal question; then the affirmative answer is, of course, analytic and trivial. Or it may be meant in the external sense: "Shall we introduce such and


such forms into our language?"; in this case it is not a theoretical but a practical question, a matter of decision rather than assertion, and hence the proposed formulation would be misleading. Or finally, it may be meant in the following sense: "Are our experiences such that the use of the linguistic forms in question will be expedient and fruitful?" This is a theoretical question of a factual, empirical nature. But it concerns a matter of degree; therefore a formulation in the form "real or not?" would be inadequate.

3 What Does Acceptance of a Kind

of Entities Mean?

Let us now summarize the essential characteristics of situations involving the introduction of a new kind of entities, characteristics which are common to the various examples outlined above.

The acceptance of a new kind of entities is represented in the language by the introduction of a framework of new forms of expressions to be used according to a new set of rules. There may be new names for particular entities of the kind in question; but some such names may already occur in the language before the introduction of the new framework. (Thus, for example, the thing language contains certainly words of the type of "blue" and "house" before the framework of properties is introduced; and it may contain words like "ten" in sentences of the form "I have ten fingers" before the framework of numbers is introduced.) The latter fact shows that the occurrence of constants of the type in question -regarded as names of entities of the new kind after the new framework is introduced - is not a sure sign of the acceptance of the new kind of entities. Therefore the introduction of such constants is not to be regarded as an essential step in the introduction of the framework. The two essential steps are rather the following. First, the introduction of a general term, a predicate of higher level, for the new kind of entities, permitting us to say for any particular entity that it belongs to this kind (e.g., "Red is a property," "Five is a number"). Second, the introduction of variables of the new type. The new entities are values of these variables; the constants (and the closed compound expressions, if any) are substitutable for the variables.3 With the help of the

variables, general sentences concerning the new entities can be formulated.

After the new forms are introduced into the language, it is possible to formulate with their help internal questions and possible answers to them. A question of this kind may be either empirical or logical; accordingly a true answer is either factually true or analytic.

From the internal questions we must clearly distinguish external questions, i.e., philosophical questions concerning the existence or reality of the total system of the new entities. Many philosophers regard a question of this kind as an ontological question which must be raised and answered before the introduction of the new language forms. The latter introduction, they believe, is legitimate only if it can be justified by an ontological insight supplying an affirmative answer to the question of reality. In contrast to this view, we take the position that the introduction of the new ways of speaking does not need any theoretical justification because it does not imply any assertion of reality. We may still speak (and have done so) of the "acceptance of the new entities" since this form of speech is customary; but one must keep in mind that this phrase does not mean for us anything more than acceptance of the new framework, i.e., of the new linguistic forms. Above all, it must not be interpreted as referring to an assumption, belief, or assertion of "the reality of the entities." There is no such assertion. An alleged statement of the reality of the system of entities is a pseudo-statement without cognitive content. To be sure, we have to face at this point an important question; but it is a practical, not a theoretical question; it is the question of whether or not to accept the new linguistic forms. The acceptance cannot be judged as being either true or false because it is not an assertion. It can only be judged as being more or less expedient, fruitful, conducive to the aim for which the language is intended. Judgments of this kind supply the motivation for the decision of accepting or rejecting the kind of entities.4

Thus it is clear that the acceptance of a linguistic framework must not be regarded as implying a metaphysical doctrine concerning the reality of the entities in question. It seems to me due to a neglect of this important distinction that some contemporary nominalists label the admission of variables of abstract types as "Platonism."5 This

362 RUDOLF CARNAP

is, to say the least, an extremely misleading terminology. It leads to the absurd consequence, that the position of everybody who accepts the language of physics with its real number variables (as a language of communication, not merely as a calculus) would be called Platonistic, even ifhe is a strict empiricist who rejects Platonic metaphysics.

A brief historical remark may here be inserted. The non-cognitive character of the questions which we have called here external questions was recognized and emphasized already by the Vienna Circle under the leadership of Moritz Schlick, the group from which the movement of logical empiricism originated. Influenced by ideas of Ludwig Wittgenstein, the Circle rejected both the thesis of the reality of the external world and the thesis of its irreality as pseudo-statements;6

the same was the case for both the thesis of the reality of universals ( abstract entities, in our present terminology) and the nominalistic thesis that they are not real and that their alleged names are not names of anything but merely flatus

vocis. (It is obvious that the apparent negation of a pseudo-statement must also be a pseudostatement.) It is therefore not correct to classify the members of the Vienna Circle as nominalists, as is sometimes done. However, ifwe look at the basic anti-metaphysical and pro-scientific attitude of most nominalists ( and the same holds for many materialists and realists in the modern sense), disregarding their occasional pseudo-theoretical formulations, then it is, of course, true to say that the Vienna Circle was much closer to those philosophers than to their opponents.

[ . . .]

. . . Generally speal<ing, if someone accepts a framework for a certain kind of entities, then he is bound to admit the entities as possible designata. Thus the question of the admissibility of entities of a certain type or of abstract entities in general as designata is reduced to the question of the acceptability of the linguistic framework for those entities. Both the nominalistic critics, who refuse the status of designators or names to expressions like "red," "five," etc., because they deny the existence of abstract entities, and the skeptics, who express doubts concerning the existence and demand evidence for it, treat the question of existence as a theoretical question. They

do, of course, not mean the internal question; the affirmative answer to this question is analytic and trivial and too obvious for doubt or denial, as we have seen. Their doubts refer rather to the system of entities itself; hence they mean the external question. They believe that only after mal<ing sure that there really is a system of entities of the kind in question are we justified in accepting the framework by incorporating the linguistic forms into our language. However, we have seen that the external question is not a theoretical question but rather the practical question whether or not to accept those linguistic forms. This acceptance is not in need of a theoretical justification ( except with respect to expediency and fruitfulness), because it does not imply a belief or assertion. Ryle says that the "Fido"-Fido principle is "a grotesque theory." Grotesque or not, Ryle is wrong in calling it a theory. It is rather the practical decision to accept certain frameworks. Maybe Ryle is historically right with respect to those whom he mentions as previous representatives of the principle, viz. John Stuart Mill, Frege, and Russell. If these philosophers regarded the acceptance of a system of entities as a theory, an assertion, they were victims of the same old, metaphysical confusion. But it is certainly wrong to regard my semantical method as involving a belief in the reality of abstract entities, since I reject a thesis of this kind as a metaphysical pseudo-statement.

The critics of the use of abstract entities in semantics overlook the fundamental difference between the acceptance of a system of entities and an internal assertion, e.g., an assertion that there are elephants or electrons or prime numbers greater than a million. Whoever malzes an internal assertion is certainly obliged to justify it by providing evidence, empirical evidence in the case of electrons, logical proof in the case of the prime numbers. The demand for a theoretical justification, correct in the case of internal assertions, is sometimes wrongly applied to the acceptance of a system of entities. Thus, for example, Ernest Nagel in his review7 asks for "evidence relevant for affirming with warrant that there are such entities as infinitesimals or propositions." He characterizes the evidence required in these cases - in distinction to the empirical evidence in the case of electrons - as "in the broad sense logical and dialectical." Beyond this


no hint is given as to what might be regarded as relevant evidence. Some nominalists regard the acceptance of abstract entities as a kind of superstition or myth, populating the world with fictitious or at least dubious entities, analogous to the belief in centaurs or demons. This shows again the confusion mentioned, because a superstition or myth is a false (or dubious) internal statement.

Let us take as example the natural numbers as cardinal numbers, i.e., in contexts like "Here are three books." The linguistic forms of the framework of numbers, including variables and the general term "number," are generally used in our common language of communication; and it is easy to formulate explicit rules for their use. Thus the logical characteristics of this framework are sufficiently dear (while many internal questions, i.e., arithmetical questions, are, of course, stillopen). In spite of this, the controversy concerning the external question of the ontological reality of the system of numbers continues. Supposethat one philosopher says: "I believe that thereare numbers as real entities. This gives me theright to use the linguistic forms of the numericalframework and to make semantical statementsabout numbers as designata of numerals." Hisnominalistic opponent replies: "You are wrong;there are no numbers. The numerals may still beused as meaningful expressions. But they are notnames, there are no entities designated by them.Therefore the word "number" and numerical variables must not be used (unless a way were foundto introduce them as merely abbreviating devices,a way of translating them into the nominalisticthing language)." I cannot think of any possibleevidence that would be regarded as relevant byboth philosophers, and therefore, if actually found,would decide the controversy or at least makeone of the opposite theses more probable than the other. (To construe the numbers as classes or properties of the second level, according to the Frege-Russell method, does, of course, not solvethe controversy, because the first philosopherwould affirm and the second deny the existenceof the system of classes or properties of the second level.) Therefore I feel compelled to regardthe external question as a pseudo-question, untilboth parties to the controversy offer a commoninterpretation of the question as a cognitive question; this would involve an indication of possibleevidence regarded as relevant by both sides.

There is a particular kind of misinterpretation of the acceptance of abstract entities in various fields of science and in semantics, that needs to be cleared up. Certain early British empiricists (e.g., Berkeley and Hume) denied the existence of abstract entities on the ground that immediate experience presents us only with particulars, not with universals, e.g., with this red patch, but not with Redness or Color-in-General; with this scalene triangle, but not with Scalene Triangularity or Triangularity-in-General. Only entities belonging to a type of which examples were to be found within immediate experience could be accepted as ultimate constituents of reality. Thus, according to this way of thinking, the existence of abstract entities could be asserted only if one could show either that some abstract entities fall within the given, or that abstract entities can be defined in terms of the types of entity which are given. Since these empiricists found no abstract entities within the realm of sense-data, they either denied their existence, or else made a futile attempt to define universals in terms of particulars. Some contemporary philosophers, especially English philosophers following Bertrand Russell, think in basically similar terms. They emphasize a distinction between the data (that which is immediately given in consciousness, e.g., sense-data, immediately past experiences, etc.) and the constructs based on the data. Existence or reality is ascribed only to the data; the constructs are not real entities; the corresponding linguistic expressions are merely ways of speech not actually designating anything (reminiscent of the nominalists' flatus vocis). We shall not criticize here this general conception. (As far as it is a principle of accepting certain entities and not accepting others, leaving aside any ontological, phenomenalistic and nominalistic pseudo-statements, there cannot be any theoretical objection to it.) But if this conception leads to the view that other philosophers or scientists who accept abstract entities thereby assert or imply their occurrence as immediate data, then such a view must be rejected as a misinterpretation. References to space-time points, the electromagnetic field, or electrons in physics, to real or complex numbers and their functions in mathematics, to the excitatory potential or unconscious complexes in psychology, to an inflationary trend in economics, and the like, do not imply the assertion that entities

364 RUDOLF CARNAP

of these kinds occur as immediate data. And the same holds for references to abstract entities as designata in semantics. Some of the criticisms

by English philosophers against such references give the impression that, probably due to the misinterpretation just indicated, they accuse the semanticist not so much of bad metaphysics (as some nominalists would do) but of bad psychology. The fact that they regard a semantical method involving abstract entities not merely as doubtful and perhaps wrong, but as manifestly absurd, preposterous and grotesque, and that they show a deep horror and indignation against this method, is perhaps to be explained by a misinterpretation of the kind described. In fact, of course, the semanticist does not in the least assert or imply that the abstract entities to which he refers can be experienced as immediately given either by sensation or by a kind of rational intuition. An assertion of this kind would indeed be very dubious psychology. The psychological question as to which kinds of entities do and which do not occur as immediate data is entirely irrelevant for semantics, just as it is for physics, mathematics, economics, etc., with respect to the examples mentioned above.8

5 Conclusion

For those who want to develop or use semantical methods, the decisive question is not the alleged ontological question of the existence of abstract entities but rather the question whether the rise of abstract linguistic foms or, in technical terms, the use of variables beyond those for things (or phenomenal data), is expedient and fruitful for the purposes for which semantical analyses are made, viz. the analysis, interpretation, clarification, or construction of languages of communication, especially languages of science. This question is here neither decided nor even discussed. It is not a question simply of yes or no, but a matter of degree. Among those philosophers who have carried out semantical analyses and thought about suitable tools for this work, beginning with Plato and Aristotle and, in a more technical way on the basis of modern logic, with C. S. Peirce and Frege, a great majority accepted

abstract entities. This does, of course, not provethe case. After all, semantics in the technical

sense is still in the initial phases of its development, and we must be prepared for possible fundamental changes in methods. Let us therefore admit that the nominalistic critics may possibly be right. But if so, they will have to offer better arguments than they have so far. Appeal to ontological insight will not carry much weight. The critics will have to show that it is possible to construct a semantical method which avoids all references to abstract entities and achieves by simpler means essentially the same results as the other methods.

The acceptance or rejection of abstract linguistic forms, just as the acceptance or rejection of any other linguistic forms in any branch of science, will finally be decided by their efficiency as instruments, the ratio of the results achieved to the amount and complexity of the efforts required. To decree dogmatic prohibitions of certain linguistic forms instead of testing them by their success or failure in practical use, is worse than futile; it is positively harmful because it may obstruct scientific progress. The history of science shows examples of such prohibitions based on prejudices deriving from religious, mythological, metaphysical, or other irrational sources, which slowed up the developments for shorter or longer periods of time. Let us learn from the lessons of history. Let us grant to those who work in any special field of investigation the freedom to use any form of expression which seems useful to them; the work in the field will sooner or later lead to the elimination of those forms which have no useful function. Let us be cautious in making assertions and critical in examining them, but tolerant in permitting linguistic forms.

Notes

The terms "sentence" and "statement" are here used

synonymously for declarative (indicative proposi

tional) sentences.

2 In my book Meaning and Necessity (Chicago,

1947) I have developed a semantical method which

takes propositions as entities designated by sentences

(more specifically, as intensions of sentences). In

order to facilitate the understanding of the system

atic development, I added some informal, extra

systematic explanations concerning the nature of

propositions. I said that the term "proposition" "is


used neither for a linguistic expression nor for a subjective, mental occurrence, but rather for something objective that may or may not be exemplified in nature ... We apply the term 'proposition' to any entities of a certain logical type, namely, those that may be expressed by (declarative) sentences in a language" (p. 27). After some more detailed discussions concerning the relation between propositions and facts, and the nature of false propositions, I added: "It has been the purpose of the preceding remarks

to facilitate the understanding of our conception of propositions. If, however, a reader should find these explanations more puzzling than clarifying, or even unacceptable, he may disregard them" (p. 31) (that is, disregard these extra-systematic explanations, not the whole theory of the propositions as intensions of sentences, as one reviewer understood). In spite of this warning, it seems that some of those readers who were puzzled by the explanations, did not disregard them but thought that by raising objections against them they could refute the theory. This is analogous to the procedure of some laymen who by (correctly) criticizing the ether picture or other visualizations of physical theories, thought they had refuted those theories. Perhaps the discussions in the present paper will help in clarifying the role of the system of linguistic rules for the introduction of a framework for entities on the one hand, and that of extra-systematic explanations concerning the nature of the entities on the other.

3 W. V. 0. Quine was the first to recognize the importance of the introduction of variables as indicating the acceptance of entities. "The ontology to which one's use of language commits him comprises simply the objects that he treats as falling .. . within the range of values of his variables." "Notes on Existence and Necessity," Journal of Philosophy, Vol. 40 (1943), pp. 113-127; compare also his "Designation and Existence," Journal of Philosophy,

Vol. 36 (1939), pp. 702-9, and "On Universals," The Journal of Symbolic Logic, Vol. 12 (1947), pp. 74-84.

4 For a closely related point of view on these questions see the detailed discussions in Herbert Feig!,

"Existential Hypotheses," Philosophy of Science, 17 (1950), pp. 35-62.

5 Paul Bernays, "Sur le platonisme clans les mathematiques" (L'Enseignement math., 34 (1935), 52-69). W. V. 0. Quine, see previous footnote and a recent paper ["On What There Is," Review of Metaphysics,

Vol. 2 (1948), pp. 21-38). Quine does not acknowledge the distinction which I emphasize above, because according to his general conception there are no sharp boundary lines between logical and factual truth, between questions of meaning and questions of fact, between the acceptance of a language structure and the acceptance of an assertion formulated in the language. This conception, which seems to deviate considerably from customary ways of thinking, is explained in his article "Semantics and Abstract Objects," Proceedings of the

American Academy of Arts and Sciences, 80 (1951), 90-6. When Quine in the article ["On What ThereIs") classifies my logistic conception of mathematics (derived from Frege and Russell) as "platonicrealism" (p. 33), this is meant (according to a personal communication from him) not as ascribingto me agreement with Plato's metaphysical doctrineof universals, but merely as referring to the factthat I accept a language of mathematics containingvariables of higher levels. With respect to the basicattitude to take in choosing a language form (an"ontology" in Quine's terminology, which seems tome misleading), there appears now to be agreementbetween us: "the obvious counsel is tolerance andan experimental spirit" (["On What There Is,"] p. 38).

6 See Carnap, Scheinprobleme in der Philosophie; das Fremdpsychische und der Realismusstreit, Berlin, 1928. Moritz Schlick, Positivismus und Realismus,

reprinted in Gesammelte Aufsatze, Wien, 1938. 7 Ernest Nagel, "Review of Meaning and Necessity,"

Journal of Philosophy 45 (1948): 467-72. 8 Wilfrid Sellars ("Acquaintance and Description

Again", in Journal of Philosophy, 46 (1949), 496-504; see pp. 502 f.) analyzes clearly the roots of the mistake "of taking the designation relation of semantic theory to be a reconstruction of being present to an experience."

5.4

The Pragmatic Vindication of Induction

Hans Reichenbach

Hans Reichenbach (1891-1953) was the founder of the Berlin Society, positivism's second point of origin (after the Vienna Circle). In this selection from his Experience and Prediction, he agrees with David Hume that we cannot justify belief in the reliability of the inductive method. Reichenbach argues that we can, nevertheless, justify our use of induction in guiding our expectations about the future. For it follows from the very concept of a limit of a frequency- the estimation of which is an induction's aim - that, if there is such a limit, the inductive method will succeed in uncovering it in the long run, and is the simplest rule that will do so.

§38 The Problem of Induction

So far we have only spoken of the useful qualities of the frequency interpretation. It also has dangerous qualities.

The frequency interpretation has two functions within the theory of probability. First, a frequency is used as a substantiation for the probability statement; it furnishes the reason why we believe in the statement. Second, a frequency is used for the verification of the probability statement; that is to say, it is to furnish the meaning of the statement. These two functions are not identical. The observed frequency from which

we start is only the basis of the probability inference; we intend to state another frequency which concerns future observations. The probability inference proceeds from a known frequency to one unknown; it is from this function that its importance is derived. The probability statement sustains a prediction, and this is why we want it.

It is the problem of induction which appears with this formulation. The theory of probability involves the problem of induction, and a solution of the problem of probability cannot be given without an answer to the question of induction. The connection of both problems is well known; philosophers such as Peirce have expressed the idea

From Hans Reichenbach, Experience and Prediction (Chicago: University of Chicago Press, 1961), pp. 339-42, 348-57.

© 1961 by Hans Reichenbach. Reprinted with permission from Maria Reichenbach.

THE PRAGMATIC VINDICATION OF INDUCTION 367

that a solution of the problem of induction is to be found in the theory of probability. The inverse relation, however, holds as well. Let us say, cautiously, that the solution of both problems is to be given within the same theory.

In uniting the problem of probability with that of induction, we decide unequivocally in favor of that determination of the degree of probability which mathematicians call the determination a posteriori. We refuse to acknowledge any so-called determination a priori such as some mathematicians introduce in the theory of the games of chance; on this point we refer to our remarks in §33, where we mentioned that the so-called determination a priori may be reduced to a determination a posteriori. It is, therefore, the latter procedure which we must now analyze.

By "determination a posteriori" we understand a procedure in which the relative frequency observed statistically is assumed to hold approximately for any future prolongation of the series. Let us express this idea in an exact formulation. We assume a series of events A and A (non-A); let n be the number of events, m the number of events of the type A among them. We have then the relative frequency

h11= mn

The assumption of the determination a posteriori may now be expressed:

For any further prolongation of the series as far as s events (s > n), the relative frequency will remain within a small interval around h0 i.e., we assume the relation

h11-£�h'�h11 +£

where £ is a small number. This assumption formulates the principle of

induction. We may add that our formulation states the principle in a form more general than that customary in traditional philosophy. The usual formulation is as follows: induction is the assumption that an event which occurred n times will occur at all following times. It is obvious that this formulation is a special case of our formulation, corresponding to the case h11 = 1. We cannot restrict our investigation to this special case because the general case occurs in a great many problems.

The reason for this is to be found in the fact that the theory of probability needs the definition of probability as the limit of the frequency. Our formulation is a necessary condition for the existence of a limit of the frequency near h11;

what is yet to be added is that there is an h11 of the kind postulated for every £ however small. If we include this idea in our assumption, our postulate of induction becomes the hypothesis that there is a limit to the relative frequency which does not differ greatly from the observed value.

If we enter now into a closer analysis of this assumption, one thing needs no further demonstration: the formula given is not a tautology. There is indeed no logical necessity that h' remains within the interval h11 ± £; we may easily imagine that this does not take place.

The nontautological character of induction has been known a long time; Bacon had already emphasized that it is just this character to which the importance of induction is due. If inductive inference can teach us something new, in opposition to deductive inference, this is because it is not a tautology. This useful quality has, however, become the center of the epistemological difficulties of induction. It was David Hume who first attacked the principle from this side; he pointed out that the apparent constraint of the inductive inference, although submitted to by everybody, could not be justified. We believe in induction; we even cannot get rid of the belief when we know the impossibility of a logical demonstration of the validity of inductive inference; but as logicians we must admit that this belief is a deception - such is the result of Hume's criticism. We may summarize his objections in two statements:

1. We have no logical demonstration for thevalidity of inductive inference.

2. There is no demonstration a posteriori for theinductive inference; any such demonstrationwould presuppose the very principle which itis to demonstrate.

These two pillars of Hume's criticism of the principle of induction have stood unshaken for two centuries, and I think they will stand as long as there is a scientific philosophy.

[ . . . ]

368 HANS REICHENBACH

39 The Justification of the Principle of Induction

We shall now begin to give the justification of induction which Hume thought impossible. In the pursuit of this inquiry, let us ask first what has been proved, strictly speaking, by Hume's objections.

Hume started with the assumption that a justification of inductive inference is only given if we can show that inductive inference must lead to success. In other words, Hume believed that any justified application of the inductive inference presupposes a demonstration that the conclusion is true. It is this assumption on which Hume's criticism is based. His two objections directly concern only the question of the truth of the conclusion; they prove that the truth of the conclusion cannot be demonstrated. The two objections, therefore, are valid only in so far as the Humean assumption is valid. It is this question to which we must turn: Is it necessary, for the justification of inductive inference, to show that its conclusion is true?

A rather simple analysis shows us that this assumption does not hold. Of course, if we were able to prove the truth of the conclusion, inductive inference would be justified; but the converse does not hold: a justification of the inductive inference does not imply a proof of the truth of the conclusion. The proof of the truth of the conclusion is only a sufficient condition for the justification of induction, not a necessa1y condition.

The inductive inference is a procedure which is to furnish us the best assumption concerning the future. If we do not know the truth about the future, there may be nonetheless a best assumption about it, i.e., a best assumption relative to what we know. We must ask whether such a characterization may be given for the principle of induction. If this turns out to be possible, the principle of induction will be justified.

An example will show the logical structure of our reasoning. A man may be suffering from a grave disease; the physician tells us: "I do not know whether an operation will save the man, but if there is any remedy, it is an operation." In such a case, the operation would be justified. Of course, it would be better to know that the operation will save the man; but, if we do not know this, the

knowledge formulated in the statement of the physician is a sufficient justification. If we cannot realize the sufficient conditions of success, we shall at least realize the necessary conditions. If we were able to show that the inductive inference is a necessary condition of success, it would be justified; such a proof would satisfy any demands which may be raised about the justification of induction.

Now obviously there is a great difference between our example and induction. The reasoning of the physician presupposes inductions; his knowledge about an operation as the only possible means of saving a life is based on inductive generalizations, just as are all other statements of empirical character. But we wanted only to illustrate the logical structure of our reasoning. If we want to regard such a reasoning as a justification of the principle of induction, the character of induction as a necessary condition of success must be demonstrated in a way which does not presuppose induction. Such a proof, however, can be given.

If we want to construct this proof, we must begin with a determination of the aim of induction. It is usually said that we perform inductions with the aim of foreseeing the future. This determination is vague; let us replace it by a formulation more precise in character:

The aim of induction is to find series of events whose frequency of occurrence converges toward a limit.

We choose this formulation because we found that we need probabilities and that a probability is to be defined as the limit of a frequency; thus our determination of the aim of induction is given in such a way that it enables us to apply probability methods. If we compare this determination of the aim of induction with determinations usually given, it turns out to be not a confinement to a narrower aim but an expansion. What we usually call "foreseeing the future" is included in our formulation as a special case; the case of knowing with certainty for every event A the event B following it would correspond in our formulation to a case where the limit of the frequency is of the numerical value 1. Hume thought of this case only. Thus our inquiry differs from that of Hume in so far as it conceives the aim of induction in a generalized form. But we do not omit any possible applications if we


determine the principle of induction as the means of obtaining the limit of a frequency. If we have limits of frequency, we have all we want, including the case considered by Hume; we have then the laws of nature in their most general form, including both statistical and so-called causal laws - the latter being nothing but a special case of statistical laws, corresponding to the numerical value 1 of the limit of the frequency. We are entitled, therefore, to consider the determination of the limit of a frequency as the aim of the inductive inference.

Now it is obvious that we have no guaranty that this aim is at all attainable. The world may be so disorderly that it is impossible for us to construct series with a limit. Let us introduce the term "predictable" for a world which is sufficiently ordered to enable us to construct series with a limit. We must admit, then, that we do not know whether the world is predictable.

But, if the world is predictable, let us ask what the logical function of the principle of induction will be. For this purpose, we must consider the definition of limit. The frequency h" has a limit at p, if for any given £ there is an n such that h'' is within p ± £ and remains within this interval for all the rest of the series. Comparing our formulation of the principle of induction (§38) with this, we may infer from the definition of the limit that, if there is a limit, there is an element of the series from which the principle of induction leads to the true value of the limit. In this sense the principle of induction is a necessary condition for the determination of a limit.

It is true that, if we are faced with the value h" for the frequency furnished by our statistics, we do not know whether this n is sufficiently large to be identical with, or beyond, the n of the "place of convergence" for £. It may be that our n is not yet large enough, that after n there will be a deviation greater than£ from p. To this we may answer: We are not bound to stay at h"; we may continue our procedure and shall always consider the last h" obtained as our best value. This procedure must at sometime lead to the true value p, if there is a limit at all; the applicability of this procedure, as a whole, is a necessary condition of the existence of a limit at p.

To understand this, let us imagine a principle of a contrary sort. Imagine a man who, if h" is reached, always makes the assumption that the

limit of the frequency is at h" + a, where a is a fixed constant. If this man continues his procedure for increasing n, he is sure to miss the limit; this procedure must at sometime become false, if there is a limit at all.

We have found now a better formulation of the necessa1y condition. We must not consider the individual assumption for an individual h"; we must take account of the procedure of continued assumptions of the inductive type. The applicability of this procedure is the necessary condition sought.

If, however, it is only the whole procedure which constitutes the necessary condition, how may we apply this idea to the individual case which stands before us? We want to know whether the individual h" observed by us differs less than£ from the limit of the convergence; this neither can be guaranteed nor can it be called a necessary condition of the existence of a limit. So what does our idea of the necessary condition imply for the individual case? It seems that for our individual case the idea turns out to be without any application.

This difficulty corresponds in a certain sense to the difficulty we found in the application of the frequency interpretation to the single case. It is to be eliminated by the introduction of a concept already used for the other problem: the concept of posit.

If we observe a frequency h" and assume it to be the approximate value of the limit, this assumption is not maintained in the form of a true statement; it is a posit such as we perform in a wager. We posit h" as the value of the limit, i.e., we wager on h", just as we wager on the side of a die. We know that h" is our best wager, therefore we posit it. There is, however, a difference as to the type of posit occurring here and in the throw of the die.

In the case of the die, we know the weight belonging to the posit: it is given by the degree of probability. If we posit the case "side other than that numbered l," the weight of this posit is 5/6. We speak in this case of a posit with appraised weight, or, in short, of an appraised posit.

In the case of our positing h", we do not know its weight. We call it, therefore, a blind posit. We know it is our best posit, but we do not know how good it is. Perhaps, although our best, it is a rather bad one.

370 HANS REICHENBACH

The blind posit, however, may be corrected. By continuing our series, we obtain new values h";

we always choose the last h". Thus the blind posit is of an approximative type; we know that the method of making and correcting such posits must in time lead to success, in case there is a limit of the frequency. It is this idea which furnishes the justification of the blind posit. The procedure described may be called the method of anticipa

tion; in choosing h" as our posit, we anticipate the case where n is the "place of convergence." It may be that by this anticipation we obtain a false value; we know, however, that a continued anticipation must lead to the true value, if there is a limit at all.

An objection may arise here. It is true that the principle of induction has the quality of leading to the limit, if there is a limit. But is it the only principle with such a property? There might be other methods which also would indicate to us the value of the limit.

Indeed, there might be. There might be even better methods, i.e., methods giving us the right value p of the limit, or at least a value better than ours, at a point in the series where h" is still rather far from p. Imagine a clairvoyant who is able to foretell the value p of the limit in such an early stage of the series; of course we should be very glad to have such a man at our disposal. We may, however, without knowing anything about the predictions of the clairvoyant, make two general statements concerning them: (1) The indications of the clairvoyant can differ, if they are true, only in the beginning of the series, from those given by the inductive principle. In the end there must be an asymptotical convergence between the indications of the clairvoyant and those of the inductive principle. This follows from the definition of the limit. (2) The clairvoyant might be an imposter; his prophecies might be false and never lead to the true value p of the limit.

The second statement contains the reason why we cannot admit clairvoyance without control. How gain such control? It is obvious that the control is to consist in an application of the inductive principle: we demand the forecast of the clairvoyant and compare it with later observations; if then there is a good correspondence between the forecasts and the observations, we shall infer, by induction, that the man's prophecies will also be true in the future. Thus it is the principle of

induction which is to decide whether the man is a good clairvoyant. This distinctive position of the principle of induction is due to the fact that we know about its function of finally leading to the true value of the limit, whereas we know nothing about the clairvoyant.

These considerations lead us to add a correction to our formulations. There are, of course, many necessary conditions for the existence of a limit; that one which we are to use however must be such that its character of being necessary must be known to us. This is why we must prefer the inductive principle to the indications of the clairvoyant and control the latter by the former: we control the unknown method by a known one.

Hence we must continue our analysis by restricting the search for other methods to those about which we may know that they must lead to the true value of the limit. Now it is easily seen not only that the inductive principle will lead to success but also that every method will do the same if it determines as our wager the value

h"+ c,,

where c,, is a number which is a function of n, or also of h", but bound to the condition

,\iP:;c,,=0

Because of this additional condition, the method must lead to the true value p of the limit; this condition indicates that all such methods, including the inductive principle, must converge asymptotically. The inductive principle is the special case where

c,, = 0

for all values of n. Now it is obvious that a system of wagers

of the more general type may have advantages. The "correction" c,,may be determined in such a way that the resulting wager furnishes even at an early stage of the series a good approximation of the limit p. The prophecies of a good clairvoyant would be of this type. On the other hand, it may happen also that c,, is badly determined, i.e., that the convergence is delayed by the correction. If the term c,, is arbitrarily formulated, we know


nothing about the two possibilities. The value c,, = 0 - i.e., the inductive principle - is therefore the value of the smallest risk; any other determination may worsen the convergence. This is a practical reason for preferring the inductive principle.

These considerations lead, however, to a more precise formulation of the logical structure of the inductive inference. We must say that, if there is any method which leads to the limit of the frequency, the inductive principle will do the same; if there is a limit of the frequency, the inductive principle is a sufficient condition to find it. If we omit now the premise that there is a limit of the frequency, we cannot say that the inductive principle is the necessary condition of finding it because there are other methods using a correction c

"' There is a set of equivalent con

ditions such that the choice of one of the members of the set is necessary if we want to find the limit; and, if there is a limit, each of the members of the set is an appropriate method for finding it.

We may say, therefore, that the applicability of the inductive principle is a necessary condition of the existence of a limit of the frequency.

The decision in favor of the inductive principle among the members of the set of equivalent means may be substantiated by pointing out its

quality of embodying the smallest risk; after all, this decision is not of a great relevance, as all these methods must lead to the same value of the limit if they are sufficiently continued. It must not be forgotten, however, that the method of clair

voyance is not, without further ado, a member

of the set because we do not know whether the correction c,, occurring here is submitted to the condition of convergence to zero. This must be proved first, and it can only be proved by using the inductive principle, viz., a method known to be a member of the set: this is why clairvoyance, in spite of all occult pretensions, is to be submitted to the control of scientific methods, i.e., by the principle of induction.

It is in the analysis expounded that we see the solution of Hume's problem. 1 Hume demanded

too much when he wanted for a justification of the inductive inference a proof that its conclusion is true. What his objections demonstrate is only that such a proof cannot be given. We do not perform, however, an inductive inference with the pretension of obtaining a true statement. What we obtain is a wager; and it is the best wager we can lay because it corresponds to a procedure the applicability of which is the necessary condition of the possibility of predictions. To fulfil the conditions sufficient for the attainment of true predictions does not lie in our power; let us be glad that we are able to fulfil at least the conditions necessary for the realization of this intrinsic aim of science.

Note

This theory of induction was first published by the author in Erkenntnis, III (1933), 421-5. A more detailed exposition was given in the author's Wehrscheinlichkeitslehre, §80.

5.5

Dissolving the Problem of Induction

Peter Strawson

Peter Strawson (1919-2006) was a prominent British philosopher in the "ordinary language" tradition, according to which philosophical disputes arise from conceptual misunderstandings that can be resolved by careful analysis of ordinary language. In this selection, he argues that David Hume's problem of induction rests on such a misunderstanding. Hume insists that we provide a reason to believe that induction is rational. But, Strawson responds, the inductive method precisely is the standard of rationality when it comes to reasoning from experience; to say that such reasoning is rational just is to say that it is in accord with the inductive method. The question whether the inductive method itself is rational therefore manifests a misunderstanding of the concept of rationality and of its range of application.

[ ... ]

II The 'Justification' of Induction

7 We have seen something, then, of the nature

of inductive reasoning; of how one statement or

set of statements may support another statement, S, which they do not entail, with varying degrees of strength, ranging from being conclusive evidence for S to being only slender evidence for it; from

making S as certain as the supporting statements,

to giving it some slight probability. We have seen,

too, how the question of degree of support is com

plicated by consideration of relative frequencies

and numerical chances. There is, however, a residual philosophical

question which enters so largely into discussion of the subject that it must be discussed. It can

be raised, roughly, in the following forms. What

reason have we to place reliance on inductive procedures? Why should we suppose that the

accumulation of instances of As which are Es, how

ever various the conditions in which they are

From P. F. Strawson, Introduction to Logical Theory (New York: John Wiley & Sons, 1952), pp. 248-52, 256-63. © 1952. Reproduced by permission of Taylor & Francis Books UK.

DISSOLVING THE PROBLEM OF INDUCTION 373

observed, gives any good reason for expecting the next A we encounter to be a B? It is our habit to form expectations in this way; but can the habit be rationally justified? When this doubt has entered our minds it may be difficult to free ourselves from it. For the doubt has its source in a confusion; and some attempts to resolve the doubt preserve the confusion; and other attempts to show that the doubt is senseless seem altogether too facile. The root-confusion is easily described; but simply to describe it seems an inadequate remedy against it. So the doubt must be examined again and again, in the light of different attempts to remove it.

If someone asked what grounds there were for supposing that deductive reasoning was valid, we might answer that there were in fact no grounds for supposing that deductive reasoning was always valid; sometimes people made valid inferences, and sometimes they were guilty of logical fallacies. If he said that we had misunderstood his question, and that what he wanted to know was what grounds there were for regarding deduction in general as a valid method of argument, we should have to answer that his question was without sense, for to say that an argument, or a form or method of argument, was valid or invalid would imply that it was deductive; the concepts of validity and invalidity had application only to individual deductive arguments or forms of deductive argument. Similarly, if a man asked what grounds there were for thinking it reasonable to hold beliefs arrived at inductively, one might at first answer that there were good and bad inductive arguments, that sometimes it was reasonable to hold a belief arrived at inductively and sometimes it was not. If he, too, said that his question had been misunderstood, that he wanted to know whether induction in general was a reasonable method of inference, then we might well think his question senseless in the same way as the question whether deduction is in general valid; for to call a particular belief reasonable or unreasonable is to apply inductive standards, just as to call a particular argument valid or invalid is to apply deductive standards. The parallel is not wholly convincing; for words like 'reasonable' and 'rational' have not so precise and technical a sense as the word 'valid'. Yet it is sufficiently powerful to make us wonder how the second question could be raised at all, to wonder why, in contrast with the corresponding question about deduction,

it should have seemed to constitute a genuine problem.

Suppose that a man is brought up to regard formal logic as the study of the science and art of reasoning. He observes that all inductive processes are, by deductive standards, invalid; the premises never entail the conclusions. Now inductive processes are notoriously important in the formation of beliefs and expectations about everything which lies beyond the observation of available witnesses. But an invalid argument is an unsound argument; an unsound argument is one in which no good reason is produced for accepting the conclusion. So if inductive processes are invalid, if all the arguments we should produce, if challenged, in support of our beliefs about what lies beyond the observation of available witnesses are unsound, then we have no good reason for any of these beliefs. This conclusion is repugnant. So there arises the demand for a justification, not of this or that particular belief which goes beyond what is entailed by our evidence, but a justification of induction in general. And when the demand arises in this way it is, in effect, the demand that induction shall be shown to be really a kind of deduction; for nothing less will satisfy the doubter when this is the route to his doubts.

Tracing this, the most common route to the general doubt about the reasonableness of induction, shows how the doubt seems to escape the absurdity of a demand that induction in general shall be justified by inductive standards. The demand is that induction should be shown to be a rational process; and this turns out to be the demand that one kind of reasoning should be shown to be another and different kind. Put thus crudely, the demand seems to escape one absurdity only to fall into another. Of course, inductive arguments are not deductively valid; if they were, they would be deductive arguments. Inductive reasoning must be assessed, for soundness, by inductive standards. Nevertheless, fantastic as the wish for induction to be deduction may seem, it is only in terms of it that we can understand some of the attempts that have been made to justify induction.

8 The first kind of attempt I shall consider might be called the search for the supreme premise of inductions. In its primitive form it is quite a crude attempt; and I shall make it cruder by

374 PETER STRA WSON

caricature. We have already seen that for a particular inductive step, such as 'The kettle has been on the fire for ten minutes, so it will be boiling by now', we can substitute a deductive argument by introducing a generalization (e.g., 'A kettle always boils within ten minutes of being put on the fire') as an additional premise. This manceuvre shifted the emphasis of the problem of inductive support on to the question of how we established such generalizations as these, which rested on grounds by which they were not entailed. But suppose the manceuvre could be repeated. Suppose we could find one supremely general proposition, which taken in conjunction with the evidence for any accepted generalization of science

or daily life (or at least of science) would entail that generalization. Then, so long as the status of the supreme generalization could be satisfactorily explained, we could regard all sound inductions

to unqualified general conclusions as, at bottom, valid deductions. The justification would be found, for at least these cases. The most obvious difficulty in this suggestion is that of formulating the supreme general proposition in such a way that it shall be precise enough to yield the desired entailments, and yet not obviously false or arbitrary. Consider, for example, the formula: 'For all f, g, wherever n cases off g, and no cases off - g, are observed, then all cases off are cases of g.' To turn it into a sentence, we have only to replace 'n' by some number. But what number? If we tal<e the value of 'n' to be 1 or 20 or 500, the resulting statement is obviously false. Moreover, the choice of any number would seem quite arbitrary; there is no privileged number of favourable instances which we take as decisive in establish

ing a generalization. If, on the other hand, we phrase the proposition vaguely enough to escape these objections - if, for example, we phrase it as 'Nature is uniform' - then it becomes too vague to provide the desired entailments. It should be noticed that the impossibility of framing a general proposition of the kind required is really a special case of the impossibility of framing precise rules for the assessment of evidence. If we could frame a rule which would tell us precisely

when we had conclusive evidence for a generalization, then it would yield just the proposition

required as the supreme premise. Even if these difficulties could be met, the

question of the status of the supreme premise

would remain. How, if a non-necessary proposition, could it be established? The appeal to experience, to inductive support, is clearly barred on pain

of circularity. If, on the other hand, it were a necessary truth and possessed, in conjunction with the evidence for a generalization, the required logical power to entail the generalization (e.g., if the latter were the conclusion of a hypothetical syllogism, of which the hypothetical premise was the necessary truth in question), then the evidence would entail the generalization independently, and the problem would not arise: a conclusion unbearably paradoxical. In practice, the extreme vagueness with which candidates for the role of supreme premise are expressed prevents their acquiring such logical power, and at the same time

renders it very difficult to classify them as analytic or synthetic: under pressure they may tend to tautology; and, when the pressure is removed, assume an expansively synthetic air.

In theories of the kind which I have here caricatured the ideal of deduction is not usually so blatantly manifest as I have made it. One finds the 'Law of the Uniformity of Nature' presented less as the suppressed premise of crypto-deductive

inferences than as, say, the 'presupposition of the validity of inductive reasoning'. I shall have more to say about this in my last section.

[ . . . ]

10 Let us turn from attempts to justify induc

tion to attempts to show that the demand for a justification is mistalzen. We have seen already that what lies behind such a demand is often the absurd wish that induction should be shown to be some kind of deduction - and this wish is clearly traceable in the two attempts at justification which we have examined. What other sense could we give to the demand? Sometimes it is expressed in the form of a request for proof that induction is a reasonable or rational procedure, that we have good grounds for placing reliance upon it. Consider the uses of the phrases 'good grounds', 'justification', 'reasonable', &c. Often we say such things as 'He has every justification for believing that p'; 'I have very good reasons for believing it'; 'There are good grounds for the

view that q'; 'There is good evidence that r'. We often tall<, in such ways as these, of justification, good grounds or reasons or evidence for certain


beliefs. Suppose such a belief were one expressible in the form 'Every case off is a case of g'. And suppose someone were asked what he meant by saying that he had good grounds or reasons for holding it. I think it would be felt to be a satisfactory answer if he replied: 'Well, in all my wide and varied experience I've come across innumerable cases of f and never a case of f which wasn't a case of g.' In saying this, he is clearly claiming to have inductive support, inductive evidence, of a certain kind, for his belief; and he is also giving a perfectly proper answer to the question, what he meant by saying that he had ample justification, good grounds, good reasons for his belief. It is an analytic proposition that it is reasonable to have a degree of belief in a statement which is proportional to the strength of the evidence in its favour; and it is an analytic proposition, though not a proposition of mathematics, that, other things being equal, the evidence for a generalization is strong in proportion as the number of favourable instances, and the variety of circumstances in which they have been found, is great. So to ask whether it is reasonable to place reliance on inductive procedures is like asking whether it is reasonable to proportion the degree of one's convictions to the strength of the evidence. Doing this is what 'being reasonable' means in such a context.

As for the other form in which the doubt may be expressed, viz., 'Is induction a justified, or justifiable, procedure?', it emerges in a still less favourable light. No sense has been given to it, though it is easy to see why it seems to have a sense. For it is generally proper to inquire of a particular belief, whether its adoption is justified; and, in asking this, we are asking whether there is good, bad, or any, evidence for it. In applying or withholding the epithets 'justified', 'well founded', &c., in the case of specific beliefs, we are appealing to, and applying, inductive standards. But to what standards are we appealing when we ask whether the application of inductive standards is justified or well grounded? If we cannot answer, then no sense has been given to the question. Compare it with the question: Is the law legal? It makes perfectly good sense to inquire of a particular action, of an administrative regulation, or even, in the case of some states, of a particular enactment of the legislature, whether or not it is legal. The question is answered by an appeal to a

legal system, by the application of a set of legal (or constitutional) rules or standards. But it makes no sense to inquire in general whether the law of the land, the legal system as a whole, is or is not legal. For to what legal standards are we appealing?

The only way in which a sense might be given to the question, whether induction is in general a justified or justifiable procedure, is a trival one which we have already noticed. We might interpret it to mean 'Are all conclusions, arrived at

inductively, justified?', i.e., 'Do people always have adequate evidence for the conclusions they draw?' The answer to this question is easy, but uninteresting: it is that sometimes people have adequate evidence, and sometimes they do not.

11 It seems, however, that this way of showing the request for a general justification of induction to be absurd is sometimes insufficient to allay the worry that produces it. And to point out that 'forming rational opinions about the unobserved on the evidence available' and 'assessing the evidence by inductive standards' are phrases which describe the same thing, is more apt to produce irritation than relief. The point is felt to be 'merely a verbal' one; and though the point of this protest is itself hard to see, it is clear that something more is required. So the question must be pursued further. First, I want to point out that there is something a little odd about talking of 'the inductive method', or even 'the inductive policy', as if it were just one possible method among others of arguing from the observed to the unobserved, from the available evidence to the facts in question. If one asked a meteorologist what method or methods he used to forecast the weather, one would be surprised if he answered: 'Oh, just the inductive method.' If one asked a doctor by what means he diagnosed a certain disease, the answer 'By induction' would be felt as an impatient evasion, a joke, or a rebuke. The answer one hopes for is an account of the tests made, the signs taken account of, the rules and recipes and general laws applied. When such a specific method of prediction or diagnosis is in question, one can ask whether the method is justified in practice; and here again one is asking whether its employment is inductively justified, whether it commonly gives correct results. This question would normally seem an admissible one. One might be tempted to conclude

376 PETER STRA WSON

that, while there are many different specific methods of prediction, diagnosis, &c., appropriate to different subjects of inquiry, all such methods could properly be called 'inductive' in the sense that their employment rested on inductive support; and that, hence, the phrase 'non-inductive method of finding out about what lies deductively beyond the evidence' was a description without

meaning, a phrase to which no sense had been given; so that there could be no question of justifying our selection of one method, called 'the inductive', of doing this.

However, someone might object: 'Surely it is possible, though it might be foolish, to use methods utterly different from accredited scientific ones. Suppose a man, whenever he wanted to form

an opinion about what lay beyond his observation or the observation of available witnesses, simply shut his eyes, asked himself the appropriate question, and accepted the first answer that came into his head. Wouldn't this be a noninductive method?' Well, let us suppose this. The man is asked: 'Do you usually get the right answer by your method?' He might answer: 'You've mentioned one of its drawbacks; I never do get the right answer; but it's an extremely easy method.' One might then be inclined to think that it was not a method of finding things out at all. But suppose he answered: Yes, it's usually (always) the right answer. Then we might be willing to call it a method of finding out, though a strange one. But, then, by the very fact of its success, it would be an inductively supported method. For each application of the method would be an application of the general rule, 'The first answer that comes into my head is generally (always) the right one'; and for the truth of this generalization there would be the inductive evidence of a long run of favourable instances with no unfavourable ones (if it were 'always'), or of a sustained high proportion of successes to trials (if it were 'generally').

So every successful method or recipe for finding out about the unobserved must be one which

has inductive support; for to say that a recipe is successful is to say that it has been repeatedly applied with success; and repeated successful application of a recipe constitutes just what we mean by inductive evidence in its favour. Pointing out this fact must not be confused with saying that 'the inductive method' is justified by its

success, justified because it works. This is a mistake, and an important one. I am not seeking to 'justify the inductive method', for no meaning

has been given to this phrase. A fortiori, I am not saying that induction is justified by its success in finding out about the unobserved. I am saying, rather, that any successsful method of finding out about the unobserved is necessarily justified by induction. This is an analytic proposition. The phrase 'successful method of finding things out which has no inductive support' is selfcontradictory. Having, or acquiring, inductive support is a necessary condition of the success of

a method. Why point this out at all? First, it may have

a certain therapeutic force, a power to reassure. Second, it may counteract the tendency to think of 'the inductive method' as something on a par with specific methods of diagnosis or prediction and therefore, like them, standing in need of (inductive) justification.

12 There is one further confusion, perhaps the most powerful of all in producing the doubts, questions, and spurious solutions discussed in this Part. We may approach it by considering the claim that induction is justified by its success in practice. The phrase 'success of induction' is by no means clear and perhaps embodies the confusion of induction with some specific method of prediction, &c., appropriate to some particular line of inquiry. But, whatever the phrase may mean,

the claim has an obviously circular look. Presumably the suggestion is that we should argue from the past 'successes of induction' to the continuance of those successes in the future; from the fact that it has worked hitherto to the conclusion that it will continue to work. Since an argument of this kind is plainly inductive, it will not serve as a justification of induction. One cannot establish a principle of argument by an argument which uses that principle. But let us go a little deeper. The argument rests the justification of induction on a matter of fact (its 'past successes'). This is characteristic of nearly all attempts to find a justification. The desired premise of Section 8 was to be some fact about the constitution of the universe which, even if it could not be used as a suppressed premise to give inductive arguments a deductive turn, was at any rate a 'presupposition of the validity of induction'. Even the mathematical argument of Section 9 required


buttressing with some large assumption about the make-up of the world. I think the source of this general desire to find out some fact about the constitution of the universe which will 'justify induction' or 'show it to be a rational policy' is the confusion, the running together, of two fundamentally different questions: to one of which the answer is a matter of non-linguistic fact, while to the other it is a matter of meanings.

There is nothing self-contradictory in supposing that all the uniformities in the course of things that we have hitherto observed and come to count on should cease to operate to-morrow; that all our familiar recipes should let us down, and that we should be unable to frame new ones because such regularities as there were were too complex for us to make out. (We may assume that even the expectation that all of us, in such circumstances, would perish, were falsified by someone surviving to observe the new chaos in which, roughly speaking, nothing foreseeable happens.) Of cuurse, we do not believe that this will happen. We believe, on the contrary, that our inductively supported expectation-rules, though some of them will have, no doubt, to be dropped or modified, will continue, on the whole, to serve us fairly well; and that we shall generally be able to replace the rules we abandon with others similarly arrived at. We might give a sense to the phrase 'success of induction' by calling this vague belief the belief that induction will continue to be successful. It is certainly a factual belief, not a

necessary truth; a belief, one may say, about the constitution of the universe. We might express it as follows, choosing a phraseology which will serve the better to expose the confusion I wish to expose:

I. (The universe is such that) induction willcontinue to be successful.

I is very vague: it amounts to saying that there are, and will continue to be, natural uniformities and regularities which exhibit a humanly manageable degree of simplicity. But, though it is vague, certain definite things can be said about it. (1) It is

not a necessary, but a contingent, statement; for chaos is not a self-contradictory concept. (2) We have good inductive reasons for believing it, good inductive evidence for it. We believe that some of our recipes will continue to hold good because they have held good for so long. We

believe that we shall be able to frame new and useful ones, because we have been able to do so repeatedly in the past. Of course, it would be absurd to tty to use I to 'justify induction', to show that it is a reasonable policy; because I is a conclusion inductively supported.

Consider now the fundamentally different statement:

IL Induction is rational (reasonable).

We have already seen that the rationality of induction, unlike its 'successfulness', is not a fact about the constitution of the world. It is a matter of what we mean by the word 'rational' in its application to any procedure for forming opinions about what lies outside our observations or that of available witnesses. For to have good reasons for any such opinion is to have good inductive support for it. The chaotic universe just envisaged, therefore, is not one in which induction would cease to be rational; it is simply one in which it would be impossible to form rational expectations to the effect that specific things would happen. It might be said that in such a universe it would at least be rational to refrain from forming specific expectations, to expect nothing but irregularities. Just so. But this is itself a

higher-order induction: where irregularity is the rule, expect further irregularities. Learning not to count on things is as much learning an inductive lesson as learning what things to count on.

So it is a contingent, factual matter that it is sometimes possible to form rational opinions concerning what specifically happened or will happen in given circumstances (I); it is a noncontingent, a priori matter that the only ways of doing this must be inductive ways (II). What people have done is to run together, to conflate, the question to which I is answer and the quite different question to which II is an answer, producing the muddled and senseless questions: 'Is the universe such that inductive procedures are rational?' or 'What must the universe be like in order for inductive procedures to be rational?' It is the attempt to answer these confused questions which leads to statements like 'The uniformity of nature is a presupposition of the validity of induction'. The statement that nature is uniform might be taken to be a vague way of expressing what we expressed by I; and certainly this fact is

378 PETER STRA WSON

a condition of, for it is identical with, the likewise contingent fact that we are, and shall continue to be, able to form rational opinions, of the kind we are most anxious to form, about the unobserved. But neither this fact about the world, nor any other, is a condition of the necessary truth that, if it is possible to form rational opinions of this kind, these will be inductively supported opinions. The discordance of the conflated questions manifests itself in an uncertainty about the status to be accorded to the alleged presupposition of the 'validity' of induction. For it was dimly, and

correctly, felt that the reasonableness of inductive

procedures was not merely a contingent, but a necessary, matter; so any necessary condition of their reasonableness had likewise to be a necessary matter. On the other hand, it was uncomfortably clear that chaos is not a self-contradictoty concept; that the fact that some phenomena do exhibit a tolerable degree of simplicity and repetitiveness is not guaranteed by logic, but is a contingent affair. So the presupposition of induction had to be both contingent and necessary: which is absurd. And the absurdity is only lightly veiled by the use of the phrase 'synthetic a priori' instead of 'contingent necessary'.

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