سکوت

PIERRE WAGNER

CARNAP’S THEORIES OF CONFIRMATION

The first theory of confirmation that Carnap developed in detail is to be found in “Testability and Meaning”.1 In this paper, he addressed the issue of a definition  of empiricism, several years after abandoning the quest for a unique and univer- sal logical framework supposed to be the basis of a clear distinction between the meaningful sentences of science and the pseudo-sentences of metaphysics. The principle of tolerance (according to which everyone is free to build up his own form of language as he wishes) had been adopted near the end of 1932, at a time when it was already obvious to Carnap that a strictly verificationist criterion of meaning was inadequate. He therefore considered a variety of empiricisms and a variety of choices for the language of science. As Carnap put it, “there are many different possibilities in framing an empiricist language”2 and, correspondingly, several degrees of liberalization of the criterion of meaning. It was in this con- text that Carnap provided both a logical (syntactical) and an empirical analysis   of confirmation (and of testing), before distinguishing requirements of different strengths which served the purpose of defining several versions of empiricism.

This use of confirmation and testing as a substitute for verification in the for- mulation of a criterion of meaning was a far cry from the theory of confirma-  tion that Carnap would begin to elaborate a few years later in the quite different context of his work on a system of inductive logic. In the meantime, however,   an intermediate stage of his thought on confirmation was expressed in the paper he read at the fifth International Congress for the Unity of Science (Cambridge, Mass., 1939), in which he borrowed a distinction from Charles Morris and divided the theory of language—and the analysis of science—into syntax, semantics, and pragmatics. At that time, Carnap insisted that the concept degree of confirmation was a pragmatic concept, not a logical (semantic) one. By this, he meant that the confirmation of a sentence, in contrast to its truth, is relative to some particular state of knowledge: “a statement of a degree of confirmation does not characterize an objective situation but rather the state of knowledge of a certain person with respect to a certain situation”.3 The need to make a sharp distinction between truth and confirmation had already been pointed out by Carnap in one of the papers he read4 at the Paris Congress on Scientific Philosophy in 1935, after Tarski had ex-

plained to him his work on the definability of a truth predicate in the framework of a semantic (as opposed to a purely syntactical) metalanguage. In 1939, the further category of a pragmatic analysis of the language of science was added, as well as the idea of considering degrees of confirmation: “If we wish to say that a certain sentence, e.g. the law of gravitation, has a high degree of confirmation, then we must add for whom and at what time, e.g. for Newton in the year such and such, or for the Chinese physicists in the year 1900.” “A statement of degree of confirma- tion (…) belongs to pragmatics.”5 Such statements are in sharp contrast both with the analysis which had been offered in “Testability and Meaning”, where the main issue was the formulation of a criterion of meaning and no concept of degree of confirmation had been considered, and with Carnap’s mature theory of confirma- tion, in which degree of confirmation would become a logical (semantic), not a pragmatic concept, formulated in terms of conditional probability: the confirma- tion of some hypothesis h would then be relativized to some evidence e, thus al- lowing a logical definition of the confirmation of h as the probability of h given e. From 1945 onward, in the context of his theory of inductive logic, Carnap distinguished a logical and a methodological concept of confirmation, devoting

most of his work to the former:

to decide to what degree h is confirmed by e—a question in logic, but here inductive, not in deductive logic—we need not know whether e is true or false, whether h is true or false, whether anybody believes in e, and, if so, whether on the basis of observation or of im- agination or of anything else. All we need is a logical analysis of the meanings of the two sentences. For this reason we call our problem the logical or semantical problem of confir- mation, in distinction to what might be called the methodological problems of confirmation, e.g., how best to construct and arrange an apparatus for certain experiments in order to test a given hypothesis, how to carry out the experiments, how to observe the results, etc.6

He then further distinguished between a positive concept of confirmation (“h is confirmed by e”), a comparative one (“h is more confirmed by e than h′ by e′”), and a quantitative one (“h is confirmed by e to the degree r”), devoting most of his work to the latter.

Carnap’s mature theory of confirmation depended on another distinction he made between two concepts of probability: the first one (“probability1”) was logi-

cal and had the meaning of degree of confirmation whereas the other one (prob-

ability2) was statistical and had the meaning of a relative frequency. More pre- cisely, in Logical Foundations of Probability7 (LFP), his explication of the logical

concept of probability was based on a theory of the quantitative concept of con- firmation. He defined confirmation functions, called c-functions, in such a way that “c(h,e) = r” meant that hypothesis h is confirmed or supported by evidence   e to the degree r. He also introduced some particular c-function called c*, which he regarded as the best candidate for an explication of the concept degree of con- firmation. Today, the theory of confirmation of 1950 and function c* constitute Carnap’s most well known explication of confirmation.

At the time he wrote LFP, Carnap thought it should be possible to isolate just one inductive method and one confirmation function for such an explication. In 1952, however, he defined a family of functions, the cλ functions (where index  λ

was a real number ranging from 0 to infinity), thus displaying an infinite number

of candidates for an explication of the same concept. In this way, he showed that his theory of confirmation actually was a general approach which led to an infinite number of inductive methods, each one corresponding to some particular con- firmation function. Later in the fifties, his theory of confirmation was improved in different ways (e.g. Carnap tried to extend it to the case of more complex lan- guages than the ones he had initially considered), and this led to the German book Induktive Logik und Wahrscheinlichkeit.8 Still later, in the sixties, he began to elaborate a different approach of the concept degree of confirmation, taking the theory of rational choice as a new basis and taking into account the recent de- velopments of semantics to which his friend and collaborator John Kemeny had contributed.9 Carnap’s new ideas in inductive logic and confirmation theory were developed in a series of papers which started with “The Aim of Inductive Logic”10 and ended with “A Basic System of Inductive Logic”, a book-length unfinished text which was to be published posthumously.11

As we can see, during more than thirty years, Carnap’s theory of confirmation evolved in different ways, offering a variety of approaches, from the logical and the empirical analyses of “Testability and Meaning” to the later theory of confir- mation expounded in “A Basic System of Inductive Logic”. Today,  this variety  of approaches is hardly known and in philosophical discussions about confirma- tion, if Carnap’s views are mentioned at all, the references are usually only to LFP and to the c*-function. And because the second (unrevised) edition of this book—a landmark in the history of philosophy of science—was printed in  1962,

the evolution of Carnap’s thought in the sixties is usually overlooked.12 The lack of interest in Carnap’s theory of confirmation is also due to its link with his logical interpretation of probability, an interpretation which is generally regarded as dead today (mainly because of the problem of assigning a priori probabilities), although some authors have recently evinced a renewed interest in logical probability.13 Carnap’s heroic efforts to find a solution to the general problem of defining a priori probabilities have often been considered as a desperate endeavour. Recently, how- ever, a more careful reading of his work on inductive logic—including his latest papers—has allowed more balanced judgements about his influence on contem- porary views on probability and confirmation: “he largely shaped the way current philosophy views the nature and role of probability, in particular its widespread acceptance of the Bayesian paradigm”.14

Because there is no place here for a detailed discussion of the different views Carnap espoused on confirmation, I shall now have to leave aside what he called the “classificatory” (or “positive”) and the “comparative” concepts of confirma- tion, and focus on his work on the “quantitative” one, from 1945 onward.

Carnap analysed degree of confirmation as a logical concept, and at that time, this meant a semantic concept, formulated in terms of conditional probability with a logical, epistemic interpretation. In this analysis, the confirmation of hypothesis

h being defined as p(h|e)— the probability of h given e—the problem was to find a

method for assigning values to such a probabilistic function, from a purely logical viewpoint. The logical character of function c (with arguments h and e) meant two things. First, it meant that the values of this function did not depend on any empiri- cal tests or pragmatic conditions, or on the factual truth or falsity of h and e, but only on the logical (semantic) analysis of these sentences, whatever their logical form and their truth value. Second, that the c-function was a logical function also meant that its arguments were sentences of some precisely defined language L. As a consequence, the logical analysis of h and e depended on the structure and the complexity of this language, and this was the reason why, in 1950, Carnap’s early work on inductive logic was restricted to quite simple languages. For example, all the individuals had to have a name in the language and two different individual constants of the language had to refer to two different objects. Also, in these sim- ple languages, predicates and relations had to be logically independent.15 Later, Carnap and other people tried to cover the case of more complex languages and

this is one direction in which he tried to improve his theory of confirmation later in the fifties.

As a consequence, the fact that the c-functions were regarded as logical func- tions also implied that the evaluation of c(h,e) depended on some particular se- mantics. From an historical viewpoint, it is no accident that Carnap’s inductive logic and his quantitative theories of confirmation were developed in the forties, after his semantic turn, and not before, during his syntactical period. The first systematic exposition of his semantic theory was published in 194216, and it is important to note that Carnap’s semantics was quite different from what has be- come standard semantics nowadays. In particular, when Carnap wrote LFP, he had neither the notion of an L-structure for language L nor the notion of truth in  a structure, and our standard notions of a model and of validity were also alien to his theory.

In Carnap’s semantics, the basic notions were state description and range, which were also fundamental for his theories of confirmation. Let us consider the simple example of a language L with three individual constants a, b and c and two one-place predicates P and Q, and let us take the conjunction of all the atomic sentences:

P(a)∧P(b) ∧P(c)∧Q(a)∧Q(b)∧Q(c).

This is one state description for language L. The basic idea is that this conjunction describes one of the possible states of the universe of discourse. If we take all the different ways of putting a negation before some or all the atomic sentences in this conjunction, we get all the other state descriptions for this language. The state de- scriptions for language L are descriptions of the different possible states of the uni- verse from the viewpoint of language L. If we now consider that a sentence A of L may be true for all, for some, or for none of the state descriptions, we can define the range of sentence A as the set of all the state descriptions which make it true. It is not difficult to discern some analogy between the state descriptions in Carnap’s semantics and the L-structures of our standard semantic theory. One of the main differences, however, is that in Carnap’s semantics the domain of discourse is fixed and reflected in the language by the individual constants. Moreover, whereas the L-structures of the standard Tarskian semantics are defined in a metalinguis- tic set-theoretic framework, Carnap’s state descriptions are given by formulas of the object language L itself. The reason for imposing restrictions on the object language is then obvious: complications arise as soon as an object of the universe does not have a name in the language or if two predicates such as “green( )” and “red(  )” occur, which are not logically independent.17 Another difficulty arises if

state descriptions are to be formulated as sentences and the language is intended for an infinite domain of objects. One of the main issues for Carnap in the fifties was to find ways of extending his theory of confirmation to the case of more real- istic languages than the simplistic ones he had considered in 1950.

How does Carnap define a confirmation function on this basis? Let us con- sider the simple case of languages LN with a finite number of one-place predicates

(which are logically independent from one another), and a finite number N of indi-

vidual constants. First, some metric is defined on the sentences of LN by a function m ascribing a real number m(A) between 0 and 1 to each formula A. This metric (to be interpreted as a probability function) is said to be regular if the following conditions are satisfied:

1. – if A is a state description, then m(A)>0 (this means that each state of the universe is possible);
2. – the sum of all the m(Ai) (where i is an index for all the state descriptions) is equal to 1 (this means that the state descriptions describe all the pos-

sible states of the universe);

3. – if the range of a formula A is null (i.e. if A is false for every state descrip- tion), then m(A)=0 (this means that if a sentence is not possible, its prob- ability is equal to zero);
4. – if the range of A is not null, then m(A) is the sum of the m(Ai) where i is an index for all the state descriptions in the range of A (this means that the

probability of a sentence depends on the number of state descriptions for which it is true).

The metric defined by the regular function m is a way of ascribing some prob- ability value to all the sentences of language LN, in such a way that the laws of probability theory apply.

The requirements imposed on the regular functions m are extremely weak: they leave room for an infinite number of possible m-functions, and one of the main issues is to find a way of assigning precise values so as to get a realistic func- tion m. For example, we may be tempted to proceed in assigning the same value to each state description, arguing that each state description has an equal a priori probability to obtain. If we proceed in this way, we get some particular function, which Carnap calls mw (because he attributes it to Wittgenstein). But he argues that this method actually produces a confirmation function which does not match the intuitive idea we have of the properties that a confirmation method should satisfy. The function cw does not have the properties we expect.

From an epistemological viewpoint, there are here two important issues:

1– Which function m do we have to choose? Some particular language L be- ing given, which values do we have to assign to each sentence of L?

2– What kind of argument shall we put forward to justify our choice of one function rather than another? What will be the basis of our choice?

In any case, the reasons we may give for our choice actually apply to the c-func- tions rather than to the m-functions, so let us first see how the confirmation c- functions are defined on the basis on the m-functions.

The definition is quite simple. Given some regular m-function—call it m—, if we see it as a function assigning probabilities to each sentence of the given lan- guage L, the confirmation function—call it c—with arguments h and e is defined as a conditional probability. For this definition, we assume that evidence e is true for at least one state description, so that m(e) is not zero:

c(h,e) = df m(h.e) / m(e).

This means that for arguments h and e, the value of the confirmation function c is defined as the conditional probability of h given e (p(h|e), which is nothing but p(h.e) / p(e)), where the values of the probability function p are just those of the measure function m. In the special case where e L-implies h, we can easily check that c(h,e) = 1, so that logical implication is a special case of confirmation. Clearly, Carnap’s theory of confirmation is a logical one; for any arguments h and e, c(h,e) depends on the logical form of h and e and on the logical structure of the language, not on any empirical testing.

The choice of the m-function determines the definition of the c-function but this still leaves room for an infinite number of possibilities for m. In 1950, Carnap tried to narrow down this number and argued for the choice of one special c- function that he called c*. Two reasons for this choice were put forward. The first idea was that a logical theory should make no discrimination between individuals. This meant that if some state description results from some other state description by a permutation of individual constants, they actually describe the same structure of the world, and an m-function should assign the same value to both of them. An m-function which satisfies this propriety is said to be a symmetrical function. The second idea is that an m-function should assign the same value not to each state description, but to each set of state descriptions which express the same structure of the universe of discourse. This is an application of the principle of indifference not to state descriptions, but to structure descriptions.

Applying these two principles, we get the metric m*, and, applying the fore- going definition of a c-function to m*, we get the confirmation function c*. For some time, Carnap seemed to consider function c* as the best candidate for an ex- plication of confirmation. It happens, however, that c* has serious shortcomings, of which I shall mention just one. Suppose h has a universal form, (this is the case, for instance, if h is a universal law of the form “for all x, if P(x) then Q(x)”) and suppose that the universe is infinite. In this case, we get the unexpected result that

c*(h,e) = 0, which was regarded as a serious objection to the adoption of c* as an explication of confirmation. In defence of c*, Carnap gave an ad hoc argument— which few people found really convincing—according to which the use of laws is not indispensible for making predictions and that science can actually do without universal laws.18

On the issue of the choice of a confirmation function, important progress was made in The Continuum of Inductive Methods.19 In this important book, Carnap managed to prove a fundamental theorem to the effect that it is possible to reduce the number of parameters on which the choice of a regular and symmetrical con- firmation function depends. It is even possible to reduce this number to just one parameter which Carnap called λ, each value of λ defining one particular inductive method. To give just a hint at the intuitive meaning of λ, notice first that the values of some confirmation function cλ depend both on an empirical factor (the evidence

we have) and on a logical factor (the language system we use). What the parameter

λ determines is the precise balance between these two factors in the computation of the values of the c-function. In the book, Carnap also examined some properties of the c-functions we get when we choose some particular values for λ.

From an epistemological viewpoint, one of the conclusions Carnap drew was that the choice of one precise confirmation function does not depend only on a purely logical basis, but also on its application and on the success we achieve when using some particular inductive method. A conclusion which might seem somewhat surprising for such a supporter of a logical theory of confirmation as Carnap was. Two remarks, however, should be added here.

First, we should not confuse what Carnap called a theory of confirmation and what he meant by a method of confirmation. The theory is purely logical and a priori, whereas the choice of a method also depends of the particular application we want to make of the theory. Second, the conclusion I have just mentioned was actually not maintained by Carnap in the sixties, after he adopted a quite different approach to the theory of confirmation. At that time, he insisted that “in principle it is never necessary to refer to experiences in order to judge the rationality of     a C-function”.20 Here, he mentions the “rationality” of the c-function, and this denotes a change in his vocabulary. In 1950, Carnap did not hesitate to identify logical probability and degree of confirmation, or to explicate the former by the latter, even though he also mentioned other explanations for the concept of logi- cal probability, such as a fair betting quotient or an estimate of relative frequency. Later, he realized that the phrase “degree of confirmation” was somewhat ambigu- ous and even misleading: it could mean either “degree of support” or “increase of the degree of support” and he therefore preferred avoiding using that phrase when

explicating logical probability. Two historical reasons why Carnap changed his approach in the sixties should also be mentioned.

First, an important turn took place in the history of logic at the time Kemeny, who had worked with Carnap in Princeton, introduced a new approach to seman- tics, thereby proposing to replace Carnap’s notion of state descriptions by a new notion of models for logical systems.21 In his later papers on inductive logic, Car- nap used the abstract notion of a model and used propositions and events rather than sentences as the basic concepts of his system.

Second, an important turn also occurred in the history of probability theory in the fifties, when more and more people became interested in subjective interpreta- tions of probability and in the works of Ramsey and de Finetti. Though Carnap did not adopt a psychological interpretation of probability in terms of subjective degree of belief, he then used the setting and the concepts of decision theory. He defined a credibility function CredX which represented the disposition of some

person X for having beliefs. The values CredX (h,e) of this function were measured

by X’s behaviour in situations where she had to bet on hypothesis h while know-

ing e. More precisely, CredX (h,e) was defined as the highest betting quotient on which X would be willing to bet on h, if her total knowledge were e. Now, how can we define a confirmation function on the basis of such a subjective credibility function? Carnap’s idea was to lay down axioms that the credibility function must satisfy in order to represent a rational disposition for having beliefs. Instead of considering some real person X, he therefore proposed to adopt the viewpoint of some idealized entity, some robot or machine able to learn, to have beliefs, and  to bet on a hypothesis, and to state axioms which would characterize a purely rational behaviour and, as a consequence, a purely rational credibility function. A confirmation function was then defined as a function c such that c (h,e) was exactly CredX (h,e), where X was a purely rational entity. Using the setting of decision theory, Carnap distinguished descriptive decision theory, which states psychologi- cal laws and “normative decision theory, which states conditions of rationality for decisions”.22 But the axioms Carnap proposed still left open the possibility of an infinite number of confirmation functions and for him, it was not clear how far we could go in narrowing down the choice to a smaller number by stating new axioms.

This raised the issue of the justification of the axioms for defining a rational c-function. In the context I have just described, Carnap criticized the idea of a justification based on past experiences, or even on general synthetic principles such as the principle of the uniformity of the world. He preferred the idea of a jus- tification by what he called “the ability of inductive intuition”.23 When using this