For Carnap, the importance of constructing an inductive logic on the basis of an adequate concept of degree of confirmation, e.g., a function like c∗, is not merely to explicate the logi- cal concept of probability. We may also be interested in explicating other inductive concepts, including concepts of relevance, estimation, information and even of entropy which are concep- tually tied to the logical concept of probability.171 Specifically, once an adequate explicatum
for the logical concept of probability is found, this explicatum can then be used to construct adequate explicata for a host of related inductive explicanda. As Carnap puts it, the concept of degree of confirmation, understood as an explicatum for the logical concept of probability, is “the fundamental concept of inductive logic” (513). It is in this wider sense of explication – of explicating an entire system of concepts based on the explication of a single concept at the conceptual core of this system – that Carnap’s work on finding an adequate quantitative induc- tive logic is an explication of inductive reasoning. Yet finding an adequate explication of logical probability which could then be used to explicate an entire system of inductive concepts is not a trivial task; as Carnap puts it, we can only find such a concept by providing the right sort of reasons for adopting it, e.g., reasons like “the fact that in many actual or imagined knowledge situations the values of c are sufficiently in agreement with the inductive thinking of a careful scientist” (540).
Turning our attention to the problem of estimation in theoretical statistics, Carnap says that the state of the field of theories of estimation, at least from the point of view from “treatises on probability and statistics” is
a startling spectacle of unsolved controversies and mutual misunderstandings, all the more disturbing when we compare it with the exactness, clarity and possibility of coming to a general agreement in other fields of mathematics. (LFP 513)
The problem of estimation is basically the problem of finding an adequate estimation, based on both an estimation function and past observations, of the value of some unknown physical quantity, or rather, an estimate of some parameter representing a physical quantity.172 As
Carnap puts the point, one can think of an estimate given by an estimation function for a physical quantity as a sort of guess – not an arbitrary guess but rather a reasonable guess (512).
171 I don’t discuss Carnap’s work on relevance in any detail in this dissertation; see LFP, chapter VI. 172
Once found, such a concept will not only play an important role in everyday scientific activity but also a foundational role in any theory of rational decision making (LFP §§49-51; also see chapter 5 of this dissertation). But the problem, at least according to Carnap in 1950, is that there is no general theory of statistical estimation. Rather, as Carnap notes, there are instead several, competing, theoretical accounts of statistical inference and estimation, including R. A. Fisher’s work on maximal likelihoods, Abraham Wald’s work on statistical decision functions, Jerzy Neyman and Egon Pearson’s statistical hypothesis testing relative to type I and II errors and Neyman’s confidence intervals (515-518). Moreover, these statistical accounts of estimation functions are all based on a frequentist or statistical concept of probability. But then “[w]hy did statisticians,” asks Carnap, “spend so much effort in developing methods of estimation, i.e., methods not based on a [logical concept of probability - CFF]?” (518). The short answer, according to Carnap, is that because of the historical association of a principle of indifference (or principle of insufficient reason) with the logical concept of probability – a principle found to lead to contradictions by scientists as early as Carl Gauss – only a theory of estimations based on a frequentist concept of probability could possibly be adequate (518).
In response, Carnap articulates two possible options. The first is to suppose that no adequate quantitative inductive logic will be found; then
the methods developed by Fisher, Neyman, Pearson, and Wald or new methods of a similar nature are presumably the best instruments for estimating parameter values and testing hypotheses. They are ingenious devices for achieving these ends without making use of any general explicatum for [the logical concept of probability - CFF], as far as the ends can be achieved under this restricting condition. (518)
Alternatively, however, suppose that an adequate inductive logic is found, viz. an inductive logic which does not depend on any unrestricted application of the principle of indifference. Then, according to Carnap,
the main reason for developing independent methods of estimation and testing would vanish. Then it would seem more natural to take the degree of confirmation as the basic concept for all of inductive statistics. (518)
This question of which of these two alternatives is more likely is connected to a problem Carnap had raised a few pages earlier in LFP. The unsatisfactory state of the theory of estimation is due to a problem that besets most theoretical fields in science: “any procedure of estimation depends upon a choice, which is a matter of practical decision and not uniquely determined by purely
theoretical, logico-mathematical considerations” (514). As Carnap points out, many procedures of science involve such a choice, like choosing a geometry for physical space. However, what is advantageous about the question of whether we can find an adequate inductive logic that could be used as a basis for a theory of estimation is that, says Carnap, “only one fundamental decision is required” (514). As Carnap continues to say:
As soon as anybody makes this decision, that is to say, chooses a concept of degree of confirmation which seems to him adequate, then he is in the possession of a general method of which makes it possible to deal with all the various problems of inductive logic in a coherent and systematic way, including the problems of estimation. Thus this method helps to overcome what seems to me the greatest weakness in the contemporary statistical theory of estimation, namely, the lack of a general method. (514)
This is an insightful passage into Carnap’s understanding of the theoretical issues at hand. By reconstructing the results of theoretical statistics and probability as depending on the choice of a single inductive concept of degree of confirmation, Carnap suggests that one could provide a grand foundation for all of statistics and probability – a general method capable of clarifying and systematizing inductive reasoning, including reasoning about how to construct non-arbitrary “guesses” or estimations for physical, but unknown, quantitative properties. It is in this way that Carnap hopes to contribute to the foundations of theoretical statistics.
Now that we have a better sense of the potential import for Carnap’s work on estimation functions, I next turn to the details of his work on estimation functions. Suppose, firstly, that R(u) is a discrete random variable representing the result of observing some physical magnitude, relative to the physical input u, which ranges over the possible values r1, r2, ..., rn and, secondly, that one of the ri is really the actual value of this physical magnitude. Provided we have evidence for previous instances of R(u), call it e, and that the sentences h1, ..., hn denote the (logically exclusive) hypotheses that the actual value of the unknown quantity is r1, ...,
rn, respectively, then Carnap suggests we can define the estimate of R(u) as a weighted mean (where the weights are confirmation values). More specifically, assuming ‘e’ logically implies ‘h1∨ · · · ∨ hn’, the estimate e is defined as follows, (see D100-1)
e(R, u, e) = n ∑
i=1
[ri× c(hi, e)]. (4.2)
unique estimation functions based on a particular class of confirmation functions. Specifically, supposing we had a continuum of different confirmation functions to choose from and that we could define a unique estimation function based on each such confirmation function, we could then investigate how well particular estimation functions behave for different “states of the universe,” or to use a more formal mode of speech, for different state descriptions. Of course, then Carnap would need to have some notion of how “reliable” different estimation functions are. Although Carnap considers several different ways of explicating such a notion, I will cut to the chase and quickly discuss the explicatum Carnap focuses on (see LFP §100B and §102). Assuming that ˆr is the actual but unknown value of the physical quantity measured by R(u), the error of the estimate e, or v, is defined as
v(R, u, e) = e(R, u, e)− ˆr. (4.3)
As is standard (because the estimation of this error term is always zero), Carnap takes for the explicatum of the reliability of estimation functions the estimate of the squared error, f2, i.e., the weighted average of these error functions, squared, where the weights are given, like above, in terms of confirmation functions.173 Importantly, the estimation of squared error is useful if
the actual value of R(u) is genuinely unknown. However, one can easily calculate a value of ˆ
r relative to some fixed state description. For example, suppose we assume that a single state description in L is the actual one, then, if R(u) is a measure of the frequency of individuals in that state description which hold of M , ˆr is simply the actual frequency of M ’s in this state description. Then instead of explicating the reliability of an estimation function in terms of f2, we can instead use the mean squared error, m2, defined relative to ˆr to investigate the relative reliability of estimation functions relative to a fixed, completely known, state description.174
In section 4.5 of this chapter, we will see that Carnap uses this notion of the mean squared error and his λ-system to try and find “optimal” estimation functions. This work constitutes, I argue, one of the clearest examples of how Carnap uses his work on inductive logic to solve a foundational problem and that this process resembles a kind of conceptual engineering activity.
173
Specifically, f2(R, u, e) =Dfe(v2, R, u, e) = ∑
i[(e(R, u, e)− ri)2× c(hi, e)].
174 Relative to our current observed sample of s-many individuals, the mean squared error of e is defended as m2(e, ˆr) = v2 (Carnap, 1952, 56-59).
However, before we can discuss that example we first need to examine the λ-system in detail.