The Deductivist Theory of Probability and Statistics

Abstract

This chapter outlines one of three common views of probability and statistics: the deductivist view, discussed through a key proponent, Rudolf Carnap. This theory suggests that logic is the basis of mathematics and hence of probability and statistics. It requires absolute certainty in intuitions and deductions, thus not allowing for flexibility and adjustment. The chapter ultimately concludes that the deductive theory fails because of lack of applications. Today the view has few proponents. However, the limitations of this theory point to what is needed: more viable ways of comprehending uses of digital logic, the nature of starting points of inquiry, and the roles of consequences, applications, and problem-solving within inquiry. Carnap’s view also provides key incipient insights into how “context” or “conditions” need to play a key role in understanding statistics and probability.

Cultural xenophobia is a frequent sequel to a society’s decline from cultural vigor. Someone has aptly called self-imposed isolation a fortress mentality. Armstrong describes it as a shift from faith in logos, reason, with its future-oriented spirit, “always…seeking to know more and to extend…areas of competence and control of the environment,” to mythos, meaning conservatism that looks backward to fundamentalist beliefs for guidance and a worldview. A fortress or fundamentalist mentality not only shuts itself off from dynamic influences originating outside but also, as a side effect, ceases influencing the outside world. (Jacobs, Jane, 2004, The Dark Age Ahead, New York: Random House, p. 17)

Ideas in modern Russian [the Soviet Union] are machine-cut blocks coming in solid colors; the nuance is outlawed, the interval walled up, the curve grossly stepped. (From p. 243, by Vladimir Nabokov, Pale Fire, 1962, New York: Perigee Books)

Addendum 1: Carnap’s “L”-Language with Only Denumerable Values and His Admitted Limitations for Scientific Applications

As indicated in the text, on the one hand, one might expect Carnap to be arguing that one can justify his logic in terms of its success, and this might include a lengthy discussion of how information systems have evolved from logic. In contrast, Carnap maintains that logic is what it is, apart from its applications, and that it should be universal for any system of concepts that fit the particular language in question. Put in other terms, Carnap’s logic forbids that should not commit the “fallacy of consequence” but should instead justify matters in a linear fashion—one step at a time.

In Logical Foundations of Probability, Carnap presents two languages: L_n and L_∞. The former consists of a finite system and the latter consists of a denumerably large system. In general, one must form the system from

• Countably many logically independent individuals, i₁, i₂, …

• Clear-cut, independent, and binary (either applicable or not) primitive classes/predicates/properties/modalities (presumably color, shape, and so on)

• Atomic sentences—which ascribe primitive properties to an individual

• Extensional combinations of atomic sentences (for everything complex or “molecular,” one uses an inclusive “or,” a tilde for negation, or an “and”)

• Binary assignments (“T” or “F”) to all sentences so formed.

Finite or denumerably infinite systems are basically in terms of information technology “digital” as opposed to “analog.” Only a countable number of variables, sentences, individuals, and state descriptions exist. In this digital world, irrational numbers become truncated. Statistical distributions become a disjunction of a countable number of individual distributions. So, how does one apply such a language L (whether L_n or L_∞)? If individuals are logically independent and primitive properties are logically independent, how can probabilistic statements be logical? Suppose that a probabilistic statement is of the Carnap/ Keynes formulation P(H|E) or the probability of H given E. Suppose further that both H and E are atomic sentences. Then, the sentences H and E therefore imply some value ip(H|E), a conditional probabilistic estimate. To that extent, H and E are not independent. H and E are thus relevant to each other or else ip(H|E) = 0, still a logical implication. Thus, at a minimum, there are serious challenges for deriving probabilistic estimates for two atomic sentences in Carnap’s theory of probability.

Digital information systems—along with many related products such as digital pictures—has had enormous successes—but with some obvious sacrifices. To what world does this digital logic apply without sacrifices? Carnap’s response appears to be that there is no world that he can think of that fits languages L_n or L_∞ perfectly. He states:

It seems best to imagine as individuals in a system L, not extended regions like physical bodies or events in our actual world, but rather positions like the space-time points in our actual world, hence unextended, indivisible entities. Since, however, the number of individuals in a system L is either finite or denumerably infinite, they cannot form a continuum… the qualities and relations with which we are acquainted in our actual world cannot, strictly speaking be applied. For instance, a color occurs in the actual world only as a property of extended, continuous area.

Later, Carnap rules out length, mass, and temperature. One might hope that a statement of the form “at point (x,y,z) at time t the temperature is T,” as being suitably atomic, but not even this is the case. Carnap admits that the actual language of science and even that of elementary physics has, of course, a much more complex structure.³⁰ The whole language of science has “its great complexities, its large variety of forms of expression, and its variables of higher levels (e.g., for real numbers).” This, though, requires a much more thorough rational reconstruction than Carnap pursues in his Logical Foundations of Probability. The simpler languages so far constructed have no applications whatsoever, and so the null set of applications is all that remains at this point of this “universal” logic.^{31 , 32}

The status of the metalanguage may be reconsidered at this stage. As with other proposed languages and their elements, the metalanguage is clearly not an L-language. The metalanguage is one may presume supra-logical, that is, beyond the restrictions of the L-language logic. Statements about the author’s goals—while to be taken at face value—as well as statements about other’s misconceptions stretch well beyond the sphere of risk-free logic.

Note that with sacrifices or tradeoffs, this logic could have immense applications and hence successes. Alternatively, Carnap in his later writings and his followers may pursue a course of a risk-free logic that is also universal.