Consideration is given mathematical problems arising in two learning theories—one developed by Bush and Mosteller, the other developed by Estes. The theory of Bush and Mosteller leads to a class of Markov processes which have been studied in considerable detail (see [1] and [7]). The Estes model can be treated as a Markov chain, i.e., a Markov process with a finite number of states. For an important class of special cases, it is shown that the Bush-Mosteller model is, in a sense, a limiting form of the Estes model. The limiting probability distributions are derived for the cases treated in both models.

I will give precise versions of the foregoing verbal definitions, by formalizing them within a formal metalanguage.

In order to avoid the necessity of formulating a meta-language, I will make use of VI 112, (i.e., Gödel's The consistency of the continuum hypothesis). Actually, the system there developed is unnecessarily strong for the limited purpose of this illustration. It will suffice to choose for the system of VI 112 without axioms C4 and E (axioms of replacement and choice); this system will be the logical basis of , and to this I will add some set-constants and some meaning postulates. The logical basis is roughly as strong as Zermelo's system with an axiom of infinity.

Within I will set up analogues of the previous definitions, applicable to a wide variety of object-languages. For the sake of concreteness it will be convenient to think of as having Zermelo set theory, without the axiom of infinity, as its logical basis – but it must be kept in mind that this is not an essential feature of the forthcoming definitions.

Our is certainly strong enough to have all primitive recursive functions calculable within it. This supplies us with the needed tool for talking about , within , by introducing a number of primitive recursive functions, properties, and relations describing the structure of . The exact definitions of these will depend on the exact form of , but for the forthcoming definitions it will suffice to know that these functions, properties, and relations are representable in for the given . This makes the resulting definitions most useful, since they can be applied to any of the type described.

This is the first in a series of articles outlining a new approach to Semantics. The novelty in the approach is that the concept of an interpretation of a logical system is taken as the central concept of Semantics. I hope to show that by means of this approach a satisfactory definition can be given for such controversial concepts as analyticity, and at the same time the approach leads to a unified foundation for formalized Semantics.

As the possibility of such definitions has been questioned in recent years, it is an important task to try to give precise definitions. Clearly, this is the task of those philosophers who believe that concepts like analyticity should play a fundamental role in Semantics. On the other hand, the philosophers who have criticized these concepts will now be able to tell just exactly why these definitions are unacceptable to them — instead of being forced to talk in generalities. It seems to me that no further progress is possible until we have precise definitions available for discussion.

The fundamental semantic concepts fall into two classes: those, like truth, for which Tarski has offered definitions; and those additional ones, like analytic truth, for which we have Carnap's proposed definitions.

The question of what constitutes fairness in betting quotients has been studied by Ramsey, deFinetti, and Shimony. Thanks to their combined efforts we now have a satisfactory definition of fairness.

On the other hand, the explication of the concept of degree of confirmation (inductive probability) has progressed rapidly in recent years, thanks primarily to Carnap. This explication has usually proceeded by laying down the axioms for frequency-probabilities, and elaborating on these. While in the case where a frequency interpretation is intended these axioms are clearly justified, in our case they have been laid down without any justification. Carnap's presentation has been criticized for just this reason.

The purpose of this paper is to show that the probability axioms are necessary and sufficient conditions to assure that the degrees of confirmation form a set of fair betting quotients. In addition it will be shown that one additional, highly controversial, axiom is precisely the condition needed to assure that not only deFinetti's weaker criterion but Shimony's criterion of fairness is also satisfied.

In 1948 Hempel and Oppenheim proposed an explicatum for the concept of systematic power 1, pp. 164–167. Since that time some shortcomings have been found in this first attempt. It is the purpose of this paper to show that one can keep the basic approach of the ‘48 paper, and overcome the known disadvantages by means of changes in the details of the definition. In this improvement certain tools will be used that were not available in 1948.