Taking for the relation of confirmation the following obvious axioms, we obtain several more or less well-known theorems and are able to solve in a definite and strict manner several problems concerning confirmation.

Let a, b, and c be variable names of sentences belonging to a certain class, the operations a·b, a + b, and ā the (syntactical) product, the sum, and the negation of them. Let us further assume the existence of a real non-negative function c(a, b) of a and b, when b is not self-contradictory. Let us read ‘c(a, b)’ ‘degree of confirmation of a with respect to b' and take the following axioms:

Axiom I. If a is a consequence of b, c(a, b) = 1.

Axiom II. If is a consequence of c,

Axiom III. c(a·b, c) = c(a, c)·c(b, a·c).

Axiom IV. If b is equivalent to c, c(a, b) = c(a, c).

As may be easily seen, the interval of variation for c is (0, 1); this is quite conventional.

The theory of relations was discussed, along with the theory of classes, by Peirce and Schröder. While the calculus of classes has subsequently been presented (by Couturat, and by Huntington) as an abstract mathematical system, no similar formulation has been published of the calculus of relations. Professor Alfred Tarski has proposed to me, the problem of developing a similar formulation for the theory of relations. In this paper I present a set of postulates for the calculus of relations.

The postulates have been chosen so as to enable us to prove Theorem A, below, which asserts a kind of completeness of the system. A set of postulates is called semi-categorical, if every two realizations which involve the same number of elements are isomorphic. From Theorem A one can easily prove that my postulates are semi-categorical, if one assumes the following theorem from Mengenlehre: if α and β are cardinal numbers such that , then α=β. This theorem has not been proved, however, except on the assumption of the axiom of choice and the generalized continuum hypothesis, and is not assumed in the present paper.

It should be pointed out, that the sort of completeness I have chosen is not the only sort, or even necessarily the most interesting sort, of completeness which a set of postulates for the calculus of relations could possess. For example, the question arises, whether it would be possible to find a set of postulates from which all true equations (involving free variables and the operations used by Schröder) could be derived, and which was the weakest possible such set. The present work should therefore be regarded only as a first step in the investigation of the axiomatization of the calculus of relations.

An individual or whole we understand to be whatever is represented in any given discourse by signs belonging to the lowest logical type of which that discourse makes use. What is conceived as an individual and what as a class is thus relative to the discourse within which the conception occurs. One task of applied logic is to determine which entities are to be construed as individuals and which as classes when the purpose is the development of a comprehensive systematic discourse.

The concept of an individual and that of a class may be regarded as different devices for distinguishing one segment of the total universe from all that remains. In both cases, the differentiated segment is potentially divisible, and may even be physically discontinuous. The difference in the concepts lies in this: that to conceive a segment as a whole or individual offers no suggestion as to what these subdivisions, if any, must be, whereas to conceive a segment as a class imposes a definite scheme of subdivision—into subclasses and members.

The relations of segments of the universe are treated in traditional logistic at two places, first in its theorems concerning the identity and diversity of individuals, and second in its calculus of membership and class-inclusion. But further relations of segments and of classes frequently demand consideration. For example, what is the relation of the class of windows to the class of buildings? No member of either class is a member of the other, nor are any of the segments isolated by the one concept identical with segments isolated by the other. Yet the classes themselves have a very definite relation in that each window is a part of some building. We cannot express this fact in the language of a logistic which lacks a part-whole relation between individuals unless, by making use of some special physical theory, we raise the logical type of each window and each building to the level of a class—say a class of atoms—such that any class of atoms that is a window will be included (class-inclusion) in some class that is a building. Such an unfortunate dependence of logical formulation upon the discovery and adoption of a special physical theory, or even upon the presumption that such a suitable theory could in every case be discovered in the course of time, indicates serious deficiencies in the ordinary logistic. Furthermore, a raising of type like that illustrated above is often precluded in a constructional system by other considerations governing the choice of primitive ideas.

Bei dem Bestreben, die Mathematik zu logisieren, erschienen die von Russell u.a. entdeckten Paradoxieen als das größte Hindernis. Zu ihrer Überwindung wurde die Typentheorie von Russell aufgestellt, die aber auch in ihrer gemilderten Form die Mathematiker nicht befriedigt. Ich bin diesen Schwierigkeiten nie begegnet, und zwar deshalb nicht, weil logisieren für mich immer bedeutet hat: in Schröderschen Relativkalkul einkleiden.

Ich bedaure es aus mehr als einem Grunde schwer, daß man von dem eleganten Peirce-Schröderschen Kalkul abgewichen ist und die Peano-Russellschen Zeichen benutzt, in denen aus bestimmten (m.A.n. nicht stichhaltigen) Gründen die sonst selbstverständliche Harmonie und Schönheit der Mathematik preisgegeben wurde, wodurch die Forschung ziemlich einseitig in eine bestimmte Richtung gedrängt wurde (in der freilich Hochbedeutendes geleistet wurde) während beim Peirce-Schröderschen Kalkul auf Schönheit besonderer Wert gelegt und mit wunderbarer Instinktsicherheit gerade das mathematisch Bedeutsame und Fruchtbare getroffen wurde. Ich bin der Überzeugung, daß in Wissenschaft und Technik das Zweckmäßige immer auch zugleich das Schöne ist. Mit Russells Zeichen hätte ich manches nicht entdeckt, was ich gefunden habe, und besäße jedenfalls nicht meine grosse Kalkulfreudigkeit. Ungünstige Zeichen lähmen die Produktivität. Der heutige Logikkalkul entbehrt daher jeder Eleganz und Geschmeidigkeit, was ganz und gar nicht in der Natur der Sache liegt. Meine Bemühungen, bestimmte Teile des Kalkuls flüssiger, geschmeidiger zu handhaben, sind unbeachtet geblieben.

The purpose of the present paper is to give a formulation of the simple theory of types which incorporates certain features of the calculus of λ-conversion. A complete incorporation of the calculus of λ-conversion into the theory of types is impossible if we require that λx and juxtaposition shall retain their respective meanings as an abstraction operator and as denoting the application of function to argument. But the present partial incorporation has certain advantages from the point of view of type theory and is offered as being of interest on this basis (whatever may be thought of the finally satisfactory character of the theory of types as a foundation for logic and mathematics).

For features of the formulation which are not immediately connected with the incorporation of λ-conversion, we are heavily indebted to Whitehead and Russell, Hilbert and Ackermann, Hilbert and Bernays, and to forerunners of these, as the reader familiar with the works in question will recognize.

The class of type symbols is described by the rules that ı and o are each type symbols and that if α and β are type symbols then (αβ) is a type symbol: it is the least class of symbols which contains the symbols ı and o and is closed under the operation of forming the symbol (αβ) from the symbols α and β.

The purpose of this note is to call attention to a minor error in Lewis and Langford's Symbolic logic. On page 221, in discussing the Tarski-Łukasiewicz three-valued logic, the authors make the following assertion: “Let T(p) be any proposition, involving only one element, whose analogue holds in the two-valued system; if T(p) does not hold in the Three-valued Calculus, then pC.T(p) and Np.C.T(p) both hold.”

I shall show, by means of a counter-example, that this assertion is not true. Let T(p) be the sentence:

It is then easily verified that T(0) = T(1) = 1, and that T(½) = 0. Thus T(p) holds in the two-valued calculus, but not in the three-valued calculus. On the other hand, pC.T(p) does not hold, since ½.CT(½) = ½C0 = ½; similarly, Np.C.T(p) does not hold, since N½.C.T(½) = ½C0 = ½.

Il s'agit d'un problème combinatoire de logique formelle, formulé par Jevons; il sera expliqué en détails dans ce qui suit (voir no. 1). Jevona luimême n'a traité le problème que dans les cas les plus simples (n = 1, 2, 3); un cas plus difficile (n = 4) a été traité par Clifford; le cas général (n quelconque) a été à peine abordé.

Le but de ce travail est de faire remarquer que ce problème de Jevons et de Clifford est contenu comme cas particutier dans un problème combinatoire général que j'ai traité ailleurs. La méthode générale ramène le problème présent à l'étude d'un certain groupe de permutations d'ordre n!2n, étroitement lié au groupe symétrique d'ordre n!. J'ai fait les calculs nécessaires pour n = 1, 2, 3, 4. Mes résultats numèriques sont complètement en accord avec les résultats de Jevons, mais ils ne s'accordent qu'en partie avec les résultats de Clifford.

Une proposition peut être vraie ou fausse. On peut exprimer la même chose en disant que nous pouvons attribuer à une proposition l'une ou l'autre des deux “valeurs logiques” qui s'excluent mutuellement: la “vérité” et la “fausseté.”

The object of this note is to show that there is no finite characteristic matrix for any one of Lewis and Langford's systems.

Theorem I. There is no finite characteristic matrix for any one of the systems S1–S5.

Proof. Let M be a matrix with less than n elements for which all the provable formulas in S1, or S2, or S3, or S4, or S5, are satisfied. Let Fn represent the formula

where ∑ stands for a ∨-chain, and the pi, are variables in any one of the calculi. Using M, there is always at least one summand in Fn where pi and pk have the same value. Therefore, Fn can always be written in the form (a = a)∨ B, and thus will give, for any B, a “designated” value, since the formula (p = p) ∨ q is provable in any one of the systems S1–S5.

Give to any one of the systems S1–S5, the following matrix, due to Henle:

1. Elements: all possible classes formed from the integers 1, 2, 3, …, n.

2. “Designated” element: the class {1, 2, 3, …,n}.

3. Boole-Schröder algebra on the elements.

4.

Under certain circumstances a postulate can be eliminated in favor of a mere definition, or convention of notational abbreviation. Suppose e.g. that a given system presupposes the machinery of ordinary logic and contains in addition a single extra-logical primitive: the relation Pt, “is a spatial part of.” Suppose this relation is governed by a single extra-logical postulate, to the effect that Pt is transitive:

Now instead of Pt, the relation O of spatial overlapping might be taken as primitive; Pt could then be defined in terms of O as follows:

The transitivity of Pt then follows by purely logical principles from the definition. The statement (1) with its ‘Pt’ clauses expanded according to (2) is a purely logical theorem, demonstrable independently of any stipulations or assumptions concerning the properties of O. Thus, through a change involving neither increase nor decrease of primitive ideas, the need of adopting (1) as a postulate is removed.

Or again, consider a system comprising ordinary logic and the sole extra-logical primitive S, the relation of simultaneity, governed by a sole postulate to the effect that S is symmetrical:

Here instead of S the relation N, “is no later than” might be taken as primitive; S could then be defined in terms of N as follows:

Postulation of (3) then becomes unnecessary, for (3) is an abbreviation, according to (4), of a theorem which is demonstrable within pure logic.

In this note I show, by means of an infinite matrix M, that the number of irreducible modalities in Lewis's system S2 is infinite. The result is of some interest in view of the fact that Parry has recently shown that there are but a finite number of modalities in the system S2 (which is the next stronger system than S2 discussed by Lewis).

I begin by introducing a function θ which is defined over the class of sets of signed integers, and which assumes sets of signed integers as values. If A is any set of signed integers, then θ(A) is the set of all signed integers whose immediate predecessors are in A; i.e., , so that n ϵ θ(A) is true if and only if n − 1 ϵ A is true.

Thus, for example, θ({−10, −1, 0, 3, 14}) = {−9, 0, 1, 4, 15}. In particular we notice that θ(V) = V and θ(Λ) = Λ, where V is the set of all signed integers, and Λ is the empty set of signed integers.

It is clear that, if A and B are sets of signed integers, then θ(A+B) = θ(A)+θ(B).

It is also easily proved that, for any set A of signed integers we have . For, if n is any signed integer, then