Inasmuch as my discussion of the formal properties of “System L,” presented elsewhere, is substantially the same as that presented by Lewis in a previous issue of this Journal, I shall confine my remarks in this brief rejoinder to the problem of deducibility with respect to necessary and impossible propositions—since it is specifically the solution of this problem which will determine the adequacy of the calculus of logical implication as a canon and critique of deductive inference.

The problem, as originally presented, can be stated succinctly in the following inconsistent triad: (1) All necessarily true principles are logically dependent on or deducible from one another, i.e., are equivalent, (2) All principles of logic and mathematics are necessarily true, and (3) Some principles of logic and mathematics are not logically dependent on or deducible from one another, i.e., are independent. These three propositions are logically inconsistent in the sense that the conjunction of any two of them will logically imply the falsity of the third. Now if Lewis should affirm the first and third of these propositions, then he should concede the falsity of the second. This concession, it was believed, he was not likely to make, inasmuch as it would seem to involve a repudiation of a doctrine (the tautological character of every logical and mathematical truth) with which he has particularly identified himself. If, in view of this fact, he should accept, instead, the second and third of these propositions, then he should concede the falsity of the first.

All notions of mathematical logic, as measured e.g. by Principia mathematica, can be constructed from a basis which embraces variables and just two further primitives: the familiar notions of inclusion and abstraction. Inclusion will be expressed by the usual form of notation “(x⊂y)”, which may be read “x is included in y.” Abstraction will be expressed by the form of notation “x3 …” as with Peano, the blank being filled by any formula (statement or statement form); the whole may be read “the class of all objects x such that ….”

A system of logic based on these primitives will be presented in this paper. The system involves the familiar theory of class types, which, for metamathematical convenience, will be recorded in the system by use of distinctive styles of variables; thus the variables “x”, “y”, and “z”, also with subscripts when further variety is needed, will take individuals as values; the variables “x′”, “y′”, and “z′”, also with subscripts, will take classes of individuals as values; the variables “x″”, “y″”, and “z″”, also with subscripts, will take classes of classes of individuals as values; and so on. The metamathematical notion of term, covering all variables and also all expressions of abstraction, is now describable recursively thus: a variable is a term, and the number of accents which it bears is called its rank; and if terms of equal rank are put for “ζ” and “η” and a variable of rank n is put for “α” in “α3(ζ ⊂ η)”, the result is a term of rank n+1. A formula finally, is any result of putting terms of equal rank for “ζ” and “η” in “(ζ ⊂ η)”.

1. The problem. The aim of the following considerations is to introduce a new type of non-Aristotelian logic by generalizing the truth-table methods so far employed for establishing non-Aristotelian sentential calculi. We shall expound the intended generalization by applying it to the particular set of pluri-valued systems introduced by J. Łukasiewicz. One will remark that the points of view illustrated by this example may serve to generalize quite analogously any other plurivalued systems, such as those originated by E. L. Post, by H. Reichenbach, and by others.

2. J. Łukasiewicz's plurivalued systems of sentential logic. First of all, we consider briefly the structure of the Łukasiewicz systems themselves.

As to the symbolic notation in which to represent those systems, we make the following agreements: For representing the expressions of the (two- or plurivalued) calculus of sentences, we make use of the Principia mathematica symbolism; however, we employ brackets instead of dots. We call the small italic letters “p”, “q”, “r”, … sentential variables or elementary sentences, and employ the term “sentence” as a general designation of both elementary sentences and the composites made up of elementary sentences and connective symbols (“~”, “ν” “.”, “⊃” “≡”).

Now, the different possible sentences (or, properly speaking, the different possible shapes of sentences, such as “p”, “p∨q”, “~p.(q∨ r)”, etc.) are the objects to which truth-values are ascribed; and just as in every other case one wants a designation for an object in order to be able to speak of it, we want now a system of designations for the sentences with which we are going to deal in our truth-table considerations.

Einleitung. Das “Auswahlaxiom” oder “multiplicative axiom” fordert in der gewöhnlichen, von B. Russell und Zermelo angegebenen Fassung, daß zu jeder Menge M, deren Elemente paarweise fremde und nicht-leere Mengen sind, mindestens eine “Auswahlmenge” existiere, die mit jedem Element von M genau éin Element gemeinsam hat. Die nächstliegende und mehrfach verwendete Methode, um ein schwächeres Postulat als die vorstehende Fassung zu formulieren, besteht darin, daß man entweder über die Mächtigkeit der Menge M, oder über die Mächtigkeiten ihrer Elemente, oder über beides gleichzeitig, einschränkende Bedingungen macht, also nur in diesen eingeschränkten Fällen die Existenz einer Auswahlmenge verlangt. Hinsichtlich der Mächtigkeit von M ist da sogleich die Bemerkung zu machen, daß sie jedenfalls transfinit (genauer: nicht endlich im induktiven Sinn) anzusetzen ist. Vermöge allgemeiner logischer Sachverhalte und des (eventuell zu beweisenden) Prinzips, das zu einem gegebenen Objekt a die Bildung der Menge {a} erlaubt, ist nämlich die Existenz einer Auswahlmenge von M beweisbar, falls M die Mächtigkeit 1 besitzt, und diese Beweisbarkeit überträgt sich mittels gewöhnlicher Induktion auf den Fall einer beliebigen endlichen Mächtigkeit. Dagegen ist die Existenz einer Auswahlmenge nicht mehr beweisbar, wenn (bei transfinitem M) die Elemente von M endliche Mengen mit Mächtigkeiten > 1 sind, z. B. also schon in dem Fall, wo M eine transfinite Menge von Paaren ist—natürlich “im allgemeinen,” d. h. wenn die Elemente von M nicht eine besondere Beschaffenheit haben, die die Auszeichnung gewisser Elemente in ihnen ermöglicht. Die Tatsache dieser Nichtbeweisbarkeit, d. h. der Unabhängigkeit des Auswahlaxioms sogar in diesem einfachsten Fall, ist vor längerer Zeit auf der Basis des Zermeloschen Axiomensystems der Mengenlehre nachgewiesen worden.

Several definitions have been given to express an exact meaning corresponding to the intuitive idea of ‘effective calculability’ as applied for instance to functions of positive integers. The purpose of the present paper is to show that the computable functions introduced by the author are identical with the λ-definable functions of Church and the general recursive functions due to Herbrand and Gödel and developed by Kleene. It is shown that every λ-definable function is computable and that every computable function is general recursive. There is a modified form of λ-definability, known as λ-K-definability, and it turns out to be natural to put the proof that every λ-definable function is computable in the form of a proof that every λ-K-definable function is computable; that every λ-definable function is λ-K-definable is trivial. If these results are taken in conjunction with an already available proof that every general recursive function is λ-definable we shall have the required equivalence of computability with λ-definability and incidentally a new proof of the equivalence of λ-definability and λ-K-definability.

A definition of what is meant by a computable function cannot be given satisfactorily in a short space. I therefore refer the reader to Computable pp. 230–235 and p. 254. The proof that computability implies recursiveness requires no more knowledge of computable functions than the ideas underlying the definition: the technical details are recalled in §5.

1. The notion of derivability. Italic capitals, with or without subscripts, will be used as variables. They are to take as values some manner of elements which may for the present be left undetermined. Now let us consider abstractly the notion of the derivability of an element X from one or more specified elements by a series of steps of a specified kind. This involves reference to two conditions upon elements. One of these conditions, expressible by some statement form containing a single free variable, determines the elements from which X is said to be derivable. The other condition, expressed say by a statement form containing k + 1 free variables, determines the kind of steps by which the derivation is to proceed; it is the condition which any elements Z1, … Zk, Y must fulfill if progress from Zi, …, Zk to Y is to constitute a step of derivation in the intended sense. Supposing “f(Y)” supplanted by the first of these statement forms, whatever it may be, and “g(Z1, …, Zk, Y)” supplanted by the other, let us adopt the form of notation

to express derivability of X in the suggested sense. The meaning of (1) can be formulated more accurately as follows:

(i) There are elements Y1 to Ym (for some m) such that Ym = X and, for each i≦m, either f(Yi) or else there are numbers j1 to Ym, each less than i, for which g(Yj1, …, Yjk, Yi).

(Variable subscripts are to be understood, here and throughout the paper, as referring only to positive integers.)

The notion (1) is illustrated in the ancestral R* of a relation R;1 for,

Another illustration is afforded by metamathematics. Suppose our elements are the expressions used in some formal system; suppose we have defined “Post(Y)”, meaning that Y is a postulate of that system; and suppose we have defined “Inf(ZI, …, Zk, Y)” (for some fixed k large enough for all purposes of the system in question), meaning that Y proceeds from Z1, …, Zk by one application of one or another of the rules of inference of the system. Then

would mean that X is a theorem of the system.

The Peanese convention for the use of dots as brackets has the disadvantage that it gives only an awkward method for representing chains of indefinite length, such as the compound implication

Such chains occur frequently in logical investigations of a metatheoretic nature, and it is convenient to have a systematic method of abbreviating them. The most obvious method of doing this would be to leave the parentheses out entirely, and to understand that in such cases the implication sign or other operation appearing on the extreme left is the most inclusive; but this method, which has been followed by Quine, is at variance with our ordinary algebraic usage, in which we write a − b + c − d for ((a − b) + c) − d, and consequently leads to confusion. It is desirable to have a modification of the Peanese convention which gives a simple method of representing such chain implications, and at the same time avoids this difficulty.

Such a modification is easily obtained by generalizing somewhat a procedure of Church. The latter author uses only a single dot, which he writes on the right of an operator to signify a bracket extending from that point to the end of the formula (or parenthesized expression). If dots are used only on the right of operations this is all that can be desired. But the essential idea of this device can be extended to the case where dots are used also on the left, as follows: let us suppose that a group of dots on the right of an operation or prefix denotes the beginning of a bracket which extends to the right until it meets a group with an equal or larger number of dots on the left of an operation; and that the scope of a group of dots on the left of an operation shall extend to the left until it reaches a larger group of dots on the right of some operation.

In the theory of conversion it is important to have a formally defined function which assigns to any positive integer n the least integer not less than n which has a given property. The definition of such a formula is somewhat involved: I propose to give the corresponding formula in λ-K-conversion, which will (naturally) be much simpler. I shall in fact find a formula þ such that if T be a formula for which T(n) is convertible to a formula representing a natural number, whenever n represents a natural number, then þ(T, r) is convertible to the formula q representing the least natural number q, not less than r, for which T(q) conv 0.2 The method depends on finding a formula Θ with the property that Θ conv λu·u(Θ(u)), and consequently if M→Θ(V) then M conv V(M). A formula with this property is,

The formula þ will have the required property if þ(T, r) conv r when T(r) conv 0, and þ(T, r) conv þ(T, S(r)) otherwise. These conditions will be satisfied if þ(T, r) conv T(r, λx·þ(T, S(r)), r), i.e. if þ conv {λptr·t(r, λx·p(t, S(r)), r)}(þ). We therefore put,

This enables us to define also a formula,

such that (T, n) is convertible to the formula representing the nth positive integer q for which T(q) conv 0.

Among various unnatural and technically burdensome effects of the theory of types, one is the unstable character of meaningfulness: a mere permutation of variables is capable of reducing a significant context to meaninglessness. Another effect, and perhaps the most conspicuous, is the systematic reduplication to which the logical constants are subjected; the calculi of classes and relations and even arithmetic lose their unity and generality, and are reproduced anew within each type. The elaborate compensatory manoeuvres which are thus made necessary are familiar to all readers of Principle, mathematica.

In a recent publication, to be cited henceforth as New foundations, I proposed an alternative course which avoids these consequences, but which would seem to offer less assurance of consistency. My efforts to derive a contradiction have delivered none, but they have lent a strange aspect to Cantor's proof that every class has more subclasses than members. This result is the topic of the present paper.

Preparatory to sketching the new system, which I shall call S′, I shall sketch a very similar but contradictory system S. By way of primitives S involves just membership, universal quantification, and alternative denial (Sheffer's stroke function), together with general variables “x”, “y”, …; the adequacy of this equipment as a basis for mathematical logic is made evident by Wiener's and Kuratowski's discovery of methods of constructing relation theory in terms of classes. Thus the formulae or statements and statement forms of S are describable recursively as follows: if a variable is put in each blank of “(ϵ)”, the result is a formula; if a variable enclosed in parentheses is prefixed to a formula, the result is a formula; and if a formula is put in each blank of “(∣)”, the result is a formula. The theorems of S are determined by a postulate and five rules, called P1 and R1-5 in New foundations. P1 and R1-3 specify various formulae as initial theorems, and R4-5 specify inferential connections for deriving further theorems. R1–2 and R4–5 are so fashioned as to provide the “theory of deduction”: they provide as theorems all those formulae which are valid by virtue merely of their structure in terms of alternative denial and quantification.

1. The theory of classes presented in this paper is a simplification of that presented by J. von Neumann in his paper Die Axiomatisierung der Mengenlehre. However, this paper is written so that it can be read independently of von Neumann's. The principal modifications of his system are the following.

(1) The idea of ordered pair is defined in terms of the other primitive concepts of the system. (See Axiom 4.3 below.)

(2) A much simpler proof of the well-ordering theorem, based on von Neumann's equivalence axiom (Axiom 2.2 below) is given. (More exactly, the theory of ordinal numbers, on which the proof of the well-ordering theorem is based, is simplified—see §8.)

(3) Functions are not assumed to be defined for all arguments. In place of “ = A” we have “is undefined.” This makes possible a constructive interpretation of the system. (See §2.)

2. We wish first to give a rough picture of the system which we are trying to construct. Suppose that we have the idea “class,” but no material from which to construct classes. Nevertheless, we can construct the class 0 having no elements. And then the class 1 having 0 as its only element. And then the classes {1}, and {0, 1}, etc., using always the previously constructed classes as elements. To extend this method to infinite classes, we must give rules telling what elements are to be included. Since the nature of the rules which we can use is not altogether clear, we try to formalize the whole system; that is, to set up a system of axioms which seems to characterize this system of classes. Now in this set of axioms we find it necessary to use the idea of function (in the “Axiom of Replacement”).

In this paper it will be shown that the fundamental modal functions (“… is necessary,” “… is possible,” etc.) may be represented in the ordinary two-valued logic of propositions, relatively to a subset S of the set K of all propositions of the logic. The only restriction on S is that no member of S can occur in any other member of S. The representation concerns the set SK consisting of all members of K in which occur members of S. Every member of SK may be regarded as a truth-function of the members of S. If S consists of only one member, P, and if f(P) and g(P) are any members of SK, then we define ∣f(P)∣ as (f(P)·f(~P)). The proposition “f(P) is S-necessary” may then be expressed by ∣f(P)· Furthermore we may express “f(P) is S-impossible” by ∣~f(P)∣, abbreviated to f(P)*; we may express “f(P) is S-possible” by abbreviated to f(P); and we may express “f(P) S-strictly implies g(P)” by (f(P)·~g(P))*, abbreviated to f(P)⊰g(P). Thus is obtained a novel analysis of modality, contrasting, on the one hand, with Carnap's syntactic theory of modality, and, on the other hand, with Lewis's view of modal concepts as underivable from other logical concepts. We would maintain that modal concepts are not absolute, but always relative to some S, generally unspecified.

In the formal exposition which follows, we shall assume tacitly but not explicitly the important distinction between the symbols we are discussing and symbols denoting or designating the former.

The object of this note is to close a slight gap, hitherto seemingly over-looked, in Nicod's derivation from his postulates of the primitives of Principia mathematica. The gap consists in this: Nicod's propositions corresponding to *1·1−*1·6 of the Principia, though they bear the same names and have the same symbol-forms as the Principia propositions, are not precisely *1·1−1·6. Indeed, if we denote Nicod's “p ⊃ q” and “p · q” by “p < q” and “pq” respectively, and if we write p′ for p∣p and p″ for (p′)′ then in terms of the “stroke,” we have

But p ⊃ q and p · q of the Principia are

Accordingly, the Nicod-Principia propositions are not *1·1−*1·6. They are *1·1−*1·6 in which < takes the place of ⊃, and Nicod has not shown us explicitly how to pass from < to ⊃.

It is true that Nicod also proves the following:

i.e.,

This yields: If ⊢· p < q then ⊢· p ⊃ q, and conversely. This would seem to allow passing from < to ⊃ throughout a proposition. But, it is clear, T1 permits this change only in the case of the principal <. The change in the case of a minor < still has to be justified.