This paper deals with the notion of direct product in the theory of decision problems.

Elementary mathematical theories are always concerned with certain functions defined in a set I (called the universe of discourse of the theory) and certain relations with the common domain I. The notions of functions and relations are known to be reducible to each other. To avoid duplication of definitions and proofs we shall eliminate the former notion in favor of the latter and consider only theories in which all the primitive terms are of the type of relations (with an arbitrary number of arguments).

It is easy to define for relations the notion of a direct product in the way which is usual in abstract algebra. We are led to this definition in a very natural way when we reflect that relations are but a particular case of functions, and the direct product of an arbitrary number of functions is defined in algebra.

A relation which is representable as a product of an arbitrary number of pairwise identical relations is called the power-relation, and the relations which occur as factors are called the base-relations.

The content of the present paper can now be characterized more precisely as follows.

We shall discuss a theory of which the primitive notions are representable as powers of certain base-relations, and shall try to reduce all the problems concerning this theory (in particular the decision problem) to problems concerning the theory of the base-relations. It will be seen that this reduction is in fact possible, and that several particular cases of the decision problem, the solutions of which are known from the literature, can be included in the general scheme established in the present paper.

In this paper we continue the study of inner models of the type studied in Inner models for set theory—Part I. The present paper is concerned exclusively with a particular kind of model, the ‘super-complete models’ defined in section 2.4 of I (page 186). The condition (c) of 2.4 and the completeness condition 1.42 imply that such a model is uniquely determined when its universal class Vm is given. Writing condition (c) and the completeness conditions 1.41, 1.42 in terms of Vm, we may state the definition in the form:

3.1. Dfn. A class Vmis said to determine a super-complete model if the model whose basic notions are defined by, satisfies axioms A, B, C.

N. B. This definition is not necessarily metamathematical in nature. If desired, it could be written out quite formally as the definition of a notion ‘SCM(U)’ (‘U determines a super-complete model’) thus:

where ψ(U) is the propositional function expressing in terms of U the fact that the model determined by U according to 3.1 satisfies the relativization of axioms A, B, C. E.g. corresponding to axiom A1m, i.e., , ψ(U) contains the equivalent term . All the relativized axioms can be similarly expressed in this way by first writing out the relativized form (after having replaced all defined symbols which occur by the corresponding formulae in primitive notation) and then replacing ‘(Am)ϕ(Am) by which is in turn replaced by , and similarly replacing ‘(xm)ϕ(xm)’ by ‘(xm)ϕ(xm)’ by ‘(X)(X ϵ U ▪ ⊃ ▪ ϕ (X)), and Thus ψ(U) is obtained in primitive notation.

In this paper the symbolism, definitions and results of my two papers, namely A test for the existence of tautologies according to many-valued truth-tables (hereafter referred to as ET) and Note on truth-tables (hereafter referred to as TT) will be presupposed.

The problem whether two arbitrary truth-tables with finite number of elements are equal or not was reduced in TT to the question of equality of two identical tables in which different elements have been designated. However, no general method for testing the equality of truth-tables was given there, although some cases were discussed for which such a method is available. In the present paper we shall describe a general method using the considerations of ET and TT.

Unter der Peano-Arithmetik verstehe ich im Folgenden eine über dem Prädikatenkalkül der ersten Stufe formalisierte Arithmetik der natürlichen Zahlen, in der als außerlogische Konstanten nur Symbole für die Null, die Nachfolgeroperation und die Gleichheitsbeziehung (die als mathematischer Grundbegriff aufgefaßt wird) vorkommen.

Ein Ausdruck H(c), in dem c die einzige freie Variable ist, heiße: ∀-defekt in Bezug auf ein (deduktives) System S, wenn (a) für jede Ziffer , und (b) nicht ∀xH(x) ϵ S.

Im allgemeinen wird das deduktive System durch ein Axiomensystem in Verbindung mit einer Syntax gegeben sein. Auf diesen Fall soil die Redeweise sinngemäß übertragen werden.

Ein System S heiße ω-unvollständig, wenn es einen in Bezug auf S ∀-defekten Ausdruck gibt.

Ein Axiomensystem A heiße kategorisch, wenn beliebige absolute Modelle (s. Abschnitt IV) von A isomorph sind.

Die Kategorizität stellt eine gewisse Vollständigkeit des Axiomensystems dar. Daß solche Axiomensysteme ω-unvollstandig sein können (bzw. im Fall der ausdrucksreicheren Arithmetik nach Gödel sogar müssen), ist eben das Überraschende.

Ich werde im Folgenden für zwei kategorische Axiomensysteme der Peano-Arithmetik die ω-Unvollständigkeit in Bezug auf die Prädikatenlogik der ersten Stufe nachweisen. In beiden Fallen gelingt der Nachweis der ω-Unvollständigkeit durch Diskussion eines ∀-defekten Ausdrucks H(c).

It is well known that negation can be defined by postulating a fixed proposition F and specifying that

where “≡” indicates identity by definition. I propose to discuss this way of introducing negation in relation to the inferential systems of my Notre Dame lectures of 1948. Acquaintance with these lectures is presupposed; they will be referred to as TFD. The definition (1) was used there, essentially, in connection with the T-systems; but it was not used in connection with the L-systems because of a certain difficulty which I was not able to overcome at that time. This paper contains proofs of results concerning L-systems which have been announced elsewhere.

There has recently been developed a system of computational logic to which was given an interpretation in terms of the 2-valued propositional calculus. The object of the present paper is to give the corresponding theory for 3-valued logic. We use the same notation as Levin, except that instead of using the numeral “2” as a constant, we use “3”.

In a series of five papers, Fitch has constructed a system of combinatory logic K′ which is adequate for much of classical analysis, and is demonstrably consistent if we assume the validity of transfinite induction up to a certain ordinal (at present undetermined). The system has the following peculiarities:

1.1. It is non-finitary, i.e. the class of Gödel-numbers of its theorems does not form the range of values of any recursive function (nor, as a matter of fact, of any function definable in elementary number theory).

1.2. It does not permit quantification over real numbers; i.e. it contains no theorems of the form

1.3. Because of 1.1, we cannot, except in trivial cases, construct actual proofs in K′; we have to resort to a metalanguage (of undetermined strength) in order to show that certain formulae are theorems of K′. Also because of 1.2, many important theorems of classical mathematics are not forthcoming in K′ itself, but only in the aforementioned metalanguage, e.g. in the form

where ‘x’ is a syntactical rather than a numerical variable, a U-real is an expression of K′ rather than a number, and ‘ … x– – –’ ascribes a syntactical rather than a real number-theoretic property to x.

The purpose of this paper is to construct a finitary metalanguage for K′ in which all of Fitch's important theorems may be proved.

We use NF to designate the system known as Quine's New Foundations (see [1] and [2]), and NF + AF to designate the same system with a suitable axiom of infinity adjoined. We use ML to designate the revised system appearing in the third printing of Quine's “Mathematical Logic” (see [3]). This system ML is just the system P proposed by Wang in [4], and essentially includes NF as a part.

The pripcipal results of the present paper are:

A. In NF the axiom of infinity is equivalent to the definability of an ordered pair having the same type as its constituents.

B. Considering the formulas of NF as having translations in the formalism of ML in the sense defined in [4], we find that a formula of NF is provable in NF if and only if its translation is provable in ML. From the fact that the axiom of infinity is provable in ML, one might be tempted to conclude that the axiom of infinity is provable in NF. It is explained why this is not a valid inference. Consequently, it would appear that there is a sense in which NF + AF is stronger than ML.

Certain axiomatic systems involve more than one category of fundamental objects; for example, points, lines, and planes in geometry; individuals, classes of individuals, etc. in the theory of types or in predicate calculi of orders higher than one. It is natural to use variables of different kinds with their ranges respectively restricted to different categories of objects, and to assume as substructure the usual quantification theory (the restricted predicate calculus) for each of the various kinds of variables together with the usual theory of truth functions for the formulas of the system. An axiomatic theory set up in this manner will be called many-sorted. We shall refer to the theory of truth functions and quantifiers in it as its (many-sorted) elementary logic, and call the primitive symbols and axioms (including axiom schemata) the proper primitive symbols and proper axioms of the system. Our purpose in this paper is to investigate the many-sorted systems and their elementary logics.

Among the proper primitive symbols of a many-sorted system Tn (n = 2, …, ω) there may be included symbols of some or all of the following kinds: (1) predicates denoting the properties and relations treated in the system; (2) functors denoting the functions treated in the system; (3) constant names for certain objects of the system. We may either take as primitive or define a predicate denoting the identity relation in Tn.

In lectures at Notre Dame University in 1948 I presented what was essentially a semantical analysis of the propositions formed from the elementary propositions of a formal system by compounding with propositional connectives and quantifiers. The system LD is one of the systems treated there in the chapter on negation. It is substantially the minimal calculus with excluded middle, and was first considered by Johansson. It corresponds semantically to the situation where we have refutability with completeness, and so it may be regarded as the natural system of strict implication. This paper gives the details of certain results concerning this system which have been reported elsewhere. Some related remarks concerning other systems will also be included. Acquaintance with the lectures cited is presupposed; they are referred to as TFD.

In this paper a theorem about numerical relations will be established and shown to have certain consequences concerning decidability in quantification theory, as well as concerning the foundation of number theory. The theorem is that relations of natural numbers are reducible in elementary fashion to symmetric ones; i.e.:

Theorem I. For every dyadic relation R of natural numbers there is a symmetric dyadic relation H of natural numbers such that R is definable in terms of H plus just truth-functions and quantification over natural numbers.

To state the matter more fully, there is a (well-formed) formula ϕ of pure quantification theory, or first-order functional calculus, which meets these conditions:

(a) ϕ has ‘x’ and ‘y’ as sole free individual variables;

(b) ϕ contains just one predicate letter, and it is dyadic;

(c) for every dyadic relation R of natural numbers there is a symmetric dyadic relation H of natural numbers such that, when the predicate letter in ϕ is interpreted as expressing H, ϕ comes to agree in truth-value with ‘x bears R to y’ for all values of ‘x’ and ‘y’.

In the paper On the logic of quantification Prof. W. V. Quine showed that for the theory of quantification we have a mechanical process to determine whether or not a monadic expression is a valid logical formula, and that to deduce the polyadic theory from the monadic theory we need only the generalized modus ponens, which reads:

If a conditional is valid, and its antecedent consists of zero or more quantifiers followed by a valid schema, then its consequent is valid, i.e.:

That we have a mechanical process to determine the validity of a monadic expression was pointed out by Löwenheim long ago; Quine's method, however, has the merit that it is very simple and hence practical. The present paper is to show that we may deduce the polyadic theory by means of only a mechanical process and the ordinary modus ponens alone, i.e., the schema,

or even by means of only the weakened modus ponens, i.e., the schema,

For, if an expression becomes monadic when we omit the initial all-quantifiers (i.e., those preceding every existence-quantifier) and regard the corresponding variables as constants, then evidently we may determine by a mechanical process whether or not it is a valid formula by examining the resulting monadic expression. For example, the expression

is of the same validity as the expression

and hence as the expression

In a current paper, Church and Quine show how any dyadic relation of natural numbers can be defined in terms of a symmetric dyadic relation of natural numbers together with the truth functions and quantification over natural numbers. The purpose of the present note is to extend their result to the following: Where R1, R2, …, ad infinitum are any classes or relations (of any degree) of natural numbers, they can all be defined in terms of a single symmetric dyadic relation of natural numbers together with the truth functions and quantification over natural numbers.

Let pi, for each i, be the tth prime > 1. Let the degree of Ri, for each i, be di. Let R be the dyadic relation which each number x bears to each number y such that either x is a factor of y or else x is 0 and y is for some such that . Since R is dyadic, we know from Church and Quine's result that R is definable in terms of a symmetric dyadic relation. Moreover, as seen in the incidental constructions in §1 of Church and Quine, ‘y = x + 1’ is expressible on the same basis.

Since the Hilbert substitution method has been applied by Ackermann to the formalism Z of number theory in [3], we shall give the no-counter-example interpretation for extensions (c) of Z, e.g. Zμ. The work required on top of Ackermann's investigations to check conditions (α) and (β), consists only in a suitable definition of a class of recursive functional. The verification of (γ) and (δ) is much more difficult, and requires the ideas of paras. 32 and 33 in the preceding section.

We consider any set of Gentzen-type rules in a formal system of the following sort. The terms of the system are propositions, generally to be thought of as the propositions of some underlying theory. The elementary statements are of the form

where and are finite sequences of propositions and (1) expresses a relation, called entailment, between such sequences. These sequences will be called prosequences; they may be void, or consist of a single proposition; and they may contain repetitions of the same proposition. The members of a prosequence will be called its constituents. Two prosequences which are permutations of one another will be regarded as the same. Prosequences will be designated by German letters, single propositions by Latin ones.

The rules of the system are to be of the following form:

This notation is to be understood as indicating p premises and a conclusion, each of the form (1). The constituents of and occur in all premises and conclusion; they are called the parametric constituents, or simply the parameters, of the inference.

The present paper contains an emendation to the theory of formal deducibility which was presented in lectures at the University of Notre Dame in the spring of 1948 [4]. The method of proof developed in these lectures for the elimination theorem, which corresponds to the “Hauptsatz” of Gentzen's thesis [6], failed when modal connectives were present. This deficiency was remedied soon thereafter; but it was not possible to include the new proof in those lectures. This new proof is presented below.

Before proceeding to this proof, the elimination theorem will be formulated, and its proof outlined, in an abstract form. Certain general conditions on the rules are formulated in §1; then, in §2, it is shown that the first two stages of the elimination theorem go through for any rules satisfying the conditions. The modifications necessary to extend this proof to cases where modal rules are present will then be taken care of in the later sections. A section on the extension of the elimination theorem to the singular form of LC is added at the end.

The treatment of simplicity in my book and in earlier articles needs one important amplification and certain minor amendments. The basic method for computing complexity values nevertheless remains the same.

New studies show that so to adjust the actual technique of computation as to take full account of the factor of reflexivity would result in unwieldy rules. Fortunately, however, this is unnecessary; for the method already prescribed works for thoroughly irreflexive predicates, and every other predicate is replaceable by such predicates. Thus we need only require that before the complexity of a basis is evaluated every primitive be replaced by the simplest adequate set of irreflexive predicates.

Consider, for example, two-place predicates that are not irreflexive. Any two-place (or indeed any n-place) predicate that is satisfied only by identity pairs (or identity n-ads) can obviously be replaced by a one-place predicate. Any two-place predicate that is satisfied by some non-identity pairs and that is either exactly reflexive (i.e. satisfied also by just the identity pairs of elements in such non-identity pairs) or totally reflexive (i.e. satisfied also by all identity pairs) can be replaced by a two-place irreflexive predicate. In the former case, for instance, we replace “P” by “Q”, so explained that Qx, y if and only if Px, y and x ≠ y; and then define:

Finally, a non-irreflexive two-place predicate “P” that is of none of these three kinds can be replaced by a one-place and an irreflexive two-place predicate as follows:—let Qx if and only if Px,x, and let Rx,y if and only if Px,y andx ≠ y; then define:

Predicates of more than two places offer much greater variety since they may exhibit any sort of reflexivity with respect to any selection of places.