This paper extends earlier work by its authors on formal aspects of the processes of contracting a theory to eliminate a proposition and revising a theory to introduce a proposition. In the course of the earlier work, Gärdenfors developed general postulates of a more or less equational nature for such processes, whilst Alchourrón and Makinson studied the particular case of contraction functions that are maximal, in the sense of yielding a maximal subset of the theory (or alternatively, of one of its axiomatic bases), that fails to imply the proposition being eliminated.

In the present paper, the authors study a broader class, including contraction functions that may be less than maximal. Specifically, they investigate “partial meet contraction functions”, which are defined to yield the intersection of some nonempty family of maximal subsets of the theory that fail to imply the proposition being eliminated. Basic properties of these functions are established: it is shown in particular that they satisfy the Gärdenfors postulates, and moreover that they are sufficiently general to provide a representation theorem for those postulates. Some special classes of partial meet contraction functions, notably those that are “relational” and “transitively relational”, are studied in detail, and their connections with certain “supplementary postulates” of Gàrdenfors investigated, with a further representation theorem established.

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 logical theory which is called the calculus of (binary) relations, and which will constitute the subject of this paper, has had a strange and rather capricious line of historical development. Although some scattered remarks regarding the concept of relations are to be found already in the writings of medieval logicians, it is only within the last hundred years that this topic has become the subject of systematic investigation. The first beginnings of the contemporary theory of relations are to be found in the writings of A. De Morgan, who carried out extensive investigations in this domain in the fifties of the Nineteenth Century. De Morgan clearly realized the inadequacy of traditional logic for the expression and justification, not merely of the more intricate arguments of mathematics and the sciences, but even of simple arguments occurring in every-day life; witness his famous aphorism, that all the logic of Aristotle does not permit us, from the fact that a horse is an animal, to conclude that the head of a horse is the head of an animal. In his effort to break the bonds of traditional logic and to expand the limits of logical inquiry, he directed his attention to the general concept of relations and fully recognized its significance. Nevertheless, De Morgan cannot be regarded as the creator of the modern theory of relations, since he did not possess an adequate apparatus for treating the subject in which he was interested, and was apparently unable to create such an apparatus. His investigations on relations show a lack of clarity and rigor which perhaps accounts for the neglect into which they fell in the following years.

We are interested in studying the length of the shortest proof of a propositional tautology in various proof systems as a function of the length of the tautology. The smallest upper bound known for this function is exponential, no matter what the proof system. A question we would like to answer (but have not been able to) is whether this function has a polynomial bound for some proof system. (This question is motivated below.) Our results here are relative results.

In §§2 and 3 we indicate that all standard Hilbert type systems (or Frege systems, as we call them) and natural deduction systems are equivalent, up to application of a polynomial, as far as minimum proof length goes. In §4 we introduce extended Frege systems, which allow introduction of abbreviations for formulas. Since these abbreviations can be iterated, they eliminate the need for a possible exponential growth in formula length in a proof, as is illustrated by an example (the pigeonhole principle). In fact, Theorem 4.6 (which is a variation of a theorem of Statman) states that with a penalty of at most a linear increase in the number of lines of a proof in an extended Frege system, no line in the proof need be more than a constant times the length of the formula proved.

The present paper attempts to state and prove a completeness theorem for the system S5 of [1], supplemented by first-order quantifiers and the sign of equality. We assume that we possess a denumerably infinite list of individual variables a, b, c, …, x, y, z, …, xm, ym, zm, … as well as a denumerably infinite list of n-adic predicate variables Pn, Qn, Rn, …, Pmn, Qmn, Rmn,…; if n=0, an n-adic predicate variable is often called a “propositional variable.” A formula Pn(x1, …,xn) is an n-adic prime formula; often the superscript will be omitted if such an omission does not sacrifice clarity.

The first order functional calculus was proved complete by Gödel in 1930. Roughly speaking, this proof demonstrates that each formula of the calculus is a formal theorem which becomes a true sentence under every one of a certain intended class of interpretations of the formal system.

For the functional calculus of second order, in which predicate variables may be bound, a very different kind of result is known: no matter what (recursive) set of axioms are chosen, the system will contain a formula which is valid but not a formal theorem. This follows from results of Gödel concerning systems containing a theory of natural numbers, because a finite categorical set of axioms for the positive integers can be formulated within a second order calculus to which a functional constant has been added.

By a valid formula of the second order calculus is meant one which expresses a true proposition whenever the individual variables are interpreted as ranging over an (arbitrary) domain of elements while the functional variables of degree n range over all sets of ordered n-tuples of individuals. Under this definition of validity, we must conclude from Gödel's results that the calculus is essentially incomplete.

It happens, however, that there is a wider class of models which furnish an interpretation for the symbolism of the calculus consistent with the usual axioms and formal rules of inference. Roughly, these models consist of an arbitrary domain of individuals, as before, but now an arbitrary class of sets of ordered n-tuples of individuals as the range for functional variables of degree n. If we redefine the notion of valid formula to mean one which expresses a true proposition with respect to every one of these models, we can then prove that the usual axiom system for the second order calculus is complete: a formula is valid if and only if it is a formal theorem.

It is a scientific platitude that there can be neither precise control nor prediction of phenomena without measurement. Disciplines are diverse as cosmology and social psychology provide evidence that it is nearly useless to have an exactly formulated quantitative theory if empirically feasible methods of measurement cannot be developed for a substantial portion of the quantitative concepts of the theory. Given a physical concept like that of mass or a psychological concept like that of habit strength, the point of a theory of measurement is to lay bare the structure of a collection of empirical relations which may be used to measure the characteristic of empirical phenomena corresponding to the concept. Why a collection of relations? From an abstract standpoint a set of empirical data consists of a collection of relations between specified objects. For example, data on the relative weights of a set of physical objects are easily represented by an ordering relation on the set; additional data, and a fortiori an additional relation, are needed to yield a satisfactory quantitative measurement of the masses of the objects.

The major source of difficulty in providing an adequate theory of measurement is to construct relations which have an exact and reasonable numerical interpretation and yet also have a technically practical empirical interpretation. The classical analyses of the measurement of mass, for instance, have the embarrassing consequence that the basic set of objects measured must be infinite. Here the relations postulated have acceptable numerical interpretations, but are utterly unsuitable empirically. Conversely, as we shall see in the last section of this paper, the structure of relations which have a sound empirical meaning often cannot be succinctly characterized so as to guarantee a desired numerical interpretation.

Consider a system of formal notations for ordinal numbers in the first and second number classes, with the following properties. Given a notation for an ordinal, it can be decided effectively whether the ordinal is zero, or the successor of an ordinal, or the limit of an increasing sequence of ordinals. In the second case, a notation for the preceding ordinal can be determined effectively. In the third case, notations for the ordinals of an increasing sequence of type ω with the given ordinal as limit can be determined effectively.

Are there systems of this sort which extend farthest into the second number class? When the conditions for the systems have been made precise, the question will be answered in the affirmative. There is an ordinal ω1 in the second number class such that there are systems of notations of the sort described which extend to all ordinals less than ω1, but none in which ω1 itself is assigned a notation.

1. An effective or constructive operation on the objects of an enumerable class is one for which a fixed set of instructions can be chosen such that, for each of the infinitely many objects (or n-tuples of objects), the operation can be completed by a finite process in accordance with the instructions. This notion is made exact by specifying the nature of the process and set of instructions. It appears possible to do so without loss of generality.

One task of metamathematics is to relate suggestive but nonelementary modeltheoretic concepts to more elementary proof-theoretic concepts, thereby opening up modeltheoretic problems to proof-theoretic methods of attack. Herbrand's Theorem (see [8] or also [9], vol. 2) or Gentzen's Extended Hauptsatz (see [5] or also [10]) was first used along these lines by Beth [1]. Using a modified version he showed that for all first-order systems a certain modeltheoretic notion of definability coincides with a certain proof theoretic notion. In the present paper the Herbrand-Gentzen Theorem will be applied to generalize Beth's results from primitive predicate symbols to arbitrary formulas and terms.

This may be interpreted as showing that (apart from some relatively minor exceptions which will be made apparent below) the expressive power of each first-order system is rounded out, or the system is functionally complete, in the following sense: Any functional relationship which obtains between concepts that are expressible in the system is itself expressible and provable in the system.

A second application is concerned with the hierarchy of second-order formulas. A certain relationship is shown to hold between first-order formulas and those second-order formulas which are of the form (∃T1)…(∃Tk)A or (T1)…(Tk)A with A being a first-order formula. Modeltheoretically this can be regarded as a relationship between the class AC and the class PC⊿ of sets of models, investigated by Tarski in [12] and [13].

T0 will denote Gödel's theory T[3] of functionals of finite type (f.t.) with intuitionistic quantification over each f.t. added. T1 will denote T0 together with definition by bar recursion of type o, the axiom schema of bar induction, and the schema

of choice. Precise descriptions of these systems are given below in §4. The main results of this paper are interpretations of T0 in intuitionistic arithmetic U0 and of T1 in intuitionistic analysis is U1. U1 is U0 with quantification over functionals of type (0,0) and the axiom schemata AC00 and of bar induction.

In a recent paper the author has proposed a definition of the commonly used term “effectively calculable” and has shown on the basis of this definition that the general case of the Entscheidungsproblem is unsolvable in any system of symbolic logic which is adequate to a certain portion of arithmetic and is ω-consistent. The purpose of the present note is to outline an extension of this result to the engere Funktionenkalkul of Hilbert and Ackermann.

In the author's cited paper it is pointed out that there can be associated recursively with every well-formed formula a recursive enumeration of the formulas into which it is convertible. This means the existence of a recursively defined function a of two positive integers such that, if y is the Gödel representation of a well-formed formula Y then a(x, y) is the Gödel representation of the xth formula in the enumeration of the formulas into which Y is convertible.

Consider the system L of symbolic logic which arises from the engere Funktionenkalkül by adding to it: as additional undefined symbols, a symbol 1 for the number 1 (regarded as an individual), a symbol = for the propositional function = (equality of individuals), a symbol s for the arithmetic function x+1, a symbol a for the arithmetic function a described in the preceding paragraph, and symbols b1, b2, …, bk for the auxiliary arithmetic functions which are employed in the recursive definition of a; and as additional axioms, the recursion equations for the functions a, b1, b2, …, bk (expressed with free individual variables, the class of individuals being taken as identical with the class of positive integers), and two axioms of equality, x = x, and x = y →[F(x)→F(y)].

The present note introduces a constructible interpretation for the logical connectives of number theory which is divergent from that of the intuitionists. Under the intuitionistic interpretation, the principle of excluded middle and certain other classically acceptable principles of logic must be rejected. Under the present interpretation, while some classical principles may be reinstated, other principles, acceptable both classically and intuitionistically, may be shown to be invalid. Among these is the principle of contradiction.

In [6, p. 317] Curry described a formal system assigning types to terms of the type-free λ-calculus. In [11] Scott gave a natural semantics for this type assignment and asked whether a completeness result holds.

Inspired by [4] and [5] we extend the syntax and semantics of the Curry types in such a way that filters in the resulting type structure form a domain in the sense of Scott [12]. We will show that it is possible to turn the domain of types into a λ-model, among other reasons because all λ-terms possess a type. This model gives the completeness result for the extended system. By a conservativity result the completeness for Curry's system follows.

Independently Hindley [8], [9] has proved both completeness results using term models. His method of proof is in some sense dual to ours.

For λ-calculus notation see [1].

Alonzo Church suggested to the writer that a certain problem of Thue [6] might be proved unsolvable by the methods of [5]. We proceed to prove the problem recursively unsolvable, that is, unsolvable in the sense of Church [1], but by a method meeting the special needs of the problem.

Thue's (general) problem is the following. Given a finite set of symbols α1, α2, …, αμ, we consider arbitrary strings (Zeichenreihen) on those symbols, that is, rows of symbols each of which is in the given set. Null strings are included.

In Herbrand's Theorem [2] or Gentzen's Extended Hauptsatz [1], a certain relationship is asserted to hold between the structures of A and A′, whenever A implies A′ (i.e., A ⊃ A′ is valid) and moreover A is a conjunction and A′ an alternation of first-order formulas in prenex normal form. Unfortunately, the relationship is described in a roundabout way, by relating A and A′ to a quantifier-free tautology. One purpose of this paper is to provide a description which in certain respects is more direct. Roughly speaking, ascent to A ⊃ A′ from a quantifier-free level will be replaced by movement from A to A′ on the quantificational level. Each movement will be closely related to the ascent it replaces.

The new description makes use of a set L of rules of inference, the L-rules. L is complete in the sense that, if A is a conjunction and A′ an alternation of first-order formulas in prenex normal form, and if A ⊃ A′ is valid, then A′ can be obtained from A by an L-deduction, i.e., by applications of L-rules only. The distinctive feature of L is that each L-rule possesses two characteristics which, especially in combination, are desirable. First, each L-rule yields only conclusions implied by the premisses.

In this paper we shall prove theorems about some systems of sentential calculus, by making use of results we have established elsewhere regarding closure algebras and Brouwerian albegras. We shall be concerned mostly with the Lewis system and the Heyting system. Some of the results here are new (in particular, Theorems 2.4, 3.1, 3.9, 3.10, 4.5, and 4.6); others have been stated without proof in the literature (in particular, Theorems 2.1, 2.2, 4.4, 5.2, and 5.3).

The proofs to be given here will be found to be mostly very simple; generally speaking, they amount to drawing conclusions from the theorems established in McKinsey and Tarski [10] and [11]. We have thought it might be worth while, however, to publish these rather elementary consequences of our previous work—so as to make them readily available to those whose main interest lies in sentential calculus rather than in topology or algebra.

Although several proofs have been published showing the completeness of the propositional calculus (cf. Quine (1)), for the first-order functional calculus only the original completeness proof of Gödel (2) and a variant due to Hilbert and Bernays have appeared. Aside from novelty and the fact that it requires less formal development of the system from the axioms, the new method of proof which is the subject of this paper possesses two advantages. In the first place an important property of formal systems which is associated with completeness can now be generalized to systems containing a non-denumerable infinity of primitive symbols. While this is not of especial interest when formal systems are considered as logics—i.e., as means for analyzing the structure of languages—it leads to interesting applications in the field of abstract algebra. In the second place the proof suggests a new approach to the problem of completeness for functional calculi of higher order. Both of these matters will be taken up in future papers.

The system with which we shall deal here will contain as primitive symbols

and certain sets of symbols as follows:

(i) propositional symbols (some of which may be classed as variables, others as constants), and among which the symbol “f” above is to be included as a constant;

(ii) for each number n = 1, 2, … a set of functional symbols of degree n (which again may be separated into variables and constants); and

(iii) individual symbols among which variables must be distinguished from constants. The set of variables must be infinite.

We prove an exponential lower bound on the length of cutting plane proofs. The proof uses an extension of a lower bound for monotone circuits to circuits which compute with real numbers and use nondecreasing functions as gates. The latter result is of independent interest, since, in particular, it implies an exponential lower bound for some arithmetic circuits.

The purpose of this article is to introduce the notion of “recursive realizability.”

Let P be some property of natural numbers. Consider the existential statement, “There exists a number n having the property P.” To explain the meaning which this has for a constructivist or intuitionist, it has been described as a partial judgement, or incomplete communication of a more specific statement which says that a certain given number n, or the number n obtainable by a certain given method, has the property P. The meaning of the existential statement thus resides in a reference to certain information, which it implies could be stated in detail, though the trouble is not taken to do so. Perhaps the detail is suppressed in order to convey a general view of some fact.

The information to which reference is made should be thought of as possibly comprising other items besides the value of n or method for obtaining it, namely such items as may be necessary to complete the communication that that n has the property P.

The present formulation should prove significant in the development of symbolic logic along the lines of Gödel's theorem on the incompleteness of symbolic logics and Church's results concerning absolutely unsolvable problems.

We have in mind a general problem consisting of a class of specific problems. A solution of the general problem will then be one which furnishes an answer to each specific problem.

In the following formulation of such a solution two concepts are involved: that of a symbol space in which the work leading from problem to answer is to be carried out, and a fixed unalterable set of directions which will both direct operations in the symbol space and determine the order in which those directions are to be applied.

In the present formulation the symbol space is to consist of a two way infinite sequence of spaces or boxes, i.e., ordinally similar to the series of integers …, − 3, − 2, − 1, 0, 1, 2, 3, …. The problem solver or worker is to move and work in this symbol space, being capable of being in, and operating in but one box at a time. And apart from the presence of the worker, a box is to admit of but two possible conditions, i.e., being empty or unmarked, and having a single mark in it, say a vertical stroke.