bluuebunny/crossref_metadata_2025 huggingface dataset:
On an implication function in many-valued systems of logic
Authors: Dienes Z. P.
Published: May 2006

In a paper J. Łukasiewicz and A. Tarski study the systems arising out of the implication function C (x, y) = min(1, 1 – x + y) and the negation function Nx = 1 – x, where x, y are real numbers between 0 and 1 inclusive. Let us then write
Сxy = min(1, 1 – x + y) and Nx = 1 – x. (1)

In this note we shall consider another implication function, namely
С′xy = ССNxyy, (2)
The word problem for semigroups with two generators
Authors: Hall Marshall
Published: May 2006

Emil Post has shown that the word problem for Thue systems is unsolvable. A Thue system is a system of strings of letters x1, x2, …, xr, where a string or word is either void or a succession of letters a1a2 … at with each ai some xi. We are given a finite number of pairs of strings Ai, Bii = 1, …, m. Operations in the Thue system consist of the replacements or “productions.”
The elimination theorem when modality is present
Authors: Curry Haskell B.
Published: May 2006

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.
Elimination of extra-logical postulates
Authors: Quine W. V. , Goodman Nelson
Published: May 2006

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.
Proof that there are infinitely many modalities in Lewis's system S2
Authors: McKinsey J. C. C.
Published: May 2006

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
Solution of a problem of Leon Henkin
Authors: Löb M. H.
Published: May 2006

If Σ is any standard formal system adequate for recursive number theory, a formula (having a certain integer q as its Gödel number) can be constructed which expresses the proposition that the formula with Gödel number q is provable in Σ. Is this formula provable or independent in Σ? [2].

One approach to this problem is discussed by Kreisel in [4]. However, he still leaves open the question whether the formula (Ex)(x, a), with Gödel-number a, is provable or not. Here (x, y) is the number-theoretic predicate which expresses the proposition that x is the number of a formal proof of the formula with Gödel-number y.

In this note we present a solution of the previous problem with respect to the system Zμ [3] pp. 289–294, and, more generally, with respect to any system whose set of theorems is closed under the rules of inference of the first order predicate calculus, and satisfies the subsequent five conditions, and in which the function (k, l) used below is definable.

The notation and terminology is in the main that of [3] pp. 306–326, viz. if is a formula of Zμ containing no free variables, whose Gödel number is a, then ({}) stands for (Ex)(x, a) (read: the formula with Gödel number a is provable in Zμ); if is a formula of Zμ containing a free variable, y say, ({}) stands for (Ex)(x, g(y)}, where g(y) is a recursive function such that for an arbitrary numeral the value of g() is the Gödel number of the formula obtained from by substituting for y in throughout. We shall, however, depart trivially from [3] in writing (), where is an arbitrary numeral, for (Ex){x, ).
Axiom schemes for m-valued propositions calculi
Authors: Rosser J. B. , Turquette A. R.
Published: May 2006

In an m-valued propositional calculus, or a formalization of such a calculus, truth-value functions are allowed to take any truth-value t where 1 ≦ t ≦ m and m ≧ 2. In working with such calculi, or formalizations thereof, it has been decided to distinguish those truth-values which it is desirable for provable formulas to have from those which it is not desirable for provable formulas to have. The first class of truth-values is called designated and the second undesignated. This specification of certain of the m truth-values as designated and the remainder as undesignated is one of the distinguishing characteristics of m-valued propositional calculi, and it should be observed at the outset that two m-valued propositional calculi will be considered to differ even if they differ only in respect to the number of truth-values which are taken as designated.
The completeness of the first-order functional calculus
Authors: Henkin Leon
Published: May 2006

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.
A syntactical characterization of S5
Authors: Bergmann Gustav
Published: May 2006

This note should be considered as an appendix to a paper by McKinsey. Familiarity with this paper is assumed; its terminology and notation will be used without explanation. McKinsey's main result (theorem 1) is that every class of sentences fulfilling P1–P5, with a set of substitutions satisfying A1–A4, is a system of modal logic in the sense that the “associated” set, T2, contains all theorems of S4. It is also proved (theorem 2) that for T2 to contain all theorems of S5, it is sufficient that the elements of S form a group. The basic idea of the construction is the formalization, by means of D5, of our intuitive notion of possibility.

In view of the great generality of S, it may be of some interest to give a condition that is both necessary and sufficient for T2 to contain S5 after some restriction suggested by our intuitive notions about modalities has first been imposed upon S. The following postulate is related to the idea that if a sentence is possible, then its negation is not necessary. Informally speaking, it requires that S be sufficiently “comprehensive” for this idea to become formalizable in terms of S.

P. If ♢α is in T1, then there exists an element sm of S such that ♢∼sm(α) is not in T1.

I shall now prove that if P is satisfied, the following condition is both necessary and sufficient for T2 to contain all theorems of S5.

C. If α is in T1, then ♢s(α) is in T1 for all elements s of S.
A solution of the decision problem for the Lewis systems S2 and S4, with an application to topology
Authors: McKinsey J. C. C.
Published: May 2006

In this paper I shall give a solution of the decision problem for the Lewis systems S2 and S4; i.e., I shall establish a constructive method for deciding whether an arbitrary given sentence of one of these systems is provable. The method is laborious to apply, since, in order to decide by means of it whether a given sentence is provable, it is necessary to construct a (usually very large) finite matrix. The argument will perhaps be of general interest, however, because it does not seem to depend too closely on the special features of these particular systems, so that it may be possible to apply it in order to solve the decision problem for other such systems.

Section II presents the decision method for S2, and Section III for S4. In Section IV, I shall establish a certain correspondence between S4 and topology, which will provide a solution for a decision problem in topology; this correspondence also enables us to settle a previously unsolved problem with regard to the Lewis systems.

In treating of this system, I shall use the notation of Lewis, with the single exception that I shall use the symbol “≣”, instead of Lewis's symbol “=”, for strict equivalence. I shall use the symbol “=” to denote identity. Thus “p≣q” is a formula of S2, while “x=y” is the statement asserting that x and y are identical. I shall refer to the rules, primitive sentences and theorems of S2 by the names and numbers used by Lewis. Whenever a theorem stated by Lewis involves the symbol “=”, I shall of course suppose that this symbol has been replaced throughout by “≣”: thus, for example, I shall take 19.82 to be “(◇p∨◇q) ≣ ◇(p∨q)”, instead of “(◇p∨◇q) ≣ ◇(p∨q)”.
Practical forms of type theory
Authors: Turing A. M.
Published: May 2006

Russell's theory of types, though probably not providing the soundest possible foundation for mathematics, follows closely the outlook of most mathematicians. The present paper is an attempt to present the theory of types in forms in which the types themselves only play a rather small part, as they do in ordinary mathematical argument. Two logical systems are described (called the “nested-type” and “concealed-type” systems). It is hoped that the ideas involved in these systems may help mathematicians to observe type theory in proofs as well as in doctrine. It will not be necessary to adopt a formal logical notation to do so.
Strict implication, deducibility and the deduction theorem
Authors: Marcus Ruth Barcan
Published: May 2006

Lewis and Langford state, “… it appears that the relation of strict implication expresses precisely that relation which holds when valid deduction is possible. It fails to hold when valid deduction is not possible. In that sense, the system of strict implication may be said to provide that canon and critique of deductive inference which is the desideratum of logical investigation.” Neglecting for the present other possible criticisms of this assertion, it is plausible to maintain that if strict implication is intended to systematize the familiar concept of deducibility or entailment, then some form of the deduction theorem should hold for it. The purpose of this paper is to analyze and extend some results previously established which bear on the problem.

We will begin with a rough statement of some relevent considerations. Let the system S contain among its connectives an implication connective ‘I’ and a conjunction connective ‘&’. Let A1, A2, …, An ⊦ B abbreviate that B is provable on the hypotheses A1, A2, …, An for a suitable definition of “proof on hypotheses”, where A1, A2, …, An, B are well-formed expressions of S.
Systems of syntactic analysis
Authors: Chomsky Noam
Published: May 2006

During the past several decades, linguists have developed and applied widely techniques which enable them, to a considerable extent, to determine and state the structure of natural languages without semantic reference. It is of interest to inquire seriously into the formality of linguistic method and the adequacy of whatever part of it can be made purely formal, and to examine the possibilities of applying it, as has occasionally been suggested, to a wider range of problems. In order to pursue these aims it is first necessary to reconstruct carefully the set of procedures by which the linguist derives the statements of a linguistic grammar from the behaviour of language users, distinguishing clearly between formal and experimental in such a way that grammatical notions, appearing as definienda in a constructional system, will be formally derivable for any language from a fixed sample of linguistic material upon which the primitives of the system are experimentally defined. The present paper will be an attempt to formalize a certain part of the linguist's generalized syntax language.

From another point of view, this paper is an attempt to develop an adequate notion of syntactic category within an inscriptional nominalistic framework. The inscriptional approach seems natural for linguistics, particularly in view of the fact that an adequate extension of the results of this paper will have to deal with the problem of homonymity, i.e., with a statement of the conditions under which tokens of the same type must be assigned to different syntactic classes.
On finite and infinite modal systems
Authors: Churchman C. West
Published: May 2006

In Oskar Becker's Zur Logik der Modalitäten four systems of modal logic are considered. Two of these are mentioned in Appendix II of Lewis and Langford's Symbolic logic. The first system is based on A1–8 plus the postulate,

From A7: ∼◊p⊰∼p we can prove the converse of C11 by writing ∼◊p for p, and hence derive

The addition of this postulate to A1–8, as Becker points out, allows us to “reduce” all complex modal functions to six, and these six are precisely those which Lewis mentions in his postulates and theorems: p, ∼p, ◊p, ∼◊p, ∼◊∼p, and ◊∼p This reduction is accomplished by showing

where ◊n means that the modal operator ◊ is repeated n times; e.g., ◊3p = ◊◊◊p. Then it is shown that

By means of (1), (2), and (3) any complex modal function whatsoever may be reduced to one of the six “simple” modals mentioned above.

It might be asked whether this reduction could be carried out still further, i.e., whether two of the six “irreducible” modals could not be equated. But such a reduction would have to be based on the fact that ◊p = p which is inconsistent with the set B1–9 of Lewis and Langford's Symbolic logic and independent of the set A1–8. Hence for neither set would such a reduction be possible.
Logical paradoxes for many-valued systems
Authors: Shaw-Kwei Moh
Published: May 2006

As we know, the system of material implication is led into an inconsistency by the Russellian class, defined as λx. Nϵxx. This class, however, does no harm to many other systems, for example, the three-valued system L3 given by J. Łukasiewicz. A natural question is whether or not there exist classes which affect some of these other logical systems. The main result of the present paper is to answer this question affirmatively. At the end of this paper we point out that the interpretation given by J. Łukasiewicz for the system L3 is not satisfactory, and propose a new interpretation.

B. Russell deduced the mentioned inconsistency by the aid of the notion of negation. Later on, H. B. Curry pointed out that we could get the same result without the aid of that notion. None of these results affects the system L3 and other similar systems. But these systems may be involved.

To show this, we need the following definitions.

A function of two variables Cpq will be called an implication when the following “implication rule” is valid:

Under this definition we should note that material equivalence Epq, for example, is an implication.

Let C be such an implication. Then the symbol “(Cp)iq” is defined recursively by

The class an is defined as λx. (Cϵxx)np, where p is a propositional variable. The rule of absorption of order n, denoted by (An), is:
On the restricted ordinal theorem
Authors: Goodstein R. L.
Published: May 2006

The proposition that a decreasing sequence of ordinals necessarily terminates has been given a new, and perhaps unexpected, importance by the rôle which it plays in Gentzen's proof of the freedom from contradiction of the “reine Zahlentheorie.” Gödel's construction of non-demonstrable propositions and the establishment of the impossibility of a proof of freedom from contradiction, within the framework of a certain type of formal system, showed that a proof of freedom from contradiction could be found only by transcending the axioms and proof processes of that formal system. Gentzen's proof succeeds by utilisingtransfinite inductionto prove that certain sequences of reduction processes, enumerated by ordinals less than ε (the first ordinal to satisfy ε = ὡ) are finite. Were it possible to provethe restricted ordinal theorem, that a descending sequence of ordinals, less thanε, is finite, in Gentzen's “reine Zahlentheorie,” then it would be possible to determine a contradiction in that number system. In his paper, Gentzen proves the theorem of transfinite induction, which he requires, by an intuitive argument. There is also a method of reducing transfinite induction, for ordinals less than ε, to a number-theoretic principle given by Hilbert and Bernays, and a similar method by Ackermann. None of these proofs of transfinite induction isfinitist.
Some theorems about the sentential calculi of Lewis and Heyting
Authors: McKinsey J. C. C. , Tarski Alfred
Published: May 2006

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.
Note on a property of matrices for Lewis and Langford's calculi of propositions
Authors: Dugundji James
Published: May 2006

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.

◊N = N (N the null class)

◊A = {1, 2, …, n} (A any non-null class).5
Definability and simplicity
Authors: Svenonius Lars
Published: May 2006

In the following pages we wish to clarify some questions concerning definitions of relations in terms of other relations. A study is made of how a definition of a relation R1 in terms of another relation R2 works in a given specified universe U. The definition is regarded as a correlation rule between the “possible extensions” of R1 and R2. There are no difficulties in formulating necessary, and in a certain sense also sufficient, formal conditions for the existence of a definition relation between R1 and R2.

In particular, we are concerned with questions such as: “Is every relation of the form s1 definable in terms of some relation of the form s2?” (§ IV) A related question concerns the criterion used by Goodman1 for degrees of complexity, namely, whether for every relation R2 of the form s1 there is some relation R2 of the form s2 such that R1 and R2 are mutually definable (§ V). Some concrete results concerning this condition are stated in the theorems 22–24.

The “complexity function” which is defined (definition 17) and which very naturally suggests itself from the discussion of the definability conditions, seems to furnish an adequate measure of the sort of complexity discussed by Goodman.
Widerspruchsfreier Aufbau der Logik I
Authors: Ackermann Wilhelm
Published: May 2006

Die vorliegenden Untersuchungen haben sich aus solchen entwickelt, die ich an andrer Stelle früher veröffentlichte. Die Kenntnis dieser früheren Arbeit wird in keiner Weise hier vorausgesetzt und ist für das folgende unwesentlich. Es sei mir jedoch gestattet, für den Kenner der früheren Arbeit einige Bemerkungen voraufzuschicken. In der genannten Arbeit hatte ich, angeregt ursprünglich durch Gedankengänge von Herrn Behmann, ein typenfreies System der Logik aufgebaut, das allein auf dem Gedanken fußte, daß der Definitionsbereich der Prädikate im allgemeinen beschränkt ist, sodaß nur bestimmte Zusammenstellungen von Prädikaten und Dingen sinnvolle, d.h. wahre oder falsche Aussagen ergeben, und das diesen Gesichtspunkt meiner Meinung nach konsequent durchführte. Eine anschließende Arbeit sollte dann die Widerspruchsfreiheit des Systems untersuchen. Diese Untersuchung ist inzwischen geschehen. Es zeigte sich, daß sich die Widerspruchsfreiheit des Systems nur dann beweisen ließ, falls die in ihm enthaltene Vollständigkeit von → die sich durch das Bestehen eines Deduktionstheorems zeigte, fallen gelassen wurde. Die Grundformeln (13) und (14) mußten wegbleiben. Die Widerspruchsfreiheit des restlichen Systems ließ sich dann durch eine dem System eigentümliche Methode der Beweistransformation zeigen.

Diese anschließienden Untersuchungen, die ihrer Zeit nicht mehr veröffentlicht werden konnten, sind inzwischen durch die vorliegenden überholt. Es war das Unbefriedigende bei dem genannten Aufbau, daß hinterher doch wieder eine Einschränkung gemacht werden mußte, die sich nicht aus den zugrunde gelegten Prinzipien ergab, sondern von ihnen aus unvorhergesehen war. Diese Einschränkung bewirkte übrigens, daß keine volle Zahlentheorie in dem System zu gewinnen war.
Another variant of natural deduction
Authors: Copi Irving M.
Published: May 2006

Since 1934 various different techniques for natural deduction have been developed by Gentzen, Jaśkowski, Rosser, Quine, and others (see [1], pp. 147–167; [2], especially footnotes 1, 3, and 4; and [3], pp. 75-83, 96-107, and 289-294). It has been pointed out to me by Professor Donald Kalish of U.C.L.A. that the restrictions placed upon Universal Generalization (UG) and Existential Instantiation (EI) in [3] force one to construct a less natural proof than seems desirable for such arguments as

(cf. [3], p. 139, and [1], pp. 175f.) It is my purpose in this note to formulate an alternative restriction on UG which will permit a more natural proof for such arguments, and to prove the consistency of the altered rule.

In the notation of [3], “The expression ‘Φμ’ will denote any propositional function in which there is at least one free occurrence of the variable denoted by ‘μ’. The expression ‘Φν’ will denote the result of replacing all free occurrences of μ in Φμ by ν, with the added proviso that when ν is a variable it must occur free in Φν at all places at which μ occurs free in Φμ.” ([3], p. 100.) The statement of EI is relatively unrestricted; being

provided that ν is a variable which occurs free in no earlier step.” ([3], p. 104.)
On the definition of ‘formal deduction’
Authors: Montague Richard , Henkin Leon
Published: May 2006

The following remarks apply to many functional calculi, each of which can be variously axiomatized, but for clarity of exposition we shall confine our attention to one particular system Σ. This system is to have the usual primitive symbols and formation rules of the pure first-order functional calculus, and the following formal axiom schemata and formal rules of inference.

Axiom schema 1. Any tautologous wff (well-formed formula).

Axiom schema 2. (a) A ⊃ B, where A is any wff, a and b are any individual variables, and B arises from A by replacing all free occurrences of a by free occurrences of b.

Axiom schema 3. (a)(A ⊃ B)⊃(A⊃ (a)B). where A and B are any wffs, and a is any individual variable not free in A.

Rule of Modus Ponens: applies to wffs A and A ⊃ B, and yields B.

Rule of Generalization: applies to a wff A and yields (a)A, where a is any individual variable.

A formal proof in Σ is a finite column of wffs each of whose lines is a formal axiom or arises from two preceding lines by the Rule of Modus Ponens or arises from a single preceding line by the Rule of Generalization. A formal theorem of Σ is a wff which occurs as the last line of some formal proof.
On the interpretation of Aristotelian syllogistic
Authors: Shepherdson J. C.
Published: May 2006

The main purpose of this note is to prove (theorem 11, § 5) that, in any interpretation of the formalisation of Aristotelian syllogistic given by Łukasiewicz [4], it is always possible to associate with each element a a non-null sub-class φ(a) of some ‘universal’ class V in such a way that ‘Aab’ (all a are b), ‘Iab’ (some a are b) are equivalent respectively to ‘φ(a) is contained in φ(b)’, ‘φ(a) has a non-null intersection with φ(b)’. Similarly (theorem 6, §4) we show that in Wedberg's system [14] with primitives ‘Aab’, ‘a’ (not a), it is possible to find a mapping a → φ(a) as above such that ‘Aab’ is equivalent to ‘φ(a) is contained in φ(b)’ and φ(a‘ is equal to φ(a)’, the complement of φ(a) with respect to V. Thus, if we make the preliminary step of identifying elements a, b such that Aab and Aba both hold (i.e. taking equivalence classes with respect to the relation Aab & Aba), we are left with essentially only one kind of interpretation for these systems, namely the ‘normal’ interpretation by classes. Slupecki [11], [12] has proved that Łukasiewicz's system is a complete and decidable theory of the relations of inclusion and intersection of non-null classes, and Wedberg [14] has proved that his system is a complete and decidable theory of the relation of inclusion and the operation of complementation for nonnull, non-universal classes. Using the above-mentioned embedding theorem, we are able to obtain (theorems 9, 6, §§ 5, 4) very simple proofs of these results.
Investigation of the equivalence of the axiom of choice and Zorn's lemma from the viewpoint of the hierarchy of types
Authors: Büchi J. Richard
Published: May 2006

It is a well known fact that Zermelo's Axiom of Choice and Zorn's Lemma are equivalent logical assumptions. However, an investigation from the viewpoint of the hierarchy of types reveals a complication in the nature of this equivalence. In a type-theoretical formalism both postulates enter as spectra (ZAα) and (ZLα) of formulas, where ZAα is Zermelo's Axiom and ZLα is Zorn's Lemma stated for variables of a fixed type α. The complication has its origin in the fact that, although the formula ZAα implies ZLα, in order to deduce ZAα it seems to be necessary to assert Zorn's Lemma ZLβ for a type β which is higher than α. In other words, we don't know a proof assuring the equivalence of the two formulas ZAα and ZLα. Therefore the equivalence of Zermelo's Axiom and Zorn's Lemma has to be understood in the sense that the assertion of the formulas ZAα for all types α implies every one of the formulas ZLβ and conversely the assertion of the spectrum of formulas (ZLα) implies every one of the formulas ZAβ.

In § 3 we shall indicate a type β which makes the deduction of ZAα from ZLβ possible. For this purpose a proof by G. Birkhoff [3] is translated into type-theoretical language.

In § 2 it is shown that ZAα implies ZLα. In this proof no use is made of the notion of well-ordering. Specifically, we do not first deduce the Well-Ordering Theorem as it is usually done in proving Zorn's Lemma from Zermelo's Axiom. However, the method is suggested by Zermelo's second proof of the Well-Ordering Theorem [2].
Foundations of mathematics for the working mathematician
Authors: Bourbaki N.
Published: May 2006

I am very grateful to the Association for Symbolic Logic for inviting me to give this address—an honor which I am conscious of having done very little to deserve. My efforts during the last fifteen years (seconded by those of a number of younger collaborators, whose devoted help has meant more to me than I can adequately express) have been directed wholly towards a unified exposition of all the basic branches of mathematics, resting on as solid foundations as I could hope to provide. I have been working on this as a practical mathematician; in matters pertaining to pure logic, I must confess to being self-taught, and laboring under all the handicaps that this implies; and if, after no little self-questioning, I am speaking here today, I am doing so chiefly in order to enjoy the benefit of your professional advice and criticism, by which I hope to correct my views before I venture into print with them.
On the interpretation of intuitionistic number theory
Authors: Kleene S. C.
Published: May 2006

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.
Correction to A note on the Entscheidungsproblem
Authors: Church Alonzo
Published: May 2006

In A note on the Entscheidungsproblem the author gave a proof of the unsolvability of the general case of the Entscheidungsproblem of the engere Funktionenkalkül. This proof, however, contains an error, in order to correct which it is necessary to modify the “additional axioms” of the system L so that they contain no free variables (either free individual variables or free propositional function variables).

The additional axioms of L other than x=y→[F(x)→F(y)] contain no free propositional function variables, and hence it is sufficient to replace each one by an expression obtained from it by quantifying all the individual variables by means of universal quantifiers initially placed—thus, in particular, x = x is replaced by (x)[x=x]. Moreover the axiom x=y→[F(x)→F(y)] may be replaced by the following set of axioms:

and similar axioms for each of the functions b1, b2, …, bk.
Identity, variables, and impredicative definitions
Authors: Jaakko K. , Hintikka J.
Published: May 2006

Recent discussion serves to bring out, amply and convincingly, the utility of observing the ordinary correct use of words and phrases for the purpose of clearing up philosophical problems. In this paper, I shall endeavour to show, by means of an example, that the reverse method may have its interest, too. An attempt will be made to cultivate a minor deviation from the accepted ways of using certain words and phrases in idiomatic English as well as in the formalized “languages” of the logicians. The words and phrases in question are those for the formalization of which a logician employs (free or bound) variables. Cases in point are the words customarily called quantifiers. The deviation I have in mind affects the relation of these words to the notion of identity. The deviation is illustrated by the following sentences:

(1a) Any two points of a straight line completely determine that line;

(2a) He is John's brother if he has the same parents as John;

(3a) Mazzini did more for the emancipation of his country than any living man of his time.

These examples may be contrasted with the following closely related sentences:

(1b) Any two distinct points of a straight line completely determine that line;

(2b) He is John's brother if he has the same parents as John and if he is not John himself;

(3b) Mazzini did more for the emancipation of his country than any other living man of his time.
A Functional calculus of first order based on strict implication
Authors: Barcan Ruth C.
Published: May 2006

The following system is an extension of the Lewis calculus S2 to include quantification.
The inconsistency of certain formal logics
Authors: Curry Haskell B.
Published: May 2006

A proof that certain systems of formal logic are inconsistent, in the sense that every formula expressible in them is provable, was published by Kleene and Rosser under the above title in 1935. For the case where the underlying system satisfies an additional condition—viz. the possession of an operator analogous to Schönfinkel's K—I gave a simpler derivation of this condition in a previous paper. That argument, like the original one of Kleene and Rosser, was a refinement of the Richard paradox. The object of the present paper is to show that, if we use other paradoxes, an inconsistency will result from a very much simpler argument and on much less restrictive hypotheses. The contradiction can no longer be called “the paradox of Kleene and Rosser,” because it is based on an entirely different principle; but, in deference to the work of the original discoverers of the inconsistency, the paper is given the same title as that which their paper bears. The central idea of the new derivation was suggested by some work of R. Carnap.

This paper is based on the one above cited, which will be referred to as PKR. However, acquaintance with that paper will be presupposed only through 3.4, i.e., through the statement of the basic hypotheses and conventions for a combinatorially complete system—except that for the second (alternative) method of construction given below the substance of PKR through 9.6 is needed.
Logical truth and logical implication
Authors: Hatcher William S.
Published: May 2006

In [1], page 85, Quine states: “… ponentials of logically true statements are logically true.” It is perhaps of some interest that this assertion is not completely precise as can be seen from counterexamples. One simple example is the following:

Let L be some language in which ‘c’ and ‘b’ are atomic sentential constants. Let ‘c’ be false and ‘b’ true. Suppose that L includes parentheses and the material conditional ‘⊃’ as logical notation, but that ‘b’, at least, counts as extra-logical.
Algebraic semantics for modal logics II
Authors: Lemmon E. J.
Published: May 2006

This paper is a sequel to [7], and the terminology of [7] is largely presupposed here. In [7], the algebraic methods of McKinsey-Tarski were employed and extended to yield semantic results of a Kripke kind for a class of relatively weak modal logics, the strongest of which was the Feysvon Wright system T. Deontic versions of both T and E2, called T(D) and D2, and even weaker systems, were handled. The main aim of the present paper is to extend these results to stronger systems of modal logic. Thus the Lewis systems S2–S5, the Brouwersche system B of Kripke [4], the systems E3–E5 of [5], and Łukasiewicz's modal logic, as well as certain new systems, are considered.

Certain modifications of the method of [7] have proved convenient. Thus in Section I, some further results concerning model structures are proved in order that the relationship between S2 and E2, S3 and E3, can be properly stated; in particular, the notions of a refined and connected model structure play a pervasive role throughout.
A remark on infinitary languages
Authors: Flum Jörg
Published: May 2006

In his paper [1] Chang provides among other things answers to questions of the following type: Given two models and of powers α and β, respectively, what is the least λ such that implies His proofs are by induction on the quantifier rank of formulas and they use an idea which in the case of ordinary first-order language goes back to Ehrenfeucht and Fraïssé. But, as we show, one can easily prove that if λ is big compared with κ and with the cardinality of the universe of the structure , then every L∞κ-formula is equivalent modulo the set of all Lλκ-sentences which hold in to a Lλκ-formula. From this, Chang's results follow immediately. The same method can be applied to similar problems concerning generalized languages.
Effectively extensible theories
Authors: Boykan Pour-El Marian
Published: May 2006

It is well known that Gödel's famous undecidability result may be viewed in the following strong form. Suppose we are given a specific presentation (i.e., a specific formulation in terms of axioms and rules of inference) of number theory. Then there exists an effective method which, when applied to a consistent axiomatizable extension of the theory yields an undecidable sentence of this extension. For distinct presentations the undecidable sentences obtained would be distinct. This is because the sentence constructed depends upon the notion of proof and hence ultimately upon the axioms and rules of inference—i.e., upon the specific presentation.
Analytic natural deduction
Authors: Smullyan Raymond M.
Published: May 2006

We consider some natural deduction systems for quantification theory whose only quantificational rules involve elimination of quantifiers. By imposing certain restrictions on the rules, we obtain a system which we term Analytic Natural Deduction; it has the property that the only formulas used in the proof of a given formula X are either subformulas of X, or negations of subformulas of X. By imposing further restrictions, we obtain a canonical procedure which is bound to terminate, if the formula being tested is valid. The procedure (ultimately in the spirit of Herbrand [1]) can be thought of as a partial linearization of the method of semantical tableaux [2], [3]. A further linearization leads to a variant of Gentzen's system which we shall study in a sequel.

The completeness theorem for semantical tableaux rests essentially on König's lemma on infinite graphs [4]. Our completeness theorem for natural deduction uses as a counterpart to König's lemma, a lemma on infinite “nest structures”, as they are to be defined. These structures can be looked at as the underlying combinatorial basis of a wide variety of natural deduction systems.

In § 1 we study these nest structures in complete abstraction from quantification theory; the results of this section are of a purely combinatorial nature. The applications to quantification theory are given in § 2.
Axiomatizability by a schema
Authors: Vaught Robert L.
Published: May 2006

A theory T is axiomatizable by a schema if there is a formula Γ, involving symbols of T plus a new relation symbol R, such that the set of all (universal closures of) instances of Γ in T is a set of axioms for T. (It is understood that, if R has n places, an instance of Γ in T is obtained by properly substituting for R in Γ a formula of T which has n selected free variables and is allowed to have any number of other free variables as parameters.) Obviously, the notion is unchanged if finitely many Γ's, each involving several new R's, are allowed instead. All theories we consider are assumed to be theories in the first-order logic with equality (as in [8]), to have finitely many nonlogical symbols, and to be recursively axiomatizable.
On effectively discontinuous type-2 objects
Authors: Grilliot Thomas J.
Published: May 2006

The notion of a recursive function with type-2 arguments was introduced by Kleene [4]. Soon E, the type-2 object that introduces numerical quantification, began to play a central role in the development of the theory. First, Kleene [4, XLVIII] proved that the hyperarithmetical functions are exactly the functions recursive in E. Later, Gandy [1, p. 14] proved the existence of a selection operator recursive in E (that is, some ψ(х, ), partial recursive in E, having the property that, if {х}(y, ) converges for some y, then ψ(х, ) converges and is such a y).
Degrees of recursively enumerable sets which have no maximal supersets
Authors: Lachlan A. H.
Published: May 2006

The purpose of this paper is to present two new theorems concerning the degrees of coinfinite recursively enumerable (r.e.) sets which have no maximal supersets Let the class of all such degrees be denoted by A. Martin in [2] conjectured that there was some equality or inequality involving a′ or a″ characterizing the degrees a in A. Martin himself proved ([2, Corollary 4.1]) that a′ = 0″ is sufficient for ar r.e. degree a to be in A, and Robinson [3] announced that a′ ≥ 0″ is necessary. In this paper we improve both of these theorems by a factor of the jump, i.e., we shall show that a″ = 0″ is sufficient for an r.e. degree a to be in A, and that a″ ≥ 0″ is necessary.
Results concerning same modal systems that contain S2
Authors: Åqvist Lennart
Published: May 2006

It is well known that if a postulate LLCpp or a rule of procedure ├α→├Lα (from a thesis α of a considered system to infer a thesis Lα of that system) is added to Lewis's modal system S3, we get his system S4 (see [7], p. 148). It is also known that the addition of this rule to S2 and SI has the effect of converting these systems into the Gödel-Feys-von Wright system T (M). My purpose in this paper is first to draw attention to some other ways of converting S3 and S2 into S4 and T respectively, as well as to extend a theorem of Halldén on S3, S4 and S7 to S2, T and S6. The results reached will also apply to the systems S3.5, S5 and S7.5, where S3.5 and S7.5 are obtained, respectively, from S3 and S7 by the addition of the postulate CNLpLNLp; an answer will be given to the question of irreducible modalities in S3.5. Moreover, two results will be proved that bear on the problem whether the systems S2 and T can be axiomatized by means of a finite number of axiom schemata and material detachment as the only primitive rule of inference.
Pseudo-complements and ordinal logics based on consistency statements
Authors: Di Paola Robert A.
Published: May 2006

Following [1] we write {n} for the nth recursively enumerable (re) set; that is, {n} = {x|VyT(n, x, y)}. By a “pair (T, α)” we mean a consistent re extension T of Peano arithmetic P and an RE-formula α which numerates the non-logical axioms of T in P [4]. Given a pair (T, α) and a particular formula which binumerates the Kleene T predicate in P, there can be defined a primitive recursive function Nα such that and which has the additional property that {Nα(Nα(n))} = ø for all n.
On analytic well-orderings
Authors: Tanaka Hisao
Published: May 2006

Since about 1955, recursive, (hyper-)arithmetic and well-orderings have been investigated by many authors. Now we shall generally consider analytic well-orderings and compare their representation-capacities for ordinals.

Let K be a class of predicates. Then, R is called a Kwell-ordering if R is a binary relation of natural numbers belonging to the class K and satisfies the following conditions:

(i) R(x, y) ∧ R(y, x) → x = y;

(ii) x, y ∈ D(R) → R(x, y) ∨ R(y, x), where D(R) is the domain of R, that is to say, the set {x ∣ (∃y)[R(x, y) ∨ R(y, x)]};

(iii) R(x, y) ∧ R(y, z) → R(x, z);

(iv) .
The decision problem for formulas in prenex conjunctive normal form with binary disjunctions
Authors: Krom M. R.
Published: May 2006

In [8] S. J. Maslov gives a positive solution to the decision problem for satisfiability of formulas of the form

in any first-order predicate calculus without identity where h, k, m, n are positive integers, αi, βi are signed atomic formulas (atomic formulas or negations of atomic formulas), and ∧, ∨ are conjunction and disjunction symbols, respectively (cf. [6] for a related solvable class). In this paper we show that the decision problem is unsolvable for formulas that are like those considered by Maslov except that they have prefixes of the form ∀x∃y1 … ∃yk∀z. This settles the decision problems for all prefix classes of formulas for formulas that are in prenex conjunctive normal form in which all disjunctions are binary (have just two terms). In our concluding section we report results on decision problems for related classes of formulas including classes of formulas in languages with identity and we describe some special properties of formulas in which all disjunctions are binary including a property that implies that any proof of our result, that a class of formulas is a reduction class for satisfiability, is necessarily indirect. Our proof is based on an unsolvable combinatorial tag problem.
Postulates for implicational calculi
Authors: Meredith C. A.
Published: May 2006

I. (Bull's system OIC, defined in [1] as positive implication plus ССССpqqrСССpqrr, can be axiomatised by adding this to a weaker base.)

1. СpСqp

2. ССpqССqrСpr

3. ССССpqqrСССpqrr

4. ССССqrСprsССpqs = D22

5. ССrСpqСССpqСrqСrq = D43

6. СССССpqССrqСrqsССrСpqs = D25

7. ССpСqrССpqСpr = D64