bluuebunny/crossref_metadata_2025 huggingface dataset:
Community Versus Market: A Note on Corporate Villages
Authors: Bates Robert H. , Curry Amy Farmer
Published: May 2006

We return to the literature on collective villages and reexamine its central arguments. In doing so, we focus on an institution for allocating land that we call the Rule. As claimed by its advocates, the Rule secures land allocations that result in outcomes differing from those that would be achieved by markets. The outcomes are not constrained to be efficient. But the economic costs paid fail to secure the attainment of guarantees of subsistence, greater equality, or a greater sense of community resulting from the elimination of envy. However, in the face of high levels of risk, the form of collective property rights embodied in the Rule may create what we call “communities of (mis)fortune,” in which no one can envy others' gains or losses resulting from the random shocks of nature.
The New Civil-Military Relations
Authors: Lyons Gene M.
Published: May 2006

Historically the character of civil-military relations in the United States has been dominated by the concept of civilian control of the military. This has largely been a response to the fear of praetorianism. As recently as 1949, for example, the first Hoover Commission asserted that one of the major reasons for strengthening the “means of exercising civilian control” over the defense establishment was to “safeguard our democratic traditions against militarism.” This same warning was raised in the report of the Rockefeller Committee on defense organization in 1953. While the overriding purpose of the committee's recommendations was to provide “the Nation with maximum security at minimum cost,” the report made it clear that this had to be achieved “without danger to our free institutions, based on the fundamental principle of civilian control of the Military Establishment.” Finally, during the debate on the reorganization proposals of 1958, senators and congressmen used the theme of a “Prussianized” military staff to attempt to slow down the trend towards centralization in the military establishment.

Despite this imposing support, the concept of civilian control of the military has little significance for contemporary problems of national security in the United States. In the first place, military leaders are divided among themselves, although their differences cannot be reduced to a crass contrast between dichomatic doctrines. Air Force leaders who are gravely concerned over the need to maintain a decisive nuclear retaliatory force are by now acknowledging the need to develop a limited war capability.
Mao Tse-tung's Revolutionary Strategy and Peking's International Behavior
Authors: Tsou Tang , Halperin Morton H.
Published: May 2006

Despite its political implications, the recent explosion of an atomic device has not greatly altered China's present military positionvis-à-visthe West. By all standards except population and size, Communist China is still not a first-rate power. But she has nevertheless proceeded to engage the two superpowers simultaneously in a contest from her position of military and economic weakness. What is equally undeniable is that the success of Peking's foreign policy in the struggle with both superpowers, though limited and perhaps only temporary, has considerably exceeded anticipations based on her military and economic strengths. It is the contention of this paper that an explanation of these two striking facts must be sought in the nature of Mao's revolutionary strategy in the Chinese internal political-military struggle and his belief in the applicability of this strategy to the international arena and to other countries, particularly those in the underdeveloped areas. Mao dared to challenge the militarily and economically strong United States because his revolutionary experience proved to his own satisfaction that his integrated and comprehensive strategy would enable him presently to score political gains from a position of military inferiority, and ultimately enable him to achieve highly ambitious objectives with initially meager means in a protracted struggle.
Partisan Dealignment in the Postwar South
Authors: Beck Paul Allen
Published: May 2006

This study attempts to explain post-World War II southern electoral politics by examining the party identifications of southerners between 1952 and 1972. Pronounced decreases in Democratic loyalties and increases in Independent leanings appear during this period and constitute a dealignment of the southern electorate. While interregional population exchanges have diluted Democratic strength, their effects are almost counterbalanced by the mobilization of blacks into politics. Instead, the principal source of dealignment is the generational replacement of the native white electorate. Its youngest members, who entered the electorate after World War II, have come to favor political independence increasingly in recent years. This behavior seems partially attributable to a tendency for young native whites in particular to bring their partisan loyalties into line with their attitudes and party images on racial issues. Even so, there are clear signs that the racial question is losing its place as the major determinant of the region's politics. For the future, one can expect a continuation of dealignment politics and little chance of a partisan realignment.
A New Guarantee on Earth: Hannah Arendt on Human Dignity and the Politics of Human Rights
Authors: Isaac Jeffrey C.
Published: May 2006

While the writings of Hannah Arendt have received an extraordinary amount of scholarly attention, few commentators have seen her as a theorist concerned with questions of human rights. I argue that the problem of human rights was central to Arendt's political theory. While she does not elaborate a theory of human rights as such, and while she avoids the juridical approaches so common among human rights theorists and advocates, her conception of political action is intended to secure an elemental human dignity that is systematically jeopardized by the imperatives of national sovereignty.
Buying Supermajorities
Authors: Groseclose Tim , Snyder James M.
Published: May 2006

Minimal winning coalitions have appeared as a key prediction or as an essential assumption of virtually all formal models of coalition formation, vote buying, and logrolling. Notwithstanding this research, we provide a model showing that supermajority coalitions may be cheaper than minimal winning coalitions. Specifically, if vote buyers move sequentially, and if the losing vote buyer is always granted a last chance to attack the winner's coalition, then minimal winning coalitions will generally not be cheapest, and equilibrium coalitions will generally not be minimal winning. We provide results relating equilibrium coalition size with preferences of the legislators and vote buyers, and we show that minimal winning coalitions should occur in only rare cases. We discuss these results in light of empirical work on coalition size and suggest other possible avenues for testing our model.
On ternary logic
Authors: Dienes Paul
Published: May 2006

1. The problem

In this paper we give the complete list of the functions of one variable with some of their properties, and lists of functions corresponding to various properties of sum, product, implication, and equivalence. The range of the variables as well as that of functional values will be о (false), ½, 1 (true). As an application we consider Frege's, Russell's and Heyting's systems in ternary logic. The proofs are mostly omitted as obvious if sometimes laborious.

Functions will be denned by matrices: e.g. F(p, q)
The axiom of infinity in Quine's New Foundations
Authors: Rosser J. Barkley
Published: May 2006

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 systemPproposed 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.
The permutability of rules in the classical inferential calculus
Authors: Curry Haskell B.
Published: May 2006

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.
A note on two-place predicates and fitting sequences of measure functions
Authors: Rubin Herman , Suppes Patrick
Published: May 2006

Carnap (in [1], p. 566) has remarked that his measure function m* is fitting for finite languages using a fixed number of one-place predicates, i.e., for any sentence i, m*(i) is the same in all such restricted finite languages in which i occurs. The main purpose of this brief note is to show by means of a counter-example that m* does not have the intuitively desirable property of fittingness when we consider languages using two-place predicates.

We first state for finite languages a general theorem (related to the results in [2]) which guides the construction of large numbers of counter-examples.

Theorem. Let be the symmetric group of n letters. In a language which consists of a finite number of predicates and n individual names, the number of state descriptions in a structure description is equal to the index of some subgroup G in .

Proof. Let Σ be a structure description (of a language satisfying our hypothesis) and let S be an arbitrary state description in Σ. Let π be a permutation in , and let π*S be the state description which results from S by applying the permutation π to the individual names of . It is easily verified that for every π in π*S ∊ Σ. Also, for π, ψ in ,

We define the set G of permutations as follows :

.
Computational logic
Authors: Levin Nathan P.
Published: May 2006

We make certain notational conventions. These are referred to as CL (Computational Logic). The relation of interchangeability is introduced as the basic connection between logical formulas. This approach lends perspicuity to the results of sentential calculus. With its technical devices CL is able to rephrase logical theorems in rather succinct manner. Our exposition tries to steer a middle course between informality and strict rigor.

It is felt that the method of CL offers advantages for the teaching of logic. Proofs are algorithmic and resemble those of elementary algebra. All inferences are reversible and practically non-tentative.

Sign is a primitive term of CL. Its denning property is a capacity for entering into binary combination with other signs, or with itself, according to this convention:

x and y are signs if, and only if, (x y) is a sign. Bracketing is looked upon as an operation on signs in terms of which other operations are definable. The practice of some authors in classifying brackets under the heading of symbols, seems to us questionable; for unlike symbols or signs brackets are never used to denote anything. Brackets enter into the composition of signs, not to denote a grouping, but rather to exhibit it, in the manner of a diagram.

An unending list of letters, with or without subscripts,

serve to denote arbitrary signs. An arbitrary sign may or may not have other signs as parts. The numeral “2” is used as a constant. Bracketing abbreviation is as follows:

and so on. Outermost brackets will ordinarily be omitted. This kind of bracketing may be termed left-associative. For convenience, these bracketing conventions are crystallized into a rule.
Axiom schemes for m-valued functional calculi of first order. Part I. Definition of axiom schemes and proof of plausibility
Authors: Rosser J. B. , Turquette A. R.
Published: May 2006

A calculus, or its formalization, is m-valued when its truth-functions are allowed to take truth-values ranging from 1 through m. It is customary to divide the m truth-values that are possible into those that are “designated” and those that are “undesignated.” Furthermore, it is usually desired of a formalization that provable formulas and only provable formulas take designated truth-values exclusively. For our present purposes, we shall suppose that the truth-values 1, …, 8 are designated and the truth-values 8+1, …, m are undesignated. When calculi differ in respect to the number of their designated truth-values, we shall consider them different calculi, even if they are otherwise similar.
The problem of interpreting modal logic
Authors: Quine W. V.
Published: May 2006

There are logicians, myself among them, to whom the ideas of modal logic (e. g. Lewis's) are not intuitively clear until explained in non-modal terms. But so long as modal logic stops short of quantification theory, it is possible (as I shall indicate in §2) to provide somewhat the type of explanation desired. When modal logic is extended (as by Miss Barcan1) to include quantification theory, on the other hand, serious obstarles to interpretation are encountered—particularly if one cares to avoid a curiously idealistic ontology which repudiates material objects. Such are the matters which it is the purpose of the present paper to set forth.
A problem in recursive function theory
Authors: Goodstein R. L.
Published: May 2006

In a recent paper [4] on mean value theorems in recursive function theory we proved the theorem that

(A) iff(n, x) is relatively differentiable with a relative derivative f1(n, x), for a ≤ x ≤ b, and if f(n, a) = f(n, b) = 0 relative to n,

then there is a recursive function ck, a < ck < b, and a recursive R(k) such that f1(n, ck) = 0(k) for n ≥ R(k); and we showed further that the added condition

(B) f(n, x) is either relatively variable or relatively constant

suffices to ensure that ck is uniformly contained in (a, b), i.e. that there exist α, β such that

A comparison with the conditions under which Rolle's theorem is established in classical analysis suggests that clause (A) itself might suffice to ensure that ck is uniformly contained in (a, b); for in the classical theory there is a single point c, a < c < b, for which lim f1(n, c) = 0, and therefore f1(n, c) = 0(k) for sufficiently great values of n, where of course c is independent of k.

The object of the present note is to show that this is not in fact the case, and we shall construct a recursive function f(n, x) satisfying condition (A) in the interval (0, 1) and such that any sequence ck for which f1(n, ck) = 0(k) for large enough values of n is not uniformly contained in (0, 1).
New sets of postulates for combinatory logics
Authors: Rosser Barkley
Published: May 2006

Sets of postulates for at least three different combinatory logics have been given ([2], [3], [4], [5], [6], [7]). The logics and their sets of postulates are quite similar. In all cases it is the proof of the completeness of the set of postulates which causes the difficulty and makes the main papers so long ([3], [7]). It is the purpose of this paper to present a method of writing down sets of postulates for combinatory logics for which the proof of completeness will be fairly simple. We assume acquaintance with certain descriptive portions of [1], namely sections 1–6, 12, 13, 15. Otherwise the present paper is self-contained.

We first solve our problem for the particular logic studied by Rosser ([7]) and then indicate how the method would apply to other logics.

We start with a set, s, of primitive terms. We define s-combination as in [1], p. 43. The precise contents of s need not concern us. It suffices that there be s-combinations I, J, and Q, with no variables in them, and having properties as described below. I and J are to have the same meaning as in [1], p. 43, namely IA is the same as A and JABCD is the same as AB(ADC). QAB is to have the meaning that A and B, considered as functions, are the same.
A formal theorem in Church's theory of types
Authors: Newman M. H. A. , Turing A. M.
Published: May 2006

This note is concerned with the logical formalism with types recently introduced by Church [1] (and called (C) in this note) It was shewn in his paper (Theorem 26α) that if Yα stands for

(a form of the “axiom of infinity” for the type α), Yα can be proved formally, from Yι and the axioms 1 to 7, for all types α of the forms ι′, ι″, …. For other types the question was left open, but for the purposes of an intrinsic characterisation of the Church type-stratification given by one of us, it is desirable to have the remaining cases cleared up. A formal proof of Yα is now given for all types α containing ι, but the proof uses, in addition to Axioms 1 to 7 and Yι, also Axiom 9 (in connection with Def. 4), and Axiom 10 (in Theorem 9).
A system of completely independent axioms for the sequence of natural numbers
Authors: Wang Shianghaw
Published: May 2006

Peano's five axioms for the sequence of natural numbers run as follows: (1) 1 is a number, (2) Any number α has a unique sequent a′ (3) Different numbers have different sequents, (4) 1 is not a sequent, and (5) The principle of mathematical induction.

Now, in a system of axioms, it seems desirable, in the first place, to avoid symbols for particular individuals and symbols for functions, and, secondly, to secure the property of complete independence. In fact, Peano's system can be replaced by an equivalent one which fulfills both requirements. In such a system, the symbol “1” and the function “sequent” will not appear, although they may be introduced afresh through definitions. Besides, the principle of mathematical induction will no more play the rôle of an axiom, because, under the second requirement, it can hardly be stated in a simple and elegant form.
Remark on Nicod's reduction of Principia mathematica
Authors: Bernstein B. A.
Published: May 2006

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.
A new approach to semantics – Part I
Authors: Kemeny John G.
Published: May 2006

This is the first in a series of articles outlining a new approach to Semantics. The novelty in the approach is that the concept of an interpretation of a logical system is taken as the central concept of Semantics. I hope to show that by means of this approach a satisfactory definition can be given for such controversial concepts as analyticity, and at the same time the approach leads to a unified foundation for formalized Semantics.

As the possibility of such definitions has been questioned in recent years, it is an important task to try to give precise definitions. Clearly, this is the task of those philosophers who believe that concepts like analyticity should play a fundamental role in Semantics. On the other hand, the philosophers who have criticized these concepts will now be able to tell just exactly why these definitions are unacceptable to them — instead of being forced to talk in generalities. It seems to me that no further progress is possible until we have precise definitions available for discussion.

The fundamental semantic concepts fall into two classes: those, like truth, for which Tarski has offered definitions; and those additional ones, like analytic truth, for which we have Carnap's proposed definitions.
Ideographic computation in the propositional calculus
Authors: Standley Gerald B.
Published: May 2006

This paper will be concerned with an adaptation of Parry's trapezoid symbolism to the solving of problems.

In Parry's notation ‘pq’, in particular, sets forth the relationship that obtains when p is true, whether q be true or false. The symbol may also be regarded as manifesting the matrix analysis of p. It is this aspect which offers promise for manipulation. We propose to write the symbol ‘’ for p itself. It is to be kept in mind, however, that ‘’ is the symbol of p when there are but two variables. How to proceed when there are more than two will be seen later.

The symbol for q will in like manner be ‘⌋’.

Let ‘’ and ‘’, then, be regarded as elemental symbols setting forth the matrix analysis of the variables. Suppose further that in transcribing from conventional to the trapezoid symbolism we substitute for the variables only, retaining the constants (antithetically to Parry's procedure). Thus ‘p ∨ q’ becomes, not ‘pq’ as with Parry, but ‘ ∨ ’. Problem-solving consists of merging the trapezoidal symbols according to procedures specified by the constants. As symbols are merged, the constants are eliminated.

The resultant symbol constitutes the solution.

The rules for merging symbols are derived from the elementary processes of forming truth-tables:

Disjunction of two symbols consists of superimposing them.

Conjunction consists of preserving only such sides as are common to both the conjoined symbols.
Ueber eine abgeschwaechte Fassung des Auswahlaxioms
Authors: Fraenkel A.
Published: May 2006

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.
Some formal relative consistency proofs
Authors: McNaughton Robert
Published: May 2006

These systems are roughly natural number theory in, respectively, nth order function calculus, for all positive integers n. Each of these systems is expressed in the notation of the theory of types, having variables with type subscripts from 1 to n. Variables of type 1 stand for natural numbers, variables of type 2 stand for classes of natural numbers, etc. Primitive atomic wff's (well-formed formulas) of Tn are those of number theory in variables of type 1, and of the following kind for n > 1: xi ϵ yi+1. Other wff's are formed by truth functions and quantifiers in the usual manner. Quantification theory holds for all the variables of Tn. Tn has the axioms Z1 to Z9, which are, respectively, the nine axioms and axiom schemata for the system Z (natural number theory) on p. 371 of [1]. These axioms and axiom schemata contain only variables of type 1, except for the schemata Z2 and Z9, which are as follows:

where ‘F(x1)’ can be any wff of Tn. Identity is primitive for variables of type 1; for variables of other types it is defined as follows:
A theorem on deducibility for second-order functions
Authors: Langford C. H.
Published: May 2006

It is known that the usual definition of a dense series without extreme elements is complete with respect to first-order functions, in the sense that any first-order function on the base of a set of postulates defining such a series either is implied by the postulates or is inconsistent with them. It is here understood, in accordance with the usual convention, that when we speak of a function on the base , the function shall be such as to place restrictions only upon elements belonging to the class determined by f; or, more exactly, every variable with a universal prefix shall occur under the hypothesis that its values satisfy f, while every variable with an existential prefix shall have this condition categorically imposed upon it.

Consider a set of postulates defining a dense series without extreme elements, and add to this set the condition of Dedekind section, to be formulated as follows. Let the conjunction of the three functions,

be written H(ϕ), where the free variables f and g, being parameters throughout, are suppressed. This is the hypothesis of Dedekind's condition, and the conclusion is

which may be written C(ϕ).
Definition by induction in quine's new foundations for mathematical logic
Authors: Bosser Barkley
Published: May 2006

In Quine's New foundations the axiom of infinity does not appear to be provable. In a certain stronger system, very closely related to Quine's New foundations, the axiom of infinity is provable. One of the peculiarities of this latter system is that even unstratified propositions can be proved by induction (this is used in the proof of the axiom of infinity). It would seem that definition by induction should also be possible quite irrespective of any conditions of stratification in this latter system. In this paper it is shown that such is the case.

Theorem. If α, α1, α2, …, αs are any classes, and if there is no confusion of bound variables in {Sxp}(v) or {Sxq}(v) and neither z nor v occurs free in p or q, and if

⊦ (y1, y1, … ys):. y1ϵα1 . y2ϵα2 . … . ysϵαs : ⊃ : (Ez) : zϵα : (v) : v . ≡ . {Sxp}(v),

⊦ (y, n, y1, y2, …, ys) :. yϵα . n ϵ Fin . y1ϵα1 . y2ϵα2. … . ysϵαs : ⊃ : (Ex) :zxα : (v) : v=z . ≡ . {Sxq}(v),

then there is an r such thai there is no confusion of bound variables in {Sxr}(v) and neither z nor v occurs free in r, and

⊦ (n, y1, y2, …, ys) :. nϵ Fin . y1ϵα1 . y2ϵα2 . … . ysϵαs : ⊃ : (Ez) : zϵα : (v) : v=z . ≡ . {Sxr}(v),

⊦ (y1, y2, …, ys) : y1ϵα1 . y2ϵα2 . … . ysϵαs . ⊃ . ιx{Snr}(0) = ιxp,

⊦ (n, y1, y2, …, ys) : nϵ Fin . y1ϵα1 . y2ϵα2 . … . ysϵαs . ⊃ . ιx{Snr}(n+1) = ιx{Syq}(ιxr).
Independent axiom schemata for S5
Authors: Anderson Alan Ross
Published: May 2006

Leo Simons has shown that H1—H6 below constitute a set of independent axiom schemata for S3, with detachment for material implication “→” as the only primitive rule. He also showed that addition of the scheme (◇ ◇ α ⥽ ◇ α) yields S4, and that these schemata for S4 are independent. The question for S5 was left open. We shall show (presupposing familiarity with Simons' paper) that H1—H6 and S, below, constitute a set of independent axiom schemata for S5, with detachment for material implication as the only primitive rule.

Let S5′ be the system generated from H1—H6 and S with the help of the primitive rule. It is easy to see that Simons' derivations of the rules (a) adjunction, (b) detachment for strict implication, and (c) intersubstitutability of strict equivalents, may be carried out for S5′. We know that (1) (∼ ◇ ∼ α ⥽ ◇ α) is provable in S2, hence also in S3 and S5′; and (1) and S yield (2) (α ⥽ ∼ ◇ ∼ ◇ α). Perry has shown that addition of (2) to S3 yields S5, so S5 is a subsystem of S5′. And it is easy to prove S in S5; hence the systems are equivalent.
Some applications of infinitely long formulas
Authors: Keisler H. Jerome
Published: May 2006

Introduction. This paper is a sequel to our paper [3]. In that paper we introduced the notion of a finite approximation to an infinitely long formula, in a language L with infinitely long expressions of the type considered by Henkin in [2]. The results of the paper [3] show relationships between the models of an infinitely long sentence and the models of its finite approximations. In the present paper we shall apply the main result of [3] to prove a number of theorems about ordinary finitely long sentences; these theorems are of a type which one might call “preservation theorems”. The following two known results are typical preservation theorems: A sentence φ is preserved under substructures if and only if φ is (logically) equivalent to some universal sentence (Łoś [7] and Tarski [11]); φ is preserved under homomorphic images if and only if φ is equivalent to some positive sentence (Lyndon [8]). An expository account of preservation theorems may be found in Lyndon [9].

We shall use freely all of the notation introduced in [3], and shall not attempt to make this paper selfcontained. However, the reader should have no difficulty following this paper after he has read [3]. In § 1 we cover some preliminary topics. In § 2 and § 3 we shall illustrate our method in familiar situations by proving anew the two preservation theorems mentioned above. We shall then obtain four new preservation theorems in §§ 4, 5, and 6, involving respectively direct powers and roots, strong homomorphisms and retracts, and direct factors. The result in § 6 was stated in the abstract [4], while the results in § 5 were stated in the abstract [5].
Constructive order types, II
Authors: Crossley John N.
Published: May 2006

In this paper we continue the investigations begun in [2] of equivalence classes of (denumerable) well-orderings under one-one partial recursive, order-preserving maps (recursive isotonisms). The numbering and notation are continued from [2] except that we now use lower case Greek letters for classical ordinals and upper case Roman letters for sets and we slightly modify the definition of (see §IX.3).

In section IX, after proving some lemmata which we use repeatedly in this paper, we establish the existence of Cantor Normal Forms for all co-ordinals less than a co-ordinal of the form WA (a fortiori for all co-ordinals less than a principal number for exponentiation).
The undecidability of the Turing machine immortality problem
Authors: Hooper Philip K.
Published: May 2006

A Turing Machine (TM) is an abstract, synchronous, deterministic computer with a finite number of internal states. It operates on the set of infinite words, or tapes, over some finite alphabet, scanning exactly one symbol of the tape at a time. (Only a 2-symbol alphabet, consisting of “0” and “|”, will be considered here, and the scanned symbol of a tape will be distinguished by an underscore.) depending upon its internal state and the symbol under scan, it can perform one or more of the following operations: replace the scanned symbol with a new symbol, focus its attention on an adjacent square, and transfer control to a new state.
Resolution in type theory
Authors: Andrews Peter B.
Published: May 2006

In [8] J. A. Robinson introduced a complete refutation procedure called resolution for first order predicate calculus. Resolution is based on ideas in Herbrand's Theorem, and provides a very convenient framework in which to search for a proof of a wff believed to be a theorem. Moreover, it has proved possible to formulate many refinements of resolution which are still complete but are more efficient, at least in many contexts. However, when efficiency is a prime consideration, the restriction to first order logic is unfortunate, since many statements of mathematics (and other disciplines) can be expressed more simply and naturally in higher order logic than in first order logic. Also, the fact that in higher order logic (as in many-sorted first order logic) there is an explicit syntactic distinction between expressions which denote different types of intuitive objects is of great value where matching is involved, since one is automatically prevented from trying to make certain inappropriate matches. (One may contrast this with the situation in which mathematical statements are expressed in the symbolism of axiomatic set theory.).
Countable vector spaces with recursive operations Part II
Authors: Dekker J. C. E.
Published: May 2006

We use the same terminology and notations as in Part I of this paper [3], but numerals in square brackets refer to the references at the end of the present part. The word “space” will be used in the sense of “subspace of ŪF”. A. G. Hamilton [6] proved that any two α-bases of any α-space are recursively equivalent (see also [5]). This means that dimαV, introduced in [3] only for an isolic α-space V, can be defined for anyα-space V. Several results of [3] can therefore be strengthened. These and some other improvements are listed in §8. It is proved in §9 that every α-subspace of an isolic α-space is again an isolic α-space.
On the completeness of some transfinite recursive progressions of axiomatic theories
Authors: Erik Fenstad Jens
Published: May 2006

The well-known incompleteness results of Gödel assert that there is no recursively enumerable set of sentences of formalized first order arithmetic which entails all true statements of that theory. It is equally well known that by introducing sufficiently nonconstructive rules, such as the ω-rule of induction, completeness can be re-established.

Starting from the work of Turing [4] Feferman in [1] developed another method, viz. the study of transfinite recursive progressions of theories, for closing the gap between Gödel (recursively enumerable sets of axioms yield incompleteness) and Tarski (number-theoretic truth is not arithmetically definable).
Comments on a variant form of natural deduction
Authors: Parry William Tuthill
Published: May 2006

This note shows that the system of natural deduction proposed by Copi in this Journal (1956), made by varying one restriction on Universal Generalization (UG) of the system of his Symbolic logic (1954), is incorrect. The original Symbolic logic system, also incorrect, was corrected in the third printing (1958) by modification of another restriction on UG; but combining this modification with that of the Journal article does not give a correct system.

1. In Symbolic logic — cited as SL (or as SL54 to indicate the first printing) — Copi precedes his formal statement of the quantification rules by a set of conventions applying to all the rules.

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 provision that [a] when ν is a variable it must occur free in Φν at all places at which μ occurs free in Φμ.

The italicized provision we call Restriction a. In the third printing of Symbolic logic — cited as SL58 — Restriction a appears explicitly as part of each of the four quantification rules.
Recursively enumerable sets which are uniform for finite extensions
Authors: Alton Donald A.
Published: May 2006

Let W0, W1 … be one of the usual enumerations of recursively enumerable (r.e.) subsets of the set N of nonnegative integers. (Background information will be given later.) Suggestions of Anil Nerode led to the following

Definitions. Let B be a subset of N and let ψ be a partial recursive function.
On definability of ordinals in logic with infinitely long expressions
Authors: Kino Akiko
Published: May 2006

Let Ω be an infinite cardinal larger than ω. By Lω we mean a language with infinitely long expressions having no individual constants, and such that the only predicates are < and =, and the length of ν or ∃ in a formula in Lω is smaller than Ω (cf. §2).
A model theoretic characterization of effective operations
Authors: Löb M. H.
Published: May 2006

Let be a standard formalization of type theory. In [6] Kreisel introduced N-models for . These are general models in the sense of [1] in which the numerals have their standard denotations.

If ξ is a closed term of we shall denote the object denoted by ξ in the N-model M by ξM. In particular, ξ* is the object denoted by ξ in the standard model of .
On the logic of incomplete answers
Authors: Cresswell M. J.
Published: May 2006

I have argued in [1] that a concept bearing some resemblance to ‘p is the answer to d’ (p a proposition and d a question) can be defined wherever d has the form,

‘For which a's is it the case that A (a)?’ (Qa)A(a)

where a is a variable and A a wff containing a. To say that p is the true and complete answer to (Qa)A(a) is expressed as saying that p is logically equivalent to the true conjunction of A(a) or ~A(a) for each a. It is defined as;

Such a concept of answer is like Belnap's [2] direct true answer to a complete list question, or like Harrah's use [3] (p. 43) of the notion of a state description. The main difference between my approach and that of Belnap and Harrah is that while they are concerned to develop a formal metalanguage for discussion of questions and answers I am concerned to express, as far as possible in existing systems, certain interrogative statements; in particular statements of the form ‘— is the (an) answer to —’.

While the account in [1] does give a formal analysis of one ‘answer’ concept there are respects in which it is inadequate.

1. Since it uses entailment (or strict implication) to define the relation between p the answer and d the question we can shew that if p is the answer to d and q is logically equivalent to p then q is the answer to d.
On formulas in which no individual variable occurs more than twice
Authors: Jaśkowski Stanisław
Published: May 2006

In Jaśkowski [1], some properties of the consequences of formulas of the sentential calculus with no more than two occurrences of any variable have been settled. Analogous problems may be raised concerning the consequences of equalities in which the same restriction relates to individual variables. In the present paper, problems of this kind are solved and more general theorems are proved about the logical consequences of some formulas, called here multiconditionals.
There exist two regressive sets whose intersection is not regressive
Authors: Appel K. I.
Published: May 2006

§1. In [2], as part of an analogy between the concepts of recursive emimerability and regressiveness, Dekker showed that the intersection of any two regressive sets which are recursively equivalent is a regressive set. In [1], McLaughlin and the author showed that the intersection of two regressive sets, if infinite, has an infinite regressive subset. However, the following theorem shows that the analogy is not complete in this instance.

Theorem. There exist two regressive sets whose intersection is not regressive.
Free L-algebras
Authors: Horn Alfred
Published: May 2006

Dummett's LC [1] is a system which characterizes all formulas of the propositional calculus which are valid in every chain (for definitions and notation see the first section of [2]). An L-algebra is a Heyting algebra in which (x → y) + (y → x) = 1 for all x, y. L-algebras bear the same relation to LC as Boolean algebras to the classical propositional calculus and Heyting algebras to the intuitionist propositional calculus.
Predicate calculus with free quantifier variables
Authors: Thomason Richmond H. , Randolph Johnson D.
Published: May 2006

In the literature of symbolic logic there are many examples of systems having free variables ranging over truth-values, individuals, or predicates. But, though many such systems are equipped with universal and existential quantifiers (and though many other quantifiers, e.g., for exactly one, are thereby definable), the problem of free variables corresponding to such constants has been neglected.
A second order logic of existence
Authors: Cocchiarella Nino B.
Published: May 2006

A. N. Prior in [9] has suggested an approach towards a second order logic of existence where, following medieval logicians, we distinguish “between predicates (like ‘is red’, ‘is hard’, etc.) which entail existence, and predicates (like ‘is thought to be red’, ‘is thought of’, etc.) which do not.”2 Let us refer to attributes (including relational attributes) which are designated by the former kind of predicate as existence attributes, or for brevity, e-attributes.
On the solvability of a subclass of the surányi reduction class
Authors: Goldberg Richard
Published: May 2006

In [1] Dreben showed that the subclass K′ (described below) of the Suranyi reduction class is recursively solvable by showing that the subclass is finitely controllable; that is, by showing that any member S of K′ is satisfiable only if it is finitely satisfiable. Dreben's argument is very complex, but much of the complexity is due to his proving not merely solvability, but the deeper property of finite controllability. In the present note, by exploiting certain features of Dreben's technique, a simpler, direct proof of the solvability of K′ is obtained — that is, a proof in which the question of satisfiability in a finite domain plays no role.
Representation of symmetric probability models
Authors: Krauss Peter H.
Published: May 2006

This paper is a sequel to the joint publication of Scott and Krauss [8] in
which the first aspects of a mathematical theory are developed which might
be called “First Order Probability Logic”. No attempt will be made to
present this additional material in a self-contained form. We will use the
same notation and terminology as introduced and explained in Scott and
Krauss [8], and we will frequently refer to the theorems stated and proved
in the preceding paper.
On a question of G. E. Sacks
Authors: Ohashi Kempachiro
Published: May 2006

In [1], Sacks points out that there is one fundamental question: which true statements of ordinary recursion theory remain true when appropriately extended to metarecursion theory?

A particular interest is taken in the question [1]:

Q6. How does one define the jump operator for metarecursion theory? (A satisfactory definition should have the property that if A is metarecursive in B, then the jump of A is metarecursive in the jump of B.)

In [2], Kreisel and Sacks give some definitions of predicates and functions analogous to those of Kleene as follows:

The T-predicate of [2] is analogous to that of Kleene [3]. Its definition is

where e is the Gödel number of a finite system of equations E and t(e, s) is a special metarecursive function which indexes “deductions” from E.

U(e, s) is a metarecursive function such that if t(e, s) = 〈e, M, N, x, y〉, then U(e,s) = y.

Two partial functions {e}s and {e} are

Then e is intrinsically consistent if for all x, s1 and s2, if t(e, S1) = 〈e, M1N1, x, y1〉, t(e, s2) = 〈e, M2, N2, x, y2〉 and (M1 ∪ M2) ∩ (N1 ∪ N2) = ∅, then y1 = y2.
Almost combinatorial Skolem functions
Authors: Ellentuck Erik
Published: May 2006

Let be a version of class set theory admitting urelemente, and with AC (= axiom of choice) replaced by AC0 (= axiom of choice for sets of finite sets), ω = nonnegative integers, and Δ = Dedekind cardinals. Let be an arbitrarily quantified positive first order sentence in functors for + and ·. Let ƒ0, … , ƒκ - 1 be function variables and the universal sentence obtained from by replacing existential quantifiers by the ƒ1 as Skolem functions.
A weak completeness theorem for infinite valued first-order logic
Authors: Belluce L. P. , Chang C. C.
Published: May 2006

This paper contains some results concerning the completeness of a first-order system of infinite valued logic

There are under consideration two distinct notions of completeness corresponding to the two notions of validity (see Definition 3) and strong validity (see Definition 4). Both notions of validity, whether based on the unit interval [0, 1] or based on linearly ordered MV-algebras, use the element 1 as the designated truth value. Originally, it was thought by many investigators in the field that one should be able to prove that the set of valid sentences is recursively enumerable. It was first proved by Rutledge in [9] that the set of valid sentences in the monadic first-order infinite valued logic is recursively enumerable.
Finite partially-ordered quantification
Authors: John Walkoe Wilbur
Published: May 2006

In [3] Henkin made the observation that certain second-order existential formulas may be thought of as the Skolem normal forms of formulas of a language which is first-order in every respect except its incorporation of a form of partially-ordered quantification. One formulation of this sort of language is the closure of a first-order language under the formation rule that Qφ is a formula whenever φ is a formula and Q, which is to be thought of as a quantifier-prefix, is a system of partial order whose universe is a set of quantifiers. Although he introduced this idea in a discussion of infinitary logic, Henkin went on to discuss its application to finitary languages, and he concluded his discussion with a theorem of Ehrenfeucht that the incorporation of an extremely simple partially-ordered quantifier-prefix (the quantifiers ∀x, ∀y, ∃v, and ∃w, with the ordering {〈∀x, ∃v〉, 〈∀y, ∃w〉}) into any first-order language with identity gives a language capable of expressing the infinitary quantifier ∃zκ0x.