bluuebunny/crossref_metadata_2025 huggingface dataset:
The Language of Equality in a Constitutional Order
Authors: O'Neill Timothy J.
Published: May 2006

Like all languages, the language of American law can liberate or confine thinking. Its confining power is illustrated by the absence of the radical “group rights” claim in the Bakke litigation despite the prominence of that argument in the popular debate over affirmative discrimination. This absence establishes the limitations of the metaphor developed to give meaning to the concept “persons” in the equal protection context. While capable of investing the corporation with many of the attributes of “personhood” as defined by the Fourteenth Amendment, the metaphor makes absurd the claim of a racial group to exercise rights or privileges distinct from those of its members. This analysis illuminates the metaphorical structure of law language and concludes that the restricted range of metaphorical thinking in law weakens the law's capacity to mediate struggles over social goals.
Machiavelli's Political Science
Authors: Mansfield Harvey C.
Published: May 2006

Machiavelli is presented as the founder of modern political science, with due regard to the fact that he never spoke of “political science.” His usage of “prudence” and “art” in The Prince is examined to see whether, as founder, he was a teacher or a ruler of future generations. His comprehensive attack on classical political science is outlined and developed through two essential points, the cycle and the soul.
Policy-Oriented Voting in Response to Economic Issues
Authors: Kiewiet D. Roderick
Published: May 2006

This study explores the hypothesis that voting in response to economic problems is policy-oriented: voters concerned about unemployment ore predicted to give greater support to Democratic candidates, while those concerned about inflation are predicted to vote more Republican. In light of evidence from previous research, this study investigates the electoral effects of inflation and unemployment as (1) problems directly experienced by the individual, and (2) problems deemed serious for the nation as a whole. Support is strongest for the unemployment side of the hypothesis. Voters personally affected by unemployment gave a modest boost to Democratic candidates in virtually every election. And in years of high unemployment the large percentage of voters who fell it was a serious national problem voted heavily Democratic as well. This study concludes by discussing the important implications these findings have for our understanding of how economic conditions influence voting behavior in American national elections.
Freedom, Recognition, and Obligation: A Feminist Approach to Political Theory
Authors: Hirschmann Nancy J.
Published: May 2006

I argue that a feminist methodology can help liberal political theory get beyond the problems that it has been recycling since the seventeenth century. Taking political obligation as the focus for my analysis, I show how feminist psychoanalytic and psychological theory can help uncover the structural sexism of liberal theory and epistemology and point the way toward more consistent—and less biased—theoretical formulations. Rejecting the essentialist view of gender difference that has been attributed to this literature, I argue that it is more interesting and appropriate to read it as a symbolic language of power and as a heuristic device for uncovering the gendered dimensions of supposedly “neutral” concepts like obligation.
Liberal Virtues
Authors: Galston William A.
Published: May 2006

I explore possible links between liberalism and conceptions of individual virtue. I first distinguish between virtue seen as instrumental to the preservation of societies and institutions and virtue seen as intrinsically valuable, that is, as an end in itself. I argue that certain distinctive instrumental virtues are required for well-functioning liberal societies, economies, and political institutions. I then sketch different versions of liberal intrinsic virtue and explore the tensions among them and between them and liberal instrumental virtue. I conclude with some competing conjectures as to what extent necessary liberal virtues are being fostered through current institutions.
Explaining African Military Coups d'Etat, 1960-1982
Authors: Johnson Thomas H. , Slater Robert O. , McGowan Pat
Published: May 2006

The purpose of this article is to contribute to the theoretical understanding of African military coups d'etat. We begin by replicating a well-known model (Jackman, 1978) that purports to identify the structural determinants of coups d'etat within the states of Sub-Saharan Black Africa. When the research problem is changed slightly to focus exclusively on military coups, we find major weaknesses in the original Jackman model. We then extend and refine this model and thereby account in a theoretically meaningful fashion for 91% of the variation in military coups within 35 Black African states from 1960 through 1982. Our major substantive findings indicate that Black African states with relatively dynamic economies whose societies were not very socially mobilized before independence and which have maintained or restored some degree of political participation and political pluralism have experienced fewer military coups, attempted coups, and coup plots than have states with the opposite set of characteristics.
Bureaucratic Agenda Control: Imposition or Bargaining?
Authors: Eavey Cheryl L. , Miller Gary J.
Published: May 2006

Niskanen's economic model of bureaucratic behavior assumes that bureaucratic agenda control allows the bureau to impose upon a passive legislature its most preferred alternative from among the set of alternatives that dominate the status quo. But does a monopoly on the agenda give bureaus this degree of influence over a voting body? This article reports on experiments that demonstrate that a monopoly on the agenda puts the agenda monopolist in a position to bargain with a voting body, without being able to impose his or her most preferred alternative on that body. This evidence supports theories of bureaucratic-legislative bargaining at the expense of theories of bureaucratic imposition.
Arms Control Verification and Other Games Involving Imperfect Detection
Authors: Wittman Donald
Published: May 2006

This article presents an analysis of the strategic behavior of countries when there is imperfect verification of an arms control agreement. It provides a framework for determining whether an arms control agreement is desirable, shows which factors are needed for the agreement to be maintained in the absence of third-party enforcers, and develops propositions relating changes in verification capabilities to changes in the likelihood of cheating and the use of verification technology. These propositions yield several paradoxes of information (for example, the better the verification technology, the less often it will be employed). Since the analysis incorporates both simultaneous and sequential moves by the players, it provides new insights into other applied areas as well as game theory.
Dimensions in Congressional Voting
Authors: Koford Kenneth
Published: May 2006

While dimensional studies of congressional voting find a single, ideological dimension, regression estimates find several constituency and party dimensions in addition to ideology. I rescale several unidimensional studies to show their increased classification success over the null hypothesis that votes are not unidimensional. Several null hypotheses are explored. With these null hypotheses, 66%–75% of nonunidimensional roll call votes are nevertheless correctly classified by one dimension. After the rescaling, one dimension succeeds in correctly classifying 25%–50% of the votes, and second and third dimensions are important.
Public Opinion and the U.S. Supreme Court: FDR's Court-packing Plan
Authors: Caldeira Gregory A.
Published: May 2006

I show the intimate connection between the actions of the justices and support for the Supreme Court during one of the most critical periods of U.S. political history, the four months of 1937 during which Franklin D. Roosevelt sought legislation to “pack” the high bench with friendly personnel. Over the period from 3 February through 10 June 1937, the Gallup Poll queried national samples on 18 separate occasions about FDR's plan. These observations constitute the core of my analyses. I demonstrate the crucial influence of judicial behavior and the mass media in shaping public opinion toward the Supreme Court. This research illuminates the dynamics of public support for the justices, contributes to a clearer understanding of an important historical episode, shows the considerable impact of the mass media on public attitudes toward the Court, and adds more evidence on the role of political events in the making of public opinion.
The Dynamics of Structural Realignment
Authors: Macdonald Stuart Elaine , Rabinowitz George
Published: May 2006

Governments render decisions on how resources and values are allocated in a society. In the United States, Congress is the institution in which most of the key allocating decisions are made. To the extent the U.S. political system is integrated, the coalitions that form around the issues debated in Congress should be reflected in the coalitions that support presidential candidates and those that support the major political parties. We formulate a spatial theory of political change in which new ideological cleavages appear in congressional behavior and presidential elections and gradually reorganize the mass party base. The theory leads us explicitly to consider the question of dealignment and to specify conditions under which the parties will lose support from voters.
State Political Culture and Public Opinion
Authors: Erikson Robert S. , McIver John P. , Wright Gerald C.
Published: May 2006

Do the states of the United States matter (or are they of no political consequence)? Using a data set with over 50 thousand respondents, we demonstrate the influence of state political culture on partisanship and ideology. For individuals, we find that the state of residence is an important predictor of partisan and ideological identification, independent of their demographic characteristics. At the aggregate level, state culture dominates state demography as a source of state-to-state differences in opinion. In general, geographic location may be a more important source of opinion than previously thought. One indication of the importance of state culture is that state effects on partisanship and ideology account for about half of the variance in state voting in recent presidential elections.
The Expected Utility Theory of Conflict: Measuring Theoretical Progress
Authors: Simowitz Roslyn , Price Barry L.
Published: May 2006

We examine the “progressivity” of Bruce Bueno de Mesquita's theory of conflict as originally developed inThe War Trapand as extended later. Bueno de Mesquita offers the progressivity of the expected utility theory relative to other theories and approaches to conflict as his major defense in responding to critics. Bueno de Mesquita essentially relies on Imre Lakatos' definition oftheoretical progressin presenting his argument. A review of the literature addressing the concept of theoretical progress indicates that Bueno de Mesquita's application of Lakatosian criteria is incomplete and that Lakatos' criteria are themselves flawed. We review the philosophy of science literature dealing with theoretical progress or rational criteria for theory choice and evaluate the progressivity of the expected utility theory of conflict in light of criteria other than Lakatos'. While we do recommend further elaboration of Bueno de Mesquita's theory, we do not find it more progressive than its rivals.
The Strategic Agenda in Legislative Politics
Authors: Mouw Calvin J. , MacKuen Michael B.
Published: May 2006

We examine the politics of the strategic agenda. Abstracting a politics on the liberal-conservative dimension, we analyze Key Vote roll call data from the U.S. House of Representatives during the Eisenhower and Reagan administrations. The data suggest that politicians set the policy agenda in a strategic fashion. Because they consider such factors as long-term political goals, the changing institutional setting, and plebiscitary presidential politics, agenda-setters propose legislation that only imperfectly reflects their and the membership's wishes on the issue at hand. Thus, as the final stage in the political process, the strategic selection of an agenda provides a means by which factors other than policy preferences affect policy outcomes. The analyses affirm the strategic agenda as a core element in political life.
Eric Voegelin's Theory of Consciousness
Authors: McCarl Steven R.
Published: May 2006

The self stands in the way of understanding and appreciating consciousness. The self is a reflexive, asymmetrical bit of consciousness that displaces the whole of consciousness and the open nature of consciousness. Such displacements of consciousness are expressions of gnosticism. Some political movements (e.g., fascism) are expressions of such restrictions of the horizon of consciousness. To understand and appreciate consciousness requires a degree of selflessness on the part of the inquirer. Philosophy and myth are modes of discourse that embody and cultivate the selflessness necessary for participating in, appreciating, and understanding consciousness. Such philosophizing enhances the possibility of more inclusive joint political action.
Ideology, Status, and The Differential Success of Direct Parties Before the Supreme Court
Authors: Sheehan Reginald S. , Mishler William , Songer Donald R.
Published: May 2006

A substantial literature on lower federal courts and state courts suggests that the “haves” usually come out ahead in litigation because they possess superior resources for it and they reap advantages from their repeat player status. We investigate the success of 10 categories of litigants before the Warren, Burger, and Rehnquist Courts to determine whether the resources or experience of litigants has effects on Supreme Court outcomes paralleling those found in the courts below. While different categories of litigants are found to have very different rates of success, those differences do not consistently favor litigants with greater resources. A time series analysis of the success of different categories of litigants over the 36 years studied suggests that the changing ideological complexion of the Court has a greater impact on the success of litigants than differences among litigants in resources and experience.
Types of Primary and Party Responsibility
Authors: Telford Ira Ralph
Published: May 2006

A chief criticism of the American party system is the lack of party responsibility. In the view of some students, one characteristic of our political system that contributes to this irresponsibility is the practice in some states of allowing individuals to vote in primaries without regard to their partisan allegiances. In such an open primary Republicans may, if they wish, vote in the Democratic primary, and vice versa. The contrasting, and more common, practice is the closed primary, in which participation is restricted to party “members.” Some political scientists think that the closed primary, by subjecting legislators to the presumed discipline of periodic scrutiny by their party's members, induces a greater measure of party regularity than the open primary, in which the official has to satisfy a more motley clientele. This position was taken in the best-known statement of the “party government” school, the 1950 report of the APS A Committee on Political Parties:

The closed primary deserves preference because it is more readily compatible with the development of a responsible party system…. on the other hand, the open primary tends to destroy the concept of membership as the basis of party organization.

Other political scientists have expressed doubts about this presumed relationship between primaries and party responsibility, but there has been no systematic empirical evidence on the point. This paper will examine the relationship between primaries and party responsibility by comparing the party regularity of senators from open primary and closed primary states.
Hume's Constitutionalism and the Identity of Constitutional Democracy
Authors: Manzer Robert A.
Published: May 2006

Modern constitutional democracy entails a particular kind of political self-understanding that uniquely centers on a constitution. While many recent studies have focused on how constitutional text shapes this self-understanding, little attention has been paid to the implications of different views of constitutional authority. This is a critical consideration, however, because constitutional authority has always been intrinsically fragile within constitutional democracy, and never more so than at present. In this article, I explore the potential of constitutional science to generate a conception of constitutional authority and collective identity. I focus on David Hume's effort to use constitutional science to shape opinion about liberty and the nature of the political community. This analysis also provides a basis for reflecting on the problematic relation of democracy to constitutionalism and on the peculiar problem of constitutional opinion in constitutional democracy.
Information Aggregation, Rationality, and the Condorcet Jury Theorem
Authors: Austen-Smith David , Banks Jeffrey S.
Published: May 2006

The Condorcet Jury Theorem states that majorities are more likely than any single individual to select the “better” of two alternatives when there exists uncertainty about which of the two alternatives is in fact preferred. Most extant proofs of this theorem implicitly make the behavioral assumption that individuals vote “sincerely” in the collective decision making, a seemingly innocuous assumption, given that individuals are taken to possess a common preference for selecting the better alternative. However, in the model analyzed here we find that sincere behavior by all individuals is not rational even when individuals have such a common preference. In particular, sincere voting does not constitute a Nash equilibrium. A satisfactory rational choice foundation for the claim that majorities invariably “do better” than individuals, therefore, has yet to be derived.
Global Political Order, Economic Change, and Armed Conflict: Coevolving Systems and the Use of Force
Authors: Pollins Brian M.
Published: May 2006

Several recent explanations for major-power war focus on purported cycles in global economic activity or in global political order. I shall argue that a better understanding of interrelationships among the economic long wave, the global leadership cycle, and armed conflict can be gained if we (1) expand the study of interstate conflict beyond the limited domain of great power or systemic wars and (2) treat the long wave and the leadership cycle as quasi-independent and interrelated processes, each contributing to the conflict dynamics of the interstate system. Theoretical foundations for these two recommendations are drawn from the core works in this field. A model based on this new approach is developed and tested empirically along with four formal representations of the core frameworks. Poisson regression is employed using data on interstate disputes (1816–1976) to test resulting hypotheses. Analysis shows that broadening the explanatory domains of existing frameworks is valid and that the “coevolving systems” model is measurably superior to all tested competitors. I conclude that these two global processes are best viewed as coevolving systems and that future studies of systemic conflict should take this approach.
A Way of Life and Law: Presidential Address, American Political Science Association, 1994
Authors: Jones Charles O.
Published: May 2006

This address stresses the importance of the study of lawmaking. Three advantages in particular are emphasized: lawmaking is the core decision-making process in a democracy, its study offers an opportunity for Americanists to overcome concentrations on a single institution, and it provides a basis for comparative analysis. The discussion focuses on statute making as a primary phase of lawmaking. Four concepts—iteration, inquiry, speculation, and declaration—are identified as key, unexplored characteristics of statute making that hold substantial promise for research.
The Concept of a Critical Realignment, Electoral Behavior, and Political Change
Authors: Nardulli Peter F.
Published: May 2006

The realignment perspective has exerted an enormous amount of influence on thinking about American politics, but recently it has fallen into disfavor. As a theory of political change, this dissatisfaction is warranted. However, in rejecting the realignment perspective, scholars risk losing a valuable concept, the notion of a critical realignment. My thesis is that, properly defined, the concept of a critical realignment can be a powerful tool in the study of electoral behavior and an important component of a broader theory of political change. This thesis derives from an analysis of presidential elections between 1828 and 1984. This analysis provides dramatic evidence for the proposition that critical realignments are important electoral phenomena. The evidence is equally clear, however, that critical realignments are subnational phenomena that vary considerably in form, not the majestic national movements some believed them to be. The analyses reported here reveal broadly based electoral eruptions of 40 to 50 points that endure for decades.
Über ω-Unvollständigkeit in der Peano-Arithmetik
Authors: Hasenjaeger G.
Published: May 2006

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

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

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

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

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

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

Ich werde im Folgenden für zwei kategorische Axiomensysteme der Peano-Arithmetik die ω-Unvollständigkeit in Bezug auf die Prädikatenlogik der ersten Stufe nachweisen. In beiden Fallen gelingt der Nachweis der ω-Unvollständigkeit durch Diskussion eines ∀-defekten Ausdrucks H(c).
Eine Bemerkung zu Henkin's Beweis für die Vollständigkeit des Prädikatenkalküls der ersten Stufe
Authors: Hasenjaeger G.
Published: May 2006

Im Folgenden wird eine Vereinfachung der Beweisanordnung für Henkin's Theorem [2] p. 162 angegeben (Abschnitt I). – Die Vereinfachung zeigt sich u.a. darin, daß für den abgeänderten Beweis eine Abschätzung der “projektiven Klasse” (im Sinne der Kleene-Mostowski'schen Theorie der projektiven Mengen von natürlichen Zahlen) für das zu konstruierende Modell gelingt (Abschnitte II–IV). Die Kenntnis der Henkin'schen Arbeit wird vorausgesetzt.

Um eine maximale widerspruchsfreie Menge Γω von geschlossenen Formeln, in der zu jeder Existenzaussage (∃x)A(x) eine Beispielaussage A(uij) vorhanden ist, zu erhalten, führt Henkin einen Turm von Kalkülen S0, S1, …, Si, … und dessen Vereinigung Sω ein. Abwechselnd wird dann die gegebene Menge von Formeln in Si zu einer maximalen widerspruchsfreien Menge Γi erweitert und in Si+1 durch Hinzunahme von Beispielaussagen für alle Existenzaussagen aus Γi ergänzt, für die das nicht schon in einem früheren Schritt geschehen ist. Γω ist dann die Vereinigung aller Γi.

Die Vereinfachung besteht nun darin, daß – nach Erweiterung von S0 zu einem Kalkül S* durch Hinzunahme einer einfachen Folge u1, u2, … von “neuen” Konstanten – in S* zu jeder Existenzaussage eine bedingte Beispielaussage gebildet wird. K sei die Klasse dieser Aussagen, Λ die gegebene widerspruchsfreie Menge; dann wird Λ* = K ◡ Λ in S* einmal zu einer maximalen widerspruchsfreien Menge ergänzt.

Unter Verwendung der Standard-Anordnung aller geschlossenen Formeln von S* erhalten wir für diejenigen von der Form (∃x)A(x) eine Aufzählung (∃x)Ai(x), dazu eine Indexfolge ji (i =1, 2, …), definiert durch:
Strong completeness of fragments of the propositional calculus
Authors: Rose Alan
Published: May 2006

There has recently been developed a method of formalising any fragment of the propositional calculus, subject only to the condition that material implication is a primitive function of the fragmentary system considered. Tarski has stated, without proof, that when implication is the only primitive function a formulation which is weakly complete (i.e., has as theorems all expressible tautologies) is also strongly complete (i.e., provides for the deduction of any expressible formula from any which is not a tautology). The methods used by Henkin suggest the following proof of the

Theorem. If in a fragment of the propositional calculus material implication can be defined in terms of the primitive functions, then any weakly complete formalisation of the fragmentary system which has for rules of procedure the substitution rule and modus ponens is also strongly complete.
Note on modal functions
Authors: Fitch Frederic B.
Published: May 2006

In a previous paper a method was given for representing modal functions in two-valued logic. The purpose of the present note is to indicate how the results of that paper can be improved in certain respects and to point out a simple Boolean representation of Lewis's system of strict implication. The definitions of the previous paper will be presupposed.
Steps toward a constructive nominalism
Authors: Goodman Nelson , Quine W. V.
Published: May 2006

We do not believe in abstract entities. No one supposes that abstract entities—classes, relations, properties, etc.— exist in space-time; but we mean more than this. We renounce them altogether.

We shall not forego all use of predicates and other words that are often taken to name abstract objects. We may still write ‘x is a dog,’ or ‘x is between y and z’; for here ‘is a dog’ and ‘is between … and’ can be construed as syncate-gorematic: significant in context but naming nothing. But we cannot use variables that call for abstract objects as values. In ‘x is a dog,’ only concrete objects are appropriate values of the variable. In contrast, the variable in ‘x is a zoölogical species’ calls for abstract objects as values (unless, of course, we can somehow identify the various zoological species with certain concrete objects). Any system that countenances abstract entities we deem unsatisfactory as a final philosophy.
On relations as coextensive with classes
Authors: Quine W. V.
Published: May 2006

In a previous note I showed a new way to define the ordered pair. I made use of the notations ‘Nn’ (for class of natural numbers) and ‘Sv’ (for successor of v), remarking that they are readily defined without appeal to ordered pairs or relations. Adopting the auxiliary abbreviation:

I defined the ordered pair thus:
On ordered pairs
Authors: Quine W. V.
Published: May 2006

Wiener, in 1914, reduced the theory of relations to that of classes by construing relations as classes of ordered pairs and defining the ordered pair in turn on the basis of class theory alone.1 The definition, as improved by Kuratowski,2 identifies the ordered pair x;y with uxyi(ix U iy).

In terms of Russell's theory of types, x;y in the above sense is two types higher than x and y. Even when we abandon Russell's theory of fixed types of objects in favor of a theory of stratified formulae,3 there is still significance in saying that ‘x;y’ is of type 2 relative to ‘x’ and ‘y’—meaning that a test of the stratification of any context involves assigning a higher number by 2 to ‘x;y’ than to ‘x’ and ‘y’.
A problem concerning the notion of definability
Authors: Tarski Alfred
Published: May 2006

We are inclined to believe that, by means of an argument entirely analogous to that which leads to the Richard antinomy, the notion of definability as applied to entities discussed in a formal system can easily be shown not to be itself definable in this system. It will be seen from this discussion that actually the situation is not quite so simple as it would appear at first glance. Our discussion will have a rather sketchy and informal character.
Quantification theory and empty individual-domains
Authors: Hailperin Theodore
Published: May 2006

In a recent paper by Mostowski [1] we find an investigation of those formulas of quantification theory which are valid in all domains of individuals, including the empty domain. Mostowski gives a complete set of axioms for such a first order functional calculus (the system is called “”) and a comparison is made with a form of the usual calculus, Church's in [2]. It is pointed out that is much less elegant; in particular, the distributivity laws for quantifiers (e.g., (x){A . B) .{x)A . (x)B) do not hold in general, and likewise the rule of modus ponens does not preserve validity in all cases.

In this paper we show that a not inelegant system is obtained if one modifies Mostowski's approach in two respects; and once this is done a somewhat neater proof of completeness can be given.

The first respect in which we diverge from Mostowski is in the treatment of vacuous quantifiers. For him if p has no free x, then (x)p and (∃x)p are both to have the same value (interpretation) as p1. But this is not the only way to assign values to vacuous quantifications. For when universal quantification is viewed as a generalized conjunction, the formula (x)Fx has the significance of Fa . Fb…. for as many conjunctands as there are individuals in the domain, and if Fx should have the “constant” value p, then (x)p is to mean the conjunction of p with itself for as many times as there are individuals in the domain (compare the arithmetical ).
Banishing the rule of substitution for functional variables
Authors: Henkin Leon
Published: May 2006

Let and be (well-formed) formulas of the functional calculus, let c be an n-adic functional variable, and let a1, …, an be distinct individual variables. Church has defined the metalogical notation to indicate the formula resulting from when each part of of the form c{1, …, n) (such that the occurrence of c is free in ) is replaced by the formula which arises from by replacing every free occurrence of ai by i, i, = 1, …, n. (Here 1, …, n may be any individual variables or constants, not necessarily all distinct.) The notation is not defined for all , , c, a1, …, an, however, but only for those cases (specified in detail by Church) when the resulting formula constitutes a valid substitution instance of the formula according to the standard interpretation of the functional calculus.

The full and correct syntactical statement of the conditions (under which this type of substitution is permissible) has proved so arduous, that it seems to have been rendered in error more often than not. Unfortunately the functional calculi are often set up as deductive systems in which this type of substitution occurs as one of the primitive rules of inference, or in one of the axiom schemata. Thus the beginning student who is introduced to the calculi through such a formulation is forced to cope from the outset with details which have proved treacherous even to the initiate. For this reason it is desirable to seek alternative formulations of the functional calculi in which this type of substitution is not mentioned.
A relative consistency proof
Authors: Shoenfield Joseph R.
Published: May 2006

LetCbe an axiom system formalized within the first order functional calculus, and letC′ be related toCas the Bernays-Gödel set theory is related to the Zermelo-Fraenkel set theory. (An exact definition ofC′ will be given later.) Ilse Novak [5] and Mostowski [8] have shown that, ifCis consistent, thenC′ is consistent. (The converse is obvious.) Mostowski has also proved the stronger result that any theorem ofC′ which can be formalized inCis a theorem ofC.

The proofs of Novak and Mostowski do not provide a direct method for obtaining a contradiction inCfrom a contradiction inC′. We could, of course, obtain such a contradiction by proving the theorems ofCone by one; the above result assures us that we must eventually obtain a contradiction. A similar process is necessary to obtain the proof of a theorem inCfrom its proof inC′. The purpose of this paper is to give a new proof of these theorems which provides a direct method of obtaining the desired contradiction or proof.

The advantage of the proof may be stated more specifically by arithmetizing the syntax ofCandC′.
The Sheffer functions of 3-valued logic
Authors: Martin Norman M.
Published: May 2006

In previous papers, Post, Webb, Götlind and the present author have described some Sheffer functions (in Swift's terminology, “independent binary generators”) in m-valued logic. Professor J. Dean Swift has recently isolated the symmetric Sheffer functions of 3-valued logic. In the present paper, we will prove some properties of Sheffer functions in m-valued logic and isolate all of the Sheffer functions of 3-valued logic.

Before we proceed we will define some terms which we will find convenient. A set of functions in m-valued logic is functionally complete, if the set of the functions which can be defined explicitly from the functions of the set is exactly the set of all functions of m-valued logic. A function is functionally complete, if its unit set is functionally complete. A Sheffer function is a two-place functionally complete function. If i and j are truth values (1 i, j ≤ m), we will say i ~ j (D), if D is a decomposition of the truth values 1, …, m into 2 or more disjoint non-empty classes and i and j are elements of the same class. A binary function f(p, q) satisfies the substitution law for a decomposition D, if for any truth values h, i, j, k, whenever h ~ j (D) and i~k(D), then f(h, i) ~ f(j, k) (D). The function f(p,q) satisfies the co-substitution law for D, if for any truth values h, i, j, k, whenever f(h, i) ~ f(j, k) (D), then h ~ j (D) or i ~ k (D). We will say f(p, q) has the proper substitution property, if there is a decomposition of the truth values into less than m classes for which it satisfies the substitution law.
Computability and λ-definability
Authors: Turing A. M.
Published: May 2006

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

A definition of what is meant by a computable function cannot be given satisfactorily in a short space. I therefore refer the reader to Computable pp. 230–235 and p. 254. The proof that computability implies recursiveness requires no more knowledge of computable functions than the ideas underlying the definition: the technical details are recalled in §5.
Alternative postulate sets for Lewis's S5
Authors: Lemmon E. J.
Published: May 2006

Professor Prior (Formal logic (1955), pp. 305–307) gives alternative postulate sets for Lewis's S5, one substantially Lewis's own set (p. 305, 6.1) and another, considerably simpler, due to Prior himself (pp. 306–307, 6.5). This note aims at showing that the systems based on these two postulate sets are in fact equivalent. We label the system based on Lewis's postulates ‘S5’, as usual, and that on Prior's postulates ‘P’.

The form of the postulate set for P considered here is as follows. We add to the full propositional calculus one definition:

Df. M: M = NLN;

and two special rules:

L1: ⊦Cαβ → ⊦CLαβ;

L2: ⊦Cαβ → ⊦CαLβ, if α is fully modalized (a propositional formula is fully modalized, if and only if all occurrences of propositional variables in it lie within the scope of a modal operator).

Prior's text (pp. 201–205) indicates sufficiently how it could be shown that any thesis of S5 is a thesis of P and that the rules and definitions of S5 are obtainable as derived rules of P. We turn, therefore, to the problem of showing that any thesis of P is a thesis of S5 and that the rules and definition of P are obtainable as derived rules of S5.
Quantification and the empty domain
Authors: Quine W. V.
Published: May 2006

Quantification theory, or the first-order predicate calculus, is ordinarily so formulated as to provide as theorems all and only those formulas which come out true under all interpretations in allnon-emptydomains. There are two strong reasons for thus leaving aside the empty domain.

(i) WhereDis any non-empty domain, any quantificational formula which comes out true under all interpretations in all domains larger thanDwill come out true also under all interpretations inD. (Cf. [4], p. 92.) Thus, though any small domain has a certain triviality, all but one of them, the empty domain, can be included without cost. To include the empty one, on the other hand, would mean surrendering some formulas which are valid everywhere else and thus generally useful.

(ii) An easy supplementary test enables us anyway, when we please, to decide whether a formula holds for the empty domain. We have only to mark the universal quantifications as true and the existential ones as false, and apply truth-table considerations.

Incidentally, the existence of that supplementary test shows that there is no difficulty in framing aninclusivequantification theory (i.e., inclusive of the empty domain) if we so desire. A proof in this theory can be made to consist simply of a proof in the exclusive (or usual) theory followed by a check by the method of (ii). We may, however, be curious to see a more direct or autonomous formulation: one which does not consist, like the above, of the exclusive theory plus a rule of expurgation. And, in fact, such formulations have of late been forthcoming: Mostowski [5], Hailperin [3], and, as part of a broader context, Church [2]. I shall not presuppose acquaintance with these papers, except in my final paragraph (and then only with [3]).
On the independence of Henkin's axioms for fragments of the propositional calculus
Authors: L'abbé Maurice
Published: May 2006

A general system of axioms has been given by Henkin for a fragment of the propositional calculus having as primitive symbols, in addition to the usual parentheses, variables, and implication sign ⊃, an arbitrarily given truth function symbol ϕ. This system of axioms, which we shall denote by S(⊃, ϕ), contains the following three axiom schemata

plus the 2m further axiom schemata involving the symbol ϕ

where ϕ is an m-placed function symbol. We refer to Henkin's paper, p. 43, for the detailed description of the axiom schemata (4).

The remark was made in the above mentioned paper that each of the 2m axiom schemata of (4) is trivially independent of the rest of the axioms of S(⊃, ϕ), and it was conjectured that the axiom schemata (1), (2) and (3) are also independent. In this note, we prove the general independence of the axiom schemata (1) and (2). As for (3), we show on the one hand its independence in the systems S(⊃) and S(⊃, f), and, on the other hand, its dependence in the system S(⊃, ∼). The net result is, therefore, that in any of these systems of axioms S(⊃, ϕ) all the axiom schemata are independent, except possibly the axiom schema (3).
On formulas with valid cases
Authors: Quine W. V.
Published: May 2006

Commonly, when we succeed in showing a formula of quantification theory to be consistent, we do so by producing a true interpretation. Sometimes we achieve the same effect without even exceeding the resources of quantification theory itself: we show a formula consistent by producing a valid formula from it by substitution. Example: ‘(∃x)Fx ⊃ (x)(∃y) (Gxy ▪ Fy)’ is seen consistent by noting its valid substitution case ‘(∃x)Fx ⊃ (x)(∃y)(Fx ∨ Fy ▪ Fy)’. How generally available is this latter method? I shall show that it is available if and only if the formula whose consistency is shown is satisfiable in a one-member universe.

The “only if” part is immediate. For, if ψ is a substitution case of ϕ, then ϕ is satisfiable wherever ψ is; and ψ, if valid, is satisfiable in a one-member universe.

Conversely, suppose a true interpretation of ϕ in a one-member universe. Each predicate letter of ϕ is thereby interpreted outright as true or false, in effect, since there is no varying the values of ‘x’, ‘y’, etc. Now form ψ from ϕ by substitution as follows: change each atomic formula ϕi(e.g. ‘Fx’, ‘Fy’, ‘Gxy’) to ⌜ϕi⊃ϕi⌝ if its predicate letter is one that was interpreted as true, or to ⌜ϕi ▪ − ⊃ϕi⌝ if its predicate letter is one that was interpreted as false. Clearly ψ under all interpretations even in large universes, will receive the truth value that ϕ received under the supposed interpretation in the one-member universe. So ψ is valid.
The logical simplicity of predicates
Authors: Goodman Nelson
Published: May 2006

In an earlier article, I proposed a way of determining the relative simplicity of different sets of extralogical primitives. The calculations assumed a fully platonistic logic, committed to an indefinite hierarchy of classes, with sequences and relations defined as classes. Recently it has been shown that a nominalistic logic, countenancing no entities other than individuals, can be made to serve many of the purposes for which a platonistic logic had been thought necessary. The question naturally arises how we are to determine the simplicity of extralogical bases of systems founded upon a nominalistic logic. In such bases, the only extralogical predicates will be predicates (of one or more places) of individuals. The present paper offers a general method of measuring the simplicity of such bases.
Fragments of the propositional calculus
Authors: Henkin Leon
Published: May 2006

Of the several methods for proving the completeness of sets of axioms for the prepositional calculus perhaps the simplest is due to Kalmár, although it does not appear to be widely known. In this paper we generalize Kalmár's method to indicate how to obtain a complete axiomatization of any fragment of the propositional calculus which includes material implication. We shall carry through the description and proofs for the case where, in addition to a symbol for implication, there is just one other primitive truth-function symbol. For systems in which there are no other function symbols, or more than one other such symbol, notational changes but not conceptual changes will be required.
A formal proof of Gödel's theorem
Authors: Chwistek L.
Published: May 2006

I. The present paper contains a formal proof of the following theorem of the elementary system ⊨ [ 1 0 ] c:

This theorem means that if there were a theorem of ⊢0 ( 3 2 ) 0 stating that there is no contradiction in ⊢2 ( 5 4 ) 2, we would have a contradiction in ⊢2 ( 5 4 ) 2. (This is in accordance with the second theorem of Gödel.)

Our proof is based on the calculus of the system ⊨ [ 1 0 ] c and of the metasystem ⊢0 ( 3 2 ) 0.

Note that we could take

instead of Ax E .1 ( L ) (.1 ( L ) .0 ( L ) L) Z. We would then have to do with the simple systems and metasystems of Hetper, which do not contain propositional variables or the logico-semantical axiom. Our method applies equally to the systems and metasystems of New Foundation and to the simple systems and metasystems of Hetper.

Our proof can be considerably simplified by using a recent result of Hetper concerning ancestral functions. Hetper introduces the following abbreviations:

If A ( Z1c), B ( x1cy1cZ1c) are propositions, the expression will be called an ancestral function (we prove without difficulty that this expression is a proposition). The expressions will be called respectively the principal term and the term of derivation of this function.
Implication and deducibility
Authors: Emch Arnold F.
Published: May 2006

Lewis, in the presentation of his calculus of strict implication, contends that this calculus accords with the usual meaning of “implies” such that “p strictly implies q” is synonymous with “q is deducible from p” (pp. 126–127, 235–262). It is the purpose of this paper to present, in 1, certain considerations in the light of which this statement does not hold, and in 2, a new implicative relation upon which a calculus of propositions can be based such that it will accord with the relation of deducibility and from which it is possible to derive the calculi of strict and material implications.

1. Strict implication regarded as synonymous with deducibility. In the presentation of his calculus of strict implication Lewis observes that despite inclusion of both intensional and extensional propositions in this calculus, the meaning of its elements, p, q, r, etc., “remain fixed, whether it is their extensional or their intensional relations which are in question” (p. 120). But if the meaning or content of the elements of this calculus is irrelevant to the logical truth which it contains, then, as Weiss has already remarked, the propositions of this calculus “are really extensionally treated, involving multiple values,” e.g., possibility, on the analogy of truth and falsity. The importance of this fact is that, in consequence, the relation of strict implication is subject to certain paradoxes, such as (19.74) “a proposition which is impossible strictly implies any proposition” or (19.75) “a proposition which is necessarily true is strictly implied by any proposition” (p. 174) or (19.84) “any two necessarily true propositions are strictly equivalent” (p. 176).
Some consequences of the axiom of power-set
Authors: Abian Alexander , Lamacchia Samuel
Published: May 2006

In this paper we prove:

Theorem 1. Any finite model of the axiom of power-set also satisfies the axioms of extensionality, sum-set and choice.

Clearly, it will follow from (2) below that in a finite model the axiom of power-set is satisfied if and only if every set is a power-set. Thus, Theorem 1 follows immediately from Theorem 2 below, where by a theory of sets we mean a first-order theory without identity and with only one binary predicate symbol ∈.

Theorem 2. If in a theory of sets every set is a power-set and if the axiom of power-set is valid, then the axioms of extensionality, sum-set and choice are valid.

The proof of Theorem 2 will follow from the lemmas which we establish below.

We mean by x = y that x and y have the same elements. We denote a power-set of x by P(x) when it exists; similarly, we denote a sum-set of x by Ux.

Clearly, in every theory of sets we have:

(1) (x ⊂ y) ↔ (P(x) ⊂ P(y)),

(2) (x = y) ↔ (P(x) = P(y)),

(3) (x = y) → ((x ∈ P(z)) → (y ∈ P(z))),

(4) ⋃P(x) = x.

In view of (2), (3) and the definition of equality, we have:

Lemma 1. If in a theory of sets every set is a power-set, then equal sets are elements of the same sets.

We have also, in view of (4):

Lemma 2. If in a theory of sets every set is a power-set, then every set has a sum-set.