bluuebunny/crossref_metadata_2025 huggingface dataset:
Behavioral Correlates of Political Support
Authors: Muller Edward N.
Published: May 2006

Theories of the behavioral consequences of political support coincide in the prediction that political support will correlate positively with indices of conventional behavior, negatively with indices of unconventional behavior. Survey data drawn from three communities in the Federal Republic of Germany show that an index of support for the structure of political authority is negatively correlated both with an index of actual participation in aggressive political behavior and with an index of participation in conventional electoral/pressure-group politics. Since the political behavior indices are themselves positively correlated, it is useful to construct a typology which differentiates between “pure” types – no participation, participation only in conventional, participation only in aggressive – and “mixed” types – participation in conventional and moderately aggressive, participation in conventional and highly aggressive. When the relationship between political support and the political-action type index is examined, it turns out that two of the types are associated with medium political support, while four of them occur at low support. To achieve more accurate explanation of types of political behavior, a model for prediction of each action type is proposed, taking into account interaction between political support, sense of personal political influence, and belief in the efficacy of past collective political aggression. The test of the model yields positive results, suggesting that it represents a fruitful beginning toward development of a theory of behavioral consequences of political support.
The Cabal in Kabul: Great-Power Interaction in Afghanistan
Authors: Dil Shaheen F.
Published: May 2006

The July 17, 1973 coup serves as a case study of the nature and extent of great-power interest and involvement in Afghanistan. The dynamics of American, Soviet, and Chinese interaction are multifaceted and volatile, and imply that no one great power had outright control. Thus, this treatment concerns influence rather than control, and multilateral interaction rather than unilateral or bilateral action. The differing interests of the great powers in Afghanistan are outlined. Next, the possibility of great-power involvement in the coup is examined. Finally, the impact of the coup upon Afghanistan's relations with the three great powers is considered. Available material suggests that neither the United States nor the People's Republic of China had sufficient interest or influence to instigate the coup. Nor is there any concrete evidence that the Soviet Union played a significant role, although it did have the opportunity, influence, and interests to do so.
Authority and Community: The Contributions of Carl Friedrich, Yves R. Simon, and Michael Polanyi
Authors: Cochran Clarke E.
Published: May 2006

The problem of authority has a practical and a theoretical side. Practically, the decline of authority contributes to the crisis of legitimacy in contemporary governments and to difficulties in the practice of morality. Theoretically, authority is often confused with power, force, or coercion. Thus, it is viewed with suspicion. The thesis of this paper is that authority is not a form of power; rather, it transforms power. Like power, force, and coercion, authority is directive, but it is so in quite a different way. An examination of the concept of authority in three very different theorists – Carl Friedrich, Yves R. Simon, and Michael Polanyi – reveals that a sound concept of authority must be rooted in community as a system of shared beliefs, experiences, and traditions and in transcendent standards referred to by such beliefs, experiences, and traditions. Authority is that which directs a community to its proper end.
The Relative Importance of Socioeconomic and Political Variables for Public Policy
Authors: Lewis-Beck Michael S.
Published: May 2006

Since Dawson and Robinson, a dominant issue in the quantitative study of public policy has been the relative importance of socioeconomic and political variables for determining policy outcomes. It is argued here that past efforts to resolve this issue have been unsatisfactory, largely because they relied on inadequate statistical techniques, i.e., simple correlation, partial correlation, or multiple regression. Coefficients from these techniques are irrelevant for all but the most peculiar models of public policy. In general, if the researcher wishes to assess the relative importance of independent variables, it will be necessary to resort to path analysis of a formally constructed causal model. The comparison of “effects coefficients,” derived from path analysis, is offered as the preferred means of evaluating independent variables, superior to comparisons of coefficients from simple correlation, partial correlation, or multiple regression. When the effects coefficients are actually calculated for a popular model of welfare policy, socioeconomic variables appear much more important than political variables, contrary to interpretations coming from the more traditional statistical techniques.
Thomas Hobbes: Skeptical Moralist
Authors: Chabot Dana
Published: May 2006

Thomas Hobbes is usually held to have been a skeptic in matters of religion and morality. I accept the claim that there is a distinctive skeptical strain in Hobbes' thought but argue that his skepticism informs his moral vision, rather than depriving him of a conception of morality. As evidence for this reading, I situate Hobbes in a tradition of “skeptical moralism,” along with Montaigne and certain other Renaissance figures. As opposed to moral skeptics, skeptical moralists think of moral agents as divided selves, pulled in one direction by law and another by conscience. Skeptical moralists use skepticism to make people aware of this tension, and I argue that (especially in his remarks on religion) Hobbes was doing just that.
Beyond Tocqueville, Please!
Authors: Stevens Jacqueline , Smith Rogers
Published: May 2006

Rogers Smith in “Beyond Tocqueville, Myrdal and Hartz: The Multiple Traditions in America” in the September 1993 issue of thisReviewargues that ascriptive inequalities (eg. racism, sexism, nativism) are neither mere deviations from liberalism nor only symptoms of liberalism. Rather, multiple ideologies coexist in uneasy tension. Jacqueline Stevens criticizes Smith for failing to attend to the work of W. E. B. Du Bois and others—whose descriptions of American ideological history, she says, provide the same insight that Smith claims as his own. She goes on to discuss how defining a “mainstream” of scholarship shapes inferences as to what counts as knowledge and further suggests that liberalism and exclusion betray an underlying consistency. In his reply, Smith recognizes the contributions of Du Bois and others, but argues that although they do foreshadow his work, they do not offer the same critique or do the same job. And he asserts that the linkages between liberalism and ascriptive inequality are political, economic, social and psychological—not logical—ones.
Political Inclusion and the Dynamics of Democratization
Authors: Dryzek John S.
Published: May 2006

Once universal adult citizenship rights have been secured in a society, democratization is mostly a matter of the more authentic political inclusion of different groups and categories, for which formal political equality can hide continued exclusion or oppression. It is important, however, to distinguish between inclusion in the state and inclusion in the polity more generally. Democratic theorists who advocate a strategy of progressive inclusion of as many groups as possible in the state fail to recognize that the conditions for authentic as opposed to symbolic inclusion are quite demanding. History shows that benign inclusion in the state is possible only when (a) a group's defining concern can be assimilated to an established or emerging state imperative, and (b) civil society is not unduly depleted by the group's entry into the state. Absent such conditions, oppositional civil society may be a better focus for democratization than is the state. A flourishing oppositional sphere, and therefore the conditions for democratization itself, may actually be facilitated by a passively exclusive state, the main contemporary form of which is corporatism. Benign inclusion in the state can sometimes occur, but any such move should also produce exclusions that both facilitate future democratization and guard against any reversal of democratic commitment in state and society. These considerations have substantial implications for the strategic choices of social movements.
Adaptive Signal Processing, Hierarchy, and Budgetary Control in Federal Regulation
Authors: Carpenter Daniel P.
Published: May 2006

Control over agency budgets is a critical tool of political influence in regulatory decision making, yet the causal mechanism of budgetary control is unclear. Do budgetary manipulations influence agencies by imposing resource constraints or by transmitting powerful signals to the agency? I advance and test a stochastic process model of adaptive signal processing by a hierarchical agency to address this question. The principal findings of the paper are two. First, presidents and congressional committees achieve budgetary control over agencies not by manipulating aggregate resource constraints but by transmitting powerful signals through budget shifts. Second, bureaucratic hierarchy increases the agency's response time in processing budgetary signals, limiting the efficacy of the budget as a device of political control. I also show that the magnitude of agency response to budgetary signals increased for executive-branch agencies after 1970 due to executive oversight reforms. I conclude by discussing the limits of budgetary manipulations as a device of political control and the response of elected authorities to adaptive signal processing by agencies.
Thick-Skinned Liberalism: Redefining Civility
Authors: Sinopoli Richard C.
Published: May 2006

Anorm of civilitydefines a standard of conduct that citizens can rightfully expect from strangers. What are appropriate norms of civility for citizens of liberal states? I argue that two approaches to civility are prominent in our political culture, one requiring “mere” tolerance, the other, that we affirm the worth of others' pursuits (and thereby the worth of those others). This split parallels a division in liberal theory between an interest-based account of liberalism (represented primarily by J. S. Mill) and a status-based account (represented principally by John Rawls and Ronald Dworkin), respectively. The exploration of this theoretical divide and how it relates to contending notions of civility helps to clarify disputes in the broader culture. I conclude that interest-based liberalism offers a more satisfactory approach to the issue at hand.
Waltz, Durkheim, and International Relations: The International System as an Abnormal Form
Authors: Barkdull John
Published: May 2006

Drawing on Emile Durkheim's Division of Labor in Society, I offer a typology of international systems. Previous uses of Durkheim to describe international systems suffer a number of conceptual errors and therefore are at variance with the spirit and intention of Durkheim's work. A deeper reading of Durkheim usefully draws attention to the moral basis for society and thus the problems with defining international systems solely in terms of power distributions. Further, rereading Durkheim offers a much richer typology than the simple distinction between mechanical and organized societies, affording in turn fresh insights into change in the international system. The abnormal forms of the division of labor offer the best description of the contemporary international system.
The Empirical Evidence for Citizen Information and a Local Market for Public Goods
Authors: Lowery David , Lyons W. E. , DeHoog Ruth Hoogland , Teske Paul , Schneider Mark , Mintrom Michael , Best Samuel
Published: May 2006

In their 1993 article in thisReview, Paul Teske, Mark Schneider, Michael Mintrom, and Samuel Best sought to establish the microfoundations for a model of a competitive market for public services between local governments in polycentric regions. An important part of their model focused on subgroups of informed citizens, especially recent movers. Theoretical analysis was supplemented by an empirical study of the factors shaping accuracy of Long Island homeowners' information about relative expenditures and tax rates of their school districts. David Lowery, W. E. Lyons and Ruth Hoogland DeHoog criticize the relevance of this empirical evidence, suggesting the atypical nature of education as a service (especially in this site) and challenging the sufficiency of the demonstrated levels of information for generating a competitive market. Teske and his colleagues reply by pointing out the general importance of education throughout American local policymaking and by defending the relevance of their measures and conclusions for their market model.
Disentangling Patterns of State Debt Financing
Authors: Clingermayer James C. , Dan Wood B.
Published: May 2006

We examine the determinants of change in state government indebtedness from 1961 through 1989 using a pooled time series cross-sectional analysis. The analysis reveals that debt is primarily a function of economic conditions reflecting both the need to borrow and the capacity of states to repay debt. However, political factors such as culture, partisan competition, and electoral cycles also affect state debt. We also find very weak evidence that tax and expenditure limitations, ironically, may increase state indebtedness, while constitutional debt limitations have no effect upon slowing the growth of state debt.
Condorcet Winners and the Paradox of Voting: Probability Calculations for Weak Preference Orders
Authors: Jones Bradford , Radcliff Benjamin , Taber Charles , Timpone Richard
Published: May 2006

That individual preferences may he aggregated into a meaningful collective decision using the Condorcet criterion of majority choice is one of the central tenets of democracy. But that individual preferences may not yield majority winners is one of the classic findings of the social choice literature. Given this problem, social choice theorists have attempted to estimate the probability of Condorcet winners, given certain empirical or theoretical conditions. We shall estimate the probabilities of Condorcet winners and intransitive aggregate orders for various numbers of individuals with strong or weak preference orders across various numbers of alternatives. We find, using computer simulation, a stark contrast between these estimates assuming strong individual preferences and the estimates allowing for individuals' indifference between pairs of alternatives. In contrast to earlier work, which depends on the strong-preference assumption, we suggest that the problem is most acute for small committee decision making and least acute for mass elections with few alternatives.
Aristotle's Science of the Best Regime
Authors: Nichols Mary P. , Bartlett Robert C.
Published: May 2006

In this Controversy, Mary Nichols and Robert Bartlett both enlist Aristotle in support of a liberal politics. Nichols finds that the Aristotle analyzed by Bartlett in his remarks in this Review in 1994 leads us away from politics into a life of perfect contemplation. The real Aristotle, she avers, by calling attention to the “limits of reason,” leaves room for the gods, personal freedom, and shared power. Bartlett responds that the life of contemplative reflection that Aristotle praises is the foundation of knowledge of virtue, without which freedom is license, democracy, dangerous.
Logic of many-sorted theories
Authors: Wang Hao
Published: May 2006

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

Among the proper primitive symbols of a many-sorted system Tn (n = 2, …, ω) there may be included symbols of some or all of the following kinds: (1) predicates denoting the properties and relations treated in the system; (2) functors denoting the functions treated in the system; (3) constant names for certain objects of the system. We may either take as primitive or define a predicate denoting the identity relation in Tn.
Modal functions in two-valued logic
Authors: Fitch Frederic B.
Published: May 2006

In this paper it will be shown that the fundamental modal functions (“… is necessary,” “… is possible,” etc.) may be represented in the ordinary two-valued logic of propositions, relatively to a subset S of the set K of all propositions of the logic. The only restriction on S is that no member of S can occur in any other member of S. The representation concerns the set SK consisting of all members of K in which occur members of S. Every member of SK may be regarded as a truth-function of the members of S. If S consists of only one member, P, and if f(P) and g(P) are any members of SK, then we define ∣f(P)∣ as (f(P)·f(~P)). The proposition “f(P) is S-necessary” may then be expressed by ∣f(P)· Furthermore we may express “f(P) is S-impossible” by ∣~f(P)∣, abbreviated to f(P)*; we may express “f(P) is S-possible” by abbreviated to f(P); and we may express “f(P) S-strictly implies g(P)” by (f(P)·~g(P))*, abbreviated to f(P)⊰g(P). Thus is obtained a novel analysis of modality, contrasting, on the one hand, with Carnap's syntactic theory of modality, and, on the other hand, with Lewis's view of modal concepts as underivable from other logical concepts. We would maintain that modal concepts are not absolute, but always relative to some S, generally unspecified.

In the formal exposition which follows, we shall assume tacitly but not explicitly the important distinction between the symbols we are discussing and symbols denoting or designating the former.
Gödel Theorems for Non-Constructive Logics
Authors: Rosser Barkley
Published: May 2006

Introduction. In setting up his definition of analyticity, Carnap uses various non-constructive rules, of which we shall be concerned with only one. This one we shall call simply “Carnap's rule.” It can be roughly stated thus: If f(0), f(1), f(2), … are all provable, then (x)f(x) shall be provable.

In this paper will be briefly considered the logics got by starting with the system of Principia plus Peano's axioms and allowing one, two, …, ω, ω+l, and so on up to, but not including ω uses of Carnap's rule (interspersed with uses of the ordinary rules), as well as the system which includes Principia and Peano's axioms and is closed under application of Carnap's rule and the ordinary rules. Under suitable assumptions as to consistency, it is shown that in each of these logics there occur undecidable propositions and that a formula which states the consistency of the logic exists in the logic but is not provable in the logic. An interesting side result is that the logic got by allowing co applications of Carnap's rule is not closed under Carnap's rule.
The combinatory foundations of mathematical logic
Authors: Curry Haskell B.
Published: May 2006

In investigations of the foundations of mathematics we can distinguish two separate tendencies. On the one hand, one may seek to define his subject with greatest possible explicitness: to obtain a formulation which satisfies the most exacting demands for precision, and which is at the same time free from paradoxes and adequate for the purpose. On the other hand, besides the problem of formulation, there is that of simplification; one can seek to find systems based upon processes of greater and greater primitiveness. The reduction of a piece of mathematics to a formal system, and still further to a completely formalized system (as explained, for example, in my New York address a year ago), is a step toward the first of these objectives. But it is evident that one can formalize in various ways, and that some of these ways constitute a more profound analysis than others. Although from some points of view one way of formalization is as good as any other, yet a certain interest attaches to the problem of simplification, as is shown by the attention which some of the greatest mathematicians have devoted to it.

The researches about which I am reporting today are directed toward the second of these objectives. In fact we are concerned with constructing systems of an extremely rudimentary character, which analyze processes ordinarily taken for granted. This is properly part of the business of mathematical logic. Of course there are those, even among logicians, who doubt the utility of this sort of thing—who profess to have no interest in improvements which do not lead to increases in deductive power or what not. However that may be, this second objective certainly has some interest in its own right; and it is this interest which has formed the primary motivation for these researches.
Bemerkungen zum Begriff der inhaltlichen Widerspruchsfreiheit
Authors: Mostowski Andrzej
Published: May 2006

In dem vorstehenden Aufsatz hat A. Tarski den Begriff der inhaltlichen Widerspruchsfreiheit (content-consistency) einer Aussagenmenge eingeführt. Der Verfasser hebt hervor, daß die inhaltliche Widerspruchsfreiheit keine innere Eigenschaft der Aussagenmenge selbst ist, sondern vielmehr auf die jeweilige Definition dieser Menge bezogen werden soll. Es ist also a priori möglich, daß wir, von zwei verschiedenen Definitionen einer und derselben Aussagenmenge E ausgehend, das eine Mal feststellen, daß E inhaltlich widerspruchsfrei ist, und das andere Mal, daß E nicht inhaltlich widerspruchsfrei ist. Daß dies tatsächlich in vielen Fällen vorkommt, zeigt folgender einfacher Satz:

I. Eine definierbare Satzmenge E, die alle gültigen Formeln des Aussagenkalküls enthält und in Bezug auf die Abtrennungsregel abgeschlossen ist, ist dann und nur dann in Bezug auf jede ihrer Definitionen inhaltlich widerspruchsfrei, wenn sie ausschließlich aus wahren Aussagen besteht.

Daß diese Bedingung hinreichend ist, hat Tarski (a. a. O. Satz 7.1) bewiesen. Nehmen wir nun an, E enthalte einen falschen Satz f (ohne freie Variable), und es sei ϕ irgendeine Definition von E. Die Aussagefunktion ψ = f¯→ϕ ist offenbar ebenfalls eine Definition von E. Wird ψ als Definition von E angenommen, so hat x(E) die Form einer Implikation mit dem Vorderglied f¯, welche auch die Aussage x sein könnte. x(E) gehört daher immer der Menge E an, woraus wir mit Rücksicht auf Bedingung 3.4 aus dem vorstehenden Artikel schließen, daß E in Bezug auf die Definition ψ nicht inhaltlich widerspruchsfrei ist, w. z. b. w.
Transfinite ordinals in recursive number theory
Authors: Goodstein R. L.
Published: May 2006

The possibility of constructing a numerical equivalent of a system of trans-finite ordinals, in recursive number theory, was briefly indicated in a previous paper, where consideration was confined to ordinals less than ε (the first to satisfy ε = ω). In the present paper we construct a representation, by functions of number-theoretic variables, for ordinals of any type.

In addition to definite numerals, and numeral variables, we introduce majorant variables σ, ω, ωr, r ≧ 1. A relation containing a single majorant variable σ is required to hold, not necessarily for all non-negative integral values of σ, but for all values greater than some assigned constant.
Two semantical paradoxes
Authors: Yuting Shen
Published: May 2006

Consider the statement:

(1) What I am saying cannot be proved.

Let us suppose that this statement can be proved. Then it must be true, i.e., in its own words, it cannot be proved, which contradicts our supposition.

Therefore the statement cannot be proved, since our supposition has led to a contradiction. In other words, the statement is true. In this way we have proved the statement.

Therefore the statement both can and cannot be proved.

In the second part of the argument, if we speak of proving (1) in a given formal system S, then we cannot draw the conclusion that we have proved the statement in S. For it is possible that the argument cannot be formalized in S. Indeed, as we know, Gödel shows in his famous article of 1931 that for suitable systems S, it is possible to construct in S a statement saying in effect that the statement itself is not provable in S. The conclusion Gödel draws, we may recall, is that the statement thus constructed is true but not provable in S. Thus no contradictions arise in this way if we are confined to provability in a given system.

A dual of (1) is the following:

(2) What I am saying can be refuted.

Let us suppose that this statement is true, or, in its own words, that it can be refuted. Then it must be false, which contradicts our supposition.

Therefore the statement is false, since our supposition has led to a contradiction. In this way, we have refuted the statement. We see that the statement can be refuted; in other words, it is true.
Simplified foundations for mathematical logic
Authors: Stanley Robert L.
Published: May 2006

A system SF, closely related to NF, is outlined here. SF has several novel points of simplicity and interest, (a) It uses only one basic notion, from which all the other concepts of logic and mathematics may be built definitionally. Three-notion systems are common, but Quine's two-notion IA has for some time represented the extreme in conceptual economy, (b) The theorems of SF are generated under just three rules of analysis, which unify into a single postulational principle, (c) SF is built solely in terms of what is commonly, known as the “natural deduction” method, under which each theorem is attacked primarily as it stands, by means of a very small body of rules, rather than less directly, through a very large, potentially infinite backlog of theorems. Although natural deduction is by no means new as a method, its exclusive applications have previously been relatively limited, not even reaching principles of identity, much less set theory, relations, or mathematics proper, (d) SF is at least as strong as NF, yielding all of its theorems, which are expressed here in forms analogous to those of the metatheorems in ML. If NF is consistent, so is SF. The main points in the relative consistency proof are set forth below in section seven.
Non-standard models for formal logics
Authors: Rosser J. Barkley , Wang Hao
Published: May 2006

In his doctor's thesis [1], Henkin has shown that if a formal logic is consistent, and sufficiently complex (for instance, if it is adequate for number theory), then it must admit a non-standard model. In particular, he showed that there must be a model in which that portion of the model which is supposed to represent the positive integers of the formal logic is not in fact isomorphic to the positive integers; indeed it is not even well ordered by what is supposed to be the relation of ≦.

For the purposes of the present paper, we do not need a precise definition of what is meant by a standard model of a formal logic. The non-standard models which we shall discuss will be flagrantly non-standard, as for instance a model of the sort whose existence is proved by Henkin. It will suffice if we and our readers are in agreement that a model of a formal logic is not a standard model if either:

(a) The relation in the model which represents the equality relation in the formal logic is not the equality relation for objects of the model.

(b) That portion of the model which is supposed to represent the positive integers of the formal logic is not well ordered by the relation ≦.

(c) That portion of the model which is supposed to represent the ordinal numbers of the formal logic is not well ordered by the relation ≦.
On the logic of quantification
Authors: Quine W. V.
Published: May 2006

The notation that I shall use here for the logic of quantification is of the familiar kind wherein the letters ‘p’, ‘q’, etc. stand in place of unspecified statements and the letters ‘ƒ’, ‘g’, etc. stand in place of unspecified predicates. The present section will deal with the significance of this notation; the purpose and scope of the paper as a whole can better be indicated afterward, in §2.

By “predicates” I mean, not properties (or classes) and relations, but merely certain notational expressions. As a first approach they may be thought of as expressions like ‘walks,’ ‘is red,’ ‘touches,’ ‘gives to.’ Where ‘ƒ’, ‘g’, ‘h’, and ‘k’ represent these four predicates, we may read ‘ƒx’ ‘gy’, ‘hxy’, and ‘κxyz’ as ‘x walks,’ ‘y is red,’ ‘x touches y,’ ‘x gives y to z.’
A set of axioms for logic
Authors: Hailperin Theodore
Published: May 2006

One of the preeminent problems confronting logicians is that of constructing a system of logic which will be adequate for mathematics. By a system's being adequate for mathematics, we mean that all mathematical theorems in general use can be deduced within the system. Several distinct logical systems, all having this end in view, have been proposed. Among these perhaps the best known are the systems referred to as “Principia Mathematica” and “set theory.” In both of these systems (we refer to the revised and simplified versions) there is a nucleus of propositions which can be derived by using only the axioms and rules of the restricted predicate calculus. However, if anything like adequacy for mathematics is to be expected, additional primitives and axioms must be added to the restricted predicate calculus. It is in their treatment of the additional primitive ε, denoting class or set membership, that the above-mentioned systems differ.

In addition to these two, a third and a stronger system has been proposed by W. V. Quine in his paper New foundations for mathematical logic. It is with this system of Quine's that our work is concerned and of which we now give a brief description.
On the interpretation of non-finitist proofs–Part II
Authors: Kreisel G.
Published: May 2006

Since the Hilbert substitution method has been applied by Ackermann to the formalism Z of number theory in [3], we shall give the no-counter-example interpretation for extensions (c) of Z, e.g. Zμ. The work required on top of Ackermann's investigations to check conditions (α) and (β), consists only in a suitable definition of a class of recursive functional. The verification of (γ) and (δ) is much more difficult, and requires the ideas of paras. 32 and 33 in the preceding section.
Remarks on identity and description in first-order axiom systems
Authors: Hailperin Theodore
Published: May 2006

Hilbert and Ackermann ([1], p. 107) define a first order axiom system as one in which the axioms contain one or more predicate constants, but no predicate variables. Here “axiom” refers to the specific subject-matter axioms and not to the rules of the restricted predicate calculus (quantification theory), which rules are presupposed for each first-order system. It is pointed out by them that an exception could be made for the predicate of identity; for the axiom scheme for this predicate, namely

which has in (b) the variable predicate F, could nevertheless be replaced, in any given first-order system, by a finite set of axioms without predicate variables. Thus, for example, if Φ[x, y) is the one constant predicate of such a system then PId(b) could be replaced by

Thus one postulates, in addition to the reflexivity, symmetry, and transitivity of identity, the substitutivity of identical entities in each of the possible “atomic” contexts of a variable (occurrences in the primitive predicates). In this method of introducing identity it has to be taken as an additional primitive predicate and further axioms are consequently needed. In such a system having PId(a), (b1)−(b3) as axioms, the scheme PId(b) can be derived as a meta-theorem of the system, F(x) then being any formula of the system.
A new theory of element and number
Authors: Wang Hao
Published: May 2006

Quine remarked that the system of his book exclusive of *200 is a completely safe basal logic. Axioms of this basal logic are given by *100-*105 and *201-*202 in the book. For relation theory and number theory, we need some further axioms to guarantee elementhood of certain entities. In the article, Quine proposed to adopt as axioms f610 and the following two statements:
The gergonne relations
Authors: Faris J. A.
Published: May 2006

In this paper I am going to set forth a formal system based on five inter-class relations. These relations exist respectively between a class of a's and a class of b's.

(i) if and only if every a is a b and every b is an a,

(ii) if and only if every a is a b and not every b is an a,

(iii) if and only if it is not the case that either every a is a b or every b is an a or no a is a b,

(iv) if and only if every b is an a and not every a is a b,

(v) if and only if no a is a b.

These relations between classes, which correspond, as will be seen, to the five relations between two circles a and b shown in the well-known Eulerian diagrams,

are of course connected in an intimate way with the four forms of proposition, A, E, I, O, of the traditional syllogistic logic. The French mathematician, J. D. Gergonne, seems to have been the first to recognize these relations explicitly and to understand their importance in syllogistic theory. It is therefore appropriate that they should be called by his name.

Gergonne first of all showed with reference to these relations what are the sufficient and necessary conditions of the truth of propositions of each of the four traditional forms: for example, an A proposition, ‘All a is b’ is true if and only if either the first or the second relation exists between the class of a's and the class of b's.
On confirmation and rational betting
Authors: Lehman R. Sherman
Published: May 2006

The purpose of this paper is to analyze rational betting. In particular, we concentrate on one necessary feature of rational betting, the avoidance of certainty of losing to a clever opponent. If a bettor is quite foolish in his choice of the rates at which he will bet, ah opponent can win money from him no matter what happens.

This phenomenon is well known to professional bettors—especially bookmakers, who must as a matter of practical necessity avoid its occurrence. Such a losing book is called by them a “dutch book.” Our investigations are thus concerned with necessary and sufficient conditions that a book not be “dutch.”

De Finetti [3] has started with the same idea and used it as a foundation for the theory of probability. It is our aim to consider the same subject more precisely and attempt to answer some questions about desirable features of a confirmation function. We wish to connect the ideas of De Finetti with those of Carnap [1] and Hossiasson-Lindenbaum [6]. The results expressed by Theorem 1 are essentially contained in De Finetti's work. Theorems 3 and 4 seem to be new.

The confirmation functions which we consider will be functions with two sentences as arguments taking real numbers as values. Intuitively, C(h, e) will represent the rate at which a bettor would be willing to bet on the hypothesis h if he knew the information expressed by the sentence e, the evidence.
Fair bets and inductive probabilities
Authors: Kemeny John G.
Published: May 2006

The question of what constitutes fairness in betting quotients has been studied by Ramsey, deFinetti, and Shimony. Thanks to their combined efforts we now have a satisfactory definition of fairness.

On the other hand, the explication of the concept of degree of confirmation (inductive probability) has progressed rapidly in recent years, thanks primarily to Carnap. This explication has usually proceeded by laying down the axioms for frequency-probabilities, and elaborating on these. While in the case where a frequency interpretation is intended these axioms are clearly justified, in our case they have been laid down without any justification. Carnap's presentation has been criticized for just this reason.

The purpose of this paper is to show that the probability axioms are necessary and sufficient conditions to assure that the degrees of confirmation form a set of fair betting quotients. In addition it will be shown that one additional, highly controversial, axiom is precisely the condition needed to assure that not only deFinetti's weaker criterion but Shimony's criterion of fairness is also satisfied.
On weakened quantification
Authors: Heyting A.
Published: May 2006

In this note I give an account of the relations between different notions of weakened quantification, which can be defined in intuitionistic logic. The reason for publishing it is the fact that the last gap in our knowledge of these relations has been filled in an article by Kleene [5].

For the intuitionistic calculus of propositions I refer to my article [2]; besides the rules of substitution and of inference I need only a few formulas from this calculus, which I shall repeat below. The simplest way of formalizing the intuitionistic predicate calculus is obtained by adjoining to the calculus of propositions the symbols, rules, and axioms as given by Hilbert and Ackermann [4], Chapter 3, §§4-5, pp. 53-57. (Reference is made to this book as HA.) I repeat only the axioms (e) and (f) and the rule γ.
On sentences which are true of direct unions of algebras
Authors: Horn Alfred
Published: May 2006

It is well known that certain sentences corresponding to similar algebras are invariant under direct union; that is, are true of the direct union when true of each factor algebra. An axiomatizable class of similar algebras, such as the class of groups, is closed under direct union when each of its axioms is invariant. In this paper we shall determine a wide class of invariant sentences. We shall also be concerned with determining sentences which are true of a direct union provided they are true of some factor algebra. In the case where all the factor algebras are the same, a further result is obtained. In §2 it will be shown that these criteria are the only ones of their kind. Lemma 7 below may be of some independent interest.

We adopt the terminology and notation of McKinsey with the exception that the sign · will be used for conjunction. Expressions of the form ∼∊, where ∊ is an equation, will be called inequalities. In accordance with the analogy between conjunction and disjunction with product and sum respectively, we shall call α1, …, αn the terms of the disjunction

and the factors of the conjunction

Every closed sentence is equivalent to a sentence in prenez normal form,

where x1, …, xm distinct individual variables, Q1, …, Qm are quantifiers, and the matrix S is an open sentence in which each of the variables x1, …, xm actually occurs. The sentence S may be written in either disjunctive normal form:

where αi,j is either an equation or an inequality, or in conjunctive normal form:

.
Interpretations of sets of conditions
Authors: Quine W. V.
Published: May 2006

The celebrated theorem of Löwenheim and Skolem tells us that every consistent set S of quantificational schemata (i.e., every set of well-formed formulas of the lower predicate calculus admitting of a true interpretation in some non-empty universe) admits of a true numerical interpretation (i.e., an interpretation of predicate letters such that all schemata of S come out true when the variables of quantification are construed as ranging over just the positive integers).

Later literature goes farther, and shows how, given S, actually to produce a numerical interpretation which will fit S in case S is consistent. The general case is covered by Kleene (see Bibliography). The special case where S contains just one schema (or any finite number, since we can form their conjunction) had been dealt with by Hilbert and Bernays. Certain extensions, along lines not to be embarked on here, have been made by Kleene, Kreisel, Hasenjäger, and Wang.

My present purpose is expository: to make the construction of the numerical interpretation, and the proof of its adequacy, more easily intelligible than they hitherto have been. The reasoning is mainly Kleene's, though closer in some ways to earlier reasoning of Gödel.
On propositions neither necessary nor impossible
Authors: Prior A. N.
Published: May 2006

There occurs in Aristotle a thesis which is symbolised in I. M. Bocheński's La Logique de Théophraste, Chapter V, by the formula EMpMNp. We might read this as “It is possible that P (Mp) if and only if (E) it is possible that not P (MNp).” But as Bocheński explains, Aristotle is here using the word “possible” to mean, not merely “not impossible”, but “neither necessary nor impossible”. And if we take “ It is impossible that P” to mean “It is necessary that not P”, Aristotle's thesis seems evidently true. For “It is neither necessary nor impossible that P” (Mp) amounts to “It is not necessary that P, and it is not necessary that not P” (KNSpNSNp, where S is the symbol for “It is necessary that”); and “It is neither necessary nor impossible that not P” (MNp) amounts to “It is not necessary that not P, and it is not necessary that not not P” (KNSNpNSNNp); and this reduces easily to the other by the law of double negation (ENNpp) and the commutativeness of conjunction (EKpqKqp).

Despite the apparently unexceptionable character of this thesis EMpMNp, when properly understood, Bocheński informs us, in paragraph 42 of this chapter, that Łukasiewicz has shown that it “implies a contradiction”. He goes on to explain that what he means by this is that it follows from this thesis, together with the assumption that some propositions are “possible” in the Aristotelian sense, that any proposition whatever is “possible” (in that sense). And this, when one thinks what it means, is a most extraordinary contention.
On ω-consistency and related properties
Authors: Orey Steven
Published: May 2006

1. This paper grew out of an attempt to answer a question raised in [4]. Let a logic L containing “numerals” z1, z2, … and a certain statement N(x) (intended to express the proposition that x is a natural number) be called ω-inconsistent if there is a statement such that ⊦ F(zk) for k = 1, 2, …, and ⊦ ∼(x)·N(x) ⊃ F(x); then it is evident that L cannot have a model in which N(x) is satisfied by the images of the numerals and nothing else if L is ω-inconsistent.

Question: If L is ω-consistent, i.e. not ω-inconsistent, must there be such a model? Calling a model of the kind just described a special model, we ask for necessary and sufficient conditions on L to insure the existence of a special model. We give several sets of such conditions, applicable to a certain very inclusive class of logics, in Theorem 1 and Theorems 3 and 4. Theorem 2 shows that a logic may be ω-consistent but still not have a special model.

This paper was close to completion when [3] appeared. For systems with only denumerably many symbols our results include Henkin's, for, by adjoining a new predicate N(x) to each of the systems considered in [3] which have only a denumerable number of constant symbols and then adding as an axiom (x)N(x), these systems become special cases of the systems we consider. It is easily seen that Henkin's Theorem 7 essentially proves the equivalence of conditions (2) and (3) in our Theorem 1, and Theorem 3 of [3] corresponds to our Theorem 2. Incidentally, our argument of Theorem 2 could also serve to prove Henkin's Theorem 6.
A formalization of the theory of ordinal numbers
Authors: Takeuti Gaisi
Published: May 2006

Although Peano's arithmetic can be developed in set theories, it can also be developed independently. This is also true for the theory of ordinal numbers. The author formalized the theory of ordinal numbers in logical systems GLC (in [2]) and FLC (in [3]). These logical systems which contain the concept of ‘arbitrary predicates’ or ‘arbitrary functions’ are of higher order than the first order predicate calculus with equality. In this paper we shall develop the theory of ordinal numbers in the first order predicate calculus with equality as an extension of Peano's arithmetic. This theory will prove to be easy to manage and fairly powerful in the following sense: If A is a sentence of the theory of ordinal numbers, then A is a theorem of our system if and only if the natural translation of A in set theory is a theorem of Zermelo-Fraenkel set theory. It will be treated as a natural extension of Peano's arithmetic. The latter consists of axiom schemata of primitive recursive functions and mathematical induction, while the theory of ordinal numbers consists of axiom schemata of primitive recursive functions of ordinal numbers (cf. [5]), of transfinite induction, of replacement and of cardinals. The latter three axiom schemata can be considered as extensions of mathematical induction.

In the theory of ordinal numbers thus developed, we shall construct a model of Zermelo-Fraenkel's set theory by following Gödel's construction in [1]. Our intention is as follows: We shall define a relation α ∈ β as a primitive recursive predicate, which corresponds to F′ α ε F′ β in [1]; Gödel defined the constructible model Δ using the primitive notion ε in the universe or, in other words, using the whole set theory.
Systems of predicative analysis
Authors: Feferman Solomon
Published: May 2006

This paper is divided into two parts. Part I provides a resumé of the evolution of the notion of predicativity. Part II describes our own work on the subject.

Part I

§1. Conceptions of sets.Statements about sets lie at the heart of most modern attempts to systematize all (or, at least, all known) mathematics. Technical and philosophical discussions concerning such systematizations and the underlying conceptions have thus occupied a considerable portion of the literature on the foundations of mathematics.
A note on a theorem of Vaught
Authors: Rosenstein Joseph G.
Published: May 2006

In [2] Vaught showed that if T is a complete theory formalized in the first-order predicate calculus, then it is not possible for T to have exactly (up to isomorphism) two countable models. In this note we extend his methods to obtain a theorem which implies the above.

First some definitions. We define Fn(T) to be the set of well-formed formulas (wffs) in the language of T whose free variables are among x1 x2, …, xn. An n-type of T is a maximal consistent set of wffs of Fn(T); equivalently, a subset P of Fn(T) is an n-type of T if there is a model M of T and elements a1, a2, …, an of M such that M ⊧ ϕ(a1, a2, …, an) for every ϕ ∈ P. In the latter case we say that 〈a1, a2, …, an〉 ony realizes P in M. Every set of wffs of Fn(T) which is consistent with T can be extended to an n-type of T.
An algebraic study of tense logics with linear time
Authors: Bull R. A.
Published: May 2006

In [2] Prior puts forward a tense logic, GH1, which is intended to axiomatise tense logic with time linear and rational; he also contemplates the tense logic with time linear and real. The purpose of this paper is to give completeness proofs for three axiom systems, GH1, GHlr, GHli, with respect to tense logic with time linear and rational, real, and integral, respectively.1 In a fourth section I show that GH1 and GHlr have the finite model property, but that GHli lacks it.

GH1 has the operators of the classical propositional calculus, together with operators P, H, F, G for ‘It has been the case that’, ‘It has always been the case that’, ‘It will be the case that’, ‘It will always be the case that’, respectively.