bluuebunny/crossref_metadata_2025 huggingface dataset:
Hegel's Critique of Liberalism
Authors: Smith Steven B.
Published: May 2006

A recent and perhaps surprising development in political theory has been the revival of neo-Kantian liberalism, with its doctrines of the neutral state and equal concern and respect. Critics of liberalism have suggested, however, that Kantian notions of rights and rationality are too thin a foundation on which to build satisfactory forms of community and political life. In this paper I examine the critique of rights-based liberalism by returning to the philosophy of Hegel. Hegel's position, I suggest, provides us with a much needed middle ground between liberalism and its contemporary critics. Like the modern communitarians he is critical of the individualistic and ahistorical conceptions of rights underlying the liberal polity, but like many liberals he is skeptical of the claims to recreate a democratic, participatory Gemeinschaft that would leave citizens defenseless before their particular communities. I conclude that like Montesquieu before him and Tocqueville after, Hegel looked to the “corporations” or intermediary associations to skirt the extremes of the market place and civic virtue.
Freedom and Authority: Beyond the Precepts of Liberalism
Authors: Preston Larry M.
Published: May 2006

Both liberal theorists and their critics have been unable to reconcile successfully respect for individual freedom and autonomy with a system of legitimate political authority. Liberal theory fails because its system of rights and representative institutions is particularistic, favoring the interests of restricted individuals and groups. Nonliberal theory threatens individuals' freedom by having political authority exercised in the name of a dubious, real will. Present liberal and nonliberal theory continues to reflect these problems.

This article is an effort to present a view of political authority that respects individual freedom and is legitimate, supporting the interests of all citizens. The failure of liberal and nonliberal theory to achieve this reconciliation is believed to result from their excessively ambiguous, incomplete views of central issues—freedom, authority, and normative reasoning—and a related tendency to examine these problems in isolation. The analysis presented here proposes a more complete view of their meaning and integrates this understanding into a conception of legitimate political authority. The conclusion seriously challenges the view of political legitimacy presented by liberal theory and suggests that a different set of problems must be addressed if political authority is to be legitimate and support individuals' freedom.
Explaining Senate Election Outcomes
Authors: Abramowitz Alan I.
Published: May 2006

Aggregate-level data are used in this analysis to explain the outcomes of Senate elections between 1974 and 1986. Using the individual Senate contest as the unit of analysis permits estimating the relative influence of a wide variety of factors on Senate election results including political characteristics of states, characteristics of the candidates, and national political conditions. Of these factors candidate characteristics had the strongest impact on the outcomes of Senate elections. The importance of candidate characteristics has had two major consequences for Senate elections. First, two-party competition has spread to every region of the country: in Senate elections, no state can be considered safe for either party. Second, money is probably now more important than ever, especially for challengers and candidates for open seats.
Liberation Theology: The Recovery of Biblical Radicalism
Authors: Roelofs H. Mark
Published: May 2006

Liberation theology is not a Marxism in Christian disguise. It is the recovery of a biblical radicalism that has been harbored in the Judeo-Christian tradition virtually from its founding. As such, liberation theology is the revitalization of one of the most profound social, political, economic, and religious challenges to established hierarchies that the West has ever known. Liberation theologians turn to modern Marxism chiefly to gain a comprehensive understanding of contemporary class conflict and poverty. Their primary concerns remain to define themselves as an “emergent church” in the biblical tradition, and to reflect upon the praxis of the “base community” movement they have spawned throughout Latin America. However, the principal liberation theologians of our day exhibit persistent ambiguities about how individuals should relate to their communities. The biblical materials suggest certain solutions to them, but ones so radical that not even the most extreme of the liberation theologians appear ready to embrace them.
On Anarchism In An Unreal World: Kramnick's View of Godwin and the Anarchists
Authors: Clark John P.
Published: May 2006

In his political theory, William Godwin, the founder of philosophical anarchism, defends a form of decentralized democracy as a transition stage toward his ideal of a stateless society. Godwin sanctioned temporary alliance with liberal political factions, and was a staunch defender of freedom of thought and expression. Contrary to Isaac Kramnick's interpretation, he bitterly opposed authoritarian measures such as Grenville's and Pitt's bills, although he also rejected violence as a valid means toward reform. Kramnick's reduction of anarchist praxis to education and theater constitutes a serious distortion of historical fact, and ignores the significance of revolutionary struggle, syndicalist organization, intentional communities, and nonviolent resistance as anarchist tactics. His assertion that anarchism is elitist involves a misunderstanding of the anarchist principles of popular participation and self-management, and his position that anarchism serves to perpetuate the status quo exhibits a failure to deal with the issues presented by the anarchist analysis of the effects of government and the nature of social change.
Justice and Rationality: Some Objections to the Central Argument in Rawls's Theory
Authors: Fishkin James
Published: May 2006

The argument is made that Rawls's proposed principles of justice would not rationally be chosen in his proposed version of the “original position.” First of all, Rawls's own account of the information available in the original position provides no basis for the conclusions about “primary goods” which he believes would imply his proposed principles. Second, even if those conclusions about primary goods were to be accepted, they would not imply the proposed principles (the “general” and the “special” conceptions of justice) because the claims which Rawls cites in support of “maximin” actually imply a different conception. Lastly, an alternative version of the original position is suggested which would avoid these difficulties and which would, in addition, successfully support a maximin conception of justice. This is not meant, however, as a conclusive argument for maximin, but only as an indication that the objections advanced here do not apply to “social contract theory” as such, but only to the particular version of it which Rawls has proposed.
Helvétius and the Roots of the “Closed” Society
Authors: Campbell Blair
Published: May 2006

The argument of this essay is that Talmon and Popper are mistaken in their suggestion that speculation about human affairs is governed by an inexorable logic of political consequences: that there exist certain broad perspectives or ‘paradigms” that impel men willy-nilly to pathological extremes in their political views, apart from intention or historical circumstance.

I seek to demonstrate that the general perspective which informed the thought of Helvétius—unquestionably one of the most manipulative of thinkers in his conception of politics—was simply the framework of early-modern science, as it was understood in France. It was the same philosophy which served his unequivocally libertarian contemporaries, such as Voltaire and Diderot, as well as their predecessors. Helvétius' political conclusions resulted, not from pathological attitudes or doctrines, but rather from his attempt to resolve a problem engendered within the new science, a fundamental dilemma in French thinking concerning the relationship of the individual to society and the state.
Are Social Background Models Time-Bound?
Authors: Ulmer S. Sidney
Published: May 2006

In this research note I seek to determine whether a significantly predicting social background model for analyzing the votes of Supreme Court justices is time-bound. I argue that an affirmative result poses serious questions for past uses of such models, none of which has controlled for the possibility that time is a confounding variable. A model that significantly predicted the votes of the justices in the Court's 1903–1968 terms was constructed. Analysis with this model for two periods—from 1903 to 1935, and from 1936 to 1968—established that the model was not timeneutral. Appropriate theoretical implications are drawn.
Recruitment Contrasts in A Divided Charismatic Movement
Authors: Madsen Douglas , Snow Peter G.
Published: May 2006

This research report analyzes the evolution of the bond between a charismatic leader and his mass following. Its empirical focus is the case of Juan Peron and the Peronist movement in Argentina. After discussing stages in the evolution of charisma, we review our earlier findings showing that by 1965 there was clear separation between those Peronists devoted to the movement and those Peronists devoted to the man. We note distinctive features in the social and attitudinal profiles of each and then turn to our principal concern, the recruitment of new blood into each Peronist camp. Personalist youth turn out to be much like their older counterparts, but organizationalist youth demonstrate social and attitudinal features that set them dramatically apart from the personalists and apart from the older organizationalists as well. We discuss the implications of this ever widening rift.
Do Cooperators Exit More Readily than Defectors?
Authors: Orbell John M. , Schwartz-Shea Peregrine , Simmons Randy T.
Published: May 2006

Standard prisoners' dilemma games offer players the binary choice between cooperating and defecting, but in a related game there is the third possibility of leaving the game altogether. We conceptualize exiting as taking the individual beyond the reach of externalities generated in the original group, and on that basis—together with the assumption of self-interested (dollar-maximizing) behavior on the part of all players—we derive the prediction that the exit option will drain the community or group more of cooperators than of defectors.

But experimental data do not support this prediction; cooperators do not leave more frequently than defectors and, in fact, there is evidence that defectors are more prone to leave than cooperators. We consider and reject the possibility that this failure of prediction results from the (admitted) greater optimism of cooperators about the incidence of cooperation “here,” and present data supporting the hypothesis that cooperators often stay when their personal interest is with exiting because of the same ethical or group-regarding impulse that (presumably) led them to cooperate in the first place. Cooperation can be produced for a group or community either by inducing people to cooperate or by inducing those who are going to cooperate to stay in the game, and ethical considerations seem to underlie the decision to stay as well as the decision to cooperate while staying.
Politics, Markets, and the Organization of Schools
Authors: Chubb John E. , Moe Terry M.
Published: May 2006

We offer a comparative analysis of public and private schools, presenting data from a new national study—the Administrator and Teacher Survey—that expands on the pathbreaking High School and Beyond survey. We find that public and private schools are distinctively different in environment and organization. Most importantly, private schools are more likely to possess the characteristics widely believed to produce effectiveness. We argue throughout that the differences across the sectors are anchored in the logic of politics and markets. This argument derives from our belief that environmental context has pervasive consequences for the organization and operation of all schools and specifically that the key differences between public and private environments—and thus between public and private schools—derive from their characteristic methods of social control: the public schools are subordinates in a hierarchic system of democratic politics, whereas private schools are largely autonomous actors “controlled” by the market.
Reckless Rationalism and Heroic Reverence in Homer's Odyssey
Authors: Dobbs Darrell
Published: May 2006

A decision-theoretic analysis of the central incident of Homer's Odyssey reveals the insufficiency of rational calculation as a guide for political prudence. Surprisingly, the poet distinguishes between two rational and formally identical calculations in no uncertain terms; he condemns one as utter recklessness and praises the other as consummate wisdom. I maintain that this discrepancy is neither an artifact of sloppy editorial patchwork nor the result of a “homeric nod” but instead points toward a politically significant distinction as yet obscured by a merely rationalistic perspective. The recklessness of Odysseus' crewmen, who deliberately slaughter sacred cattle to forestall starvation, consists in their rationalistic transgression of the limits of reason. These limits are most evident in the defiance of commensurability that characterizes the sacred. The wisdom of Odysseus, by contrast, is manifest in his learning to temper reason with respect for the sacred. By virtue of reverence, Odysseus wins his struggle to preserve his psychê, home and regime.
Measuring the Southern Contribution to the Democratic Coalition
Authors: Abramson Paul R.
Published: May 2006

Measuring the southern contribution to the Democratic coalition is an important task. To measure this contribution one must choose appropriate data and the appropriate unit of analysis for studying party coalitions in the United States. Two recent studies of party coalitions use the National Election Studies to estimate the southern contribution to the Democratic party, and these studies illustrate the problems one may encounter. This note demonstrates two points. First, survey research results may lead to erroneous estimates and it is preferable, where possible, to rely upon official election statistics. Second, the contribution of demographic groups to party coalitions should be assessed within the context of the political rules that make such coalitions meaningful.
The Role of Research in the Unification of a Discipline
Authors: Miller Warren E.
Published: May 2006

Large-scale complex research project designs are providing a new impetus to the elimination of subfield boundaries within political science. Major projects are taking advantage of the methodology and technology of contemporary social research to include comparisons among institutions, across cultural boundaries, and extending through time. As a consequence, traditionally narrow field and subfield concentrations on segments of the political process are giving way to intellectual interests that bring together hitherto separate concerns. The full potential for discipline-unifying research will, however, not be realized until there is a strengthening of the organizational infrastructures for research, a broadening of training in research design and administration, and an increase in funding for large-scale projects. The execution and subsequent intellectual exploitation of large research projects will carry additional problems that will be solved only with substantial changes in the workways of the political scientist, but those problems are greatly outweighed by the positive contribution that such research will make to the future of the discipline.
Language and Its Abuses in Hobbes' Political Philosophy
Authors: Whelan Frederick G.
Published: May 2006

As in Hobbes' view it is principally the capacity for speech that distinguishes men from even the social animals, so it is in verbal and doctrinal controversies that he usually finds the sources of conflict and sedition. Hobbes analyzes—in the hope of doing away with them—a variety of what he regards as abuses of language, such as metaphor, equivocation, eloquence, and absurdity, which are especially productive of political disorder. He also offers models and, in his own political philosophy, examples of the proper uses of language as science and counsel, which he believes are necessary to the establishment and governance of well-ordered commonwealths in the modern world, characterized as it is by widespread learning and disputatious habits. In the pursuance of his project, however, Hobbes himself is paradoxically forced to resort to the eloquence which he otherwise condemns, and his own observations on language provide grounds for doubts about the success of his enterprise.
Democratic Representation and Partisan Bias in Congressional Elections
Authors: King Gary , Browning Robert X
Published: May 2006

The translation of citizen votes into legislative seats is of central importance in democratic electoral systems. It has been a longstanding concern among scholars in political science and in numerous other disciplines. Throughout this literature, two fundamental tenets of democratic theory, partisan bias and democratic representation, have often been confused. We develop a general statistical model of the relationship between votes and seats and separate these two important concepts theoretically and empirically. In so doing, we also solve several methodological problems with the study of seats, votes, and the cube law. An application to U.S. congressional districts provides estimates of bias and representation for each state and demonstrates the model's utility. Results of this application show distinct types of representation coexisting in U. S. states. Although most states have small partisan biases, there are some with a substantial degree of bias.
The Triple Entente and the Triple Alliance 1880–1914: A Collective Goods Approach
Authors: Conybeare John A. C. , Sandler Todd
Published: May 2006

The distribution of burdens in alliances may be explained in terms of public and private outputs. A joint product model is applied to the Triple Alliance and Triple Entente, using generalized least squares-auto regressive moving average (GLS-ARMA) techniques and time series data. Results indicate that countries regarded allies' military effort more as complements than as substitutes, though several examples of free-riding behavior are noted. The method used here may provide more accurate estimation of publicness problems than is found in the usual static tests.
Missing the Archimedean Point: Liberalism's Institutional Presuppositions
Authors: Grafstein Robert
Published: May 2006

Though liberalism has been widely criticized for its attempt to frame a detached judgment of society based on an asocial conception of individuals, insufficient attention has been paid to the particular social and political relationships this search for an Archimedean point presupposes. Using collective choice theory, I show that liberalism has adopted two distinct kinds of Archimedean points reflecting different and unjustified presuppositions about the true institutional relation between politics and society. Liberalism's Archimedean search is not merely unsuccessful but biased in a way that is significant even for positions critical of liberalism. It is possible, I argue, to have a normative political theory that avoids an asocial conception of individuals without falling victim to liberalism's specific biases concerning institutional relations. The implications for both Rawlsian- or Nozickian-style liberalism are discussed, including the possibility of a political philosophy that avoids their “analytical extremism.“
Political Obligation, Freedom and Feminism
Authors: Pateman Carole , Hirschmann Nancy J.
Published: May 2006

How should feminist theorists approach the analysis of political obligation and freedom? In her article in the December 1989 issue of thisReview, Nancy Hirschmann proposed a fundamental reconstruction of these concepts. Carole Pateman agrees that liberal obligation theory inadequately treats the coercion of women and other groups under the guise of voluntary obligations. But she argues that Hirschmann goes too far, particularly in rejecting the priority of freedom. Hirschmann explicates their disagreements and her approach.
Political Science and Prevision
Authors: de Jouvenel Bertrand
Published: May 2006

The political scientist is a teacher of public men in the making, and an adviser of public men in activity; “public men,” that is, men who are taught, invited or assumed to feel some responsibility for the exercise of political power; “political power,” that is, concentrated means of affecting the future.

Obviously we can not affect the past, or that present moment which is now passing away, but only what is not yet: the future alone is sensitive to our actions, voluntary if aimed at a pictured outcome, rational if apt to cause it, prudently conceived if we take into account circumstances outside our control (known to decision theorists as “states of nature”), and the conflicting moves of others (known in game theory as opponents' play). A result placed in the future, conditions intervening in the future, need we say more to stress that decisions are taken “with an eye to the future,” in other terms, with foresight?
British M.P.S and their Local Parties: The Suez Cases
Authors: Epstein Leon D.
Published: May 2006

Most students of British parties have accepted the view that the mass organizations are decidedly subordinate to parliamentary leadership. Mainly this has meant rejection of the idea that policy is imposed by a party conference, or its delegated executive, in the Labour party as in the Conservative party. But it may also lead one to ignore or depreciate the role of the constituency units which compose the national organizations. That it is a mistake to do so is now suggested by the activities of local party associations during the Suez crisis of 1956–57. Research material derived from this experience provides the bases for altering the common model of the constituency party as a unit in the mass service organizations sustaining the parliamentary leadership. True enough, the association of dues-paying partisans is primarily service rather than policymaking. However, the Suez experience indicates that this service includes a partly selfgenerating function in relation to the maintenance of parliamentary party cohesion, going beyond the well-known earlier instances of local Labour units simply following national orders to drop candidates who were suspected Communists.
The Primacy of Labor in American Constitutional Development
Authors: Orren Karen
Published: May 2006

As demonstrated in landmark decisions of the U.S. Supreme Court, American constitutional development has been fueled since the framing by disputes arising from changing labor relations in both private and public settings. This pattern is explained by the original provisions of the Constitution, the English background of its emergence, and the primacy of labor as a theoretical concept for studying political change. The Court's decisions protecting property express the status quo, the establishment against which transformations proceed. However, the property cases may also be reinterpreted along the lines indicated.
Choosing the Best Social Order: New Principles of Justice and Normative Dimensions of Choice
Authors: Lissowski Grzegorz , Swistak Piotr
Published: May 2006

There are many standard rules of aggregating individual preferences; majority rule is but one example. These rules determine what we typically consider to be a fair or a just solution to the problem of social choice. Yet we know very little, either about how these solutions relate to what a person, not a rule, would choose as a fair outcome or about what ethical or political beliefs would guide people in their choices. An empirical study was conducted to address these problems. As standard normative solutions fail to explain choices obtained in the study, we propose a new set of solutions, which generalize two classical principles of justice: Rawlsian and conservative. These generalized solutions fit the data remarkably well. More important, they uncover two normative dimensions within which choices turn out to be very consistent. These dimensions, we conjecture, indicate subjects' underlying attitudes. We use this theory to compare ethical-political attitudes in samples from Poland, Japan, and the United States.
On the reduction of the decision problem. First paper. Ackermann prefix, a single binary predicate
Authors: Kalmár László
Published: May 2006

1. Although the decision problem of the first order predicate calculus has been proved by Church to be unsolvable by any (general) recursive process, perhaps it is not superfluous to investigate the possible reductions of the general problem to simple special cases of it. Indeed, the situation after Church's discovery seems to be analogous to that in algebra after the Ruffini-Abel theorem; and investigations on the reduction of the decision problem might prepare the way for a theory in logic, analogous to that of Galois.

It has been proved by Ackermann that any first order formula is equivalent to another having a prefix of the form

(1) (Ex1)(x2)(Ex3)(x4)…(xm).

On the other hand, I have proved that any first order formula is equivalent to some first order formula containing a single, binary, predicate variable. In the present paper, I shall show that both results can be combined; more explicitly, I shall prove the

Theorem. To any given first order formula it is possible to construct an equivalent one with a prefix of the form (1) and a matrix containing no other predicate variable than a single binary one.

2. Of course, this theorem cannot be proved by a mere application of the Ackermann reduction method and mine, one after the other. Indeed, Ackermann's method requires the introduction of three auxiliary predicate variables, two of them being ternary variables; on the other hand, my reduction process leads to a more complicated prefix, viz.,

(2) (Ex1)…(Exm)(xm+1)(xm+2)(Exm+3)(Exm+4).
On Cantor's theorem
Authors: Quine W. V.
Published: May 2006

Among various unnatural and technically burdensome effects of the theory of types, one is the unstable character of meaningfulness: a mere permutation of variables is capable of reducing a significant context to meaninglessness. Another effect, and perhaps the most conspicuous, is the systematic reduplication to which the logical constants are subjected; the calculi of classes and relations and even arithmetic lose their unity and generality, and are reproduced anew within each type. The elaborate compensatory manoeuvres which are thus made necessary are familiar to all readers of Principle, mathematica.

In a recent publication, to be cited henceforth as New foundations, I proposed an alternative course which avoids these consequences, but which would seem to offer less assurance of consistency. My efforts to derive a contradiction have delivered none, but they have lent a strange aspect to Cantor's proof that every class has more subclasses than members. This result is the topic of the present paper.

Preparatory to sketching the new system, which I shall call S′, I shall sketch a very similar but contradictory system S. By way of primitives S involves just membership, universal quantification, and alternative denial (Sheffer's stroke function), together with general variables “x”, “y”, …; the adequacy of this equipment as a basis for mathematical logic is made evident by Wiener's and Kuratowski's discovery of methods of constructing relation theory in terms of classes. Thus the formulae or statements and statement forms of S are describable recursively as follows: if a variable is put in each blank of “(ϵ)”, the result is a formula; if a variable enclosed in parentheses is prefixed to a formula, the result is a formula; and if a formula is put in each blank of “(∣)”, the result is a formula. The theorems of S are determined by a postulate and five rules, called P1 and R1-5 in New foundations. P1 and R1-3 specify various formulae as initial theorems, and R4-5 specify inferential connections for deriving further theorems. R1–2 and R4–5 are so fashioned as to provide the “theory of deduction”: they provide as theorems all those formulae which are valid by virtue merely of their structure in terms of alternative denial and quantification.
Definability and decision problems in arithmetic
Authors: Robinson Julia
Published: May 2006

In this paper, we are concerned with the arithmetical definability of certain notions of integers and rationals in terms of other notions. The results derived will be applied to obtain a negative solution of corresponding decision problems.

In Section 1, we show that addition of positive integers can be defined arithmetically in terms of multiplication and the unary operation of successorS(whereSa=a+ 1). Also, it is shown that both addition and multiplication can be defined arithmetically in terms of successor and the relation of divisibility | (wherex|ymeansxdividesy).
Inner models for set theory—Part II
Authors: Shepherdson J. C.
Published: May 2006

In this paper we continue the study of inner models of the type studied inInner models for set theory—Part I.The present paper is concerned exclusively with a particular kind of model, the ‘super-complete models’ defined in section 2.4 of I (page 186). The condition (c) of 2.4 and the completeness condition 1.42 imply that such a model is uniquely determined when its universal class Vmis given. Writing condition (c) and the completeness conditions 1.41, 1.42 in terms of Vm, we may state the definition in the form:

3.1. Dfn.A classVmis said to determine a super-complete model if the model whose basic notions are defined by,satisfies axiomsA, B, C.

N. B. This definition is not necessarily metamathematical in nature. If desired, it could be written out quite formally as the definition of a notion ‘SCM(U)’ (‘Udetermines a super-complete model’) thus:

whereψ(U) is the propositional function expressing in terms ofUthe fact that the model determined byUaccording to 3.1 satisfies the relativization of axioms A, B, C. E.g. corresponding to axiom A1m, i.e.,,ψ(U) contains the equivalent term. All the relativized axioms can be similarly expressed in this way by first writing out the relativized form (after having replaced all defined symbols which occur by the corresponding formulae in primitive notation) and then replacing ‘(Am)ϕ(Am) bywhich is in turn replaced by, and similarly replacing ‘(xm)ϕ(xm)’ by ‘(xm)ϕ(xm)’ by ‘(X)(X ϵ U ▪ ⊃ ▪ ϕ(X)), andThusψ(U) is obtained in primitive notation.
A note on the theory of quantification
Authors: Shaw-Kwei Moh
Published: May 2006

In the paper On the logic of quantification Prof. W. V. Quine showed that for the theory of quantification we have a mechanical process to determine whether or not a monadic expression is a valid logical formula, and that to deduce the polyadic theory from the monadic theory we need only the generalized modus ponens, which reads:

If a conditional is valid, and its antecedent consists of zero or more quantifiers followed by a valid schema, then its consequent is valid, i.e.:

That we have a mechanical process to determine the validity of a monadic expression was pointed out by Löwenheim long ago; Quine's method, however, has the merit that it is very simple and hence practical. The present paper is to show that we may deduce the polyadic theory by means of only a mechanical process and the ordinary modus ponens alone, i.e., the schema,

or even by means of only the weakened modus ponens, i.e., the schema,

For, if an expression becomes monadic when we omit the initial all-quantifiers (i.e., those preceding every existence-quantifier) and regard the corresponding variables as constants, then evidently we may determine by a mechanical process whether or not it is a valid formula by examining the resulting monadic expression. For example, the expression

is of the same validity as the expression

and hence as the expression
Extensions of the Lewis system S5
Authors: Scroggs Schiller Joe
Published: May 2006

Dugundji has proved that none of the Lewis systems of modal logic, S1 through S5, has a finite characteristic matrix. The question arises whether there exist proper extensions of S5 which have no finite characteristic matrix. By an extension of a sentential calculus S, we usually refer to any system S′ such that every formula provable in S is provable in S′. An extension S′ of S is called proper if it is not identical with S. The answer to the question is trivially affirmative in case we make no additional restrictions on the class of extensions. Thus the extension of S5 obtained by adding to the provable formulas the additional formula p has no finite characteristic matrix (indeed, it has no characteristic matrix at all), but this extension is not closed under substitution—the formula q is not provable in it. McKinsey and Tarski have defined normal extensions of S4* by imposing three conditions. Normal extensions must be closed under substitution, must preserve the rule of detachment under material implication, and must also preserve the rule that if α is provable then ~◊~α is provable. McKinsey and Tarski also gave an example of an extension of S4 which satisfies the first two of these conditions but not the third. One of the results of this paper is that every extension of S5 which satisfies the first two of these conditions also satisfies the third, and hence the above definition of normal extension is redundant for S5. We shall therefore limit the extensions discussed in this paper to those which are closed under substitution and which preserve the rule of detachment under material implication. These extensions we shall call quasi-normal. The class of quasi-normal extensions of S5 is a very broad class and actually includes all extensions which are likely to prove interesting. It is easily shown that quasi-normal extensions of S5 preserve the rules of replacement, adjunction, and detachment under strict implication. It is the purpose of this paper to prove that every proper quasi-normal extension of S5 has a finite characteristic matrix and that every quasi-normal extension of S5 is a normal extension of S5 and to describe a simple class of characteristic matrices for S5.
A demonstrably consistent mathematics—Part II
Authors: Fitch Frederic B.
Published: May 2006

13.1. If ‘[p f a]’ is in K′, we will say that ‘p’ is a value of ‘f’ at ‘a’.

13.2. If ‘a’ and ‘b’ are rational U-reals and if ‘b’ is not less than ‘a’, then ‘{a, b}’ will be called a rational U-interval.

13.3. If ‘f’ has a value at ‘d’, if ‘d’ is within a U-interval ‘{c, e}’, and if all values of ‘f’ at U-reals with ‘{c, e}’ are within the U-interval ‘{r, t}’, then we will say that ‘{c, e}’ restricts ‘f’ to ‘{r, t}’ at ‘d’, and that ‘{c, e}’ is an ‘f’-restrictive correlate of ‘d’ with respect to ‘{r, t}’.

13.4. If ‘d’ is within ‘{a, b}’, we will say that ‘{a, b}’ “contains” ‘d’.

13.5. If ‘f’ has a value at ‘d’ and if corresponding to every U-interval ‘{r, t}’ that contains a value of ‘f’ at ‘d’ there is an ‘f’-restrictive correlate of ‘d’ with respect to ‘{r, t}’, then ‘f’ will be said to be “continuous” at ‘d’. Without loss of generality the ‘f’-restrictive correlate can be required to be a rational U-interval.

13.6. If ‘f’ is continuous at every U-real in ‘{a, b}’, we will say that ‘f’ is continuous in ‘{a, b}’.

13.7. Theorem. If ‘k’ is a positive U-real and ‘f’ is continuous in a U-interval ‘{a, b}’, then ‘{a, b}’ is covered by the class S of rational U-intervals that are ‘f’-restrictive correlates of rational U-reals in ‘{a, b}’ with respect to those rational U-intervals ‘{r, t)’ that contain values of ‘f’ at rational U-reals in ‘{a, b}’ and that are such that’∣t − r∣‘ is equal to ‘k’ (in the sense of 10.10).
Correction to a paper on modal logic
Authors: Anderson Alan Ross
Published: May 2006

In a recent paper (here referred to as “IDP”) the writer outlined a decision procedure for Lewis's system S4 of modal logic. One of the clauses in definition 3.1 of IDP requires correction. Clause II of 3.1 (2) should read as follows.

II. Some constituent of the form ◊β, of degree n1 ≤ n, has the value T in Row (i), and some constituents of the forms ◊δ1, … ◊δh, and ◊η1, …, ◊ηm, all have the value F in Row (i) (h ≥ 0, m ≥ 0, h+m ≥ 1), where β → (δ1 ∨ … ∨ δh ∨ ◊η1 ∨ … ∨ ◊ηm) is an (n1 − 1)-tautology of S4.

This change is required in order to carry out the proof of Metathcorcm 3.19. In particular, the change guarantees the following. If the expression η of the second paragraph of 3.20 is of degree n − 1, then the antecedent λ of formula ζ on page 210 of IDP is also of degree n − 1; and consequently the formula ζ of 3.19 is of degree n (since ◊λ is a constituent of ζ). (If we fail to make the correction, then it might be the case that both ζ and (3) are of degree n − 1, in which case Row (i) would not satisfy clause II as originally stated, contrary to the claim at the end of 3.20.) The proofs for the remaining cases of 3.20 can then be carried out, using the revised clause II, in the way originally indicated.

The proof of Metatheorem 3.2 requires only trivial corrections for case 2.
On the syntactical construction of systems of modal logic
Authors: Mckinsey J. C. C.
Published: May 2006

When C. I. Lewis developed his theory of strict implication, he left open the question which of his various systems should be regarded as being closest to our intuitions—though he was inclined to favor the system S2. There are to be found in the literature numerous discussions of this question; most of these have condemned S2 as being too strong, and have proposed ways of weakening it.

In the present paper I shall attempt to throw some light on this question by setting up a syntactical definition of “possibility.” I shall show that every system of modal logic constructed on the basis of this definition is at least as strong as the Lewis system S4.

As the intuitive basis for the syntactical definition of possibility, I take the position that to say a sentence is possible means that there exists a true sentence of the same form. Thus, for example, it would be said that the sentence, “Lions are indigenous to Alaska,” is possible, because of the fact that the sentence, “Lions are indigenous to Africa,” has the same form and is true.
On absolute properties of relations
Authors: Mostowski Andrzej
Published: May 2006

We shall be concerned in this paper with properties of relations. For simplicity we shall consider only two relations, a binary relation R and a ternary relation S, but the generalization to the case of any number of any relations presents no difficulty. We shall denote by A0 the field of the relations R and S and assume that A0 is infinite (not inductive).

The properties to be dealt with are expressible in a symbolic language L whose principal features will be sketched below.
New foundation of formal metamathematics
Authors: Chwistek L. , Hetper W.
Published: May 2006

Our method is based on the same principles which we have assumed in a paper about the foundations of metamathematics which we published in collaboration with J. Herzberg some years ago. Nevertheless we have introduced some important modifications which enable us to use a very restricted preliminary language and to avoid any artificial construction.

The problem of a restricted language in mathematics appears to be a very important one, as our ordinary language is really ambiguous and obscure. On the other hand there is the problem of restricting the intuitive logic to be used in our construction. We shall see that for this purpose we need only substitutions and modus ponens.

If we avoided any abbreviation, we could reduce our preliminary language to propositions of the following forms:

X is a theorem.

If P, then Q.

But we have to do with very long symbolic expressions and as there is not yet any machine to build up such expressions we must use abbreviations. For this purpose we shall use propositions of the following forms in the preliminary language:

X is an expression.

X is an abbreviation of Y.

Any proposition which does not belong to one of our four forms will be excluded from our construction.
On reduction to a symmetric relation
Authors: Craig William , Quine W. V.
Published: May 2006

In a current paper, Church and Quine show how any dyadic relation of natural numbers can be defined in terms of a symmetric dyadic relation of natural numbers together with the truth functions and quantification over natural numbers. The purpose of the present note is to extend their result to the following: Where R1, R2, …, ad infinitum are any classes or relations (of any degree) of natural numbers, they can all be defined in terms of a single symmetric dyadic relation of natural numbers together with the truth functions and quantification over natural numbers.

Let pi, for each i, be the tth prime > 1. Let the degree of Ri, for each i, be di. Let R be the dyadic relation which each number x bears to each number y such that either x is a factor of y or else x is 0 and y is for some such that . Since R is dyadic, we know from Church and Quine's result that R is definable in terms of a symmetric dyadic relation. Moreover, as seen in the incidental constructions in §1 of Church and Quine, ‘y = x + 1’ is expressible on the same basis.
A correction to Lewis and Langford's Symbolic logic
Authors: McKinsey J. C. C.
Published: May 2006

The purpose of this note is to call attention to a minor error in Lewis and Langford's Symbolic logic. On page 221, in discussing the Tarski-Łukasiewicz three-valued logic, the authors make the following assertion: “Let T(p) be any proposition, involving only one element, whose analogue holds in the two-valued system; if T(p) does not hold in the Three-valued Calculus, then pC.T(p) and Np.C.T(p) both hold.”

I shall show, by means of a counter-example, that this assertion is not true. Let T(p) be the sentence:

It is then easily verified that T(0) = T(1) = 1, and that T(½) = 0. Thus T(p) holds in the two-valued calculus, but not in the three-valued calculus. On the other hand, pC.T(p) does not hold, since ½.CT(½) = ½C0 = ½; similarly, Np.C.T(p) does not hold, since N½.C.T(½) = ½C0 = ½.
The deduction theorem in a functional calculus of first order based on strict implication
Authors: Barcan Ruth C.
Published: May 2006

In a previous paper, a functional calculus based on strict implication was developed. That system will be referred to as S2. The system resulting from the addition of Becker's axiom will be referred to as S4. In the present paper we will shw that a restricted deduction theorem is provable in S4 or more precisely in a system equivalent to S4. We will also show that such a deduction theorem is not provable in S2.

The following theorems not derived in Symbolic logic will be required for the fundamental theorems XXVIII* and XXIX* of this paper. We will state most of them without proofs.
A formal system of logic
Authors: Wang Hao
Published: May 2006

The main purpose of this paper is to present a formal systemPin which we enjoy a smooth-running technique and which countenances a universe of classes which is symmetrical as between large and small. More exactly,Pis a system which differs from the inconsistent system of [1] only in the introduction of a rather natural new restrictive condition on the defining formulas of the elements (sets, membership-eligible classes). It will be proved that if the weaker system of [2] is consistent, thenPis also consistent.

After the discovery of paradoxes, it may be recalled, Russell and Zermelo in the same year proposed two different ways of safeguarding logic against contradictions (see [3], [4]). Since then various simplifications and refinements of these systems have been made. However, in the resulting systems of Zermelo set theory, generation of classes still tends to be laborious and uncertain; and in the systems of Russell's theory of types, complications in the matter of reduplication of classes and meaningfulness of formulas remain. In [2], Quine introduced a system which seems to be free from all these complications. But later it was found out that in it there appears to be an unavoidable difficulty connected with mathematical induction. Indeed, we encounter the curious situation that although we can prove in it the existence of a class V of all classes, and we can also prove particular existence theorems for each of infinitely many classes, nobody has so far contrived to prove in it that V is an infinite class or that there exists an infinite class at all.
Modality and quantification in S5
Authors: Prior A. N.
Published: May 2006

In the first of her papers on functional calculi based on strict implication, Ruth Barcan Marcus takes as her starting point the Lewis systems S2 and S4, supplemented by one of the normal bases for quantification theory, and by one special axiom for the mixture, asserting that if possibly something φ's then something possibly φ's. In the symbolism of Łukasiewicz, which will be used here, this axiom is expressible as CMΣxφxΣxMφx. In the present note I propose to show that if S5 had been taken as a startingpoint rather than S2 or S4, this formula need not have been laid down as an axiom but could have been deduced as a theorem.

It has been shown by Gödel that a system equivalent to S5 may be obtained if we add to any complete basis for the classical propositional calculus a pair of symbols for ‘Necessarily’ and ‘Possibly,’ which here will be ‘L’ and ‘M’; the axioms

the rule

RL: If α is a thesis, so is Lα;

and the definition

Df. M: M = NLN.
Reduction to a dyadic predicate
Authors: Quine W. V.
Published: May 2006

Consider any interpreted theory Θ, formulated in the notation of quantification theory (or lower predicate calculus) with interpreted predicate letters. It will be proved that Θ is translatable into a theory, likewise formulated in the notation of quantification theory, in which there is only one predicate letter, and it a dyadic one.

Let us assume a fragment of set theory, adequate to assure the existence, for all x and y without regard to logical type, of the set {x, y) whose members are x and y, and to assure the distinctness of x from {x, y} and {{x}}. ({x} is explained as {x, x}.) Let us construe the ordered pair x; y in Kuratowski's fashion, viz. as {{x}, {x, y}}, and then construe x;y;z as x;(y;z), and x;y;z;w as x;(y;z;w), and so on. Let us refer to w, w;w, w;w;w, etc. as 1w, 2w, 3w, etc.

Suppose the predicates of Θ are ‘F1’, ‘F2’, …, finite or infinite in number, and respectively d1-adic, d2-adic, …. Now let Θ′ be a theory whose notation consists of that of quantification theory with just the single dyadic predicate ‘F’, interpreted thus:

The universe of Θ′ is to comprise all objects of the universe of Θ and, in addition, {x, y) for every x and y in the universe of Θ′. (Of course the universe of Θ may happen already to comprise all this.)

Now I shall show how the familiar notations ‘x = y’, ‘x = {y, z}’, etc., and ultimately the desired ‘’, ‘’, etc. themselves can all be defined within Θ′.