Proposition Of System Research Articles

Reconstruction theorems in quantum mechanics

Given a physical system, one knows that there is a logical duality between its properties and its states. In this paper, we choose its states as the undefined notions of our axiomatic construction. In fact, by means of well-motivated assumptions expressed in terms of a transition probability function defined on the set of all pure states of the system, we construct a system of elementary propositions, i.e., a complete orthomodular atomic lattice satisfying the covering law. We also study in this framework the important notion of compatibility of propositions, and we define the superpositions and the mixtures of the states of the physical system.

A Strategy for Reformulating the Dialectical and Functional Theories of Conflict

This paper attempts to examine the assumptions and major propositions of Ralf Dahrendorf's and Lewis Coser's theories of conflict. Particular attention is drawn to the divergence in their respective schemes and how such divergence actually makes the schemes highly complementary. By formalizing the propositions of Dahrendorf and Coser, then examining how the theoretical statements of each inventory correct for omissions in the other, a strategy for synthesizing the propositions on the causes and form of is suggested. Such synthesis is presumed to improve upon previous attempts at theoretical reconciliation, since it is couched in propositional rather than assumptive terms. The growing disenchantment with structuralfunctional has been marked by the rise of alternative theoretical perspectives over the last two decades. One of the most conspicuous of these alternatives has been conflict theory which has presumably rediscovered for the discipline such phenomena as power, force, coercion, constraint, and change in social systems. Despite the excessive polemics which have often accompanied this rediscovery '(Dahrendorf, 1958b; Horowitz, 1962; Lockwood, 1956; Rex, 1961) there have been a number of impressive attempts at developing systems of theoretical statements on certain processes (e.g., Blalock, 1967; Coser, 1967a; Dahrendorf, 1957; Mack and Snyder, 1957; Williams, 1947). One of the drawbacks of this accumulated body of theoretical statements is that attempts at synthesizing, reconciling, and integrating them into a more adequate system of propositions have not been often undertaken. The result is that sets of propositions exist side by side in the literature yet in virtual isolation from one another. In this paper, I seek to begin redressing this oversight by examining the propositional inventories of two prominent schemes, the of Lewis Coser (1956; 1957; 1962; 1966; 1967a; 1967b; 1968; 1969a; 1969b) and the dialectical perspective of Ralf Dahrendorf (1958a; 1958b; 1957; 1961; 1967). The and dialectical schemes of these thinkers are singled out for examination because they are often presumed to be contradictory; and thus, if some tentative guidelines for synthesis can be suggested for these perspectives, then other schemes should be more readily reconciled-thereby allowing for the development of a more unified of processes. In this effort, I will draw attention to how Coser's and Dahrendorf's assumptions have been translated into highly suggestive propositions. Emphasis is placed upon the respective propositions of these two theorists, because it is in this form that: (a) the causal relations between concepts can be accurately discerned, (b) the points of compatibility and incompatibility between schemes can be readily visualized, and (c) the promise of operationalization and empirical investigation are greatest. Thus, I have chosen to focus on propositions because it is only when assumptions are translated into statements of covariance among explicitly stated variables that theoretical schemes become sufficiently clear to allow for tentative attempts at synthesis. Such a synthesis is, of course, only an uncertain first step in what will be a long process of converting theoretical schemes into propositional inventories, and then, critically examining them to see what they have to offer sociological theory. DIVERGENT ASSUMPTIONS In his efforts to direct sociological out of a functional utopia, Dahrendorf has reformulated in even more extreme form some of Marx's key assumptions: (1) social life is typified by opposed interests cohering around differ* This paper offers a revised version of propositions presented in an earlier work (Turner, 1974a) to suggest potential lines of synthesis for building a general of processes, a difficult task which was not undertaken in my earlier work.

The Political Economy of Public Organizations

The search for a theory of public administration often takes on aspects of a quest for Hold Grail or a hunt for mythical unicorn. Public administration theory has meant variously: a search for scientific principles; broad ruminations on what phenomena are included within the field; and general orientations of students of subject, both professional and academic.1 Seldom has theory referred to systematic, empirically based explanations of a phenomenon; a system of related and proven propositions that answer question Why? Though this article cannot begin to present such a theory, hopefully, it does more than issue another pious call for one. It is intended to set forth a framework with roots in organizational analysis that is simple but has enough heuristic power to make its application appealing to a wide range of students of public administration; that can pose questions for those areas still in need of exploration, and conceptually link them with those areas already well defined. A framework that can perform such an integrative role would represent a major step toward explanations of individuals, groups, or organizations behave as they do in that part of political system we have analytically abstracted and labeled public administration; and it would tell us something about how that behavior affects public policy. If we can better answer why questions,

The Power Theory of Takada Yasuma

1. The sociological, economic, and political theories of Takada Yasuma (1883-1972) are organized around an integrative focus and thus constitute a synthesis. While Takada was all-round social scientist, the synthesis is clearly centered on sociology and the integrative focus is his power theory. 2. Takada's power concept started from the idea of “the desire for force” in his early works such as The Division of Labor (1913) and Essays on Social Class (1922, but substantially writen around 1910 in most essays). He noted that the origin of this idea came from works of Ward, Ribot, McDougall and Simmel. However, the concept of “(social) force” is not neccessarily identical with that of “power” which designates the possibility of control over others.Thus his power concept per se was established in Social Class and the Third View of History (1925), whereas his systematic formulation of power theory was produced in the later stage, that is, in Power (1940). 3. The substance of Takada's power theory consists of a system of propositions : the law of correlation and transfer of different power elements, the law of accereration in the growth of power, the antagonistic relation between “official” and “wild” power, the law of levelling in the distribution of power, the law of cyclical change between centralization and decentralization of power, and so on. These propositions were not the aggregate of mere empirical generalizations, but were derived from the combination of his basic assumptions such as the desire for force, population increase, the trend of Vergesellschaf tung, etc. 4. The application of his power theory to the price determination and other parts of econcmics, known as “the economics of power theory”, was formulated in The New Lecture on Economics (5 Vols., 1929-1932). Observing the wide spread of unemployment at that time (the Great Depression of the world) he argued that the Walrasian system of general equilibrium equations does not explain this phenomenon. Unemployment, he insisted, can be explained when we see that the price of labor (wages) is not determined in terms of endogeneous variables in the equation system, but is determined from “outside.” That is to say that it is, apart from the equation system, determined by the combination of marginal productivity of labor on the demand side and power elements of workers and their organizations on the supply side. 5. Takada's power theory was formulated under the background of the development of sociological theories up to 1920s, especially Tarde, Wieser, and Max Weber. When he came to know, later, the new development of power theory of 1950s and 60s such as Lasswell, Parsons, Homans, BI-au and so on, he was too old to absorb them. But it must be emphasized that Takada's theory is not entirely heterogeneous with these recent development. 6. Takada's economics of power theory was not accepted by the main stream of economic theory. As the results of severe disputes with Nakayama, Kimura, and others he himself at last cut his power theory from the main body of his principles of economics after Treatise of Economics (1938). However, it seems that his idea of power theory is highly suggestive for the possible explanation of, for example, contemporary inflationary process based on the expansion revolution of the expectations of people as organized masses. Most of the economists have a strict tendency to distinguish the economic factors from the non-economic, and thus rejected Takada's hypothesis ; most of the sociologists other than Takada lacked sufficient interests and knowledges on economic analysis. So Takada could not get supporters and failed. But as an economic sociology we would have to reevaluate it as really an original and useful explanatory hypothesis concerning our contemporary economy.

The decision problem for mathematical structures of quantum theory

The purpose of this letter is to enunciate some results concerning the ~!ecision problem relative to a well-known mathematical structure of quantum theory. Such a structure, originally proposed by BIRKHOFF and yon NEU~ANN (1) and later developed by PI toN (3) and EMCH (3) together with JAVCH (4,5), is essentially the structure of the <( propositions ~ (*) on a given quantum system and turns out to be the same of a complete, atomic, orthocomplemented, weakly modular, satisfying-the-covering-law lattice ((( system of propositions ~, in the nomenclature introduced by PIRON). Let us denote with TQ~ the theory of the above systems of propositions ; we will prove that TQM is undecidable, in a sense that will be specified below. To begin with, it is useful to recall some definitions which are of common use in mathematical logic; they arc essentially due to TARSKI (6). The theories discussed in the following are all formalized within the lower predicate calculus, and they have the same logical constants (i.e. quantifiers, connectives, symbol of equivalence) and rules of inference. Let us call T anyone of these theories; we assume further tha t T is specified by its nonlogical constants (i.e. symbols of operations, and so on) and nonlogical axioms (i.e. relations). We suppose known to the reader the definitions of a formula, a sentence (i.e. a formula without free variables), a provable sentence (in terms of nonlogical constants and nonlogical axioms), together with the definitions of consistency, completeness, and finite axiomatizabili ty of a theory T. Moreover, we suppose also known the concept of a general recursive set.

Observables and the Field in Quantum Mechanics

Corresponding to any irreducible proposition system L in general quantum mechanics there is a division ring D with an anti-automorphism * and a vector space (V, D) over D with a definite sesquilinear form φ such that L is isomorphic to the set of φ closed subspaces of (V, D). The main task remaining in connecting the general quantum mechanics to the conventional quantum theory in a complex Hilbert space is to give physical arguments which force D to be the complex field. In this paper it is shown that if L admits a certain type of observable (together with other structure which seems to be physically justified), then D contains the real field as a subfield. Steps are then indicated that can be taken to move from the reals to the complexes or quaternions.

Boolean properties of observables in axiomatic quantum mechanics

The aim of this paper is to investigate those properties of observables in axiomatic quantum mechanics which are connected with properties of Boolean algebras, in particular, with direct limits and the amalgamation property. An axiom system for quantum mechanics, which is a modification of Mackey's axioms, is proposed and the logic of this system is interpreted as the Lindenbaum-Tarski algebra of some system of propositions. The theorem on compatible observables is discussed. A set of observables is defined to be Boolean representable iff it can be embedded into the set of observables on a Boolean algebra preserving all functional relationships between observables. It is shown that a set of observables is Boolean representable iff there exists a non-degenerate direct limit of the set of Boolean algebras which are ranges of these observables. The Boolean representability of the set of all observables is discussed and examples of sets of observables which are Boolean representable, but not compatible, are given. The relation between Boolean representability and hidden variables is discussed.

The Anatomy of Leviathan (review)

,178 HISTORY OF PHILOSOPHY in accord with Thomas on certain important points, but his own theology is more in the spirit of Plato and Augustine. Professor Kristeller's vast learning is at the service of admirably balanced conclusions . Not everyone will agree with all his interpretations; Ficino perhaps did not imagine himself to be "constructing a system of philosophy" (p. 96) since the very title of his work, Theologia platonica, suggests that he continues, with new materials, the labor of mediaeval theolo#ans. And Ficino's doctrine of the continuity of the supernatural life of the soul in this world and the next (p. 120) seems to be completely in accord with Thomas. But the author's very careful scholarly method and sound general conclusions are beyond criticism. Two hitherto unpublished Latin texts are printed in critical edition at the end of the French lecture, the Opus aureum in Thomistas by Baptists Mantuanus, which is anti-Thomas, and the Opusculum ad Laurentium Medicem on beatitude, by Vincentius Bandellus, which is pro-Thomas. Like Professor Kristeller's fine lecture, these new texts are a permanent contribution to scholarship. PAUL J. W. MILLER University of Colorado The Anatomy of Leviathan. By F. S. McNeilly. (New York: St. Martin's Press, 1968. Pp. 264. $8.50) The one thing about which Hobbes's commentators have agreed is the absence of any significant process of development in the several statements of his philosophic views. This has been most evident in their willingness to use the various texts as a common repository from which evidence for their interpretations could be drawn with little regard for its specific textual origin. Notoriously, no consensus has emerged from such endeavors and McNeilly believes this method of interpreting Hobbes far too indiscriminate. He contends that careful examination of the differing expressions and contexts of apparently recurrent arguments will show, in many cases, not the same argument repeated several times, but several different arguments. And these, properly discriminated , reveal a definite pattern of development culminating, in Leviathan, with a political theory substantially different from that of the earlier works. While the political theory of the Elements of Law and De Cive consists of a series of predictions founded on the known psychological tendencies of men, the theory of Leviathan, MeNeilly contends, is independent of any particular concept of human nature. Rather it is a "formal system of necessarily true propositions" (p. 213) constituting "a systematic analysis of value systems generally" (p. 245). Excepting two introductory chapters, the whole of McNeilly's argument is given over to substantiating this radical thesis, his point of departure being Hobbes's allegedly shifting views on method which are taken to vary somewhat inconsistently in the earlier works among self evidence, conventionalist and hypothetico-deductive theories and combinations of these. By contrast, in Leviathan, McNeilly argues, Hobbes expounds a conventionalist view with "clarity and persistence" (pp. 84-85). When this BOOK REVIEWS 479 conventionalism is combined with the adoption of mathematics as the paradigm of sound science, it leads Hobbes to the view that pohtical theory can be constructed as a formal system of necessary propositions because he believes that mathematics is about the world and extends this mistake to formal systems generally. These methodological commitments then, render any concept of human nature irrelevant to political theory. McNeilly's task in the second main section of the book is to show that in contrast to the earlier works, and contrary to all appearances, the Leviathan's account of human nature is completely neutral with regard to any possible set of motives and hence, wholly formal. Specifically he argues that materialism and mechanism play no active role in Hobbes's account of the passions and that the psychological egoism characteristic of the Elements and the fourth part of De Corpore is abandoned. (Throughout it is assumed that De Corpore was substantially complete before Leviathan was composed; this may not be true of the fourth part, as McNeilly recognizes, but he is undeterred.) The formal account of the passions in Leviathan then is formally compatible with either egoistic or nonegoistic interpretations of human motivation. Similarly, it is argued that Hobbes gives up his pessimism about men as...

Soil genesis, soil classification and soil survey

Mutual relationships exist between soil genesis, soil classification and soil survey. The first two subjects are dealt with separately. Some of the general lines in soil genetic investigation are traced. Next trends in soil classification are dealt with. The concepts of Knox (1965), with some additions, form the terminology of the paper. Soil classification systems are valid within certain boundary conditions. With many of the existing systems, little attention is given to this, although it is essential for judging and comparing. The method followed in compiling traditional classification systems is based on trial and error. By sampling of the universe according to the method of stratified sampling of (soil-)landscape bodies, a system suitable to soil survey can be compiled more quickly than by the method of simple random sampling. The system should be tested on the objective and by application in soil survey. The development of a system is very complex and usually takes on an iterative character. The lower and higher levels can be developed separately, but under a continual interplay. This interplay also exists with soil scientific knowledge. The lack of sharply defined and reproducible methods is striking. The broadly described numerical taxonomy is a reproducible method for parts of the process. However, this method has not yet been sufficiently tested. In spite of the precision of numerical taxonomy, a great many subjective choices have to be made. Only within the framework of these choices is this method reproducible. At the conclusion of the article, the connection between soil genesis, soil classification and soil survey is represented in a system of propositions and conclusions. These cover: (1) the connection between permanent soil characteristics, state factors of soil formation and genetic processes; (2) the existence of soil-landscape bodies; (3) the forming of a classification system from a compromise between the optimizing of classes of pedons, of soil-landscape bodies and of classes directed towards application; and (4) the recognition of delineated soil bodies on the basis of soil observations, landscape observations and the relevant theory. In the discussion, some ideas based on the relations outlined are given on soil research which is important to soil survey.

Complexity and Ambiguity: Some Observations on Professor Danto's "Complex Events"

This joint meeting of American and philosophers is the first of its kind, and for this first time we perhaps could not hope to reach a complete level of philosophical dialogue. Perhaps what we have had in its place is only an alternative succession of tentative positions. In my opinion, this is only natural, and we need not regret it. Each of us is already restricted in his thought by his basic presuppositions which we rarely have the opportunity to reflect upon or be more fully aware of. But in this regard, this philosophical meeting is a challenge of alternative possibilities: several problematical positions are presented on each side. It is such confrontation and dialogue that we surely need. My task in this paper cannot be to examine the philosophy of Professor Danto in its entirety. It would be enough if I can make clear his initial system of propositions by way of contrast, and by the same means, if I can express my own set of philosophical judgments, at least, if I dare say, as an instance of French philosophy. In his Philosophical Investigations, Wittgenstein wrote:

Duhem, Quine and a New Empiricism

As in the case of great books in all branches of philosophy, Pierre Duhem's Le Théorie Physique, first published in 1906, can be looked to as the progenitor of many different and even conflicting currents in subsequent philosophy of science. On a superficial reading, it seems to be an expression of what later came to be called deductivist and instrumentalist analyses of scientific theory. Duhem's very definition of physical theory, put forward early in the book, is the quintessence of instrumentalism:A physical theory is not an explanation. It is a system of mathematical propositions, deduced from a small number of principles, which aim to represent as simply, as completely, and as exactly as possible a set of experimental laws [p. 19].

Duhem, Quine and a New Empiricism

As in the case of great books in all branches of philosophy, Pierre Duhem's Le Théorie Physique, first published in 1906, can be looked to as the progenitor of many different and even conflicting currents in subsequent philosophy of science. On a superficial reading, it seems to be an expression of what later came to be called deductivist and instrumentalist analyses of scientific theory. Duhem's very definition of physical theory, put forward early in the book, is the quintessence of instrumentalism:A physical theory is not an explanation. It is a system of mathematical propositions, deduced from a small number of principles, which aim to represent as simply, as completely, and as exactly as possible a set of experimental laws [p. 19].

Coordinate and Momentum Observables in Axiomatic Quantum Mechanics

An axiomatic model for quantum mechanics is formulated using physically significant axioms. The model contains a slight strengthening of Mackey's first six axioms, together with two axioms which ensure the existence of coordinate and momentum observables. The symmetries or rigid motions are an essential part of the structure, and a link is constructed between these and the quantum proposition system. Coordinate and momentum observables are defined in terms of abstract coordinate systems and one-parameter groups of motions. It is then shown that as far as the statistical properties of these observables in certain canonical states are concerned, the abstract model may be represented by the usual Hilbert space formulation. Spectral properties of σ-homomorphisms are also investigated. Also included are two appendices of a technical nature: the first considers one-parameter groups of derivables, and the second absolutely continuous σ-homomorphisms.

Conflict and Criminality

Discussions of crime and conflict have been characterized by imprecision in defining and relating the two concepts. This has contributed to confusion about the subject matter and research aims of criminology, to a tendency to introduce new theoretical formulations before older alternatives have been adequately explored, and to an investment of resources in piecemeal instead of cumulative research. Since formalization is an aid to the clearer statement and more thorough exploration of theoretical alternatives, an effort is made to develop a system of formal propositions as a basis for research. This paper includes (1) a summary of conceptualizations found in the literature, (2) discussions and suggested resolutions of the basic conceptual issues, and (3) a presentation of the theoretical scheme as it has so far been developed, i.e., of a set of propositions stating conditions under which the probability of conflict between legal authorities and subjects varies and, given a conflict, conditions under which the probability of criminalization of subjects varies. It is concluded that further elaboration of propositions requires additional exploratory and descriptive research, and that rigorous formalization would be premature until the various hypothesized relationships have been more thoroughly examined in empirical research.

The Theory of Assertoric Consequences in Albert of Saxony

THE THEORY OF ASSERTORIC CONSEQUENCES IN ALBERT OF SAXONY* CHAPTER III QUANTIFICA TION In the preceding chapter we examined Albert's theory of consequences among unanalyzed propositions, and we shall be concerned in the three following chapters with Albert's exposition of consequences among assertoric analyzed propositions. The present chapter will deal with the analysis of propositions in regard to their quality, quantity, opposition, and ampliation. The fourth chapter will study consequences from one analyzed assertoric proposition to another. These analyzed assertoric propositions are called 'quantified propositions'. The fifth chapter will study consequences from two quantified propositions to a third. These three chapters might be called the theory of the categorical assertoric syllogism ; the first two chapters being the theory of quantity of propositions, and the third may be regarded properly as the analyzation of the syllogistic. In our times some modern logicians have remarked that there are several errors in the so-called traditional theory of the syllogism; for instance, the contention of the traditional position that a particular proposition follows from a universal. The following examples are given to prove that such a contention leads to errors, and hence, the theory is inconsistent in some aspects. 'AU my dimes are shiny, therefore some of my dimes are shiny', 'AU centaurs are Greek, therefore some centaurs are Greek' ; If there are no centaurs, and I have no dime, then, the universal propositions are true, but the particulars are false; and, so, the inferences are invalid. If round squares are impossible, then, the following universal is true: 'AU round squares are round,' but the following particular is false: 'Some round squares are round;' consequently, to infer the last proposition from the first is invalid. Sometimes the universal proposition will be true whether there are, or there are not objects such as the propositions indicates ; but if there are not, then the partieSee Francican Studies 18 (1958) pp. 290—354. 13 14A.GONZÁLEZ ular wiU be false; and, therefore, the inference from the former to the latter is invalid, as the following examples show : 'All golden mountains are mountains, therefore some golden mountains are mountains/ 'AU Pegasi are winged horses, therefore some Pegasus is a winged horse,' 'AU brakeless trains are dangerous, therefore some brakeless train is dangerous,' 'AU dragons breathe flame, therefore some dragon breathes flame.' The preceding and similar instances prove that if V is the subject of a universal proposition, and 'x' is an empty term or null class; then, the universal having 'x' as subject is true, but the particular with 'x' as subject is false. Hence, the inference from universal to particular is invalid. Furthermore, all the laws of the logical square, except those of contradiction, are invalid; the same holds for the laws of conversion per accidens, and for all syUogisms with two universal premises and a particular conclusion.1 Lukasiewicz has proven in an historical and systematic analysis that Aristotle's theory of the categorical syUogism, is free from the errors that have been imputed to it regarding the existential import of propositions. The Aristotehan theory is exceedingly exact, although it has a narrow frame. The categorical syllogism is a deductive system of true propositions or theses concerning the constants: A (every), E (none), I (some), O (some-not). Almost aU theses of the Aristotelian doctrine are implications, or 'if . . . then' propositions. The variables of the system are term-variables, the values of which are exclusively positive, universal , non-empty terms. Thus, singular, negative, and empty terms are not introduced into the theory; which is the case also with the quantifiers . Another great and fundamental shortcoming of Aristotle's system is the supposition that all problems can be expressed by the four types of syUogistic premises: viz., universal affirmative, particular affirmative , universal negative, and particular negative; and that the syUogism is the only instrument of proof. The logic of propositions is absent from the system, although some of its laws are used intuitively. The important invention of the propositional calculus was due to the Stoics.2 After an historical and textual analysis of Aristotle's doctrine, Lukasiewicz builds up a system of the Aristotelian non-modal categorical syUogism according to the requirements of...

The Secured Transactions Article of the Commercial Code

Article 9, Secured Transactions, of the Uniform Commercial Codet covers a great deal of territory. To use a jargon which was fashionable not long ago, it proposes to integrate, under a single system of legal propositions and a single system of terminology, the entire range of transactions in which money debts are secured by personal property. The organizing concept which underlies Article 9 is that security transactions should be differentiated with reference to the status of the person whose property secures the debt and the kind of property put up as collateral-thus: consumer goods;2 farm products;3 and property used in business enterprises, which is subdivided according to whether it is intended for use in the business (equipment)4 or for sale (inventory)5 or represents the proceeds of the sale of inventory (accounts receivable,6 paper,7 etc.). Armed with this battery of definitions, the draftsmen have found themselves able to make necessary distinctions between the unlike elements of different security transactions and at the same time to formulate a single set of rules to cover the very large area of likeness common to all such transactions. The negative side of this key concept is the proposal to abandon-almost completely, so far as terminology is concerned, and to varying degrees so far as substantive law is concerned-the traditional differentiation of security transactions according to the form of the security agreement, such as chattel mortgage, conditional sale, trust receipt, and

Public Opinion and Public Opinion Polling

T HIS paper presents some observations on and on polling as currently performed. It is hoped that these observations will provoke the discussion for which, I understand, this meeting has been arranged. The observations are not along the line of what seems to be the chief preoccupation of students of polling, to wit, the internal improvement of their technique. Instead, the -observations are designed to invite attention to whether polling actually deals with opinion. The first observations which I wish to make are in the nature of a prelude. They come from a mere logical scrutiny of polling as an alleged form of scientific investigation. What I note is the inability of polling to isolate public opinion as an abstract or generic concept which could thereby become the focal point for the formation of a system of propositions. It would seem needless to point out that in an avowed scientific enterprise seeking to study a class of empirical items and to develop a series of generalizations about that class it is necessary to identify the class. Such identification enables discrimination between the instances which fall within the class and those which do not. In this manner, the generic character of the object of study becomes delineated. When the generic object of study is distinguishable, it becomes possible to focus study on that object and thus to learn progressively more about that object. In this way the ground is prepared for cumulative generalizations or propositions relative to the generic object of investigation. As far as I can judge, the current study of by polling ignores the simple logical point which has just been made. This

Economic Analysis and Agricultural Policy

By “agricultural policy” I mean the pattern of those acts of government, especially of national governments, which specifically concern the agricultural segment of society. The discussion of agricultural policy, like that of any other social problem, involves all the social sciences, perhaps one should say all the sciences. Economic analysis, dealing as it does with a limited range of abstractions, cannot pretend to give an “answer” to any social problem, agricultural policy included. Sociological, political, legal, historical, psychological, technical considerations obviously enter into the discussion. Nevertheless it may be useful to ask what are the insights yielded by economic analysis, as a discipline of thought and as a system of related propositions, in approaching a problem of this degree of generality. The case for economics, or for any other discipline of thought, is the case for specialization. The problems involved in the study of even so relatively simple a problem as agricultural policy are beyond the wit of any single mind to explore fully. To appreciate the problems of agriculture fully one would need to be an economist, sociologist, political scientist, social psychologist, biologist, chemist, physicist, botanist, zoologist, and much more; and in these days no one man can hope to be expert in all fields. The only way in which we can even begin to encompass any problem is by specialization; but, according to what is almost the first proposition in economics, specialization is useless without exchange. The aim of this paper is a little intellectual trade, a small offering of an economist to other disciplines, in the hope perhaps of some enlightenment in return.

The New Logic

The rapid development of physics, the result of observations made and ideas introduced within the last few decades, has brought about a change in the whole system of physical concepts. This fact is common knowledge, and has already attracted the attention of philosophers. It is less well known that geometry too has had its crises, and undergone a reconstruction. For centuries, so-called “geometrical intuition” was used as a method of proof. In geometrical demonstrations, certain steps were allowed because they were “self-evident,” because the correctness of the conclusion was “shown by the adjoined diagram,” etc. A crisis occurred in geometry because such intuition proved to be untrustworthy. Many of the propositions regarded as self-evident or based upon the consideration of diagrams turned out to be false. And so Euclidean geometry was reconstructed by methods free from all intuitive elements and strictly logical in nature. Moreover, for more than a century, various other geometries have been devised purely as logical constructions. Since they start with assumptions different from those of Euclid, and lead to conclusions partly in contradiction with his theorems, they are called “non-Euclidean.” Nevertheless, each one of these geometries is a closed system of propositions exempt from contradiction. Recently, some of them have even found application in physics.

On the Foundations of Mathematics

Mathematics.-The notion within a given domain of defining the objects of consideration rather by a body of properties than by particular expressions or intuitions is as old a5 mathematics itself. And yet the central importance of the notion appeared only during the last century-in a host of researches on special theories and on the foundations of geometry and analysis. Thus has arisen the general point of view of what may be called abstract mathematics. One comes in touch with the literature very conveniently by the mediation of Peano's Revue des Mathematiques. The Italian school of Peano and the Formulaire Mathematique, published in connection with the Revue, * Presidential address delivered before The American Mathematical Society at its ninth annual meeting, December 29, 1902. This content downloaded from 50.193.31.206 on Sat, 21 Dec 2013 10:21:01 AM All use subject to JSTOR Terms and Conditions [N. S. VOL. XVII. No. 428. are devoted to the codification in Peano's symbolic language of the principal mathematical theories, and to researches on abstract mathematics. General interest in abstract mathematics was aroused by Hil-mathematics was aroused by Hilbert's Gauss-Weber Festschrift of 1899: 'Ueber die Grundlagen der Geometrie,' a memoir rich in results and suggestive in methods; I refer to the reviews by Sommer,* Poincare,t Halsted,j Hedrick? and Veblen. 11 We have as a basal science logic, and as depending upon it the special deductive sciences which involve undefined symbols and whose propositions are not all capable of proof. The symbols denote either classes of 'elements or relations amongst elements. In any such science one may choose in various ways the system of undefined symbols and the system of undemonstrated or primitive propositions, or postulates. Every proposition follows from the postulates by a finite number of logical steps. A careful statement of the fundamental generalities is given by Padoa in a paperT? before the Paris Congress of Philosophy, 1900. Having in mind a definite system of undefined symbols and a definite system of postulates, we have first of all the notion of the compatibility of these postulates; that is, that it is impossible to prove by a finite number of logical steps the simultaneous validity of a statement and its contradictory statement; in the next place, the question of the independence of the postu* Bull. Amer. Math. Soc. (2), vol. 6 (1900), p. 287. t Bull. Sciences Matherm., vol. 26 (1902), p. 249. t The Open Court, September, 1902. ?Bull. Amer. Math. Soc. (2), vol. 9 (1902), p. 158. 11 The Monist, January, 1903. [ 'Essai d'une theorie algebrique des nombres entiers, priced6 d'une Introduction logique a une thdorie deductive quelconque,' Bibliotheque du Congres International de Philosophic, vol. 3, p. 309. lates or the irreducibility of the system of postulates; that is, that no postulate is provable from the remaining postulates. Padoa introduces the notion of the irreducibility of the system of undefined symbols. A system of undefined symbols is said to be reducible if for one of the symbols, X, it is possible to establish, as a logical consequence of the assumption of the validity of the postulates, a nominal or symbolic definition of the form X A, where in the expression A there enter only the undefined symbols distinct from X. For the purpose of practical application, it seems to be desirable to modify the definition so as to call the system -of undefined symbols reducible if there is a nominal definition X -A of one of them X in terms of the others such that in any interpretation of the science the postulates retain their validity when instead of the initial interpretation of the symbol X there is placed the interpretation A of that symbol. If the system of symbols is reducible in the sense of the original definition it is in the sense' of the new definition, but not necessarily conversely, as appears for instance from the following example, occur-ring in the foundations of geometry. Hilbert uses the following undefined symbols: 'point, ' line,' 'plane,' ' incidence' of point and line, 'incidence' of point and plane, 'between,' and 'congruent.' Now it is possible to give for the symbol 'plane' a symbolic definition in terms of the other, undefined symbols-for instance, a plane is a certain class of points (as Peano showed in 1892), or again, a plane is a certain class of lines; while the notion 'incidence' of point and plane receives convenient definition. It is apparent from the fact that these definitions may be given in these two ways that Hilbert's system of undefined symbols is not in Padoa 's sense irreducible, at least, in so far as the symbols 'plane,' 'incidence' of point and plane are 402 SCIENCE. This content downloaded from 50.193.31.206 on Sat, 21 Dec 2013 10:21:01 AM All use subject to JSTOR Terms and Conditions