Les débats menés autour de l’État africain, ces trente dernières années, ont reflété les grandes idées qui ont agité le monde contemporain, mais ils sont, dans l’ensemble, restés muets sur des questions soulevées au cours des deux décennies des Nations Unies pour la femme. Or une approche de genre des questions du politique s’avère indispensable sur bien des plans. Elle aide désormais à mieux saisir la manière dont l’inégalité entre les sexes est instituée, prolongée, voire renforcée au cœur du politique. (...) Cet article vise à montrer l’importance des travaux sur le genre, l’État et le politique et la logique dont ils procèdent. Il vise également à faire comprendre en quoi consiste l’interrogation sur le sexe de l’État en Afrique, sur la dimension masculine et patriarcale de l’État et son impact sur les politiques, sur les femmes et sur les rapports et les rôles sociaux des sexes. On ne saurait tendre vers les objectifs d’égalité dans le développement en Afrique sans reconsidérer le sexe de l’État et de ses politiques. (shrink)

Les débats menés autour de l’État africain, ces trente dernières années, ont reflété les grandes idées qui ont agité le monde contemporain, mais ils sont, dans l’ensemble, restés muets sur des questions soulevées au cours des deux décennies des Nations Unies pour la femme. Or une approche de genre des questions du politique s’avère indispensable sur bien des plans. Elle aide désormais à mieux saisir la manière dont l’inégalité entre les sexes est instituée, prolongée, voire renforcée au cœur du politique. (...) Cet article vise à montrer l’importance des travaux sur le genre, l’État et le politique et la logique dont ils procèdent. Il vise également à faire comprendre en quoi consiste l’interrogation sur le sexe de l’État en Afrique, sur la dimension masculine et patriarcale de l’État et son impact sur les politiques, sur les femmes et sur les rapports et les rôles sociaux des sexes. On ne saurait tendre vers les objectifs d’égalité dans le développement en Afrique sans reconsidérer le sexe de l’État et de ses politiques. (shrink)

Background Ensuring individual free and informed decision‐making for research participation is challenging. It is thought that preliminarily informing communities through ‘community sensitization’ procedures may improve individual decision‐making. This study set out to assess the relevance of community sensitization for individual decision‐making in research participation in rural Gambia. Methods This anthropological mixed‐methods study triangulated qualitative methods and quantitative survey methods in the context of an observational study and a clinical trial on malaria carried out by the Medical Research Council Unit Gambia. (...) Results/discussion Although 38.7% of the respondents were present during sensitization sessions, 91.1% of the respondents were inclined to participate in the trial when surveyed after the sensitization and prior to the informed consent process. This difference can be explained by the informal transmission of information within the community after the community sensitization, expectations such as the benefits of participation based on previous research experiences, and the positive reputation of the research institute. Commonly mentioned barriers to participation were blood sampling and the potential disapproval of the household head. Conclusion Community sensitization is effective in providing first‐hand, reliable information to communities as the information is cascaded to those who could not attend the sessions. However, further research is needed to assess how the informal spread of information further shapes people's expectations, how the process engages with existing social relations and hierarchies (e.g. local political power structures; permissions of heads of households) and how this influences or changes individual consent. (shrink)

A layman's guide to the mechanics of Gödel's proof together with a lucid discussion of the issues which it raises. Includes an essay discussing the significance of Gödel's work in the light of Wittgenstein's criticisms.

This paper critically engages Philip Mirowki's essay, "The scientific dimensions of social knowledge and their distant echoes in 20th-century American philosophy of science." It argues that although the cold war context of anti-democratic elitism best suited for making decisions about engaging in nuclear war may seem to be politically and ideologically motivated, in fact we need to carefully consider the arguments underlying the new rational choice based political philosophies of the post-WWII era typified by Arrow's impossibility theorem. A distrust (...) of democratic decision-making principles may be developed by social scientists whose leanings may be toward the left or right side of the spectrum of political practices. (shrink)

In 1931 the mathematical logician Kurt Godel published a revolutionary paper that challenged certain basic assumptions underpinning mathematics and logic. A colleague of Albert Einstein, his theorem proved that mathematics was partly based on propositions not provable within the mathematical system and had radical implications that have echoed throughout many fields. A gripping combination of science and accessibility, Godel’s Proof by Nagel and Newman is for both mathematicians and the idly curious, offering those with a taste for logic and (...) philosophy the chance to satisfy their intellectual curiosity. (shrink)

Arrow’s axiomatic foundation of social choice theory can be understood as an application of Tarski’s methodology of the deductive sciences—which is closely related to the latter’s foundational contribution to model theory. In this note we show in a model-theoretic framework how Arrow’s use of von Neumann and Morgenstern’s concept of winning coalitions allows to exploit the algebraic structures involved in preference aggregation; this approach entails an alternative indirect ultrafilter proof for Arrow’s dictatorship result. This link also connects Arrow’s seminal result (...) to key developments and concepts in the history of model theory, notably ultraproducts and preservation results. (shrink)

A new approach to quantum theory is proposed in this paper. The basis is taken to be theoretical variables, variables that may be accessible or inaccessible, i.e., it may be possible or impossible for an observer to assign arbitrarily sharp numerical values to them. In an epistemic process, the accessible variables are just ideal observations connected to an observer or to some communicating observers. Group actions are defined on these variables, and group representation theory is the basis for developing the (...) Hilbert space formalism here. Operators corresponding to accessible theoretical variables are derived, and in the discrete case, it is proved that the possible physical values are the eigenvalues of these operators. The focus of the paper is some mathematical theorems paving the ground for the proposed foundation of quantum theory. It is shown here that the groups and transformations needed in this approach can be constructed explicitly in the case where the accessible variables are finite-dimensional. This simplifies the theory considerably: To reproduce the Hilbert space formulation, it is enough to assume the existence of two complementary variables. The interpretation inferred from the proposed foundation here may be called a general epistemic interpretation of quantum theory. A special case of this interpretation is QBism; it also has a relationship to several other interpretations. (shrink)

Halin in 1965 proved that if a graph has [Formula: see text] many pairwise disjoint rays for each [Formula: see text] then it has infinitely many pairwise disjoint rays. We analyze the complexity of this and other similar results in terms of computable and proof theoretic complexity. The statement of Halin’s theorem and the construction proving it seem very much like standard versions of compactness arguments such as König’s Lemma. Those results, while not computable, are relatively simple. They only (...) use arithmetic procedures or, equivalently, finitely many iterations of the Turing jump. We show that several Halin-type theorems are much more complicated. They are among the theorems of hyperarithmetic analysis. Such theorems imply the ability to iterate the Turing jump along any computable well ordering. Several important logical principles in this class have been extensively studied beginning with work of Kreisel, H. Friedman, Steel and others in the 1960s and 1970s. Until now, only one purely mathematical example was known. Our work provides many more and so answers Question 30 of Montalbán’s Open Questions in Reverse Mathematics in 2011. Some of these theorems including ones in Halin in 1965 are also shown to have unusual proof theoretic strength as well. (shrink)

Northern Ireland physicist John Stewart Bell's possible understanding of quantum theory.

This short book has two main purposes. The first is to explain Kurt Gödel's first and second incompleteness theorems in informal terms accessible to a layperson, or at least a non-logician. The author claims that, to follow this part of the book, a reader need only be familiar with the mathematics taught in secondary school. I am not sure if this is sufficient. A grasp of the incompleteness theorems, even at the level of ‘the big picture’, might require some experience (...) with the rigor of mathematical proof. Moreover, since the incompleteness theorems concern formal deductive systems, it would help for a potential reader of this book to have some familiarity with at least elementary logic.There are, of course, a number of informal expositions of the incompleteness theorems. I do not see the need to engage in a comparative study. The second, and more important, goal of the present book is to discuss some alleged consequences of the incompleteness theorems and, in particular, to debunk thoroughly claims in philosophy, religion, and literary criticism that are supposed to be established, or at least bolstered, by incompleteness. The execution of this part of the book is masterly. The arguments are clear and compelling.The book has eight chapters, each between eight and twenty-eight pages. The opening chapter gives a very brief account of the incompleteness theorems and a sketch of Gödel's life and work. Chapter 2 gives the main overview of the incompleteness theorems, plus …. (shrink)

The main theorem of this paper is: If ƒ is a partial function from ℵ 1 to ℵ 1 which is ∑ 1 1 -bounded, then there is a weakly finite primitive recursive dilator D such that for all infinite αϵdom , ƒ ⩽ D . The proof involves only elementary combinatorial constructions of trees. A generalization to ptykes is also given.

A set S of degrees is said to be an initial segment if c ≤ d ∈ S→-c∈S. Shoenfield has shown that if P is the lattice of all subsets of a finite set then there is an initial segment of degrees isomorphic to P. Rosenstein [2] (independently) proved the same to hold of the lattice of all finite subsets of a countable set. We shall show that “countable set” may be replaced by “set of cardinality at most that of (...) the continuum.” This result is also an extension of [3, Corollary 2 to Theorem 15], which states that there is a sublattice of degrees isomorphic to the lattice of all finite subsets of 2N. (A sublattice of degrees is a subset closed under ∪ and ∩; an initial segment closed under ∪ is necessarily a sublattice, but not conversely.)It seems worth noting that the proof of the present result was preceded in time by our proof [4] of the analogous theorem for hyperdegrees, and is in fact an adaptation of that proof. Thus the present work has been influenced much more directly by the Gandy-Sacks forcing construction of a minimal hyperdegree [1] than by previous work on initial segments of degrees. (shrink)

This is a critical analysis of the first part of Go¨del’s 1951 Gibbs lecture on certain philosophical consequences of the incompleteness theorems. Go¨del’s discussion is framed in terms of a distinction between objective mathematics and subjective mathematics, according to which the former consists of the truths of mathematics in an absolute sense, and the latter consists of all humanly demonstrable truths. The question is whether these coincide; if they do, no formal axiomatic system (or Turing machine) can comprehend the mathematizing (...) potentialities of human thought, and, if not, there are absolutely unsolvable mathematical problems of diophantine form. (shrink)

In a fluent, clear, and lively style this translation by two of Maslov's junior colleagues brings the work of the late Soviet scientist S. Yu. Maslov to a wider audience. Maslov was considered by his peers to be a man of genius who was making fundamental contributions in the fields of automatic theorem proving and computational logic. He published little, and those few papers were regarded as notoriously difficult. This book, however, was written for a broad audience of readers (...) and describes elegant examples of applications in such fields as computer science, artificial intelligence, operations research, economic modeling, and biological modeling, among others. The book also brings to light the work by the American mathematician E. L. Post, which inspired Maslov's own work in the development of a general theory and which has been long neglected by mathematicial logicians and systems theorists in the United States. The book's first chapter introduces the Rules of the Game. Part I, Mathematics of Calculi, covers E. L. Post's canonical systems, deductive systems and algorithms, and probabilistic calculi and deductive information. Part II, Horizonal Modeling, takes up a "toy" economy, the calculi of technological possibilities, and the development of rules. Part III, Vertical Modeling, deals with the topics of "to fight and to search" and the consequences of the asymmetry of cognitive mechanisms. Vladimir Lifschitz is affiliated with the Department of Computer Science at Stanford University, and Michael Gelfond with the Department of Electrical Engineering and Computer Science at the University of Texas, El Paso. Theory of Deductive Systems and Its Applicationsis included in the Foundation of Computing Series, edited by Michael Garey. (shrink)

This book discusses how rational choice theory grew out of RAND's work for the US Air Force. It concentrates on the work of William J. Riker, Kenneth J. Arrow, James M. Buchanan, Russel Hardin, and John Rawls. It argues that within the context of the US Cold War with its intensive anti-communist and anti-collectivist sentiment, the foundations of capitalist democracy were grounded in the hyper individualist theory of non-cooperative games.

By indirect-deduction theorems introduced in the present paper we mean the theorems that allow to formalize indirect reasonings occurring in deductive practice in general and in mathematics in particular. We discuss the relationship between the introduced theorems and some logical calculi being virtually confined to propositional calculi with implication and negation. It is worth to notice that the above theorems are very handy and effective in proving logical theses.

Charles Chihara's new book develops and defends a structural view of the nature of mathematics, and uses it to explain a number of striking features of mathematics that have puzzled philosophers for centuries. The view is used to show that, in order to understand how mathematical systems are applied in science and everyday life, it is not necessary to assume that its theorems either presuppose mathematical objects or are even true. Chihara builds upon his previous work, in which he presented (...) a new system of mathematics, the constructibility theory, which did not make reference to, or presuppose, mathematical objects. Now he develops the project further by analysing mathematical systems currently used by scientists to show how such systems are compatible with this nominalistic outlook. He advances several new ways of undermining the heavily discussed indispensability argument for the existence of mathematical objects made famous by Willard Quine and Hilary Putnam. And Chihara presents a rationale for the nominalistic outlook that is quite different from those generally put forward, which he maintains have led to serious misunderstandings.A Structural Account of Mathematics will be required reading for anyone working in this field. (shrink)

It is demonstrated that a non Abelian Stokes Theorem is necessary to describe the B3 field of radiation. A simple form of the theorem is build up from the fundamental definition of B3 in O(3) gauge field theory, which is a gauge field theory applied to electrodynamics with an O(3) internal gauge symmetry bases on a complex basis ((1), (2), (3)). The indices (1) and (2) are complex conjugate pairs based on circular polarization, and the index (3) is (...) aligned with the propagation axis of radiation. (shrink)

The aim of this paper is to distinguish between, and examine, three issues surrounding Humphreys's paradox and interpretation of conditional propensities. The first issue involves the controversy over the interpretation of inverse conditional propensities — conditional propensities in which the conditioned event occurs before the conditioning event. The second issue is the consistency of the dispositional nature of the propensity interpretation and the inversion theorems of the probability calculus, where an inversion theorem is any theorem of probability that (...) makes explicit (or implicit) appeal to a conditional probability and its corresponding inverse conditional probability. The third issue concerns the relationship between the notion of stochastic independence which is supported by the propensity interpretation, and various notions of causal independence. In examining each of these issues, it is argued that the dispositional character of the propensity interpretation provides a consistent and useful interpretation of the probability calculus. (shrink)

Intuitionistic logic is presented here as part of familiar classical logic which allows mechanical extraction of programs from proofs. to make the material more accessible, basic techniques are presented first for propositional logic; Part II contains extensions to predicate logic. This material provides an introduction and a safe background for reading research literature in logic and computer science as well as advanced monographs. Readers are assumed to be familiar with basic notions of first order logic. One device for making this (...) book short was inventing new proofs of several theorems. The presentation is based on natural deduction. The topics include programming interpretation of intuitionistic logic by simply typed lambda-calculus (Curry-Howard isomorphism), negative translation of classical into intuitionistic logic, normalization of natural deductions, applications to category theory, Kripke models, algebraic and topological semantics, proof-search methods, interpolation theorem. The text developed from materal for several courses taught at Stanford University in 1992-1999. (shrink)

Using recent results in topos theory, two systems of higher-order logic are shown to be complete with respect to sheaf models over topological spaces- so -called "topological semantics." The first is classical higher-order logic, with relational quantification of finitely high type; the second system is a predicative fragment thereof with quantification over functions between types, but not over arbitrary relations. The second theorem applies to intuitionistic as well as classical logic.

In this paper we review some properties of fuzzy observables, mainly as realized by commutative positive operator valued measures. In this context we discuss two representation theorems for commutative positive operator valued measures in terms of projection valued measures and describe, in some detail, the general notion of fuzzification. We also make some related observations on joint measurements.

Using recent results in topos theory, two systems of higher-order logic are shown to be complete with respect to sheaf models over topological spaces-so-called "topological semantics". The first is classical higher-order logic, with relational quantification of finitely high type; the second system is a predicative fragment thereof with quantification over functions between types, but not over arbitrary relations. The second theorem applies to intuitionistic as well as classical logic.

We prove the following three theorems on the enumeration degrees of ∑20 sets. Theorem A: There exists a nonzero noncuppable ∑20 enumeration degree. Theorem B: Every nonzero Δ20enumeration degree is cuppable to 0′e by an incomplete total enumeration degree. Theorem C: There exists a nonzero low Δ20 enumeration degree with the anticupping property.

Some sufficient conditions on a Symmetric Monoidal Closed category K are obtained such that a diagram in a free SMC category generated by the set A of atoms commutes if and only if all its interpretations in K are commutative. In particular, the category of vector spaces on any field satisfies these conditions . Instead of diagrams, pairs of derivations in Intuitionistic Multiplicative Linear logic can be considered . Two derivations of the same sequent are equivalent if and only if (...) all their interpretations in K are equal. In fact, the assignment of values to atoms is defined constructively for each pair of derivations. Taking into account a mistake in R. Voreadou's proof of the “abstract coherence theorem” found by the author, it was necessary to modify her description of the class of non-commutative diagrams in SMC categories; our proof of S. Mac Lane conjecture also proves the correctness of the modified description. (shrink)

Two postulates concerning observables on a quantum logic are formulated. By Postulate 1 compatibility of observables is defined by the strong topology on the set of observables. Postulate 2 requires that the range of the sum of observables ought to be contained in the smallestC-closed sublogic generated by their ranges. It is shown that the Hilbert space logicL(H) satisfies the two postulates. A theorem on the connection between joint distributions of types 1 and 2 on the logic satisfying Postulate (...) 2 is proved. (shrink)

In [10] Friedman showed that is a conservative extension of <ε0for-sentences wherei= min, i.e.,i= 2, 3, 4 forn= 0, 1, 2 +m. Feferman [5], [7] and Tait [11], [12] reobtained this result forn= 0, 1 and even with instead of. Feferman and Sieg established in [9] the conservativeness of over <ε0for-sentences for alln. In each paper, different methods of proof have been used. In particular, Feferman and Sieg showed how to apply familiar proof-theoretical techniques by passing through languages with Skolem (...) functionals.In this paper we study the same choice principles in the presence of theBar Rule, which permits one to infer the scheme of transfinite induction on a primitive recursive relation ≺ when it has been proved that ≺ is wellfounded. The main result characterizes + as a conservative extension of a system of the autonomously iterated-comprehension axiom for-sentences. Forn= 0 this has been proved by Feferman in the form that + is a conservative extension of <Γ0; this was first done in [8] by use of the Gödel functional interpretation for the stronger systemZω+μ+ + and then more recently by the simpler methods of [9]. Jäger showed how the latter methods could also be used to obtain the general result of Theorem 1 below. (shrink)

Varieties of the Fan Theorem have recently been developed in reverse constructive mathematics, corresponding to different continuity principles. They form a natural implicational hierarchy. Some of the implications have been shown to be strict, others strict in a weak context, and yet others not at all, using disparate techniques. Here we present a family of related Kripke models which separates all of the as yet identified fan theorems.

Theorems of hyperarithmetic analysis occupy an unusual neighborhood in the realms of reverse mathematics and recursion-theoretic complexity. They lie above all the fixed iterations of the Turing jump but below ATR $_{0}$. There is a long history of proof-theoretic principles which are THAs. Until the papers reported on in this communication, there was only one mathematical example. Barnes, Goh, and Shore [1] analyze an array of ubiquity theorems in graph theory descended from Halin’s [9] work on rays in graphs. They (...) seem to be typical applications of ACA $_{0}$ but are actually THAs. These results answer Question 30 of Montalbán’s Open Questions in Reverse Mathematics [19] and supply several other natural principles of different and unusual levels of complexity.This work led in [25] to a new neighborhood of the reverse mathematical zoo: almost theorems of hyperarithmetic analysis. When combined with ACA $_{0}$ they are THAs but on their own are very weak. Denizens both mathematical and logical are provided. Generalizations of several conservativity classes are defined and these ATHAs as well as many other principles are shown to be conservative over RCA $_{0}$ in all these senses and weak in other recursion-theoretic ways as well. These results answer a question raised by Hirschfeldt and reported in [19] by providing a long list of pairs of principles one of which is very weak over RCA $_{0}$ but over ACA $_{0}$ is equivalent to the other which may be strong or very strong going up a standard hierarchy and at the end being stronger than full second-order arithmetic. (shrink)