Study of logical paradoxes

By a paradox we understand a seemingly true statement or set of statements which lead by valid deduction to contradictory statements. Logical paradoxes - paradoxes which involve logical concepts - are in fact as old as the history of logic. The Liar paradox, for instance, goes back to Epimenides (6th century B.C.?). In the late 19th century a new impetus v/as given to the investigation of logical paradoxes by the discovery of new logico-mathematical paradoxes such as those of Russell and Burali- Porti. This came about in the course of attempts to give mathematics a rigorous axiomatic foundation. Sometimes a distinction is maintained between a paradox and an antinomy. In a paradox, it is said, semantical notions are involved and a certain "oddity", "strangeness", or what may be called "paradoxical situation", resides in its construction. The resolution of a paradox is therefore not simply a matter of removing contradiction, but also requires clarifying and removing the "oddity". On the other hand, an antinomy is said to consist in the derivation of a contradiction in an axiomatic system and its resolution lies in revising the system so as to avoid the contradiction. In discussing paradoxes and antinomies, we shall not be strictly bound by this usage of these terms: we use "paradox" and "antinomy" interchangeably. Indeed, from our point of view, even antinomies in an axiomatic system ultimately need semantic clarification and thus removal of paradoxical situations

The City as Refuge: Constructing Urban Blackness in Paul Laurence Dunbar’s \u27The Sport of the Gods\u27 and James Weldon Johnson’s \u27Autobiography of an Ex-Colored Man.\u27

This essay analyzes the narrative strategies that Paul Laurence Dunbar and James Weldon Johnson used to represent black characters in The Sport of the Gods and The Autobiography of an Ex-Colored Man as a means of examining the authors\u27 construction of the city as an alternative space for depicting African Americans. In late nineteenth- and early twentieth-century fiction, the majority of African American images in popular fiction were confined to Southern-based pastoral depictions that restricted black identity to stereotypically limited and historically regressive ideas, exemplified in such characters as Zip Coon, Sambo, Uncle Tom, Jim Crow, and Mammy Jane. The plantation tradition inherently connected blacks to the country by marking them as rustic, and blacks were seen as simple, primitive people who needed the protection of the benevolent whites they served. Positive depictions of African Americans in urban settings were neither prevalent nor acceptable to the literary establishment; as Dickson Bruce, Jr., states, African American writers “could talk about themselves, their hopes, their aspirations, only in the language of mainstream America.” With exceedingly few exceptions, African American characters who were placed in urban spaces were portrayed using the pastoral identities that had been defined by Southern, post-Reconstruction authors

Experiences of Anglo-Burmese migrants in Perth, Western Australia : a substantive theory of marginalisation, adaptation and community

The experience of migration and adaptation of ethnically mixed migrants; like the Anglo-Burmese migrants, has received little attention. This group began migrating to Australia, in particular Western Australia, in the 1960s due to changing socio-political circumstances in Burma. The examination of cultural issues in psychological research has operated in a number of different perspectives including cross-cultural psychology, cultural psychology and more recently, community psychology in Australia. The development of community psychology in Australia has led to the development of a community research approach by Bishop, Sonn, Drew and Contos (2002). This approach requires the exploration of the substantive domain using the iterative~ reflective- generative process. This leads to the development of tacit knowledge which is reflected upon and influenced by the conceptual domain. Over subsequent iterations, the conceptual domain develops, resulting in a substantive theory. Three substantive questions were addressed in this series of studies:(l) What, if any, have been the experiences of cultural and social marginalisation of Anglo-Burmese migrants over time? (2) What relationship exists between acculturation outcomes, psychological well-being and psychological sense of community for the Anglo-Burmese migrants? (3) How have the Anglo-Burrnese migrants interpreted their own experience of acculturation within their own unique set of contextual circumstances

Events in computation

SIGLEAvailable from British Library Document Supply Centre- DSC:D36018/81 / BLDSC - British Library Document Supply CentreGBUnited Kingdo

Boundary Algebra: A Simpler Approach to Boolean Algebra and the Sentential Connectives

Boundary algebra [BA] is a algebra of type , and a simplified notation for Spencer-Brown’s (1969) primary algebra. The syntax of the primary arithmetic [PA] consists of two atoms, () and the blank page, concatenation, and enclosure between ‘(‘ and ‘)’, denoting the primitive notion of distinction. Inserting letters denoting, indifferently, the presence or absence of () into a PA formula yields a BA formula. The BA axioms are A1: ()()= (), and A2: “(()) [abbreviated ‘⊥’] may be written or erased at will,” implying (⊥)=(). The repeated application of A1 and A2 simplifies any PA formula to either () or ⊥. The basis for BA is B1: abc=bca (concatenation commutes & associates); B2, ⊥a=a (BA has a lower bound, ⊥); B3, (a)a=() (BA is a complemented lattice); and B4, (ba)a=(b)a (implies that BA is a distributive lattice). BA has two intended models: (1) the Boolean algebra 2 with base set B={(),⊥}, such that () ⇔ 1 [dually 0], (a) ⇔ a′, and ab ⇔ a∪b [a∩b]; and (2) sentential logic, such that () ⇔ true [false], (a) ⇔ ~a, and ab ⇔ a∨b [a∧b]. BA is a self-dual notation, facilitates a calculational style of proof, and simplifies clausal reasoning and Quine’s truth value analysis. BA resembles C.S. Peirce’s graphical logic, the symbolic logics of Leibniz and W.E. Johnson, the 2 notation of Byrne (1946), and the Boolean term schemata of Quine (1982).Boundary algebra; boundary logic; primary algebra; primary arithmetic; Boolean algebra; calculation proof; G. Spencer-Brown; C.S. Peirce; existential graphs

A synthetic axiomatization of Map Theory

Includes TOC détaillée, index et appendicesInternational audienceThis paper presents a subtantially simplified axiomatization of Map Theory and proves the consistency of this axiomatization in ZFC under the assumption that there exists an inaccessible ordinal. Map Theory axiomatizes lambda calculus plus Hilbert's epsilon operator. All theorems of ZFC set theory including the axiom of foundation are provable in Map Theory, and if one omits Hilbert's epsilon operator from Map Theory then one is left with a computer programming language. Map Theory fulfills Church's original aim of introducing lambda calculus. Map Theory is suited for reasoning about classical mathematics as well ascomputer programs. Furthermore, Map Theory is suited for eliminating thebarrier between classical mathematics and computer science rather than just supporting the two fields side by side. Map Theory axiomatizes a universe of "maps", some of which are "wellfounded". The class of wellfounded maps in Map Theory corresponds to the universe of sets in ZFC. The first version MT0 of Map Theory had axioms which populated the class of wellfounded maps, much like the power set axiom et.al. populates the universe of ZFC. The new axiomatization MT of Map Theory is "synthetic" in the sense that the class of wellfounded maps is defined inside MapTheory rather than being introduced through axioms. In the paper we define the notion of kappa- and kappasigma-expansions and prove that if sigma is the smallest strongly inaccessible cardinal then canonical kappasigma expansions are models of MT (which proves the consistency). Furthermore, in the appendix, we prove that canonical omega-expansions are fully abstract models of the computational part of Map Theory