Formal systems are standardly envisaged in terms of a grammar specifying well-formed formulae together with a set of axioms and rules. Derivations are ordered lists of formulae each of which is either an axiom or is generated from earlier items on the list by means of the rules of the system; the theorems of a formal system are simply those formulae for which there are derivations. Here we outline a set of alternative and explicitly visual ways of envisaging and analyzing (...) at least simple formal systems using fractal patterns of infinite depth. Progressively deeper dimensions of such a fractal can be used to map increasingly complex wffs or increasingly complex 'value spaces', with tautologies, contradictions, and various forms of contingency coded in terms of color. This and related approaches, it turns out, offer not only visually immediate and geometrically intriguing representations of formal systems as a whole but also promising formal links (1) between standard systems and classical patterns in fractal geometry, (2) between quite different kinds of value spaces in classical and infinite-valued logics, and (3) between cellular automata and logic. It is hoped that pattern analysis of this kind may open possibilities for a geometrical approach to further questions within logic and metalogic. (shrink)
Zeno's paradoxes of motion and the semantic paradoxes of the Liar have long been thought to have metaphorical affinities. There are, in fact, isomorphisms between variations of Zeno's paradoxes and variations of the Liar paradox in infinite-valued logic. Representing these paradoxes in dynamical systems theory reveals fractal images and provides other geometric ways of visualizing and conceptualizing the paradoxes.
Resumo As festas, durante o século XVIII, desempenhavam um importante papel no cotidiano das associações de leigos e religiosas. As ordens terceiras franciscanas organizavam distintas celebrações no intuito de promover a instituição no campo religioso local, difundir suas devoções e, ao mesmo tempo, ampliar o seu recrutamento. Este artigo analisa alguns elementos constituintes das celebrações realizadas pelas ordens terceiras de São Francisco em diferentes cidades do império português (Braga e São Paulo), visando compreender o significado e a valorização atribuídos às (...) celebrações no interior das igrejas por essas agremiações. Para realizar esse estudo foram utilizados, principalmente, os livros contábeis das instituições com o objetivo de vislumbrar o investimento nas celebrações e as suas características particulares. A partir das informações compulsadas constata-se um dispêndio avultado com as cerimônias e com a manutenção do culto pelas ordens terceiras franciscanas, evidenciando a importância dedicada por essas associações às celebrações e às festas executadas em seus templos. Palavras-chave: ordem terceira de São Francisco; festa; império português.The festivals, during the eighteenth century, played an important role in the daily religious and lay associations. The Third Orders of St. Francis organized distinct celebrations in order to promote the institution in the local religious field, spread their devotions, and at the same time broaden their recruitment. This article examines some elements of the celebrations held by the Third Orders of St. Francis in different cities of the Portuguese empire (Braga and São Paulo), aiming to understand the meaning and value attributed to the celebrations inside the churches by these associations. To perform this study were primarily used the accounting books of the institutions in order to glimpse the investment in the celebrations and their characteristics. From the gathered information we find a compulsive spending with the ceremonies and the maintenance of worship by the Third Orders of St. Francis, indicating the importance given by these associations to celebrations and festivals performed in their temples. Keywords : Third Order of St. Francis; festival; Portuguese Empire. (shrink)