El objetivo de este artículo es analizar qué quiere decir capturar una teoría (se utilizará la aritmética como ejemplo paradigmático) por medio de un sistema axiomático formal. Se considerarán para ello dos enfoques, que pueden denominarse “semántico” y “sintáctico”, tanto en términos de sus ventajas como de sus limitaciones. El enfoque semántico (presupuesto por Barrio y Da Ré en este volumen y expuesto en la primera sección), entiende la expresabilidad como una restricción de la clase de los modelos; se muestra (...) que esta concepción lleva a tres clases distintas de limitaciones expresivas. En la segunda sección se examinan -y finalmente rechazan- algunos posibles caminos de solución a estos problemas. En la última sección se ofrecen objeciones a las alternativas restantes y en base a ellas se elabora el enfoque sintáctico que liga la noción de captura a la de prueba de las oraciones relevantes. También se argumenta que desde este marco pueden evitarse algunas de las limitaciones del enfoque semántico. (shrink)

This paper is a conceptual study in the philosophy of logic. The question considered is 'How may formulae of the propositional calculus be brought into a representational relation to the world?'. Four approaches are distinguished: (1) the denotational approach, (2) the abbreviational approach, (3) the truth-conditional approach, and (4) the modelling approach. (2) and (3) are very familiar, so I do not discuss them. (1), which is now largely obsolete, led to some interesting twists and turns in early analytic philosophy (...) which will come as news to many contemporary readers, so I discuss it in some detail. The modelling approach is, to the best of my knowledge, newly introduced here. I am not presenting it as a rival to the other approaches, but as a philosophically interesting possibility. (shrink)

This paper has a two-fold objective: to provide a balanced, multi-faceted account of the origins of logicism; to rehabilitate Richard Dedekind as a main logicist. Logicism should be seen as more deeply rooted in the development of modern mathematics than typically assumed, and this becomes evident by reconsidering Dedekind's writings in relation to Frege's. Especially in its Dedekindian and Fregean versions, logicism constitutes the culmination of the rise of ?pure mathematics? in the nineteenth century; and this rise brought with it (...) an inter-weaving of methodological and epistemological considerations. The latter aspect illustrates how philosophical concerns can grow out of mathematical practice, as opposed to being imposed on it from outside. It also sheds new light on the legacy and the lasting significance of logicism today. (shrink)