Online research in philosophy

In Part III of his 1879 logic Frege proves a theorem in the theory of sequences on the basis of four definitions. He claims in Grundlagen that this proof, despite being strictly deductive, constitutes a real extension of our knowledge, that it is ampliative rather than merely explicative. Frege furthermore connects this idea of ampliative deductive proof to what he thinks of as a fruitful definition, one that draws new lines. My aim is to show that we can make good (...) sense of these claims if we read Frege’s notation diagrammatically, in particular, if we take that notation to have been designed to enable one to exhibit the (inferentially articulated) contents of concepts in a way that allows one to reason deductively on the basis of those contents. (shrink)

In his Locke Lectures Brandom proposes to extend what he calls the project of analysis to encompass various relationships between meaning and use. As the traditional project of analysis sought to clarify various logical relations between vocabularies so Brandom’s extended project seeks to clarify various pragmatically mediated semantic relations between vocabularies. The point of the exercise in both cases is to achieve what Brandom thinks of as algebraic understanding. Because the pragmatist critique of the traditional project of analysis was precisely (...) to deny that such understanding is appropriate to the case of natural language, the very idea of an analytic pragmatism is called into question by that critique. My aim is to clarify the prospects for Brandom’s project, or at least something in the vicinity of that project, through a comparison of it with what I will suggest we can think of as Kant’s analytic pragmatism as developed by Peirce. (shrink)

Russell’s theory of descriptions in “On Denoting” has long been hailed as a paradigm of the sort of analysis that is constitutiue of philosophical understanding. It is not the only model of logical analysis available to us, however. On Frege’s quite different view, analysis provides not a reduction of some problematic notion to other, unproblematic ones -- as Russell’s analysis does -- but instead a deeper, clearer articulation of the very notion with which we began. This difference, I suggest, is (...) grounded in their two very different conceptions of the nature of language / thought; and it grounds in turn two very different conceptions of the nature of philosophical understanding. (shrink)

My aim is to understand the practice of mathematics in a way that sheds light on the fact that it is at once a priori and capable of extending our knowledge. The account that is sketched draws first on the idea, derived from Kant, that a calculation or demonstration can yield new knowledge in virtue of the fact that the system of signs it employs involves primitive parts (e.g., the ten digits of arithmetic or the points, lines, angles, and areas (...) of Euclidean geometry) that combine into wholes (numerals or drawn Euclidean figures) that are themselves parts of larger wholes (the array of written numerals in a calculation or the diagram of a Euclidean demonstration). Because wholes such as numerals and Euclidean figures both have parts and are parts of larger wholes, their parts can be recombined into new wholes in ways that enable extensions of our knowledge. I show that sentences of Frege's Begriffsschrift can also be read as involving three such levels of articulation; because they have these three levels, we can understand in essentially the same way how a proof from concepts alone can extend our knowledge. (shrink)

The most enlightening examination to date of the developments of Frege's thinking about his logic, this book introduces a new kind of logical language, one that ...