6 November 2012, Logic Tea, Vlasta Sikimic

Abstract

This talk is devoted to the famous Lewis Carroll's paradox about deduction entitled “Achilles and the Tortoise”, first published in Mind 1900. It might not be so well known that the author of the book Alice in Wonderland was a logician in Oxford. One of his greatest contributions and part of his legacy to logic is paradox about Achilles and the Tortoise and every logician should be familiar with it. Over the years Carroll's article has evoked numerous responses. After presenting the paradox, I will try to provide an empirical classification of these responses. This classification should be more extensive than the existing ones, such as the one offered by Pascal Engel. After systematizing responses to the paradox, I will argue, from philosophical and proof-theoretical point of view, in favor of the group of interpretations and solutions that emphasizes the difference between rules of inference and premises.

Following the lines of Kosta Došen's article about inferential semantics, it will be shown that modified proof-theoretic semantics has potential to overcome the so-called dogmas of semantics. The second dogma is formulated as follows: “The correctness of the hypothetical notions reduces to the preservation of the correctness of the categorical ones”. This problematic reduction of inference to implication is prominent both in verificationism in science and in constructivism in mathematics. However, inference cannot be reduced to consequence relations. Now, we will see that we should have already learned this from the most prominent interpretation of Carroll's paradox. In parallel, some explanations will be given why category theory might be the most appropriate “tool” when reasoning about this improved proof-theoretic semantics.

For more information, please contact Johannes Marti (johannes.marti at gmail.com), Sebastian Speitel (sebastian.speitel at gmail.com), or Matthijs Westera (M.Westera at uva.nl).