Curtis Franks

The papers where Gerhard Gentzen introduced natural deduction and sequent calculi suggest that his conception of logic differs substantially from the now dominant views introduced by Hilbert, Gödel, Tarski, and others. Specifically, (1) the definitive features of natural deduction calculi allowed Gentzen to assert that his classical system nk is complete based purely on the sort of evidence that Hilbert called ?experimental?, and (2) the structure of the sequent calculi li and lk allowed Gentzen to conceptualize completeness as a question about the relationships among a system's individual rules (as opposed to the relationship between a system as a whole and its ?semantics?). Gentzen's conception of logic is compelling in its own right. It is also of historical interest, because it allows for a better understanding of the invention of natural deduction and sequent calculi.

There is an ambiguity in the concept of deductive validity that went unnoticed until the middle of the twentieth century. Sometimes an inference rule is called valid because its conclusion is a theorem whenever its premises are. But often something different is meant: The rule's conclusion follows from its premises even in the presence of other assumptions. In many logical environments, these two definitions pick out the same rules. But other environments are context-sensitive, and in these environments the second notion is stronger. Sorting out this ambiguity has led to profound mathematical investigations with applications in complexity theory and computer science. The origins of this ambiguity and the history of its resolution deserve philosophical attention, because our understanding of logic stands to benefit from their details. I am eager to examine together with you, Crito, whether this argument will appear in any way different to me in my present circumstances, or whether it remains the same, whet...

The exegesis of sacred rites in the Talmud is subject to a restriction on the iteration and composition of inference rules. In order to determine the scope and limits of that restriction, the sages of the Talmud deploy those very same inference rules. We present the remarkable features of this early use of self-reference to navigate logical constraints and uncover the hidden complexity behind the sages? arguments. Appendix 11 contains a translation of the relevant sugya. 1Hebrew and Aramaic transliteration approximates traditional Sephardic pronunciation, which is closer to academic standard transliteration than the various Ashkenazic pronunciations, yet is legible. Specific references follow the convention of folio (number), side (a or b), number of lines from the top or (if negative) bottom of the page

Most scholars think of David Hilbert's program as the most demanding and ideologically motivated attempt to provide a foundation for mathematics, and because they see technical obstacles in the way of realizing the program's goals, they regard it as a failure. Against this view, Curtis Franks argues that Hilbert's deepest and most central insight was that mathematical techniques and practices do not need grounding in any philosophical principles. He weaves together an original historical account, philosophical analysis, and his own development of the meta-mathematics of weak systems of arithmetic to show that the true philosophical significance of Hilbert's program is that it makes the autonomy of mathematics evident. The result is a vision of the early history of modern logic that highlights the rich interaction between its conceptual problems and technical development.

I attribute an 'intensional reading' of the second incompleteness theorem to its author, Kurt G del. My argument builds partially on an analysis of intensional and extensional conceptions of meta-mathematics and partially on the context in which G del drew two familiar inferences from his theorem. Those inferences, and in particular the way that they appear in G del's writing, are so dubious on the extensional conception that one must doubt that G del could have understood his theorem extensionally. However, on the intensional conception, the inferences are straightforward. For that reason I conclude that G del had an intensional understanding of his theorem. Since this conclusion is in tension with the generally accepted view of G del's understanding of mathematical truth, I explain how to reconcile that view with the intensional reading of the theorem that I attribute to G del. The result is a more detailed account of G del's conception of meta-mathematics than is currently available

In his Philosophical Investigations, Wittgenstein said that “turning our whole investigation around” is the only way to shake the illusion of a “preconceived idea of crystalline purity.” Commentators have built sweeping descriptions of Wittgenstein’s general approach to philosophy out of their interpretations of this slogan. For his own part, Wittgenstein specified a particular subject. “For,” he wrote, “the crystalline purity of logic was…not a result of investigation: it was a requirement.” In asking what Wittgenstein could have meant by reversing the direction of the investigation of logic and of what he called its apparent “hardness,” I propose to follow his own advice: to look around before thinking. Two episodes from logic’s history serve as illustrative examples of concepts being dislodged from their original position as theorems or discoveries and relocated as starting points of an investigation (and vice-versa). The details of the cases showcase how “turning our whole investigation around” has proven its practical utility in the growth and development of logic in the last century. Further, it suggests that Wittgenstein might have intended something at once humbler and, for logic, more important than a grand reconceptualization of the nature of necessity: the science of logic itself relies on a continuous inversion of discoveries and requirements.