Curtis Franks

The Theory Pamphlet presents Gerhard Gentzen's natural deduction and sequent calculi with emphasis on the theory behind the formalism. Its five chapters serve as an advanced logic textbook, introducing universal properties, proof normalization techniques, and decision procedures for classical, intuitionistic, and linear logics. The same material serves also as a philosophical treatise, describing the meaning of, significance of, and relationships among three different ways to conceptualize the idea of logical completeness. Most of the material featured has never before been presented in a systematic and accessible manner.

The inference pattern known as disjunctive syllogism (DS) appears as a derived rule in Gentzen’s natural deduction calculi ni and nk. This is a paradoxical feature of Gentzen’s calculi in so far as DS is sometimes thought of as appearing intuitively more elementary than the rules $$\vee $$ E, $$\lnot $$ E, and EFQ that figure in its derivation. For this reason, many contemporary presentations of natural deduction depart from Gentzen and include DS as a primitive rule. However, such departures violate the spirit of natural deduction, according to which primitive rules are meant to relationally define logical connectives via universal properties (§2). This situation raises the question: Can disjunction be relationally defined with DS instead of with Gentzen’s $$\vee $$ I and $$\vee $$ E rules? We answer this question in the affirmative and explore the duality between Gentzen’s definition and our own (§3). We argue further that the two universal characterizations, rather than provide competing relational definitions of a single disjunction operator, disambiguate natural language’s “or” (§4). Finally, this disambiguation is shown to correspond exactly with the additive and multiplicative disjunctions of linear logic (§5). The hope is that this analysis sheds new light on the latter connective, so often deemed mysterious in writing about linear logic.

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...

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

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

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.