•  43
    The Deduction Theorem
    History and Philosophy of Logic 42 (2): 129-159. 2021.
    Attempts to articulate the real meaning or ultimate significance of a famous theorem comprise a major vein of philosophical writing about mathematics. The subfield of mathematical logic has supplie...
  •  41
    The Context of Inference
    History and Philosophy of Logic 39 (4): 365-395. 2018.
    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 envi…Read more
  •  31
    The Gödelian Inferences
    History and Philosophy of Logic 30 (3): 241-256. 2009.
    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 extensiona…Read more
  •  67
    Cut as Consequence
    History and Philosophy of Logic 31 (4): 349-379. 2010.
    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…Read more
  •  36
    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 tran…Read more
  •  40
    Logical Nihilism (review)
    In Andrés Villaveces, Roman Kossak, Juha Kontinen & Åsa Hirvonen (eds.), Logic Without Borders: Essays on Set Theory, Model Theory, Philosophical Logic and Philosophy of Mathematics, De Gruyter. pp. 147-166. 2015.
  •  36
    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…Read more