FLAP Journal 2025 Journal Article
Staged Logic: A Unified Framework Against Paradox
- Daniel Rowe
The ideas and much of the content here is adapted from a Chapter of my PhD thesis (forthcoming). There (and in an appendix) I present some of the results in more technical detail, including fully worked through soundness and completeness theorems for propositional staged logic. This Chapter is more condensed, but hopefully suffices to express the core idea, as well as to see how it delivers a paradox-resistant logic that remains nevertheless powerful enough to deliver the types of mathematical results we would want from logic. Dov Gabbay’s contributions to logic consistently exemplify the search for systematicity and unification. Among his many unificatory projects, two stand out as especially resonant with the staged framework: Labelled Deductive Systems (1996), where the inferential behaviour of formulas is made sensitive to labels (much as staged logic makes it sensitive to stage indices), and Fibring Logics (1999), which provides a systematic method for combining logics while preserving discipline at their points of interaction. These works provide the closest formal analogues to the staged approach, and show how Gabbay’s unificatory vision continues in new guises. Professor Gabbay has a more direct hand in this too. In 2023 whilst working on a joint paper in Talmudic logic, he encouraged me to explore whether any of the Talmudic logic approaches we were studying might help to dissolve the Liar paradox. That served as the inspiration for this logic. He also offered important feedback and insight at an earlier stage of the development of this logic. Other important feedback came from Alex Paseau, James Studd and Carl Posy as well as audiences at three conferences in which early versions were presented: the Jean Paul van Bendegem Masterclass in the Philosophy of Mathematical Practice, University of Brussels, June 2023; the Israel Philosophical Association annual conference, Ben Gurion University, July 2023; and the ‘Perspectives About Truth’ online conference, September 2023.