This is an intensive class on logical and philosophical issues in proof theory, taught by Shawn Standefer and me at nasslli 2016. We cover cut elimination, some substructural logics, and hypersequents, with a bit of inferentialism and bilateralism mixed in. Here are the slides for each day of the course.

**Day 1**:*Foundations*. An introduction to the sequent calculus for intuitionistic and classical logic, cut elimination and some of its consequences.**Day 2**:*Substructural Logics*. Structural rules in sequent systems; the case of distribution; different substructural logics and their applications; revisiting cut elimination in substructural logics.**Day 3**:*Beyond Sequents*. Sequent sytstems for basic modal logics; three ways to move beyond traditional sequent systems—display logic, labelled sequents and tree hypersequents.**Day 4**:*Hypersequents for S5, Actuality and 2D Modal Logics*. From tree hypersequents to simple hypersequents for S5; extending simple hypersequents to model actuality and two-dimensional modal logic.**Day 5**:*Semantics*. Normative pragmatics; the scope of rules and definitions; between proofs and models; moving beyond propositional logics.

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- Francesca Poggiolesi, “A Cut-Free Simple Sequent Calculus for Modal Logic S5,”
*Review of Symbolic Logic*1:1, 3–15, 2008. - Greg Restall, “Proofnets for S5: sequents and circuits for modal logic,”
*Logic Colloquium 2005*, ed. C. Dimitracopoulos, L. Newelski, D. Normann and J. R. Steel, Cambridge University Press, 2008. - Kaja Bednarska and Andrzej Indrzejczak, “Hypersequent Calculi for S5: the methods of cut elimination”
*Logic and Logical Philosophy*, 24, 277–311, 2015. - Greg Restall, “A Cut-Free Sequent System for Two Dimensional Modal Logic, and why it matters,”
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*Logic, Methodology and Philosophy of Science*:*Proceedings of the Twelfth International Congress*, edited by P. Hájek, L. Valdés-Villanueva and D. Westerståhl, KCL Publications, 2005. - Greg Restall, “Truth Values and Proof Theory”
*Studia Logica*, 92(2):241–264, 2009. - Raymond Smullyan,
*First-Order Logic*. Springer-Verlag, 1968. - Gaisi Takeuti,
*Proof Theory*, 2nd edition. Elsevier, 1987. - Michael Kremer “Kripke and the Logic of Truth.”
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*Basic Proof Theory*, 2nd ed. Cambridge University Press, 2000.

- nasslli 2016, Rutgers, July 2016.
- Class Proposal, containing our draft class outline.

I’m *Greg Restall*, and this is my *personal website*. ¶ I am the Shelby Cullom Davis Professor of Philosophy at the University of St Andrews, and the Director of the Arché Philosophical Research Centre for Logic, Language, Metaphysics and Epistemology ¶ I like thinking about – and helping other people think about – logic and philosophy and the many different ways they can inform each other.

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