Proof Theory: Logical and Philosophical Aspects

August 2026 COMING SOON

at The 2026 European Summer School in Logic, Language and Information (ESSLLI 2026), Prague

This is an intensive class on logical bilateralism, taught by Sara Ayhan and me at esslli 2026. Logical bilateralism is an approach to meaning and consequence that foregrounds a symmetry between certain notions, like assertion and denial, proof and refutation or truth and falsity. Bilateralist approaches to such dual distinctions take both sides as primitive rather than—as in conventional ‘unilateralist’ approaches—taking one notion to be fundamental and defining the other in its terms. In recent years different bilateralist logical systems have been developed, displaying a wide variety in their specific orientation. What is missing so far is a systematic framework putting these very different approaches on a map and thereby highlighting similarities, distinctions and connections between them. Though giving a complete picture will also not be possible within the scope of the intended course, our aim is to familiarize the attendants with many different forms of logical bilateralism and use these to exemplify both the variety of approaches and the coherence of the underlying principles.

This page will host slides for each day of the course, and pointers to relevant references.

• Day 1: Foundations: Natural deduction, sequent calculus and models.
• Day 2: Signed Bilateralist Natural Deduction: Signed bilateralist natural deduction and multi-conclusion sequent calculus.
• Day 3: Non-Classical Frameworks: Proofs and models for non-classical bilateralist logics, like N4, FDE, and 2Int.
• Day 4: Bilateralist Term Calculus: Parigot’s \(\lambda\mu\) calculus, the Curien and Herbelin’s symmetric \(\overline{λ}μ\tilde{μ}\) calculus, and their computational and dialogical interpretations.
• Day 5: Recent Devlopments

Readings and References

Foundations

• Gerhard Gentzen, “Untersuchungen über das logische Schließen—I”, Mathematische Zeitschrift, 39(1):176–210, 1935.

• Gerhard Gentzen, The Collected Papers of Gerhard Gentzen, Translated and Edited by M. E. Szabo, North Holland, 1969.

• Peter Milne, “Harmony, Purity, Simplicity and a ‘Seemingly Magical Fact’”, The Monist, 85(4):498–534, 2002

• Greg Restall, “Multiple Conclusions” pp. 189–205 in 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.