CODE 107014 ACADEMIC YEAR 2022/2023 CREDITS 6 cfu anno 1 METODOLOGIE FILOSOFICHE 8465 (LM-78) - GENOVA 6 cfu anno 1 MATEMATICA 9011 (LM-40) - GENOVA 6 cfu anno 2 MATEMATICA 9011 (LM-40) - GENOVA SCIENTIFIC DISCIPLINARY SECTOR MAT/01 TEACHING LOCATION GENOVA SEMESTER 2° Semester TEACHING MATERIALS AULAWEB OVERVIEW Proof theory is one of the four pillars of mathematical logic, and is of fundamental interest to mathematicians, computer scientists, philosophers, and linguists. Proof theory has historical roots in Aristotle's syllogisms and in Leibniz's dream of a calculus ratiocinator for the formalization and automation of reasoning. In the 20th century proof theory was strongly called for by Hilbert's program and with the groundbreaking contributions by Gentzen and the invention of natural deduction and sequent calculus was developed into an independent field of study. Proof theory is concerned with the development of formal systems of deduction, conceived as syntactically defined formal objects, the study of their properties, and their applicability to concrete problems beyond the discipline. AIMS AND CONTENT LEARNING OUTCOMES In this course we will study deductive systems for classical and intuitionistic logic and their extensions for mathematical theories. In particular, we will study the relationship between axiomatic systems, of natural deduction and sequent calculus, with particular emphasis on their use for the search for proofs. AIMS AND LEARNING OUTCOMES The purpose of the course is, on the one hand, to present the central results of proof theory and, on the other hand, to enable the students to autonomously define and use deductive systems with good properties for the meta-theoretical study of logical systems, their applications, and the automation of proofs. TEACHING METHODS Traditional lectures SYLLABUS/CONTENT 0. Introduction to proof theory 1. From natural deduction to sequent calculus 2. An invertible classical calculus for classical propositional logic 3. Constructive reasoning 4. Intuitionistic sequent calculus 5. Admissibility of contraction and cut 6. Consequences of cut elimination 7. Completeness 8. Quantifiers in natural deduction and in sequent calculus 9. Completeness of classical predicate logic 10. Variants of sequent calculi 11. Structural proof analysis of axiomatic theories, with examples from algebra, geometry, lattice theory. 12. The constructive content of classical proofs 13. Labelled calculi for modal and non-classical logics. RECOMMENDED READING/BIBLIOGRAPHY In addition to articles and notes that will be made available on AulaWeb, the following textbooks will be used: S. Negri and J. von Plato, Structural Proof Theory, Cambridge University Press 2001. S. Negri and J. von Plato, Proof Analysis, Cambridge University Press 2011. A.S. Troelstra and H. Schwichtenberg, Basic Proof Theory, second edition. Cambridge University Press 2000. TEACHERS AND EXAM BOARD SARA NEGRI Ricevimento: By appointment Exam Board SARA NEGRI (President) RICCARDO CAMERLO GIUSEPPE ROSOLINI (President Substitute) LESSONS Class schedule The timetable for this course is available here: Portale EasyAcademy EXAMS ASSESSMENT METHODS Assessment based on assignments during the course and a final presentation on a topic to be agreed with the docent.
