CODE  107014 

ACADEMIC YEAR  2022/2023 
CREDITS 

SCIENTIFIC DISCIPLINARY SECTOR  MAT/01 
TEACHING LOCATION 

SEMESTER  2° Semester 
TEACHING MATERIALS  AULAWEB 
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.
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.
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 metatheoretical study of logical systems, their applications, and the automation of proofs.
Traditional lectures
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 nonclassical logics.
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.
Office hours: By appointment
SARA NEGRI (President)
RICCARDO CAMERLO
GIUSEPPE ROSOLINI (President Substitute)
All class schedules are posted on the EasyAcademy portal.
Assessment based on assignments during the course and a final presentation on a topic to be agreed with the docent.