# Structural Proof Theory

It is available on Amazon for about $35, and the emphasis is on structural proof theory. The treatment is terse and encyclopedic, which can make it hard to read as an introductory text. There is an online review here.

## Structural Proof Theory

This dissertation pertains to algebraic proof theory, a research field aimed at solving problems in structural proof theory using results and insights from algebraic logic, universal algebra, duality and representation theory for classes of algebras. The main contributions of this dissertation involve the very recent theory of multi-type calculi on the proof-theoretic side, and the well established theory of heterogeneous algebras on the algebraic side. Given a cut-admissible sequent calculus for a basic logic (e.g. the full Lambek calculus), a core question in structural proof theory concerns the identification of axioms which can be added to the given basic logic so that the resulting axiomatic extensions can be captured by calculi which are again cut-admissible. This question is very hard, since the cut elimination theorem is notoriously a very fragile result. However, algebraic proof theory has given a very satisfactory answer to this question for substructural logics, by identifying a hierarchy (Nn, Pn) of axioms in the language of the full Lambek calculus, referred to as the substructural hierarchy, and guaranteeing that, up to the level N2, these axioms can be effectively transformed into special structural rules (called analytic) which can be safely added to a cut-admissible calculus without destroying cut-admissibility. The research program of algebraic proof theory can be exported to arbitrary signatures of normal lattice expansions, to the study of the systematic connections between algebraic logic and display calculi, and even beyond display calculi, to the study of the systematic connections between the theory of heterogeneous algebras and multi-type calculi, a proof-theoretic format generalizing display calculi, which has proven capable to encompass logics which fall out of the scope of the proof-theoretic hierarchy, and uniformly endow them with calculi enjoying the same excellent properties which (single-type) proper display calculi have. The defining feature of the multi-type calculi format is that it allows entities of different types to coexist and interact on equal ground: each type has its own internal logic (i.e. language and deduction relation), and the interaction between logics of different types is facilitated by special heterogeneous connectives, primitive to the language, and treated on a par with all the others. The fundamental insight justifying such a move is the very natural consideration, stemming from the algebraic viewpoint on (unified) correspondence, that the fundamental properties underlying this theory are purely order-theoretic, and that as long as maps or logical connectives have these fundamental properties, there is very little difference whether these maps have one and the same domain and codomain, or bridge different domains and codomains. This enriched environment is specifically designed to address the problem of expressing the interactions between entities of different types by means of analytic structural rules. In the present dissertation, we extend the semantic cut elimination and finite model property from the signature of residuated lattices to arbitrary signatures of normal lattice expansions, and build or refine the multi-type algebraic proof theory of three logics, each of which arises in close connection with a well known class of algebras (semi De Morgan algebras, bilattices, and Kleene algebras) and is problematic for standard proof-theoretic methods.

The aim of the course is to acknowledge the listener with the main tools of structural proof theory. The course is a concise introduction to structural proof theory: its origins, its main tools, and the consequences of the famous admissibility results (cut elimination). The focus is on sequent calculi for classical, intuitionistic, and minimal logic; however, natural deduction basics are also introduced. An emphasis is placed also on the techniques of proving cut elimination.

An Introduction to Proof Theory provides an accessible introduction to the theory of proofs, with details of proofs worked out and examples and exercises to aid the reader's understanding. It also serves as a companion to reading the original pathbreaking articles by Gerhard Gentzen. The first half covers topics in structural proof theory, including the GÃ¶del-Gentzen translation of classical into intuitionistic logic (and arithmetic), natural deduction and the normalization theorems (for both NJ and NK), the sequent calculus, including cut-elimination and mid-sequent theorems, and various applications of these results. The second half examines ordinal proof theory, specifically Gentzen's consistency proof for first-order Peano Arithmetic. The theory of ordinal notations and other elements of ordinal theory are developed from scratch, and no knowledge of set theory is presumed. The proof methods needed to establish proof-theoretic results, especially proof by induction, are introduced in stages throughout the text. Mancosu, Galvan, and Zach's introduction will provide a solid foundation for those looking to understand this central area of mathematical logic and the philosophy of mathematics.

Sergio Galvan is emeritus Professor of Logic at the Catholic University of Milan, Italy. His main areas of research are proof-theory and philosophical logic. In the first area he focuses on sequent calculi and natural deduction, the metamathematics of arithmetic systems (from Q to PA), GÃ¶del's incompleteness theorems, and Gentzen's cut-elimination theorem. In the second area, his major interest is in the philosophical interpretations (deontic, epistemic and metaphysical) of modal logic. Recently, he has been working on the relationships between formal proof and intuition in mathematics, and on the metaphysics of essence and the ontology of possibilia.

Richard Zach is Professor of Philosophy at the University of Calgary, Canada. He works in logic, history of analytic philosophy, and the philosophy of mathematics. In logic, his main interests are non-classical logics and proof theory. He has also written on the development of formal logic and historical figures associated with this development such as Hilbert, GÃ¶del, and Carnap. He has held visiting appointments at the University of California, Irvine, McGill University, and the University of Technology, Vienna.

The Proof Society supports the notion of proof in its broadest sense, through a series of suitable activities; to be therefore inclusive in reaching out to all scientific areas which consider proof as an object in their studies; to enable the community to shape its future by identifying, formulating and communicating it most important goals; to actively promote proof to increase its visibility and representation.

It is the aim of the summer school to cover basic and advanced topics in proof theory. The focus of the second edition will be on philosophy of proof theory, proof theory of impredicative theories, structural proof theory, proof mining, reverse mathematics, type theory and bounded arithmetic.

SD'19 is the fifth in a series of workshops aiming to gather various communities of structural proof theorists. The main interest is in new algebraic and geometric results in proof theory which expand our abilities to manipulate proofs, help to reduce bureaucracy in deductive systems, and ultimately lead to new methods for proof search and new kinds of proof certificates. 041b061a72