Algebra of proofs
type: Item
instance of: scholarly article
title: Algebra of proofs
zbMATH Open document ID: 0532.03030
published in: Studies in Logic and the Foundations of Mathematics
publication date: 1978-01-01T00:00:00Z
review text: The author studies the algebraic properties of the proof theory of intuitionistic first-order logic in a categorical setting. The conceptual basis of the book is the realization that the Lindenbaum-Tarski algebras of formulas may be viewed as categories and that the formal proofs of the associated deductive systems determine structured categories as their canonical algebras which are of the same type as the Lindenbaum-Tarski algebras of formulas of the underlying languages. Following the Introduction (Chapter I), there are twelve additional chapters, in which the author studies twelve theories of varied linguistic and deductive strength. The theories are divided into two main types: the monoidal type, in which theories based on the common algebraic properties of conjunction and disjunction are investigated, and the Cartesian type, in which conjunction and disjunction have their proper meanings. In every chapter the author follows the same scheme. He first constructs a category of a certain type as an algebraic model for the class of formal proofs being considered. Then he proves a completeness theorem to the effect that the arrows of the constructed category can be represented by formal proofs in a Gentzen-style sequent calculus with cut elimination. In the propositional cases the algorithmic character of the cut- elimination process is used to provide an effective description of the arrows of the category constructed and to develop decision procedures, in the form of Church-Rosser theorems, for the commutativity of the finite diagrams of these categories. In the last chapter, the author also shows how to accommodate quantifiers in the calculus of adjoints and describes the topos-theoretic setting required in order to develop the proof theory of intuitionistic first-order logic. The contents of the chapters are as follows: Chapter I: Introduction. The author provides the categorical and logical notions to be used in the rest of the book. Chapter II: Monoidal categor…
Mathematics Subject Classification ID: 03G30
Mathematics Subject Classification ID: 03F50
Mathematics Subject Classification ID: 03F55
Mathematics Subject Classification ID: 03-02
Mathematics Subject Classification ID: 18-02
Mathematics Subject Classification ID: 18D10
Mathematics Subject Classification ID: 18D05
Mathematics Subject Classification ID: 18D15
Mathematics Subject Classification ID: 18B25
Mathematics Subject Classification ID: 18A15
zbMATH DE Number: 3843747
zbMATH Keywords: proof theory
zbMATH Keywords: intuitionistic first-order logic
zbMATH Keywords: Lindenbaum-Tarski algebras
zbMATH Keywords: functional description of quantifiers
zbMATH Keywords: completeness
zbMATH Keywords: algebraic model
zbMATH Keywords: cut-elimination
author: Manfred E. Szabo
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3843747