# Algebra of Proofs

## 1st Edition

**Authors:**M. E. Szabo

**Editors:**K. J. Barwise D. Kaplan H. J. Keisler

**eBook ISBN:**9781483275420

**Imprint:**North Holland

**Published Date:**1st January 1978

**Page Count:**310

## Description

*Algebra of Proofs* deals with algebraic properties of the proof theory of intuitionist first-order logic in a categorical setting. The presentation is based on the confluence of ideas and techniques from proof theory, category theory, and combinatory logic. The conceptual basis for the text is the Lindenbaum-Tarski algebras of formulas taken as categories. 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 the formulas of underlying languages). Gentzen's theorem, which asserts that provable formulas code their own proofs, links the algebras of formulas and the corresponding algebras of formal proofs. The book utilizes the Gentzen's theorem and the reducibility relations with the Church-Rosser property as syntactic tools. The text explains two main types of theories with varying linguistic complexity and deductive strength: the monoidal type and the Cartesian type. It also shows that quantifiers fit smoothly into the calculus of adjoints and describe the topos-theoretical setting in which the proof theory of intuitionist first-order logic possesses a natural semantics. The text can benefit mathematicians, students, or professors of algebra and advanced mathematics.

## Table of Contents

Preface

Contents

Chapter 1. Introduction

1.1. Categorical Preliminaries

1.2. Logical Preliminaries

Chapter 2. Monoidal Categories

2.1. Definition

2.2. Examples

2.3. The Category Fm(X)

2.4. The Deductive System mΔ(X)

2.5. The Semantics of Der(mΔ(X))

2.6. The Syntax of Fm(X)

Chapter 3. Symmetric Monoidal Categories

3.1. Definition

3.2. Examples

3.3. The Category Fsm(X)

3.4. The Deductive System smΔ(X)

3.5. The Semantics of Der(smΔ(X))

3.6. The Syntax of Fsm(X)

Chapter 4. Cartesian Categories

4.1. Definition

4.2. Examples

4.3. The Category Fc(X)

4.4. The Deductive System cΔ(X)

4.5. The Semantics of Der(cΔ(X))

4.6. The Syntax of FcΔ(X)

Chapter 5. Bicartesian Categories

5.1. Definition

5.2. Examples

5.3. The Category Fbc(X)

5.4. The Deductive System bcΔ(X)

5.5. The Semantics of Der(bcΔ(X))

5.6. The syntax of Fbc(X)

Chapter 6. Distributive Bicartesian Categories

6.1. Definition

6.2. Examples

6.3. The Category Fdbc(X)

6.4. The Deductive System dbcΔ(X)

6.5. The Semantics of Der(dbcΔ(X))

6.6. The Syntax of Fdbc(X)

Chapter 7. Monoidal Closed Categories

7.1. Definition

7.2. Examples

7.3. The Category Fmcl(X)

7.4. The Deductive System mclΔ(X)

7.5. The Semantics of Der(mclΔ(X))

7.6. The Syntax of Fmcl(X)

Chapter 8. Symmetric Monoidal Closed Categories

8.1. Definition

8.2. Examples

8.3. The Category Fsmcl(X)

8.4. The Deductive System smclΔ(X)

8.5. The Semantics of Der(smclΔ(X))

8.6. The Syntax of Fsmcl(X)

Chapter 9. Cartesian Closed Categories

9.1. Definition

9.2. Examples

9.3. The Category Fccl(X)

9.4. The Deductive System cclΔ(X)

9.5. The Semantics of Der(cclΔ(X))

9.6. The Syntax of Fccl(X)

Chapter 10. Bicartesian Closed Categories

10.1. Definition

10.2. Examples

10.3. The Category Fbccl(X)

10.4. The Deductive System bcclΔ(X)

10.5. The Semantics of Der(bcclΔ(X))

10.6. The Syntax of Fbccl(X)

Chapter 11. Residuated Categories

11.1. Definition

11.2. Examples

11.3. The Category Fr(X)

11.4. The Deductive System rΔ(Χ)

11.5. The Semantics of Der(rΔ(X))

11.6. The Syntax of Fr(X)

Chapter 12. Monoidal Biclosed Categories

12.1. Definition

12.2. Examples

12.3. The Category Fmbcl(X)

12.4. The Deductive System mbclΔ(X)

12.5. The Semantics of Der(mbclΔ(X))

12.6. The Syntax of Fmbcl(X)

Chapter 13. Quantifier-Complete Categories

13.1. Categorical Preliminaries

13.2. The Language L

*(X)*

13.3. The Deductive System Δ(Χ)

13.3. The Deductive System Δ

13.4. The Semantics of Der(Δ

*(X))*

13.5. The Syntax of Fqc(AtL(X))

13.5. The Syntax of Fqc(AtL

Appendix A. The Labelled Deductive System Δ(Χ)

A.1. The Class Lb(Δ(X))

A.2. The Axioms of Δ(Χ)

A.3. The Rules of Inference of Δ(Χ)

A.4. The Class Der(Δ(X))

Appendix B. The Unlabelled Deductive System Δ(Χ)

B.1. The Axioms of Δ(Χ)

B.2. The Rules of Inference of Δ(Χ)

B.3. The Class Der(Δ(X))

Appendix C. The Cut Elimination Algorithm

Appendix D. The Normalization Algorithm

Bibliography

Index of Symbols

Index of Subjects

## Details

- No. of pages:
- 310

- Language:
- English

- Copyright:
- © North Holland 1978

- Published:
- 1st January 1978

- Imprint:
- North Holland

- eBook ISBN:
- 9781483275420