Algebra of Proofs
Published on 3. June 2016
310 pages
978-1-4832-7542-0 (ISBN)
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.
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PrefaceContentsChapter 1. Introduction 1.1. Categorical Preliminaries 1.2. Logical PreliminariesChapter 2. Monoidal Categories 2.1. Definition 2.2. Examples 2.3. The Category Fm(X) 2.4. The Deductive System m¿(X) 2.5. The Semantics of Der(m¿(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 sm¿(X) 3.5. The Semantics of Der(sm¿(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 c¿(X) 4.5. The Semantics of Der(c¿(X)) 4.6. The Syntax of Fc¿(X)Chapter 5. Bicartesian Categories 5.1. Definition 5.2. Examples 5.3. The Category Fbc(X) 5.4. The Deductive System bc¿(X) 5.5. The Semantics of Der(bc¿(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 dbc¿(X) 6.5. The Semantics of Der(dbc¿(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 mcl¿(X) 7.5. The Semantics of Der(mcl¿(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 smcl¿(X) 8.5. The Semantics of Der(smcl¿(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 ccl¿(X) 9.5. The Semantics of Der(ccl¿(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 bccl¿(X) 10.5. The Semantics of Der(bccl¿(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 r¿(¿) 11.5. The Semantics of Der(r¿(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 mbcl¿(X) 12.5. The Semantics of Der(mbcl¿(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 ¿*(¿) 13.4. The Semantics of Der(¿*(X)) 13.5. The Syntax of Fqc(AtL*(X))Appendix A. The Labelled Deductive System ¿(¿) A.1. The Class Lb(¿(X)) A.2. The Axioms of ¿(¿) A.3. The Rules of Inference of ¿(¿) A.4. The Class Der(¿(X))Appendix B. The Unlabelled Deductive System ¿(¿) B.1. The Axioms of ¿(¿) B.2. The Rules of Inference of ¿(¿) B.3. The Class Der(¿(X))Appendix C. The Cut Elimination AlgorithmAppendix D. The Normalization AlgorithmBibliographyIndex of SymbolsIndex of Subjects
