A Philosophical Introduction to Higher-order Logics

✍ Scribed by Andrew Bacon

• Publisher: Routledge
• Year: 2023
• Tongue: English
• Leaves: 483
• Edition: 1
• Category: Library

✦ Synopsis

This is the first comprehensive textbook on higher-order logic that is written specifically to introduce the subject matter to graduate students in philosophy. The book covers both the formal aspects of higher-order languages―their model theory and proof theory, the theory of λ-abstraction and its generalizations―and their philosophical applications, especially to the topics of modality and propositional granularity. The book has a strong focus on non-extensional higher-order logics, making it more appropriate for foundational metaphysics than other introductions to the subject from computer science, mathematics, and linguistics.

A Philosophical Introduction to Higher-order Logics assumes only that readers have a basic knowledge of first-order logic. With an emphasis on exercises, it can be used as a textbook though is also ideal for self-study.

Author Andrew Bacon organizes the book's 18 chapters around four main parts:
I. Typed Language
II. Higher-Order Languages
III. General Higher-Order Languages
IV. Higher-Order Model Theory

In addition, two appendices cover the Curry-Howard isomorphism and its applications for modeling propositional structure. Each chapter includes exercises that move from easier to more difficult, strategically placed throughout the chapter, and concludes with an annotated suggested reading list providing graduate students with most valuable additional resources.

Key Features:

• Is the first comprehensive introduction to higher-order logic as a grounding for addressing problems in metaphysics
• Introduces the basic formal tools that are needed to theorize in, and model, higher-order languages
• Offers an abundance of
  - Simple exercises throughout the book, serving as comprehension checks on basic concepts and definitions
  - More difficult exercises designed to facilitate long-term learning
• Contains annotated sections on further reading, pointing the reader to related literature, learning resources, and historical context

✦ Table of Contents

Cover
Half Title
Title Page
Copyright Page
Table of Contents
Nomenclature
Preface
0 Introduction
0.1 Typed Languages
0.2 Generalizations
0.3 Higher-order Generalizations
0.4 Abstraction
0.5 Some Things that Higher-order Generalizations are Not
0.6 Higher-order Generalizations in Philosophy
0.7 Semantics and Model Theory for Higher-order Languages
0.8 Glossing Higher-order Generalizations in English
0.9 How to Read This Book
0.10 Other Resources
Endnotes
Part I: Typed Languages
1 Typed Languages
1.1 Types
1.2 Typed Languages
1.3 The Concept Horse Problem
1.4 Alternative Type Systems
Endnotes
2 An Informal Introduction to Abstraction
2.1 Abstraction
2.2 Introducing λ
2.3 Multiple Abstraction and Currying
2.4 Getting More Abstract
Endnotes
3 λ-languages
3.1 The Full λ-language
3.2 Combinators
3.3 Synonymy, α, β and η
3.4 Reduction
3.5 Combinatory Languages
3.6 More Efficient Definitions of Ersatz Abstraction
Endnotes
Part II: Higher-order Languages
4 Higher-order Languages
4.1 Higher-order Languages
4.2 Quantifiers and Variable Binding
Endnote
5 Higher-order Logics
5.1 Higher-order Logics
5.2 Higher-order Logics in Other Logical Signatures
5.3 Inductive Definitions in Higher-order Logic
Endnotes
6 Application: Higher-order Theories of Granularity
6.1 Propositional Individuation: Propositional Booleanism
6.2 Propositional Individuation: Weaker Theories
6.3 Individuating Properties and Relations: Booleanism and Weakenings
6.4 Individuating Properties and Relations: Classicism
6.5 Functionality Principles
Endnotes
7 Application: Modal Logicism
7.1 Modal Logicism
7.2 Necessity
7.3 Entailment
7.4 Necessity in the Highest Degree
7.5 Possible Worlds
7.6 Reducing the Intensional to the Extensional
Endnotes
8 Application: Consequences and Strengthenings of Classicism
8.1 The Modal Logic of Broad Necessity
8.2 Some Strengthenings of Classicism and their Modal Consequences
8.3 Logical Necessity
8.4 Further Reading
Endnotes
Part III: General Higher-order Languages
9 General λ-languages
9.1 Higher-order Ontology and λ-languages
9.2 General λ-languages
9.3 Relevant, Affine, Linear and Ordered Languages
9.4 Quantifiers in General λ-languages
9.5 General Higher-order Logics
9.6 Application: Propositional Aboutness and Constituency
9.7 General λ-languages Without Combinators
9.8 Variable Free Approaches
Endnotes
10 Curry Typing
10.1 Curry Typing
10.2 Substructural Curry Typing
10.3 Curry Typing for Logical Operations
Endnotes
11 Application: Structure I
11.1 Quasi-syntactic Accounts of Structure
11.2 Pictorial Accounts of Structure
11.3 Relational Diagrams
11.4 Translating Between Diagrams and λ-terms
11.5 Unique Decomposition
Endnotes
12 Application: Structure II
12.1 Converses, Reflexizations, Vacuous λ-abstraction
12.2 Logical Modes of Combination
12.3 Combinators and Pure Entities
12.4 Positionalism
Endnotes
13 Application: Structure III
13.1 Theoretical Primitives
13.2 A General Logical Framework
13.3 Further Reading
Endnotes
Part IV: Higher-order Model Theory
14 Applicative Structures
14.1 Applicative Structures
14.2 Functional Interpretations
14.3 The Environment Model Condition
14.4 Congruences and Quotients
14.5 Homomorphisms
14.6 Isomorphisms
14.7 Initial Structures
Endnotes
15 Models of Higher-order Languages
15.1 General Models of Higher-order Logic
15.2 Soundness
15.3 Completeness
15.4 The Interpretation of Identity and Granularity
15.5 Philosophical Issues Surrounding Model Theory
15.6 Incompleteness and Higher-order Logic
Endnotes
16 Logical Relations
16.1 Logical Relations
16.2 The Fundamental Theorem of Logical Relations
16.3 Logical Partial Functions
16.4 Applications to Equational Theories
16.5 Logical Partial Equivalence Relations
16.6 λ-definability
16.7 Kripke Logical Relations
Endnotes
17 Modalized Sets, M-sets and Cartesian Closed Categories
17.1 Modalized Applicative Structures
17.2 Substitution Structures
17.3 Applications of Substitution Structures
17.4 Abstract Operation Spaces
17.5 Categories
17.6 Actions
Endnotes
18 The Model Theory of Classicism
18.1 Modal Models of Classicism
18.2 Soundness of Modal Models
18.3 Standard Models, Modal Completeness and Higher-order Incompleteness
18.4 Completeness of Modal Models
18.5 the Disjunction and Coherence Properties in Extensions of Classicism
18.6 Coalesced Sums
Endnotes
Part V: Appendices
Appendix A The Curry-howard Isomorphism
A.1 Implicational Propositional Logics
A.2 Combinatory Languages and Hilbert Systems
A.3 Correspondences Between Hilbert and Natural Deduction Systems
Appendix B Definability Semantics
B.1 Definability Semantics and Metaphysical Definability
B.2 Validity and Frame Conditions
B.3 Logics with Weakening
B.4 Completeness
B.5 Identity and Associativity
B.6 Definability Structures for General λ-languages
Endnotes
Bibliography
Index