A Philosophical Introduction to Higher-order Logics

Cover\nHalf Title\nTitle Page\nCopyright Page\nTable of Contents\nNomenclature\nPreface\n0 Introduction\n	0.1 Typed Languages\n	0.2 Generalizations\n	0.3 Higher-order Generalizations\n	0.4 Abstraction\n	0.5 Some Things that Higher-order Generalizations are Not\n	0.6 Higher-order Generalizations in Philosophy\n	0.7 Semantics and Model Theory for Higher-order Languages\n	0.8 Glossing Higher-order Generalizations in English\n	0.9 How to Read This Book\n	0.10 Other Resources\n	Endnotes\nPart I: Typed Languages\n	1 Typed Languages\n		1.1 Types\n		1.2 Typed Languages\n		1.3 The Concept Horse Problem\n		1.4 Alternative Type Systems\n		Endnotes\n	2 An Informal Introduction to Abstraction\n		2.1 Abstraction\n		2.2 Introducing λ\n		2.3 Multiple Abstraction and Currying\n		2.4 Getting More Abstract\n		Endnotes\n	3 λ-languages\n		3.1 The Full λ-language\n		3.2 Combinators\n		3.3 Synonymy, α, β and η\n		3.4 Reduction\n		3.5 Combinatory Languages\n		3.6 More Efficient Definitions of Ersatz Abstraction\n		Endnotes\nPart II: Higher-order Languages\n	4 Higher-order Languages\n		4.1 Higher-order Languages\n		4.2 Quantifiers and Variable Binding\n		Endnote\n	5 Higher-order Logics\n		5.1 Higher-order Logics\n		5.2 Higher-order Logics in Other Logical Signatures\n		5.3 Inductive Definitions in Higher-order Logic\n		Endnotes\n	6 Application: Higher-order Theories of Granularity\n		6.1 Propositional Individuation: Propositional Booleanism\n		6.2 Propositional Individuation: Weaker Theories\n		6.3 Individuating Properties and Relations: Booleanism and Weakenings\n		6.4 Individuating Properties and Relations: Classicism\n		6.5 Functionality Principles\n		Endnotes\n	7 Application: Modal Logicism\n		7.1 Modal Logicism\n		7.2 Necessity\n		7.3 Entailment\n		7.4 Necessity in the Highest Degree\n		7.5 Possible Worlds\n		7.6 Reducing the Intensional to the Extensional\n		Endnotes\n	8 Application: Consequences and Strengthenings of Classicism\n		8.1 The Modal Logic of Broad Necessity\n		8.2 Some Strengthenings of Classicism and their Modal Consequences\n		8.3 Logical Necessity\n		8.4 Further Reading\n		Endnotes\nPart III: General Higher-order Languages\n	9 General λ-languages\n		9.1 Higher-order Ontology and λ-languages\n		9.2 General λ-languages\n		9.3 Relevant, Affine, Linear and Ordered Languages\n		9.4 Quantifiers in General λ-languages\n		9.5 General Higher-order Logics\n		9.6 Application: Propositional Aboutness and Constituency\n		9.7 General λ-languages Without Combinators\n		9.8 Variable Free Approaches\n		Endnotes\n	10 Curry Typing\n		10.1 Curry Typing\n		10.2 Substructural Curry Typing\n		10.3 Curry Typing for Logical Operations\n		Endnotes\n	11 Application: Structure I\n		11.1 Quasi-syntactic Accounts of Structure\n		11.2 Pictorial Accounts of Structure\n		11.3 Relational Diagrams\n		11.4 Translating Between Diagrams and λ-terms\n		11.5 Unique Decomposition\n		Endnotes\n	12 Application: Structure II\n		12.1 Converses, Reflexizations, Vacuous λ-abstraction\n		12.2 Logical Modes of Combination\n		12.3 Combinators and Pure Entities\n		12.4 Positionalism\n		Endnotes\n	13 Application: Structure III\n		13.1 Theoretical Primitives\n		13.2 A General Logical Framework\n		13.3 Further Reading\n		Endnotes\nPart IV: Higher-order Model Theory\n	14 Applicative Structures\n		14.1 Applicative Structures\n		14.2 Functional Interpretations\n		14.3 The Environment Model Condition\n		14.4 Congruences and Quotients\n		14.5 Homomorphisms\n		14.6 Isomorphisms\n		14.7 Initial Structures\n		Endnotes\n	15 Models of Higher-order Languages\n		15.1 General Models of Higher-order Logic\n		15.2 Soundness\n		15.3 Completeness\n		15.4 The Interpretation of Identity and Granularity\n		15.5 Philosophical Issues Surrounding Model Theory\n		15.6 Incompleteness and Higher-order Logic\n		Endnotes\n	16 Logical Relations\n		16.1 Logical Relations\n		16.2 The Fundamental Theorem of Logical Relations\n		16.3 Logical Partial Functions\n		16.4 Applications to Equational Theories\n		16.5 Logical Partial Equivalence Relations\n		16.6 λ-definability\n		16.7 Kripke Logical Relations\n		Endnotes\n	17 Modalized Sets, M-sets and Cartesian Closed Categories\n		17.1 Modalized Applicative Structures\n		17.2 Substitution Structures\n		17.3 Applications of Substitution Structures\n		17.4 Abstract Operation Spaces\n		17.5 Categories\n		17.6 Actions\n		Endnotes\n	18 The Model Theory of Classicism\n		18.1 Modal Models of Classicism\n		18.2 Soundness of Modal Models\n		18.3 Standard Models, Modal Completeness and Higher-order Incompleteness\n		18.4 Completeness of Modal Models\n		18.5 the Disjunction and Coherence Properties in Extensions of Classicism\n		18.6 Coalesced Sums\n		Endnotes\nPart V: Appendices\n	Appendix A The Curry-howard Isomorphism\n		A.1 Implicational Propositional Logics\n		A.2 Combinatory Languages and Hilbert Systems\n		A.3 Correspondences Between Hilbert and Natural Deduction Systems\n	Appendix B Definability Semantics\n		B.1 Definability Semantics and Metaphysical Definability\n		B.2 Validity and Frame Conditions\n		B.3 Logics with Weakening\n		B.4 Completeness\n		B.5 Identity and Associativity\n		B.6 Definability Structures for General λ-languages\n		Endnotes\nBibliography\nIndex