Treatise on Intuitionistic Type Theory [Hardcover]

Intuitionistic type theory can be described, somewhat boldly, as a partial fulfillment of the dream of a universal language for science. This book expounds several aspects of intuitionistic type theory, such as the notion of set, reference vs. computation, assumption, and substitution. Moreover, the book includes philosophically relevant sections on the principle of compositionality, lingua characteristica, epistemology, propositional logic, intuitionism, and the law of excluded middle. Ample historical references are given throughout the book.Intuitionistic type theory can be described, somewhat boldly, as a fulfillment of the dream of a universal language for science.? In particular, intuitionistic type theory is a foundation for mathematics and a programming language.Contents.- List of Figures.- List of Tables.- Introduction.- Chapter I. Prolegomena. 1. A treefold correspondence.- 2. The acts of the mind.- 3. The principle of compositionality.- 4. Lingua characteristica.- Chapter II. Truth of Knowledge. 1. The meaning of meaning.- 2. A division of being.- 3. Mathematical entities.- 4. Judgement and assertion.- 5. Reasoning and demonstration.- 6. The proposition.- 7. The laws of logic.- 8. Variables and generality.- 9. Division of definitions.- Chapter III. The Notion of Set. 1. A History of set-like notions.- 2. Set-theoretical notation.- 3. Making universal concepts ito objects of thought.- 4. Canonical sets and elements.- 5. How to define a canonical set.- 6. More canonical sets.- Chapter IV. Reference and Computation. 1. Functions, algorithms, and programs.- 2. The concept of function.- 3. A formalization of computation.- 4. Noncanonical sets and elements.- 5. Nominal definitions.- 6. Functions as objects.- 7. Families of sets.- Chapter V. Assumption and Substitution. 1. The concept of function revisited.- 2. Hypothetical assertions.- 3. The calculus of substitutions.- 4. Sets and elements in hypothetical assertions.- 5. Closures and ? -calculas.- 6. The dislT