A Modern Perspective on Type Theory: From its Origins until Today
Springer Science & Business Media, Mar 10, 2006 - Mathematics - 360 pages
`Towards the end of the nineteenth century, Frege gave us the abstraction principles and the general notion of functions. Self-application of functions was at the heart of Russell's paradox. This led Russell to introduce type theory in order to avoid the paradox. Since, the twentieth century has seen an amazing number of theories concerned with types and functions and many applications. Progress in computer science also meant more and more emphasis on the use of logic, types and functions to study the syntax, semantics, design and implementation of programming languages and theorem provers, and the correctness of proofs and programs. The authors of this book have themselves been leading the way by providing various extensions of type theory which have been shown to bring many advantages. This book gathers much of their influential work and is highly recommended for anyone interested in type theory. The main emphasis is on:
- Types: from Russell to Ramsey, to Church, to the modern Pure Type Systems and some of their extensions.
- Functions: from Frege, to Russell to Church, to Automath and the use of functions in mathematics, programming languages and theorem provers.
- The role of types in logic: Kripke's notion of truth, the evolution and role of the propositions as types concept and its use in logical frameworks.
- The role of types in computation: extensions of type theories which can better model proof checkers and programming languages are given.
The first part of the book is historical, yet at the same time, places historical systems (like Russell's RTT) in the modern setting. The second part deals with modern type theory as it developed since the 1940s, and with the role of propositions as types (or proofs as terms), but at the same time, places another historical system (the proof checker Automath) in the modern setting. The third part uses this bridging in the first two parts between historical and modern systems to propose new systems that bring more advantages together. This book has much to offer to mathematicians, logicians and to computer scientists in general. It will have considerable influence for many years to come.' - Henk Barendregt
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Type theory in Principia Mathematica 2a Principias propositional functions 2a1 2a2 2a3 Definition Principias propositional functions as Principias pf...
Propositions as Types Pure Type Systems AUTOMATH
The prePAT RTT and STT in PATstyle 5a RTT in PAT style 5a1 5a2 5a3 5a4 5a5 5a6 An introduction to The system Metaproperties of Interpreting ...
rameter types The different treatment of constants and variables The definition system and the translation using 7c 7c1 7c2 7c3 7c4 7c5 Definition a...
7d1 with parameters and
Pure Type Systems with definitions
A Type systems in this book
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abstraction argument assume AUT-QE AUTOMATH Axiom of Reducibility Begriffsschrift Bruijn Calculus of Constructions Chapter Church-Rosser Church-Rosser Theorem Church’s construction Corollary Correctness of Types declaration define denoted described DPTSs equivalent example expression extend Fairouz first-order formal formalisation free variables Frege fully applied give Hence hierarchy implementation impredicative induction hypothesis introduce variables intuition Kripke’s legal pfs Lemma mathematics Moreover natural number notation notion Nuprl objects obtain paradox parameter mechanism parametric constants parametric definitions parametric specification parametric terms possible predicative type present Principia Mathematica properties proposition of order propositional functions prove PTSs Pure Type Systems quantifier Ramified Theory ramified type theory ramified types real numbers Remark restricted Russell Russell’s Section simple types strong normalisation Subject Reduction Substitution Lemma subterm syntactic syntax term of type Theorem Theory of Types tion transitive relation translation typable type theory Unicity of Types Vicious Circle Principle write