scholarly journals Denotational Semantics of the Simplified Lambda-Mu Calculus and a New Deduction System of Classical Type Theory

2016 ◽  
Vol 213 ◽  
pp. 11-23
Author(s):  
Ken Akiba
Keyword(s):  
Type Theory ◽  
Denotational Semantics ◽  
Classical Type ◽  
Deduction System

scholarly journals Intensional models for the theory of types

2007 ◽  
Vol 72 (1) ◽  
pp. 98-118 ◽  
Author(s):  
Reinhard Muskens
Keyword(s):  
Type Theory ◽  
Possible Worlds ◽  
Sequent Calculus ◽  
Propositional Attitude ◽  
Essential Elements ◽  
Classical Type ◽  
Logical Omniscience ◽  
Small Fragment ◽  
Model Existence ◽  
Model Existence Theorem

AbstractIn this paper we define intensional models for the classical theory of types, thus arriving at an intensional type logic ITL. Intensional models generalize Henkin's general models and have a natural definition. As a class they do not validate the axiom of Extensionality. We give a cut-free sequent calculus for type theory and show completeness of this calculus with respect to the class of intensional models via a model existence theorem. After this we turn our attention to applications. Firstly, it is argued that, since ITL is truly intensional, it can be used to model ascriptions of propositional attitude without predicting logical omniscience. In order to illustrate this a small fragment of English is defined and provided with an ITL semantics. Secondly, it is shown that ITL models contain certain objects that can be identified with possible worlds. Essential elements of modal logic become available within classical type theory once the axiom of Extensionality is given up.

The calculus of dependent lambda eliminations

2017 ◽  
Vol 27 ◽  
Author(s):  
AARON STUMP
Keyword(s):  
Type Theory ◽  
Lambda Calculus ◽  
Denotational Semantics ◽  
Prototype Implementation ◽  
Constructive Type Theory ◽  
Typed Lambda Calculus ◽  
Constructive Type

AbstractModern constructive type theory is based on pure dependently typed lambda calculus, augmented with user-defined datatypes. This paper presents an alternative called the Calculus of Dependent Lambda Eliminations, based on pure lambda encodings with no auxiliary datatype system. New typing constructs are defined that enable induction, as well as large eliminations with lambda encodings. These constructs are constructor-constrained recursive types, and a lifting operation to lift simply typed terms to the type level. Using a lattice-theoretic denotational semantics for types, the language is proved logically consistent. The power of CDLE is demonstrated through several examples, which have been checked with a prototype implementation called Cedille.

Transfinite Constructions in Classical Type Theory

2015 ◽  
pp. 391-404 ◽  
Author(s):  
Gert Smolka ◽  
Steven Schäfer ◽  
Christian Doczkal
Keyword(s):  
Type Theory ◽  
Classical Type

Classical Type Theory

2001 ◽  
pp. 965-1007 ◽  
Author(s):  
Peter B. Andrews
Keyword(s):  
Type Theory ◽  
Classical Type
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