Results 1 
4 of
4
Metalogical Frameworks
, 1992
"... In computer science we speak of implementing a logic; this is done in a programming language, such as Lisp, called here the implementation language. We also reason about the logic, as in understanding how to search for proofs; these arguments are expressed in the metalanguage and conducted in the me ..."
Abstract

Cited by 57 (15 self)
 Add to MetaCart
In computer science we speak of implementing a logic; this is done in a programming language, such as Lisp, called here the implementation language. We also reason about the logic, as in understanding how to search for proofs; these arguments are expressed in the metalanguage and conducted in the metalogic of the object language being implemented. We also reason about the implementation itself, say to know it is correct; this is done in a programming logic. How do all these logics relate? This paper considers that question and more. We show that by taking the view that the metalogic is primary, these other parts are related in standard ways. The metalogic should be suitably rich so that the object logic can be presented as an abstract data type, and it must be suitably computational (or constructive) so that an instance of that type is an implementation. The data type abstractly encodes all that is relevant for metareasoning, i.e., not only the term constructing functions but also the...
A New Paradox in Type Theory
 Logic, Methodology and Philosophy of Science IX : Proceedings of the Ninth International Congress of Logic, Methodology, and Philosophy of Science
, 1994
"... this paper is to present a new paradox for Type Theory, which is a typetheoretic refinement of Reynolds' result [24] that there is no settheoretic model of polymorphism. We discuss then one application of this paradox, which shows unexpected connections between the principle of excluded middle and ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
this paper is to present a new paradox for Type Theory, which is a typetheoretic refinement of Reynolds' result [24] that there is no settheoretic model of polymorphism. We discuss then one application of this paradox, which shows unexpected connections between the principle of excluded middle and the axiom of description in impredicative Type Theories. 1 Minimal and Polymorphic HigherOrder Logic
Data Types, Infinity and Equality in System AF2
 In CSL ’93, volume 832 of LNCS
, 1995
"... This work presents an extension of system AF 2 to allow the use of infinite data types. We extend the logic with inductive and coinductive types, and show that the "programming method" is still correct. Unlike previous work in other typesystems, we only use the pure calculus. Propositions about no ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
This work presents an extension of system AF 2 to allow the use of infinite data types. We extend the logic with inductive and coinductive types, and show that the "programming method" is still correct. Unlike previous work in other typesystems, we only use the pure calculus. Propositions about normalization and unicity of the representation of data have no equivalent in other systems. Moreover, the class of data types we consider is very large with some unusual ones. 1 Introduction Since the work of Curry, a lot of typesystems have been created (e.g., De Bruijn's Automath [4]; Girard's system F [5]; MartinLof's type theory [10]; CoquandHuet's Calculus of construction [3]; etc). One of their purposes is program extraction via the CurryHoward isomorphism [6], which establishes a correspondence between programs and proofs of specifications. One of these systems is AF 2 (second order functional arithmetic) due to Leivant and Krivine [9, 7, 8]. It uses equations as algorithmic specif...
Impredicative Representations of Categorical Datatypes
, 1994
"... this document that certain implications are not based on a well stated formal theory but require a certain amount of handwaving. ..."
Abstract
 Add to MetaCart
this document that certain implications are not based on a well stated formal theory but require a certain amount of handwaving.