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Explicit Provability And Constructive Semantics
 Bulletin of Symbolic Logic
, 2001
"... In 1933 G odel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that G odel's provability calculus is noth ..."
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In 1933 G odel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that G odel's provability calculus is nothing but the forgetful projection of LP. This also achieves G odel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a BrouwerHeytingKolmogorov style provability semantics for Int which resisted formalization since the early 1930s. LP may be regarded as a unified underlying structure for intuitionistic, modal logics, typed combinatory logic and #calculus.
An intuitionistic theory of types
"... An earlier, not yet conclusive, attempt at formulating a theory of this kind was made by Scott 1970. Also related, although less closely, are the type and logic free theories of constructions of Kreisel 1962 and 1965 and Goodman 1970. In its first version, the present theory was based on the strongl ..."
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An earlier, not yet conclusive, attempt at formulating a theory of this kind was made by Scott 1970. Also related, although less closely, are the type and logic free theories of constructions of Kreisel 1962 and 1965 and Goodman 1970. In its first version, the present theory was based on the strongly impredicative axiom that there is a type of all types whatsoever, which is at the same time a type and an object of that type. This axiom had to be abandoned, however, after it was shown to lead to a contradiction by Jean Yves Girard. I am very grateful to him for showing me his paradox. The change that it necessitated is so drastic that my theory no longer contains intuitionistic simple type theory as it originally did. Instead, its proof theoretic strength should be close to that of predicative analysis.
Induction principles formalized in the Calculus of Constructions
 Programming of Future Generation Computers. Elsevier Science
, 1988
"... The Calculus of Constructions is a higherorder formalism for writing constructive proofs in a natural deduction style, inspired from work of de Bruijn [2, 3], Girard [12], MartinLöf [14] and Scott [18]. The calculus and its syntactic theory were presented in Coquand’s thesis [7], and an implementa ..."
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The Calculus of Constructions is a higherorder formalism for writing constructive proofs in a natural deduction style, inspired from work of de Bruijn [2, 3], Girard [12], MartinLöf [14] and Scott [18]. The calculus and its syntactic theory were presented in Coquand’s thesis [7], and an implementation by the author was used to mechanically verify a substantial number of proofs demonstrating the power of expression of the formalism [9]. The Calculus of Constructions is proposed as a foundation for the design of programming environments where programs are developed consistently with formal specifications. The current paper shows how to define inductive concepts in the calculus. A very general induction schema is obtained by postulating all elements of the type of interest to belong to the standard interpretation associated with a predicate map. This is similar to the treatment of D. Park [16], but the power of expression of the formalism permits a very direct treatment, in a language that is formalized enough to be actually implemented on computer. Special instances of the induction schema specialize to Nœtherian induction and Structural induction over any algebraic type. Computational Induction is treated in an axiomatization of Domain Theory in Constructions. It is argued that the resulting principle is more powerful than LCF’s [13], since the restriction on admissibility is expressible in the object language. Notations We assume the reader is familiar with the Calculus of Constructions, as presented in [7, 9, 10, 11]. More precisely, we shall use in the present paper the extended system defined in Section 11 of [8]. The notation [x: A]B stands for the algorithm with formal parameter x of type A and body B, whereas (x: A)B stands for the product of types B indexed by x ranging over A. Thus square brackets are used for λabstraction, whereas parentheses stand for product formation. The atom P rop is the type of logical propositions. The atom T ype stands for the first level in the predicative hierarchy of types (and thus we have P rop: T ype). We abbreviate (x: A)B into A → B whenever x does not occur in B. When B: P rop, we think of (x: A)B as the universally quantified proposition ∀x: A·B. When x does not occur in B and A: P rop,
Proof Assistants: history, ideas and future
"... In this paper we will discuss the fundamental ideas behind proof assistants: What are they and what is a proof anyway? We give a short history of the main ideas, emphasizing the way they ensure the correctness of the mathematics formalized. We will also briefly discuss the places where proof assista ..."
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In this paper we will discuss the fundamental ideas behind proof assistants: What are they and what is a proof anyway? We give a short history of the main ideas, emphasizing the way they ensure the correctness of the mathematics formalized. We will also briefly discuss the places where proof assistants are used and how we envision their extended use in the future. While being an introduction into the world of proof assistants and the main issues behind them, this paper is also a position paper that pushes the further use of proof assistants. We believe that these systems will become the future of mathematics, where definitions, statements, computations and proofs are all available in a computerized form. An important application is and will be in computer supported modelling and verification of systems. But their is still along road ahead and we will indicate what we believe is needed for the further proliferation of proof assistants.
N.G. de Bruijn’s Contribution to the Formalization of Mathematics
"... N.G. de Bruijn was one of the pioneers to explore the idea of using a computer to formally check mathematical proofs. The Automath project, that started in 1967 and ran until 1980, was the first in developing computer programs to actually check mathematical proofs. But Automath is more than that: it ..."
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N.G. de Bruijn was one of the pioneers to explore the idea of using a computer to formally check mathematical proofs. The Automath project, that started in 1967 and ran until 1980, was the first in developing computer programs to actually check mathematical proofs. But Automath is more than that: it is a language for doing mathematics and it has philosophical implications for the way we look at logic and the
Constructive notions of set Part I Sets in Martin–Löf type theory
"... This is the first of two articles dedicated to the notion of constructive set. In them we attempt a comparison between two different notions of set which occur in the context of the foundations for constructive mathematics. We also put them under perspective by stressing analogies and differences ..."
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This is the first of two articles dedicated to the notion of constructive set. In them we attempt a comparison between two different notions of set which occur in the context of the foundations for constructive mathematics. We also put them under perspective by stressing analogies and differences with the notion of set as codified in the classical theory Zermelo–Fraenkel. In the current article we illustrate in some detail the notion of set as expressed in Martin–Löf type theory and present the essential characters of this theory. In a second article we shall explore a distinct notion of set, as arising in the context of intuitionistic versions of Zermelo–Fraenkel set theory. The theory we shall analyse there is Aczel’s CZF (Constructive Zermelo– Fraenkel) and we shall supplement its exposition by a succinct account of Aczel’s interpretation of CZF in type theory. This will enable us to compare the two notions in a more precise sense. 1.