Constructive Type Theory: Foundation and Formalization

Part of the Logic, Epistemology, and the Unity of Science book series (LEUS, volume 10)

Constructive Type Theory has been developed by Per Martin-Löf in a series of papers and lectures since the 1970s: its first formulation, known as Intuitionistic Type Theory, was based on a strong impredicative axiom which allowed a type of all types being at the same time a type and an object of that type; it was abandoned after it was shown to lead to contradiction by Jean Yves Girard; the reformulation of the entire framework led to a strong predicative theory, which is now known as Constructive Type Theory (CTT). The theory has its theoretical core in the contribution by Brouwer and Heyting to Intuitionistic logic, and it is therefore built on a constructive epistemic framework, providing a new interpretation to many of the central notions of classical logic, such as those of proposition, truth, and proof. I will begin by presenting in this section some general aspects of the constructive type-theoretical approach, analysing in the next sections its formal structure. To start with, only a general theoretical description of such a logical approach will be given and later fully explained, especially in connection with the notions of judgement and proof. The main aim of the present chapter is thus to present the theoretical, logical, and formal basis of CTT: a philosophical analysis of the theory and the explanation of the elements allowing to reconsider the problem of analyticity in the light of the constructive framework will in turn justify the introduction of the notion of information within the epistemic description.

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© Springer Science + Business Media B.V 2008

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