The proofs of universally quantified statements, in mathematics, are given as "schemata" or as "prototypes" which may be applied to each specific instance of the quantified variable. Type Theory allows to turn into a rigorous notion this informal intuition described by many, including Herbrand. In this constructive approach where propositions are types, proofs are viewed as terms of \Gammacalculus and act as "proof-schemata", as for universally quantified types. We examine here the critical case of Impredicative Type Theory, i.e. Girard's system F, where type-quantification ranges over all types. Coherence and decidability properties are proved for prototype proofs in this impredicative context.

ABSTRACT: Here is a crude list, possibly summarizing the role of paradoxes within the framework of mathematical logic: 1. directly motivating important theories (e.g. type theory, axiomatic set theory, combinatory logic); 2. suggesting methods of proving fundamental metamathematical results (fixed point theorems, incompleteness, undecidability, undefinability); 3. applying inductive definability and generalized recursion; 4. introducing new semantical methods (e. g. revision theory, semi-inductive definitions, which require non-trivial set theoretic results); 5. (partly) enhancing new axioms in set theory: the case of anti-foundation AFA and the mathematics of circular phenomena; 6. suggesting the investigation of non-classical logical systems, from contraction-free and many-valued logics to systems with generalized quantifiers; 7. suggesting frameworks with flexible typing for the foundations of Mathematics and Computer Science; 8. applying forms of self-referential truth and in Artificial Intelligence, Theoretical Linguistics, etc. Below we attempt to shed some light on the genesis of the issues 1–8 through the history of the paradoxes in the twentieth century, with a special emphasis on semantical aspects.

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