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A Reflection on Russell's Ramified Types and Kripke's Hierarchy of Truths
 Journal of the Interest Group in Pure and Applied Logic 4(2
, 1996
"... Both in Kripke's Theory of Truth ktt [8] and Russell's Ramified Type Theory rtt [16, 9] we are confronted with some hierarchy. In rtt, we have a double hierarchy of orders and types. That is, the class of propositions is divided into different orders where a propositional function can only depend on ..."
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Both in Kripke's Theory of Truth ktt [8] and Russell's Ramified Type Theory rtt [16, 9] we are confronted with some hierarchy. In rtt, we have a double hierarchy of orders and types. That is, the class of propositions is divided into different orders where a propositional function can only depend on objects of lower orders and types. Kripke on the other hand, has a ladder of languages where the truth of a proposition in language Ln can only be made in Lm where m ? n. Kripke finds a fixed point for his hierarchy (something Russell does not attempt to do). We investigate in this paper the similarities of both hierarchies: At level n of ktt the truth or falsehood of all ordernpropositions of rtt can be established. Moreover, there are ordernpropositions that get a truth value at an earlier stage in ktt. Furthermore, we show that rtt is more restrictive than ktt, as some type restrictions are not needed in ktt and more formulas can be expressed in the latter. Looking back at the dou...
Ramified HigherOrder Unification
, 1996
"... While unification in the simple theory of types (a.k.a. higherorder logic) is undecidable, we show that unification in the pure ramified theory of types with integer levels is decidable. Since pure ramified type theory is not very expressive, we examine the impure case, which has an undecidable uni ..."
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While unification in the simple theory of types (a.k.a. higherorder logic) is undecidable, we show that unification in the pure ramified theory of types with integer levels is decidable. Since pure ramified type theory is not very expressive, we examine the impure case, which has an undecidable unification problem even at order 2. However, the decidability result for the pure subsystem indicates that unification terminates more often than general higherorder unification. We present an application to ACA 0 and other expressive subsystems of secondorder Peano arithmetic.
Journal of the IGPL
"... In categorical proof theory, propositions and proofs are presented as objects and arrows in a category. It thus embodies the strong constructivist paradigms of propositionsastypes and proofsasconstructions, which lie in the foundation of computational logic. Moreover, in the categorical setting, ..."
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In categorical proof theory, propositions and proofs are presented as objects and arrows in a category. It thus embodies the strong constructivist paradigms of propositionsastypes and proofsasconstructions, which lie in the foundation of computational logic. Moreover, in the categorical setting, a third paradigm arises, not available elsewhere: logicaloperationsasadjunctions. It offers an answer to the notorious question of the equality of proofs. So we chase diagrams in algebra of proofs. On the basis of these ideas, the present paper investigates proof theory of regular logic: the f; 9gfragment of the first order logic with equality. The corresponding categorical structure is regular fibration. The examples include stable factorisations, sites, triposes. Regular logic is exactly what is needed to talk about maps, as total and singlevalued relations. However, when enriched with proofsasarrows, this familiar concept must be supplied with an additional conversion rule, conne...