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A formalization of the Ramified Type Theory
, 1994
"... In "Principia Mathematica " [17], B. Russell and A.N. Whitehead propose a type system for higher order logic. This system has become known under the name "ramified type theory". It was invented to avoid the paradoxes, which could be conducted from Frege's &am ..."
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In &quot;Principia Mathematica &quot; [17], B. Russell and A.N. Whitehead propose a type system for higher order logic. This system has become known under the name &quot;ramified type theory&quot;. It was invented to avoid the paradoxes, which could be conducted from Frege's &quot;Begriffschrift&quot; [7]. We give a formalization of the ramified type theory as described in the Principia Mathematica, trying to keep it as close as possible to the ideas of the Principia. As an alternative, distancing ourselves from the Principia, we express notions from the ramified type theory in a lambda calculus style, thus clarifying the type system of Russell and Whitehead in a contemporary setting. Both formalizations are inspired by current developments in research on type theory and typed lambda calculus; see e.g. [3]. In these formalizations, and also when defining &quot;truth&quot;, we will need the notion of substitution. As substitution is not formally defined in the Principia, we have to define it ourselves. Finally, the reaction by Hilbert and Ackermann in [10] on the
Towards a unified treatment of induction, I: the general recursion theorem, unfinished draft manuscript
, 1996
"... The recursive construction of a function f: A → Θ consists, paradigmatically, of finding a functor T and maps α: A → TA and θ: TΘ → Θ such that f = α; Tf; θ. The role of the functor T is to marshall the recursive subarguments, and apply the function f to them in parallel. This equation is called pa ..."
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The recursive construction of a function f: A → Θ consists, paradigmatically, of finding a functor T and maps α: A → TA and θ: TΘ → Θ such that f = α; Tf; θ. The role of the functor T is to marshall the recursive subarguments, and apply the function f to them in parallel. This equation is called partial correctness of the recursive program, because we have also to show that it terminates, i.e. that the recursion (coded by α) is well founded. This may be done by finding another map g: A → N, called a loop variant, where N is some standard well founded srtucture such as the natural numbers or ordinals. In set theory the functor T is the covariant powerset; in the study of the free algebra for a free theory Ω (such as in proof theory) it is the polynomial Σr∈Ω(−)ar(r), and it is often something very crude. We identify the properties of the category of sets needed to prove the general recursion theorem, that these data suffice to define f uniquely. For any pullbackpreserving functor T, a structure similar to the von Neumann hierarchy is developed which analyses the free Talgebra if it exists, or deputises for it otherwise. There is considerable latitude in the choice of ambient category, the functor T and the class of predicates admissible in the induction scheme. Free algebras, set theory, the familiar ordinals and novel forms of them which have arisen in theoretical computer science are treated in a uniform fashion. The central idea in the paper is a categorical definition of well founded coalgebra α: A. TA, namely that any pullback diagram of the form
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 ..."
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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...
REVIEW ARTICLE Tracking the origins of transformational generative grammar 1
"... Tracking the main influences of 19th and 20thcentury mathematics, logic and philosophy on pre1958 American linguistics and especially on early Transformational Generative Grammar (TGG) is an ambitious crossdisciplinary endeavour. Ideally it would call for expertise in the methods of intellectual ..."
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Tracking the main influences of 19th and 20thcentury mathematics, logic and philosophy on pre1958 American linguistics and especially on early Transformational Generative Grammar (TGG) is an ambitious crossdisciplinary endeavour. Ideally it would call for expertise in the methods of intellectual historiography, the history and content of 20thcentury American linguistics, the history and philosophy of science (including logic and mathematics), the tools and results of mathematical logic, and the theory of computable functions. Scholars fully versed in all of these fields are rare indeed. If Marcus Tomalin makes some mistakes in his book (henceforth LFS), that should not be surprising. What is surprising is how much progress he makes in furthering intellectually serious work on the history of modern linguistics, and how wide his reading in the relevant technical literature has been. LFS locates the intellectual roots of TGG in the methods developed by 19th and 20thcentury mathematics and logic for exhibiting the conceptual structure of theories and constructing rigorous proofs of theorems. Tomalin discusses the methods developed by AugustinLouis Cauchy for the rigorisation of the calculus in the 1820s; Whitehead & Russell’s use of the axiomatic method in Principia Mathematica (1910–1913); the Hilbert program (in the 1920s) to prove all of mathematics consistent; Bloomfield’s early axiomatisation of a general linguistic theory (1926); Carnap’s logical